Multiscale gyrokinetics for rotating tokamak plasmas: II. Reduced models for electron dynamics

In this section, we introduce the assumptions and resulting equations of this formalism. The detailed derivation of these results can be found in [1].

First, we split all physical quantities into mean and fluctuating parts:

$$\begin{equation} {\widetilde{\boldsymbol{B}}} = {\boldsymbol{B}} + { {\delta\boldsymbol{B}}}, \quad {\boldsymbol{B}} = { {\langle {\widetilde{\boldsymbol{B}}} \rangle}_{\mathrm{turb}}}, \end{equation} \tag{ 1 }$$

$$\begin{equation} {\widetilde{\boldsymbol{E}}} = {\boldsymbol{E}} + { {\delta\boldsymbol{E}}}, \quad {\boldsymbol{E}} = { {\langle {\widetilde{\boldsymbol{E}}} \rangle}_{\mathrm{turb}}}, \end{equation} \tag{ 2 }$$

$$\begin{equation} f_s = F_s + \delta f_s, \quad F_s = { {\left\langle f_s \right\rangle}_{\mathrm{turb}}}, \end{equation} \tag{ 3 }$$

where ${\widetilde {\boldsymbol {B}}}$ is the magnetic field, ${\widetilde {\boldsymbol {E}}}$ the electric field, f_s the distribution function of species s and ${ {\left \langle \cdot \right \rangle }_{\mathrm {turb}}}$ is some average over the fluctuations.

We assume that the fluctuations obey the standard gyrokinetic ordering⁶:

$$\begin{equation} \frac{\omega}{\Omega_s} \sim \frac{k_\parallel}{k_\perp} \sim \frac{|{ {\delta\boldsymbol{E}}}|}{|{\boldsymbol{E}}|} \sim \frac{|{ {\delta\boldsymbol{B}}}|}{|{\boldsymbol{B}}|} \sim\frac{\delta f_s}{F_s} \sim \frac{\nu_s}{\Omega_s} \sim \frac{\rho_s}{a} = \epsilon \ll 1, \end{equation} \tag{ 4 }$$

where ω is the typical fluctuation frequency in the frame rotating with the plasma, Ω_s = Z_seB/m_sc is the cyclotron frequency of species s with charge Z_se and mass m_s in the mean magnetic field B = |B|, k_∥ and k_⊥ are typical wavenumbers parallel and perpendicular to the mean magnetic field, ν_s is a typical collision frequency, ρ_s = v_ths/Ω_s is the thermal Larmor radius, ${v_{\mathrm {th}_s}} = \sqrt {2T_s / m_s}$ is the thermal speed of species s with mean temperature T_s and a is a typical equilibrium length scale. The mean quantities (B,E,F_s) are assumed to vary on the long length scale a and the long (transport) timescale τ_E, the energy confinement time. This long timescale is ordered so that

$$\begin{equation} \tau_{\mathrm{E}}^{-1} \sim \epsilon^{3} \Omega_s. \end{equation} \tag{ 5 }$$

Typical values for the timescales and length scales in a tokamak are given in table 1, which shows the very large scale separation, characterized by the small parameter . This scale separation in time and space of the mean and fluctuating quantities motivates us to define the average ${ {\left \langle \cdot \right \rangle }_{\mathrm {turb}}}$ in terms of temporal and spatial averages over intermediate time and length scales. This is done precisely in equations (14)–(18) of [1].

Table 1. Typical length and time scales in selected fusion devices.

Parameter	JET	D-IIID	ITER
ne (m−3)	1–5×1019	1018	1020
Ti (keV)	5–15	1–5	15–25
B (T)	2–4	1–2	3–5
βi (%)	0.01–2.0	1.0–4.0	1.0–4.0
ρi (m)	5.1×10−3	6.0×10−3	3.2×10−3
ρe (m)	8.2×10−5	9.7×10−5	5.2×10−5
a (m)	1	0.5	2
νee (Hz)	1.6×103	4.5×103	1.3×103
ω≈vthi/a (Hz)	2.0×104	2.8×104	5.5×105
Ωi (Hz)	2.6×107	1.52×107	3.8×107
τE (s)	0.5	0.1	3.5
=ρi/a	5×10−3	1.2×10−2	1.6×10−3
δ (%)	1.5–1.6	1.5–1.6	1.5–1.6

It is shown in section 3.2 of [1] that if we introduce the cylindrical coordinate system (R,z,ϕ) and assume axisymmetry with respect to ϕ, the mean magnetic field can be written (to lowest order in ) as

$$\begin{equation} {\boldsymbol{B}} = I(\psi,t) \nabla \phi + \nabla\psi \times\nabla\phi, \end{equation} \tag{ 6 }$$

where the poloidal flux function ψ is given in terms of the mean vector potential A by

$$\begin{equation} \psi(R,z,t) = A_\phi = R^2{\boldsymbol{A}}\cdot\nabla\phi, \end{equation} \tag{ 7 }$$

and I(ψ,t) = R²B·∇ϕ is the toroidal component of the mean magnetic field. We will also need the following alternate form of the magnetic field [18, 19]:

$$\begin{equation} {\boldsymbol{B}} = \nabla\psi\times\nabla \alpha, \end{equation} \tag{ 8 }$$

where α is a variable that labels different field lines within a flux surface. We cut the toroidal surface on the inboard side to force α to be single valued. This cut forms an axisymmetric ribbon with one edge the magnetic axis and the other the plasma boundary or separatrix. Letting l be the distance along a field line, (ψ, α, l) is a set of field-aligned coordinates for B. There are many different forms of field aligned coordinates and none of our results depend on a particular choice. Nevertheless, we will use (ψ, α, l) for specificity. Axisymmetric functions are functions of ψ and l but not α. Note that both ψ(R,z) and I(ψ) are mean quantities and therefore vary on the transport timescale.

We assume that the mean electric field is large, ${\boldsymbol {E}}\sim ( { {v_{\mathrm {th}_s}}}/{c} ) {\boldsymbol {B}}$, which can be shown to imply that the plasma rotates toroidally with a velocity that is species-independent and whose angular velocity only depends on the flux label ψ [20] and the slow transport timescale:

$$\begin{equation} \boldsymbol{u} = \omega(\psi,t) R^2\nabla\phi. \end{equation} \tag{ 9 }$$

If we express the mean electric field in terms of potentials using Gaussian units and Coulomb gauge,

$$\begin{equation} {\boldsymbol{E}} = -\nabla\varphi - \frac{1}{c} { \frac{\partial {\boldsymbol{A}}} {\partial t} } , \end{equation} \tag{ 10 }$$

then the scalar potential takes the form

$$\begin{equation} \varphi = \Phi(\psi, t) + \varphi_0, \quad \varphi_0 \sim \epsilon \Phi, \end{equation} \tag{ 11 }$$

and we have the following relationship between ω(ψ,t) and Φ(ψ,t):

$$\begin{equation} \omega(\psi,t) = c \frac{\mathrm{d}\Phi}{\mathrm{d}\psi}. \end{equation} \tag{ 12 }$$

With these results for the fields, our orderings imply that the distribution function f_s of species s takes the following form:

$$\begin{equation} f_s = F_s + \delta f_s, \end{equation} \tag{ 13 }$$

$$\begin{equation} F_s = F_{0s} \left( \psi(\boldsymbol{R}_s), {{\varepsilon}_s}, t \right) + F_{1s}\left( \boldsymbol{R}_s,{{\varepsilon}_s},\mu_s,\sigma, t\right) + \Or(\epsilon^2 f), \end{equation} \tag{ 14 }$$

$$\begin{equation} \delta f_s = - \frac{Z_s e}{T_s} \delta \varphi'(\boldsymbol{r},t) F_{0s} + h_s\left(\boldsymbol{R}_s,{{\varepsilon}_s},\mu_s,\sigma,t\right) + \Or(\epsilon^2 f). \end{equation} \tag{ 15 }$$

Let us explain this notation. We have followed [1] and changed phase-space variables from (r,v) to (R_s,ε_s,μ_s,ϑ,σ):

$$\begin{equation} \boldsymbol{R}_s = \boldsymbol{r} - \frac{1}{\Omega_s} {\boldsymbol{b}}\times\boldsymbol{w}, \end{equation} \tag{ 16 }$$

$$\begin{equation} {{\varepsilon}_s}= \frac{1}{2} m_s v^2 + Z_s e \left( \Phi(\psi,t) + \varphi_0 \right) - Z_s e\Phi({\psi^*_s},t), \end{equation} \tag{ 17 }$$

$$\begin{equation} \mu_s= \frac{m_s w_\perp^2}{2{B}}, \end{equation} \tag{ 18 }$$

where b = B/B is the unit vector along the mean magnetic field, the peculiar velocity w is

$$\begin{equation} \boldsymbol{w} = \boldsymbol{v} - \boldsymbol{u}= w_\parallel {\boldsymbol{b}} + w_\perp \left(\cos \vartheta\,\boldsymbol{e}_2 - \sin\vartheta\,\boldsymbol{e}_1 \right), \end{equation} \tag{ 19 }$$

where e₁ and e₂ are two arbitrary orthogonal unit vectors perpendicular to the magnetic field, the direction of the parallel motion is σ = w_∥/|w_∥|, and we have defined

$$\begin{equation} {\psi^*_s}(\boldsymbol{r},\boldsymbol{v},t) = \psi(\boldsymbol{r},t) + \frac{m_s c}{Z_s e} R^2\boldsymbol{v}\cdot\nabla\phi. \end{equation} \tag{ 20 }$$

The quantity ψ*_s(r,v,t) is equal to c/(Z_se) times the canonical toroidal angular momentum and is thus conserved in an exactly axisymmetric system—i.e. in the absence of fluctuations. In these variables, (14) splits the mean distribution function into the near Maxwellian equilibrium

$$\begin{equation} F_{0s} = N_{s}(\psi(\boldsymbol{R}_s,t),t) \left[\frac{m_s}{2\pi T_s(\psi(\boldsymbol{R}_s,t),t)}\right]^{3/2} \mathrm{e}^{- {{\varepsilon}_s}/ T_s(\psi(\boldsymbol{R}_s,t),t)}, \end{equation} \tag{ 21 }$$

and the neoclassical distribution function F_1s, which describes large-scale $\Or(\epsilon )$ deviations from a Maxwellian. The function N_s is related to the mean density n_s by

$$\begin{equation} n_s= N_{s}(\psi(\boldsymbol{r},t),t) \exp\left[- \frac{ Z_s e \varphi_0(\boldsymbol{r},t)}{T_s(\psi(\boldsymbol{r},t),t)} + \frac{m_s\omega^2(\psi(\boldsymbol{r},t),t) R^2}{2T_s(\psi(\boldsymbol{r},t),t)}\right]. \end{equation} \tag{ 22 }$$

We now drop the explicit slow time dependence of all quantities—this is for notational clarity only and is not an assumption that the mean quantities are constant.

The fluctuating distribution function, given by (15), is composed of the Boltzmann response (the first term of (15)), where the fluctuating potentials δφ and δA have been combined to form the electrostatic potential in the frame rotating with the plasma

$$\begin{equation} \delta \varphi' = \delta\varphi - \frac{1}{c} \boldsymbol{u}\cdot{ {\delta\boldsymbol{A}}}, \end{equation} \tag{ 23 }$$

and the gyrokinetic distribution function h_s. Assuming that all mean quantities are known, then the gyrokinetic distribution function h_s is given by the gyrokinetic equation [1, 2]

$$\begin{eqnarray} &&\fl \left[ { \frac{\partial } {\partial t} } +\boldsymbol{u}(\boldsymbol{R}_s)\cdot { \frac{\partial } {\partial \boldsymbol{R}_{s}} } \right]h_s + ( w_\parallel {\boldsymbol{b}} + \boldsymbol{V}_{\mathrm{D}s}+ {{\langle \boldsymbol{V}_\chi \rangle}_{\boldsymbol{R}}} )\cdot { \frac{\partial h_s} {\partial \boldsymbol{R}_{s}} } - {{\langle {{C}[h_s]} \rangle}_{\boldsymbol{R}}} \nonumber\\ &&=\frac{Z_s e F_{0s}}{T_s}\left[ { \frac{\partial } {\partial t} } + \boldsymbol{u}(\boldsymbol{R}_s)\cdot { \frac{\partial } {\partial \boldsymbol{R}_{s}} } \right] {{\left\langle \chi \right\rangle}_{\boldsymbol{R}}}- \left\{ { \frac{\partial F_{0s}} {\partial \psi} } + \frac{m_s F_{0s}}{T_s}\right. \nonumber\\ &&\quad \times \left. \left[ \frac{I(\psi) w_\parallel}{{B}} + \omega(\psi)R^2 \right]\frac{\mathrm{d}\omega}{\mathrm{d}\psi}\right\} {{\langle \boldsymbol{V}_\chi \rangle}_{\boldsymbol{R}}}\cdot\nabla\psi, \end{eqnarray} \tag{ 24 }$$

where the guiding-centre drift velocity is

$$\begin{eqnarray} &&\fl \boldsymbol{V}_{\mathrm{D}s}= \frac{{\boldsymbol{b}}}{\Omega_s} \times \left[ w_\parallel^2 {\boldsymbol{b}}\cdot\nabla{\boldsymbol{b}} + \frac{1}{2}w_\perp^2\nabla \ln {B}- \omega^2(\psi,t) R\nabla R - 2 w_\parallel \omega(\psi,t) {\boldsymbol{b}}\times\nabla z +\frac{Z_s e}{m_s} \nabla\varphi_0\right], \nonumber\\ \end{eqnarray} \tag{ 25 }$$

and where we have defined the gyrokinetic potential

$$\begin{equation} \chi = \delta \varphi - \frac{1}{c}\boldsymbol{v}\cdot{ {\delta\boldsymbol{A}}} = \delta \varphi' - \frac{1}{c} \boldsymbol{w}\cdot{ {\delta\boldsymbol{A}}}, \end{equation} \tag{ 26 }$$

and the fluctuating velocity due to χ⁷

$$\begin{equation} \boldsymbol{V}_{\!\chi} = \frac{c}{{B}}{\boldsymbol{b}}\times\nabla\chi,\quad {{\langle \boldsymbol{V}_{\! \chi} \rangle}_{\boldsymbol{R}}} = \frac{c}{{B}} {\boldsymbol{b}}\times { \frac{\partial {{\left\langle \chi \right\rangle}_{\boldsymbol{R}}}} {\partial \boldsymbol{R}_{s}} } + \Or(\epsilon^2 {v_{\mathrm{th}_s}}). \end{equation} \tag{ 27 }$$

We have also introduced the gyroaverage at constant R_s, ε_s and μ_s:

$$\begin{equation} \fl {{\left\langle \chi(\boldsymbol{r},\boldsymbol{w},t) \right\rangle}_{\boldsymbol{R}}} = \frac{1}{2\pi} \oint \mathrm{d}\vartheta \chi\left(\boldsymbol{r}(\boldsymbol{R}_s,{{\varepsilon}_s},\mu_s,\vartheta),\boldsymbol{w}(\boldsymbol{R}_s,{{\varepsilon}_s},\mu_s,\vartheta,\sigma),t\right). \end{equation} \tag{ 28 }$$

The equation for h_s is closed through Maxwell's equations for the fluctuating fields. The fluctuating potential δφ' obeys the quasineutrality condition

$$\begin{equation} \sum_s \frac{Z_s^2 e^2 n_s\delta \varphi'}{T_s} = \sum_s Z_s e \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}, \end{equation} \tag{ 29 }$$

where the integral over velocities is performed at constant r and the gyroaverage at constant r, w_∥, and w_⊥ is defined by

$$\begin{equation} \fl { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}} = \frac{1}{2\pi} \oint \mathrm{d}\vartheta h_s(\boldsymbol{R}_s(\boldsymbol{r},w_\parallel,w_\perp,\vartheta),{{\varepsilon}_s}(\boldsymbol{r},w_\parallel,w_\perp,\vartheta),\mu_s(\boldsymbol{r},w_\perp),\sigma,t). \end{equation} \tag{ 30 }$$

The fluctuating magnetic field is determined from δA_∥ = b·δA and δB_∥ = b·δB, which obey the parallel component of Ampère's law

$$\begin{equation} -\nabla_\perp^2 { {\delta A_\parallel}} = \frac{4\pi}{c} \sum_s Z_s e \int \mathrm{d}^3 \boldsymbol{w} w_\parallel { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}, \end{equation} \tag{ 31 }$$

and the perpendicular part of Ampère's law (see the discussion in section 7.5 of [1])

$$\begin{equation} \nabla_\perp^2 \frac{{ {\delta B_\parallel}} {B}}{4\pi} + \boldsymbol{\nabla}_\perp\boldsymbol{\nabla}_\perp\boldsymbol{:} \sum_s \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle m_s\boldsymbol{w}_\perp\boldsymbol{w}_\perp h_s \right\rangle}_{\boldsymbol{r}}} = 0, \end{equation} \tag{ 32 }$$

respectively. The evolution equation (24) and constituent relations (29), (31) and (32) form a closed system of equations for determining h_s, δφ', δA_∥, δB_∥ on the turbulent timescale. Typically, the system (24), (29), (31) and (32) is solved until statistical equilibrium is reached (a few nonlinear turnover times). The slowly-varying equilibrium is treated as constant during this evolution. The slow time dependence of F_0s is determined by transport equations for ω(ψ,t), N_s(ψ,t), T_s(ψ,t) and I(ψ,t) and the equilibrium condition determining ψ(r,t). Such equations are given in [1].

In order to express the scale separation of the electron and ion scales, we introduce the secondary small parameter $\delta = \sqrt { {m_{\mathrm {e}}}/{m_{\mathrm {i}}} } \ll 1$. We will use a subscript i to denote any ion species, of which there may be many. We consider the expansion in δ ≪ 1 to be a subsidiary expansion of the gyrokinetic system (equations (24), (29), (31), and (32))⁸. Thus, we assume that  = ρ_i/a, i.e. that the fundamental gyrokinetic expansion parameter is that of the ions, and that

$$\begin{equation} \epsilon \ll \delta \ll 1. \end{equation} \tag{ 33 }$$

All other dimensionless parameters are treated as finite—i.e. independent of δ. Therefore, we are assuming that T_i ∼ T_e and β = 8πp/B² ∼ 1, where p is the plasma pressure. In typical fusion plasmas, β is often small and one might worry about this assumption. A more detailed analysis shows that our expansion in δ requires that [21]

$$\begin{equation} \beta \gg \frac{m_{\mathrm{e}}}{m_{\mathrm{i}}} = \delta^2, \end{equation} \tag{ 34 }$$

which is equivalent to requiring that the Alfvén speed is smaller than the electron thermal speed⁹. In plasmas of interest, this is indeed satisfied and we are justified in treating β as finite.

We assume that fluctuating quantities vary on ion rather than electron scales. Thus, we order the timescales of fluctuating quantities in the rotating frame and the size of the fluctuations by

$$\begin{eqnarray} &&\fl { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla \sim { {v_{\mathrm{th}_{\mathrm{i}}}}}{\boldsymbol{b}} \cdot { \frac{\partial } {\partial \boldsymbol{R}_{s}} } \sim \frac{c}{B} |( \nabla \delta \varphi' \times {\boldsymbol{b}} ) \cdot\nabla_\perp| \nonumber\\ &&\sim\frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{B} |( \nabla { {\delta A_\parallel}}\times {\boldsymbol{b}} ) \cdot\nabla_\perp|\sim \frac{cT}{e{B}^2} |( \nabla { {\delta B_\parallel}} \times {\boldsymbol{b}} ) \cdot\nabla_\perp| \sim \frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{a}, \end{eqnarray} \tag{ 35 }$$

so that the typical fluctuation timescale, the ion parallel streaming time, and the ion nonlinear timescale due to the fluctuating fields are all comparable. We assume that the toroidal rotation velocity u is comparable to the ion thermal speed u ∼ v_thi, which implies that the electric field is ordered as

$$\begin{equation} {\boldsymbol{E}} \sim \frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{c} {\boldsymbol{B}}. \end{equation} \tag{ 36 }$$

The perpendicular length scales are of course ordered as

$$\begin{equation} |\nabla_\perp| \sim k_\perp \sim \rho_{\mathrm{i}}^{-1}. \end{equation} \tag{ 37 }$$

Clearly, the turbulent dynamics of the ions are not simplified by the subsidiary expansion in δ and hence h_i is determined by the gyrokinetic equation (24).

The electron dynamics, however, are simplified because

$$\begin{equation} k_\parallel{ {v_{\mathrm{th}_{\mathrm{e}}}}} \sim \delta^{-1} \frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{a} \sim \frac{ {v_{\mathrm{th}_{\mathrm{e}}}}}{B} |( \nabla { {\delta A_\parallel}} \times {\boldsymbol{b}} ) \cdot\nabla_\perp| \gg { \frac{\partial } {\partial t} } + {\boldsymbol{u}}\cdot\nabla, \end{equation} \tag{ 38 }$$

so the parallel electron motion is much faster than the evolution of the fluctuations¹⁰ and also

$$\begin{equation} k_\perp \rho_{\mathrm{e}} \sim \delta \ll 1, \end{equation} \tag{ 39 }$$

so we can neglect finite-electron-gyroradius effects. We choose the electron collision rate ν_e to be comparable to the fluctuation frequency, i.e.

$$\begin{equation} \nu_{\mathrm{e}} \sim \frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{a}\sim \frac{1}{\delta}\nu_{\mathrm{i}}. \end{equation} \tag{ 40 }$$

Note that although electron collisions are comparable to the fluctuation frequency, and, therefore, important in the electron fluctuation dynamics, this would conventionally be considered a low collisionality plasma since ν_eff = ν_ia/v_thi ∼ ν_ea/v_the ∼ δ ≪ 1. Under these assumptions, we expand h_e, δφ', δA_∥ and δB_∥ as

$$\begin{eqnarray} &&\begin{array}{l} h_{\mathrm{e}} = h_{\mathrm{e}}^{(0)} + h_{\mathrm{e}}^{(1)} + \cdots, \quad \delta \varphi' = \delta \varphi'^{(0)} + \delta \varphi'^{(1)}+ \cdots, \\ { {\delta A_\parallel}} = { {\delta A_\parallel^{(0)}}} + { {\delta A_\parallel^{(1)}}}+ \cdots, \quad {\mathrm{and}} \quad { {\delta B_\parallel}} = { {\delta B_\parallel^{(0)}}} + { {\delta B_\parallel^{(1)}}}+ \cdots, \end{array} \end{eqnarray} \tag{ 41 }$$

where h⁽¹⁾_e ∼ δ h⁽⁰⁾_e, etc¹¹. In sections 3 and 4, we systematically expand the gyrokinetic system of equations, (24), (29), (31) and (32) in powers of δ to obtain h⁽⁰⁾_e and close the system at first order in δ.

We can now find particular solutions of (46). To do this, we write δf_e in the following form:

$$\begin{equation} \delta f_{\mathrm{e}} = \lambda(\boldsymbol{r},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\sigma)F_{0\mathrm{e}} + g_{\mathrm{e}}(\boldsymbol{r},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\sigma) + \Or\left(\epsilon\delta F_{0\mathrm{e}} \right), \end{equation} \tag{ 48 }$$

where g_e satisfies the homogeneous equation

$$\begin{equation} w_\parallel {\widetilde{\boldsymbol{b}}}\cdot\nabla g_{\mathrm{e}} = 0. \end{equation} \tag{ 49 }$$

This can clearly be done for any δf_e by suitable choice of λ. Substituting (48) into (46) we obtain

$$\begin{eqnarray} &&\fl {\widetilde{\boldsymbol{b}}}\cdot\nabla \left( \lambda - \frac{e\delta \varphi'}{T_{\mathrm{e}}} \right) = \frac{e}{cT_{\mathrm{e}}} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right) { {\delta A_\parallel^{(0)}}} - {\widetilde{\boldsymbol{b}}}\cdot\nabla \ln N_{\mathrm{e}} - \left(\frac{{{\varepsilon}_{\mathrm{e}}}}{T_{\mathrm{e}}} - \frac{3}{2}\right){\widetilde{\boldsymbol{b}}}\cdot\nabla \ln T_{\mathrm{e}}. \end{eqnarray}\noindent \tag{ 50 }$$

To find the form of λ, we take the derivative of this equation with respect to μ_e and the second derivative with respect to ε_e to obtain

$$\begin{equation} {\widetilde{\boldsymbol{b}}}\cdot\nabla { \frac{\partial \lambda} {\partial \mu_{\mathrm{e}}} } = 0 \quad \mbox{and}\quad {\widetilde{\boldsymbol{b}}}\cdot\nabla { \frac{\partial ^2\lambda} {\partial {{\varepsilon}_{\mathrm{e}}}^2} } = 0, \end{equation} \tag{ 51 }$$

respectively. Clearly, any solution to these equations is either independent of μ_e and a linear function of ε_e or satisfies ${\widetilde {\boldsymbol {b}}}\cdot \nabla \lambda = 0$. Any solution of the second kind can be absorbed into the function g_e, leaving us with the general solution for λ given by

$$\begin{equation} \lambda = \frac{\overline{\delta n}_{\mathrm{e}}(\boldsymbol{r})}{n_{\mathrm{e}}} + \left(\frac{{{\varepsilon}_{\mathrm{e}}}}{T_{\mathrm{e}}} - \frac{3}{2}\right)\frac{\overline{\delta T}_{\!\mathrm{e}}(\boldsymbol{r})}{T_{\mathrm{e}}}, \end{equation}\noindent \tag{ 52 }$$

where we have written the linear function of ε_e in the natural way as a perturbed Maxwellian. However, it should be noted that $\overline {\delta n}_{\mathrm {e}}(\boldsymbol {r})$ is not the perturbed electron density and $\overline {\delta T}_{\!\mathrm {e}}(\boldsymbol {r})$ is not the perturbed electron temperature because g_e can have both density and energy moments. Inserting (52) into (50) and taking a derivative with respect to ε_e, we find the following equation for $\overline {\delta T}_{\!\mathrm {e}}$:

$$\begin{equation} {\widetilde{\boldsymbol{b}}}\cdot\nabla\left(\ln T_{\mathrm{e}} + \frac{\overline{\delta T}_{\!\mathrm{e}}}{T_{\mathrm{e}}}\right) = 0. \end{equation} \tag{ 53 }$$

Finally, substituting both (52) and (53) into (50), we obtain

$$\begin{equation} {\widetilde{\boldsymbol{b}}}\cdot\nabla \left(\frac{\overline{\delta n}_{\mathrm{e}}}{n_{\mathrm{e}}} - \frac{e \delta \varphi'}{T_{\mathrm{e}}} + \ln N_{\mathrm{e}}\right) = \frac{e}{cT_{\mathrm{e}}} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right) { {\delta A_\parallel}}. \end{equation} \tag{ 54 }$$

Familiar physics is contained in these two equations. It will be shown in section 3.5 that the contribution to g_e from passing electrons is a function only of $\tilde {\psi }$ and not of $\tilde {\alpha }$ or $\tilde {l}$, and, therefore, so are its density and temperature. Thus, for the purposes of interpreting (54) and (53), we can add the density and temperature of the passing part of g_e to $\overline {\delta n}_{\mathrm {e}}$ and $\overline {\delta T}_{\!\mathrm {e}}$ respectively. Hence, (53) describes how (part of) the temperature of the electrons is equalized along field lines, due to rapid thermal conduction. The only part of the temperature that does not equalize along the field line, as we shall see when we solve (49), is the part of the temperature of the trapped particles contained in g_e. This is to be expected; the trapped particles do not traverse the entire field line and so cannot equalize their temperature along it.

Equation (54) describes how the pressure gradient of the electrons (again, not including the trapped particle pressure) along the field line is balanced by a parallel electric field. As we shall demonstrate, it is useful to think of (54) as an evolution equation for δA_∥ where δφ' and $\overline {\delta n}_{\mathrm {e}}$ are given.

To solve (49), (53) and (54) we will need to know the spatial structure of the total magnetic field. As (54) determines δA_∥, it determines δB_⊥ and thence ${\widetilde {\boldsymbol {b}}}$. In this section, we will prove that (54) implies that magnetic flux is conserved and consequently that the evolution of the magnetic perturbation cannot change the structure of the magnetic field lines.

Magnetic flux is said to be conserved if there exists an effective velocity field u_eff such that closed curves moving with this velocity always enclose the same amount of magnetic flux. It is proved by Newcomb in [25] that if such a u_eff exists, it also preserves magnetic field lines and thus the magnetic topology is fixed. Now, if the parallel electric field satisfies

$$\begin{equation} {\widetilde{\boldsymbol{E}}}\cdot{\widetilde{\boldsymbol{b}}} = -{\widetilde{\boldsymbol{b}}}\cdot\nabla\xi, \end{equation} \tag{ 55 }$$

where ξ is a single-valued scalar function, then the velocity field

$$\begin{equation} \boldsymbol{u}_{\mathrm{eff}} = \frac{c}{{\widetilde{B}}^2} ({\widetilde{\boldsymbol{E}}} + \nabla\xi )\times{\widetilde{\boldsymbol{B}}} + U(\boldsymbol{r}) {\widetilde{\boldsymbol{b}}}, \end{equation} \tag{ 56 }$$

(where ${\widetilde {B}} = |{\widetilde {\boldsymbol {B}}}|$ and U(r) is an arbitrary single-valued function) satisfies ${\widetilde {\boldsymbol {E}}} + {\boldsymbol {u}_{\mathrm {eff}}\times {\widetilde {\boldsymbol {B}}}}/{c} = - \nabla \xi$. Thus Faraday's law becomes ${\partial {\widetilde {\boldsymbol {B}}} }/{\partial t} = \nabla \times (\boldsymbol {u}_{\mathrm {eff}}\times {\widetilde {\boldsymbol {B}}})$. The familiar proof of the frozen-in theorem (that is reproduced in almost all MHD textbooks) clearly applies if we recognize u_eff as the flux- and field-line-preserving velocity. The condition (55) is a special case of the general conditions for the existence of frozen flux that were investigated in [25].

In appendix B, we prove that we can write ${\widetilde {\boldsymbol {E}}}\cdot {\widetilde {\boldsymbol {b}}}$ in terms of potentials as

$$\begin{eqnarray} &&\fl {\widetilde{\boldsymbol{E}}}\cdot{\widetilde{\boldsymbol{b}}} = - {\widetilde{\boldsymbol{b}}}\cdot\nabla \varphi_0 - {\widetilde{\boldsymbol{b}}}\cdot\nabla \delta \varphi' - \frac{1}{c} { \frac{\partial {\boldsymbol{A}}} {\partial t} } \cdot{\boldsymbol{b}} - \frac{1}{c} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right){ {\delta A_\parallel}} + \Or\left(\epsilon^3\frac{ {v_{\mathrm{th}_{\mathrm{i}}}}}{c} {B}\right). \end{eqnarray} \tag{ 57 }$$

In appendix C, we prove that the mean magnetic flux is conserved irrespective of the fluctuations, and so (see (C.7))

$$\begin{equation} { \frac{\partial {\boldsymbol{A}}} {\partial t} } \cdot{\boldsymbol{b}} = {\boldsymbol{b}} \cdot\nabla \frac{T_{\mathrm{e}} n_{1\mathrm{e}}}{e {n}_{e}}, \end{equation} \tag{ 58 }$$

where n_1e is the $\Or(\epsilon )$ correction to the mean electron density (see discussion before (C.7)). Finally, using (54) and (58) in (57), we obtain

$$\begin{equation} {\widetilde{\boldsymbol{E}}}\cdot{\widetilde{\boldsymbol{b}}} = -{\widetilde{\boldsymbol{b}}}\cdot\nabla\left[ \frac{T_{\mathrm{e}}\left( \overline{\delta n}_{\mathrm{e}} + n_{1\mathrm{e}} \right)}{e n_{\mathrm{e}}} + \frac{T_{\mathrm{e}}}{e} \ln N_{\mathrm{e}} + \varphi_0\right], \end{equation} \tag{ 59 }$$

which is indeed in the form (55).¹² Thus, we can define

$$\begin{equation} \xi = \frac{T_{\mathrm{e}}}{e} \left(\ln N_{\mathrm{e}} + \frac{\overline{\delta n}_{\mathrm{e}} + n_{1\mathrm{e}}}{n_{\mathrm{e}}}\right) +\varphi_0 + \tilde{\xi}, \end{equation} \tag{ 60 }$$

where $\tilde {\xi }$ is an arbitrary single-valued function that satisfies

$$\begin{equation} {\widetilde{\boldsymbol{b}}}\cdot\nabla \tilde{\xi} = 0, \end{equation} \tag{ 61 }$$

and with this definition, (56) defines the field-line-preserving velocity u_eff. In (56), we are free to choose U(r) and $\tilde {\xi }$ subject to the constraint ${\widetilde {\boldsymbol {b}}}\cdot \nabla \tilde {\xi } = 0$. We postpone making choices for these parameters until section 3.5. Since the field is frozen to u_eff the ion-scale turbulence cannot alter the magnetic field structure on the turbulent timescale.

In the previous section we have demonstrated that ion-scale turbulence preserves field lines. The topology of the field is thus fixed. But what is this topology? Electron-scale fluctuations can alter the topology and so can generate a stochastic field¹³ from an initially regular one. Such fluctuations may arise 'locally' from electron-scale instabilities (e.g. electron temperature gradient instabilities or microtearing), but electron-scale fluctuations may also be driven 'non-locally' by a cascade from ion-scale fluctuations. In both cases, some tangling of the magnetic field is possible, indeed probable. Thus, at some intuitive level, the magnetic field is expected to be stochastic. A stochastic field will be advected by u_eff and remain stochastic.

If the magnetic field were stochastic and stationary then the heat diffusivity of the electrons would be given by the well-known Rechester–Rosenbluth formula [26]

$$\begin{equation} \chi_{T_{\mathrm{e}}} = D_{\mathrm{st}} {v_{\mathrm{th}_{\mathrm{e}}}} \sim {v_{\mathrm{th}_{\mathrm{e}}}} l_{\mathrm{c}}\left(\frac{\delta B_{\mathrm{stoch}}}{B}\right)^2, \end{equation} \tag{ 62 }$$

where D_st is a species-independent diffusion coefficient characterizing the stochasticity of the field¹⁴ and l_c is the parallel correlation length of the stochastic component of the field δB_stoch. Let us assume the stochastic component of the field is comparable to the ion-scale field perturbations (i.e. δB_stoch/B ∼ ) and the correlation length is the system size (i.e. l_c ∼ a). Then, (62) predicts that the electron heat diffusivity in a tokamak should be δ⁻¹ larger than the ion heat diffusivity (both the electrostatic and electromagnetic parts of the ion heat diffusivity). This is not backed up by experimental observations; ion and electron heat diffusivities are observed to be of the same order of magnitude. Thus, we take this as experimental justification for assuming that the amount of magnetic field stochasticity is limited¹⁵.

We therefore assume good flux surfaces in solving equations (49), (53) and (54)—these surfaces will be approximations to the exact field that we will refer to as the 'ion-scale field'. The ion-scale turbulence will distort the ion-scale field but the surfaces will remain intact. A smaller stochastic field component with l_c ∼ a and δB_stoch/B ∼ δ^1/2 would cause observable levels of transport and we cannot rule out a component of this size. Indeed, such a field would change the final electron equations in this paper. However, we will assume for simplicity that the stochastic component of the field is zero to order $\Or (\epsilon \delta )$ (i.e. δB_stoch/B ≪ δ) in this paper.

If we write the ion-scale magnetic field in the Clebsch form

$$\begin{equation} {\widetilde{\boldsymbol{B}}} = \nabla\tilde{\psi} \times\nabla\tilde{\alpha}, \end{equation} \tag{ 63 }$$

then $\tilde {\psi }$ is a single-valued function that labels the flux surfaces and $\tilde {\alpha }$ is a function that labels different field lines within a given flux surface (see figure 1). To ensure that $\tilde {\alpha }$ is single-valued, we make a cut in the poloidal domain on the inboard side of the tokamak. This cut makes a ribbon forming a toroidal loop with one edge of the ribbon being the magnetic axis and the other the plasma boundary or separatrix. We allow the cut to move with the velocity u_eff. This is legitimate as u_eff is continuous across the cut. Since the cut is perturbed by the turbulence, it will not now be axisymmetric.

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The expression (63) for the magnetic field allows us (see e.g. [19]) to introduce a system of field-aligned coordinates ($\tilde {\psi }$, $\tilde {\alpha }$, $\tilde {l}$) where $\tilde {l}$ measures distance along a given field line. By definition, $\tilde {l}$ runs from one side of the cut in the poloidal plane to the other side of the cut. As δB ≪ B, the variables aligned to the exact field only deviate slightly from those aligned to the background field:

$$\begin{equation} |\tilde{\psi} - \psi | \sim \epsilon \psi, \quad |\tilde{\alpha} - \alpha | \sim \epsilon \alpha, \quad |\tilde{l} - l| \sim \epsilon l. \end{equation}\noindent \tag{ 64 }$$

Importantly, this means that all axisymmetric mean quantities are, to lowest order in , functions of $\tilde {\psi }$ and $\tilde {l}$ but not of $\tilde {\alpha }$.

We need expressions for the spatial derivative operators ${\widetilde {\boldsymbol {b}}}\cdot \nabla$ and the Poisson bracket $\left \{ {a} , {b} \right \} = {\boldsymbol {b}}\cdot \left ( \nabla a\times \nabla b \right )$ in our new coordinates. By using the fact that ${\widetilde {\boldsymbol {b}}}\cdot \nabla \tilde {\psi } = {\widetilde {\boldsymbol {b}}}\cdot \nabla \tilde {\alpha } = 0$, we discover that

$$\begin{equation} {\widetilde{\boldsymbol{b}}} \cdot\nabla a = {\widetilde{\boldsymbol{b}}}\cdot\nabla \tilde{l} \left. { \frac{\partial a} {\partial \tilde{l}} } \right|_{\tilde{\psi},\tilde{\alpha}} = \left. { \frac{\partial a} {\partial \tilde{l}} } \right|_{\tilde{\psi},\tilde{\alpha}} \end{equation}\noindent \tag{ 65 }$$

for any function a(r). Similarly, by using the fact that ${\widetilde {\boldsymbol {b}}}\cdot ( \nabla \tilde {\alpha }\times \nabla \tilde {\psi } ) = {B}$ to lowest order, we see that the Poisson bracket takes the form

$$\begin{equation} \fl \left\{ {a} , {b} \right\} = {\widetilde{\boldsymbol{b}}}\cdot\left( \nabla a\times\nabla b \right) +\Or(\epsilon|\nabla a||\nabla b|) = {B}\left( { \frac{\partial a} {\partial \tilde{\alpha}} } { \frac{\partial b} {\partial \tilde{\psi}} } - { \frac{\partial a} {\partial \tilde{\psi}} } { \frac{\partial b} {\partial \tilde{\alpha}} } \right) + \Or( \epsilon |\nabla a||\nabla b|). \end{equation}\noindent \tag{ 66 }$$

Since u_eff is a field-line-preserving velocity, we have that

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla \right) \tilde{\psi} = 0 \end{equation}\noindent \tag{ 67 }$$

and

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla \right)\tilde{\alpha} = 0. \end{equation} \tag{ 68 }$$

As $\tilde {l} \sim l$ and ∇·u_eff ∼ ω, u_eff also approximately preserves the length of field lines and we have, to lowest order in ,

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla \right) \tilde{l} = 0. \end{equation}\noindent \tag{ 69 }$$

It is of course convenient to choose the cut in the definition of $\tilde {\alpha }$ such that it is convected with u_eff.

Finally, the time derivative at constant $\tilde {\psi }$, $\tilde {\alpha }$ and $\tilde {l}$ is related to the time derivative at constant r by

$$\begin{equation} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}= { \frac{\partial } {\partial t} } + \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla. \end{equation} \tag{ 70 }$$

We now use the magnetic coordinates introduced in section 3.4 to solve (49) and (53). We split velocity space (ε_e and μ_e) into two regions: the passing region where, for a given $\tilde {\psi }$, the equation

$$\begin{equation} w_\parallel(\tilde{l},\tilde{\psi},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}) = \sqrt{{\frac{2}{m_{\mathrm{e}}}}\left[ {{\varepsilon}_{\mathrm{e}}} - \mu_{\mathrm{e}} {B} + e\varphi_0 \right]} = 0 \end{equation} \tag{ 71 }$$

has no solutions for any $\tilde {l}$ (i.e. ε_e > μ_eB − eφ₀ for all $\tilde {l}$) and the trapped region in which there are two values of $\tilde {l}$ at which w_∥ = 0. The points where w_∥ vanishes are called the bounce points of the electron's orbit. In the passing region, (49) implies that

$$\begin{equation} { \frac{\partial g_{\mathrm{e}}} {\partial \tilde{l}} } = 0, \end{equation}\noindent \tag{ 72 }$$

so g_e is constant along field lines. As most magnetic surfaces are irrational, one field line spans the entire surface and so, for passing electrons, g_e cannot depend on $\tilde {\alpha }$:

$$\begin{equation} g_{\mathrm{e}} = g_{\mathrm{pe}} ( \tilde{\psi}, {{\varepsilon}_{\mathrm{e}}}, \mu_{\mathrm{e}}, \sigma, t). \end{equation} \tag{ 73 }$$

In the trapped region of velocity space, (49) implies that

$$\begin{equation} \mbox{either} \quad w_\parallel(\tilde{l},\tilde{\psi},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}) = 0, \quad\mbox{or}\; { \frac{\partial g_{\mathrm{e}}} {\partial \tilde{l}} } = 0. \end{equation} \tag{ 74 }$$

Thus, g_e must be constant along magnetic field lines, but only between the bounce points. So g_e can depend on the location of the bounce points, and, therefore, on $\tilde {\alpha }$:

$$\begin{equation} g_{\mathrm{e}} = g_{\mathrm{te}} (\tilde{\psi},\tilde{\alpha},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},t), \end{equation} \tag{ 75 }$$

where the dependence on σ has been dropped as g_e must be continuous as w_∥ passes through zero and changes sign. For convenience, we define g_pe to be zero in the trapped region and g_te to be zero in the passing region, so the complete solution is just the sum of the two functions:

$$\begin{equation} g_{\mathrm{e}} = g_{\mathrm{pe}}(\tilde{\psi},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\sigma,t) + g_{\mathrm{te}}(\tilde{\psi},\tilde{\alpha},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},t). \end{equation} \tag{ 76 }$$

The solution of (53) in our new coordinates is

$$\begin{equation} T_{\mathrm{e}}(\psi) + \overline{\delta T}_{\!\mathrm{e}} = \tilde{T}_{\mathrm{e}} (\tilde{\psi}), \end{equation}\noindent \tag{ 77 }$$

for some function $\tilde {T}_{\mathrm {e}}$. As we can absorb any part of $\overline {\delta T}_{\mathrm {e}}$ that is only a function of $\tilde {\psi }$ into g_e, we can pick $\tilde {T}_{\mathrm {e}}(\tilde {\psi }) = T_{\mathrm {e}}(\tilde {\psi })$ and so

$$\begin{equation} \overline{\delta T}_{\!\mathrm{e}} = (\tilde{\psi} - \psi) \frac{\mathrm{d} T_{\mathrm{e}}}{\mathrm{d}\psi}. \end{equation} \tag{ 78 }$$

We can also now pick the arbitrary function $\tilde {\xi }$ in the definition (60) of ξ to be any function of $\tilde {\psi }$. In appendix E, we find convenient forms for the arbitrary functions $\tilde {\xi }(\tilde {\psi })$ and U(r) to obtain (see (E.6))

$$\begin{equation} \boldsymbol{u}_{\mathrm{eff}} = \boldsymbol{u} + \frac{c}{{B}}{\widetilde{\boldsymbol{b}}} \times \nabla\left( \delta \varphi' - \zeta \right), \end{equation} \tag{ 79 }$$

where we have defined

$$\begin{equation} \zeta = \frac{T_{\mathrm{e}}}{e}\left[ \frac{\overline{\delta n}_{\mathrm{e}}}{n_{\mathrm{e}}} - (\tilde{\psi} - \psi) \frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi}\right]. \end{equation}\noindent \tag{ 80 }$$

With this definition, we can express the general solution of (46) as

$$\begin{eqnarray} &&\fl \delta f_{\mathrm{e}} = \frac{e\zeta}{T_{\mathrm{e}}} F_{0\mathrm{e}} + ( \tilde{\psi} - \psi ) { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } + g_{\mathrm{pe}}(\tilde{\psi},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\sigma) + g_{\mathrm{te}} (\tilde{\psi},\tilde{\alpha},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}) + \Or\left(\epsilon\delta F_{0\mathrm{e}} \right), \end{eqnarray}\noindent \tag{ 81 }$$

and the evolution equation (54) for δA_∥ becomes

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla \right){ {\delta A_\parallel}} = c {\widetilde{\boldsymbol{b}}}\cdot\nabla\left(\zeta - \delta \varphi'\right) + \Or\left(\epsilon \delta \Omega_{\mathrm{i}} { {\delta A_\parallel}} \right). \end{equation}\noindent \tag{ 82 }$$

This equation allows us to solve for δA_∥ correctly to lowest order in δ. Equations for the evolution of ζ, g_pe and g_te at the next order in our expansion are obtained in section 4.

At this juncture, we have enough information to determine δB_⊥ in two ways. Firstly, we have the evolution equations (67) and (68) for $\tilde {\psi }$ and $\tilde {\alpha }$, with u_eff given by (79). This enables us to calculate $\tilde {\psi }$ and $\tilde {\alpha }$ correctly to first order in . In appendix D, we demonstrate that this gives us enough information to construct δA_∥ from the equations

$$\begin{equation} { \frac{\partial { {\delta A_\parallel}}} {\partial \tilde{\psi}} } = - { \frac{\partial \tilde{\alpha}} {\partial l} } \quad\mbox{and}\quad { \frac{\partial { {\delta A_\parallel}}} {\partial \tilde{\alpha}} } = { \frac{\partial \tilde{\psi}} {\partial l} } . \end{equation} \tag{ 83 }$$

Secondly, we could solve for δA_∥ from the evolution equation (82) and then use (83) to determine $\tilde {\psi }$ and $\tilde {\alpha }$. Either of these methods is valid, and provides a solution for δA_∥ correct to the requisite order. For the purposes of simulating our equations, a third option presents itself: determine δA_∥ from (82), $\tilde {\psi }$ and $\tilde {\alpha }$ from (67) and (68), and then use (83) as consistency checks upon the solutions thus obtained. We choose the first of these options, i.e. to eliminate δA_∥ in favour of the fluctuating fields $\tilde {\psi }-\psi$ and $\tilde {\alpha }-\alpha$.

One point remains to be resolved. It would appear that knowing $\tilde {\psi }$ and $\tilde {\alpha }$ determines δB completely (via (63)). However, to determine δB_∥ from $\tilde {\psi }$ and $\tilde {\alpha }$, one must know $\tilde {\psi }$ and $\tilde {\alpha }$ to $\Or(\epsilon ^2)$ or, equivalently, the compressible part of u_eff to higher order ($\Or (\epsilon ^2 {v_{\mathrm {th}_{\mathrm {i}}}})$; see appendix D), but we do not. Therefore, we determine δB_∥ from (32) which involves only lowest-order quantities—quantities that we do calculate.

As explained in the previous section, we have obtained a solution for δA_∥. However, δA_∥ is usually determined from the parallel component of Ampère's Law, (31). Substituting our expansion for h_e (41) into the parallel component of Ampère's law (31), we find

$$\begin{equation} \nabla^2 { {\delta A_\parallel}} = \frac{4\pi e}{c} \left( \int \mathrm{d}^3 \boldsymbol{w} w_\parallel { {\left\langle h_{\mathrm{e}}^{(0)} + h_{\mathrm{e}}^{(1)} \right\rangle}_{\boldsymbol{r}}} \right) - \frac{4\pi}{c} \sum_{s=\mathrm{i}} Z_s e \int \mathrm{d}^3 \boldsymbol{w} w_\parallel { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}, \end{equation} \tag{ 84 }$$

where a sum over s = i denotes a sum over all ion species. In this equation, the term due to h⁽⁰⁾_e is larger than the rest by δ⁻¹. Thus, to lowest order,

$$\begin{equation} \int \mathrm{d}^3 \boldsymbol{w} w_\parallel h_{\mathrm{e}}^{(0)} = \int \mathrm{d}^3 \boldsymbol{w} w_\parallel g_{\mathrm{pe}} = 0. \end{equation} \tag{ 85 }$$

This is a constraint on g_pe in the lowest-order solution given by (81). Note that the trapped particles do not contribute to the current constraint. The parallel electron flow (and current) is comparable to that of the ions and contained in h⁽¹⁾_e. Defining the parallel electron flow

$$\begin{equation} \delta u_{\parallel \mathrm{e}} = \int \mathrm{d}^3 \boldsymbol{w} w_\parallel h_{\mathrm{e}}^{(1)} \end{equation} \tag{ 86 }$$

reduces (84) to an equation for δu_∥e:

$$\begin{equation} \delta u_{\parallel \mathrm{e}} = \frac{c}{4\pi e} \nabla^2 { {\delta A_\parallel}} + \sum_{s=\mathrm{i}} Z_s \int \mathrm{d}^3 \boldsymbol{w} w_\parallel { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}. \end{equation} \tag{ 87 }$$

Thus, instead of determining δA_∥ from the electron current, Ampère's law determines the electron flow from δA_∥ [21]. As we are no longer using δA_∥ to describe δB_⊥ in our system of equations, we consider ∇²_⊥δA_∥ to be a shorthand for a complicated expression involving derivatives of $\tilde {\psi }-\psi$ and $\tilde {\alpha }-\alpha$ via the relations (D.10) or (D.11). We require δu_∥e to complete the solution in the next order, but we do not need any information about h⁽¹⁾_e other than δu_∥e.

The one piece of information that is known about h⁽¹⁾_e is its parallel velocity moment δu_∥e, determined by (87). Thus, we integrate (92) over all velocities and find

$$\begin{eqnarray} &&\fl\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left[\frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}} - \frac{{ {\delta B_\parallel}}}{{B}} n_{\mathrm{e}}\right] + {B} { \frac{\partial } {\partial \tilde{l}} } \frac{\delta u_{\parallel \mathrm{e}}}{{B}} + \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla g_{\mathrm{e}} \nonumber\\ &&\quad+\, \frac{c}{{B}}\left\{ {\zeta} , {\int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}}} \right\}- \frac{c}{e}\left\{ {\frac{{ {\delta B_\parallel}}}{{B}}} , {\int \mathrm{d}^3 \boldsymbol{w} \mu_{\mathrm{e}} g_{\mathrm{e}}} \right\} \nonumber\\ &&= - \int \mathrm{d}^3 \boldsymbol{w} {\boldsymbol{V}}_{\!\mathrm{D}}\cdot\nabla (\tilde{\psi} - \psi) { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } + \left( \widehat{\boldsymbol{V}}_{\mathrm{D}} + \boldsymbol{V}_{\!\varphi} \right) \cdot\nabla\left( \delta \varphi'-\zeta\right) \frac{e n_{\mathrm{e}}}{T_{\mathrm{e}}} \nonumber\\ &&\quad +\,cn_{\mathrm{e}} { \frac{\partial } {\partial \tilde{\alpha}} } \left(\zeta - { {\delta B_\parallel}} \frac{T_{\mathrm{e}}}{e{B}} \right) \left(\frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi} - e\varphi_0 \frac{\mathrm{d}\ln T_{\mathrm{e}}}{\mathrm{d}\psi}\right) - c { \frac{\partial { {\delta B_\parallel}}} {\partial \tilde{\alpha}} } \frac{n_{\mathrm{e}} T_{\mathrm{e}}}{e{B}} \frac{\mathrm{d}\ln T_{\mathrm{e}}}{\mathrm{d}\psi} \nonumber\\ &&\quad -\frac{In_{\mathrm{e}}}{{B}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi} - \psi), \end{eqnarray} \tag{ 94 }$$

where we have used ${B} {\widetilde {\boldsymbol {b}}} = {\widetilde {\boldsymbol {B}}} + \Or(\epsilon {B})$, the fact that h⁽⁰⁾_e carries no current, and

$$\begin{equation} \int \mathrm{d}^3 \boldsymbol{w} = \sum_\sigma \int \frac{B\, \mathrm{d}{{\varepsilon}_{\mathrm{e}}}\,\mathrm{d}\mu_{\mathrm{e}} \,\mathrm{d}\vartheta}{m_{\mathrm{e}}^2|w_\parallel|}. \end{equation} \tag{ 95 }$$

We have also defined

$$\begin{equation} \widehat{\boldsymbol{V}}_{\mathrm{D}} = -\frac{cT_{\mathrm{e}}}{e B} {\boldsymbol{b}} \times \left( {\boldsymbol{b}}\cdot\nabla{\boldsymbol{b}} + \nabla\ln {B} \right) \end{equation} \tag{ 96 }$$

and

$$\begin{equation} \boldsymbol{V}_{\!\varphi} = \frac{c}{B} {\boldsymbol{b}}\times\nabla\varphi_0 \end{equation} \tag{ 97 }$$

so that

$$\begin{equation} \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{V}_{\mathrm{D}\mathrm{e}} F_{0\mathrm{e}} = \left( \widehat{\boldsymbol{V}}_{\mathrm{D}} + \boldsymbol{V}_{\!\varphi} \right)n_{\mathrm{e}}. \end{equation} \tag{ 98 }$$

We have also used $\nabla \cdot {\widetilde {\boldsymbol {B}}} = 0$ and (D.11) to eliminate ${\widetilde {\boldsymbol {b}}}$ and δA_∥ from our equation. It is important to note that this equation is written in the moving coordinate system. It relates $\zeta (\tilde {\psi },\tilde {\alpha },\tilde {l},t)$ to g_e and the other fluctuating fields, also given as functions of $\tilde {\psi }$, $\tilde {\alpha }$ and $\tilde {l}$. In this context, $\tilde {\psi } - \psi$ should be viewed as just another fluctuating field. Therefore, all gradients should be thought of in terms of derivatives with respect to $\tilde {\psi }$ and $\tilde {\alpha }$, for the Poisson bracket this is done via (66).

That (94) is essentially the fluctuating continuity equation is not immediately obvious. Let us attempt to make this more transparent. Clearly, the first two terms under the time derivative in (94) are the perturbed electron density due to the fluctuations; the time derivative of δB_∥ adds to this the fluctuation in the electron density due to the compression of the flux surfaces¹⁶. This total density is changed by various flows—the compression in the parallel flow δu_∥e, the flow of electrons past the flux surface given by $(c/B){\widetilde {\boldsymbol {b}}}\times \nabla \zeta$, the grad-B drift due to δB_∥, and the magnetic drifts. Indeed, even the final term in (94) can be interpreted as a compression—it is the compression of the electron fluid due to the perturbed flux surfaces 'sticking out' into regions with different toroidal angular velocities.

Equation (94) provides us with an evolution equation for ζ if we know g_e and the perturbed fields. The evolution equations for g_e given ζ are found in the next two sections.

In order to derive an equation for g_e in the trapped region of velocity space, we need to define an averaging operator that eliminates the fast electron bounce motion (i.e. annihilates $w_\parallel {\widetilde {\boldsymbol {b}}}\cdot \nabla$ and thereby eliminates h⁽¹⁾_e). The appropriate average is the bounce average along the perturbed field i.e. at fixed $\tilde {\psi }$ and $\tilde {\alpha }$. For an arbitrary function $a(\tilde {\psi },\tilde {\alpha },\tilde {l},{{\varepsilon }_{\mathrm {e}}},\mu _{\mathrm {e}},\sigma )$ we define the bounce average to be

$$\begin{equation} \left\langle a\right\rangle_{\parallel} = \oint \frac{\mathrm{d}\tilde{l}}{w_\parallel} a \left/ \oint \frac{\mathrm{d}\tilde{l}}{w_\parallel}\right., \end{equation} \tag{ 99 }$$

where the integration is carried out from one bounce point where w_∥ = 0, along the magnetic field, to the second bounce point where w_∥ passes through zero and changes sign, and then back to the first bounce point. This integral is carried out at constant $\tilde {\psi }$, $\tilde {\alpha }$, ε_e, μ_e and ϑ. Thus, equivalently, the bounce average is

$$\begin{equation} \left\langle a\right\rangle_{\parallel} = \left. \int \nolimits_{l_1}^{l_2} \frac{\mathrm{d}\tilde{l}}{|w_\parallel|} \left[ a(\sigma = 1) + a(\sigma = -1) \right] \right/ 2\int\nolimits_{l_1}^{l_2} \frac{\mathrm{d}\tilde{l}}{|w_\parallel|}, \end{equation} \tag{ 100 }$$

where l₁ and l₂ are the bounce points of the particle orbit.

Clearly, the bounce average commutes with the time derivative at fixed moving coordinates $\tilde {\psi }$, $\tilde {\alpha }$ and $\tilde {l}$. In appendix F, we prove that

$$\begin{equation} \left\langle w_\parallel {\widetilde{\boldsymbol{b}}}\cdot\nabla h_{\mathrm{e}}^{(1)}\right\rangle_{\parallel} = 0 \end{equation} \tag{ 101 }$$

and that, to lowest order in ,

$$\begin{equation} \langle \boldsymbol{V}_{\!\mathrm{D}\mathrm{e}}\cdot\nabla\tilde{\psi}\rangle_{\parallel} = 0. \end{equation} \tag{ 102 }$$

Using these results, we take (92) in the trapped region of velocity space and perform the bounce average to find

$$\begin{eqnarray} &&\fl \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{g_{\mathrm{te}}} + \left\langle \boldsymbol{V}_{\!\mathrm{D}\mathrm{e}}\cdot\nabla g_{\mathrm{te}}\right\rangle_{\parallel} + \frac{c}{e{B}} \{ {e\left\langle \zeta\right\rangle_{\parallel}-\mu_{\mathrm{e}}\left\langle { {\delta B_\parallel}}\right\rangle_{\parallel}} , {g_{\mathrm{te}}} \} - \left\langle {{\left\langle {\,{C}\hspace{-0.5mm}\left[g_{\mathrm{te}}\right]} \right\rangle}_{\boldsymbol{R}}}\right\rangle_{\parallel}\nonumber\\ &&\,= - \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left(\left\langle \zeta\right\rangle_{\parallel} - \mu_{\mathrm{e}} \left\langle { {\delta B_\parallel}}\right\rangle_{\parallel}\right) \frac{F_{0\mathrm{e}}}{T_{\mathrm{e}}} - \langle \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla(\tilde{\psi} - \psi)\rangle_{\parallel} { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } \nonumber\\ &&\quad +\,\left\langle \boldsymbol{V}_{\!\mathrm{D}\mathrm{e}}\cdot\nabla\left(\delta \varphi'-\zeta\right)\right\rangle_{\parallel}\frac{e F_{0\mathrm{e}}}{T_{\mathrm{e}}} + \frac{c}{e}\, { \frac{\partial } {\partial \tilde{\alpha}} } \left( e\left\langle \zeta\right\rangle_{\parallel} - \mu_{\mathrm{e}}\left\langle { {\delta B_\parallel}}\right\rangle_{\parallel} \right) { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } \nonumber\\ &&\quad +\, \left\langle \frac{m_{\mathrm{e}} w_\parallel^2 }{{B}} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi}-\psi) \right\rangle_{\!\parallel} \frac{I(\psi)}{T_{\mathrm{e}}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} F_{0\mathrm{e}}, \end{eqnarray} \tag{ 103 }$$

where the last line in (92) has vanished under the bounce average because it is antisymmetric in σ (the sign of the parallel velocity). This equation is a closed equation for g_te, once the fields and ζ are given. Note that all quantities must be expressed in the moving coordinates, in order to perform the bounce average. Again, as with (94), all gradients should be interpreted in terms of derivatives with respect to the moving coordinates as should the Poisson bracket (via (66)).

In the passing region of velocity space, the fast streaming of electrons along perturbed field lines causes a passing electron to sample an entire perturbed flux surface ($\tilde {\psi }=\mbox {const.}$ surface). Thus, the averaging procedure we will need is the average over a perturbed flux surface. For any function a, this is defined by

$$\begin{equation} \left\langle a(\boldsymbol{r},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\vartheta,\sigma)\right\rangle_{\tilde\psi} = \lim_{\Delta \psi \rightarrow 0} \left\{ \int\nolimits_\Delta \mathrm{d}^3{\boldsymbol{r}} a(\boldsymbol{r},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\vartheta,\sigma) \right/ \left. \int\nolimits_\Delta \mathrm{d}^3{\boldsymbol{r}}\right\}, \end{equation} \tag{ 104 }$$

where the region of integration Δ is the volume between the surface labelled by $\tilde {\psi }$ and that labelled by $\tilde {\psi } + \Delta \psi$, and the integral is taken at constant ε_e, μ_e and ϑ. In appendix G, we prove that

$$\begin{equation} \left\langle {\widetilde{\boldsymbol{B}}}\cdot\nabla h_{\mathrm{e}}^{(1)}\right\rangle_{\tilde\psi} = 0, \end{equation} \tag{ 105 }$$

that the flux surface average commutes with the time derivative following the magnetic field, that

$$\begin{equation} \left\langle \frac{{B}}{|w_\parallel|}\boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla\psi\right\rangle_{\tilde\psi} = 0, \end{equation} \tag{ 106 }$$

and that, for any function a(r),

$$\begin{equation} \left\langle \frac{{B}}{|w_\parallel|}\left( {\widetilde{\boldsymbol{b}}}\times\nabla a \right) \cdot \nabla\tilde{\psi}\right\rangle_{\tilde\psi} = 0. \end{equation} \tag{ 107 }$$

We also prove that, for any fluctuating function λ,

$$\begin{equation} \left\langle \frac{{B}}{|w_\parallel|} \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla\lambda\right\rangle_{\tilde\psi} = 0. \end{equation} \tag{ 108 }$$

Taking (92) in the passing region of velocity space, multiplying by B/|w_∥|, and flux-surface averaging (to eliminate $w_\parallel {\widetilde {\boldsymbol {b}}}\cdot \nabla h_{\mathrm {e}}^{(1)}$), we obtain

$$\begin{eqnarray} &&\fl \left\langle \frac{{B}}{|w_\parallel|}\right\rangle_{\tilde\psi}\left. { \frac{\partial g_{\mathrm{pe}}} {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}} - \left\langle \frac{{B}}{|w_\parallel|} {{\left\langle {\,{C}\hspace{-0.5mm}\left[g_{\mathrm{pe}}\right]} \right\rangle}_{\boldsymbol{R}}}\right\rangle_{\tilde\psi}\nonumber\\ &&= -\frac{F_{0\mathrm{e}}}{T_{\mathrm{e}}}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left\langle \frac{{B}}{|w_\parallel|}\left(e\zeta-\mu_{\mathrm{e}}{ {\delta B_\parallel}}\right)\right\rangle_{\tilde\psi} - \left\langle {m_{\mathrm{e}} |w_\parallel| } { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi} - \psi)\right\rangle_{\tilde\psi} \frac{I(\psi)}{T_{\mathrm{e}}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} F_{0\mathrm{e}}\nonumber\\ &&\quad +\, \sigma \left\langle {B} F_{0\mathrm{e}}\left( \frac{e}{c{T_{\mathrm{e}}}} { \frac{\partial { {\delta A_\parallel^{(1)}}}} {\partial t} } + \frac{1}{{B}}\left\{ { { {\delta A_\parallel^{(1)}}}} , {\ln F_{0s}} \right\}\right)\right\rangle_{\tilde\psi}, \end{eqnarray} \tag{ 109 }$$

where the final term in (92) vanishes because in the passing region g_e is only a function of $\tilde {\psi }$ and we can use (107). We still have to eliminate the final line of (109). We note that we only require the part of g_pe that is even in σ—we require its density in the quasineutrality condition (29) and its pressure in perpendicular Ampère's law (32). Thus, we average (109) over the two signs of σ and henceforth g_pe denotes only the even part. Hence we obtain

$$\begin{eqnarray} &&\fl \left\langle \frac{{B}}{|w_\parallel|}\right\rangle_{\tilde\psi}\left. { \frac{\partial g_{\mathrm{pe}}} {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}} - \left\langle \frac{{B}}{|w_\parallel|} {{\langle {{C}[g_{\mathrm{pe}}]} \rangle}_{\boldsymbol{R}}}\right\rangle_{\tilde\psi} = -\frac{F_{0\mathrm{e}}}{T_{\mathrm{e}}}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left\langle \frac{{B}}{|w_\parallel|}\left(e\zeta-\mu_{\mathrm{e}}{ {\delta B_\parallel}}\right)\right\rangle_{\tilde\psi}\nonumber\\ &&- \left\langle {m_{\mathrm{e}} |w_\parallel| } { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi} - \psi)\right\rangle_{\tilde\psi} \frac{I(\psi)}{T_{\mathrm{e}}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} F_{0\mathrm{e}}. \end{eqnarray} \tag{ 110 }$$

This completes our solution for g_e, as (110) is a closed equation for the part of g_pe that is even in σ. It is useful to note that only the zonal (i.e. $\tilde {\alpha }$-independent) parts of ζ and δB_∥ will give a significant drive for g_pe—the $\tilde {\alpha }$-dependent parts will tend to cancel under the flux-surface average.

In this section, we finally close our system of equations by writing the field equations in terms of our solution for δf_e and ion quantities.

We find δφ' by using the solution (81) for δf_e in the quasineutrality condition (29):

$$\begin{equation} \fl \frac{1}{n_{\mathrm{e}}} \sum_{s=\mathrm{i}} \frac{Z_s^2 e n_s \delta \varphi'}{T_s} = \frac{e\zeta}{T_{\mathrm{e}}} + (\tilde{\psi}-\psi)\frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi} + \frac{1}{n_{\mathrm{e}}}\int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}} +\sum_{s=\mathrm{i}} \frac{Z_s}{n_{\mathrm{e}}} \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}. \end{equation} \tag{ 111 }$$

Similarly, we find δB_∥ by using (81) in (32) to find

$$\begin{eqnarray} &&\fl-\nabla^2_\perp \frac{{ {\delta B_\parallel}}{B}}{4\pi} = \sum_{s=\mathrm{i}}\nabla_\perp\nabla_\perp: \int m_s { {\left\langle \boldsymbol{w}_\perp \boldsymbol{w}_\perp h_s \right\rangle}_{\boldsymbol{r}}} + n_{\mathrm{e}}\nabla_\perp^2 \left(\zeta-\delta \varphi'\right) \nonumber\\ &&+\, n_{\mathrm{e}} \nabla_\perp^2(\tilde{\psi}-\psi) \left[T_{\mathrm{e}} \frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi} + \left( T_{\mathrm{e}} - e \varphi_0 \right) \frac{\mathrm{d}\ln T_{\mathrm{e}}}{\mathrm{d}\psi}\right]+ \nabla^2_\perp \int \mathrm{d}^3 \boldsymbol{w} \frac{m_{\mathrm{e}} w_\perp^2}{2} g_{\mathrm{e}}. \end{eqnarray} \tag{ 112 }$$

These two equations determine δφ' and δB_∥.

It is instructive to note that (111) and (112) are not naturally solved in the moving coordinate system. It is because of this coupling to the fixed coordinates (and the ion physics) that we require the mapping between the coordinate systems (from the solution of (67)–(69)).

In our collisional expansion, ${\,{C}\hspace {-0.5mm}\left [g_{\mathrm {e}}\right ]}$ is the dominant term in (92). Thus, up to corrections that are small in ω/ν_ee ≪ 1, g_e is given by

$$\begin{equation} {\,{C}\left[g_{\mathrm{e}}\right]} = 0. \end{equation}\noindent \tag{ 114 }$$

Hence, g_e is a perturbed Maxwellian to leading order:

$$\begin{equation} g_{\mathrm{e}} = \left\{\frac{\delta n_H}{N_{\mathrm{e}}} + \left( \frac{{{\varepsilon}_{\mathrm{e}}}}{T_{\mathrm{e}}} - \frac{3}{2} \right) \frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} \right\}F_{0\mathrm{e}} +\Or\left(\frac{\delta}{\nu^{*}_{\mathrm{e}}}\epsilon f_{\mathrm{e}}\right), \end{equation}\noindent \tag{ 115 }$$

where δn_H and δT_e are arbitrary functions of r – δn_H will shortly be shown to be the homogeneous (i.e. a function only of $\tilde {\psi }$) part of the electron density perturbation and δT_e the perturbed electron temperature.

Now, we know from section 3.1 that g_e is a solution of (49). Substituting this into (49), we see that δn_H/N_e and δT_e/T_e are functions of $\tilde {\psi }$ only (the derivatives in (49) being taken at constant ε_e). Thus, in the this limit, collisions trap and de-trap electrons rapidly enough that there is no longer a distinction between passing and trapped electrons ($\tilde {\alpha }$-dependence being the signature of kinetic trapped electrons)—all electrons sample the entire flux surface on the fluctuation timescale.

Let us now further simplify our solution for g_e. Substituting (115) into (81), we obtain

$$\begin{equation} \fl \delta f_{\mathrm{e}} = \left(\frac{e \zeta}{T_{\mathrm{e}}} + \frac{\delta n_H}{N_{\mathrm{e}}}\right) F_{0\mathrm{e}} + ( \tilde{\psi} - \psi ) { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } + \left( \frac{{{\varepsilon}_{\mathrm{e}}}}{T_{\mathrm{e}}} - \frac{3}{2} \right) \frac{\delta T_{\mathrm{e}}(\tilde{\psi})}{T_{\mathrm{e}}} F_{0\mathrm{e}} + \Or\left( \epsilon \delta F_{0\mathrm{e}} \right). \end{equation}\noindent \tag{ 116 }$$

From this expression, we observe that we were justified in our notation for δT_e—it really is the perturbed electron temperature relative to the moving surfaces. We also note that ζ and δn_H both contribute to the perturbed density. As we have the freedom to add an arbitrary function of $\tilde {\psi }$ to the definition of ζ—it does not change the velocity of the flux surface (see the end of section 3.2), we redefine ζ to be ζ − T_eδn_H/eN_e and so eliminate δn_H entirely. Thus, our solution for δf_e becomes

$$\begin{equation} \fl \delta f_{\mathrm{e}} = \frac{e \zeta}{T_{\mathrm{e}}} F_{0\mathrm{e}} + ( \tilde{\psi} - \psi ) { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } + \left( \frac{{{\varepsilon}_{\mathrm{e}}}}{T_{\mathrm{e}}} - \frac{3}{2} \right) \frac{\delta T_{\mathrm{e}}(\tilde{\psi})}{T_{\mathrm{e}}} F_{0\mathrm{e}} + \Or\left( \frac{\delta}{\nu^{*}_{\mathrm{e}}} \epsilon F_{0\mathrm{e}} \right), \end{equation}\noindent \tag{ 117 }$$

where the entire density perturbation is contained in the Boltzmann-like first term.

We now derive an equation for ζ in the collisional limit.

Using the solution (115) for g_e in (94), we obtain, to lowest order in δ/ν*_e,

$$\begin{eqnarray} &&\fl \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left( \frac{e\zeta}{T_{\mathrm{e}}} - \frac{e\varphi_0}{T_{\mathrm{e}}}\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} -\frac{{ {\delta B_\parallel}}}{{B}} \right) n_{\mathrm{e}} + {B} { \frac{\partial } {\partial \tilde{l}} } \frac{ \delta u_{\parallel \mathrm{e}}}{{B}} - \frac{cT_{\mathrm{e}}}{e{B}}\left\{ {\frac{{ {\delta B_\parallel}}}{{B}}} , {\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}}} \right\}n_{\mathrm{e}}\left(1-\frac{e\varphi_0}{T_{\mathrm{e}}}\right)\nonumber\\ &&=-\widehat{\boldsymbol{V}}_{\mathrm{D}}\cdot\nabla (\tilde{\psi}-\psi) \left(\frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi} + \frac{\mathrm{d}\ln T_{\mathrm{e}}}{\mathrm{d}\psi}\right)n_{\mathrm{e}} - \boldsymbol{V}_{\!\varphi}\cdot\nabla(\tilde{\psi}-\psi)\frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi}n_{\mathrm{e}} \nonumber\\ &&\quad +\, \frac{ec\varphi_0}{{B} T_{\mathrm{e}}}\left\{ {\zeta} , {\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}}} \right\}- \widehat{\boldsymbol{V}}_{\mathrm{D}}\cdot\nabla\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} n_{\mathrm{e}} + \left(\widehat{\boldsymbol{V}}_{\mathrm{D}} + \boldsymbol{V}_{\!\varphi}\right) \cdot\nabla\left(\delta \varphi'-\zeta\right) \frac{e n_{\mathrm{e}}}{T_{\mathrm{e}}} \nonumber\\ &&\quad +\,c { \frac{\partial } {\partial \tilde{\alpha}} } \left(\zeta - { {\delta B_\parallel}} \frac{T_{\mathrm{e}}}{e{B}} \right) { \frac{\partial \ln N_{\mathrm{e}}} {\partial \psi} } n_{\mathrm{e}} - c { \frac{\partial { {\delta B_\parallel}}} {\partial \tilde{\alpha}} } \frac{n_{\mathrm{e}} T_{\mathrm{e}}}{e{B}} { \frac{\partial \ln T_{\mathrm{e}}} {\partial \psi} } -\frac{In_{\mathrm{e}}}{{B}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi} - \psi),\nonumber\\ \end{eqnarray} \tag{ 118 }$$

where we have used the fact that, as we have absorbed the density perturbation into ζ, $\int \mathrm {d}^3 \boldsymbol {w} g_{\mathrm {e}} = \Or( (\delta /\nu ^{*}_{\mathrm {e}})\epsilon n_{\mathrm {e}} )$. As in section 4.1, this is just the fluctuating continuity equation for electrons.

To close our system of collisional equations, we now need the fluctuating energy equation from which we will determine δT_e. Thus, we multiply (92) by m_ew²/(2T_e) − 3/2 and integrate over all velocities to find

$$\begin{eqnarray} &&\fl \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left(\frac{3}{2}\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} - \frac{{ {\delta B_\parallel}}}{B}\right) n_{\mathrm{e}}+ \nabla\cdot\left({\widetilde{\boldsymbol{b}}} q_{\parallel \mathrm{e}}\right) +\frac{c}{{B}}\left\{ {\zeta - \frac{T_{\mathrm{e}}}{e}\frac{{ {\delta B_\parallel}}}{{B}}} , {\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}}} \right\}n_{\mathrm{e}} \nonumber\\ &&= - \frac{1}{2}\left(7\widehat{\boldsymbol{V}}_{\mathrm{D}}+3\boldsymbol{V}_{\!\varphi}\right)\cdot\nabla ( \tilde{\psi}-\psi ) { \frac{\partial } {\partial \tilde{\psi}} } \left( \frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} + \ln T_{\mathrm{e}} \right) n_{\mathrm{e}} \nonumber\\ &&\quad -\, \widehat{\boldsymbol{V}}_{\mathrm{D}}\cdot\nabla( \tilde{\psi} -\psi ) { \frac{\partial \ln N_{\mathrm{e}}} {\partial \tilde{\psi}} } n_{\mathrm{e}} + \widehat{\boldsymbol{V}}_{\mathrm{D}} \cdot\nabla\left(\delta \varphi'-\zeta\right) \frac{e n_{\mathrm{e}}}{T_{\mathrm{e}}}\nonumber\\ &&\quad -\, { \frac{\partial { {\delta B_\parallel}}} {\partial \tilde{\alpha}} } \frac{cT_{\mathrm{e}} n_{\mathrm{e}}}{e{B}} \left( { \frac{\partial \ln N_{\mathrm{e}}} {\partial \psi} } + { \frac{\partial \ln T_{\mathrm{e}}} {\partial \psi} } \right) -\frac{In_{\mathrm{e}}}{{B}}\frac{\mathrm{d}\omega}{\mathrm{d}\psi} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi}-\psi), \end{eqnarray} \tag{ 119 }$$

where $q_{\parallel e} = \int \mathrm {d}^3 \boldsymbol {w} \left ( {m_{\mathrm {e}} w^2}/{2T_{\mathrm {e}}} - {3}/{2} \right ) h_{\mathrm {e}}^{(1)}$. As we do not know q_∥e, we wish to eliminate it from this equation. Therefore, we average (119) over the perturbed flux surfaces and obtain

$$\begin{eqnarray} &&\fl { \frac{\partial } {\partial t} } \left(\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}}\left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi} + \left\langle \frac{2{ {\delta B_\parallel}}}{3B}n_{\mathrm{e}}\right\rangle_{\tilde\psi}\right) + \left\langle \boldsymbol{V}_{\!\varphi}\cdot\nabla\left(\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} n_{\mathrm{e}}\right)\right\rangle_{\tilde\psi}= \nonumber\\ &&-\,\left\langle \left(\frac{7}{3}\widehat{\boldsymbol{V}}_{\!\mathrm{D}}+\boldsymbol{V}_{\!\varphi}\right)\cdot\nabla (\tilde{\psi}-\psi) { \frac{\partial \ln T_{\mathrm{e}}} {\partial \tilde{\psi}} } n_{\mathrm{e}}\right\rangle_{\tilde\psi} -\frac{2}{3}\left\langle \widehat{\boldsymbol{V}}_{\mathrm{D}}\cdot\nabla(\tilde{\psi}-\psi) { \frac{\partial \ln N_{\mathrm{e}}} {\partial \tilde{\psi}} } n_{\mathrm{e}}\right\rangle_{\tilde\psi} \nonumber\\ &&-\, \frac{2}{3}\left\langle \frac{n_{\mathrm{e}}}{{B}} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi}-\psi)\right\rangle_{\tilde\psi} I(\psi) \frac{\mathrm{d}\omega}{\mathrm{d}\psi}, \end{eqnarray} \tag{ 120 }$$

where we have used (G.6) to eliminate some terms.

Let us examine the terms in (120). Other than the last term on the right-hand side, which is due to the viscous heating of electrons, these terms seem fairly opaque. However, they all vanish if we apply the low-Mach-number ordering from section 11.1 of [1]—we can move n_e outside the flux-surface averages and then use the properties of the flux-surface average and the drift velocities to eliminate the terms in question. Thus, we conclude that these terms, which disappear if the Mach number is small, are to do with the exchange of potential energy of the electrons with thermal energy.

Equation (120) completes our solution for δf_e. In the next section, we close the system by working out the electron contributions to the field equations in the collisional limit.

We now use our solution for δf_e to simplify the field equations. Substituting the solution (117) into (111) gives the following equation for δφ':

$$\begin{eqnarray} &&\fl \frac{1}{n_{\mathrm{e}}} \sum_{s=\mathrm{i}} \frac{Z_s^2 e n_s \delta \varphi'}{T_s} = \frac{e\zeta}{T_{\mathrm{e}}} + (\tilde{\psi}-\psi)\frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi}- \frac{e\varphi_0}{T_{\mathrm{e}}}\frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} +\sum_{s=\mathrm{i}} \frac{Z_s}{n_{\mathrm{e}}} \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}. \end{eqnarray} \tag{ 121 }$$

Similarly, by substituting (117) into (112), we obtain an equation for δB_∥ in the collisional limit:

$$\begin{eqnarray} &&\fl - \nabla^2_\perp \frac{{ {\delta B_\parallel}}{B}}{4\pi} = \sum_{s=\mathrm{i}}\nabla_\perp\nabla_\perp: \int m_s { {\left\langle \boldsymbol{w}_\perp \boldsymbol{w}_\perp h_s \right\rangle}_{\boldsymbol{r}}} + n_{\mathrm{e}}\nabla_\perp^2 \left(\zeta-\delta \varphi'\right)\nonumber\\ &&+\, n_{\mathrm{e}} \nabla_\perp^2(\tilde{\psi}-\psi) \left[T_{\mathrm{e}} \frac{\mathrm{d}\ln N_{\mathrm{e}}}{\mathrm{d}\psi} + \left( T_{\mathrm{e}} - e \varphi_0 \right) \frac{\mathrm{d}\ln T_{\mathrm{e}}}{\mathrm{d}\psi}\right] + \left(1-\frac{e\varphi_0}{T_{\mathrm{e}}}\right)n_{\mathrm{e}} \nabla^2_\perp \delta T_{\mathrm{e}}. \end{eqnarray} \tag{ 122 }$$

The field equations (121) and (122) close the collisional electron model, which we now summarize.

The collisional electron model is a simplified version of the model given in section 5. We have replaced the distribution functions g_pe and g_te by a single perturbed electron density and temperature. The complete set of equations describing this model is:

• (67) and (68) to evolve $\tilde {\psi }$ and $\tilde {\alpha }$ which define the moving coordinate system,
• (87) to determine δu_∥e,
• ζ is given by (118),
• δT_e is given by (119),
• (121) and (122) determine the fields δφ' and δB_∥ respectively,
• and the ion dynamics (h_i) are given by (24).

In section H.3 of the appendix, we derive the electron particle transport equation, correct to lowest order in δ:

$$\begin{equation} \frac{1}{V'}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} V'\left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi} + \frac{1}{V'} { \frac{\partial } {\partial \tilde{\psi}} } V'\left\langle \Gamma_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle S^{({n})}_{\mathrm{e}}\right\rangle_{\tilde\psi}, \end{equation}\noindent \tag{ 123 }$$

where S⁽ⁿ⁾_e is defined by (164) and the particle flux is given by

$$\begin{equation} \left\langle \Gamma_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle { {\left\langle c \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} \frac{\delta B_\parallel}{B}\right) \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}. \end{equation}\noindent \tag{ 124 }$$

Note that whilst we use the same notation for the particle flux Γ_e as in [1], it is not the same quantity. In this paper the transport equations are derived for transport across a $\tilde {\psi }=\mathrm {const.}$ surface not a ψ = const. surface.

Several important features of this flux immediately present themselves. Firstly, only the trapped electrons contribute—the passing particles play no part in the transport. Secondly, there is no 'flutter' transport (transport due to correlations between δu and δB_⊥ despite the fact that the contribution to (170) of [1] from δA_∥ is clearly not zero. This is, in fact, the primary advantage of writing our transport equations with respect to the moving surfaces, we observe this cancellation analytically rather than having to recover it via a very careful evaluation of (166) and (170) of [1]. Finally, we see that drifts that do cause transport are the drift due to ζ and the ∇B drift due to δB_∥. This supports our interpretation of the drift due to ζ being the drift of electrons across the moving surfaces.

7.1.1. Electron transport in the collisional model

Similarly, in section I.4 of the appendix, we derive the electron heat transport equation, to lowest order in δ:

$$\begin{eqnarray} &&\fl \frac{1}{V'}\frac{3}{2}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} V' \left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi}T_{\mathrm{e}} + \frac{1}{V'} { \frac{\partial } {\partial \tilde{\psi}} } V' \left\langle q_{\mathrm{e}}\right\rangle_{\tilde\psi} =P^{\mathrm{turb}}_{\mathrm{e}} + P^{\mathrm{comp}}_s+ P^{\mathrm{pot}}_s+ C^{{(E)}}_s+ S^{({E})}_{s}, \end{eqnarray} \tag{ 125 }$$

where C^(E)_s and S^(E)_s are defined by (192) and (193) of [1] respectively, the heat flux is given by

$$\begin{equation} \fl \left\langle q_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle { {\left\langle {c}\int \mathrm{d}^3 \boldsymbol{w} \left( \frac{1}{2} m_{\mathrm{e}} w^2 - e \varphi_0 \right) g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} \frac{{ {\delta B_\parallel}}}{B}\right) \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}, \end{equation} \tag{ 126 }$$

and the heat sources are given by

$$\begin{equation} P^{\mathrm{comp}}_s= -T_{\mathrm{e}} \left\langle n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi}, \end{equation} \tag{ 127 }$$

$$\begin{equation} P^{\mathrm{pot}}_s= \left\langle e\varphi_0 \left(\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}n_{\mathrm{e}} -S^{({n})}_{\mathrm{e}}\right) \right\rangle_{\tilde\psi} + \left\langle e\varphi_0 n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi} + \left\langle e\varphi_0 { {\left\langle \left( \frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}} \right) \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} \end{equation} \tag{ 128 }$$

and

$$\begin{equation} \fl P^{\mathrm{turb}}_{\mathrm{e}} = -\left\langle \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle g_{\mathrm{e}}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left(e\zeta - \mu_{\mathrm{e}}{ {\delta B_\parallel}}\right) \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} + e\left\langle n_{\mathrm{e}} { {\left\langle \zeta\nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}. \end{equation} \tag{ 129 }$$

In these expressions for the heating, the divergence of u_eff is given by

$$\begin{equation} \fl\nabla\cdot\boldsymbol{u}_{\mathrm{eff}} =\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{{ {\delta B_\parallel}}}{{\widetilde{B}}} +\left( {\widetilde{\boldsymbol{b}}}\cdot\nabla\psi \right) \frac{I(\psi)}{{B}} \frac{\mathrm{d}\omega}{\mathrm{d}\psi} - \frac{c}{B}\left( {\boldsymbol{b}}\cdot\nabla{\boldsymbol{b}} \right)\cdot {\boldsymbol{b}} \times \nabla\left( \zeta-\delta \varphi' \right). \end{equation} \tag{ 130 }$$

This expression is correct and closed to the required order in and δ. Again, it is crucial to note that whilst we use the same notation for q_e, P^comp_s, P^pot_s, and P^turb_e as in [1], these are not the same quantities.

Let us now interpret the heat flux and the heating terms. The heat flux parallels the particle flux: there is no flutter transport, only trapped particles contribute to the flux, and they contribute via the E × B-drift across flux surfaces and the ∇B-drift due to δB_∥.

Of the heating terms, the first, P^comp_s is just the compressional heating due to the mean motion of the flux surfaces. The second, P^pot_s, is the change in the thermal energy of the electrons due to the change in their electrostatic potential energy¹⁸. Finally, we have the turbulent heat source P^turb_e. We interpret the terms in the expression (129) for P^turb_e as follows: the first term is heating due to work done against the parallel electric field (through ζ); the term involving the time-derivative of δB_∥ is in fact composed of the electron viscous heating, compressional heating due to the fluctuating motion of the $\tilde {\psi }=\mathrm {const.}$ surfaces and work done bending the magnetic field. This can be seen by substituting for δB_∥ from (130)—the term on the left-hand side of (130) is the compressional heating, the second term on the right is part of the electron viscous heating. The final term is the potential energy contribution to the compressional heating due to the fluctuating motion of the exact flux surfaces.

7.2.1. Electron heating in the collisional model

Again, in the collisional limit, the flux in (125) vanishes. We are just left with local heating terms, of which only the turbulent heating simplifies

$$\begin{eqnarray} &&P^{\mathrm{turb}}_{\mathrm{e}} = \left\langle { {\left\langle n_{\mathrm{e}}\delta T_{\mathrm{e}}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{{ {\delta B_\parallel}}}{{B}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} + e\left\langle n_{\mathrm{e}} { {\left\langle \zeta\nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}, \end{eqnarray} \tag{ 131 }$$

where ∇·u_eff is given by (130) as before. Thus, despite the lack of any transport, the electron response to ion-scale turbulence can still dissipate free energy. All other terms in (125) remain unchanged.

As the momentum transport involves all species, not merely electrons, we do not write a new equation for it in the moving frame. The transport equation for toroidal angular momentum is (equation (179) of [1])

$$\begin{equation} \frac{1}{V'}\left. { \frac{\partial } {\partial t} } \right|_{\psi}{{V'J \omega(\psi)}} + \frac{1}{V'} { \frac{\partial } {\partial \psi} } V' {\left\langle {\pi}^{(\psi\phi)}_{\mathrm{tot}} \right\rangle_{\psi}} = {\left\langle S^{({\omega})} \right\rangle_{\psi}}, \end{equation} \tag{ 132 }$$

where the moment of inertia of the plasma is now given by

$$\begin{equation} J = \sum_{s=\mathrm{i}} m_s {\left\langle n_s R^2 \right\rangle_{\psi}}, \end{equation} \tag{ 133 }$$

and the total momentum flux is

$$\begin{eqnarray} &&\fl {\langle {\pi}^{(\psi\phi)}_{\mathrm{tot}} \rangle_{\psi}} = {\langle \pi^{(\psi\phi)}_{\mathrm{e}} \rangle_{\psi}} + \sum_{s=i} \left( {\left\langle \pi^{(\psi\phi)}_{s} \right\rangle_{\psi}} + m_s\omega(\psi) {\left\langle R^2\Gamma_{s} \right\rangle_{\psi}}\right) \nonumber\\ &&- \sum_{s=i}\frac{Z_s e}{c} {\left\langle \int \mathrm{d}^3 \boldsymbol{w} { {\langle { {\langle h_s \boldsymbol{w}\cdot\nabla\psi \rangle}_{\boldsymbol{r}}}{ {\delta\boldsymbol{A}}} \rangle}_{\mathrm{turb}}}\cdot R^2\nabla\phi \right\rangle_{\psi}} \nonumber\\ &&- \frac{1}{4\pi} {\langle \nabla\psi \cdot { {\langle { {\delta\boldsymbol{B}}}{ {\delta\boldsymbol{B}}} \rangle}_{\mathrm{turb}}}\cdot R^2\nabla\phi \rangle_{\psi}}, \end{eqnarray} \tag{ 134 }$$

where the sums over s = i are taken over all ion species and fluxes π^(ψϕ)_s and Γ_s for the ions are given by (184) and (170) of [1], respectively. The electron momentum flux is now (see section H.5 in the appendix)

$$\begin{equation} {\left\langle \pi^{(\psi\phi)}_{\mathrm{e}} \right\rangle_{\psi}} = {\left\langle I(\psi) { {\left\langle \int \mathrm{d}^3 \boldsymbol{w} \left( m_{\mathrm{e}} w_\parallel^2 - \frac{1}{2} m_{\mathrm{e}} w_\perp^2\right) g_{\mathrm{e}} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi}-\psi) \right\rangle}_{\mathrm{turb}}} \right\rangle_{\psi}}. \end{equation} \tag{ 135 }$$

The three evolution equations (123), (125) and (132) close our system on the transport timescale. The fluxes are written entirely in terms of fluctuating quantities that our reduced electron model provides (note that nowhere do we require the part of g_pe that is odd in σ).

7.3.1. The collisional limit

Many previous works have introduced physically-motivated model responses for the electrons. Some are based on the linear electron response [10] and others use fluid moments of the electron distribution function to try and capture the passing particle response [11, 14, 27]. Our model improves upon this by being a rigorous consequence of the electron gyrokinetic equation, clearly demonstrating how the electron response fits into the full turbulent transport framework. Also, by retaining the bounce- and flux-surface-averaged kinetics of the electrons, we retain the distinction between the passing and trapped populations and retain drift resonances that may be lost in a fluid picture. We recover some of the results of the fluid models in the collisional limit, where there are no trapped electrons. In particular, the fact that δT_H is only a function of $\tilde {\psi }$ means that the temperature has equilibrated along perturbed field lines as expected. Even in this limit, the previous models did not allow for a rapid toroidal rotation (comparable to the ion thermal speed). Also all these models neglect δB_∥ and only retain δA_∥ in the perturbed magnetic field.

One of the original and still widely-used models for the electron response is the adiabatic electron response corrected for zonal perturbations [28]. It can be shown that this is a rigorous limit of our model in the limit of vanishing β (the electrostatic limit) and collisional electrons. For collisionless electrostatic turbulence, similar equations have been derived rigorously in [9]. Our model extends this to include fully electromagnetic turbulence and rapid toroidal rotation. The electrostatic model of [9] also neither handles general magnetic geometry nor discusses the behaviour of passing electrons. This limit is useful as an easily-implemented stepping stone between the adiabatic electron model (strictly only valid for quite collisional plasmas) and simulating the full electron gyrokinetic equation (which is either computationally prohibitive or formally invalid depending on whether one chooses to resolve the electron scales or not).

The most complete previous model is that of [12]. However, this model is derived under two restrictive assumptions. Firstly, the mass ratio is assumed to be much smaller than we assume here (δ ∼  rather than 1 ≫ δ ≫  which we assume) and secondly δB_∥ is assumed to be identically zero. We note that (81) corresponds directly to equation (54) of [12], but that the further assumption made in [12] of (in our notation) g_pe = 0 is unjustified (g_pe = 0 is not a solution of (110)) and hence neglects the electron response that is zonal with respect to the perturbed flux surfaces. In addition, the model in [12] does not include the effects of sonic toroidal rotation.

Finally, we must discuss how our model compares to the common practice of simulating the full electron gyrokinetic equation at ion scales (so-called 'kinetic electrons'). Firstly, the electron gyrokinetic equation clearly contains electron-gyroscale and electron-bounce-frequency physics so simulating only the ion scales is formally invalid—deliberately leaving the simulation unresolved in some sense. Secondly, how an unresolved simulation of the electron gyrokinetic equation behaves will be sensitive to the numerical algorithm used—it is possible to design a numerical algorithm that numerically bounce-averages the kinetic equation in the long-timestep limit, but of the commonly-used gyrokinetic codes only GS2 [29] does this. Indeed, the only way to confirm that simulating 'kinetic electrons' gives the correct physics is to either compare with full two-scale simulations that resolve the electron dynamics as well as the ion dynamics or to compare with simulations of the electron model presented in this paper. As well-resolved two-scale simulations are generally beyond current computational resources, simulations of our model are the only way to confirm the validity of this practice.

In this paper we have derived equations describing the electron response to ion-scale turbulence. The fast electron timescales have been eliminated from this system. The scale separation between electron and ion scales both motivates and enables the derivation of such a model. Starting from electron gyrokinetics [1], we systematically expand our equations in the square root of the electron-to-ion mass ratio to obtain our model. As discussed in the introduction, as studies of turbulence at non-zero β become more and more important – both for conventional and spherical tokamaks—having an electron response model that handles this rigorously will be necessary. We have derived full (β ∼ 1) collisional and arbitrary-collisionality models—summarized in sections 5 and 6 respectively.

The procedure by which we constructed this model, a rigorous expansions in $\sqrt {m_{\mathrm {e}}/m_{\mathrm {i}}}\ll 1$, allowed us to clearly delineate which effects are due to electron-scale physics and which to ion-scale. Most importantly, the elimination of the electron scales has a profound impact upon the structure of the fluctuating magnetic field. We prove in section 3.2, that purely ion-scale turbulence conserves the total magnetic flux, and, moreover that a velocity field can be found into which the magnetic field is frozen. This generalization of the usual MHD frozen-in theorem demonstrates that it is only electron-scale effects which can change the magnetic topology. The inclusion of β ∼ 1, trapped particles, sonic rotation, and arbitrary axisymmetric geometry do not change this fundamental fact.

Similarly, whilst deriving transport equations across the exact flux surfaces, we have obtained another general property of ion-scale turbulence: passing electrons do not contribute to transport fluxes (see section 7). This result could prove useful in designing magnetic geometries that have beneficial trapped-particle properties, safe in knowledge that the passing particles never contribute to the transport.

In addition to these general results, two other ancillary results have been proved in the course of this work. Firstly, in appendix C, we proved that the topology of the mean magnetic field (i.e. the q(ψ) profile) evolves on the resistive rather than the transport timescale. Importantly, this result holds independently of the nature of the turbulence. Secondly, in appendix J, we prove that the commonly-used adiabatic electron model is, in fact, the correct model for collisional (ν ≫ ω) and electrostatic (β ≪ 1) electrons.

Throughout our derivation, we have simply assumed that the magnetic stochasticity produced by electron-scale fluctuations is sufficiently small that we can neglect it. This assumption must be investigated in detail and will be explored in future work. Similarly, the potential effects of small resonant regions around rational surfaces (which would give a finite width to each $\tilde {\psi }$-constant surface) should be examined closely. Indeed, a natural extension of this work would be to include the reaction of a small amount of purely electron-scale turbulence upon the ion scales. As always, the final arbiter of the usefulness of these models will be their numerical implementation and application to simulating moderate- and high-β ion-scale turbulence, in e.g. nonlinear studies of finite-β ITG turbulence or kinetic ballooning mode turbulence.

In order to derive the transport equations, we will need some new averages. We define a new average over the fluctuations where the perpendicular spatial integrals are taken between surfaces of constant $\tilde {\psi }$ and $\tilde {\alpha }$ rather than ψ and α. Similarly, the time integral in this average is taken at constant $\tilde {\psi }$, $\tilde {\alpha }$ and $\tilde {l}$. We denote this new average by ${ {\left \langle \cdot \right \rangle }_{\mathrm {turb}}'}$

$$\begin{equation} { {\left\langle g \right\rangle}_{\mathrm{turb}}'} = \left. \int \mathrm{d}\tilde{\psi} \int \mathrm{d}\tilde{\alpha} \int\nolimits_{T/2}^{T/2} \mathrm{d}t g \right/ T\int \mathrm{d}\tilde{\psi} \int \mathrm{d}\tilde{\alpha}, \end{equation} \tag{ H.4 }$$

where the spatial integral is taken over a small patch that has dimension λ (the intermediate length scale) in each direction—for reference compare with (15) of [1].

This average commutes with time derivatives at constant $\tilde {\psi }$ and with flux-surface averages over $\tilde {\psi }$-constant surfaces.

As usual, the composition of this average and the average over $\tilde {\psi }$-constant surfaces is an average over time composed with an average over an annular region between $\tilde {\psi } = \tilde {\psi }_0$ and $\tilde {\psi }_0+\Delta$. It is important to note that this average differs from the turbulence average in [1] only by terms that are small in . This will be crucial, as we wish to replace all instances of this average by their unprimed version in the final equations.

We will want to average the transport equations over the fluctuating flux surfaces. For this we need two identities. The first is just a specific case of (G.3). If A(r,t) is a function only of space and time then

$$\begin{equation} \left\langle \nabla\cdot\boldsymbol{A}\right\rangle_{\tilde\psi} = \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \langle \tilde{V}' \boldsymbol{A}\cdot\tilde{\psi}\rangle_{\tilde\psi}, \end{equation}\noindent \tag{ H.5 }$$

where the derivative on the right is taken at constant t only.

Secondly we need to interchange flux-surface averages and time derivatives. We do this via

$$\begin{equation} \left\langle { \frac{\partial g} {\partial t} } \right\rangle_{\tilde\psi} = \frac{1}{\tilde{V}'} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}'\left\langle g\right\rangle_{\tilde\psi} - \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \left\langle \tilde{V}' g \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}, \end{equation} \tag{ H.6 }$$

which is proven in an identical fashion to (45) of [1] except that we retain the velocity explicitly rather than replacing it with ${\partial \tilde {\psi }}/{\partial t}$ via (67).

We now move to the meat of the problem. Deriving our first transport equation. Integrating (H.2) over all velocities

$$\begin{equation} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{\int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}}} + \nabla\cdot \left(\int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\right) + \left( \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right) \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}} = {S^{({n})}_{s}}, \end{equation}\noindent \tag{ H.7 }$$

where we have used the definition of the time derivative at constant $\tilde {\psi }$, $\tilde {\alpha }$ and $\tilde {l}$. Averaging this equation over the fluctuating flux surfaces we obtain

$$\begin{equation} \frac{1}{\tilde{V}'}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}}\right\rangle_{\tilde\psi} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}' \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi} = {S^{({n})}_{s}}, \end{equation}\noindent \tag{ H.8 }$$

where we have used (H.5) and (H.6). We now multiply this equation by $\tilde {V}'$, apply the new average over the fluctuations, and divide by $\tilde {V}'$ to find

$$\begin{equation} \frac{1}{\tilde{V}'}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}'\left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}'\left\langle \Gamma_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle S^{({n})}_{s}\right\rangle_{\tilde\psi}, \end{equation} \tag{ H.9 }$$

where we have used the fact that ${ { \langle \tilde {V}' \rangle }_{\mathrm {turb}}}' = V'(\tilde {\psi }) = \tilde {V}' + \Or(\epsilon V')$ and the fact that S⁽ⁿ⁾_s ∼ n_e/τ_E. The particle flux Γ_e is defined by

$$\begin{equation} \left\langle \Gamma_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} { {\left\langle f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}'} \right\rangle_{\tilde\psi}. \end{equation} \tag{ H.10 }$$

In section I.3 of the appendix, we use the kinetic equation to find the following explicit expression for the particle flux:

$$\begin{equation} \left\langle \Gamma_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} { {\left\langle g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( c\zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {{ {\delta B_\parallel}}} \right) \right\rangle}_{\mathrm{turb}}} \right\rangle_{\tilde\psi}. \end{equation} \tag{ H.11 }$$

Multiplying (H.2) by ${m_s\widetilde {{w}}^2}/{2}$, integrating over all velocities and averaging over flux-surfaces and the turbulence we obtain

$$\begin{eqnarray}&&\fl \frac{1}{\tilde{V}'}\frac{3}{2}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}' \left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi}T_{\mathrm{e}} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}' \left\langle Q_{\mathrm{e}}\right\rangle_{\tilde\psi} \nonumber\\ &&= { {\left\langle \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left(e\nabla\xi - { \frac{\partial \boldsymbol{u}_{\mathrm{eff}}} {\partial t} } - \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\right)\cdot\widetilde{\boldsymbol{w}} f_s \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} + C^{{(E)}}_s+ S^{({E})}_{s}, \end{eqnarray} \tag{ H.12 }$$

where the electron thermal energy flux is defined by

$$\begin{equation} \left\langle Q_{\mathrm{e}}\right\rangle_{\tilde\psi} = { {\left\langle \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{1}{2} m_{\mathrm{e}} \widetilde{{w}}^2 f_{\mathrm{e}}\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{equation} \tag{ H.13 }$$

We show in section I.4 of the appendix that we can rewrite this equation as

$$\begin{eqnarray}&&\fl \frac{1}{\tilde{V}'}\frac{3}{2}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}' \left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi}T_{\mathrm{e}} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}' \left\langle q_{\mathrm{e}}\right\rangle_{\tilde\psi} \nonumber\\ &&=P^{\mathrm{turb}}_{\mathrm{e}} + P^{\mathrm{comp}}_s+ P^{\mathrm{pot}}_s+ C^{{(E)}}_s+ S^{({E})}_{s}, \end{eqnarray} \tag{ H.14 }$$

The heating terms are, in order, the turbulent heating

$$\begin{eqnarray}&&\fl P^{\mathrm{turb}}_{\mathrm{e}} = -\left\langle \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle g_{\mathrm{e}}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left(e\zeta - \mu_{\mathrm{e}}{ {\delta B_\parallel}}\right) \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} + e\left\langle n_{\mathrm{e}} { {\left\langle \zeta\nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}, \end{eqnarray} \tag{ H.15 }$$

the mean compressional heating

$$\begin{equation} P^{\mathrm{comp}}_s= -T_{\mathrm{e}} \left\langle n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi}, \end{equation} \tag{ H.16 }$$

where V _ψ is the velocity of the mean flux surfaces defined in (44) of [1], and the exchange between thermal energy and electrostatic potential energy:

$$\begin{equation} P^{\mathrm{pot}}_s= \left\langle\! e\varphi_0 \left(\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}n_{\mathrm{e}} -S^{({n})}_{\mathrm{e}}\!\right)\! \right\rangle_{\tilde\psi} + \left\langle e\varphi_0 n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi} + \left\langle e\varphi_0 { {\left\langle \left( \frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}}\! \right) \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}. \end{equation} \tag{ H.17 }$$

In (H.17) and (H.15), the divergence of u_eff is given by

$$\begin{eqnarray}&&\fl \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} =\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{{ {\delta B_\parallel}}}{{\widetilde{B}}} +\left( {\widetilde{\boldsymbol{b}}}\cdot\nabla\psi \right) \frac{I}{{B}} \frac{\mathrm{d}\omega}{\mathrm{d}\psi} - \frac{c}{B}\left( {\boldsymbol{b}}\cdot\nabla{\boldsymbol{b}} \right)\cdot {\boldsymbol{b}} \times \nabla\left( \zeta-\delta \varphi' \right). \end{eqnarray} \tag{ H.18 }$$

This expression is correct and closed to the required order in and δ.

In section I.5 of the appendix, we show that the heat flux can be written in terms of g_e as

$$\begin{equation}\fl \left\langle q_{\mathrm{e}}\right\rangle_{\tilde\psi} = \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left(\frac{1}{2}m_{\mathrm{e}} w^2 - e\varphi_0 - e \zeta\right) { {\left\langle g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( c\zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {{ {\delta B_\parallel}}} \right) \right\rangle}_{\mathrm{turb}}} \right\rangle_{\tilde\psi}. \end{equation} \tag{ H.19 }$$

We start from (C.15) of [1] and drop all terms that are small in δ to obtain

$$\begin{equation}\fl \left(\nabla\psi\right)\cdot {\boldsymbol{\mathsf{\Pi}}}_s\cdot\left(R^2\nabla\phi\right) = { {\left\langle m_{\mathrm{e}} R^4 \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle \left(\boldsymbol{w}\cdot\nabla\phi\right)\left[\left(\boldsymbol{w}\times{ {\delta\boldsymbol{B}}}\right) \cdot\nabla\phi\right]\delta f_{\mathrm{e}} \right\rangle}_{\boldsymbol{r}}} \right\rangle}_{\mathrm{turb}}}, \end{equation} \tag{ H.20 }$$

where we have expanded δa_s and only kept terms that are of order ²R²Bn_iT_i, i.e. comparable to the ion momentum flux. Substituting for δf_e from (81) and performing the gyroaverage gives (134).

We now collate results from the rest of the paper to write down a convenient form for f_e. First, using the general solution (13) for f_e, we can write

$$\begin{eqnarray}&&\fl f_{\mathrm{e}} = F_{0\mathrm{e}}(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}'),{{\varepsilon}_{\mathrm{e}}})\left(1+\frac{e\delta \varphi'}{T_{\mathrm{e}}}\right) + h_{\mathrm{e}}'(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}'),\tilde{\alpha}(\boldsymbol{R}_{\mathrm{e}}'),{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}},\vartheta,t) + F_{1\mathrm{e}}(\boldsymbol{R}_{\mathrm{e}}',{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}) + f_{2\mathrm{e}}, \end{eqnarray} \tag{ I.1 }$$

where f_2e ∼ ²f_e and we have defined $h_{\mathrm {e}}' = h_{\mathrm {e}} - [\tilde {\psi }(\boldsymbol {R}_{\mathrm {e}}') - \psi (\boldsymbol {R}_{\mathrm {e}}')] {\partial F_{0\mathrm {e}}}/{\partial \tilde {\psi }}$ in order to be able to evaluate F_0e at $\tilde {\psi }(\boldsymbol {R}_{\mathrm {e}}')$. In addition, we have introduced the guiding centre position defined with respect to the exact field

$$\begin{equation} \boldsymbol{R}_{\mathrm{e}}' = \boldsymbol{r} - cm_{\mathrm{e}} \frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{e{\widetilde{B}}}, \end{equation} \tag{ I.2 }$$

which differs from the guiding centre with respect to B – R_e—by $\Or(\epsilon ^2 \boldsymbol {R}_{\mathrm {e}}')$.

We now use the solutions for the distribution function obtained in this paper. For F_1e, we have from (C.11), that

$$\begin{equation} F_{1\mathrm{e}} = \frac{n_{1\mathrm{e}}}{n_{\mathrm{e}}} F_{0\mathrm{e}} + F_{1\mathrm{e}}^{(\mathrm{H})}(\boldsymbol{r},{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}) + \Or(\delta^2 \epsilon^2 f_{\mathrm{e}}), \end{equation} \tag{ I.3 }$$

where F^(H)_1e ∼ δf_e contains the higher-order (in δ) contributions to F_1e. We also wish to write F_0e in terms of the new velocity variable $\widetilde {\boldsymbol {w}}$. To this end, we introduce the following quantity:

$$\begin{eqnarray}&&\fl F_{\mathrm{e}}' = \left( \frac{m_{\mathrm{e}}}{2\pi T_{\mathrm{e}}(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}'))} \right)^{3/2} \exp\left[ \frac{e\xi}{T_{\mathrm{e}}(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}'))} + \frac{m_{\mathrm{e}} \omega^2(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}')) R^2}{2 T_{\mathrm{e}}(\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}'))} - \frac{m_{\mathrm{e}} \widetilde{{w}}^2}{2T_{\mathrm{e}}(\psi(\boldsymbol{R}_{\mathrm{e}}'))}\right] \nonumber\\ &&+ \frac{m_{\mathrm{e}} \left(\boldsymbol{u}_{\mathrm{eff}}-\boldsymbol{u}\right)\cdot\widetilde{\boldsymbol{w}}}{T_{\mathrm{e}}} F_{0\mathrm{e}}. \end{eqnarray} \tag{ I.4 }$$

Using (60) and (80), we obtain the following expression for ξ:

$$\begin{equation} \frac{e\xi}{T_{\mathrm{e}}} = \ln N_{\mathrm{e}}(\tilde{\psi}) + \frac{e\varphi_0}{T_{\mathrm{e}}} + \frac{n_{1\mathrm{e}}}{n_{\mathrm{e}}} + \frac{e\zeta}{T_{\mathrm{e}}}, \end{equation} \tag{ I.5 }$$

in which T_e is evaluated at $\tilde {\psi }$ and we have let $\widetilde {\xi } = 0$. Substituting this into (I.4), and using the definition (21) of F_0e, we can show that

$$\begin{equation} F_{0\mathrm{e}}\left(1 + \frac{n_{1\mathrm{e}}}{n_{\mathrm{e}}} + \frac{e \delta \varphi'}{T_{\mathrm{e}}}\right) + F_{1\mathrm{e}}^{(\mathrm{H})} = F_{\mathrm{e}}' - \frac{e\left(\zeta-\delta \varphi'\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + F_{1\mathrm{e}}^{(\mathrm{H})'} + \Or(\epsilon^2 f_{\mathrm{e}}), \end{equation} \tag{ I.6 }$$

where F^(H)'_1e ∼ δf_e differs from F^(H)_1e by some small gyrophase-independent pieces—we will never need F^(H)_1e explicitly so can absorb any gyrophase-independent function into it. Because of the convenience of the form of F_e', we choose to write f_e as

$$\begin{equation} f_{\mathrm{e}} = F_{\mathrm{e}}' - \frac{e\left(\zeta-\delta \varphi'\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + F_{1\mathrm{e}}^{(\mathrm{H})'} + h_{\mathrm{e}}' + f_{2\mathrm{e}}. \end{equation} \tag{ I.7 }$$

Using our solution (81) for δf_e and the definition of h_e', we have that, to lowest order in δ,

$$\begin{equation} h_{\mathrm{e}}' = \frac{e\left(\zeta-\delta \varphi'\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + g_{\mathrm{e}}. \end{equation} \tag{ I.8 }$$

We will use this equation later to eliminate h_e' from our final results.

As mentioned above, we will need properties of f_2e to calculate the fluxes in our transport equations. We now derive such an equation by using the solution (I.7) in (H.3)—we will need to retain corrections to f_2e up to $\Or(\delta \epsilon ^2 f_{\mathrm {e}})$. We perform this substitution step-by-step. The preliminary step is to notice that all the terms in the second line of (H.3) are $\Or(\delta ^2\epsilon ^2 f_{\mathrm {e}})$ or smaller and so do not contribute in this calculation.

Next, we compute the left-hand-side of (H.3) for F_e':

$$\begin{eqnarray}&&\fl \left(\widetilde{\boldsymbol{w}}\times{\widetilde{\boldsymbol{b}}}\right)\cdot { \frac{\partial F_{\mathrm{e}}'} {\partial \widetilde{\boldsymbol{w}}} } = \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\nabla \tilde{\psi} \left\{ \frac{\mathrm{d} T_{\mathrm{e}}}{\mathrm{d} \tilde{\psi}} \left[ \frac{e\xi}{T_{\mathrm{e}}} + \frac{m_{\mathrm{e}} \left( \omega^2(\psi) R^2 - \widetilde{{w}}^2 \right)}{2T_{\mathrm{e}}} - \frac{3}{2} \right] + \frac{m_{\mathrm{e}} \omega(\tilde{\psi}) R^2}{2T_{\mathrm{e}}} \frac{\mathrm{d}\omega}{\mathrm{d}\tilde{\psi}} \right\} F_{\mathrm{e}}'\nonumber\\ &&\quad+\, \frac{m_{\mathrm{e}} \left(\boldsymbol{u}_{\mathrm{eff}} - \boldsymbol{u}\right)\cdot{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{T_{\mathrm{e}}} F_{0\mathrm{e}} + \Or(\delta^2 \epsilon^2 f_{\mathrm{e}}), \end{eqnarray} \tag{ I.9 }$$

where we have used the definition of R_e' and the fact that ${\widetilde {\boldsymbol {b}}}\cdot \nabla \tilde {\psi } = 0$. Using this result and the definition (I.4) of F_e', we can show that

$$\begin{eqnarray}&&\fl \left(\widetilde{\boldsymbol{w}}\times{\widetilde{\boldsymbol{b}}}\right)\cdot { \frac{\partial F_{\mathrm{e}}'} {\partial \widetilde{\boldsymbol{w}}} } - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\nabla F_{\mathrm{e}}' = -\frac{cm_{\mathrm{e}}}{e{\widetilde{B}} T_{\mathrm{e}}} \widetilde{\boldsymbol{w}}\cdot \left( e\nabla\xi + \frac{1}{2} m_{\mathrm{e}} \omega^2(\psi) \nabla R^2 \right) F_{\mathrm{e}}' \nonumber\\ &&+ \frac{c m_{\mathrm{e}}}{e {\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\nabla\left(\zeta-\delta \varphi'\right) F_{0\mathrm{e}} + \frac{m_{\mathrm{e}}}{T_{\mathrm{e}}} \widetilde{\boldsymbol{w}}\widetilde{\boldsymbol{w}}\boldsymbol{:}\nabla\left( \boldsymbol{u}_{\mathrm{eff}}-\boldsymbol{u} \right) F_{0\mathrm{e}}, \end{eqnarray} \tag{ I.10 }$$

where we have used (79) to show that $(\boldsymbol {u}_{\mathrm {eff}}-\boldsymbol {u})\cdot {\widetilde {\boldsymbol {b}}}\times \widetilde {\boldsymbol {w}} = (c/{\widetilde {B}}) \widetilde {\boldsymbol {w}}\cdot \nabla \left (\zeta -\delta \varphi '\right )$. However, we also have that

$$\begin{eqnarray}&&\fl \left(\frac{c}{{\widetilde{B}}}\nabla\xi - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}}\boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\right)\cdot { \frac{\partial F_{\mathrm{e}}'} {\partial \widetilde{\boldsymbol{w}}} } = -\frac{cm_{\mathrm{e}}}{e{\widetilde{B}} T_{\mathrm{e}}} \widetilde{\boldsymbol{w}}\cdot \left( e\nabla\xi + \frac{1}{2} m_{\mathrm{e}} \omega^2(\psi) \nabla R^2 \right) F_{\mathrm{e}}' \nonumber\\ &&- \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \frac{m_{\mathrm{e}}}{T_{\mathrm{e}}} \widetilde{\boldsymbol{w}}\widetilde{\boldsymbol{w}}\boldsymbol{:}\nabla\boldsymbol{u}_{\mathrm{eff}} F_{\mathrm{e}}' \end{eqnarray} \tag{ I.11 }$$

where we have used the fact that $\boldsymbol {v} = \boldsymbol {u}_{\mathrm {eff}} + \widetilde {\boldsymbol {w}}$ and the fact that, to lowest order in , u_eff = u. Thus, the contributions from F_e' to (H.3) cancel (up to corrections of order $\Or(\delta ^2\epsilon ^2 f_{\mathrm {e}})$) save for the last term in (I.11).

Then, using these results and substituting (I.7) into (H.3), we have

$$\begin{eqnarray}&&\fl \left({\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}\right)\cdot { \frac{\partial f_{2\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } = \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left(\widetilde{\boldsymbol{w}}\cdot\nabla g_{\mathrm{e}} -\left.\widetilde{\boldsymbol{w}}\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}}g_{\mathrm{e}}\right) + \frac{c}{{\widetilde{B}}}\nabla\left(\varphi_0 + \zeta\right)\cdot { \frac{\partial g_{\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } \nonumber\\ &&- \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \frac{m_{\mathrm{e}}}{T_{\mathrm{e}}}\widetilde{\boldsymbol{w}}\widetilde{\boldsymbol{w}}\boldsymbol{:}\nabla\boldsymbol{u} F_{0\mathrm{e}}+ \Or(\epsilon^2\delta^2 f_{\mathrm{e}}), \end{eqnarray} \tag{ I.12 }$$

where we have used (I.8) to write the result in terms of g_e and used the fact that, for any function H(R_e',ε_e,μ_e),

$$\begin{equation} \left({\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}\right)\cdot { \frac{\partial H} {\partial \widetilde{\boldsymbol{w}}} } = \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\left.\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} H. \end{equation} \tag{ I.13 }$$

We now convert the derivative of g_e at constant $\widetilde {\boldsymbol {w}}$ into a derivative at constant ε_e and μ_e.

$$\begin{eqnarray}&&\fl \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\nabla g_{\mathrm{e}} = \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left.\widetilde{\boldsymbol{w}}\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} g_{\mathrm{e}} - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\left( \left.\nabla\right|_{\widetilde{\boldsymbol{w}}}{{\varepsilon}_{\mathrm{e}}} { \frac{\partial g_{\mathrm{e}}} {\partial {{\varepsilon}_{\mathrm{e}}}} } + \left.\nabla\right|_{\widetilde{\boldsymbol{w}}}\mu_{\mathrm{e}} { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \right) \nonumber\\ &&= \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left.\widetilde{\boldsymbol{w}}\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} g_{\mathrm{e}} - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left( - e\widetilde{\boldsymbol{w}}\cdot\nabla\varphi_0 { \frac{\partial g_{\mathrm{e}}} {\partial {{\varepsilon}_s}} } + \widetilde{\boldsymbol{w}}\cdot\nabla\mu_{\mathrm{e}} { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \right), \end{eqnarray} \tag{ I.14 }$$

in which we have used the definition ${{\varepsilon }_{\mathrm {e}}} = (m_{\mathrm {e}}/2)\widetilde {{w}}^2 - e \varphi _0 + \Or(\delta ^2 T_{\mathrm {e}})$. Combining this equation with the term from (I.12) involving φ₀, we have

$$\begin{eqnarray}&&\fl \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \widetilde{\boldsymbol{w}}\cdot\nabla g_{\mathrm{e}} + \frac{c}{{\widetilde{B}}} \nabla\varphi_0 \cdot { \frac{\partial g_{\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } =\frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left.\widetilde{\boldsymbol{w}}\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} g_{\mathrm{e}} - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \left(\widetilde{\boldsymbol{w}}\left.\cdot \nabla\right|_{\widetilde{\boldsymbol{w}}}\mu_{\mathrm{e}} - \frac{e}{{B}}\widetilde{\boldsymbol{w}}_\perp\cdot\nabla\varphi_0\right) { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } .\nonumber\\ \end{eqnarray} \tag{ I.15 }$$

By taking the derivative of the definition of μ_e we can show that

$$\begin{equation}\fl \widetilde{\boldsymbol{w}}\cdot\nabla\mu_s- e\widetilde{\boldsymbol{w}}_\perp\cdot\nabla\varphi_0 = -m_{\mathrm{e}}\widetilde{\boldsymbol{w}} \cdot \left( \frac{w_\perp^2}{2{B}} \nabla \ln {B} + \frac{w_\parallel}{{B}} \widetilde{\boldsymbol{w}}\cdot\nabla{\boldsymbol{b}}\right) - e\widetilde{\boldsymbol{w}}_\perp\cdot\nabla\varphi_0. \end{equation} \tag{ I.16 }$$

We can rewrite this in terms of V _De as

$$\begin{eqnarray}&&\fl \widetilde{\boldsymbol{w}}\cdot\nabla\mu_s- e\widetilde{\boldsymbol{w}}_\perp\cdot\nabla\varphi_0 =m_{\mathrm{e}}\widetilde{\boldsymbol{w}}_\perp \cdot ( {\widetilde{\boldsymbol{b}}}\times\boldsymbol{V}_{\mathrm{D}\mathrm{e}}) - \frac{m_{\mathrm{e}}w_\parallel}{{B}} \left( \widetilde{\boldsymbol{w}}_\perp\widetilde{\boldsymbol{w}}_\perp \boldsymbol{:} \nabla {\boldsymbol{b}} + \frac{w_\perp^2}{2} {\boldsymbol{b}}\cdot\nabla\ln{B} \right). \end{eqnarray} \tag{ I.17 }$$

Substituting this back into (I.15) and thence back into (I.12) we arrive at

$$\begin{eqnarray}&&\fl \left({\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}\right)\cdot { \frac{\partial f_{2\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } = \frac{c}{{\widetilde{B}}}\nabla\zeta\cdot { \frac{\partial g_{\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } - \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \frac{m_{\mathrm{e}}}{T_{\mathrm{e}}}\widetilde{\boldsymbol{w}}\widetilde{\boldsymbol{w}}\boldsymbol{:}\left( \nabla\boldsymbol{u} \right) F_{0\mathrm{e}}\frac{cm_{\mathrm{e}}}{e{\widetilde{B}}} \nonumber\\ &&\times \left[ m_{\mathrm{e}}\widetilde{\boldsymbol{w}}\cdot{\widetilde{\boldsymbol{b}}}\times\boldsymbol{V}_{\mathrm{D}\mathrm{e}} - \frac{m_{\mathrm{e}}w_\parallel}{{B}} \left( \widetilde{\boldsymbol{w}}_\perp\widetilde{\boldsymbol{w}}_\perp \boldsymbol{:} \nabla {\boldsymbol{b}} + \frac{w_\perp^2}{2} {\boldsymbol{b}}\cdot\nabla\ln{B} \right) \right] { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } + \Or(\epsilon^2\delta^2 f_{\mathrm{e}}).\nonumber\\ \end{eqnarray} \tag{ I.18 }$$

Estimating f_2e from this equation, we see that f_2e ∼ δ²f_e, rather than f_2e ∼ ²f_e which we would naïvely expect—this is because it is only the gyrophase-dependent piece of f_2e we need and, to lowest order in δ, f_2e is gyrophase-independent.

Using the solution (I.7) for f_e in the definition of the particle flux, we have

$$\begin{eqnarray} &&\tilde{V}'\Gamma_{\mathrm{e}} = { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left( F_{\mathrm{e}}' - \frac{e\left(\zeta-\delta \varphi'\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.19 }$$

Substituting for F_e' from (I.4) and performing the velocity integration, we have

$$\begin{eqnarray}&&\fl \tilde{V}'\Gamma_{\mathrm{e}} = { {\left\langle \tilde{V}'{\left\langle n_{\mathrm{e}} \left(\boldsymbol{u}_{\mathrm{eff}} - \boldsymbol{u}\right)\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left( h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.20 }$$

Using (79) for u_eff and using the definition of the flux-surface average in the first term, we have

$$\begin{eqnarray}&&\fl \tilde{V}'\Gamma_{\mathrm{e}} = { {\left\langle \int \mathrm{d}\tilde{\alpha}\, \mathrm{d}\tilde{l} {\frac{ cn_{\mathrm{e}}}{{\widetilde{B}}} { \frac{\partial } {\partial \tilde{\alpha}} } \left(\zeta-\delta \varphi'\right) } \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left( h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.21 }$$

Integrating by parts with respect to $\tilde {\alpha }$ in the first term, we find that

$$\begin{eqnarray}&&\fl \tilde{V}'\Gamma_{\mathrm{e}} = { {\left\langle \int \mathrm{d}\tilde{\alpha}\, \mathrm{d}\tilde{l} { \frac{cn_{\mathrm{e}}}{{\widetilde{B}}} {(\zeta-\delta \varphi')} { \frac{\partial } {\partial \tilde{\alpha}} } \left(\frac{{ {\delta B_\parallel}}}{{\widetilde{B}}}\right) } \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left( h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.22 }$$

The calculation of the term involving h_e' is particularly involved. In section I.6 of the appendix, we prove the following identity:

$$\begin{equation} { {\left\langle \tilde{V}'\left\langle h_{\mathrm{e}}'\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = { {\left\langle \tilde{V}'\left\langle g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \frac{w_\perp^2}{2\Omega_{\mathrm{e}}}{{ {\delta B_\parallel}}} \right) \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} - { {\left\langle \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} \frac{c n_{\mathrm{e}}}{{\widetilde{B}}} (\zeta-\delta \varphi') { \frac{\partial } {\partial \tilde{\alpha}} } \left( \frac{{ {\delta B_\parallel}}}{{\widetilde{B}}} \right) \right\rangle}_{\mathrm{turb}}}. \end{equation} \tag{ I.23 }$$

Using this in (I.22), we find

$$\begin{eqnarray}&&\fl \Gamma_{\mathrm{e}} = { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \frac{w_\perp^2}{2\Omega_{\mathrm{e}}}{{ {\delta B_\parallel}}} \right) \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} {\widetilde{\boldsymbol{b}}}\times\nabla\tilde{\psi}\cdot\widetilde{\boldsymbol{w}} \left(\widetilde{\boldsymbol{w}}\times{\widetilde{\boldsymbol{b}}}\right)\cdot { \frac{\partial f_{2\mathrm{e}}} {\partial \widetilde{\boldsymbol{w}}} } \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}},\nonumber\\ \end{eqnarray} \tag{ I.24 }$$

where we have integrated by parts in the term involving f_2e.

Now, the derivative of f_2e in (I.24) is precisely the one appearing in (I.18). Thus, substituting (I.18) into (I.24), we have

$$\begin{eqnarray}&&\fl \Gamma_{\mathrm{e}} = { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \frac{w_\perp^2}{2\Omega_{\mathrm{e}}}{{ {\delta B_\parallel}}} \right) \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{c}{{\widetilde{B}}}\left( {\widetilde{\boldsymbol{b}}}\times\nabla\zeta\cdot\nabla\tilde{\psi} \right) g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&- { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}}\frac{\widetilde{{w}}_\perp^2}{2} \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla\tilde{\psi} { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}, \end{eqnarray} \tag{ I.25 }$$

where we have integrated by parts with respect to $\widetilde {\boldsymbol {w}}$ in the second line and used the gyrophase-independence of g_e in the third line. We have also used the isotropy of F_0e and the gyrotropy of g_e to eliminate terms in (I.18) that contain an even power of $\widetilde {\boldsymbol {w}}_\perp$ (we are multiplying by one power of $\widetilde {\boldsymbol {w}}_\perp$ so this becomes an odd power and thus vanishes upon integration over ϑ).

We now prove that the last line of (I.25) is small and can be dropped. Writing it in terms of μ_e and using (F.8), we obtain

$$\begin{eqnarray}&&\fl { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}}\frac{\widetilde{{w}}_\perp^2}{2} \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla\tilde{\psi} { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= \frac{c}{e} { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \mu_{\mathrm{e}} \frac{w_\parallel}{\Omega_{\mathrm{e}}} {\widetilde{\boldsymbol{B}}}\left.\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}}\Bigg[\frac{w_\parallel}{B} ({\boldsymbol{b}}\times\nabla\tilde{\psi})\cdot\nabla\tilde{l}\Bigg] { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}\nonumber\\ &&= \frac{c^2m_{\mathrm{e}}}{e^2} { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} {\widetilde{{w}}_\parallel} {\widetilde{\boldsymbol{b}}}\left.\cdot\nabla\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}}\left[\mu_{\mathrm{e}} { \frac{\partial g_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \frac{\widetilde{{w}}_\parallel}{B} ({\boldsymbol{b}}\times\nabla\tilde{\psi})\cdot\nabla\tilde{l}\right]\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}\nonumber\\ &&=0, \end{eqnarray} \tag{ I.26 }$$

where we have been able to write (F.8) in terms of $\widetilde {\boldsymbol {w}}$ because the difference between $\widetilde {\boldsymbol {w}}$ and w is $\Or(\epsilon {v_{\mathrm {th}_{\mathrm {i}}}})$ and thus negligible.

Finally, integrating by parts with respect to $\tilde {\alpha }$ in the second line of (I.25) and dropping all the references to $\widetilde {\boldsymbol {w}}$, we have

$$\begin{equation} \Gamma_{\mathrm{e}} = \left\langle \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( c\zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {{ {\delta B_\parallel}}} \right) \right\rangle}_{\mathrm{turb}}} \right\rangle_{\tilde\psi}, \end{equation} \tag{ I.27 }$$

where the passing electron response has vanished because ${\partial g_{\mathrm {pe}}}/{\partial \tilde {\alpha }} = 0$ (at constant ε_e and μ_e). This is (H.11) as required.

As in appendix C.4 of [1], it is convenient to adjust the form of the pressure evolution equation before calculating the heat flux. In particular, we subtract ${ { \langle \langle \nabla \cdot [e(\varphi _0+\zeta ) \int \mathrm {d}^{3}\widetilde {\boldsymbol {w}} \widetilde {\boldsymbol {w}} f_{\mathrm {e}}] \rangle _{\tilde \psi } \rangle }_{\mathrm {turb}}}$ from both sides of (H.12), and use

$$\begin{eqnarray}&&\fl \nabla\cdot { {\left\langle e\varphi_0\int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}} =\int \mathrm{d}^3 \boldsymbol{w} e\nabla\varphi_0 \cdot { {\left\langle \widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}} + e\varphi_0 \nabla\cdot { {\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= { {\left\langle \int \mathrm{d}^3 \boldsymbol{w} e\nabla\varphi_0 \cdot \widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}} - e\varphi_0 \left( \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{n_{\mathrm{e}}} + { {\left\langle \left( \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right)\int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}} - S^{({n})}_{s}\right),\nonumber\\ \end{eqnarray} \tag{ I.28 }$$

where we have used (H.7). This results in the following heat transport equation:

$$\begin{eqnarray}&&\fl \frac{1}{\tilde{V}'}\frac{3}{2}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}' \left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi}T_{\mathrm{e}} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}' \left\langle Q_{\mathrm{e}} - e { {\left\langle \left( \varphi_0+\zeta \right) \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}}\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} \nonumber\\ &&= { {\left\langle \left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left[e\nabla\left( \xi - \varphi_0 \right) - { \frac{\partial \boldsymbol{u}_{\mathrm{eff}}} {\partial t} } - \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\right]\cdot\widetilde{\boldsymbol{w}} f_s \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&\quad -\, { {\left\langle \left\langle \nabla\cdot\left(e\zeta\int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}}\widetilde{\boldsymbol{w}}\right)\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} + C^{{(E)}}_s+ S^{({E})}_{s}\nonumber\\ &&\quad +\, \left\langle e\varphi_0\left(\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{n_{\mathrm{e}}} + { {\left\langle \left( \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right)\int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}}- S^{({n})}_{s}\right)\right\rangle_{\tilde\psi}. \end{eqnarray} \tag{ I.29 }$$

I.4.1. Electron heating

We now evaluate the heat sources in the temperature evolution equation, from left to right. The first heating term is

$$\begin{equation} e { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla \left(\xi-\varphi_0\right)\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = e { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla \zeta\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}, \end{equation} \tag{ I.30 }$$

where we have used (I.5) for ξ. Rearranging the derivative to act on f_e, we have

$$\begin{eqnarray}&&\fl e { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla \zeta\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = e { {\left\langle \left\langle \nabla\cdot\int \mathrm{d}^3 \boldsymbol{w} \zeta \widetilde{\boldsymbol{w}} f_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} - e { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \zeta \widetilde{\boldsymbol{w}}\left.\cdot\nabla\right|_{\widetilde{\boldsymbol{w}}} f_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.31 }$$

Now, integrating (H.3) multiplied by $(e{\widetilde {B}} / cm_{\mathrm {e}})\zeta$ over $\widetilde {\boldsymbol {w}}$, we have that

$$\begin{equation} 0 = \zeta \int \mathrm{d}^3 \boldsymbol{w} \widetilde{\boldsymbol{w}}\left.\cdot\nabla\right|_{\widetilde{\boldsymbol{w}}} f_{\mathrm{e}} + \zeta n_{\mathrm{e}} \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} + \zeta \int \mathrm{d}^3 \boldsymbol{w} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}f_{\mathrm{e}}, \end{equation} \tag{ I.32 }$$

where we have integrated by parts with respect to $\widetilde {\boldsymbol {w}}$ as needed. Substituting this back into (I.31), we have

$$\begin{eqnarray}&&\fl e { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla \zeta\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = e { {\left\langle \left\langle \nabla\cdot\int \mathrm{d}^3 \boldsymbol{w} \zeta \widetilde{\boldsymbol{w}} f_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}\nonumber\\ &&+\, e { {\left\langle \left\langle \zeta n_{\mathrm{e}} \nabla\cdot\boldsymbol{u}_{\mathrm{eff}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}- e { {\left\langle \left\langle g_{\mathrm{e}} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{\zeta}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}, \end{eqnarray}\noindent \tag{ I.33 }$$

where we have integrated by parts with respect to time in the second term. The first of these terms cancels with the corresponding term in (I.29) and the second and third will become part of the turbulent heating. We will need to process terms involving ∇·u_eff several times in this section. In section I.4.2 of the appendix, we show that we can write it in a closed (if unwieldy) form—see (I.45). For conciseness we will not expand ∇·u_eff in this section.

Now, the time-derivative term is small:

$$\begin{equation} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} m_{\mathrm{e}} { \frac{\partial \boldsymbol{u}_{\mathrm{eff}}} {\partial t} } \cdot\widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = \Or(\delta \epsilon^3 \Omega_{\mathrm{i}} T_{\mathrm{e}}), \end{equation}\noindent \tag{ I.34 }$$

and can thus be neglected.

Substituting the solution (I.7) for f_e into the remaining heating term gives

$$\begin{equation}\fl - m_{\mathrm{e}} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\cdot\widetilde{\boldsymbol{w}} \left( F_{\mathrm{e}}' + h_{\mathrm{e}}' + \frac{e\left(\delta \varphi'-\zeta\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + F_{1\mathrm{e}}^{(\mathrm{H})'} + f_{2\mathrm{e}} \right)\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{equation}\noindent \tag{ I.35 }$$

Estimating the size of the contribution from the gyrophase-dependent piece of f_2e ∼ δ²f_e, we see that it is $\Or(\delta \epsilon ^3\Omega _{\mathrm {i}}n_{\mathrm {e}}T_{\mathrm {e}})$ and so we can neglect it. Now, as ${ { \langle \nabla \cdot \boldsymbol {u}_{\mathrm {eff}} \rangle }_{\mathrm {turb}}} \sim { { \langle {\widetilde {\boldsymbol {b}}}\cdot \nabla \boldsymbol {u}_{\mathrm {eff}}\cdot {\widetilde {\boldsymbol {b}}} \rangle }_{\mathrm {turb}}} \sim \Or(\epsilon ^3 \Omega _{\mathrm {i}})$, we can also neglect the contributions from F^(H)'_1e and any gyrophase-independent piece of f_2e. Of the remaining terms, we handle the F_e' term first. Using (I.4), we have

$$\begin{eqnarray}&&\fl -m_{\mathrm{e}} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\cdot\widetilde{\boldsymbol{w}} F_{\mathrm{e}}' \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= - \left\langle { {\left\langle \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}} n_{\mathrm{e}}\right\rangle_{\tilde\psi} T_{\mathrm{e}} - m_{\mathrm{e}} \left\langle { {\left\langle \left( \boldsymbol{u}_{\mathrm{eff}}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}} \right)\cdot\left( \boldsymbol{u}_{\mathrm{eff}}-\boldsymbol{u} \right) \right\rangle}_{\mathrm{turb}}} n_{\mathrm{e}}\right\rangle_{\tilde\psi}, \end{eqnarray} \tag{ I.36 }$$

where we have used the fact that, to lowest order in , the flux-surface average and the turbulence average commute. As u_eff − u ∼ v_thi we can use (79) to express u_eff in the second term. Doing this, we find that the entire second term is $\Or(\delta \epsilon ^3 n_{\mathrm {e}} T_{\mathrm {e}}\Omega _{\mathrm {i}})$ or smaller and can thus be neglected. In the first term, we cannot use (79), but instead, we interchange the divergence and the turbulence average to find that

$$\begin{equation} -m_{\mathrm{e}} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\cdot\widetilde{\boldsymbol{w}} F_{\mathrm{e}}' \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = - T_{\mathrm{e}}\left\langle n_{\mathrm{e}}{\nabla\cdot\boldsymbol{V}_{\!\psi}} \right\rangle_{\tilde\psi} , \end{equation} \tag{ I.37 }$$

where we have used ${ {\left \langle \boldsymbol {u}_{\mathrm {eff}} \right \rangle }_{\mathrm {turb}}} = \boldsymbol {V}_\psi$—the average of the velocity of the exact surfaces is just the velocity of the average surfaces. We now work out the corresponding term involving g_e. Using the gyrophase-independence of g_e, we have

$$\begin{eqnarray}&&\fl -m_{\mathrm{e}} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\cdot\widetilde{\boldsymbol{w}} g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = - { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} {B}\left( {\boldsymbol{\mathsf{I}}}-{\widetilde{\boldsymbol{b}}}{\widetilde{\boldsymbol{b}}}\right)\boldsymbol{:}\left( \nabla\boldsymbol{u}_{\mathrm{eff}} \right) \mu_{\mathrm{e}} g_{\mathrm{e}} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.38 }$$

Substituting for ∇·u_eff from (I.45), we obtain

$$\begin{equation}\fl -m_{\mathrm{e}} { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \boldsymbol{v}\cdot\nabla\boldsymbol{u}_{\mathrm{eff}}\cdot\widetilde{\boldsymbol{w}} g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = - { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} \mu_{\mathrm{e}} g_{\mathrm{e}} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{ {\delta B_\parallel}} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{equation} \tag{ I.39 }$$

Finally, we have to deal with the term involving φ₀ and the divergence of u_eff. Using our solution for f_e, we have

$$\begin{equation} e\left\langle \varphi_0 { {\left\langle \left(\nabla\cdot\boldsymbol{u}_{\mathrm{eff}}\right) \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} = e \left\langle \varphi_0 n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi} + e \left\langle \varphi_0 { {\left\langle \left( \frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}}\right) \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}. \end{equation} \tag{ I.40 }$$

Using (I.45) to substitute for ∇·u_eff in the last term, we have

$$\begin{eqnarray}&&\fl e\left\langle \varphi_0 { {\left\langle \left(\nabla\cdot\boldsymbol{u}_{\mathrm{eff}}\right) \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} = e \left\langle \varphi_0 n_{\mathrm{e}} \nabla\cdot\boldsymbol{V}_{\!\psi}\right\rangle_{\tilde\psi} \nonumber\\ &&+\, e \left\langle \varphi_0 { {\left\langle \left( \frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}}\right) \left( \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{{ {\delta B_\parallel}}}{{\widetilde{B}}} + {\widetilde{\boldsymbol{b}}}{\widetilde{\boldsymbol{b}}}\boldsymbol{:}\nabla\boldsymbol{u}_{\mathrm{eff}}\right) \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi}, \end{eqnarray} \tag{ I.41 }$$

where the parallel compression is given by (I.46).

Using (I.33), (I.37), (I.39), and (I.41) in (I.29), we obtain

$$\begin{eqnarray}&&\fl \frac{1}{\tilde{V}'}\frac{3}{2}\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi}} \tilde{V}' \left\langle n_{\mathrm{e}}\right\rangle_{\tilde\psi}T_{\mathrm{e}} + \frac{1}{\tilde{V}'} { \frac{\partial } {\partial \tilde{\psi}} } \tilde{V}' \left\langle Q_{\mathrm{e}} - e { {\left\langle \left( \varphi_0+\zeta \right) \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} f_{\mathrm{e}}\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}}\right\rangle_{\tilde\psi} \nonumber\\ &&=- { {\left\langle \left\langle \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left( e\zeta + \mu_{\mathrm{e}}{ {\delta B_\parallel}} \right) \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} +e { {\left\langle \left\langle \zeta n_{\mathrm{e}}\nabla\cdot\boldsymbol{u}_{\mathrm{eff}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}\nonumber\\ &&\quad +\left\langle e\varphi_0\left[\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}{n_{\mathrm{e}}} + n_{\mathrm{e}}\nabla\cdot\boldsymbol{V}_{\!\psi} + { {\left\langle \left(\frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}}\right) \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} \right\rangle}_{\mathrm{turb}}} - S^{({n})}_{s}\right]\right\rangle_{\tilde\psi} \nonumber\\ &&\quad - T_{\mathrm{e}}\left\langle n_{\mathrm{e}}{\nabla\cdot\boldsymbol{V}_{\!\psi}} \right\rangle_{\tilde\psi}+C^{{(E)}}_s+ S^{({E})}_{s}, \end{eqnarray} \tag{ I.42 }$$

where ∇·u_eff is a shorthand for the expression given in (I.45). Gathering the terms in this equation together according to their physical interpretation, we obtain the heat transport equation (H.14) as required.

I.4.2. The divergence of u_eff

We wish to calculate the divergence of u_eff correct to $\Or(\epsilon ^2 \Omega _{\mathrm {i}})$. We start from Faraday's Law:

$$\begin{equation} { \frac{\partial {\widetilde{\boldsymbol{B}}}} {\partial t} } = \nabla\times\left(\boldsymbol{u}_{\mathrm{eff}}\times{\widetilde{\boldsymbol{B}}}\right) + \Or(\epsilon^2\delta {\widetilde{\boldsymbol{B}}} \Omega_{\mathrm{i}}). \end{equation} \tag{ I.43 }$$

Taking the component of this equation along ${\widetilde {\boldsymbol {b}}}$, we obtain

$$\begin{equation} \left( { \frac{\partial } {\partial t} } +\boldsymbol{u}_{\mathrm{eff}}\cdot\nabla\right){ {\delta B_\parallel}} = {\widetilde{B}} \left( {\boldsymbol{\mathsf{I}}} - {\widetilde{\boldsymbol{b}}}\,{\widetilde{\boldsymbol{b}}}\right)\boldsymbol{:}\nabla\boldsymbol{u}_{\mathrm{eff}} + \Or(\epsilon^2\delta {\widetilde{\boldsymbol{B}}} \Omega_{\mathrm{i}}). \end{equation} \tag{ I.44 }$$

Rearranging this into an equation for ∇·u_eff, we have

$$\begin{equation} \nabla\cdot\boldsymbol{u}_{\mathrm{eff}} = \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{{ {\delta B_\parallel}}}{{\widetilde{B}}} + {\widetilde{\boldsymbol{b}}}\,{\widetilde{\boldsymbol{b}}}\boldsymbol{:}\nabla\boldsymbol{u}_{\mathrm{eff}}+ \Or(\epsilon^2\delta {\widetilde{\boldsymbol{B}}} \Omega_{\mathrm{i}}). \end{equation} \tag{ I.45 }$$

To the required order, we can use (79) to write the parallel compression due to u_eff as

$$\begin{equation} {\widetilde{\boldsymbol{b}}}\,{\widetilde{\boldsymbol{b}}}\boldsymbol{:}\nabla\boldsymbol{u}_{\mathrm{eff}} = \left( {\widetilde{\boldsymbol{b}}}\cdot\nabla\psi \right) \frac{I}{{B}} \frac{\mathrm{d}\omega}{\mathrm{d}\psi} - \frac{c}{B}\left( {\boldsymbol{b}}\cdot\nabla{\boldsymbol{b}} \right)\cdot {\boldsymbol{b}} \times \nabla\left( \zeta-\delta \varphi' \right) + \Or(\epsilon^2\delta \Omega_{\mathrm{i}}). \end{equation} \tag{ I.46 }$$

The derivation of the heat flux proceeds in exactly the same way as the particle flux derivation.

Using the solution (I.7) for f_e in the definition of the heat flux, we have

$$\begin{equation} \tilde{V}' q_{\mathrm{e}} = { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \left(\frac{1}{2}m_{\mathrm{e}} \widetilde{{w}}^2 - e \varphi_0 - e\zeta\right) \left( F_{\mathrm{e}} + h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}. \end{equation} \tag{ I.47 }$$

Substituting for F_e' and integrating, we have

$$\begin{eqnarray}&&\fl \tilde{V}' q_{\mathrm{e}} = { {\left\langle \tilde{V}'{\left\langle \tilde{\varepsilon}_{\mathrm{e}} \left(\boldsymbol{u}_{\mathrm{eff}} - \boldsymbol{u}\right)\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \tilde{\varepsilon}_{\mathrm{e}} \left( h_{\mathrm{e}}' + f_{2\mathrm{e}} \right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}}\nonumber\\ &&-\, { {\left\langle \tilde{V}'{\left\langle e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}}, \end{eqnarray} \tag{ I.48 }$$

where, for conciseness, we have defined $\tilde {\varepsilon }_{\mathrm {e}} = (1/2)m_{\mathrm {e}} \widetilde {{w}}^2 - e\varphi _0$.

As with the calculation of the particle flux, we use the identity (I.23) to handle the term involving h_e'. Using (I.23) in (I.48) and integrating by parts where required, we obtain

where we have used (79) and the definition of the flux-surface average to cancel the first term in (I.48) with the contribution arising from the first term of (I.23).

Substituting for f_2e from (I.18), we have

$$\begin{eqnarray}&&\fl q_{\mathrm{e}} = { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \tilde{\varepsilon}_{\mathrm{e}} g_{\mathrm{te}} { \frac{\partial } {\partial \tilde{\alpha}} } \left( \frac{w_\perp^2}{2\Omega_{\mathrm{e}}}{{ {\delta B_\parallel}}} \right) \right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} + { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \tilde{\varepsilon}_{\mathrm{e}}\frac{c}{{\widetilde{B}}}\left( {\widetilde{\boldsymbol{b}}}\times\nabla\zeta\cdot\nabla\tilde{\psi} \right) g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&+\, { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{cm_{\mathrm{e}}}{{\widetilde{B}}}\widetilde{\boldsymbol{w}}\cdot\nabla\zeta\left( {\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right) g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&-\, { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{cm_{\mathrm{e}}}{e{\widetilde{B}}}\frac{\widetilde{{w}}_\perp^2}{2} \tilde{\varepsilon}_{\mathrm{e}}\boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla\tilde{\psi} g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} - { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}},\nonumber\\ \end{eqnarray} \tag{ I.50 }$$

where we have integrated by parts with respect to $\widetilde {\boldsymbol {w}}$ and used the gyrophase independence of g_e. We have also used the isotropy of F_0e and the gyrotropy of g_e to show that most terms in (I.18) do not contribute to the heat flux. The proof that the term involving V _De can be dropped is identical to the proof for the particle flux—the manipulations of (I.26) go through in exactly the same way with an extra multiple of ε_e.

Next, we show that the third and fifth lines of (I.50) cancel. Taking the fifth line and expressing f_e via (I.7), we have

$$\begin{equation}\fl { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} = { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta \left(F_{\mathrm{e}}' + h_{\mathrm{e}}'\right) \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}}, \end{equation} \tag{ I.51 }$$

where we have been able to drop terms associated with F^(H)'_1e as it is both small and gyrophase-independent. Substituting for F_e' from (I.4), we have

$$\begin{eqnarray}&&\fl { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta \left[ \frac{m_{\mathrm{e}} \left(\boldsymbol{u}_{\mathrm{eff}}-\boldsymbol{u}\right)\cdot\widetilde{\boldsymbol{w}}}{T_{\mathrm{e}}} F_{0\mathrm{e}} + \left(\frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\right)\cdot\nabla h_{\mathrm{e}}' \right] \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}},\nonumber\\ \end{eqnarray} \tag{ I.52 }$$

where we have expanded h_e' around r = R_e' and dropped terms that are small in δ. Substituting for h_e' from (I.8) and using (79), we see that the first term in the brackets in (I.52) cancels with the contribution from the first term on the right-hand side of (I.8) leaving us with

$$\begin{eqnarray}&&\fl { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} = { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta \left( \frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}} \right)\cdot\nabla g_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.53 }$$

Performing the gyroaverage explicitly we have

$$\begin{eqnarray}&&\fl { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} e \zeta f_{\mathrm{e}} \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} = { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} {cm_{\mathrm{e}}} \zeta w_\perp^2 { \frac{\partial g_{\mathrm{e}}} {\partial \tilde{\alpha}} } \right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= - { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} {cm_{\mathrm{e}}} g_{\mathrm{e}} w_\perp^2 { \frac{\partial \zeta} {\partial \tilde{\alpha}} } \right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}}, \end{eqnarray} \tag{ I.54 }$$

where we have used the definition of the flux-surface average to integrate by parts with respect to $\tilde {\alpha }$. Now, taking the third line of (I.50) and performing the gyroaverage, we obtain

$$\begin{eqnarray}&&\fl { {\left\langle \tilde{V}'\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} \frac{cm_{\mathrm{e}}}{{\widetilde{B}}}\widetilde{\boldsymbol{w}}\cdot\nabla\zeta\left( {\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi} \right) g_{\mathrm{e}}\right\rangle_{\tilde\psi} \right\rangle}_{\mathrm{turb}}} = - { {\left\langle \tilde{V}'{\left\langle \int \mathrm{d}^{3}\widetilde{\boldsymbol{w}} {cm_{\mathrm{e}}} g_{\mathrm{e}} w_\perp^2 { \frac{\partial \zeta} {\partial \tilde{\alpha}} } \right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.55 }$$

Thus, the third and fifth lines of (I.50) cancel.

Finally, using this result and integrating by parts with respect to $\tilde {\alpha }$ in the second line of (I.50) and dropping all the references to $\widetilde {\boldsymbol {w}}$, we have

$$\begin{equation} q_{\mathrm{e}} = \left\langle \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle g_{\mathrm{te}} \left(\frac{1}{2} m_{\mathrm{e}} w^2 - e \varphi_0\right) { \frac{\partial } {\partial \tilde{\alpha}} } \left( c\zeta + \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {{ {\delta B_\parallel}}} \right) \right\rangle}_{\mathrm{turb}}} \right\rangle_{\tilde\psi}, \end{equation} \tag{ I.56 }$$

where the passing electron response has vanished because ${ \frac {\partial g_{\mathrm {pe}}} {\partial \tilde {\alpha }} } = 0$ (at constant ε_e and μ_e). This is appendix (H.19) as required.

In order to derive this result, we will need a concise notation for the following operation:

$$\begin{equation} \widetilde{\nabla}_\perp = \nabla - \nabla\tilde{l} { \frac{\partial } {\partial \tilde{l}} } = \nabla\tilde{\alpha} { \frac{\partial } {\partial \tilde{\alpha}} } + \nabla\tilde{\psi} { \frac{\partial } {\partial \tilde{\psi}} } . \end{equation} \tag{ I.57 }$$

This is an important operation as $\nabla \tilde {l}$ is not parallel to the exact field, but the strong perpendicular variation is contained in $\tilde {\psi }$ and $\tilde {\alpha }$.

Now, the quantity we have to evaluate is

$$\begin{equation} { {\left\langle \tilde{V}'{\left\langle h_{\mathrm{e}}' \widetilde{\boldsymbol{w}}\cdot\nabla\tilde{\psi}\right\rangle_{\tilde\psi}} \right\rangle}_{\mathrm{turb}}} = \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}} \widetilde{\boldsymbol{w}} \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}}. \end{equation} \tag{ I.58 }$$

First, we evaluate ${\widetilde {B}}(\tilde {\psi },\tilde {\alpha },\tilde {l})$ at $\tilde {\psi }(\boldsymbol {R}_{\mathrm {e}}')$, $\tilde {\alpha }(\boldsymbol {R}_{\mathrm {e}}')$, and $\tilde {l}$ to obtain

$$\begin{eqnarray}&&\fl \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} = \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')}\left(1 - \frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\cdot\widetilde{\nabla}_\perp\ln{\widetilde{B}}\right) \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}} \cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}}.\nonumber\\ \end{eqnarray} \tag{ I.59 }$$

In the term involving the gradient of $\ln {\widetilde {B}}$ we can perform the gyroaverage to find

We see that this term is just the ∇B drift in the exact magnetic field.

Now we handle the first term on the right-hand side of (I.60). Integrating by parts with respect to space, we have

$$\begin{eqnarray}&&\fl \int \,\mathrm{d}\tilde{\alpha}\, \mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} = \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle \widetilde{\boldsymbol{w}}_\perp\left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')}\tilde{\psi} \right\rangle}_{\mathrm{turb}}} \right\rangle}_{\boldsymbol{r}}} \nonumber\\ &&- \int \mathrm{d}\tilde{\alpha}\, \mathrm{d}\tilde{l} { {\left\langle { {\left\langle \widetilde{\boldsymbol{w}}_\perp\left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \right\rangle}_{\boldsymbol{r}}}\tilde{\psi} \right\rangle}_{\mathrm{turb}}}, \end{eqnarray} \tag{ I.61 }$$

where we have used the fact that $\nabla \tilde {\psi } = \widetilde {\nabla }_\perp \tilde {\psi }$ and the fact that we can then choose to take that derivative at constant ε_e and μ_e rather than at constant $\widetilde {\boldsymbol {w}}$.

The crucial step in this derivation is to now write $\tilde {\psi }$ inside the first bracket as $\tilde {\psi }(\boldsymbol {R}_{\mathrm {e}}') + (\boldsymbol {r} - \boldsymbol {R}_{\mathrm {e}}')\cdot \widetilde {\nabla }_\perp \tilde {\psi }$. Doing this, we obtain

$$\begin{eqnarray}&&\fl \int \mathrm{d}\,\tilde{\alpha} \mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&= \int \,\mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle \widetilde{\boldsymbol{w}}_\perp\left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')}\left[\tilde{\psi}(\boldsymbol{R}_{\mathrm{e}}') + \frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\cdot\nabla\tilde{\psi}\right] \right\rangle}_{\mathrm{turb}}} \right\rangle}_{\boldsymbol{r}}} \nonumber\\ &&\quad -\, \int \,\mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \widetilde{\boldsymbol{w}}_\perp\left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \right\rangle}_{\boldsymbol{r}}}\tilde{\psi} \right\rangle}_{\mathrm{turb}}}. \end{eqnarray} \tag{ I.62 }$$

Now, for any function $H(\tilde {\psi }(\boldsymbol {R}_{\mathrm {e}}'),\tilde {\alpha }(\boldsymbol {R}_{\mathrm {e}}'),\tilde {l},{{\varepsilon }_{\mathrm {e}}},\mu _{\mathrm {e}})$ we have that

$$\begin{equation} { {\left\langle \boldsymbol{w}_\perp \left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} H \right\rangle}_{\boldsymbol{r}}} = { {\left\langle \Omega_{\mathrm{e}} { \frac{\partial \boldsymbol{R}_{\mathrm{e}}'} {\partial \vartheta} } \cdot { \frac{\partial H} {\partial \boldsymbol{R}_{\mathrm{e}}'} } \right\rangle}_{\boldsymbol{r}}} = \Omega_{\mathrm{e}} { {\left\langle { \frac{\partial H} {\partial \vartheta} } \right\rangle}_{\boldsymbol{r}}} = 0. \end{equation} \tag{ I.63 }$$

Applying this result to (I.62), we obtain

$$\begin{eqnarray}&&\fl \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} \nonumber\\ &&=\int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle \widetilde{\boldsymbol{w}}_\perp\left.\cdot\widetilde{\nabla}_\perp\right|_{{{\varepsilon}_{\mathrm{e}}},\mu_{\mathrm{e}}} { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \left(\frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\cdot\nabla\tilde{\psi}\right) \right\rangle}_{\mathrm{turb}}} \right\rangle}_{\boldsymbol{r}}}. \end{eqnarray} \tag{ I.64 }$$

We now work on writing this term as a perpendicular divergence. First, we convert the derivative back to one at constant $\widetilde {\boldsymbol {w}}$:

$$\begin{eqnarray}&&\fl \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} \left\langle \left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle_{\boldsymbol{r}}\cdot\nabla\tilde{\psi} \right\rangle_{\mathrm{turb}} = \tilde{V}'\left\langle {\widetilde{B}} \left\langle \widetilde{\boldsymbol{w}}\cdot\widetilde{\nabla}_\perp \left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \left(\frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\cdot\nabla\tilde{\psi}\right) \right\rangle_{\mathrm{turb}} \right\rangle_{\boldsymbol{r}}\right\rangle_{\tilde\psi} \nonumber\\ &&-\tilde{V}'\left\langle \left\langle \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {\widetilde{\boldsymbol{b}}}\times\nabla\tilde{\psi}\cdot\left( { \frac{\partial h_{\mathrm{e}}} {\partial {{\varepsilon}_{\mathrm{e}}}} } \nabla {{\varepsilon}_{\mathrm{e}}} + { \frac{\partial h_{\mathrm{e}}} {\partial \mu_{\mathrm{e}}} } \nabla\mu_{\mathrm{e}}\right) \right\rangle_{\mathrm{turb}}\right\rangle_{\tilde\psi}. \end{eqnarray}\noindent \tag{ I.65 }$$

Next, using the fact that $\partial { {\left \langle \cdot \right \rangle }_{\mathrm {turb}}}/\partial \tilde {\alpha } \sim \epsilon \tilde {\alpha }$ and $({\partial {{\varepsilon }_{\mathrm {e}}}}/{\partial \tilde {\alpha }}) \sim B (\partial \mu _{\mathrm {e}} / \partial \tilde {\alpha }) \sim \epsilon T_{\mathrm {e}}$ by axisymmetry, we have

$$\begin{eqnarray}&&\fl \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} =\tilde{V}'\left\langle {\widetilde{B}} \widetilde{\nabla}_\perp\cdot \left\langle \widetilde{\boldsymbol{w}} \left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \left(\frac{{\widetilde{\boldsymbol{b}}}\times\widetilde{\boldsymbol{w}}}{\Omega_{\mathrm{e}}}\cdot\nabla\tilde{\psi}\right) \right\rangle_{\mathrm{turb}} \right\rangle_{\boldsymbol{r}}\right\rangle_{\tilde\psi} . \end{eqnarray}\noindent \tag{ I.66 }$$

We can now perform the gyroaverage inside the divergence to find that

$$\begin{eqnarray} &&\fl \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} =-\tilde{V}'\left\langle {\widetilde{B}} \widetilde{\nabla}_\perp\cdot \left\langle \left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}} \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} {\widetilde{\boldsymbol{b}}}\times\nabla\tilde{\psi} \right\rangle_{\mathrm{turb}} \right\rangle_{\boldsymbol{r}}\right\rangle_{\tilde\psi} , \end{eqnarray}\noindent \tag{ I.67 }$$

$$\begin{equation} \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle { {\left\langle \frac{h_{\mathrm{e}}'}{{\widetilde{B}}(\boldsymbol{R}_{\mathrm{e}}')} \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} = \Or(\delta\epsilon^2 {v_{\mathrm{th}_{\mathrm{i}}}} f_{\mathrm{e}}|\nabla\tilde{\psi}|B). \end{equation}\noindent \tag{ I.68 }$$

Inserting this final result back into (I.25), we obtain

$$\begin{eqnarray} &&\int \mathrm{d}\tilde{\alpha}\, \mathrm{d}\tilde{l} { {\left\langle \frac{1}{{\widetilde{B}}} { {\left\langle h_{\mathrm{e}}' \widetilde{\boldsymbol{w}}_\perp \right\rangle}_{\boldsymbol{r}}}\cdot\nabla\tilde{\psi} \right\rangle}_{\mathrm{turb}}} = \int \mathrm{d}\tilde{\alpha} \,\mathrm{d}\tilde{l} { {\left\langle \frac{w_\perp^2}{2\Omega_{\mathrm{e}}} { \frac{\partial \ln{\widetilde{B}}} {\partial \tilde{\alpha}} } h_{\mathrm{e}}' \right\rangle}_{\mathrm{turb}}}. \end{eqnarray}\noindent \tag{ I.69 }$$

Substituting for h_e' from (I.8) and using the fact that ${\partial {\widetilde {B}}}/{\partial \tilde {\alpha }} = \partial { {\delta B_\parallel }} / \partial \tilde {\alpha }$ we obtain (I.23) as required.

Formally, this is an expansion in $\sqrt {\beta _i} \ll 1$ subsidiary to our expansions in δ and . In the main body of the paper the Alfvén velocity v_A is assumed to be comparable to the ion thermal velocity v_thi; we must now distinguish the two. We assume that the fluctuations we are interested in have frequencies comparable to those of sound waves, rather than Alfvén waves:

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right) \sim k_\parallel {v_{\mathrm{th}_{\mathrm{i}}}} \end{equation} \tag{ J.1 }$$

and that typical fluctuating velocities scale with the ion thermal speed (rather than the Alfvén speed):

$$\begin{equation} \int \mathrm{d}^3 \boldsymbol{w} h_s \boldsymbol{w} \sim \epsilon {v_{\mathrm{th}_{\mathrm{i}}}}n_s, \end{equation}\noindent \tag{ J.2 }$$

and that this scaling also applies to the fluctuating E × B velocity:

$$\begin{equation} \frac{c}{B} {\boldsymbol{b}} \times \nabla \delta\varphi \sim \epsilon {v_{\mathrm{th}_{\mathrm{i}}}}. \end{equation}\noindent \tag{ J.3 }$$

In order to consider electrostatic turbulence, we assume that δB_⊥ and δA_∥ are small:

$$\begin{equation} \frac{{ {\delta\boldsymbol{B}}}_\perp}{{B}} \sim \epsilon\sqrt{\beta},\quad { {\delta A_\parallel}} \sim \frac{ {v_{\mathrm{th}_s}}}{c} \sqrt{\beta} {\delta\varphi}. \end{equation}\noindent \tag{ J.4 }$$

These orderings imply that $\tilde {\psi }$ and $\tilde {\alpha }$ differ from ψ and α by amounts that are small in $\epsilon \sqrt {\beta }$.

Estimating the size of terms in (112) we see that

$$\begin{equation} \frac{{ {\delta B_\parallel}}}{{B}} \sim \beta\epsilon, \end{equation}\noindent \tag{ J.5 }$$

and so we can drop all terms containing δB_∥.

As we wish to obtain the simplest equations possible, we will also neglect all finite-Mach-number effects. Thus, we combine this expansion with the low-Mach-number expansion of section 11 of [1]. The primary result of the low-Mach expansion is that we can neglect φ₀ and the poloidal variation of the mean density. In addition to this simplification, we can drop the distinction between δφ' and δφ—from now on only working with the electrostatic potential δφ.

In the subsequent sections we will apply this ordering to our collisionless and collisional equations. One result is common to both the collisional and collisionless electrostatic limits—the general solution for ζ. We obtain this by applying our electrostatic ordering to (82). The left-hand-side of (82) is $\sqrt {\beta }$ smaller than the right; thus, to lowest order, we have

$$\begin{equation} {\boldsymbol{b}}\cdot\nabla\left(\zeta - {\delta\varphi}\right) = 0, \end{equation}\noindent \tag{ J.6 }$$

where we have also used the fact that ${\widetilde {\boldsymbol {b}}} = {\boldsymbol {b}} + \Or(\sqrt {\beta })$. Solving for ζ, we obtain

$$\begin{equation} \zeta = {\delta\varphi} + \hat{\zeta}(\psi), \end{equation}\noindent \tag{ J.7 }$$

with $\hat {\zeta }$ an arbitrary function to be determined. Using this result in the expression for the field-line velocity u_eff, we see that u_eff is directed within the flux surfaces and merely moves one field line onto another without distorting them. Thus, this solution for ζ is consistent with our assumption that δB is small.

In the collisional limit, we have already used the freedom in the definition of ζ to eliminate δn_H and so cannot use this freedom to eliminate $\hat {\zeta }$. Instead, we use (118) to determine $\hat {\zeta }$. Dropping all terms that are small in $\sqrt {\beta }$, (118) becomes

$$\begin{eqnarray} &&\left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\frac{e\zeta}{T_{\mathrm{e}}} n_{\mathrm{e}} + {B} { \frac{\partial } {\partial \tilde{l}} } \frac{\delta u_{\parallel \mathrm{e}}}{{B}} = -\widehat{\boldsymbol{V}}_{\mathrm{D}} \cdot\nabla \frac{\delta T_{\mathrm{e}}}{T_{\mathrm{e}}} + \widehat{\boldsymbol{V}}_{\mathrm{D}} \cdot\nabla\left({\delta\varphi} - \zeta\right) + c { \frac{\partial \zeta} {\partial \tilde{\alpha}} } { \frac{\partial n_{\mathrm{e}} } {\partial \psi} } , \end{eqnarray} \tag{ J.8 }$$

where we have used the fact that $\tilde {\psi } - \psi \sim \epsilon \sqrt {\beta } \psi$. Using (J.7) for ζ and averaging over a flux surface, we obtain

$$\begin{equation} \left. { \frac{\partial } {\partial t} } \right|_{\tilde{\psi},\tilde{\alpha},\tilde{l}}\left( {\left\langle {\delta\varphi} \right\rangle_{\psi}} + \hat{\zeta} \right ) = 0, \end{equation} \tag{ J.9 }$$

where we have used (G.11) and (G.12). Assuming no constant perturbations, we have

$$\begin{equation} \hat{\zeta} = - {\left\langle {\delta\varphi} \right\rangle_{\psi}}. \end{equation} \tag{ J.10 }$$

Applying the low-Mach-number limit of [1] to (120), we obtain

$$\begin{eqnarray} &&{ \frac{\partial } {\partial t} } \left(\frac{\delta T_H}{T_{\mathrm{e}}} + \left\langle \frac{2{ {\delta B_\parallel}}}{3B}\right\rangle_{\tilde\psi}\right) n_{\mathrm{e}} = \frac{2}{3}\left\langle \frac{1}{{B}} { \frac{\partial } {\partial \tilde{l}} } (\tilde{\psi}-\psi) \right\rangle_{\tilde\psi} I n_{\mathrm{e}} \frac{\mathrm{d}\omega}{\mathrm{d}\psi}, \end{eqnarray} \tag{ J.11 }$$

where we have used the low-Mach-number result that n_e is a flux function to remove it from the flux-surface averages whereupon we used (G.11) and (G.12) to eliminate most of the terms in (120). Now, using (J.5) and (J.4) in (J.11) and again assuming that there are no constant perturbations, we have that δT_H = 0.

Substituting these results into (117) we have

$$\begin{equation} \delta f_{\mathrm{e}} = \frac{e \left({\delta\varphi} - {\left\langle {\delta\varphi} \right\rangle_{\psi}}\right)}{T_{\mathrm{e}}} F_{0\mathrm{e}} + \Or(\epsilon \sqrt{\beta} F_{0\mathrm{e}}), \end{equation} \tag{ J.12 }$$

with δφ found from

$$\begin{equation} \frac{1}{n_{\mathrm{e}}} \left( \sum_{s=\mathrm{i}} \frac{Z_s^2 e n_s}{T_s} \right) {\delta\varphi} + \frac{e}{T_{\mathrm{e}}} \left({\delta\varphi} - {\left\langle {\delta\varphi} \right\rangle_{\psi}}\right) = \frac{1}{n_{\mathrm{e}}} \sum_{s=\mathrm{i}} Z_s \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}. \end{equation} \tag{ J.13 }$$

This is just the usual adiabatic electron model including the zero response to zonal perturbations [28]. This reassures us that we can recover known physics from our complex model and that there is a physically-relevant regime where the adiabatic electron model is an exact limit of gyrokinetics and not an ad hoc prescription.

In this limit, we can use the freedom in the definition of ζ to set $\hat {\zeta } = 0$ in (J.7). Using (79) we see that this implies that

$$\begin{equation} \boldsymbol{u}_{\mathrm{eff}} = \boldsymbol{u} + \Or(\epsilon {v_{\mathrm{th}_{\mathrm{i}}}}\sqrt\beta). \end{equation} \tag{ J.14 }$$

Equation (103) for g_te then becomes

$$\begin{eqnarray}&&\fl \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right) g_{\mathrm{te}} + \left\langle \boldsymbol{V}_{\mathrm{D}\mathrm{e}}\cdot\nabla g_{\mathrm{te}}\right\rangle_{\parallel} + \frac{c}{e{B}} \left\{ {\left\langle {\delta\varphi}\right\rangle_{\parallel}} , {g_{\mathrm{te}}} \right\} - \left\langle {{\left\langle {\,{C}\hspace{-0.5mm}\left[g_{\mathrm{te}}\right]} \right\rangle}_{\boldsymbol{R}}}\right\rangle_{\parallel} \nonumber\\ &&= - \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right) \frac{ e \left\langle {\delta\varphi}\right\rangle_{\parallel} }{T_{\mathrm{e}}} F_{0\mathrm{e}} + c { \frac{\partial \left\langle {\delta\varphi}\right\rangle_{\parallel}} {\partial \alpha} } { \frac{\partial F_{0\mathrm{e}}} {\partial \psi} } , \end{eqnarray} \tag{ J.15 }$$

where we have used (J.4) and (J.5) to drop terms involving δA_∥ and δB_∥. This is the usual bounce-averaged kinetic equation originally derived in [9], again demonstrating that the preexisting exact models are contained in our model. However, we must still find a solution for the distribution function of passing electrons: g_pe (which was not discussed in [9]). This is determined from the electrostatic limit of (110):

$$\begin{equation} \left( { \frac{\partial } {\partial t} } + \boldsymbol{u}\cdot\nabla\right)\left( {\left\langle \frac{{B}}{|w_\parallel|} \right\rangle_{\psi}}g_{\mathrm{pe}}- \frac{eF_{0\mathrm{e}}}{T_{\mathrm{e}}} {\left\langle \frac{{B}{\delta\varphi}}{|w_\parallel|} \right\rangle_{\psi}}\right) = {\left\langle \frac{{B}}{|w_\parallel|} {{\left\langle {\,{C}\hspace{-0.5mm}\left[g_{\mathrm{pe}}\right]} \right\rangle}_{\boldsymbol{R}}} \right\rangle_{\psi}} . \end{equation} \tag{ J.16 }$$

From this equation we see that g_pe is driven by the competition between the shielding effects of the passing electrons that drive g_pe towards the maximally-shielding solution

$$\begin{equation} g_{\mathrm{pe}} = - \frac{1}{ {\left\langle {{B}}/{|w_\parallel|} \right\rangle_{\psi}}} {\left\langle \frac{{B} {\delta\varphi}}{|w_\parallel|} \right\rangle_{\psi}}\frac{e F_{0\mathrm{e}}}{T_{\mathrm{e}}} \end{equation} \tag{ J.17 }$$

and collisions which drive g_pe back towards a Maxwellian. The solution for g_pe is non-zero and so the original discussion in [9] is incomplete. Thus the solution for δf_e is

$$\begin{equation} \delta f_{\mathrm{e}} = \frac{e{\delta\varphi}}{T_{\mathrm{e}}} F_{0\mathrm{e}} + g_{\mathrm{e}}, \end{equation} \tag{ J.18 }$$

with g_e = g_pe + g_te found from (J.15) and (J.16) and δφ obtained from

$$\begin{equation} \frac{1}{n_{\mathrm{e}}} \left( \sum_{s=\mathrm{i}} \frac{Z_s^2 n_s e}{T_s} + \frac{e}{T_{\mathrm{e}}} \right) {\delta\varphi} = \frac{1}{n_{\mathrm{e}}} \int \mathrm{d}^3 \boldsymbol{w} g_{\mathrm{e}} + \sum_{s=\mathrm{i}} \frac{Z_s}{n_{\mathrm{e}}} \int \mathrm{d}^3 \boldsymbol{w} { {\left\langle h_s \right\rangle}_{\boldsymbol{r}}}. \end{equation} \tag{ J.19 }$$