Dependence of alpha-particle-driven Alfvén eigenmode linear stability on device magnetic field strength and consequences for next-generation tokamaks

The structure and real frequency of the AEs present in a given tokamak will be determined by its magneto-hydrodynamic (MHD) equilibrium. These equilibria will be solutions of the Grad–Shafranov equation [16–18],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:gs} R \frac{\partial}{\partial R} \left(\frac{1}{R} \frac{\partial \psi}{\partial R} \right) + \frac{\partial^2 \psi}{\partial Z^2} = - \mu_0 R^2 \frac{{\rm d}p}{{\rm d} \psi} - \frac{1}{2} \frac{{\rm d}F^2}{{\rm d} \psi}, \nonumber \end{align} \tag{ 3 }$$

where R is the distance to the torus' axis, Z is the height above the midplane, $\psi$ is the poloidal flux, p  is the plasma pressure, and the parameter $\newcommand{\e}{{\rm e}} F\equiv R B_\phi$ is the product of major radius and toroidal magnetic field, which is related to the poloidal current in the plasma and the toroidal field coils.

This equation has a symmetry that allows it to be scaled through size and magnetic field strength without affecting the shape of equilibrium functions or device figures of merit. In particular, the equation is symmetric under the transformation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:col} \left\{\begin{array}{@{}l@{}} R \rightarrow cR \nonumber \\ Z \rightarrow cZ \nonumber \\ \psi \rightarrow \psi_{\rm N} \psi \nonumber \\ p ( \psi ) \rightarrow \frac{\psi_{\rm N}^2}{c^4} p \left( \frac{\psi}{\psi_{\rm N}} \right) \nonumber \\ F^2 ( \psi) \rightarrow \frac{\psi_{\rm N}^2}{c^2} F^2 \left( \frac{\psi}{\psi_{\rm N}} \right), \end{array}\right. \nonumber \end{align} \tag{ 4 }$$

with c and $\psi_{\rm N}$ numerical scaling constants. These transformations also mean that the toroidal magnetic field, the current density, and the overall plasma current change according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:quant} \left\{\begin{array}{@{}l@{}} B_{\phi} \rightarrow \frac{\psi_{\rm N}}{c^2}B_{\phi} \nonumber \\ J_{\phi} \rightarrow \frac{\psi_{\rm N}}{c^3} J_{\phi} \nonumber \\ I_{\rm p} \rightarrow \frac{\psi_{\rm N}}{c^3} c^2 I_{\rm p} \sim \frac{\psi_{\rm N}}{c} I_{\rm p}, \end{array}\right. \nonumber \end{align} \tag{ 5 }$$

and thus that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:param} \left\{\begin{array}{@{}l@{}} \beta \equiv \frac{p}{B_0^2/ ( 2 \mu_0 )} \rightarrow \beta \nonumber \\ \beta_{\rm N} \equiv \frac{\beta a B_0}{I_{\rm p}} \rightarrow \beta_{\rm N} \nonumber \\ q(\psi ) \equiv \frac{F ( \psi )}{2 \pi} \int \frac{{\rm d}l}{R^2 B_{\rm p}} \rightarrow q\left(\psi / \psi_{\rm N}\right), \end{array}\right. \nonumber \end{align} \tag{ 6 }$$

with $B_{\rm p}$ the poloidal magnetic field. These transformations alter device size and magnetic field strength without altering device figures of merit $\beta$ and $\beta_{\rm N}$ or the shape of the pressure, current, and q profiles. However, plasma current and pressure will increase as magnetic field strength does, though the maintenance of constant $\beta_{\rm N}$ ensures that the resulting configuration is consistent with MHD stability limits. Because this transformation changes field without changing figures of merit and increases pressure and current in the way frequently implied when considering the high field approach to fusion, we take it as defining the overall trend in equilibrium quantities with field.

As pressure increases, plasma density and temperature will necessarily increase. As discussed in the following section, the maintenance of constant Greenwald fraction will ensure that the density does not rise beyond limits allowed by the current. We note that increasing plasma temperature is frequently observed to change plasma wall interaction in a way that increases the impurity accumulation in the core of the plasma [3, 19], such that increasing temperature sufficiently to keep a constant $\beta$ may be contingent on wall material or on having a core plasma which is good at removing impurities (such as an I-mode). This trend is, however, considered to be of secondary importance to the overall improved confinement at high field leading to increasing pressure at roughly constant $\beta$.

Next, we consider the magnetic field dependence of the density and temperature profiles of the species present in the plasma. In particular, we consider the magnetic field trends of the profiles of electrons, DT ions, alpha particles, and helium ash.

2.2.1. Electron and DT ion density.

The electrons in a tokamak can be characterized by the profiles $n_{\rm e} (\psi )$ and $T_{\rm e} ( \psi )$; each of the ion species may be characterized by the profiles $n_{\rm i} ( \psi )$ and $T_{\rm i} ( \psi )$. In this paper, we assume that the dominant ion species are deuterium and tritium, with $n_{\rm D}(\psi ) = n_{\rm T} (\psi )= n_{\rm e} (\psi ) /2$. We choose not to consider the effect of ion dilution $\sigma = \left(n_{\rm D} + n_{\rm T} \right)/n_{\rm e}$, as this parameter is not expected to exhibit a strong scaling with on-axis device magnetic field; a discussion of its influence on AE linear stability may be found in [9].

The dominant scaling of density with magnetic field is determined through the Greenwald fraction [20], which is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:fgw} f_{\rm GW} \equiv \frac{\bar{n}_{\rm e} [ 10^{20}~{\rm m}^{-3}]}{I_{\rm p}[{\rm MA}]/ \pi a[{\rm m}]^2}, \nonumber \end{align} \tag{ 7 }$$

with a the device minor radius and $\bar{n}_{\rm e}$ the line-averaged $n_{\rm e}$. Disruptivity limits this parameter to [20]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{\rm GW} \leqslant 1. \nonumber \end{align} \tag{ 8 }$$

When a tokamak operates in L-mode or I-mode it has great freedom to vary the Greenwald fraction. In this case, since the constant-figure-of-merit scaling given by (4) through (6) gives plasma current increasing linearly with magnetic field for fixed device size, density will likewise tend to increase linearly with device magnetic field for a given choice of $f_{\rm GW}$.

However, operation in H-mode is somewhat more complicated. In some types of H-mode, like the EDA H-mode routinely observed on Alcator C-Mod, the density pedestal height, which strongly correlates with the overall density, is found to be determined by shot parameters at the time of the L-H transition and insensitive to attempts to modify the density pedestal height after the H-mode transition [21–23]. For EDA H-mode, the Greenwald fraction defined with the density pedestal height has been experimentally found [24] to scale as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:tolscale} &f_{\rm GW,ped} \equiv \frac{n_{\rm e,ped}}{I_{\rm p} / ( \pi a^2 )} \nonumber \\ & = (0.83) \times \left(\bar{n}_{\rm e,L} [10^{20}~{\rm m}^{-3}]\right)^{0.22} \times \left( B_0 [ {\rm T} ] \right)^{-0.64 }. \nonumber \end{align} \tag{ 9 }$$

Here, $n_{\rm e,ped}$ is the density at the top of the density pedestal and $\bar{n}_{\rm e,L}$ is the line-averaged density of the L-mode that created the H-mode, which represents how strongly the shot was fueled. Because the density pedestal height is strongly correlated with the overall density, this indicates that the shot Greenwald fraction decreases strongly with magnetic field.

Some types of H-mode can be fueled during the H-mode phase, but these H-modes exhibit an H-mode density limit, which corresponds to a maximum line-averaged density, below the Greenwald limit, at which the plasma back transitions to an L-mode [25]. An H-mode can thus vary its Greenwald fraction below this limit, but not above it; if the tokamak seeks to maximize its density, the density will be determined by this limit. Regressions over a very limited magnetic field range (roughly 1 T–3 T) have found that the Greenwald fraction corresponding to this H-mode density limit declines sharply with magnetic field, but the full magnetic field behavior has not been explored [26, 27].

These behaviors may be represented together through the statement

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{\rm GW} \sim B_0^{-\xi}, \nonumber \end{align} \tag{ 10 }$$

where $\xi$ may be 0 or a finite number depending on if operating regimes with a dominant scaling of Greenwald fraction with magnetic field are to be considered. From the scaling of current with magnetic field (5) $I_{\rm p} \sim B_0$ at fixed size, we can thus state

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_{\rm e} \sim B_0^{1-\xi}. \nonumber \end{align} \tag{ 11 }$$

2.2.2. Electron and D-T ion temperature.

The pressure of a plasma composed of electrons, deuterium and tritium ions, and alpha particles is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p( \psi ) = n_\alpha ( \psi ) T_\alpha ( \psi ) + n_{\rm e} (\psi ) &T_{\rm e} ( \psi ) + n_{\rm D} (\psi ) T_{\rm D} (\psi )\nonumber \nonumber \\ &+ n_{\rm T} ( \psi ) T_{\rm T} (\psi ). \nonumber \end{align} \tag{ 12 }$$

Care must be taken in defining the alpha temperature because the alpha particles are represented by a slowing down distribution, not a Maxwellian (see section 2.2.3). We have that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_\alpha T_\alpha = \frac{1}{3} m_\alpha \int v^2 f_\alpha {\rm d}^3 \vec {v}, \nonumber \end{align} \tag{ 13 }$$

which should be evaluated for a slowing down distribution (18). This yields a temperature roughly given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:alphtemp} T_\alpha \sim E_{\alpha 0} / 6 \sim 580~{\rm keV}. \nonumber \end{align} \tag{ 14 }$$

We consider regimes with $n_{\rm D} = n_{\rm T} = n_{\rm e}/2$; let us further take that $\newcommand{\e}{{\rm e}} T_{\rm D} = T_{\rm T} = T_{\rm e} \equiv T$. As will be discussed in section 2.2.3, alpha density is determined as a function of the local density and temperature, such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_\alpha ( \psi ) = n_{\rm e}(\psi ) g\left(T( \psi )\right), \nonumber \end{align} \tag{ 15 }$$

where the function g may be found from (22). Thus, the plasma temperature is related to the plasma pressure via

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p&(\psi ) \nonumber \\ &= n_{\rm e}( \psi ) \left[ g\left(T(\psi) \right) T_\alpha+ 2T ( \psi ) \right] \equiv n_{\rm e} (\psi)\,f\left(T(\psi )\right). \nonumber \end{align} \tag{ 16 }$$

From this statement, we may solve for temperature as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:temp} T (\psi ) = f^{-1} \left( \frac{p(\psi )}{n_{\rm e}(\psi)}\right), \nonumber \end{align} \tag{ 17 }$$

which yields the temperature necessary to obtain the given pressure at a given density. Since pressure will grow with magnetic field squared at constant $\beta$, and density will grow linearly or slower than linearly with magnetic field, temperature will, over regimes of interest, grow with magnetic field strength. Given the complicated nature of the formula for the cross section, (17) is not straightforward to evaluate analytically, but may be easily found numerically. For cases where $n_\alpha T_\alpha \ll n_{\rm e} T_{\rm e}$, equation (17) reduces to the familiar $T = p/\left(2n_{\rm e}\right)$.

2.2.3. Alpha particles.

Alpha particles born with a source rate S in a D-T plasma are described by the slowing down distribution

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:origsd} f_\alpha (v, \psi ) = \frac{S \epsilon_0^2 m_{\rm equiv} m_\alpha}{4 n_{\rm e} e^4 \ln \Lambda} \left( \frac{1}{1 + v^3/v_{\rm crit}^3}\right), \, v<v_{\alpha 0}, \nonumber \end{align} \tag{ 18 }$$

with $v_{\alpha 0}$ the alpha particle birth velocity, $v_{\alpha 0 } =$ $1.3\! \times \!10^7~{\rm m~s^{-1}}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle v_{\rm crit} = 3^{1/3} ( \pi /2 )^{1/6} \left[ T_{\rm e} / \left(m_{\rm e}^{1/3} m_{\rm equiv}^{2/3} \right)\right]^{1/2}, \nonumber \end{align} \tag{ 19 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle m_{\rm equiv} \equiv \frac{2m_{\rm D} m_{\rm T}}{m_{\rm D} + m_{\rm T}}. \nonumber \end{align} \tag{ 20 }$$

A derivation of this is presented in the appendix. Sometimes, the distribution is modified to account for the small energy dispersion around 3.5 MeV; in this case, the distribution is as given in equation (43). For D-T fusion alphas, the source rate is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:S} S = n_{\rm D} n_{\rm T} \langle \sigma v \rangle_{\rm DT}. \nonumber \end{align} \tag{ 21 }$$

Thus, the alpha particle density is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:alphdens} &n_{\alpha}\left( n_{\rm e} (\psi ), T( \psi ) \right) \nonumber \\ & =\frac{n_{\rm D} n_{\rm T} \langle \sigma v \rangle_{\rm DT}\epsilon_0^2 m_{\rm equiv} m_\alpha}{4 n_{\rm e} e^4 \ln \Lambda} \int_0^{v_{\alpha 0}} \left( \frac{4 \pi v^2}{1 + v^3/v_{\rm crit}^3}\right) {\rm d}v \nonumber \\ &= \frac{\pi n_{\rm e} ( \psi ) \langle \sigma v \rangle_{\rm DT} \left(T (\psi ) \right)\epsilon_0^2 m_{\rm equiv} m_\alpha v_{\rm crit}\left( T (\psi )\right)^3}{12 e^4 \ln \Lambda} \nonumber \\ &\qquad \qquad \qquad \times \ln \left(1 + \frac{v_{\alpha 0}^3}{v_{\rm crit}\left(T(\psi ) \right)^3} \right). \nonumber \end{align} \tag{ 22 }$$

We take $\ln \Lambda = 17$; the fusion reactivity $\langle \sigma v \rangle_{\rm DT} (T)$ may be parameterized as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \langle \sigma v \rangle_{\rm DT} = 6.4 \times 10^{-20}~{\rm m}^{3}~{\rm s}^{-1} x^{-5/6}y^2 \exp{\left( -3 x^{1/3} y\right)}, \nonumber \end{align} \tag{ 23 }$$

where, with T expressed in keV,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &x =1 \nonumber \\ &-\frac{\left(1.5\times 10^{-2} \right)T+ \left(4.6\times 10^{-3}\right)T^2 - \left(1.1\times 10^{-4}\right)T^3}{1 + \left(7.5 \times 10^{-2} \right)T+ \left( 1.4 \times 10^{-2}\right)T^2 + \left(1.4 \times 10^{-5} \right)T^3} , \nonumber \end{align} \tag{ 24 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle y = \frac{6.7}{T^{1/3}}. \nonumber \end{align} \tag{ 25 }$$

The alpha contribution to the AE growth rate increases strongly with $\newcommand{\e}{{\rm e}} \beta_\alpha \equiv 2 \mu_0 n_\alpha T_\alpha /B_0^2$ (see section 3). Thus, we wish to obtain an understanding of the dominant scaling of this quantity with magnetic field strength. We consider the ratio of $\beta_\alpha$ at one field with respect to that at another field,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:alphbet} &\frac{\beta_\alpha \left(B_0' \right)}{\beta_\alpha (B_0 )} = \frac{B_0^2 T_\alpha \left(B_0' \right)n_\alpha \left( B_0 '\right)}{B_0'^2 T_\alpha (B_0) n_\alpha ( B_0 ) } \nonumber \\ &\approx \left( \frac{B_0 }{B_0'} \right)^{1+\xi} \frac{\langle \sigma v \rangle_{\rm DT} \left(\,f^{-1} \left(\frac{p(B_0)}{n_{\rm e}(B_0)} \left( \frac{B_0'}{B_0} \right)^{1+\xi} \right) \right)}{\langle \sigma v \rangle_{\rm DT}\left(\,f^{-1} \left(\frac{p(B_0)}{n_{\rm e}(B_0)}\right)\right)} \nonumber \\ &\qquad \qquad \times \left(\frac{f^{-1} \left(\frac{p(B_0)}{n_{\rm e}(B_0)} \left( \frac{B_0'}{B_0} \right)^{1+\xi} \right) }{f^{-1} \left(\frac{p(B_0)}{n_{\rm e}(B_0)}\right)}\right)^{3/2} . \nonumber \end{align} \tag{ 26 }$$

In deriving this expression, we have taken $T_\alpha \sim E_{\alpha 0} / 6$ (see equation (14)), used (22), and recalled that $p \sim B_0^2$ while $n_{\rm e} \sim B_0^{1-\xi}$, with $\xi$ representing any dominant scaling of Greenwald fraction. Plotting this quantity for both constant and decreasing Greenwald fraction cases gives figure 1, which shows the growth in $\beta_\alpha$ over fields that correspond to an increase in temperature from 5 to 13.9 keV in the case of constant $f_{\rm GW}$ and 5 to 24.6 keV in the case of decreasing $f_{\rm GW}$. This plot demonstrates that $\beta_\alpha$ increases with magnetic field over plasma temperatures of interest, indicating that higher magnetic field machines will tend to have higher $\beta_\alpha$ than lower magnetic field machines³.

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Also of note is that decreasing Greenwald fraction at fixed pressure (as the dotted orange line demonstrates) will in general increase $\beta_\alpha$. This is due to two effects. First, decreasing density increases the amount of time it takes alpha particles to slow down on the background plasma, such that the overall alpha pressure is higher. Second, decreasing plasma density at fixed pressure, such that temperature increases, will increase the fusion reactivity.

2.2.4. He ash and other impurities.

Prior to considering how growth rates of the AEs present in high magnetic field devices will tend to change with increasing magnetic field strength, we begin by showing that the structure of the modes present will not tend to change with magnetic field. The transformations (4) leave the shapes of the profiles unchanged, so the shapes and frequency normalized to on-axis Alfvén frequency (which scales with magnetic field divided by radius) of the eigenmodes existing in equilibria related by this transformation will be the same. This may be shown explicitly from the MHD eigenmode equation, which determines the mode real frequency and structure:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:eigenmode} -\rho \omega^2 \vec {\xi}_\perp = \vec {F}_\perp \left( \vec {\xi}_\perp \right) \equiv \vec {J}_1 \times \vec {B}_{\rm eq} + \vec {J}_{\rm eq} \times \vec {B}_1 + \nabla \left( \vec {\xi}_\perp \cdot \nabla p \right), \nonumber \end{align} \tag{ 27 }$$

where $\omega$ is the mode real frequency, $\vec {\xi}_\perp$ is the perturbed $\vec {E} \times \vec {B}$ displacement, $\newcommand{\e}{{\rm e}} \vec {E}_{1 \perp} \equiv {\rm i} \omega \vec {\xi}_\perp \times \vec {B}_{\rm eq}$, $\vec {F}_\perp$ is the ideal MHD force operator, $\vec {J}_1$ and $\vec {B}_1$ are the perturbed current and magnetic field, respectively, and the subscript _eq denotes equilibrium quantities. Scaling the mode frequency to the on-axis Alfvén frequency $\newcommand{\e}{{\rm e}} \omega_{A0} \equiv B_0 / \left( R_0 \sqrt{\mu_0 \rho_0}\right)$ gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle - \frac{B_0^2}{R_0^2 \mu_0 } \frac{\rho}{\rho_0} \left(\frac{\omega}{\omega_{A0}}\right)^2 \vec {\xi}_\perp = \vec {J}_1 \times \vec {B}_{\rm eq} + \vec {J}_{\rm eq} \times \vec {B}_1 + \nabla \left( \vec {\xi}_\perp \cdot \nabla p \right). \nonumber \end{align} \tag{ 28 }$$

Scaling this equation according to the transformations (4) through (6) shows that the shape of $\vec {\xi}_\perp$ and the value of $\omega /\omega_{A0}$ will be the same as long as the shape of the density profile, $\rho/ \rho_0$, is unaltered.

A given tokamak will have a variety of AEs which have the potential to become excited. The symmetry discussed in section 3.1 indicates that the set of existing modes with the potential to become excited does not depend on magnetic field. However, the linear growth rate of each of these modes will depend strongly on magnetic field.

Total linear AE growth or damping rate is determined by the contributions from each of the species in the plasma interacting with the AE:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:sum} \frac{\gamma}{\omega} = \frac{\gamma_\alpha}{\omega} + \sum_j \frac{\gamma_j}{\omega}, \nonumber \end{align} \tag{ 29 }$$

where the mode growth rate $\gamma$, expressed as a ratio of the mode frequency $\omega$, is given as a sum of the contribution due to alpha particles and due to thermal plasma species. Each of these contributions may be dependent on magnetic field.

To obtain a straightforward, simplified view of AE stability, many analytical calculations of these contributions ignore the finite width of AE modes, considering them to have one discrete radial localization, and consider only fully passing particles. In this section, we make both of these assumptions in order to understand the basic trends in AE stability with device field strength. These assumptions are not present in the numerical treatment that forms section 5; one of the main effects of the loss of these assumptions is that discrete AE resonances become spread out in phase space. However, even with the more precise numerical treatment, the framework developed in this section serves well to understand overall behavior.

Analytically, the contribution of alpha particles to the AE growth rate is estimated to be [28]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:alphagrow} \frac{\gamma_\alpha}{\omega} \sim -q_{\rm AE}^2 \beta_\alpha \left( G^\alpha ( \lambda ) - n q_{\rm AE} \delta_\alpha H^\alpha (\lambda ) \right), \nonumber \end{align} \tag{ 30 }$$

with $\lambda$ the parameter defined in (2), n the toroidal mode number, $q_{\rm AE}$ the value of q at the location of the mode, $\newcommand{\e}{{\rm e}} \delta_\alpha \equiv -(2/3) r_{\rm L\theta} \left({\rm d}p_\alpha /{\rm d}r \right)/ p_\alpha$ the ratio of the alpha particle birth poloidal gyroradius, $\newcommand{\e}{{\rm e}} r_{\rm L\theta} \equiv v_{\alpha 0}/ \Omega_{\theta \alpha}$ with $\newcommand{\e}{{\rm e}} \Omega_{\theta \alpha}\equiv q_\alpha B_{\rm p} /m_\alpha$, to the pressure scale length. The function $G^\alpha(\lambda )$ represents damping of the mode on the alpha particles due to the gradient of their distribution function with respect to energy, $\frac{\partial f_\alpha}{\partial E}$, at points where the mode is resonant with the alpha particle population. The function $H^\alpha (\lambda )$ represents growth of the mode due to the spatial gradient of the alpha particle distribution function, $\frac{\partial f_\alpha}{\partial \psi}$, at the points where the mode is resonant. The location of the resonant points depends on the type of mode. For example, the TAE has its primary resonances at alpha particle velocities $v_\parallel = v_{A0}$ and $v_\parallel = v_{A0}/3$, while the EAE has its primary resonance at $v_\parallel = v_{A0}/2$.

The alpha particle energy distribution function will be given by a slowing down distribution, (18) (or (43) if the finite birth energy dispersion around 3.5 MeV is included); the shape of the distribution function in $\psi$ will depend on the equilibrium. The slowing down distribution is pictured in figure 2(a).

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Importantly, this distribution function includes only particles with energy below roughly $E_{\alpha 0} = 3.5~{\rm MeV}$, the alpha particle birth velocity (some particles will have slightly more energy than 3.5 MeV due to retained energy from the collision that caused the fusion); furthermore, more alpha particles are at lower energies than at higher energies. This means that if the energy corresponding to one of the resonances of an AE lies above 3.5 MeV, that resonance will not contribute to the growth rate of the AE, reducing the overall drive to the AE. For the TAE, the first loss of a resonance will occur at $\newcommand{\e}{{\rm e}} \lambda \equiv v_{A0}/v_{\alpha 0} = 1$. However, since more alpha particles are at lower energies than at higher energies, the loss of this resonance is expected to remove only a fraction of the overall AE drive.

Meanwhile, the factor $\beta_\alpha$ in (30) multiplies the part of the expression dependent on resonances. As discussed in section 2.2.3, $\beta_\alpha$ will tend to increase with magnetic field, such that we expect this factor to cause a strong increase in AE drive with magnetic field. This increase, combined with any decrease due to loss of resonances, will determine the overall evolution of growth rate as a function of magnetic field.

We must also consider the damping contributions to the AE. These result from the interaction of the AE with bulk thermal species and are given by [28]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:thermaldamp} \frac{\gamma_j}{\omega} \sim -q_{\rm AE}^2 \beta_j G^{\rm T}\left(\lambda_{\rm T}\right). \nonumber \end{align} \tag{ 31 }$$

Here, $G^{\rm T}\left(\lambda_{\rm T}\right)$ represents damping due to the energy gradient of the thermal species distribution, $\partial f_j/ \partial E$, at the points where the mode is resonant with the particles; $\lambda_{\rm T} = v_{A0}/v_{{\rm T}j}$ with $v_{{\rm T}j}$ the thermal velocity of the species in question. Since bulk species have a Maxwellian distribution, there will be no complete loss of a resonance when the magnetic field becomes too strong, as there was with a slowing down distribution, which has a sharp cutoff at the birth energy. Likewise, $\beta_j$ for bulk species tends to remain constant with magnetic field. The damping of AEs will thus not display strong trends with magnetic field, and instead will change gradually only due to the changing value of $\partial f_j/ \partial E$ at the resonance location.

As discussed in section 3.2, the AE interacts with alpha particles through a set of resonances. In the case of the TAE, the higher energy resonance at $v_\parallel = v_{A0}$ will be cut off in machines with high enough field. The lower resonance $v_\parallel = v_{A0}/3$ will still provide substantial drive to the mode. However, the growth rate of a mode is not the only important result of an analysis of linear AE physics. Also important is the energy of the alpha particles that are able to resonate with the mode, as these will be the alpha particles which have the potential to be transported out of the core by AEs. The loss of the upper resonance means that only lower-energy particles can be affected by the mode. This will reduce the loss of higher energy alphas, which have more chance of damaging the machine and also carry more heating power than lower-energy alphas. A visual representation of this effect is given in figure 2(b), which shows the slowing down distribution weighted by energy, showing that although more particles exist at lower energy, the energy content is roughly evenly distributed between lower and higher energy, such that cutting off a higher resonance will greatly reduce the overall energy of the alpha particles that interact with the AE.

While all AEs in the tokamak will, regardless of mode number, experience the general trends described in sections 3.2 through 3.3, there will be another effect that modulates this trend and leads to a change in which mode is the most unstable at a given field.

The width of an AE may be estimated by taking (27) and solving for a mode of the form $\newcommand{\e}{{\rm e}} \vec {\xi}_\perp = \exp\left[ {\rm i} ( n \phi - \omega t )\right] \sum_m \vec {\xi}_{\perp, m} ( r ) \exp{\left(-{\rm i} m \theta\right)}$. Stating the result to zeroth order in inverse aspect ratio gives [29–31]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}}{{\rm d}r} \left( \frac{\omega^2}{v_A^2} - k_{\parallel m}^2\right) \frac{{\rm d} \vec {\xi}_{\perp,m }}{{\rm d}r} - \frac{m^2}{r^2} \left( \frac{\omega^2}{v_A^2} - k_{\parallel m}^2\right) \vec {\xi}{\perp, m} = 0, \nonumber \end{align} \tag{ 32 }$$

with $\newcommand{\e}{{\rm e}} k_{\parallel m} \equiv (1/R) \left[ n - m/q(r)\right]$. Balancing the terms in this equation gives that the mode width

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:modew} \Delta_{m} \sim r_{m} / m, \nonumber \end{align} \tag{ 33 }$$

with r_m the radial location of the mode. For TAEs, $q (r_m ) = ( m - 1/2 ) /n$, while for EAEs $q (r_m ) = ( m - 1 ) /n$, and similar relationships hold for other types of AE. Thus (33) may be approximated by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:modew2} \Delta_{m} \sim r_{m} / n q \left(r_{m}\right). \nonumber \end{align} \tag{ 34 }$$

Meanwhile, the width of an alpha particle orbit for a particle with $v_\parallel \gg v_\perp$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:modewidth} \Delta_{\rm orbit} \sim \frac{q v_\parallel}{\Omega_\alpha}, \nonumber \end{align} \tag{ 35 }$$

with $\Omega_\alpha$ the alpha particle cyclotron frequency.

As suggested by equation (30), TAE growth rate will roughly tend to increase with increasing n when orbit width effects do not have to be considered; the same trend is true for other types of AE. However, once the alpha particle orbit width exceeds the mode width, the growth rate will decrease with increasing n because the coupling between the mode and the particle is less effective [29, 32–34]. The result is that the maximally excited modes are those with toroidal mode numbers given roughly by $\Delta_{m} \sim \Delta_{\rm orbit}$, or

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:nmax} n_{\max} \sim \frac{r \Omega_\alpha}{v_\parallel q^2}. \nonumber \end{align} \tag{ 36 }$$

We note that experimental evidence for a similar scaling has been presented in [35]. Because $\Omega_\alpha = q_\alpha B_0/m_\alpha$, while under the trends outlined in table 1,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle v_{A0} \sim \sqrt{B_0} B_0^{\xi/2}, \nonumber \end{align} \tag{ 37 }$$

the toroidal mode number of the most unstable mode will increase with magnetic field except in the case of very significant decrease in Greenwald fraction with magnetic field. This increase will be affected as well by the changing parallel velocity of the resonant particles. For the case of the TAE, lower magnetic field machines with $\lambda < 1$ will have resonant particles with $v_\parallel \sim v_{A0}$ and $v_\parallel \sim v_{A0}/3$, while machines with $\lambda >1$ will have contributions exclusively from particles with $v_\parallel \sim v_{A0}/3$, further enhancing the increase in n.

The set of TAEs in this equilibrium is found using MISHKA, following the frequency scan and filtering process described in [13]. This process finds 77 TAEs; the set of eigenmodes is the same regardless of the magnetic field strength, in agreement with section 3.1. Four TAEs are selected in this section to demonstrate typical TAE growth rate evolution with field; these are pictured in figure 4. Note the noticeably smaller width of the higher-n modes with respect to the lower-n modes, which confirms the expected trend from (35).

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In this section, we consider how the contribution of each of the plasma species to the growth rate evolves with magnetic field. That is, we seek to verify and expand on the trends identified in section 3.2 for each of the species contributing to (29). This process is conducted with the modes shown in figure 4.

The modes are selected to represent two different classes of TAEs; figures 4(a) and (b) are modes located in the core of the device with small radial extent and a simple structure; figures 4(c) and (d) are more mid-radius modes of more complex structure. Within each of these two categories, modes of lower and higher n values are selected. CASTOR-K is used to calculate the contribution of each plasma species to the overall growth rate of the mode. In all cases, the damping of the modes on electrons is found to be negligibly small in comparison to damping on deuterium and tritium, so it is neglected. Though only four modes are shown in this section of the paper, many more modes, and many other equilibrium configurations, were examined in the course of this study; all showed similar trends.

The results are shown in figure 5. Figure 5(a) displays the alpha particle contribution to the TAE growth rate. The growth rate is normalized to the on-axis Alfvén frequency at 10 T such that growth rates can be compared between fields. Initially, the alpha particle contribution to the growth rate increases with magnetic field, which is in part a representation of increasing $\beta_\alpha$. Then, the contribution flattens out or reduces near $\lambda = 1$, indicating a loss of a TAE resonance. Also note that at lower fields, the lower-n modes have higher growth rates than their equivalent higher-n modes, while at higher fields the modes are more comparable or the higher-n modes are more unstable.

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More direct evidence of the loss of the resonance can be found by normalizing the alpha particle growth rate to mode frequency at the given field and to $\beta_\alpha$, such that comparison to equation (30) is immediate. This quantity is plotted in figure 5(b). Somewhat below $\lambda = 1$, the trend in $\gamma_\alpha / ( \omega \beta_\alpha )$ changes from whatever it was below to a notable decrease. This is due to the loss of resonance, as will be further represented by the results in section 5.3. The decrease often begins below $\lambda = 1$ because of effects not included in the simplified passing-particle, localized mode analytic treatment; these effects allow particles close to, but not exactly, fulfilling the analytic resonance condition to exchange energy with the mode.

The contribution of deuterium to mode damping is shown in figure 5(c); the trends in the tritium damping are identical. As expected, the damping does not show the strong $\lambda$-related changes that the alpha particle contribution does.

Finally, the overall growth rate of the modes is plotted in figure 5(d). For all of the modes, the initial increase with $\lambda$ due to increasing $\beta_\alpha$ is visible. For the n  =  21 mode, the loss of resonance allows the mode to achieve stability after it was unstable. For the other modes, the loss of the resonance appears to substantially reduce the growth rate near $\lambda = 1.0$ relative to what it would be if the lower-field increase were still present, such that two modes are completely stable and the other is near marginal stability.

While such alpha particle resonance loss cannot be experimentally tested in presently-existing devices, we note that the stabilization of NBI-driven AE activity due to changing effective temperature of beam ions and the resulting resonance loss has been recently experimentally observed on DIII-D [48]. Direct parallels can also be found in studies of the magnetic field dependence of NBI-driven AEs on TFTR [49]. While providing clear demonstration of resonance loss, these experimental demonstrations differ from alpha particle resonance loss in several fundamental ways, including the fast ion velocities at which they occur and the radial profiles and distribution of pitch angle characterizing the fast ions.

Next, we examine plots of regions in phase space of energy exchange between alpha particles and one of the TAEs produced by the CASTOR-K analysis. This serves to demonstrate the energy of the alpha particles which interact with the modes, to give further evidence that the trends near $\lambda = 1.0$ noted in the previous section are indeed due to the loss of resonances, and to give insight into the different roles played by different types of particle orbits.

We consider results shown in figure 6, which show energy exchange between the n  =  22 TAE pictured in figure 4(c) and the alpha particle population. These results are shown at two different values of $\newcommand{\e}{{\rm e}} \Lambda \equiv (v_\perp^2 / v^2 ) ( B_0 / B )$, $\Lambda = 0.01$ and $\Lambda = 1.0$. The lower value of $\Lambda$ corresponds to passing particles, which are the only types of particles considered by the analytical theory presented in section 3, while the higher value of $\Lambda$ corresponds to trapped particle orbits, and is usually one of the values of $\Lambda$ at which the most energy exchange between the mode and particles occurs. The results are shown as a function of alpha particle energy and of toroidal canonical angular momentum $\newcommand{\e}{{\rm e}} P_\phi \equiv q_\alpha ( \psi + v_\parallel R B_\phi / \Omega_\alpha )$ normalized to $q_\alpha B_0 R_0^2$. The plots are evaluated at 7 T, 12 T, and 16 T, which correspond in this setup to $\lambda = 0.74, 0.97,$ and 1.11.

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Consider first the lowest-$\lambda$ plot for passing particles, figure 6(a). This plot displays one primary resonance peaked roughly around 2000 keV, corresponding to the primary TAE resonance $v_\parallel = v_{A0}$. This resonance is very spread out. The lower resonance, though present, is not significant. The corresponding plot for more complex particle orbit is shown in figure 6(b); this plot has more complicated resonance structure.

Next, consider the plots at 12 T, $\lambda = 0.97$, figures 6(c) and (d). Notable in these plots is the partial cutoff of resonances near the alpha particle birth velocity, even in the case of more complex particle orbits. Lower energy resonances, including the $v_\parallel = v_{A0}/3$ resonance in figure 6(c) become noticeable as a key component of the mode's drive, reducing the overall energy of the alpha particles that interact with the mode.

Finally, plots at 16 T, $\lambda = 1.11$ are shown in figures 6(e) and (f). In these plots, the primary resonance at $v_\parallel = v_{A0}$ for passing particles has nearly completely disappeared from the map. However, another resonance at high energy has appeared (which was slightly visible in figure 6(c)), likely due to the complex, non-localized mode structure of the particular TAE examined, which has yet to be fully cut off. The lower energy resonances are very visible as a source of TAE drive. Likewise, in the $\Lambda = 1.0$ case, many resonances have been cut off and those remaining are at lower energy.

Finally, we may consider trends in the toroidal mode number of the most unstable TAEs with magnetic field. Figure 7 displays overall growth rate (including alpha particle, deuterium, and tritium contributions) as a function of toroidal mode number for modes with frequencies between $\omega = 0.45~\omega_{A0}$ and $\omega = 0.5~\omega_{A0}$ (because mode frequency is correlated to radial location, this allows a comparison between modes which experience similar background gradients and with similar structures). At 7 T, the lower n modes are closer to being unstable than the higher-n modes (because $\beta_\alpha$ is low at this low field, none of the modes is unstable). At 13 T, the higher-n modes have higher overall growth rates (though few are unstable, in part because of the cut off of resonances at this field).

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While the numerical results in this section offer significant insight into AE linear stability trends with magnetic field, they do not include finite alpha gyroradius effects. These become important when $k_\perp \rho_\alpha \sim 1$, or, equivalently, as discussed in the introduction to this section, when $n\sim 45 \left(B_0 / 10~{\rm T}\right)$, making them important to some of the low-field results we present in this section. We comment in this section on the probable effect of these terms on the trends presented in this section, guided by the analytical estimate that finite gyroradius effects reduce alpha particle drive by a factor of $1/J_0^2(k_\perp \rho_\alpha)$, with J₀ the zeroth order Bessel function [13, 50].

For a given mode, the importance of these corrections will tend to decrease with magnetic field, such that damping due to finite alpha gyroradius effects decreases, leading to an overall increase in growth rate with field. For alphas $\rho_\alpha = m_\alpha v_\perp / (q_\alpha B_0 )$, which for a given perpendicular velocity gives $\rho_\alpha \sim 1/B_0$. This decrease in gyroradius will result in decreasing $k_\perp \rho_\alpha$ through field, and hence a reduction in the factor $1/J_0^2(k_\perp \rho_\alpha)$ by which finite gyroradius effects reduce the growth rate. Since $k_\perp \sim 1 / \Delta_m \sim n$ (consider equation (34)), this trend will be stronger for higher-n modes, such that the trend identified in section 5.4 is accentuated: stronger reduction in damping with field will further increase the instability of high-n modes with respect to low-n modes as field increases.

In this paper, we have considered AE stability behavior relevant to next generation very-high-magnetic-field tokamaks which are currently experiencing significant interest in fusion research. We began in section 2 by considering the dominant trends with magnetic field of tokamak parameters relevant to AE stability, while noting that at a given magnetic field, a tokamak has flexibility to slightly modify these quantities. A result of this study was the realization in section 2.2.3 that $\beta_\alpha$ will tend to be significantly larger in high magnetic field devices.

Sections 3 and 5 then argue theoretically and computationally for trends in AE stability with magnetic field. The increase in $\beta_\alpha$ with magnetic field will increase AE drive. If an HTS device operates with sufficiently high Alfvén speed, it may be able to cut off AE resonances, reducing mode drive and the energy of alpha particles interacting with the mode. Meanwhile, damping will not depend strongly on magnetic field. The structure of AEs will not tend to change with device magnetic field, and the toroidal mode number of the most strongly driven mode will increase.

Given the strong increase in $\beta_{\alpha}$ with magnetic field, it is likely that HTS machines will experience AE activity, and planning for the management of this activity should be part of their design process. However, consideration of the source of the $\beta_\alpha$ growth with magnetic field gives context to how this challenge should be evaluated when evaluating the relative merits of low- and high-field approaches. When tokamak figures of merit are held constant as magnetic field is scaled, as we do in this paper, the fusion rate (21) increases with magnetic field strength, and this increasing fusion rate is responsible for the increasing $\beta_\alpha$. If a fusion device aims to produce a certain amount of fusion power, small, high magnetic field machines will indeed have higher $\beta_\alpha$. This is because larger, lower magnetic field machines will obtain the target amount of fusion power by operating with a smaller source rate (21), and a lower $\beta_\alpha$, over a larger volume, while smaller, high magnetic field machines will operate with high source rate (21), and a high $\beta_\alpha$ over a smaller volume. Since high $\beta_\alpha$ encourages AE instability, this is a critical consideration for high magnetic field machines.

However, if economic considerations are taken into account, this trend may not be overly punitive to high magnetic field devices. Some studies [51] assume that any device, regardless of size, will for economic viability need to achieve not a fixed amount of power, but rather a fixed value of the fusion power density given in (1). To achieve this, a device must achieve a specific value of fusion source rate (21), regardless of the device size and the resulting absolute amount of fusion power the device will produce. The tendency of AE modes to be destabilized is correlated to the ratio $\beta_\alpha/\beta$, which is roughly the ratio of AE drive to AE damping when the effect of resonance location is neglected (consider equations (31) and (30)). A figure of merit may be defined which represents the amount of $\beta_\alpha/ \beta$ per amount of fusion power density:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:fixpov} \frac{\beta_\alpha/ \beta}{n_{\rm D} n_{\rm T} \langle \sigma v \rangle_{\rm DT} } \sim \frac{\frac{n^2 T^2/\left(n/T^{3/2}\right)}{nT}}{n^2T^2} \sim \frac{T^{1/2}}{n^2}. \nonumber \end{align} \tag{ 45 }$$

(Here, we have taken the simplified expression $\langle \sigma v \rangle_{\rm DT} \sim T^2$ and neglected the influence of alpha pressure on $\beta$.) This quantity strongly decreases with B₀ due to the higher absolute densities accessible at high magnetic field. Thus, when economic considerations are taken into account, the amount of AE drive incurred to achieve a given performance decreases with field. This is a positive trend for high magnetic field machines.

Regardless of the lens through which the high field approach is evaluated, the results of this paper suggest two regimes in which the unique characteristics of HTS machines are optimally exploited to minimize and control AE activity that is present: at low density and high density. Operation in the first of these regimes attempts to cut off AE resonances, while the latter accelerates the slowing down of alphas. Intermediate densities allow neither of these advantages. Which of these regimes an HTS machine operates in is likely to be determined by operational constraints distinct from AE behavior. We consider each of these regimes in turn here.

Cutting off AE resonances, even partially, reduces the drive to AEs and the energy of alpha particles which interact with the mode. This will reduce the ability of the AEs to cause transport, though significant transport will likely still be present. However, cutting off even the first TAE resonance, i.e. achieving $\lambda \sim 1$, at practically-achievable magnetic fields, will require operating at low Greenwald fraction and potentially high values of q₉₅ (and thus lower values of current). Consider that the configuration studied in section 5, with q₉₅  =  3.5 and $f_{\rm GW} = 0.18$, only achieves $\lambda = 1$ at 13 T. We note that such low Greenwald fraction at high field corresponds to significant density; in the scenario considered, the density at 10 T (see figure 3(b)) is higher than that in the ITER baseline scenario [13]. Correspondingly, such low Greenwald fractions are within the operating space currently envisioned for high field devices [6]. However, because of favorable trends in overall tokamak performance with increasing current and density, such low Greenwald fraction operation may not be the preferred operation state for high-magnetic-field devices, and certainly does not represent the highest performance they could ideally achieve. Nevertheless, operation in such a low density state may end up being a requirement of high-magnetic-field operation if the trends in H-mode density pedestal height, mentioned in section 2, or perhaps a similar trend in H-mode density limit, are born out at high fields, or if I-mode operation is required due to some operational constraint. Even if not practical in high-performance shots, resonance cutoff could still be an interesting physics effect to observe; furthermore, given the broad nature of the resonances and the resulting beginning of resonance cutoff well before $\lambda = 1$ for some modes, it will likely occur for a few modes even in higher Greenwald-fraction shots.

If a device does not operate at low density due to operational or mission constraints, it can minimize AE drive by maximizing density at the expense of temperature. A given device will want to minimize AE drive subject to a fixed performance. Neglecting the effect of resonance locations, as in (45), this means minimizing $\beta_\alpha/\beta \sim T^{5/2}$ subject to $n^2T^2 = C$, with C a constant. This corresponds to minimizing

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta_\alpha \sim \frac{C^{5/4}}{n^{5/2}}. \nonumber \end{align} \tag{ 46 }$$

This division by density exists because higher-density plasmas slow down alpha particles more quickly, and indicates that increasing n while decreasing T at fixed $\beta$ will reduce the amount of AE drive. This trend was also noted in section 2.2.3. High field machines are better able to make this tradeoff than low field machines, because high field machines can access higher densities due to their higher currents, when compared to low-field devices with similar power densities.

The present work suggests several avenues for future work. First, as noted in section 4, further analysis of heating drive for AEs should be completed. We observe that this may be particularly important for the SPARC concept, as it has a fairly modest Q target of 2–5 [6], meaning that a significant fraction of AE drive could come from fast ions resulting from ICRH or other heating methods.

Next, this paper has focused on the TAE, and its results may be applied in a straightforward way to the EAE, NAE, and other gap modes. However, a variety of other Alfvénic fluctuations can be excited in tokamaks. They are not a focus of this paper due to their different physical nature, which prevents them from being studied using the suite of codes employed here, which is not capable of treating continuum damping or thermal plasma compressibility. In addition, some modes, like the RSAE (caused by a reversed shear q profile), are left out because they are not expected to be observed in baseline scenarios for SPARC, which are currently envisioned to use conservative profiles without reversed shear [52].

However, study of these modes and the dependence of their behavior on magnetic field strength is important and should be undertaken in detail in future work. In many cases, the framework established in this paper can be adapted to other modes; we give two examples here. Energetic particle continuum modes (EPMs) are modes at frequencies characteristic of energetic particle motion (often outside of the gaps in the continuum spectrum, where the TAE and similar modes live), which arise when drive from the spatial gradient in the fast particles is sufficient to exceed continuum damping [53]. The excitation of these modes is expected to be only weakly dependent on the velocity space fast particle distribution function [53]. The smaller amount of fast ion beta generated at a given fusion power density thus is likely to be the most important determinant of EPM stability. This will prove advantageous for high-magnetic field reactors, and indeed previous studies of high-field devices like IGNITOR and FIRE have found these devices to be stable to EPMs [54].

BAEs exist in a frequency gap caused by finite thermal plasma compressibility located below the shear Alfvén continuum spectrum, $0 < \left(\omega/\omega_{A0}\right){}^2 \lesssim \Gamma \beta$, with $\Gamma$ the ratio of specific heats [55]. Analytically, these modes can be shown to be resonant with energetic particles with transit frequencies approximately equal to the mode frequency; the source of the drive is the spatial gradient of the energetic particle distribution function [55, 56]. That the resonant velocities for these modes are significantly lower than for the TAE suggests it will not be possible to cut them off, and for this mode, again, the magnetic field dependence of $\beta_\alpha$ is the most important determinant of alpha particle contribution to the growth rate.

Finally, the topic of this paper is linear AE behavior at high field, but actual alpha particle transport and evolution results from nonlinear processes [8]. Comparable analysis of high field trends in this area is a topic of current research by the authors. In addition to trends intrinsic to nonlinear physics, this analysis could reveal nonlinear trends that result from linear physics. For example, high n modes, which tend to be narrower, are more likely than low n modes to be unstable in high field machines, while the converse is true in low field machines. The reduced width of destabilized AEs in HTS machines could influence the ability of these modes to cause radial transport; further, the interaction of AEs with periodic magnetic field ripple is a key source alpha transport [8, 41], and the changed periodicity of destabilized AEs could influence this synergy.