2. The gyrokinetic parent model

In order to derive a gyrofluid model only based on a quasi-static closure assumption, we consider as a starting point the following set of gyrokinetics equations, which corresponds to the system provided by equations (C 58), (C 60) and (C 66)–(C 68) of Kunz et al. (Reference Kunz, Schekochihin, Chen, Abel and Cowley2015) when collisions and equilibrium velocities are neglected and bi-Maxwellian distribution functions are chosen as equilibrium distribution functions for all the particle species. For simplicity, we specialize to the case of a plasma consisting of two species: electrons and one species of singly ionized particles. The equations of the resulting gyrokinetic model are given by

(2.1)\begin{align} & \frac{\partial \tilde{g}_s}{\partial t}+\frac{c}{B_0}\left[ \mathcal{J}_{0s} \tilde{\phi} - \frac{v_{\|}}{c} \mathcal{J}_{0s} \tilde{A}_{\parallel} +2 \frac{\mu_s B_0}{q_{s}}\mathcal{J}_{1s}\frac{\tilde{B}_\parallel}{B_0} , \tilde{g}_s \right] \nonumber\\ &\quad +v_{\|} \frac{\partial}{\partial z}\left( \tilde{g}_s +\frac{q_{s}}{T_{{0 }_{\parallel s}}} \mathcal{F}_{{eq}_s}\left( \mathcal{J}_{0s} \tilde{\phi} - \frac{v_{\|}}{c} \mathcal{J}_{0s}\tilde{A}_{\parallel} +2 \frac{\mu_s B_0}{q_{s}} \mathcal{J}_{1s}\frac{\tilde{B}_\parallel}{B_0}\right)\right)=0, \end{align}

(2.2)\begin{align} &\sum_{s} q_{s} \int\,\textrm{d} \mathcal{W}_s \, \mathcal{J}_{0s} \tilde{g}_s = \sum_{s} \frac{q_{s}^2 }{T_{{0 }_{\perp s}}} \int\,\textrm{d} \mathcal{W}_s \, \mathcal{F}_{{eq}_s} \left( 1 - \mathcal{J}_{0s}^2 \right) \tilde{\phi} \nonumber\\ &\quad - \sum_s q_{s} \int\,\textrm{d} \mathcal{W}_s \, 2 \frac{\mu_s B_0}{T_{{0 }_{\perp s}}} \mathcal{F}_{{eq}_s} \mathcal{J}_{0s} \mathcal{J}_{1s} \frac{\tilde{B}_\parallel}{B_0}, \end{align}

(2.3)\begin{align} &\sum_{s} q_{s} \int\,\textrm{d} \mathcal{W}_s \, v_{\|} \mathcal{J}_{0s} \left( \tilde{g}_s -\frac{q_{s}}{T_{{0 }_{\parallel s}}} \frac{v_{\|}}{c} \mathcal{F}_{{eq}_s} \mathcal{J}_{0s} \tilde{A}_{\parallel}\right) \nonumber\\ &\quad = -\frac{c}{4 {\rm \pi}} \nabla_{\perp}^2 \tilde{A}_{\parallel} + \sum_s \frac{q_{s}^2}{m_s}\int\,\textrm{d} \mathcal{W}_s \,\mathcal{F}_{{eq}_s} \left( 1 - \frac{1}{\Theta_s}\frac{v_{\|}^2}{v_{{\textrm{th}}_{\parallel s}}^2}\right)(1 - \mathcal{J}_{0s}^2) \frac{\tilde{A}_{\parallel}}{c}, \end{align}

(2.4)\begin{align} &\sum_{s} \frac{\beta_{\perp_s}}{n_0} \int\,\textrm{d} \mathcal{W}_s \, 2 \frac{\mu_s B_0}{T_{{0 }_{\perp s}}} \mathcal{J}_{1s} \tilde{g}_s= - \sum_s \frac{\beta_{\perp_s}}{n_0} \frac{q_{s}}{T_{{0 }_{\perp s}}} \int\,\textrm{d} \mathcal{W}_s \, 2 \frac{\mu_s B_0}{T_{{0 }_{\perp s}}} \mathcal{F}_{{eq}_s} \mathcal{J}_{0s} \mathcal{J}_{1s} \tilde{\phi} \nonumber\\ &\quad -\left(2 + \sum_s \frac{\beta_{\perp_s}}{n_0}\int\,\textrm{d} \mathcal{W}_s \, \mathcal{F}_{{eq}_s} \left( 2 \frac{\mu_s B_0}{T_{{0 }_{\perp s}}} \mathcal{J}_{1s} \mathcal{J}_{0s} \right)^2 \right) \frac{\tilde{B}_\parallel}{B_0}. \end{align}

The index $s \in \{e,i \}$ adopted above indicates the particle species, so that quantities labelled with $s$ refer to the electron species when $s=e$ and to the ion species when $s=i$.

In the system (2.1)–(2.4), the function $\tilde {g}_s$ is defined by

(2.5)\begin{equation} \tilde{g}_s(x,y,z,v_{\|},\mu_s,t)=\tilde{f}_{s}(x,y,z,v_{\|},\mu_s,t)+\frac{q_{s}}{T_{{0 }_{\parallel s}}}\frac{v_{\|}}{c}\mathcal{F}_{{eq}_s} (v_{\|} , \mu_s) \mathcal{J}_{0s} \tilde{A}_{\parallel} (x,y,z,t), \end{equation}

where $\tilde {f}_{s}$ is the perturbation of the gyrocentre distribution function for particles of species $s$, $q_{s}$ is the charge of these particles (so that $q_e=-e$ and $q_i=e$, with $e$ indicating the proton charge) and $m_s$ their mass. Furthermore, $c$ denotes the speed of light. The bi-Maxwellian equilibrium distribution function is given by

(2.6)\begin{equation} \mathcal{F}_{{eq}_s}(v_{\|},\mu_s)= \left(\frac{m_{s}}{{2 {\rm \pi}}}\right)^{3/2} \frac{n_0}{T_{{0 }_{\parallel s}}^{1/2} T_{{0 }_{\perp s}}}\exp\left({-\frac{m_{s} v_{\|}^2}{2 T_{{0 }_{\parallel s}}}-\frac{\mu_s B_0}{ T_{{0 }_{\perp s}} }}\right), \end{equation}

where $n_0$ is the uniform and constant equilibrium density, $T_{{0 }_{\parallel s}}$ and $T_{{0 }_{\perp s}}$ are respectively the equilibrium temperatures of the particle of species s, parallel and in a plane perpendicular to an equilibrium magnetic guide field of amplitude $B_0$, directed along the $z$ direction of a Cartesian coordinate frame $\{x,y,z\}$. We suppose that the spatial domain of the system corresponds to a box $\mathcal {D}=\{ (x,y,z): -L_x \leq x \leq L_x , -L_y \leq y \leq L_y, -L_z \leq z \leq L_z \}$, with $L_x$, $L_y$ and $L_z$ positive constants. All quantities of the system which depend on the spatial variables $x$, $y$ and $z$ are supposed to satisfy periodic boundary conditions on the domain $\mathcal {D}$, so that they can be expanded in Fourier series. We indicated with $v_{\|} \in \mathbb {R}$ the velocity coordinate parallel to the guide field and with $\mu _s=m_s v_{\perp }^2/(2 B_0) \in [0, +\infty )$ the magnetic moment of the particle of species $s$ in the unperturbed guide field, where $v_{\perp }$ corresponds to the velocity coordinate perpendicular to the guide field. We assume that all functions depending on $v_{\|}$ decay to zero as $v_{\|} \rightarrow \pm \infty$. Functions depending on $\mu _s$ are assumed to tend to zero as $\mu _s \rightarrow +\infty$ and to be bounded at $\mu _s=0$. The coordinate $t\in [0, + \infty )$ refers to time. The expressions $\mathcal {J}_{0s}$ and $\mathcal {J}_{1s}$ are related to the standard gyroaverage operators for the species $s$ in Fourier space. Their definition can be introduced explicitly in the following way: adopting the notation $\boldsymbol {x}$ to indicate a point of coordinates $(x,y,z) \in \mathcal {D}$ and, similarly, $\boldsymbol {k}$ to indicate a point $(k_x, k_y, k_z)\in \mathscr {D}$, where $\mathscr {D}$ is the lattice defined by $\mathscr {D}=\{(2{\rm \pi} l /(2 L_x), 2 {\rm \pi}m/ (2 L_y) , 2 {\rm \pi}n / (2 L_z)) : (l,m,n) \in \mathbb {Z}^3 \}$, we can consider a function $f : \mathcal {D} \times [0, + \infty )\rightarrow \mathbb {R}$, periodic over $\mathcal {D}$, so that it admits the Fourier representation $f(\boldsymbol {x} ,t)=\sum _{\boldsymbol {k} \in \mathscr {D}} f_{\boldsymbol {k}}(t) \exp (\textrm {i} \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {x})$. The action of the operators $\mathcal {J}_{0s}$ and $\mathcal {J}_{1s}$ on the function $f$ is defined by

(2.7)\begin{gather} \mathcal{J}_{0s}\, f (\boldsymbol{x} ,t)=\sum_{\boldsymbol{k} \in \mathscr{D}} J_0 (a_{s}) f_{\boldsymbol{k}}(t) \exp(\textrm{i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}), \end{gather}

(2.8)\begin{gather} \mathcal{J}_{1s}\, f (\boldsymbol{x} ,t)=\sum_{\boldsymbol{k} \in \mathscr{D}} \frac{J_1 (a_{s})}{a_{s}} f_{\boldsymbol{k}}(t) \exp(\textrm{i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}), \end{gather}

where $J_0$ and $J_1$ indicate the zeroth- and first-order $J$ Bessel functions, respectively, $a_{s}=k_{\perp } \rho _{\perp _s}$ with $k_{\perp }$ and $\rho _{\perp _s}$ corresponding to the perpendicular wave number and the Larmor radius of the particle of species $s$. The former is defined as $k_{\perp }=\sqrt {k_x^2+k_y^2}$ with $k_x=2 {\rm \pi}l/(2 L_x)$, $k_y=2 {\rm \pi}m/(2 L_y)$ for $l,m \in \mathbb {Z}$, while the latter is given by $\rho _{\perp _s}=v_{\perp } / \omega _{cs}$, where $\omega _{cs}=e B_0/ (m_s c)$ is the cyclotron frequency referred to the guide field and related to the particle of species $s$.

The leading-order expression (up to second-order terms in the perturbations) for the magnetic field is given by

(2.9)\begin{equation} \boldsymbol{B}(x,y,z,t)=\boldsymbol{\nabla} \tilde{A}_{\parallel} (x,y,z,t) \times \widehat{\boldsymbol{z}} + (\tilde{B}_\parallel ( x,y,z,t) + B_0)\widehat{\boldsymbol{z}}, \end{equation}

where $\widehat{\boldsymbol{z}}$ is the unit vector along the $z$ direction, $\tilde {A}_{\parallel }$ (referred to as magnetic flux function) corresponds to the $z$ component of the magnetic vector potential and $\tilde {B}_\parallel$ is the perturbation of the magnetic guide field, also referred to as parallel magnetic perturbation or parallel magnetic fluctuations. We remark that the guide field is assumed to be spatially homogeneous. This assumption is valid for a local description of space plasmas such as the solar wind, where the background magnetic field varies on scales so large that, in the local description, it can be assumed to be homogeneous. The situation would be different, for instance, in the case of tokamak plasmas. Indeed, gyrofluid models more oriented towards tokamak applications (as, for instance those of Brizard Reference Brizard1992; Snyder & Hammett Reference Snyder and Hammett2001; Scott Reference Scott2010; Waelbroeck & Tassi Reference Waelbroeck and Tassi2012; Madsen Reference Madsen2013; Keramidas Charidakos, Waelbroeck & Morrison Reference Keramidas Charidakos, Waelbroeck and Morrison2015) take into account background magnetic inhomogeneities. The set of electromagnetic quantities involved in the system is completed by the electrostatic potential $\tilde {\phi }=\tilde {\phi }(x,y,z,t)$. In (2.2)–(2.4) we adopted the symbol $\textrm {d} \mathcal {W}_s=(2{\rm \pi} B_0/m_s)\,\textrm {d} \mu _s\, \textrm {d}v_{\|}$ to indicate the volume element in space velocity.

The parameters $\Theta _s$ and $\beta _{\perp _s}$ are defined by

(2.10a,b)\begin{equation} \Theta_s=\frac{T_{{0 }_{\perp s}}}{T_{{0 }_{\parallel s}}}, \quad \beta_{\perp_s}=8 {\rm \pi}\frac{n_0 T_{{0 }_{\perp s}}}{B_0^2} \end{equation}

and measure, for each species $s$, the equilibrium temperature anisotropy and the ratio between equilibrium kinetic and magnetic pressure, respectively.

Equation (2.1) is the gyrokinetic equation related to the species $s$, whereas (2.2)–(2.4) relate the gyrocentre distribution functions to electromagnetic quantities in the non-relativistic limit. In particular, (2.2) corresponds to the quasi-neutrality relation, whereas (2.3) and (2.4) derive from Ampère's law projected along directions parallel and perpendicular to the guide field, respectively.

The gyrokinetic model (2.1)–(2.4) is valid for small perturbation of the equilibrium distribution function ($\delta f$-gyrokinetic approximation) and weak variations of the fields along the direction of the guide field, the equilibrium temperature anisotropy $\varTheta _s$ and the parameter $\beta _{\perp _s}$ of all the species, being kept finite in this asymptotics. Further details about the derivation the regime of validity of the model and its derivation can be found in Kunz et al. (Reference Kunz, Schekochihin, Chen, Abel and Cowley2015). Its Hamiltonian structure, on the other hand, is presented in Tassi (Reference Tassi2019).

3. The gyrofluid model

We define the following gyrofluid moments:

(3.1a,b)\begin{gather} \tilde{N}_s=\int\,\textrm{d} \mathcal{W}_s \, \tilde{f}_{s} , \quad \tilde{U}_s=\frac{1}{n_0}\int\,\textrm{d} \mathcal{W}_s \, v_{\|} \,\tilde{f}_{s}, \end{gather}

(3.2a–c)\begin{gather} \left. \begin{aligned} &\tilde{T}_{\parallel s} =\frac{T_{{0 }_{\parallel s}}}{n_0} \int\,\textrm{d} \mathcal{W}_s \, \left( \frac{v_{\|}^2}{v_{{\textrm{th}}_{\parallel s}}^2} -1\right) \tilde{f}_{s}, \quad \tilde{T}_{\perp s}=\frac{T_{{0 }_{\perp s}}}{n_0} \int\,\textrm{d} \mathcal{W}_s \, \left( \frac{\mu_s B_0}{T_{{0 }_{\perp s}}} -1 \right) \tilde{f}_{s}, \\ &\hspace{5pc}\tilde{Q}_{\parallel s} =T_{{0 }_{\parallel s}} v_{{\textrm{th}}_{\parallel s}} \int\,\textrm{d} \mathcal{W}_s \, \left(\frac{v_{\|}^3}{v_{{\textrm{th}}_{\parallel s}}^3} - 3 \frac{v_{\|}}{v_{{\textrm{th}}_{\parallel s}}}\right) \tilde{f}_{s}. \end{aligned} \right\} \end{gather}

For each particle species, the fields $\tilde {N}_s$ and $\tilde {U}_s$ represent the fluctuations of the gyrocentre densities and parallel fluid velocities, respectively. On the other hand, $\tilde {T}_{\parallel s}$ and $\tilde {T}_{\perp s}$ correspond to the fluctuations of the gyrofluid temperatures defined with respect to the parallel and perpendicular gyrocentre velocities, respectively, whereas $\tilde {Q}_{\parallel s}$ indicates the gyrocentre parallel heat flux fluctuations. In defining the parallel temperature and heat flux fluctuations, we introduced the constant $v_{{\textrm {th}}_{\parallel s}}=\sqrt {T_{{0 }_{\parallel s}} / m_s}$, corresponding to the parallel thermal velocity associated with the species $s$.

We intend to derive a gyrofluid model by taking moments of the gyrokinetic equation (2.1) and by imposing a closure relation derived from the quasi-static linear theory. In particular, the gyrofluid model, accounting for equilibrium temperature anisotropy, should be able, in the limit of vanishing finite Larmor radius effects, to reproduce the field-swelling instability criterion of Basu & Coppi (Reference Basu and Coppi1984). We restrict ourselves to the evolution of the first three moments referring to the parallel direction. Therefore, the resulting gyrofluid model should evolve the following six fields: $\tilde {N}_e, \tilde {N}_i, \tilde {U}_e, \tilde {U}_i, \tilde {T}_{\parallel e}$ and $\tilde {T}_{\parallel i}$. Also, we show that the model conserves the total energy and, moreover, that it possesses a non-canonical Hamiltonian structure, as is the case for the parent gyrokinetic model (Tassi Reference Tassi2019). We refer to such model as to the six-field gyrofluid (GF6) model.

For the sake of the comparison, carried out in § 4, of the linear gyrofluid theory with the linear gyrokinetic theory we also consider an extension of GF6 accounting for a Landau-fluid closure, analogously to that discussed in Hammett & Perkins (Reference Hammett and Perkins1990), Hammett et al. (Reference Hammett, Dorland and Perkins1992), Snyder, Hammett & Dorland (Reference Snyder, Hammett and Dorland1997), Passot & Sulem (Reference Passot and Sulem2007, Reference Passot and Sulem2015) and Tassi et al. (Reference Tassi, Grasso, Borgogno, Passot and Sulem2018). This variant of the model, which we denote as GF6L, differs from GF6 for the expression of the parallel heat flux fluctuations $\tilde {Q}_{\parallel s}$. Therefore, in order to avoid some redundancy in the exposition, in the following we present the model equations leaving $\tilde {Q}_{\parallel s}$ unspecified and we will subsequently indicate the corresponding expressions for $\tilde {Q}_{\parallel s}$ leading to GF6 and GF6L, respectively. The closure leading to GF6L, in particular, will be given in § 3.2.

A further variant of the model, denoted as GF4, will also be considered in § 4. This model evolves only the four fields $\tilde {N}_e, \tilde {N}_i, \tilde {U}_e$ and $\tilde {U}_i$ and represents a minimal Hamiltonian model, derived from the quasi-static closure, capable to reproduce the field-swelling instability criterion. This variant of the model will be introduced in § 3.2.

The six-field system, both in the Hamiltonian (GF6) and dissipative (GF6L) versions, can be written, in a dimensionless form, as

(3.3)\begin{align} &\frac{\partial N_s}{\partial t}+[G_{10s}\phi + \textrm{sgn}(q_{s}) \tau_{\perp_s} 2 G_{20s} B_\parallel, N_s] - [G_{10s} A_{\parallel}, U_s ] +\frac{\partial U_s}{\partial z}=0, \end{align}

(3.4)\begin{align} &\frac{\partial }{\partial t}\left( \frac{m_s}{m_i} U_s + \textrm{sgn}(q_{s}) G_{10s} A_{\parallel}\right) \nonumber\\ &\quad +\left[ G_{10s} \phi +\textrm{sgn}(q_{s}) \tau_{\perp_s} 2 G_{20s} B_\parallel, \frac{m_s}{m_i} U_s + \textrm{sgn}(q_{s}) G_{10s} A_{\parallel} \right] \nonumber\\ &\quad -\frac{\tau_{\perp_s}}{\Theta_s}[G_{10s} A_{\parallel} , N_s +T_{\parallel s}] \nonumber\\ &\quad +\frac{\partial}{\partial z} \left( \textrm{sgn}(q_{s}) G_{10s} \phi + \tau_{\perp_s} 2 G_{20s}B_\parallel+\frac{\tau_{\perp_s}}{\Theta_s}(N_s +T_{\parallel s}) \right)=0, \end{align}

(3.5)\begin{align} &\frac{\partial T_{\parallel s}}{\partial t}+[G_{10s}\phi + \textrm{sgn}(q_{s}) \tau_{\perp_s} 2 G_{20s} B_\parallel, T_{\parallel s}] \nonumber\\ &\quad - [G_{10s} A_{\parallel}, 2 U_s + Q_{\parallel s}] +\frac{\partial}{\partial z}( 2 U_s + Q_{\parallel s})=0, \end{align}

(3.6)\begin{align} &\sum_s \left(\textrm{sgn}( q_{s} ) G_{10s} N_s + (1 -\Theta_s) \Gamma_{0s} \frac{\phi}{\tau_{\perp_s}}+(\Theta_s G_{10s}^2 -1)\frac{\phi}{\tau_{\perp_s}} \right. \nonumber\\ &\quad \left. + \textrm{sgn} (q_{s} ) (1-\Theta_s)(\Gamma_{0s}-\Gamma_{1s}) B_\parallel +\textrm{sgn}(q_{s}) \Theta_s 2 G_{10s} G_{20s} B_\parallel\right)=0, \end{align}

(3.7)\begin{align} &\nabla_{\perp}^2 A_{\parallel}= \frac{\beta_{\perp_e}}{2}\sum_s \left(\frac{m_i}{m_s}\left(1-\frac{1}{\Theta_s}\right)(1-\Gamma_{0s}) A_{\parallel} -\textrm{sgn} (q_{s}) G_{10s} U_s \right), \end{align}

(3.8)\begin{align} B_\parallel&=-\frac{1}{2}\sum_s \beta_{\perp_s} \left(2 G_{20s} N_s+(1-\Theta_s)(\Gamma_{0s}-\Gamma_{1s})\textrm{sgn}(q_{s})\frac{\phi}{\tau_{\perp_s}} \right. \nonumber\\ &\quad \left. + \Theta_s 2 G_{10s} G_{20s}\textrm{sgn}(q_{s}) \frac{\phi}{\tau_{\perp_s}}+2 (1-\Theta_s)(\Gamma_{0s} - \Gamma_{1s})B_\parallel + \Theta_s 4 G_{20s}^2 B_\parallel \right). \end{align}

In (3.3)–(3.8), we adopted the following normalized quantities

(3.9a,b)\begin{gather} N_s=\frac{\tilde{N}_s}{n_0}, \quad U_s=\frac{\tilde{U}_s}{c_{s_{\perp}}}, \end{gather}

(3.10a,b)\begin{gather} T_{\parallel s}=\frac{\tilde{T}_{\parallel s}}{T_{{0 }_{\parallel s}}}, \quad Q_{\parallel s}=\frac{\tilde{Q}_{\parallel s}}{n_0 T_{{0 }_{\parallel s}} c_{s_{\perp}}}, \end{gather}

(3.11a–c)\begin{gather} \phi= \frac{e \tilde{\phi}}{T_{{0 }_{\perp e}}}, \quad A_{\parallel}=\frac{\tilde{A}_{\parallel}}{B_0 \rho_s}, \quad B_\parallel=\frac{\tilde{B}_\parallel}{B_0}, \end{gather}

(3.12a–d)\begin{gather} x=\frac{\bar{x}}{\rho_s}, \quad y=\frac{\bar{y}}{\rho_s}, \quad z=\frac{\bar{z}}{\rho_s}, \quad t=\omega_{ci}{\bar{t}}, \end{gather}

where the quantities with overbars in (3.12a–d) are the dimensional spatial and time coordinates. In (3.9a,b)–(3.12a–d) the quantities (note that, according to a customary notation, in the symbols $c_{s_{\perp }}$ and $\rho _s$, the subscript $s$ is to indicate sonic quantities and not the particle or gyrocentre species.)

(3.13a,b)\begin{equation} c_{s_{\perp}}=\sqrt\frac{T_{{0 }_{\perp e}}}{m_i}, \quad \rho_s=\frac{c_{s_{\perp}}}{\omega_{ci}} \end{equation}

were introduced, which indicate the sound speed and the sonic Larmor radius, respectively, based on the perpendicular equilibrium temperature.

The parameter

(3.14)\begin{equation} \tau_{\perp_s}=\frac{T_{{0 }_{\perp s}}}{T_{{0 }_{\perp e}}}, \end{equation}

for $s=e,i$, on the other hand, measures the ratio of the equilibrium perpendicular temperatures.

In (3.3)–(3.8) we introduced the operator $b_s= -\nabla _{\perp }^2 \rho _{\textrm {th}_{ \perp s}}^2$, with $\nabla _{\perp }^2$ denoting the Laplacian relatively to the transverse variables and

(3.15)\begin{equation} \rho_{\textrm{th}_{ \perp s}}=\frac{1}{\omega_{cs}}\sqrt{\frac{T_{{0 }_{\perp s}}}{m_s}} \end{equation}

indicating the perpendicular thermal Larmor radius associated with the species $s$.

The canonical bracket $[ \, , \, ]$, on the other hand, is defined as $[\,f,g]=\partial _x f \partial _y g - \partial _y f \partial _x g$, for two functions $f$ and $g$.

The gyroaverage operators $G_{10s}$, $G_{20s}$, $\Gamma _{0s}$, $\Gamma _{1s}$ in (3.3)–(3.8) can in turn be expressed in terms of the operators

(3.16a,b)\begin{gather} G_{10}(b_s)=\textrm{e}^{-b_s/2}, \quad G_{20}(b_s)=\frac{\textrm{e}^{-b_s/2}}{2}, \end{gather}

(3.17a,b)\begin{gather} \varGamma_0(b_s)=I_0( b_s)\textrm{e}^{-b_s}, \quad \varGamma_1(b_s)=I_1(b_s)\textrm{e}^{-b_s}, \end{gather}

which have to be intended as Fourier multipliers whose symbols are obtained by replacing $b_s$ by $k_{\perp }^2 \rho ^2_{\textrm {th} \perp s}$. We make this statement more precise, as an example, in the case of the operator $G_{10i}$, referred to the ion species. The expression $G_{10i} f$ is defined by $G_{10i} f (\boldsymbol {x} ,t)=\sum _{\boldsymbol {k} \in \mathscr {D}} G_{10} (b_i) f_{\boldsymbol {k}}(t) \exp (\textrm {i} \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {x})=\sum _{\boldsymbol {k} \in \mathscr {D}} \exp (-k_{\perp }^2 \rho _{\textrm {th}_{\perp _i}}^2/2) f_{\boldsymbol {k}}(t) \exp (\textrm {i} \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {x})$, for a function $f$ periodic in space. Analogous expressions are valid for the other gyroaverage operators. In (3.17a,b), in particular, the symbols $I_0$ and $I_1$ indicate the modified Bessel functions $I$ of order zero and one, respectively.

The expressions for the operators $G_{10s}$ and $G_{20s}$ given in (3.16a,b) correspond to those present in Brizard (Reference Brizard1992) and follow from assuming that the perturbation of the distribution function can be written as

(3.18)\begin{equation} \tilde{f}_{s} (x,y,z,v_{\|},\mu_s,t)=\mathcal{F}_{{eq}_s} (v_{\|} ,\mu_s)\sum_{m,n =0}^{+\infty}\frac{1}{\sqrt{m !}} H_m\left(\frac{v_{\|}}{v_{{\textrm{th}}_{\parallel s}}}\right)L_n\left(\frac{\mu_s B_0}{T_{{0 }_{\perp s}}}\right)f_{{mn}_s} (x,y,z,t), \end{equation}

where $H_m$ and $L_n$ indicate the Hermite and Laguerre polynomials, respectively, of order $m$ and $n$, with $m$ and $n$ non-negative integers. The functions $f_{{mn}_s}$ are coefficients of the expansion and are related to the moments of $\tilde {f}_{s}$, with respect to Hermite polynomials in $v_{\|} / v_{{\textrm {th}}_{\parallel s}}$ and Laguerre polynomials in $\mu _s B_0 / T_{{0 }_{\perp s}}$. Indeed, from the orthogonality properties of Hermite and Laguerre polynomials, the following relation holds:

(3.19)\begin{equation} f_{{mn}_s}=\frac{1}{n_0 \sqrt{ m !}}\int\,\textrm{d} \mathcal{W}_s \, H_m \left(\frac{v_{\|}}{v_{{\textrm{th}}_{\parallel s}}}\right)L_n\left(\frac{\mu_s B_0}{T_{{0 }_{\perp s}}}\right) \tilde{f}_{s}. \end{equation}

It can be useful, in particular, to write explicitly the following relations between the lowest-order normalized moments and the quantities $f_{{mn}_s}$:

(3.20a,b)\begin{gather} f_{{00}_s}=N_s, \quad f_{{10}_s}=\sqrt{\frac{\Theta_s}{\tau_{\perp_s}}}\sqrt{\frac{m_s}{m_i}}U_s, \end{gather}

(3.21a,b)\begin{gather} f_{{20}_s}=\frac{1}{\sqrt{2}}T_{\|s}, \quad f_{{01}_s}=-T_{\perp s}, \end{gather}

(3.22)\begin{gather} f_{{30}_s}=\frac{1}{\sqrt{6}}\sqrt{\frac{\Theta_s}{\tau_{\perp_s}}}\sqrt{\frac{m_s}{m_i}}Q_{\parallel s}, \end{gather}

where, in (3.21a,b), we introduced the normalized gyrocentre perpendicular temperature fluctuations $T_{\perp s}=\tilde {T}_{\perp s}/T_{{0 }_{\perp s}}$.

As above anticipated, in the system (3.3)–(3.8) we temporarily left unspecified the expression for the parallel heat flux fluctuations $Q_{\|s}$. In order to obtain the model GF6, the infinite hierarchy of gyrofluid equations following from the parent gyrokinetic model (2.1)–(2.4) is closed by imposing relations obtained by computing the gyrocentre moments other than $N_s$, $U_s$ and $T_{\|s}$ from a linearization of the parent gyrokinetic system about a homogeneous equilibrium, in the quasi-static limit. In the case of $Q_{\parallel s}$, this leads (consider (3.22) and (A 21)) to

(3.23)\begin{equation} Q_{\parallel s}=0, \quad s=e,i. \end{equation}

The model GF6 is thus obtained by inserting the relation (3.23) into the system (3.3)–(3.8). Details on the derivation of the closure relations originated from the quasi-static assumption can be found in appendix A. A remarkable property of this closure is that it leads to the annihilation of all the contribution of the higher-order moments in the gyrofluid equations.

We find it useful to provide also a reformulation of the evolution equations (3.3)–(3.5), which should help putting in evidence the physical nature of the terms contributing to the evolution of the various fields. Equations (3.3)–(3.5) can indeed be rewritten as

(3.24)\begin{gather} \frac{\partial N_s}{\partial t}+\boldsymbol{u}_{\perp_s}\boldsymbol{\cdot}\boldsymbol{\nabla} N_s+ \nabla_{\parallel s} U_s=0, \end{gather}

(3.25)\begin{gather} \frac{m_s}{m_i}\frac{\partial U_s}{\partial t}+\frac{m_s}{m_i}\boldsymbol{u}_{\perp_s}\boldsymbol{\cdot}\boldsymbol{\nabla} U_s-\textrm{sgn} (q_s)E_{\parallel s} + \nabla_{\parallel s} \mathcal{P}_s=0, \end{gather}

(3.26)\begin{gather} \frac{\partial T_{\parallel s}}{\partial t}+\boldsymbol{u}_{\perp_s}\boldsymbol{\cdot}\boldsymbol{\nabla} T_{\parallel s}+ 2\nabla_{\parallel s} U_s=-\nabla_{\parallel s} Q_{\parallel s}, \end{gather}

where the parallel gradient operator $\nabla _{\parallel s}$ is defined, for each species $s$, by

(3.27)\begin{equation} \nabla_{\parallel s} \, f=-[G_{10s} A_{\parallel}, f ] + \frac{\partial f}{\partial z}, \end{equation}

for a function $f$. From the formulation (3.24)–(3.26) it emerges that the gyrocentre density, parallel momentum and parallel temperature fluctuations are all advected, in the perpendicular plane, by the incompressible velocity field

(3.28)\begin{equation} \boldsymbol{u}_{\perp_s}=\widehat{\boldsymbol{z}}\times\boldsymbol{\nabla}\left(G_{10s} \phi + \textrm{sgn} (q_s) \tau_{\perp_s} 2 G_{20s} B_\parallel\right). \end{equation}

Such velocity field includes a first contribution, associated with $G_{10s} \phi$, which corresponds to the usual $\boldsymbol {E}\times \boldsymbol {B}$ drift (with E indicating the electric field based on the gyroaveraged electrostatic potential), ubiquitous in low-$\beta$ gyrofluid models such as those discussed in Waelbroeck & Tassi (Reference Waelbroeck and Tassi2012), Snyder & Hammett (Reference Snyder and Hammett2001), Keramidas Charidakos et al. (Reference Keramidas Charidakos, Waelbroeck and Morrison2015) and Waelbroeck, Hazeltine & Morrison (Reference Waelbroeck, Hazeltine and Morrison2009). When higher $\beta$ values are allowed, however, the perpendicular advection acquires a further contribution due to the parallel magnetic perturbations, as it transpires from (3.28). We remark that, similarly to the $\boldsymbol {E}\times \boldsymbol {B}$ contribution, also the latter contribution does not vanish in the limit $b_s \rightarrow 0$ of negligible FLR corrections. This is a consequence of the fact that, in the presence of parallel magnetic perturbations, gyrocentre density fluctuations differ from particle density fluctuations, even in the absence of FLR corrections (Brizard Reference Brizard1992).

From the continuity equation (3.24), we see that the evolution of gyrocentre density fluctuations has also a source in the last term on the left-hand side, which is due to the compressibility of the gyrocentre mean velocity along the direction of the magnetic field.

From (3.25) we see that parallel momentum, in addition to be advected by $\boldsymbol {u}_{\perp _s}$, evolves due to the term $-\textrm {sgn} (q_s)E_{\parallel s}$, where $E_{\parallel s}$ is defined by

(3.29)\begin{equation} E_{\parallel s}=-\frac{\partial G_{10s} A_{\parallel}}{\partial t}-[ G_{10s} \phi , G_{10s} A_{\parallel}]-\frac{\partial G_{10s} \phi}{\partial z}. \end{equation}

Such a term represents the force exerted by the gyroaveraged electric field, along the direction of the magnetic field. A further source for the parallel momentum is due to the term $\nabla _{\parallel s} \mathcal {P}_s$, where

(3.30)\begin{equation} \mathcal{P}_s=\tau_{\perp_s} \left(2 G_{20s} B_\parallel +\frac{1}{\Theta_s}(N_s+T_{\parallel s})\right). \end{equation}

This term is associated with the parallel component of the divergence of an anisotropic pressure tensor.

The parallel temperature equation (3.26) has, on its left-hand side, the same structure of the continuity equation (3.24). We just remark the presence of the coefficient $2$ multiplying $\nabla _{\parallel s} U_s$. This coefficient, of course, follows directly from taking the second order moment, in Hermite polynomials for the normalized parallel velocity, of the gyrokinetic equation (2.1), as discussed in appendix B. However, as pointed out by Keramidas Charidakos et al. (Reference Keramidas Charidakos, Waelbroeck and Morrison2015), in the presence of background magnetic curvature such coefficient, in general, has to be adjusted in order to obtain a Hamiltonian structure. Finally, we consider the term on the right-hand side of (3.26), associated with the parallel heat flux. If the expression for $Q_{\parallel s}$ is chosen according to the quasi-static closure, i.e. imposing the relation (3.23), this term vanishes and the system, as will be shown in § 3.1, is Hamiltonian. On the other hand, if the Landau fluid closure (3.35) is chosen, this term acts as a sink and the system is not energy conserving.

3.1. Hamiltonian structure of GF6

The quasi-static closure (3.23) leading to GF6, allows the resulting model to be cast in Hamiltonian form. In particular, it can be verified by direct computation that, when the electromagnetic fluctuations $\phi$, $A_{\parallel }$ and $B_\parallel$ can be expressed in terms of the dynamical variables $N_e$, $N_i$, $M_e$ and $M_i$ (where we introduced the short-hand notation $M_s=(m_s/m_i)U_s+\textrm {sgn}(q_{s})G_{10s} A_{\parallel }$ to indicate the parallel canonical momenta), by making use of the relations (3.6)–(3.8), the evolution equations (3.3)–(3.5) complemented by (3.23), can be written in the Hamiltonian form

(3.31a–c)\begin{equation} \frac{\partial N_s}{\partial t}=\{N_s , H\}, \quad \frac{\partial M_s}{\partial t}=\{M_s , H\}, \quad \frac{\partial T_{\parallel s}}{\partial t}=\{T_{\parallel s} , H\}, \end{equation}

with $s=e,i$. In (3.31a–c) $H$ is the Hamiltonian functional

(3.32)\begin{align} &H(N_e,N_i,M_e,M_i, T_{\parallel_e}, T_{\parallel_i})= \frac{1}{2}\sum_{s} \int d^3 x \, \left( \frac{\tau_{\perp_s}}{\Theta_s} N_s^2 \right. \nonumber\\ &\quad + \frac{m_i}{m_s} M_s^2+(\textrm{sgn} (q_{s}) G_{10s} \phi + 2 \tau_{\perp_s} G_{20s} B_\parallel )N_s \nonumber\\ &\quad \left. -\textrm{sgn}(q_{s}) \frac{m_i}{m_s} G_{10s} A_{\parallel} M_s + \frac{\tau_{\perp_s}}{\Theta_s} \frac{T_{\parallel s}^2}{2}\right), \end{align}

and $\{ \, , \, \}$ is a non-canonical Poisson bracket given by

(3.33)\begin{align} \{F,G\}&=- \sum_s \int d^3 x \, \left[ \textrm{sgn} (q_{s}) \left(N_s \left( [F_{N_s} , G_{N_s}]+\frac{\tau_{\perp_s}}{\Theta_s}\frac{m_s}{m_i}[F_{M_s} , G_{M_s}] + 2 [F_{T_{\parallel s}}, G_{T_{\parallel s}}]\right) \right. \right. \nonumber\\ &\quad \left. \left. +M_s([F_{M_s} , G_{N_s}] +[F_{N_s} , G_{M_s}] +2 ([F_{M_s} , G_{T_{\parallel s}}]+[F_{T_{\parallel s}} , G_{M_s}]))\right.\right. \nonumber\\ &\quad \left. \left. + T_{\parallel s} \left( \frac{\tau_{\perp_s}}{\Theta_s}\frac{m_s}{m_i} [F_{M_s} , G_{M_s}] + [F_{N_s} , G_{T_{\parallel s}}] + [F_{T_{\parallel s}} , G_{N_s}] + [F_{T_{\parallel s}} , G_{T_{\parallel s}}] \right)\right)\right. \nonumber\\ &\quad \left. +F_{N_s} \frac{\partial G_{M_s}}{\partial z} + F_{M_s}\frac{\partial G_{N_s}}{\partial z} + 2F_{T_{\parallel s}} \frac{\partial G_{M_s}}{\partial z} + 2 F_{M_s}\frac{\partial G_{T_{\parallel s}}}{\partial z} \right], \end{align}

for two functionals $F$ and $G$. For details about the non-canonical Hamiltonian formulation of fluid models one can refer, for instance, to Morrison (Reference Morrison1998). In (3.33) the subscripts on the functionals indicate functional derivatives. In order to verify the formulation (3.31a–c) it is convenient to remark that, from (3.32), one obtains

(3.34a–c)\begin{equation} H_{N_s}=\frac{\tau_{\perp_s}}{\Theta_s}N_s+\textrm{sgn} (q_{s}) G_{10s} \phi + \tau_{\perp_s} 2 G_{20s} B_\parallel, \quad H_{M_s} = U_s, \quad H_{T_{\parallel s}}=\frac{\tau_{\perp_s}}{\Theta_s}\frac{T_{\parallel s}}{2}. \end{equation}

In order to derive the relations (3.34a–c) we made use of the formal symmetry of the operators $G_{10s}$ and $G_{20s}$, i.e. $\int \textrm {d}^3x \, f G_{10s} g = \int \textrm {d}^3 x \, g G_{10s} f$ and $\int \textrm {d}^3x \, f G_{20s} g = \int \textrm {d}^3 x \, g G_{20s} f$ for two functions $f$ and $g$, as well as of the formal symmetry of the linear operators in terms of which one can express $\phi , B_\parallel$ and $A_{\parallel }$ in terms of $N_s$ and $M_s$ through (3.6)–(3.8). We did not provide the explicit expression for such operators which can, however, be obtained considering the representation in Fourier series of the fields involved, following the procedure discussed in Tassi (Reference Tassi2019).

The Hamiltonian functional (3.32) is a conserved quantity for the dynamics and corresponds to the total energy. In (3.33), the sum of all the terms with $s=e$ is a Poisson bracket in its own right. Similarly, all the terms with $s=i$ form a Poisson bracket. The sum of these two contributions is a direct sum of Poisson brackets which is in turn a Poisson bracket verifying in particular the Jacobi identity. The Poisson brackets referring to the electron and ion quantities correspond to those already discussed in other Hamiltonian reduced fluid models. The reader can in particular refer to Tassi (Reference Tassi2015) and Keramidas Charidakos et al. (Reference Keramidas Charidakos, Waelbroeck and Morrison2015) for the verification of the Jacobi identity for brackets of such form and for a discussion of the corresponding Casimir invariants. The model GF6, in the two-dimensional limit when the dependence on the $z$ coordinate is suppressed, can also be cast in the form of a system of advection of equations for Lagrangian invariants, as is the case for several other reduced fluid and gyrofluid models (see, e.g. Schep, Pegoraro & Kuvshinov Reference Schep, Pegoraro and Kuvshinov1994; Waelbroeck et al. Reference Waelbroeck, Hazeltine and Morrison2009; Waelbroeck & Tassi Reference Waelbroeck and Tassi2012; Grasso & Tassi Reference Grasso and Tassi2015; Keramidas Charidakos, Waelbroeck & Morrison Reference Keramidas Charidakos, Waelbroeck and Morrison2015; Tassi Reference Tassi2015, Reference Tassi2019).

Remark The gyroaverage operators $G_{10}$ with a form different from (3.16a,b) have been proposed in the literature (see, e.g. Dorland & Hammett Reference Dorland and Hammett1993; Snyder & Hammett Reference Snyder and Hammett2001; Scott Reference Scott2010), when the expansion (3.18) is not assumed, and are frequently adopted. In particular, $G_{10} (b_s)=\varGamma _0^{1/2} (b_s)$ was shown to provide better agreement with the linear theory at large $b_s$ (Dorland & Hammett Reference Dorland and Hammett1993). We point out, however, that many important features of the model (3.3)–(3.8), such as the total energy conservation and the Hamiltonian structure, are guaranteed whatever the form of the operators $G_{10}$ and $G_{20}$ of (3.16a,b) is, provided these operators are linear and formally symmetric, in the sense defined above. This is in particular the case with $G_{10} (b_s)=\varGamma _0^{1/2} (b_s)$. These issues are also discussed by Mandell, Dorland & Landreman (Reference Mandell, Dorland and Landreman2018).

3.2. Variants of the model

3.2.1. Six-field model with Landau closure (GF6L)

The variant GF6L of the six-field gyrofluid model, accounting for Landau damping, corresponds to the system (3.3)–(3.8) with $Q_{\parallel s}$ given by

(3.35)\begin{equation} Q_{\parallel s} =-2 \alpha_s \mathcal{L} T_{\parallel s}, \quad s=e,i. \end{equation}

In (3.35) we introduced the constant $\alpha _s=(2/{\rm \pi} )^{1/2}(m_i/m_s)^{1/2}$. The operator $\mathcal {L}$ holds for the Landau damping operator. Its modelling in the nonlinear regime is discussed in Tassi et al. (Reference Tassi, Grasso, Borgogno, Passot and Sulem2018). In the linear approximation, it reduces to the negative of the Hilbert transform in the direction of the ambient magnetic field (here taken in the $z$ direction). The presence of this Landau operator in reduced fluid models breaks the Hamiltonian structure by violating energy and Casimir conservation (Tassi et al. Reference Tassi, Grasso, Borgogno, Passot and Sulem2018; Grasso et al. Reference Grasso, Borgogno, Tassi and Perona2020). Its purpose, on the other hand, is to introduce terms that allow the linear dispersion relation of the gyrofluid model to reproduce that of the parent gyrokinetic model.

3.2.2. Four-field model with quasi-static closure (GF4)

The second variant GF4 is obtained by retaining the evolution equations for $N_s$ and $M_s$ and imposing the quasi-static closure on the parallel temperature fluctuations $T_{\parallel s}$. Considering (3.21a,b) and (A 21), this amounts to setting

(3.36)\begin{equation} T_{\parallel s}=0, \quad s=e,i. \end{equation}

The resulting model reads

(3.37)\begin{align} &\frac{\partial N_s}{\partial t}+[G_{10s}\phi + \textrm{sgn}(q_{s}) \tau_{\perp_s} 2 G_{20s} B_\parallel, N_s] - [G_{10s} A_{\parallel}, U_s ] +\frac{\partial U_s}{\partial z}=0, \end{align}

(3.38)\begin{align} &\frac{\partial }{\partial t}\left( \frac{m_s}{m_i} U_s + \textrm{sgn}(q_{s}) G_{10s} A_{\parallel}\right) \nonumber\\ &\quad +\left[ G_{10s} \phi +\textrm{sgn}(q_{s}) \tau_{\perp_s} 2 G_{20s} B_\parallel, \frac{m_s}{m_i} U_s + \textrm{sgn}(q_{s}) G_{10s} A_{\parallel} \right] \nonumber\\ &\quad -\frac{\tau_{\perp_s}}{\Theta_s}[G_{10s} A_{\parallel} , N_s ] +\frac{\partial}{\partial z} \left( \textrm{sgn}(q_{s}) G_{10s} \phi + \tau_{\perp_s} 2 G_{20s}B_\parallel+\frac{\tau_{\perp_s}}{\Theta_s}N_s \right)=0, \end{align}

(3.39)\begin{align} &\sum_s \left(\textrm{sgn}( q_{s} ) G_{10s} N_s + (1 -\Theta_s) \Gamma_{0s} \frac{\phi}{\tau_{\perp_s}}+(\Theta_s G_{10s}^2 -1)\frac{\phi}{\tau_{\perp_s}} \right. \nonumber\\ &\quad \left. + \textrm{sgn} (q_{s} ) (1-\Theta_s)(\Gamma_{0s}-\Gamma_{1s}) B_\parallel +\textrm{sgn}(q_{s}) \Theta_s 2 G_{10s} G_{20s} B_\parallel\right)=0, \end{align}

(3.40)\begin{align} &\nabla_{\perp}^2 A_{\parallel}= \frac{\beta_{\perp_e}}{2}\sum_s \left(\frac{m_i}{m_s}\left(1-\frac{1}{\Theta_s}\right)(1-\Gamma_{0s}) A_{\parallel} -\textrm{sgn} (q_{s}) G_{10s} U_s \right), \end{align}

(3.41)\begin{align} B_\parallel&=-\frac{1}{2}\sum_s \beta_{\perp_s} \left(2 G_{20s} N_s+(1-\Theta_s)(\Gamma_{0s}-\Gamma_{1s})\textrm{sgn}(q_{s})\frac{\phi}{\tau_{\perp_s}} \right. \nonumber\\ &\quad \left. + \Theta_s 2 G_{10s} G_{20s}\textrm{sgn}(q_{s}) \frac{\phi}{\tau_{\perp_s}}+2 (1-\Theta_s)(\Gamma_{0s} - \Gamma_{1s})B_\parallel + \Theta_s 4 G_{20s}^2 B_\parallel \right) \end{align}

and corresponds to taking (3.3), (3.4), (3.6), (3.7) and (3.8) of GF6 with $T_{\parallel s}=0$. Its Hamiltonian structure is given by the Hamiltonian

(3.42)\begin{align} H(N_e,N_i,M_e,M_i)&= \frac{1}{2}\sum_{s} \int d^3 x \, \left( \frac{\tau_{\perp_s}}{\Theta_s} N_s^2 \right. + \frac{m_i}{m_s} M_s^2+(\textrm{sgn} (q_{s}) G_{10s} \phi \nonumber\\ &\left. \quad + 2 \tau_{\perp_s} G_{20s} B_\parallel )N_s -\textrm{sgn}(q_{s}) \frac{m_i}{m_s} G_{10s} A_{\parallel} M_s \right) \end{align}

and by the Poisson bracket

(3.43)\begin{align} &\{F,G\}=- \sum_s \int d^3 x \, \left[ \textrm{sgn} (q_{s}) \left(N_s \left( [F_{N_s} , G_{N_s}]+\frac{\tau_{\perp_s}}{\Theta_s}\frac{m_s}{m_i}[F_{M_s} , G_{M_s}] \right) \right. \right. \nonumber\\ &\quad \left. \left. +M_s([F_{M_s} , G_{N_s}] +[F_{N_s} , G_{M_s}] ) \right)+F_{N_s} \frac{\partial G_{M_s}}{\partial z} + F_{M_s}\frac{\partial G_{N_s}}{\partial z} \right]. \end{align}

If one neglects electron FLR effects (i.e. $b_e \rightarrow 0$), parallel magnetic perturbations (i.e. $B_\parallel =0$), equilibrium temperature anisotropies (i.e. $\varTheta _e=\varTheta _i=1$) and sets $G_{10} (b_i)=\varGamma _{0}^{1/2} (b_i)$ (i.e. takes the alternative form of the ion gyroaverage operator mentioned in the above remark in § 3.1), GF4 reduces to the Hamiltonian gyrofluid model of Waelbroeck & Tassi (Reference Waelbroeck and Tassi2012), the latter taken in the limit of vanishing magnetic curvature and equilibrium density gradients. We note, however, that, once that the quasi-static closure relations are determined, as in (A 21), all the terms in GF4 (and likewise for GF6), are determined exactly. In particular no approximations of the gyroaverage operators (unlike, for instance, in Scott Reference Scott2010; Waelbroeck & Tassi Reference Waelbroeck and Tassi2012) are carried out. Terms involving gyroaverage operators are determined exactly also in Brizard (Reference Brizard1992), but without making use of the quasi-static closure. As a result, in Brizard (Reference Brizard1992), nonlinear terms involving more than one gyroaverage operator do not result in having the single canonical bracket structure (as is the case in GF4, for instance, with the term $\textrm {sgn} (q_{s}) [ G_{10s} \phi ,G_{10s} A_{\parallel }]$ appearing in the second line of (3.38)) and which is crucial for determining the Lie–Poisson Hamiltonian structure. This property, which follows from the quasi-static closure, differentiates the models presented in our paper also from its closest predecessors, i.e. the gyrofluid models constructed with the technique recently presented in Tassi (Reference Tassi2019). The latter gyrofluid models, in fact, are also Hamiltonian and account for equilibrium temperature anisotropy, but adopt a different closure. Namely, all gyrofluid moments involving finite powers of the magnetic moment $\mu _s$ (e.g. the perpendicular temperature fluctuations) are set equal to zero. This allows for a Hamiltonian structure but the terms involving gyroaverage operators are not all determined exactly from taking the moments of the gyrokinetic equations. For this reason, we think that, for situations where the quasi-static assumption $|\omega /(k_z v_{{\textrm {th}}_{\parallel s}})| \ll 1$ is satisfied, the models presented in this paper might be preferable to the models described in Tassi (Reference Tassi2019).

4. Comparison with the linearized gyrokinetic parent model

4.1. KAW dispersion relation

We first discuss the KAW dispersion relation as predicted by the gyrofluid models, with and without Landau damping (GF6L and GF4 respectively), in comparison with the predictions of the linearized parent model (Kunz et al. Reference Kunz, Abel, Klein and Schekochihin2018) which identifies with the low-frequency kinetic theory described in Kuznetsov, Passot & Sulem (Reference Kuznetsov, Passot and Sulem2012). This latter dispersion relation involves the plasma response function $R(\zeta _s)$ of the particles of species $s$, which is related to the corresponding plasma dispersion function $Z(\zeta _s)$ by $R(\zeta _s) = 1 + \zeta _s Z(\zeta _s)$ with $\zeta _s = \omega /(2^{1/2}|k_z|v_{\textrm {th}_{\| s}})$. Different Padé approximants $R_{ij}(\zeta _e)$ (for which we follow the notations of Hunana et al. Reference Hunana, Tenerani, Zank, Goldstein, Webb, Velli and Adhikari2019) are used to estimate the electron response function.

In order to test the validity of the quasi-static assumption that affects the form of the FLR terms, independently from the effects resulting from Landau damping, we are led to compare the prediction of the four-field gyrofluid model GF4 with the gyrokinetic dispersion relation obtained by choosing for the electron plasma response function, the function $R_{21}(\zeta _e)=1/(1-2\zeta _e^2)$ (model denoted GKNL). This choice directly results from the assumption ${T_{{\|e}}}=0$ (see (B7) of Passot & Sulem Reference Passot and Sulem2007) and ignores electron Landau damping.Footnote ¹ Predictions of the GF6L model will, on the other hand, be compared with the full gyrokinetic dispersion relation, referred to as GK. In the latter description, we use for the electron Padé approximant the function $R_{86}(\zeta _e)$, which shows an excellent agreement with the exact response function. For the ions, we use in all the cases the Padé approximant $R_{20}(\zeta _i)=1/(1-2\zeta _i^2-\textrm {i}{\rm \pi} ^{1/2}\zeta _i)$ (or $R_{21}(\zeta _i)$ in the absence of Landau damping). Higher-order Padé approximants give almost identical results.

Because we are making use of several acronyms to refer to the different adopted models, we summarized them in table 1. In all the examples presented in this section, $\varTheta _i=1$. Introducing the ion to electron parallel temperature ratio at equilibrium $\tau _{\|} = {T_{0_{\|i}}}/{T_{0_{\|e}}}=\tau _{\perp _ i}\varTheta _e/{\varTheta _i}$ and denoting by $\alpha$ the angle between the wave vector and the guide field (propagation angle), we compare, in figure 1(a), the (real) normalized KAW frequency $\omega /(k_z c_A)$ for $\beta _{\perp e}=1$, $\tau _{\|}=1$, $\varTheta _e=1$, $\alpha =89^{\circ }$ as a function of $k d_i$ (where $k=\sqrt {k_{\perp }^2+k_z^2}$ and $d_i$ is the ion inertial length defined by $d_i = c_A/\omega _{ci}$, with $c_A =\sqrt {B_0/(4{\rm \pi} m_i n_0)}$ the Alfvén velocity), calculated using GF4 (black diamond symbols), with the GKNL prediction (green solid line). An excellent agreement is found, the slight deviation appearing as $kd_i>40$ probably resulting from the failure of the quasi-static approximation used in the calculation of all the moments starting from the temperature. For the same plasma parameters, the case with Landau damping is displayed as a red solid line for GK and blue diamond symbols for GF6L, with a good agreement up to scales $kd_i \lessapprox 20$. The damping rate is displayed with the same symbols in figure 1(b). Interestingly, the prediction of GF6 (not shown) departs from GKNL significantly at all the scales. Indeed, in GF6, the parallel temperature fluctuations that obey a dynamical equations with zero heat flux do not approach a quasi-static dynamics. Such a non-dissipative odd-order closure rather fits an adiabatic regime (see footnote in this section). Landau damping is requested to ensure convergence to a quasi-static regime. In that sense, GF4 is preferable to GF6 for addressing a non-dissipative problem, the interest of the latter model being mainly to provide a framework where Landau damping can be supplemented to an otherwise Hamiltonian description.

Figure 1. (a) Normalized real part of the KAW frequency $\omega /(k_z c_A)$ for $\beta _{\perp e}=1$, $\tau _{\|}=1$, $\varTheta _e=1$, $\alpha =89^{\circ }$, as a function of $k d_i$, calculated using GF4 (black diamond symbols) and GF6L (blue diamond symbols) models (respectively without and with Landau damping) and with the gyrokinetic dispersion relations GKNL (green solid line) and GK (red solid line). (b) Damping rate from the GK theory (red solid line) and from the GF6L model (blue diamond symbols). (c) Normalized growth rate of the firehose instability as a function of $k d_i$, for $\beta _{\perp e}=0.5$, $\tau _{\|}=10^{-3}$, $\varTheta _e=0.16$, $\alpha =89^{\circ }$, as predicted by GF4 and GF6L models, together with GKNL and GK dispersion relations, using the same graphic conventions.

Table 1. Acronyms and corresponding models.

4.3. SW dispersion relation and field-swelling instability

Figure 2 concerns a similar comparison in the case of the field swelling instability discussed in appendix C, for $\beta _{\perp e}=1$, $\tau _{\|}=1$, $\alpha =89^{\circ }$, $\varTheta _e=2.2$ (panel a). For both the cases with and without Landau damping, the relatively good agreement found at large scale between the gyrofluid theories and the gyrokinetic ones deteriorates at smaller scales. The stabilization scale is, however, correctly captured. This discrepancy is associated with a value of $\zeta _i$, which is not small enough (in this case, $\zeta _e$ remains reasonably small). We show in figure 2(b) with $\varTheta _e=2.41$, again with $\beta _{\perp e}=1$ and $\alpha =89^{\circ }$, that a much better agreement can be found when ions are hotter ($\tau _{\|}=50$), which prescribes a small $\zeta _i$. The case with cold ions ($\tau _{\|}=10^{-5}$), for which the ion dynamics is decoupled, is displayed in figure 3, showing an even better agreement. Figure 3(a) displays the real part of the slow wave for $\varTheta _e=1$, obtained with the GKNL model (solid green line) or the GF4 model (diamond symbols). Figure 3(b) shows the growth rate of the field-swelling instability for $\varTheta _e=2.01$ (keeping unchanged the other parameters).

Figure 2. Normalized growth rate of the field-swelling instability versus $kd_i$ for $\alpha =89^{\circ }$ and $\beta _{\perp e}=1$. (a) Predictions of GF4 and GF6L, compared with those of GKNL and GK respectively, for $\tau _{\|}=1$ and $\varTheta _e=2.2$. (b) Prediction of GF4 compared with that of GKNL, for $\tau _{\|}=50$ and $\varTheta _e=2.41$. Same graphic conventions as in figure 1 are used.

Figure 3. (a) Normalized slow-wave frequency versus $kd_i$ in regimes where the ions are cold ($\tau _{\|}=10^{-5}$), with $\beta _{\perp e}=1$ and $\alpha =89^{\circ }$, as predicted by the GF4 model and the GKNL dispersion relation. (b) Normalized growth rate of the swelling instability for $\tau _{\|}=10^{-5}$ and $\varTheta _e=2.01$. Same conventions as in figure 1 are used.

Declaration of interests

The authors report no conflict of interest.