Does shaping bring an advantage for reversed field pinch plasmas?

The shape of the plasma boundary is described by the following formulae:

$$\begin{equation} R=R_{{\rm 0}} +R_{{\rm 0}} \varepsilon \cos (\theta +\delta \sin (\theta )), \end{equation} \tag{ 1 }$$

$$\begin{equation} Z=R_{0} \varepsilon \kappa \sin (\theta ), \end{equation} \tag{ 2 }$$

where R and Z are the Cartesian coordinates with the origin at the centre of the torus, R₀ is the major radius, ε is the inverse aspect ratio, κ is the elongation and δ is the triangularity. The RFP equilibria are computed by solving the Grad–Shafranov equation using the CHEASE code [23], given the plasma current and pressure profiles as the input, together with the plasma boundary specified above. Figure 1 shows examples of the computed equilibrium flux surfaces with various shaped cross-sections: circular (κ = 1.0, δ = 0.0), elliptic (κ = 1.3, δ = 0.0), triangular (κ = 1.0, δ = 0.3) and D-shaped (κ = 1.3 and δ = 0.3). In order to make sensible comparisons between different shaping parameters on the RFP physics, we keep the areas of the cross-sections invariant for different types of shaping, meaning that all types of cross-sections have the same area as that of the circular case, πa². In addition, the following parameters have the same values for various shapes in the computation: the safety factor at the magnetic axis q(0), the field reversal parameter F and the total plasma current I. Here F is defined as F = B_T(a)/〈B_T〉, where 〈···〉 is the average over the plasma volume.

Zoom In Zoom Out Reset image size

The MARS-K code numerically solves the linearized, single-fluid MHD equations with self-consistent inclusion of drift kinetic resonances in toroidal geometry [24]. For a given curvilinear flux coordinate system (s, χ, φ), and by assuming that all the perturbations have the form A(s, χ, φ, t) = A(s, χ)e^{−iωt−inφ}, the MHD equations are written in the Eulerian frame in the code:

$$\begin{eqnarray} &&\fl -{\rm i}(\omega +n\Omega ){\boldsymbol{\xi}}={\bit{v}}+({\boldsymbol{\xi}}\cdot \nabla \Omega )R^{2}\nabla \phi, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\fl -{\rm i}\rho (\omega +n\Omega ){\bit{v}}=-\nabla \cdot {\bit{p}}+{\bit{j}}\times {\bit{B}}+{\bit{J}}\times {\bit{Q}}-\rho [2\Omega {\hat{{{\bit Z}}}}\nonumber\\&&\times {\bit{v}}+({\bit{v}}\cdot \nabla \Omega )R^{2}\nabla \phi ], \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&\fl -{\rm i}(\omega +n\Omega ){\bit{Q}}=\nabla \times ({\bit{v}}\times {\bit{B}})+({\bit{Q}}\cdot \nabla \Omega )R^{2}\nabla \phi \nonumber\\ &&-\nabla \times (\eta {\bit{j}}), \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\fl -{\rm i}(\omega +n\Omega )p=-{\bit{v}}\cdot \nabla P-\Gamma P\nabla \cdot {\boldsymbol{\xi}}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\fl {\bit{j}}=\nabla \times {\bit{Q}}, \end{eqnarray} \tag{ 7 }$$

where s is the normalized radial coordinate, labelling the equilibrium flux surface, and χ is a generalized poloidal angle. ω = iγ − ω_r is the complex eigenvalue of the mode (γ being the mode growth rate, ω_r the mode rotation frequency in the laboratory frame). The mode frequency is corrected by a Doppler shift in Ω, with n being the toroidal mode number, Ω the plasma rotation frequency in the toroidal direction φ. ξ, v, Q, j and p represent the perturbed quantities: the plasma displacement, the perturbed velocity, magnetic field, current and pressure tensor, respectively. ρ is the unperturbed plasma density, η is the plasma resistivity, R is the plasma major radius and ${\hat{{{\bit Z}}}}$ is the unit vector in the vertical direction. B, J and P denote the equilibrium magnetic field, current and pressure, respectively. A conventional unit system is assumed with the vacuum permeability μ₀ = 1. For the RWM study, a set of vacuum equations for the perturbed magnetic field Q, and the resistive wall equation based on the thin-shell approximation, are solved together with equations (3)–(7) [25].

The drift kinetic effects from the thermal particles are self-consistently coupled to the MHD equations, as detailed in [24]. The key element in this formulation is the wave–particle resonance operator, expressed as follows:

$$\begin{equation} \lambda_{ml}^{\alpha} =\frac{n[\omega_{^\ast N} +(\widehat{\varepsilon}_{k} -3/2)\omega_{^\ast T} +\Omega ]+\omega}{n\omega_{{\rm d}} -[\alpha (m-nq)+l]\omega_{{\rm b}} +n\Omega +\omega +{\rm i}\nu_{{\rm eff}}}, \end{equation} \tag{ 8 }$$

where $\omega_{^\ast N}$ and $\omega_{^\ast T}$ are the diamagnetic drift frequencies due to the plasma density and temperature gradients, respectively. In this drift kinetic formulation, it has been assumed that the effect of finite radial excursion width of particles across magnetic surfaces is negligible. q is the safety factor, ν_eff is the effective collision frequency and $\widehat{\varepsilon}_{k} =\varepsilon _{k} /T$ is the particle kinetic energy normalized by the temperature. ω_d is the bounce-orbit-averaged precession drift frequency. For trapped particles, α = 0, and ω_b is the bounce frequency. For passing particles, α = 1, and ω_b represents the transit frequency. In further discussions we also use notation ω_p for the transit frequency, in order to distinguish from the bounce frequency. Equation (8) includes particle bounce, transit, as well as magnetic precession drift resonances with the mode. The imaginary part of the resonant operator represents damping physics resulted from the energy transfer between the mode and particles.

In order to gain a better physical understanding, we compute various components of the quadratic energy form, for both fluid and drift kinetic energy perturbations, from the self-consistent solution. As is well known, the quadratic energy form can be constructed by multiplying equation (4) by ${\boldsymbol{\xi}}_{\bot}^{\ast}$ and integrating over the plasma volume V^P. We define the following energy components of the fluid potential energy δW_F [21, 26] and the kinetic potential energy δW_k :

$$\begin{equation} \delta W_{{\rm F}} =\delta W_{{\rm mb}} +\delta W_{{\rm mc}} +\delta W_{{\rm pre}} +\delta W_{{\rm cur}}, \end{equation} \tag{ 9a }$$

$$\begin{equation} \delta W_{{\rm mb}} =\frac{1}{2}\int_{V^{{\rm P}}} {\left| {{\bit{Q}}_{\bot}} \right|^{2}J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi}, \end{equation} \tag{ 9b }$$

$$\begin{equation} \delta W_{{\rm mc}} =\frac{1}{2}\int_{V^{{\rm P}}} {B^{2}\vert \nabla \cdot {\boldsymbol{\xi}}_{\bot} +2{\boldsymbol{\xi}}_{\bot} \cdot {\boldsymbol{\kappa}}\vert^{2}J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi} , \end{equation} \tag{ 9c }$$

$$\begin{equation} \delta W_{{\rm pre}} =-\frac{1}{2}\int_{V^{{\rm P}}} {({\boldsymbol{\xi}}_{\bot} \cdot \nabla P)({\boldsymbol{\kappa}}\cdot {\boldsymbol{\xi}}_{\bot}^{\ast} )J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi}, \end{equation} \tag{ 9d }$$

$$\begin{equation} \delta W_{{\rm cur}} =-\frac{1}{2}\int_{V^{{\rm P}}} {J_{\parallel} ({\boldsymbol{\xi}}_{\bot}^{\ast} \times {\bit{b}})\cdot {\bit{Q}}_{\bot} J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi} , \end{equation} \tag{ 9e }$$

where J is the Jacobian of the flux coordinates. δW_mb is the magnetic bending term representing the energy required to bend magnetic field lines, δW_mc responds to the energy necessary to compress the magnetic field. Both terms are positive and give stabilizing contributions. δW_pre and δW_cur represent potential sources of instability, and are referred to as the pressure-driven and the current-driven terms, respectively. Both δW_pre and δW_cur can be negative. We consider cases with vanishing perturbed surface current, where the surface terms in the potential energy disappear. In the energy calculation, we neglect the centrifugal and the Coriolis force terms in the RHS of equation (4), assuming a slow equilibrium flow. The kinetic energy term is then obtained as

$$\begin{eqnarray} &&\fl \delta W_{{\rm k}} =\frac{1}{2}\int_{V^{{\rm P}}} J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi \bigg[ p_{\bot} \frac{1}{B}(Q_{\parallel}^{\ast} +\nabla B\cdot {\boldsymbol{\xi}}_{\bot}^{\ast} )\nonumber\\&&+\,p_{\parallel} {\boldsymbol{\kappa}}\cdot {\boldsymbol{\xi}}_{\bot}^{\ast} \bigg], \end{eqnarray} \tag{ 10 }$$

where the surface term $\frac{1}{2}\int_{S^{{\rm P}}} {\tilde{{p}}_{\bot} {\boldsymbol{\xi}}_{\bot}^{\ast} \cdot {\bit{n}}J_{s} \,{\rm d}\chi \,{\rm d}\phi}$ is negligible if we assume that the equilibrium pressure vanishes at the plasma edge P(a) = 0 (the perturbed kinetic pressure is roughly proportional to the equilibrium pressure). S^p here is the plasma surface, J_s = |∇s|J is the surface Jacobian and n is an outward normal vector to the vacuum region.

We also compute the vacuum energy $\delta W_{{\rm v}^\infty}$ and δW_vb, without wall and with an ideal wall at the minor radius b, respectively:

$$\begin{eqnarray} &&\fl \delta W_{{\rm v}^\infty} =\frac{1}{2}\int_{V^{\infty}} {\left| {{\bit{Q}}} \right|^{2}J\,{\rm d}s\,{\rm d}\chi \,{\rm d}\phi} =-\frac{1}{2}\int_{S^{{\rm p}}} {b_{1}^{n} \hat{{V}}_{1}^{\ast \infty} J_{s} \,{\rm d}\chi \,{\rm d}\phi},\nonumber\\ \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} &&\fl \delta W_{{\rm v}b} =\frac{1}{2}\int_{V^{b}} {\left| {{\bit{Q}}} \right|^{2}J\,{\rm d}s\,{\rm d}\chi} =-\frac{1}{2}\int_{S^{{\rm p}}} {b_{1}^{n} \hat{{V}}_{1}^{\ast b} J_{s} \,{\rm d}\chi \,{\rm d}\phi},\nonumber\\ \end{eqnarray} \tag{ 12 }$$

where $b_{1}^{n}$ is the normal magnetic field perturbation, and is related to the perturbed magnetic components Q¹ with $Q^{1}=J_{s} b_{1}^{n}$ [27]; the component Q¹ is essentially the perturbed flux function. $\hat{{V}}_{1}^{\ast \infty ,b}$ is the complex conjugate of the perturbed magnetic scalar potential, which is determined by the ideal wall position and $b_{1}^{n}$ at the plasma surface [21]. The two vacuum energy terms are associated with the vacuum magnetic field perturbation, induced by the plasma instability. They are always positive and play a stabilizing role for the RWM. $\delta W_{{\rm v}^\infty}$ and δW_vb can be written either in a volume integral, or in a surface integral, as shown in equations (11) and (12).

Equations (9a)–(9e) to (12) are implemented in the MARS-K code [10, 11], and applied for the energy analysis of the RWM physics in this work.

With respect to tokamaks, the RFP plasmas are easier to be 'kinking' due to the weaker toroidal field in the configuration. As a result, the RWM, being a potentially disruptive instability, grows whenever the duration of the discharge is longer than the wall penetration time, even in the absence of the plasma pressure.

Furthermore, due to the toroidal field reversal, resulted from the relaxation process, RFPs can operate in a stable regime of the 'resonant' (with rational surface being inside the plasma) ideal kink modes. The observed RWM instabilities always have their rational surfaces outside the plasma. They are the so-called 'externally non-resonant' modes (ENRM), if the rational surfaces are located at q < q(a) < 0 (q(a) is the safety factor at the plasma edge r = a), or the 'internally non-resonant' modes (INRM), if the rational surfaces are located at q > q(0) > 0 [12, 19, 20]. In the following, we use n < 0 to denote the INRM, and n > 0 to represent the ENRMs. The experimentally observed resonant modes (with the resonant surface inside the plasma) are TMs, which are generally believed to be responsible for the dynamo generation in RFPs. We use n < 0 to denote the TM with the rational surface inside the B_T reversal position, and vice versa.

Shown in figure 2 is the plot of the RWM growth rate as a function of the toroidal mode numbers n, where both INRM and ENRM are computed by MARS-K in a circular, zero β RFP plasma without plasma flow. The wall position of b/a = 1.12 and the inverse aspect ratio of ε = 0.23 are taken from the RFX-mod [28] parameters. The wall time is taken as τ_w/τ_A ≈ 6.4 × 10³, where τ_w is the penetration time of the resistive wall and τ_A is the Alfvén time calculated at the magnetic axis. As we already observed previously [20], a shallower reversal (a smaller negative F value) results in a larger growth rate of the INRM and a smaller growth of ENRM, and vice versa. The modes with n < −6 (e.g. n = −7, −8 ...) are resonant modes, and are observed experimentally as TMs.

Zoom In Zoom Out Reset image size

In RFP plasmas, all RWMs are current-driven modes, therefore the no-wall β_p limit $\beta_{{\rm p}}^{{\rm no\mbox{-}wall}} =0$ (β_p = (8π/I²R_o)〈p〉 V_tot, 〈p〉 is the volume-averaged plasma pressure, 2π V_tot is the total plasma volume [23]), whilst the ideal-wall β_p limit, $\beta_{{\rm p}}^{{\rm ideal}}$, is finite for a given wall position at the minor radius r_b = b. When $\beta_{{\rm p}} >\beta _{{\rm p}}^{{\rm ideal}}$, the non-resonant ideal kink mode becomes unstable with the ideal wall at r_b = b. For $\beta_{{\rm p}} <\beta _{{\rm p}}^{{\rm ideal}}$, the RWM instability appears if the ideal wall is replaced by a resistive wall.

When the plasma elongation and triangularity are introduced, it is found that the shaping induces a strong poloidal coupling, which increases the mode growth rate and reduces the ideal-wall β limit for both the INRM and the ENRM. Figure 3 shows the RWM eigenfunctions (the perturbed magnetic field in the normal direction) $Q_{m}^{1}$ of various poloidal harmonics for (a) a circular case, (b) a case with elongation only, κ = 1.3 and δ = 0.0, (c) a case with triangularity only, κ = 1.0, δ = 0.3. It shows that in a circular toroidal RFP plasma (figure 3(a)), the m = 1 mode has a dominant amplitude, and the m = 2, 3 ... modes appear with low amplitudes due to the weak coupling by toroidicity [11]. Note that RWMs in circular RFP plasmas have much weaker toroidal coupling than that in tokamaks due to the weaker poloidal asymmetry induced by the strong poloidal field. The elongation (figure 3(b)) induces the variation of the magnetic field strength along the poloidal angle, which in turn enhances the mode coupling between m = 1 and m = ±2. As a result, the m = −1 and m = 3 modes have much larger amplitudes with respect to the circular case. The triangularity (figure 3(c)) induces multiple coupling between m = 1 and m = ±1, m = ±2 and m = ±3 modes, such that multiple modes appear due to the coupling, but with less increments of the amplitudes compared with the elongation-induced mode coupling.

Zoom In Zoom Out Reset image size

Figure 4 shows the RWM growth rate plotted as a function of the elongation with different values of triangularity. The normalized mode growth rate (normalized by ω_A) increases with κ, showing a destabilizing role played by elongation. This destabilization is also consistent with the decrease in the ideal-wall beta limit, as will be shown in figure 5(a). The triangularity itself slightly increases the RWM growth rate in the absence of elongation (κ = 1). When the triangularity is combined with elongation, it causes a slight cancellation of the destabilizing effect due to the elongation. The net effect still leads to an increase in the RWM growth rate.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 5 plots the computed ideal-wall β_p limits with varying wall position, for different toroidal harmonics of the RWMs. The INRMs with n = −6, −5 and −4 are plotted in (a), and the ENRMs with n = 4, 3, 2 are plotted in (b). Each line indicates the stability boundary of the ideal kink mode. This boundary depends on both the β_p value and the normalized wall position b/a. The unstable domains are located above and to the right of the lines, and the (ideal mode) stable domains below and to the left of the lines. For κ = 1.3, the $\beta_{{\rm p}}^{{\rm ideal}}$ values are largely reduced with respect to the circular case. For the most unstable RWM observed in RFX-mod, with n = −6, which sets the most severe $\beta_{{\rm p}}^{{\rm ideal}}$ limit and the closest fitting wall position among all the RWMs, the circular plasma model gives $\beta_{{\rm p}}^{{\rm ideal}} =0.{\rm 16}$; the elongation reduces this value to $\beta_{{\rm p}}^{{\rm ideal}} =0.{\rm 13}$ with the same wall position. Other toroidal harmonics such as n = −5, −4 and n = 4, 3, 2 have relatively smaller growth rates than that of n = −6, thus setting higher β_p limits and moderately closer wall positions. The triangularity δ = 0.3 is also considered for the n = −6 mode in figure 5(a) and for the n = 4 mode in (b) showing a similar, but less significant destabilizing effect than the elongation. The results in figure 5 indicate that, for a given plasma β_p, the shaped RFP requires a closer fitting wall to stabilize the ideal kink mode than the circular case; this effect is found extremely significant for ENRMs, as shown in figure 5(b).

For the purposes of a better physics understanding, we perform further numerical analysis, including the calculation of the perturbed energy components. The following analysis shows that the strong poloidal mode coupling, induced by shaping in the RFP, causes the reduction of the vacuum potential energy δW_vb, which is an important stabilizing factor for the ideal kink mode. Consequently, the shaping-enhanced mode coupling destabilizes the RWM and reduces the ideal-wall β limits for the ideal kink modes.

The well-known dispersion relation for the RWM, in the absence of the kinetic effects, can be expressed as [29, 30]

$$\begin{equation} \gamma \tau_{{\rm w}}^{\ast} =-\frac{\delta W_{\infty}}{\delta W_{b}}=-\frac{\delta W_{{\rm F}} +\delta W_{v^\infty}}{\delta W_{{\rm F}} +\delta W_{{\rm v}b}}, \end{equation} \tag{ 13 }$$

where γ is the mode growth rate and $\tau_{{\rm w}}^{\ast}$ characterizes the penetration time of the resistive wall. The ideal-wall β limit corresponds to the ideal kink marginal stability, which can be determined by the following condition:

$$\begin{equation} \delta W_{{\rm F}} +\delta W_{{\rm v}b} =0, \end{equation} \tag{ 14 }$$

where δW_F < 0, dominated by the current-driven energy δW_cur and pressure-driven energy δW_pre as expressed in equations (9a)–(9e), and δW_vb > 0, which plays an important stabilizing role. As described in equation (12), δW_vb is determined by the perturbed magnetic field (including all poloidal harmonics m) in the normal direction on the plasma surfaces $\vert b_{1}^{n} \vert$, where $b_{1}^{n} =\sum\nolimits_m {(b_{1}^{n}} )_{m}$, and $(b_{1}^{n} )_{m} =(\hat{{b}}_{1}^{n} )_{m} \,\exp ({\rm i}m\chi )$. As an example, in figure 6 we plot the magnetic perturbation in the normal direction, for various poloidal harmonics $(b_{1}^{n} )_{m}$ as functions of the poloidal angle χ for the n = −6 mode and elongation κ = 1.3, at the plasma surface. Both the real and imaginary parts of each $(b_{1}^{n} )_{m}$ are shown in the figure. It is clear that the m = 2 coupling, induced by the elongation, leads to the cancellation of the imaginary parts between the m = 1, 3 and the m = −1, −3 harmonics. This ultimately results in a reduction of the total magnitude of $b_{1}^{n}$ at the plasma surface. Figure 7 shows (a) the real, and (b) the imaginary parts of $b_{1}^{n}$ as the Fourier summation of all 21 poloidal harmonics $(b_{1}^{n} )_{m}$, m from −10 to 10, for κ = 1.0, 1.3 and 1.6, respectively. The amplitude $\vert b_{1}^{n} \vert$ is plotted in (c). The elongation significantly reduces the imaginary part of the amplitude of the magnetic perturbation, Im $(b_{1}^{n} )$, at the plasma surface, whilst bringing only a slight change to the amplitude of Re $(b_{1}^{n} )$. The eventual effect is a significant decrease in $\vert b_{1}^{n} \vert$ along most part of the poloidal angle, as shown in (c). As a result, the vacuum perturbation energy δW_vb correspondingly decreases. Shown in figure 8 are the various perturbed energy components of the RWM instability under the shaping assumptions of (a) elongation κ = 1.3 and 1.6, δ = 0.0, (b) triangularity δ = 0.3, κ = 1.0, and compared with the circular plasma κ = 1.0 and δ = 0.0.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The energy components are presented in several groups of columns as marked in the figure. In each group, the first column denotes the current-driven terms δW_cur and the pressure-driven terms δW_pre. The second column presents the stabilizing terms δW_mc and δW_mb. The third group presents the vacuum energy, δW_vb, with an ideal wall at the minor radius b and vacuum energy, $\delta W_{{\rm v}^\infty}$, without a wall. The fourth group is the total plasma fluid potential energy δW_F. The last column is δW_b and δW_∞, as expressed in equation (13). Figure 8(a) presents the elongation effect on various energy components. Increasing the elongation significantly reduces δW_vb, but only slightly reduces |δW_F| (mainly due to enhancement of the magnetic bending). Therefore, the value of δW_b = δW_F + δW_vb (δW_b > 0) is reduced by the elongation. Figure 8(a) also shows that, although $\delta W_{{\rm v}^\infty}$ is slightly reduced (less notable than δW_vb), the no-wall energy perturbation $\delta W_{\infty}\ ({=}\delta W_{{\rm F}} +\delta W_{{\rm v}^\infty} )$ almost does not change with increasing κ. Following the dispersion relation (13) and (14), we find that an increase in the elongation κ leads to an increase in the RWM growth rate, and to a reduction in the ideal-wall β_p limit. Figure 8(b) presents the triangularity effect, which is similar to that by the elongation, except that the decrease in δW_vb appears less significant than that caused by the elongation.

An interesting observation is that, due to B_p ∼ B_T in RFPs, the shaping effect causes a significant variation of the field strength along the poloidal angle, resulting in multiple trapping regions along the poloidal angle. Figure 9 is a 2D plot of the values of the total magnetic field strength | B| on the shaped RFP cross-section with κ = 1.3 and δ = 0.3. The new areas where the lower magnetic strength appears due to the shaping are marked regions 1, 2 and 3, respectively, in the figure. The strength of the normal field due to the toroidicity is not shown in this figure; the trapped fraction due to the toroidicity will be shown in figure 10(a) and denoted as '0'. Three new regions of the trapped particles appear in the shaped RFP plasmas, which are shown in figure 10, where we plot the fractions of the trapped particle density (n_trap/n_total, n_trap is the local trapped particle density and n_total is the local total densities including the trapped particles and passing particles) in these regions on the cross-section: (a) the fraction of the usual trapping due to the toroidicity, which we denote as 'trap 0'. (b) and (c) present the fractions of 'traps 1 and 3' and 'trap 2' resulted from both elongation and triangularity. Obviously, the shaping effect increases the total fraction of trapped particles due to the appearance of the multiple trapping regions. Figure 11 shows the radial profiles of various fractions of the trapped particles in a shaped RFP, where the total trapped fraction and passing fraction are also presented as a function of the magnetic flux coordinates. The data of figure 11 are obtained by taking average over each magnetic flux surface. It shows that the total trapped fraction keeps increasing towards the edge of the shaped plasma, contrary to the circular cross-section case where the total trapped fraction decreases towards the plasma edge [31]. Figure 12 plots the radial profiles of the bounce frequencies of trapped particles in different regions. The new trapped regions are located in a rather narrow area along the poloidal angle near the plasma edge, having higher bounce frequency than the nominal case. In particular, the particles of traps 1 and 3 have bounce frequencies comparable to or even larger than the transit frequency of the passing particles. The precession frequencies for both ions and electrons are also shown in the figure. The new trapped particles do not have substantially different precession frequencies w.r.t. the particles in 'trap 0'.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

By taking into account the particle–wave resonance effects, the stabilization condition of RWMs can be written as [32]

$$\begin{eqnarray} &&\fl \delta W_{\infty} \delta W_{b} +\delta W_{{\rm k}}^{{\rm re}} (\delta W_{b} +\delta W_{\infty} )+(\delta W_{{\rm k}}^{{\rm re}} )^{2}+(\delta W_{{\rm k}}^{{\rm im}} )^{2}>0.\nonumber\\ \end{eqnarray} \tag{ 15 }$$

As expressed in equation (10), the kinetic energy δW_k consists of the resonant (imaginary) part $\delta W_{{\rm k}}^{{\rm im}}$ and the non-resonant (real) part $\delta W_{{\rm k}}^{{\rm re}}$, $\delta W_{{\rm k}} =\delta W_{{\rm k}}^{{\rm re}} +\rmi\delta W_{{\rm k}}^{{\rm im}}$. Generally, the imaginary part always gives a stabilizing effect. The real part can be either stabilizing or destabilizing. For RWMs, the first term in equation (15) is negative, i.e. δW_∞δW_b < 0, which is the most important destabilizing term contributed by the fluid effects. The kinetic energy δW_k usually has a smaller value than the fluid energy components. In such cases the kinetic effects can play a significant role only when δW_∞ or δW_b is very small (which leads to the first destabilizing term (δW_bδW_∞) being small). This conclusion has already been obtained by a previous analysis [11]. In RFPs, the shaping leads to a smaller δW_b, therefore, the kinetic effects become more significant than those in the circular case. Earlier studies [10, 11] found that the RWMs in RFP could be kinetically stabilized mainly by the transit resonance of passing (thermal) ions for a high-β_p plasma, due to the ion acoustic Landau damping. The required plasma rotation frequency for the stabilization is in the ion acoustic wave frequency range. Considering the shaping effects, we find that the transit resonance of passing ions is still the main mechanism for the kinetic stabilization of the RWM in RFPs. The (more significant) kinetic effects result in a lower β value, required for the full stabilization, than that required in the circular case. As for the plasma rotation needed for the stabilization, the elongation leads to a slightly smaller rotation frequency; whilst the triangularity leads to a slightly larger rotation frequency for the mode stabilization, compared with that of the circular RFP. Figure 13 plots the mode growth rate versus the normalized plasma rotation frequency Ω_o (Ω_o = Ω/ω_A, where ω_A is the Alfvén frequency calculated at the magnetic axis) under various kinetic effects, such as the transit resonance of passing ions, the precession and bounce resonances of trapped particles, and the full kinetic contributions from both passing and trapped particles. The results of the fluid theory are also plotted for comparison. Figure 13(a) is for the circular plasma and (b) is for a plasma with elongation κ = 1.3. It is found that the stabilization of the RWM for the n = −6 mode in the circular plasma requires a higher β_p value, β_p = 0.15, and a rotation at Ω_o = 0.025 under the full kinetic effects, whilst for the shaped plasma with κ = 1.3, the required β_p value is lower, with the stabilization found at β_p = 0.12. The critical rotation frequency for full stabilization is also slightly decreased from 0.025 to 0.024. The transit resonance of passing ions is the dominant kinetic stabilization mechanism for both cases. The new trapped regions enhance the damping by enhanced bounce resonances, as shown in figure 13(b). On the other hand, the stabilization by these new bounce resonance regimes requires a higher rotation frequency than the transit damping; we do not consider these new regimes as the principal mechanism for stabilizing the RWM in RFP. However, the multiple trapped regions may become a significant factor to be considered, when the fast ions produced by neutral beam injection or alpha particles are taken into account.

Zoom In Zoom Out Reset image size

Figure 14 shows a comparison between an elongated and a circular RFP on the stability windows (shaded area) induced by the kinetic effects, in the β_p − b/a plane (b/a is the normalized wall radial position). The INRMs with n = −6, −5 are presented in (a), and a typical ENRM with n = 4 is in (b). For each stability window, the solid lines represent the ideal-wall beta limit $\beta_{{\rm p}}^{{\rm ideal}}$, while the dashed lines represent the boundary of the stability windows opened by kinetic damping. Thereby, the area above and to the right of the shaded area represents the unstable region of the ideal kink (with ideal wall at r = b); the area below and to the left of the shaded area represents unstable RWMs. For a given wall position, each RWM, with toroidal harmonic n, has the stability window opened along the β_p-axis. For a given β_p value, each n mode has a stability window in the b/a-axis. Within stable windows, if both β_p and b/a are given, only one critical rotation frequency Ω_c (normalized by ω_A) will be found correspondingly. The critical rotation frequencies Ω_c marked in the figures correspond to the required rotation at the smallest b/a points (highest β_p) of the dashed lines. More complete data of each stability boundary are listed in table 1. It is shown that elongation makes the kinetic damping more efficient, and thus the stability window opened in the lower β_p region with a slightly lower rotation frequency than the circular one. For INRMs, elongation leads to narrower stability windows in both β_p and b/a parameter spaces. For ENRMs, shown in (b), the kinetic effects provide a rather wide stability window except at a very low β_p, where a rather fast plasma rotation is required for the mode stabilization. For n = 4 mode, Ω_c is around 0.17–0.285 with a wall at b/a = 1.35 for the circular RFP, and Ω_c ∼ 0.26–0.32 with b/a = 1.2 for the κ = 1.3 non-circular case. Higher critical rotation Ω_c is required if the wall is closer to the plasma. These rotation ranges are much faster than that of the currently operating RFP plasmas without external momentum sources such as that from the tangential neutral beam injection.

Zoom In Zoom Out Reset image size