1. Formulation

* Blue-colored sentences are physical assumptions used in GKV [ 1-1 ] . This manual is based on the GKV version gkvp_f0.65 .

1.1. Governing equations

One derives gyrokinetic equations based on the following gyrokinetic ordering [ 1-2 ] ,

(1.1) \[\frac{\tilde{f}}{F} \sim \frac{e\tilde{\phi}}{T} \sim \frac{\tilde{B}}{B} \sim \frac{k_\parallel}{k_\perp} \sim \frac{\omega}{\Omega} \equiv \delta \ll 1.\]

GKV follows \(\delta f\) gyrokinetics, where distribution functions are split into equilibrium and perturbed parts \(\mathcal{F}=F+\tilde{f}\) . Additionally, there are some subsidiary assumptions:

• separation of the equilibrium and perturbed scale lengths \(|\nabla F|/F \ll |\nabla \tilde{f}|/f\) decouples neoclassical physics from turbulent dynamics and treats flute-type perturbations

• low \(\beta\) value justifies neglect of compressional magnetosonic waves \(\tilde{B}_\parallel\) and higher-order correction in \(\beta\) , but retains shear Alfvénic dynamics \(\tilde{A}_\parallel\)

• the present version of GKV treat equilibrium \(\bm{E} \times \bm{B}\) flow shear effect.

• the equilibrium distribution function is to be a local Maxwellian \(F=F_\mathrm{M}=n \left(\frac{m}{2 \pi T}\right)^\frac{3}{2} e^{-\frac{mv_\parallel^2}{2T}-\frac{\mu B}{T}}\)

• the equilibrium magnetic field satisfies the MHD equilibrium \(\nabla P = \bm{J} \times \bm{B}\)

Then, the \(\delta f\) gyrokinetic Vlasov-Poisson-Ampère equations are

(1.2) \[\begin{split}&\frac{\partial \tilde{f}_\mathrm{s}}{\partial t} + \left( \bm{V}_E + v_\parallel \frac{\bm{B} + \tilde{\bm{B}}_\perp}{B} + \tilde{\bm{v}}_\mathrm{E} + \bm{v}_\mathrm{sG} + \bm{v}_\mathrm{sC} \right) \cdot \nabla \left( \tilde{f}_\mathrm{s} + \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} J_{0\mathrm{s}} \tilde{\phi} \right) \\ &- \frac{\mu \nabla_\parallel B}{m_\mathrm{s}} \frac{\partial}{\partial v_\parallel} \left( \tilde{f}_\mathrm{s} + \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} J_{0\mathrm{s}} \tilde{\phi} \right) \\ &+ \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} \left[ v_\parallel \frac{\partial J_{0\mathrm{s}} \tilde{A}_\parallel}{\partial t} - \bm{v}_{\mathrm{s}*} \cdot \nabla J_{0\mathrm{s}} (\tilde{\phi} - v_\parallel \tilde{A}_\parallel) \right] = C_\mathrm{s},\end{split}\]

(1.3) \[ \left[ \nabla_\perp^2 - \frac{1}{\varepsilon_0} \sum_\mathrm{s} \frac{e_\mathrm{s}^2 n_\mathrm{s}}{T_\mathrm{s}} \left( 1 - \Gamma_{0\mathrm{s}} \right) \right] \tilde{\phi} = - \frac{1}{\varepsilon_0} \sum_\mathrm{s} e_\mathrm{s} \int dv^3 J_{0\mathrm{s}} \tilde{f}_\mathrm{s},\]

(1.4) \[ \nabla_\perp^2 \tilde{A}_\parallel = - \mu_0 \sum_\mathrm{s} e_\mathrm{s} \int dv^3 J_{0\mathrm{s}} v_\parallel \tilde{f}_\mathrm{s},\]

where the gyrophase-average operators \(J_{0\mathrm{s}} = \oint (d\xi/2\pi) e^{\bm{\rho}_\mathrm{s} \cdot \nabla} = \oint (d\xi/2\pi) e^{-\bm{\rho}_\mathrm{s} \cdot \nabla}\) and \(\Gamma_{0\mathrm{s}} = \int dv^3 (F_\mathrm{sM}/n_\mathrm{s}) J^2_{0\mathrm{s}}\) are used with the gyroradius vector \(\bm{\rho}_\mathrm{s} = \bm{b} \times m_\mathrm{s} \bm{v} / (e_\mathrm{s}B)\) . \(\bm{V}_E = \bm{b} \times \nabla \Phi /B\) denotes the equilibrium \(\bm{E} \times \bm{B}\) flow. The perturbed electric and magnetic fields are \(\tilde{\bm{E}} = - \nabla (J_{0\mathrm{s}} \tilde{\phi}) - \bm{b} \partial \tilde{A}_\parallel /\partial t\) and \(\tilde{\bm{B}}_\perp = \nabla (J_{0\mathrm{s}} \tilde{A}_\parallel) \times \bm{b}\) . The perturbed \(\bm{E} \times \bm{B}\) , grad-B, curvature, diamagnetic drift velocities are respectively given by \(\tilde{\bm{v}}_\mathrm{E} = \bm{b} \times \nabla (J_{0\mathrm{s}} \tilde{\phi})/B\) , \(\bm{v}_\mathrm{sG} = \bm{b} \times \mu \nabla B/(e_\mathrm{s}B)\) , \(\bm{v}_\mathrm{sC} = \bm{b} \times m_\mathrm{s} v_\parallel^2 \bm{b} \cdot \nabla \bm{b}/(e_\mathrm{s}B)\) and \(\bm{v}_{\mathrm{s}*} = \bm{b} \times [T_\mathrm{s} \nabla \ln n_\mathrm{s} + ( m_\mathrm{s} v_\parallel^2 /2+ \mu B - 3 T_\mathrm{s}/2) \nabla \ln T_\mathrm{s} ] /(e_\mathrm{s}B)\) . \(C_s\) is the linearized collision term on the species \(s\) and will be explained in Section Collision operator . The nonlinear term in the Vlasov eq. (denoted \(\mathcal{N}_\mathrm{s}\) below), which originates from \(\bm{E} \times \bm{B}\) and \(v_\parallel \tilde{\bm{B}}_\perp/B\) advections of \(\tilde{f}\) and \(\tilde{\bm{E}} \cdot \tilde{\bm{B}}_\perp\) acceleration of \(F\) , can be rewritten as,

(1.5) \[\begin{split}\mathcal{N}_\mathrm{s} &= \left( \tilde{\bm{v}}_\mathrm{E} + v_\parallel \frac{\tilde{\bm{B}}_\perp}{B} \right) \cdot \nabla \tilde{f}_\mathrm{s} + \frac{e_\mathrm{s} \tilde{\bm{E}}}{m_\mathrm{s}} \cdot \frac{\tilde{\bm{B}}_\perp}{B} \left(- \frac{m_\mathrm{s} v_\parallel}{T_\mathrm{s}} F_\mathrm{Ms} \right) \\ &= \frac{\bm{b}}{B} \cdot \nabla \left( J_{0\mathrm{s}} \tilde{\phi} - v_\parallel J_{0\mathrm{s}} \tilde{A}_\parallel \right) \times \nabla \left( \tilde{f}_\mathrm{s} + \frac{e_\mathrm{s} F_\mathrm{Ms}}{T_\mathrm{s}} J_{0\mathrm{s}} \tilde{\phi} \right),\end{split}\]

respectively.

1.2. Geometry and coordinates

The following explanation is conventional flux-tube model without equilibrium flow [ 1-3 ] . For the numerical treatment of equilibrium flow shear effects in the rotating flux-tube model, please refer to [ 1-4 ] .

When an equilibrium magnetic field is known, one can construct a flux coordinate \((\rho_f, \theta_f, \varphi_f)\) such that,

(1.6) \[\bm{B} = \nabla \Psi_p(\rho_f) \times \nabla [q(\rho_f)\theta_f - \varphi_f],\]

where we use the safety factor \(q(\rho_f) = d\Psi_t/d\Psi_p\) and the toroidal and poloidal flux \(\Psi_p(\rho_f)\) and \(\Psi_t(\rho_f)\) . GKV employs Clebsch-type coordinate as

(1.7) \[\begin{split}x &= c_x (\rho_f - \rho_{f0}), \\ y &= c_y [q(\rho_f) \theta_f - \varphi_f], \\ z &= \theta_f,\end{split}\]

where \(\rho_{f0}\) , \(c_x\) and \(c_y\) are constant. We refer \((x, y, z)\) as the radial, field-line-label, and field-aligned coordinates, respectively. Using this GKV coordinates, the equilibrium magnetic field is represented by

(1.8) \[\bm{B} = c_b \nabla x \times \nabla y = \frac{c_b}{\sqrt{g}} \frac{\partial \bm{r}}{\partial z},\]

where \(c_b = (d\Psi_p/d\rho_f)/(c_x c_y)\) and \(\sqrt{g} = (\nabla x \cdot \nabla y \times \nabla z)^{-1}\) .

Simulation domain of GKV is based on the local flux-tube model [ 1-3 ] . Using flute approximation for perturbed quantities \(k_\perp \gg k_\parallel\) (consistent with the gyrokinetic ordering Eq. (1.1) , vector differential operators in gyrokinetic Eqs. (1.2) – (1.4) become

(1.9) \[\begin{split}\nabla_\parallel \tilde f &= \bm{b} \cdot \nabla \tilde f = \frac{c_b}{B \sqrt{g}} \frac{\partial \tilde f}{\partial z}, \\ \nabla^2 \tilde f &= \frac{1}{\sqrt{g}} \frac{\partial}{\partial r^i} \left[ \sqrt{g} \left( \frac{\partial \tilde f}{\partial r^j} \nabla r^j \right) \cdot \nabla r^i \right] \\ &\simeq g^{xx} \frac{\partial^2 \tilde f}{\partial x^2} + 2 g^{xy} \frac{\partial^2 \tilde f}{\partial x \partial y} + g^{yy} \frac{\partial^2 \tilde f}{\partial y^2}, \\ \bm{b} \times \nabla \tilde h \cdot \nabla \tilde f &= \bm{b} \cdot \left(\frac{\partial \tilde h}{\partial r^i} \nabla r^i \times \frac{\partial \tilde f}{\partial r^j} \nabla r^j \right) \\ &\simeq \frac{B}{c_b} \left( \frac{\partial \tilde h}{\partial x}\frac{\partial \tilde f}{\partial y} - \frac{\partial \tilde h}{\partial y}\frac{\partial \tilde f}{\partial x} \right), \\ \bm{b} \times \nabla H \cdot \nabla \tilde f &\simeq \frac{B}{c_b} \left( \frac{\partial H}{\partial x}\frac{\partial \tilde f}{\partial y} - \frac{\partial H}{\partial y}\frac{\partial \tilde f}{\partial x} \right) \\ &+ \frac{\partial H}{\partial z} \left( \frac{g^{xz}g^{yx} - g^{xx}g^{yz}}{B/c_b} \frac{\partial \tilde f}{\partial x} + \frac{g^{xz}g^{yy} - g^{xy}g^{yz}}{B/c_b} \frac{\partial \tilde f}{\partial y} \right),\end{split}\]

where \(g^{ij}=\nabla r^i \cdot \nabla r^j\) denotes the metric tensor.

Since the magnetic curvature can be replaced by

(1.10) \[\bm{b}\cdot\nabla\bm{b} = \frac{\nabla_\perp B}{B} + \frac{\nabla P}{B^2/\mu_0},\]

when the equilibrium satisfies the MHD equilibrium, \(\nabla P = \bm{J} \times \bm{B}\) and \(\nabla \times \bm{B} = \mu_0 \bm{J}\) ,

the magnetic (i.e., grad-B and curvature) drift velocity is given by

(1.11) \[\bm{v}_\mathrm{sG} + \bm{v}_\mathrm{sC} = \frac{1}{e_\mathrm{s} B} \bm{b} \times \left( \frac{m_\mathrm{s} v_\parallel^2 + \mu B}{B} \nabla B + \frac{m_\mathrm{s} v_\parallel^2}{B^2/ \mu_0} \nabla P \right),\]

and then the magnetic and diamagnetic drift terms are

(1.12) \[\begin{split}(\bm{v}_\mathrm{sG} + \bm{v}_\mathrm{sC}) \cdot \nabla (J_0 \tilde{\phi}) = &\frac{m_\mathrm{s} v_\parallel^2 + \mu B}{e_\mathrm{s} c_b} \left( K_x \frac{\partial J_0 \tilde{\phi}}{\partial x} + K_y \frac{\partial J_0 \tilde{\phi}}{\partial y} \right) \\ &+ \frac{m_\mathrm{s} v_\parallel^2}{e_\mathrm{s} c_b} \frac{dP/dx}{B^2/\mu_0} \frac{\partial J_0 \tilde{\phi}}{\partial y}, \\ \bm{v}_\mathrm{s*} \cdot \nabla (J_0 \tilde{\phi}) = &- \frac{T_\mathrm{s}}{e_\mathrm{s} c_b} \left[ \frac{1}{L_{n\mathrm{s}}} + \left( \frac{m_\mathrm{s} v_\parallel^2}{2T_\mathrm{s}} + \frac{\mu B}{T_\mathrm{s}} - \frac{3}{2} \right) \frac{1}{L_{T\mathrm{s}}} \right] \frac{\partial J_0 \tilde{\phi}}{\partial y},\end{split}\]

where

(1.13) \[K_x = - \frac{\partial \ln B}{\partial y} + \frac{g^{xz} g^{yx} - g^{xx} g^{yz}}{B^2/c_b^2} \frac{\partial \ln B}{\partial z},\]

(1.14) \[K_y = \frac{\partial \ln B}{\partial x} + \frac{g^{xz} g^{yy} - g^{xy} g^{yz}}{B^2/c_b^2} \frac{\partial \ln B}{\partial z},\]

and the density and temperature scale lengths \(L_{n\mathrm{s}} = - (d\ln n_\mathrm{s}/dx)^{-1}\) , \(L_{T\mathrm{s}} = - (d\ln T_\mathrm{s}/dx)^{-1}\) , and total pressure gradient \(dP/dx = d (\sum_\mathrm{s} n_\mathrm{s} T_\mathrm{s})/dx = - \sum_\mathrm{s} n_\mathrm{s} T_\mathrm{s} (L_{n\mathrm{s}}^{-1}+L_{T\mathrm{s}}^{-1})\) .

1.3. Local approximation

Simulation box \(-L_x \leq x < L_x\) , \(-L_y \leq y < L_y\) , \(-N_\theta \pi < z < N_\theta \pi\) gives flux-tube domain aligned to the equilibrium magnetic field.

By assuming the perpendicular scale separation of equilibrium and perturbed quantities, the equilibrium quantities can be evaluated by the value at the center of flux-tube domain \(x=0\) or equivalently \(\rho_f = \rho_{f0}\) . When one considers an axisymmetric equilibrium \(\partial_y=0\) , the equilibrium quantities are independent to \(x\) and \(y\) , i.e., \(F = F(z,v_\parallel,\mu)\) , \(B = B(z)\) , and so on. In a non-axisymmetric equilibrium case, one may treat a thin flux-tube domain not only in \(x\) but also in \(y\) direction and evaluate the equilibrium quantities at \(x=0\) and \(y=0\) .

1.4. Pseudo-periodic boundary condition along a field line

Since the equilibrium quantities are independent to perpendicular \(x\) and \(y\) directions, one expand the distribution function and electromagnetic potentials by means of Fourier basis,

(1.15) \[\begin{split}\tilde{f}_\mathrm{s}(\bm{x},v_\parallel,\mu,t) &= \sum_{k_x}\sum_{k_y} \tilde{f}_{\mathrm{s}\bm{k}} (z,v_\parallel,\mu,t) e^{i(k_xx+ik_yy)}\\ \tilde{\phi}(\bm{x}',t) &= \sum_{k_x}\sum_{k_y} \tilde{\phi}_{\bm{k}} (z,t) e^{i(k_xx'+ik_yy')}\\ J_{0\mathrm{s}} \tilde{\phi}(\bm{x},\mu,t) &= \sum_{k_x}\sum_{k_y} J_0(k_\perp \rho_{\mathrm{ts}}) \tilde{\phi}_{\bm{k}} (z,t) e^{i(k_xx+ik_yy)}\end{split}\]

where \(\bm{x}\) is the gyrocenter coordinates and \(\bm{x}'=\bm{x}+\bm{\rho}_\mathrm{s}\) is the particle-position coordinates.

Additionally, considering the torus periodicity constraint \(\tilde{\phi}(\rho_f,\theta_f+2N_\theta\pi,\varphi_f) = \tilde{\phi}(\rho_f,\theta_f,\varphi_f)\) , one finds the pseudo-periodic boundary condition along a field line,

(1.16) \[\tilde{\phi}_{k_x+\delta k_x,k_y} (z+2N_\theta\pi) C_{k_y} = \tilde{\phi}_{k_x,k_y}(z),\]

where \(\delta k_x = -2N_\theta \pi \hat{s} k_y, C_{k_y} = \exp (i2N_\theta \pi k_y c_y q_0)\) . This conversion along a field line physically means twisting of the mode by the parallel streaming in the presence of magnetic shear.

1.5. Collision operator

The present version of GKV equips three types of gyrokinetic model collision operators, operating on the non-adiabatic part of the distribution function \(\tilde{g}_{\mathrm{s}\bm{k}} = \tilde{f}_{\mathrm{s}\bm{k}} + \frac{e_\mathrm{s} F_\mathrm{Ms}}{T_\mathrm{s}} J_{0\mathrm{s}\bm{k}} \tilde{\phi}_{\bm{k}}\) .

Note

NOTE: Although the Lenard-Bernstein model collision gkvp_f0.48 operates on \(\tilde{f}_{\mathrm{s}\bm{k}}\) but not on \(\tilde{g}_{\mathrm{s}\bm{k}}\) due to historical reason, it will be modified near-future update.

Lenard-Bernstein model collision operator

(1.17) \[\begin{split}C_{\mathrm{a}\bm{k}}^\mathrm{LB} = \nu_\mathrm{a} &\Bigg[ v_\mathrm{ta}^2 \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel^2} + v_\mathrm{ta}^2 \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp^2} + v_\parallel \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel} \\ &+ \left(\frac{v_\mathrm{ta}^2}{v_\perp} + v_\perp \right) \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp} + 3 \tilde{g}_{\mathrm{a}\bm{k}} - k_\perp^2\rho_\mathrm{ta}^2 \tilde{g}_{\mathrm{a}\bm{k}} \Bigg].\end{split}\]

Lorentz model collision operator

(1.18) \[\begin{split}C_{\mathrm{a}\bm{k}}^\mathrm{Lorentz} = \nu_\mathrm{D}^\mathrm{ab} &\Bigg[ \frac{v_\perp^2}{2} \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel^2} + \frac{v_\parallel^2}{2} \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp^2} - v_\parallel v_\perp \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel \partial v_\perp} \\ &- v_\parallel \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel} + \frac{v_\perp}{2} \left(\frac{v_\parallel^2}{v_\perp^2} - 1 \right) \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp} \\ &- \frac{k_\perp^2\rho_\mathrm{ta}^2}{4v_\mathrm{ta}^2} (2v_\parallel^2+v_\perp^2) \tilde{g}_{\mathrm{a}\bm{k}} \Bigg].\end{split}\]

Sugama model collision operator [ 1-5 ]

(1.19) \[C_{\mathrm{a}\bm{k}}^\mathrm{Sugama} = \sum_\mathrm{b} \left[ C^\mathrm{V}_\mathrm{ab}(\tilde{g}_{\mathrm{a}\bm{k}}) + C^\mathrm{D}_\mathrm{ab}(\tilde{g}_{\mathrm{a}\bm{k}}) + C^\mathrm{F}_\mathrm{ab}(\tilde{g}_{\mathrm{b}\bm{k}}) \right].\]

The test-particle differential term \(C^\mathrm{V}_\mathrm{ab}\) , the test-particle non-isothermal term \(C^\mathrm{D}_\mathrm{ab}\) , and the field-particle term \(C^\mathrm{F}_\mathrm{ab}\) are given by,

(1.20) \[\begin{split}C^\mathrm{V}_\mathrm{ab}(\tilde{g}_{\mathrm{a}\bm{k}}) &= \frac{\nu_\parallel^\mathrm{ab}v_\parallel^2+\nu_\mathrm{D}^\mathrm{ab}v_\perp^2}{2}\frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel^2} + \frac{\nu_\mathrm{D}^\mathrm{ab}v_\parallel^2+\nu_\parallel^\mathrm{ab}v_\perp^2}{2}\frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp^2} \\ &+ (\nu_\parallel^\mathrm{ab}-\nu_\mathrm{D}^\mathrm{ab})v_\parallel v_\perp \frac{\partial^2 \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel \partial v_\perp} \\ &+ \nu_\mathrm{g}^\mathrm{ab} v_\parallel \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\parallel} + \left[ \nu_\mathrm{g}^\mathrm{ab} + \frac{\nu_\mathrm{D}^\mathrm{ab}}{2} \left( 1 + \frac{v_\parallel^2}{v_\perp^2} \right) \right] v_\perp \frac{\partial \tilde{g}_{\mathrm{a}\bm{k}}}{\partial v_\perp} \\ &+\left[ \frac{\nu_\mathrm{h}^\mathrm{ab} x_\mathrm{a}^2}{2} - \frac{k_\perp^2}{4\Omega_\mathrm{a}^2} \left\{ \nu_\mathrm{D}^\mathrm{ab} (2v_\parallel^2 + v_\perp^2) + \nu_\parallel^\mathrm{ab} v_\perp^2 \right\} \right] \tilde{g}_{\mathrm{a}\bm{k}}, \\ C^\mathrm{D}_\mathrm{ab}(\tilde{g}_{\mathrm{a}\bm{k}}) &= \sum_{j=1}^6 X_j^\mathrm{ab} M_j^\mathrm{ab}, \\ C^\mathrm{F}_\mathrm{ab}(\tilde{g}_{\mathrm{b}\bm{k}}) &= \sum_{j=1}^6 Y_j^\mathrm{ab} M_j^\mathrm{ba},\end{split}\]

where \(x_\mathrm{a} = v / (\sqrt{2}v_\mathrm{ta})\) , \(\alpha_\mathrm{ab}=v_\mathrm{ta}/v_\mathrm{tb}\) , \(\nu_\mathrm{g}^\mathrm{ab} = \nu_\parallel^\mathrm{ab}x_\mathrm{a}^2(1-\alpha_\mathrm{ab})\) , and \(\nu_\mathrm{h}^\mathrm{ab} = 3\sqrt{\pi}\tau^{-1}_\mathrm{ab}\alpha_\mathrm{ab}\Phi'(x_\mathrm{b})/(4x_\mathrm{a}^2)\) . The energy-diffusion and deflection frequencies are respectively given by \(\nu_\parallel^\mathrm{ab} = 3\sqrt{\pi}\tau^{-1}_\mathrm{ab}G(x_\mathrm{b})/(2x_\mathrm{a}^3)\) and \(\nu_\mathrm{D}^\mathrm{ab} = 3\sqrt{\pi}\tau^{-1}_\mathrm{ab}[\Phi(x_\mathrm{b})-G(x_\mathrm{b})]/(4x_\mathrm{a}^3)\) with the error function \(\Phi(x) = \mathrm{erf}(x)\) and \(G(x) = [\Phi(x) - x \Phi'(x)]/(2x^2)\) . Expressions of the other coefficients \(X_j^\mathrm{ab}\) and \(Y_j^\mathrm{ab}\) and of the fluid moments \(M_j^\mathrm{ab}\) are found, e.g., in the literature [ 1-6 ] .

1.6. Summary of formulation

Finally, one obtains the \(\delta f\) gyrokinetic Vlasov-Poisson-Ampère equations in a local flux-tube model, represented in perpendicular wave-number space,

(1.21) \[\begin{split}&\frac{\partial \tilde{f}_{\mathrm{s}\bm{k}}}{\partial t} + \left( v_\parallel \nabla_\parallel + i \bm{k} \cdot \bm{v}_\mathrm{sG} + i \bm{k} \cdot \bm{v}_\mathrm{sC} \right) \left( \tilde{f}_{\mathrm{s}\bm{k}} + \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} J_{0\mathrm{s}\bm{k}} \tilde{\phi}_{\bm{k}} \right) + N_{\mathrm{s}\bm{k}} \\ &- \frac{\mu \nabla_\parallel B}{m_\mathrm{s}} \frac{\partial}{\partial v_\parallel} \left( \tilde{f}_{\mathrm{s}\bm{k}} + \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} J_{0\mathrm{s}\bm{k}} \tilde{\phi}_{\bm{k}} \right) \\ &+ \frac{e_\mathrm{s} F_\mathrm{sM}}{T_\mathrm{s}} \left[ v_\parallel \frac{\partial J_{0\mathrm{s}\bm{k}} \tilde{A}_{\parallel\bm{k}}}{\partial t} - i \bm{k} \cdot \bm{v}_{\mathrm{s}*} J_{0\mathrm{s}\bm{k}} (\tilde{\phi}_{\bm{k}} - v_\parallel \tilde{A}_{\parallel\bm{k}}) \right] = C_{\mathrm{s}\bm{k}},\end{split}\]

(1.22) \[\left[ k_\perp^2 + \frac{1}{\varepsilon_0} \sum_\mathrm{s} \frac{e_\mathrm{s}^2 n_\mathrm{s}}{T_\mathrm{s}} \left( 1 - \Gamma_{0\mathrm{s}\bm{k}} \right) \right] \tilde{\phi}_{\bm{k}} = \frac{1}{\varepsilon_0} \sum_\mathrm{s} e_\mathrm{s} \int dv^3 J_{0\mathrm{s}\bm{k}} \tilde{f}_{\mathrm{s}\bm{k}},\]

(1.23) \[k_\perp^2 \tilde{A}_{\parallel\bm{k}} = \mu_0 \sum_\mathrm{s} e_\mathrm{s} \int dv^3 J_{0\mathrm{s}\bm{k}} v_\parallel \tilde{f}_{\mathrm{s}\bm{k}},\]

where \(J_{0\mathrm{s}\bm{k}}=J_0(k_\perp \rho_{\mathrm{s}})\) and \(\Gamma_{0\mathrm{s}\bm{k}}=I_0(k_\perp^2 \rho_{\mathrm{ts}}^2)e^{-k_\perp^2 \rho_{\mathrm{ts}}^2}\) with 0th-order Bessel and modified Bessel functions \(J_0\) and \(I_0\) . The included operators are again listed below,

(1.24) \[\begin{split}\nabla_\parallel &= \frac{c_b}{B \sqrt{g}} \frac{\partial}{\partial z}, \\ k_\perp^2 &= g^{xx} k_x^2 + 2 g^{xy} k_x k_y + g^{yy} k_y^2, \\ i \bm{k} \cdot (\bm{v}_\mathrm{sG} + \bm{v}_\mathrm{sC}) &= \frac{m_\mathrm{s} v_\parallel^2 + \mu B}{e_\mathrm{s} c_b} \left( i K_x k_x + i K_y k_y \right) + i \frac{m_\mathrm{s} v_\parallel^2}{e_\mathrm{s} c_b} \frac{dP/dx}{B^2/\mu_0} k_y, \\ i \bm{k} \cdot \bm{v}_\mathrm{s*} &= - i \frac{T_\mathrm{s}}{e_\mathrm{s} c_b} \left[ \frac{1}{L_{n\mathrm{s}}} + \left( \frac{m_\mathrm{s} v_\parallel^2}{2T_\mathrm{s}} + \frac{\mu B}{T_\mathrm{s}} - \frac{3}{2} \right) \frac{1}{L_{T\mathrm{s}}} \right] k_y, \\ \mathcal{N}_{\mathrm{s}\bm{k}} &= - \sum_{\bm{k}'} \sum_{\bm{k}''} \delta_{\bm{k}'+\bm{k}'',\bm{k}}\frac{\bm{b} \cdot \bm{k}' \times \bm{k}''}{c_b} J_{0\mathrm{s}\bm{k}'} \\ &\times \left( \tilde{\phi}_{\bm{k}'} - v_\parallel \tilde{A}_{\parallel\bm{k}'} \right) \left( \tilde{f}_{\mathrm{s}\bm{k}''} + \frac{e_\mathrm{s} F_\mathrm{Ms}}{T_\mathrm{s}} J_{0\mathrm{s}\bm{k}''} \tilde{\phi}_{\bm{k}''} \right).\end{split}\]

The coefficients for magnetic drift \(K_x, K_y\) are given by Eqs. (1.13) and (1.14) , and the collision operator \(C_{\mathrm{s}\bm{k}}\) is given by one of Eqs. (1.17) – (1.19) .

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