2.1. gyromoment hierarchy equation

To approach the solution of the GK model, given by (2.4) and (2.5), we use a Hermite–Laguerre moment expansion of the ion distribution function. This allows us to reduce the dimensionality of the linearized GK equation, (2.1), by projecting it onto velocity-space Hermite–Laguerre basis polynomials. More precisely, we decompose the perturbed gyrocentre distribution function as (Jorge, Ricci & Loureiro Reference Jorge, Ricci and Loureiro2017; Mandell, Dorland & Landreman Reference Mandell, Dorland and Landreman2018; Jorge et al. Reference Jorge, Frei and Ricci2019a; Frei et al. Reference Frei, Jorge and Ricci2020)

(2.6)\begin{equation} {g} = \sum_{p = 0}^{\infty} \sum_{j = 0}^{\infty} N^{pj} \frac{H_p(s_{{\parallel}}) L_j(x)}{\sqrt{2^{p} p!}} F_{M}, \end{equation}

where we define the Hermite–Laguerre velocity moments of ${g}$, as follows:

(2.7)\begin{equation} N^{pj} = \frac{1}{N} \int \,{\rm d} \mu \,{\rm d} {v_{{\parallel}}} \,{\rm d} \theta\frac{B}{m} {g} \frac{H_p(s_\parallel) L_j(x)}{\sqrt{2^{p} p!}}, \end{equation}

with $N = \int \,{\rm d} \mu \,{\rm d} {v_{\parallel }} \,{\rm d} \theta B F_M / m$. The Hermite–Laguerre coefficients $N^{pj}$ in (2.6) are referred to as the gyromoments. In (2.6), we introduce the Hermite and Laguerre polynomials, $H_p$ and $L_j$, via their Rodrigues’ formulae (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2014), that is

(2.8)\begin{equation} H_p(y) = ({-}1)^{p} {\rm e}^{y^{2}} \frac{{\rm d}^{p}}{{{\rm d} y}^{p}} ( {\rm e}^{- y^{2}} ) \end{equation}

and

(2.9)\begin{equation} L_j(y) = \frac{{\rm e}^{y}}{j!} \frac{{\rm d}^{j} }{{{\rm d} y}^{j}} ({\rm e}^{- y } y^{j}) \end{equation}

and their orthogonality relations, that is

(2.10)\begin{equation} \int_{- \infty}^{\infty} {{\rm d} y} H_p(y) H_{p'}(y) {\rm e}^{- y^{2}} = 2^{p} p! \sqrt{\rm \pi} \delta_{p}^{p'} \end{equation}

and

(2.11)\begin{equation} \int_0^{\infty} {{\rm d} y} L_j(y) L_{j'}(y) {\rm e}^{{-}y} = \delta_j^{j'}, \end{equation}

respectively. We now project (2.4) and (2.5) onto the Hermite–Laguerre polynomial basis. For this purpose, we note that the Bessel function ${\rm J}_m$ appearing in both (2.4) and (2.5) and in the GK collision operators (Frei et al. Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021) can be conveniently projected by expanding the Bessel functions, ${\rm J}_m$, onto associated Laguerre polynomials, $L^{m}_n(x) = (-1)^{m} \,{\rm d}^{m} L_{n + m}(x) / {{\rm d} x}^{m}$,

(2.12)\begin{equation} {\rm J}_m(b \sqrt{x}) = \left(\frac{b \sqrt{x}}{2}\right)^{m}\sum_{n=0}^{\infty} \frac{n!\mathcal{K}_{n}}{(n + m)!} L^{m}_n(x), \end{equation}

where we introduce the $n$th-order kernel function

(2.13)\begin{equation} \mathcal{K}_{n} = \frac{1}{n!}\left(\frac{b}{2}\right)^{2n } {\rm e}^{- b^{2} /4}, \end{equation}

with argument $b = \sqrt {2 \tau } k_\perp$ (Frei et al. Reference Frei, Jorge and Ricci2020). The projections of (2.4) and (2.5) onto the Hermite–Laguerre basis yields the linearized gyromoment hierarchy equation that states the time evolution of the gyromoments, $N^{pj}$, that is

(2.14)\begin{gather} \frac{\partial}{\partial t} N^{pj} + {\rm i} k_\parallel \sqrt{\tau} \sum_{p'} \mathcal{H}^{{\parallel} p}_{p'} N^{p'j} + {\rm i} \frac{\tau}{q}\omega_{ B } \sum_{p'j'} ( \mathcal{H}_{B p'j'}^{{\parallel} pj} + \mathcal{H}_{B p'j'}^{{\perp} pj} ) N^{p'j'} \nonumber\\ = (\mathcal{F}^{pj} - \mathcal{F}_{B}^{{\parallel} pj}) \phi + \mathcal{C}^{pj} , \end{gather}

where we introduce the phase-mixing operators, associated with the parallel and perpendicular drifts,

(2.15a)\begin{gather} \mathcal{H}^{{\parallel} p}_{p'} = \sqrt{p +1 } \delta_{p'}^{p+1}+ \sqrt{p} \delta_{p'}^{p-1}, \end{gather}

(2.15b)\begin{gather}\mathcal{H}_{B p'j'}^{{\parallel} pj} = \delta_{j'}^{j} (\sqrt{(p+1)(p+2)} \delta_{p'}^{p+2} + \sqrt{2p+1} \delta_{p'}^{p} + \sqrt{p(p-1)} \delta_{p'}^{p-2}), \end{gather}

(2.15c)\begin{gather}\mathcal{H}_{B p'j'}^{{\perp} pj} = \delta_{p'}^{p} ( (2j+1)\delta_{j'}^{j} - j \delta_{j'}^{j-1} - (j+1) \delta_{j'}^{j+1}), \end{gather}

and the operators that provide the instability drive and are proportional to the equilibrium gradients

(2.16a)\begin{gather} \mathcal{F}^{pj} = {\rm i} \omega_* \left[ \mathcal{K}_{j} \delta_p^{0} + \eta \left( \mathcal{K}_{j} \frac{\delta_p^{2}}{\sqrt{2}} + \delta_p^{0} \left(2 j\mathcal{K}_{j} - j \mathcal{K}_{j-1} - (j+1)\mathcal{K}_{j+1} \right) \right) \right] , \end{gather}

(2.16b)\begin{gather}\mathcal{F}_{B}^{{\parallel} pj} =\frac{\delta_p^{1}}{\sqrt{\tau} } {\rm i} k_\parallel \mathcal{K}_{j} + {\rm i} \omega_{ B} \delta_p^{0} \left[ 2 ( j +1) \mathcal{K}_{j} - j \mathcal{K}_{j-1} - (j+1)\mathcal{K}_{j+1} \right]+ {\rm i} \omega_B \mathcal{K}_{j} \sqrt{2} \delta_p^{2} . \end{gather}

In (2.14), the Hermite–Laguerre projection of the collision operator, $\mathcal {C}$, is defined by

(2.17)\begin{equation} \mathcal{C}^{pj} = \int \,{\rm d} \theta \,{\rm d} \mu \,{\rm d} v_\parallel \frac{B}{m} \frac{H_p(s_\parallel) L_j(x)}{ \sqrt{2^{p} p!}} \mathcal{C}. \end{equation}

In this work, different collision operator models for $\mathcal {C}$ are considered and detailed in § 2.2. The linearized gyromoment hierarchy equation is coupled with the self-consistent GK quasineutrality condition that, projected onto the Hermite–Laguerre basis, is given by

(2.18)\begin{equation} \left[ 1 + \frac{q^{2}}{\tau}\left( 1 - \sum_{n=0}^{\infty} \mathcal{K}_{n}^{2} \right)\right] \phi = q \sum_{n=0}^{\infty} \mathcal{K}_{n} N^{0n}. \end{equation}

Equations (2.14) and (2.18) constitute an infinite set of linear coupled fluid equations for the gyromoments $N^{pj}$, and depend only on the Fourier mode wavevector $\boldsymbol k$ and time $t$. In order to solve numerically (2.14) together with (2.18), we apply a simple closure by truncation, i.e. we solve (2.14) and (2.18) up to $(p,j) \leq (P,J)$ and set $N^{pj} = 0$ for all $(p,j) > (P,J)$ given $0 < P, J < \infty$. When applying the closure by truncation to the gyromoment hierarchy equation, the infinite sums appearing in the GK quasineutrality, (2.18), are consistently truncated up to $J$, that is

(2.19)\begin{equation} \left[ 1 + \frac{q^{2}}{\tau}\left( 1 - \sum_{n=0}^{J} \mathcal{K}_{n}^{2} \right)\right] \phi = q \sum_{n=0}^{J} \mathcal{K}_{n} N^{0n}. \end{equation}

Finally, we remark that a high-collisional closure of the gyromoment hierarchy is derived in § 4.1.

A gyromoment hierarchy equation, similar to the one given in (2.14), is obtained by Mandell et al. (Reference Mandell, Dorland and Landreman2018). In their formulation, the probabilist's Hermite polynomials, defined by $\hat {H}_p(y) = 2^{- p/2} H_p( y / \sqrt {2})$ and satisfying the orthogonality relation given in (2.10) but weighted by ${\rm e}^{-y^{2} / 2}$, are used, while collisional effects are modelled by the GK Dougherty collision operator model because of the simplicity of its expression on the Hermite–Laguerre basis (see Appendix C). Since the collisional effects are the main focus of the present study, we consider here more realistic collision operator models.

2.2. Gyrokinetic linearized collision operator models

While numerous collision operator models are available in the literature (see, e.g. Hirshman & Sigmar Reference Hirshman and Sigmar1976a; Abel et al. Reference Abel, Barnes, Cowley, Dorland and Schekochihin2008; Sugama, Watanabe & Nunami Reference Sugama, Watanabe and Nunami2009; Li & Ernst Reference Li and Ernst2011; Madsen Reference Madsen2013b; Sugama et al. Reference Sugama, Matsuoka, Satake, Nunami and Watanabe2019), the main focus of this work is a linearized GK Coulomb collision operator, and we leverage its formulation derived and implemented in Frei et al. (Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021). This advanced linearized GK collision operator is obtained from the nonlinear full-F GK Coulomb operator derived in Jorge et al. (Reference Jorge, Frei and Ricci2019a). To highlight the importance of FLR terms in the Coulomb collision operator, the DK limit of the GK Coulomb collision operator is also considered. In addition, collision operators found in the literature and widely used in numerical codes are also considered for comparison. In particular, we consider the GK and DK Sugama collision operators (Sugama et al. Reference Sugama, Watanabe and Nunami2009), the GK and DK momentum-conserving pitch-angle scattering operators (Helander & Sigmar Reference Helander and Sigmar2002), the zeroth-order DK HSC collision operator (Hirshman & Sigmar Reference Hirshman and Sigmar1976a) and the GK and DK Dougherty collision operators (Dougherty Reference Dougherty1964). The gyromoment expansion of the linearized GK/DK Coulomb and the GK/DK Sugama collision operators were reported in Frei et al. (Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021) where the linearized DK Coulomb and GK/DK Sugama operators are successfully benchmarked against the GK continuum code GENE (Jenko et al. Reference Jenko, Dorland, Kotschenreuther and Rogers2000). The Hermite–Laguerre expansion of the pitch-angle, HSC and Dougherty collision operators are reported in appendices A–C, respectively.

Before turning to the investigation of the impact of collisions on the ITG mode, here we recall some important conservation properties of linearized like-species collision operators. These properties are useful to derive the high-collisional limit of the gyromoment hierarchy (see § 4.1) and to verify the numerical implementation of the collision operators. A model of a linearized collision operator for ion–ion collisions, $C$, is obtained by linearizing a nonlinear collision operator, that we denote by $C^{NL}$. Since collisions act at the particle position $\boldsymbol {r}$ (rather than at the gyrocentre position $\boldsymbol {R}$), the nonlinear collision operator is most often derived as acting on the full particle distribution function and is expressed in the particle phase space $(\boldsymbol {r}, \boldsymbol v)$. While the functional form of $C^{NL}$ depends on the choice of collision model, the linearized collision operator, $C$, can always be expressed as

(2.20)\begin{equation} C = C^{T} + C^{F}, \end{equation}

where we introduce the test component of the linearized collision operator, $C^{T} = C^{NL}(f,f_{M})$, and the field component, $C^{F} = C^{NL}(f_{M},f)$, with $f_M = f_M(\boldsymbol r, \boldsymbol v)$ the particle Maxwellian distribution function and $f = f (\boldsymbol {r},\boldsymbol v)$ a small perturbation. Conservation properties of the collision operator constrain the test and field components. In fact, while particle conservation is satisfied by both components, such that

(2.21)\begin{equation} \int \,{\rm d} \boldsymbol{v} C^{T} = \int \,{\rm d} \boldsymbol{v} C^{F} = 0, \end{equation}

the momentum and energy conservations require that

(2.22)\begin{gather} \int \,{\rm d} \boldsymbol{v} \boldsymbol v C^{T} ={-} \int \,{\rm d} \boldsymbol{v}\boldsymbol v C^{F} , \end{gather}

(2.23)\begin{gather}\int \,{\rm d} \boldsymbol{v} v^{2} C^{T} ={-} \int \,{\rm d} \boldsymbol{v} v^{2} C^{F} , \end{gather}

respectively. From the conservation properties of the linearized collision operator $C$, (2.21)–(2.23), constraints on the coefficients $\mathcal {C}^{pj}$ can be derived in the zero gyroradius limit, when the gyrocentre and particle position correspond. In fact, using (2.17) combined with (2.21)–(2.23) yields the relations

(2.24a–c)\begin{equation} \mathcal{C}^{00} = 0, \quad \mathcal{C}^{10}= 0, \quad \mathcal{C}^{20} = \sqrt{2} \mathcal{C}^{01}, \end{equation}

which express the conservation of gyrocentres, momentum and energy. The constraints in (2.24a–c) are satisfied by all linearized collision operator models considered in the present work in the long wavelength limit and are fulfilled numerically to machine precision.

3.1. Collisionless GK dispersion relation

We first derive the ITG dispersion relation by solving directly the GK equation, (2.4). Fourier transforming (2.1) in time with $\mathcal {C} =0$, such that $\partial _t \to - {\rm i} \omega$, and introducing $\omega = \omega _r + {\rm i} \gamma$, with $\omega _r$ and $\gamma$ the real mode frequency and the growth rate, respectively, the normalized perturbed ion gyrocentre distribution function, ${g}/ \phi$, can be expressed as

(3.1)\begin{equation} \frac{{g}}{\phi}= \sum_{l=1}^{3} \hat{{g}}_{l} \end{equation}

with

(3.2a)\begin{gather} \hat{{g}}_{1 } ={-} \frac{q}{\tau} {\rm J}_0(b \sqrt{x}) F_{M}, \end{gather}

(3.2b)\begin{gather}\hat{{g}}_{2 } = \frac{q}{\tau} \frac{ \omega }{ \omega - \omega_{\boldsymbol{\nabla} B}- z_\parallel s_\parallel } {\rm J}_0(b \sqrt{x}) F_{M}, \end{gather}

(3.2c)\begin{gather}\hat{{g}}_{3} ={-}\frac{\omega_{T}^{*}}{\omega - \omega_{\boldsymbol{\nabla} B}- z_\parallel s_\parallel }{\rm J}_0(b \sqrt{x}) F_{M}, \end{gather}

where we introduce $z_\parallel = \sqrt {2 \tau } k_\parallel$. The $l=1$ contribution to the sum in (3.1) is the adiabatic response, while the $l=2$ and $l=3$ terms are associated with the Landau damping and equilibrium gradient drive. We remark the presence of the velocity-dependent magnetic frequency $\omega _B$ in the resonant denominator of the $l=2$ and $l=3$ contributions.

The collisionless GK dispersion relation is deduced from the linearized GK quasineutrality condition given in (2.5),

(3.3)\begin{equation} 1 + \frac{q^{2}}{\tau} \left( 1 - \varGamma_0(a) \right) - q \hat{n} =0, \end{equation}

where $\hat {n} = \sum _{l=1}^{3} \hat {n}_l$ is the density perturbation with $\hat {n}_l = \int \,{\rm d} \boldsymbol {v} {\rm J}_0 \hat {{g}}_l$ and $\hat {{g}}_l$ given in (3.2). By considering an unstable mode, i.e. $\gamma > 0$, the velocity integrals appearing in $\hat {n}_l$ can be performed explicitly by writing the resonant term, $1/(\omega - \omega _{\boldsymbol {\nabla } B} - z_\parallel s_\parallel )$, in integral form (Beer & Hammett Reference Beer and Hammett1996)

(3.4)\begin{equation} \frac{1}{ \omega - \omega_{\boldsymbol{\nabla} B} - z_\parallel s_\parallel} ={-} {\rm i} \int_0^{\infty} \,{\rm d} \sigma \exp({ {\rm i} \sigma (\omega - \omega_{\boldsymbol{\nabla} B} - z_\parallel s_\parallel)}). \end{equation}

We remark that (3.4) allows us to evaluate the velocity integrals, appearing in $\hat {n}_l$, exactly and to retain the effects of magnetic drifts in $\hat {N}_{l}^{pj}$ at arbitrary order in $\omega _B / \omega$. Using (3.4), one derives

(3.5a)\begin{align} \hat{n}_{1} & ={-} \frac{q}{\tau} \varGamma_0(a), \end{align}

(3.5b)\begin{align} \hat{n}_{2} & ={-} \frac{ {\rm i}q}{\tau} \omega \int_0^{\infty} \,{\rm d} \sigma {\rm e}^{{\rm i} \sigma \omega} {\rm I}_\perp(\sigma) {\rm I}_\parallel(\sigma), \end{align}

(3.5d)\begin{align} \hat{n}_{3} & = {\rm i} k_\perp \int_0^{\infty} \,{\rm d} \sigma {\rm e}^{{\rm i} \sigma \omega} \left[\vphantom{\frac{3}{2}} {\rm I}_\parallel (\sigma) {\rm I}_\perp (\sigma) \right. \nonumber\\ & \quad \left.+ \eta \left( {\rm I}_{{\parallel} 2} (\sigma) {\rm I}_\perp (\sigma) + {\rm I}_{{\parallel} } (\sigma) {\rm I}_{{\perp} 1} (\sigma) - \frac{3}{2} {\rm I}_\parallel (\sigma) {\rm I}_\perp (\sigma) \right) \right], \end{align}

where we introduce

(3.6a)\begin{align} {\rm I}_\perp(\sigma)& = \frac{1}{1 + {\rm i}\alpha \sigma} {\rm I}_0\left(\frac{a }{1 + {\rm i}\alpha \sigma} \right) \exp({-a /(1 + {\rm i} \alpha\sigma)}), \end{align}

(3.6b)\begin{align} {\rm I}_\parallel(\sigma) & = \frac{1}{\sqrt{1 + 2{\rm i} \alpha \sigma}} \exp({ - z_\parallel^{2} \sigma^{2} /[4 (1 + 2 {\rm i} \alpha \sigma)]}), \end{align}

\begin{align} {\rm I}_{{\perp} 1}(\sigma) & = \frac{\exp({- a / (1 + {\rm i} \alpha \sigma)})}{2 (1 + {\rm i} \alpha \sigma)^{3}} \nonumber\\ & \quad \times \left[ (2 (1 + {\rm i} \alpha \sigma) - 2 a) {\rm I}_0 \left( \frac{a}{ (1 + {\rm i} \alpha \sigma)} \right)+ 2 a {\rm I}_1 \left( \frac{a}{ (1 + {\rm i} \alpha \sigma)} \right) \right], \end{align}

(3.6c)\begin{align} {\rm I}_{{\parallel} 2}(\sigma) & = \frac{(2(1 + 2 {\rm i} \sigma \alpha ) - \sigma^{2} z_\parallel^{2}) }{4 (1 + 2 {\rm i} \sigma \alpha)^{5/2}} \exp({- z_\parallel^{2} \sigma^{2} / ( 4 (1 + 2 {\rm i} \sigma \alpha))}), \end{align}

with $\alpha = q \omega _B / \tau$. Equation (3.3), with the definitions in (3.5) and the velocity integrals in (3.6) constitute the collisionless GK dispersion relation for the unstable modes $\gamma > 0$. We note that the exponential factors ${\rm e}^{{\rm i} \sigma \omega }$, appearing in (3.5), ensure the convergence of the integrals at $\sigma \to \infty$ for $\gamma > 0$. On the other hand, the inclusion of stable modes ($\gamma < 0$) in (3.3) can be obtained by expressing the velocity integrals in $\hat {n}$ using generalized plasma dispersion functions (Gültekin & Gürcan Reference Gültekin and Gürcan2019) and efficient numerical algorithms can be used to solve the dispersion relation in the entire complex plane (Xie et al. Reference Xie, Li, Lu, Ou and Li2017; Gültekin & Gürcan Reference Gültekin and Gürcan2018). However, here we focus on the upper half of the complex plane since one of the advantages of the transformation (3.4) is that it yields one-dimensional integrals that can be easily evaluated numerically.

We finally note that the collisionless GK dispersion relation can be simplified in the sITG case. Neglecting $\omega _B$ in (3.2), the velocity integrals appearing in $\hat {n}$ and given in (3.5), can be expressed in terms of the plasma dispersion function. Then, the GK dispersion relation, (3.3), becomes

(3.7)\begin{align} & 1 + \frac{1}{\tau} + \frac{1}{\tau} \xi Z(\xi) \varGamma_0(a) - \frac{k_\perp}{k_\parallel} \frac{1}{\sqrt{2 \tau}} \left[ \varGamma_0(a) Z(\xi) \right. \nonumber\\ & \qquad \left. + \eta \left( \varGamma_0(a) \xi \left( 1 + \xi Z(\xi)\right) - \frac{\varGamma_0(a)}{2}Z(\xi) + a Z(\xi)\left(\varGamma_1(a) - \varGamma_0(a) \right) \right)\right] =0, \end{align}

where $\xi = \omega \sqrt {2 \tau }/ k_\parallel$ is the normalized mode phase velocity, $Z(\xi ) \!=\! \int _{-\infty }^{\infty } \,{\rm d} s_\parallel {\rm e}^{- s_\parallel ^{2}}/(s_\parallel \!-\! \xi ) /\sqrt {{\rm \pi} }$ is the plasma dispersion function and $\varGamma _1(a) = {\rm I}_1(a) {\rm e}^{- a}$.

3.2. Collisionless linear gyromoment response

We now obtain the collisionless expressions of the gyromoments by projecting (3.1) onto the Hermite–Laguerre basis. This yields

(3.8)\begin{equation} \frac{N^{pj}}{\phi} = \sum_{l=1}^{3} \hat{N}_{l}^{pj}, \end{equation}

where we introduce

(3.9)\begin{equation} \hat{N}_{l}^{pj} = 2 {\rm \pi}\int \,{\rm d} v_\parallel \,{\rm d} \mu \frac{B}{m} \frac{H_p(s_\parallel ) L_j(x) \hat{{g}}_l }{ \sqrt{2^{p} p !}}, \end{equation}

with $\hat {{g}}_l$ defined in (3.2). Performing the velocity integral in (3.9) using (3.4) for an unstable mode with $\gamma >0$ and identities involving hypergeometric functions and Hermite–Laguerre polynomials (described in Appendix D), we obtain

(3.10a)\begin{align} \hat{N}_1^{pj} & ={-} \frac{q}{\tau} \mathcal{K}_{j} \delta_p^{0}, \end{align}

(3.10b)\begin{align} \hat{N}_{2}^{pj}& ={-} \frac{{\rm i} q \omega}{\tau} \sum_{n=0}^{\infty} \frac{ \mathcal{K}_{n} }{\sqrt{2^{p} p!} }\mathcal{I}_{n}^{pj} , \end{align}

(3.10c)\begin{align} \hat{N}_{3}^{pj}& = \frac{{\rm i} k_\perp}{\sqrt{2^{p} p!}}\sum_{n=0}^{\infty}\mathcal{K}_{n} \left\{\vphantom{\frac{3}{2}} \mathcal{I}_n^{pj} + \eta \left[\vphantom{\frac{3}{2}} (2j+1) \mathcal{I}_n^{pj} - j \mathcal{I}_n^{pj-1}- (j+1) \mathcal{I}_n^{pj+1} \right.\right. \nonumber\\ & \quad \left. \left. + \frac{\mathcal{I}_{n}^{p+2j}}{4} + \left(p + \frac{1}{2} \right) \mathcal{I}_n^{pj} + p(p-1) \mathcal{I}_n^{p-2j}- \frac{3}{2}\mathcal{I}_n^{pj}\right] \right\}, \end{align}

where $\mathcal {I}_{n}^{pj}$ is given in (D8) as a finite sum of terms that involve velocity-independent definite integrals of hypergeometric functions with complex argument. The expressions in (3.10) are semianalytical because they still contain a real definite integrals that need to be computed numerically. We remark that generalized plasma dispersion relations can also be used to express the collisionless gyromoments $\hat {N}_{l}^{pj}$, given in (3.10), extending the formulae in Appendix D to the lower half of the complex plane of $\omega$. We use the collisionless expressions of the gyromoments in § 7 to investigate their magnitude in the presence of kinetic effects, such as Landau damping and magnetic gradient drift resonance.

In the case of the sITG branch, the velocity integrals containing the resonant term, $1 /(\omega - z_\parallel s_\parallel )$ (being $\omega _B = 0$), can be performed without using (3.4). In fact, in this case, the collisionless expression of the gyromoments becomes

(3.11)\begin{align} \frac{N^{pj}}{\phi}& ={-}\frac{q}{\tau} \mathcal{K}_{j} \delta_{p}^{0} + \frac{1}{\sqrt{2 \tau}}\frac{k_\perp}{k_\parallel} \frac{1 }{\sqrt{2^{p} p!}} \left( \mathcal{K}_{j} ({-}1)^{p} Z^{(p)} (\xi) \right. \nonumber\\ & \quad \left. + \eta \left[\!\mathcal{K}_{j} \left( \frac{({-}1)^{p+2}}{4} Z ^{(p+2)}(\xi) + (p + 2j) ({-}1)^{p} Z ^{(p)}(\xi) + p(p-1) ({-}1)^{p-2} Z ^{(p-2)}(\xi)\!\right) \right. \right.\nonumber\\ & \quad \left. \left. - \left[ j \mathcal{K}_{j-1} + (j+1) \mathcal{K}_{j+1} \right] ({-}1)^{p} Z ^{(p)}(\xi) \right) \right] - \frac{q}{\tau} \frac{ \xi \mathcal{K}_{j}}{\sqrt{2^{p} p!}} ({-}1)^{p} Z^{(p)}(\xi), \end{align}

where $Z^{(p)} (\xi )= {\rm d}^{p} Z(\xi )/ {\rm d} \xi ^{p}$ is the $p$th-order derivative of the plasma dispersion relation, which can be defined in terms of the Hermite polynomials, $H_p$, by $Z^{(p)} (\xi ) = (-1)^{p} \int _{-\infty }^{\infty } \,{\rm d} s_\parallel H_p(s_\parallel ) {\rm e}^{- s_\parallel ^{2}}/(s_\parallel - \xi )/\sqrt {{\rm \pi} }$.

As an aside, we also note that (3.8) can be related to the kinetic response of the monomial velocity-space moments, $M_{j,k} = v_T^{j + 2k}\int \,{\rm d} \boldsymbol {v} {g}s_\parallel ^{j} x^{k} / 2^{k}$, used in Beer & Hammett (Reference Beer and Hammett1996) to derive general closure expressions for toroidal gyrofluid equations, in the $k_\perp = 0$ limit, such as

(3.12)\begin{equation} N^{pj} = \sum_{p_1=0}^{\lfloor p/2 \rfloor} \sum_{j_1=0}^{j} \frac{({-}1)^{p_1} 2^{p-2p_1}}{p_1!(p-2p_1)!} \frac{({-}1)^{j_1} j !}{(j - j_1)!(j_1!)^{2}} \frac{2^{j_1}}{v_T^{2j_1 + p - 2p_1}} \frac{M_{p-2p_1,j_1}}{\sqrt{2^{p} p!}}. \end{equation}

Equation (3.12) provides a direct link between the gyromoment hierarchy equation described here and the moment equations derived in Beer & Hammett (Reference Beer and Hammett1996) and associated closure expressions.

3.3. gyromoment dispersion relation and infinite gyromoment limit

The ITG dispersion relation expressed in terms of the gyromoments $N^{pj}$ (with the closure up to $(p,j) \leq (P,J)$) is obtained from the truncated GK quasineutrality condition (2.19), yielding

(3.13)\begin{equation} 1 + \frac{q^{2}}{\tau}\left( 1 - \sum_{j=0}^{J} \mathcal{K}_{j}^{2} \right) - \sum_{j=0}^{J} \mathcal{K}_{j} \sum_{l=1}^{3 }\hat{N}^{0j}_l = 0, \end{equation}

where $\hat {N}_l^{0j}$ is given by (3.10) with $p=0$.

Focusing on the sITG branch, we now demonstrate that, in the $P \to \infty$ and $J \to \infty$ limits, the ITG gyromoment dispersion relation, (3.13), is equivalent to the GK dispersion relation given in (3.7). In order to solve the gyromoment dispersion relation (3.13), we use the expressions for the collisionless gyromoments, $N^{pj} / \phi$ given in (3.11) in the sITG case. Setting $p =0$ in (3.11) and plugging the resulting expression for $N^{p0} / \phi$ into (3.13) yields

(3.14)\begin{align} & 1 + \frac{1}{\tau} + \frac{1}{\tau} \xi Z(\xi) \hat{\varGamma_0}(a) - \frac{k_\perp}{k_\parallel} \frac{1}{\sqrt{2 \tau}} \left[\vphantom{\frac{\hat{\varGamma_0}(a,J)}{2}} \hat{\varGamma_0}(a,J) Z(\xi) + \eta\left(\vphantom{\frac{\hat{\varGamma_0}(a,J)}{2}}\hat{\varGamma_0}(a,J) \xi \left( 1 + \xi Z(\xi)\right) \right. \right. \nonumber\\ & \quad \left. \left. - \frac{\hat{\varGamma_0}(a,J)}{2}Z(\xi) +Z(\xi) \hat{\varPi}_1(a,J) \right)\right] =0, \end{align}

where we introduce the quantities

(3.15a)\begin{gather} \hat{\varGamma_0}(a,J)= \sum_{j=0}^{J} \mathcal{K}_{j}^{2}, \end{gather}

(3.15b)\begin{gather}\hat{\varPi}_1(a,J) = \sum_{j=0}^{J} \mathcal{K}_{j} \left(2j \mathcal{K}_{j} - j \mathcal{K}_{j-1} -(j+1)\mathcal{K}_{j+1} \right), \end{gather}

which are associated with FLR effects. We remark that the sITG dispersion relation, given in (3.14) obtained using (3.11) and (3.13), agrees with the one found in Mandell et al. (Reference Mandell, Dorland and Landreman2018). By comparing (3.14) with (3.7), one observes that they are equivalent in the infinite moment limits provided that $\hat {\varGamma _0}(a,\infty ) = \varGamma _0$ and $\hat {\varPi }_1(a,\infty ) = a (\varGamma _1 - \varGamma _0)$. This can be verified explicitly by considering the $J \to \infty$ limit in (3.15) and using the definition of the kernel function $\mathcal {K}_{j}$ in (2.13). In fact, we derive

(3.16)\begin{align} \hat{\varGamma_0}(a,\infty) & = \sum_{j=0}^{\infty} \frac{1}{(j!)^{2}} \left( \frac{b}{2}\right)^{4j} {\rm e}^{- b^{2} /2} \nonumber\\ & = {\rm e}^{- a} {\rm I}_0\left( a\right) = \varGamma_0(a) \end{align}

and

(3.17)\begin{align} \hat{\varPi}_1(a,\infty) & = 2 {\rm e}^{ - b^{2} /2}\sum_{j=0}^{\infty} \left[ \frac{1 }{j!(j+1)!} \left( \frac{b}{2} \right)^{4(j+1)} - \frac{1}{(j!)^{2}} \left( \frac{b}{2}\right)^{4j + 2}\right] \nonumber\\ & = a {\rm e}^{- a} \left( {\rm I}_1(a) - {\rm I}_0(a)\right) = a \left( \varGamma_1(a)- \varGamma_0(a)\right). \end{align}

Figure 1 displays the quantities, $\hat {\varGamma _0}(a,J)$ and $\hat {\varPi }_1(a,J)$ defined in (3.15), for different values of $J$, as well as their $J \to \infty$ limits. It can be observed that $\hat {\varGamma _0}(a,J)$ and $\hat {\varPi }_1(a,J)$ converge to $\varGamma _0$ and $a(\varGamma _1 - \varGamma _0)$ when $J\to \infty$. We notice that convergence is faster for low values of $a = b^{2}/2$, i.e. at long wavelength. This is in agreement with the fact that the truncation of the sums in (3.16) and (3.17) to include only $j \leq J$ terms yields an error $O( b^{4J})$ (Mandell et al. Reference Mandell, Dorland and Landreman2018).

Figure 1.

Plots of $\hat {\varGamma _0}(a,J)$ (a) and $\hat {\varPi }_1(a,J)$ (b) as a function of $a = b^{2}/2$ for increasing values of $J$. The solid black lines represent their asymptotic limits, $\varGamma _0$ and $- a (\varGamma _1 - \varGamma _0)$, respectively.

The impact on the ITG mode due to approximating $\hat {\varGamma _0}(a,\infty )$ and $\hat {\varPi }_1(a,\infty )$ with $\hat {\varGamma _0}(a,J)$ and $\hat {\varPi }_1(a,J)$ can be illustrated by solving the ITG dispersion relation, (3.14), for the mode complex frequency $\omega = \omega _r + {\rm i} \gamma$, as a function of $J$, and comparing its prediction with the one from the GK dispersion relation, given in (3.7). Figure 2 shows the ITG growth rate $\gamma$ and the corresponding real frequency $\omega _r$ solution of the gyromoment sITG dispersion relation given in (3.14), plotted as a function of the perpendicular wavenumber $k_\perp$ for increasing values of $J$. For comparison, the collisionless GK solution, obtained from (3.7), is shown. Both $\gamma$ and $\omega _r$, obtained by the dispersion relation expressed in terms of gyromoments, (3.14), approach the one predicted by (3.7) as $J \to \infty$. Convergence is achieved with small $J$ at long wavelengths (e.g. the growth rate peak at $k_\perp \simeq 0.6$ is well reproduced with $J \sim 5$). On the other hand, for small perpendicular wavelengths and as a rule of thumb, $J \gtrsim k_\perp ^{2}$ must be retained in order to retrieve accurately the GK collisionless prediction (e.g. the $k_\perp \simeq 5$ case is well reproduced with $J \gtrsim 25$). We remark that, if $J$ is not sufficiently large, the truncation of the quantities, used in (3.15), can yield significant deviations in the growth rate $\gamma$ and frequency $\omega _r$. The presence of collisions is expected to affect the convergence estimate reducing $J$ even for short wavelength modes, as numerically demonstrated in § 8.

Figure 2.

Collisionless sITG growth rate, $\gamma$ (a), and frequency, $\omega _r$ (b), as a function of the perpendicular wavenumber $k_\perp$ for different values of $J$. The coloured lines are the solution of the gyromoment hierarchy dispersion relation, given in (3.14), while the black lines are solutions of (3.7), i.e. the $J = \infty$ solution. Here, $\eta =3$ and $k_\parallel = 0.1$.

4.1. 6GM and 4GM reduced models

We start by deriving the 6GM model. We introduce the lowest-order normalized fluid gyromoments, i.e. the gyrocentre density $N = N^{00}$, the parallel velocity $V = N^{10}/\sqrt {2}$, the parallel and perpendicular temperatures, $T_\parallel = \sqrt {2} N^{20} + N$ and $T_\perp = N - N^{01}$ and, finally, the parallel and perpendicular heat fluxes, associated with the deviations of ${g}$ from the Maxwellian distribution function, $Q_\parallel = N^{30}$ and $Q_\perp = N^{11}$. The evolution equations of these gyromoments are obtained by setting $(p,j) = (0,0)$, $(1,0)$, $(2,0)$, $(0,1)$, $(3,0)$ and $(1,1)$ in (2.14), respectively, and by neglecting all higher-order gyromoments. This yields

(4.1a)\begin{align} & \frac{\partial}{\partial t } N + {\rm i} k_\parallel \sqrt{2 \tau} V + {\rm i} \tau \omega_{B} \left( T_\parallel{-} T_\perp \right) + {\rm i} \varGamma_1 \phi = 0 \end{align}

(4.1b)\begin{align} & \sqrt{2} \frac{\partial}{\partial t }V + \frac{{\rm i} k_\parallel}{\sqrt{\tau}} \mathcal{K}_{0} \phi + {\rm i} k_\parallel \sqrt{\tau} T_\parallel{+} {\rm i} \tau \omega_B \left( 4 \sqrt{2} V- Q_\perp{+} \sqrt{6} Q_\parallel \right)= 0,\end{align}

(4.1c)\begin{align} & \frac{\partial}{\partial t } T_\parallel{+} {\rm i} k_\parallel \sqrt{6\tau} Q_\parallel{+} {\rm i} \tau \omega_B \left( 5 T_\parallel{-} 4 N + T_\perp \right) \nonumber\\ & \qquad + {\rm i} (\varGamma_2 - \varGamma_1) \phi ={-} \nu \alpha \left( T_\parallel{-} T_\perp \right),\end{align}

(4.1d)\begin{align} & \frac{\partial}{\partial t }T_\perp{+} {\rm i} k_\parallel \sqrt{\tau} ( \sqrt{2 } V+Q_\perp ) - {\rm i} \tau \omega_B \left( 3 N - T_\parallel{-}3 T_\perp \right) \nonumber\\ & \qquad - {\rm i} (\varGamma_{3}- \varGamma_1) \phi = \nu \frac{\alpha}{2}\left( T_\parallel{-} T_\perp\right), \end{align}

(4.1e)\begin{align} & \frac{\partial}{\partial t } Q_\parallel{+} {\rm i} k_\parallel \sqrt{\frac{3 \tau}{2}} (T_\parallel{-} N) + {\rm i} \tau \omega_B \left( 2 \sqrt{3} V + 8Q_\parallel\right)={-} \nu \left( \frac{8}{5}\sqrt{\frac{2}{\rm \pi}} Q_\parallel{+} \frac{8}{5 \sqrt{3 {\rm \pi}}} Q_\perp \right), \end{align}

(4.1f)\begin{align} & \frac{\partial}{\partial t }Q_\perp{+} \frac{{\rm i} k_\parallel }{\sqrt{\tau}} \mathcal{K}_{1} \phi + {\rm i} k_\parallel \sqrt{\tau} ( N - T_\perp)\nonumber\\ & \qquad - {\rm i} \tau \omega_B \left( \sqrt{2} V - 6 Q_\perp \right)={-} \nu \left( \frac{28}{15}\sqrt{\frac{2}{\rm \pi}} Q_\perp{+} \frac{8}{5 \sqrt{3 {\rm \pi}}} Q_\parallel \right), \end{align}

where we introduce the numerical coefficients $\alpha = 16/15 \sqrt {2 / {\rm \pi}}$. We remark that the collisional terms in (4.1) are obtained by using the lowest-order gyromoments of the DK Coulomb collision operator (Frei et al. Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021) and the energy constraints expressed in (2.24a–c). The quantities appearing in the evolution equations, (4.1), are defined by

(4.2a)\begin{gather} \varGamma_1 = \mathcal{K}_{0} \left( 2 \omega_B - \omega_* \right) + \mathcal{K}_{1} \left( \omega_* \eta - \omega_B\right) , \end{gather}

(4.2b)\begin{gather}\varGamma_2 = \left( 2 \omega_B- \omega_* \eta \right) \mathcal{K}_{0}, \end{gather}

(4.2c)\begin{gather}\varGamma_3 = \mathcal{K}_{0} \left( \omega_* \eta - \omega_B \right) + \mathcal{K}_{1} \left( 4 \omega_B- \omega_* ( 1 + 2 \eta)\right) + 2 \mathcal{K}_{2} \left( \eta \omega_* - \omega_B \right). \end{gather}

Additionally, the electrostatic potential, $\phi$, is determined by the self-consistent quasineutrality condition obtained from (2.18), i.e.

(4.3)\begin{equation} \left[ 1 + \frac{q^{2}}{\tau} \left( 1 - \sum_{j=0}^{1} \mathcal{K}_{j}^{2} \right) \right]\phi = q(\mathcal{K}_{0} + \mathcal{K}_{1} ) N - \mathcal{K}_{1}T_\perp, \end{equation}

where the infinite sum appearing in the polarization term in (2.18) is truncated to $j = 1$ consistently with the fact that the FLR terms are represented up to $\mathcal {K}_{1}$ in the right-hand side of the same equation. The system in (4.1) with (4.3) constitutes a closed set of fluid-like equations that we refer to as the 6GM model.

To derive the 4GM model, further reducing the number of gyromoments appearing in the 6GM model, we apply the Chapman–Enskog asymptotic closure scheme (Chapman & Cowling Reference Chapman and Cowling1941; Jorge et al. Reference Jorge, Ricci and Loureiro2017) to the 6GM model. We introduce the dimensionless small parameter $\epsilon \sim \lambda _{{\rm mfp}} k_\parallel \sim \sqrt {\tau } k_\parallel / \nu \ll 1$, which quantifies the importance of the non-Maxwellian part of ${g}$. Thus, we expect that the Chapman–Enskog closure and the reduced models derived in this section to be valid when typically $\nu \gg \sqrt {\tau } k_\parallel$. The small parameter $\epsilon$ allows us to express the heat fluxes $Q_\parallel$ and $Q_\perp$ as a function of the lowest-order gyromoments, by applying the ordering $Q_\parallel \sim Q_\perp \sim \epsilon N$ (with $T_\parallel \sim T_\perp \sim V \sim N$) and $\partial _t \sim \gamma \sim \epsilon \nu$. We neglect the terms proportional to $\omega _B$, which are smaller than the terms proportional to $k_\parallel T_\parallel$ and $k_\parallel T_\perp$ since $\omega _B / k_\parallel \lesssim 1$ in the boundary region. With these assumptions, (4.1c) and (4.1d) can be solved at the leading order in $\epsilon$, yielding

(4.4a)\begin{gather} Q_{{\parallel} } = {\rm i} k_\parallel \frac{\chi_{{\perp}}^{{\parallel}} }{\tau} \mathcal{K}_{1} \phi + {\rm i} k_\parallel \chi_{{\perp}}^{{\parallel}} ( N - T_\perp) - {\rm i} k_\parallel \chi_{{\parallel}}^{{\parallel}} (T_\parallel{-} N ), \end{gather}

(4.4b)\begin{gather}Q_\perp{=}{-} {\rm i} k_\parallel \frac{\chi_\perp^{{\perp}} }{\tau} \mathcal{K}_{1} \phi - {\rm i} k_\parallel \chi_\perp^{{\perp}} ( N - T_\perp) + {\rm i} k_\parallel \chi^{{\perp}}_{{\parallel}} (T_\parallel{-} N ), \end{gather}

where we introduce the thermal conductivities,

(4.5a–d)\begin{align} \chi_{{\perp}}^{{\parallel}} = \frac{5}{16}\sqrt{\frac{\rm \pi}{3}} \frac{\sqrt{\tau}}{\nu}, \quad \chi_{{\parallel}}^{{\parallel}} = \frac{35}{32} \sqrt{\frac{2 {\rm \pi}}{3}}\frac{\sqrt{\tau}}{\nu}, \quad \chi_\perp^{{\perp}} = \frac{5\sqrt{2{\rm \pi}}} {16}\frac{\sqrt{\tau}}{\nu}, \quad \chi_\parallel^{{\perp}} = \frac{5\sqrt{\rm \pi}}{16} \frac{\sqrt{\tau}}{\nu}. \end{align}

We remark that $\chi _{\parallel }^{\parallel } > \chi _{\perp }^{\parallel }$ and that $\chi _\perp ^{\perp } > \chi _\parallel ^{\perp }$. Equation (4.1), with the closure relations expressed in (4.4), yields the 4GM for the lowest-order gyromoments fluid quantities $N$, $V$, $T_\parallel$ and $T_\perp$ and provides the rigorous $\nu \gg \sqrt {\tau } k_\parallel$ asymptotic limit of the gyromoment hierarchy. We remark that an analytical procedure to close the gyromoment hierarchy similar to the one considered here for the ITG mode, can be applied in the more general cases that include variation of equilibrium quantities along the magnetic field line and electron dynamics.

To study the limits of validity of the 6GM and 4GM models, we compare in figure 3 the estimate of the ITG growth rate $\gamma$ and real frequency $\omega _r$ with the results of the gyromoment hierarchy using the DK Coulomb collision operator (Frei et al. Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021) and the GK collisionless dispersion relation given in (3.3). To compute $\gamma$ and $\omega _r$ from the gyromoment hierarchy equation (see (2.14)) and from the 6GM and 4GM models (see (4.1) and (4.4)), an initial-value calculation as well as a direct eigenvalue solver are used, and their results are compared for consistency. The same numerical methods are used to evaluate all the results reported in the present paper. For verification purposes, the results of GENE simulations using the same operator (Jenko et al. Reference Jenko, Dorland, Kotschenreuther and Rogers2000) are also shown. Here, we consider $(P,J) = (18,6)$ with a closure by truncation (convergence studies are reported in § 8) and $k_\perp = 0.5$, which corresponds approximately to the fastest growing ITG mode. The results are shown as a function of the collisionality $\nu$, for two different normalized temperature gradients, $\eta =3$ and $\eta =5$. First, an excellent agreement is observed between the gyromoment approach and the GENE simulations at arbitrary collisionality $\nu$, using the same DK Coulomb collision operator, confirming the preliminary study reported in Frei et al. (Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021). Second, both GENE and the gyromoment ITG growth rates and real frequencies agree well with the collisionless limit (i.e. $\nu \ll 1$) and with the high-collisional predictions (i.e. $\nu \gg \sqrt {\tau } k_\parallel \sim 0.1$), obtained from the 6GM and 4GM models. Overall, the 6GM provides a better approximation to the low-collisionality solution than the 4GM, while they both agree in the high-collisional limits. It is remarkable that the 6GM retrieves surprisingly well the ITG growth rate $\gamma$ when $\eta =5$, but fails to predict correctly the mode frequency $\omega _r$. Finally, we remark that the 4GM yields an unphysical $\omega _r < 0$, when the collisionality is below $\nu \lesssim 10^{-2}$ with $\eta =3$. This is mainly due to the fact that, in the low-collisionality limit, the $1/\nu$ dependence of the thermal conductivities given in (4.5a–d) produces arbitrarily large heat fluxes. The good agreement observed in figure 3 illustrates the multi-fidelity nature of the gyromoment approach that allows the derivation of reduced models able to address both the collisionless and high-collisional limits of the GK equation, an attractive feature in the assessment of the plasma dynamics in the boundary region.

Figure 3.

The sITG growth rate $\gamma$ (a) and real frequency $\omega _r$ (b) as a function of collisionality $\nu$ obtained using the gyromoment approach with the DK Coulomb collision operator (solid blue line), GENE with the same operator (red markers), the collisionless GK dispersion relation (dotted black) and the 4GM and 6GM high-collisional limits (red dashed–dotted and red dotted lines, respectively). The solution of the dispersion relation given by (4.6), derived in the case of $Q_{\parallel, \perp } = 0$, is also plotted for comparison by the solid thin black lines. Here, we consider $k_\parallel = 0.1$, $k_\perp = 0.5$ and $\tau = 1$.

4.2. High-collisional dispersion relation

We now derive an algebraic dispersion relation using the 4GM equations and discuss the limits of the slab and toroidal branch of the ITG mode. As collisional effects enter through the parallel and perpendicular thermal conductivities such that $\chi ^{\parallel }_{\perp } \sim \chi ^{\parallel }_{\parallel } \sim \chi ^{\perp }_{\parallel } \sim \chi ^{\perp }_{\perp } \sim 1/\nu$ (see (4.5a–d)), we assume that the heat fluxes become negligible when $\nu \gg k_\parallel$, i.e. $Q_{\parallel } \simeq 0$ and $Q_\perp \simeq 0$ (see (4.4)). Hence, a dispersion relation can be derived from the 4GM equations, (4.1), yielding

(4.6)\begin{equation} a \omega^{4} + b \omega^{3} + c \omega^{2} + d \omega + e = 0, \end{equation}

where

(4.7a)\begin{align} a & = \varGamma_\phi , \end{align}

(4.7b)\begin{align} b & = \frac{3 }{2} {\rm i} \alpha \nu \varGamma_\phi - \varGamma_1 \mathcal{K}_{0} - \varGamma_3 \mathcal{K}_{1}- 16 \varGamma_\phi \tau \omega_{B}, \end{align}

(4.7c)\begin{align} c & = \frac{ {\rm i} \alpha \nu}{2} \left( \mathcal{K}_{1}\left( \varGamma_2 - 2 \varGamma_3\right) - 32 \tau \varGamma_\phi \omega_B \varGamma_1 \mathcal{K}_{0} \right) \nonumber\\ & \quad - 3 k_\parallel^{2} \tau \varGamma_\phi - k_\parallel^{2} \mathcal{K}_{0}^{2} + \tau \omega_B \left( \mathcal{K}_{0} \left( 14 \varGamma_1 - \varGamma_2 + \varGamma_3 \right) + \mathcal{K}_{1} \left(\varGamma_1 + 12 \varGamma_3 \right) \right), \end{align}

(4.7d)\begin{align} d& = {\rm i} \alpha \nu \left[ k_\parallel^{2} \left( \mathcal{K}_{0} \mathcal{K}_{1} - \frac{5}{2} \tau \varGamma_\phi - \frac{3}{2} \mathcal{K}_{0}^{2} \right)\right.\nonumber\\ & \quad \left.+ \tau \omega_B \left( \varGamma_1 \left(13 \mathcal{K}_{0} + 2 \mathcal{K}_{1} \right) - (\varGamma_2 - 2 \varGamma_3)(\mathcal{K}_{0} + 3 \mathcal{K}_{1}) \right)\vphantom{\left( \mathcal{K}_{0} \mathcal{K}_{1} - \frac{5}{2} \tau \varGamma_\phi - \frac{3}{2} \mathcal{K}_{0}^{2} \right)} \right] \nonumber\\ & \quad + \tau k_\parallel^{2} \left[ \omega _{B } \left(18 \tau \varGamma _{\phi }+\mathcal{K}_{0} \left(8 \mathcal{K}_{0} +\mathcal{K}_{1} \right)\right)+ \left(2 \varGamma _1-\varGamma _2\right) \mathcal{K}_{0} +3 \varGamma _3 \mathcal{K}_{1} \right], \end{align}

(4.7e)\begin{align} e & = {\rm i} \alpha \nu \left[ 5 k_\parallel^{2} \tau \omega_B \left( \tau \varGamma _{\phi } + \mathcal{K}_{0}^{2} \right) +k_\parallel^{2} \tau \left( \varGamma _1- \frac{\varGamma _2}{2}+ \varGamma _3\right) \left(\mathcal{K}_{0}+ \mathcal{K}_{1}\right] \right) \nonumber \\ &+ k_\parallel^{2} \tau^{2} \omega_B \left(\mathcal{K}_{1} \left(\varGamma _2-6 \varGamma _3 -2 \varGamma _1 \right) -2 \mathcal{K}_{0} \left(4 \varGamma _1-2 \varGamma _2+\varGamma _3\right) \right), \end{align}

being terms proportional to $\omega _B^{2}$ neglected ($\omega _B \ll 1$) and having defined $\varGamma _\phi = 1 + q^{2} (1 - \mathcal {K}_{0}^{2} - \mathcal {K}_{1}^{2}) / \tau$. In (4.6), collisional effects are represented through the terms proportional to ${\rm i}\alpha \nu$. The solution of (4.6) is plotted in figure 3 and approaches the 6GM and 4GM models when $\nu \gg k_\parallel$, such that $Q_\parallel = Q_\perp \simeq 0$.

We now consider the slab and toroidal limits of (4.6) separately in detail. An estimate of the sITG peak can be derived from (4.6) as a function of $k_\parallel$, $\nu$ and $\eta$. Besides neglecting the terms proportional to $\omega _B$ and FLR effects (i.e. imposing $\mathcal {K}_{0} \simeq 1$, $\mathcal {K}_{1} \simeq 0$, $\mathcal {K}_{2} \simeq 0$, such that $\varGamma _\phi \simeq 1$), we assume that the sITG mode is far from the marginal stability limit $\eta \sim \nu \gg 1$. Then, the dispersion relation in (4.6) reduces to

(4.8)\begin{align} \hat{\omega}_*^{4} + \frac{3}{2} {\rm i} \alpha \hat{\nu}_* \hat{\omega}_*^{3} - \hat{k}_{{\parallel} *}^{2} \left( 1 + 3 \tau\right) \hat{\omega}_*^{2} + \hat{k}_{{\parallel} *}^{2} \left[ \tau - \frac{3}{2} {\rm i} \alpha \hat{\nu}_* \left( 1 + \frac{5}{2} \tau\right)\right] \hat{\omega}_* + \frac{3}{2} {\rm i} \alpha \hat{k}_{{\parallel} *} \hat{\nu}_* \tau =0, \end{align}

where we introduce $\hat {\omega }_* = \omega / (\eta \omega _*)$, $\hat {\nu }_* = \nu / (\eta \omega _*)$ and $\hat {k}_{\parallel *} = \hat {k}_\parallel / (\eta \omega _*)$. Given the normalization of (4.8), the maximum growth rate is expected to be proportional to $\gamma \simeq g(\tau, \hat {\nu }_*) \eta \omega _*$ and occurs at $k_\parallel \simeq f(\tau, \hat {\nu }_*) \eta \omega _*$. In figure 4, we show the maximum ITG growth rate, $\hat {\gamma }_* = \gamma / (\eta \omega _*)$ (i.e. the imaginary part of $\hat {\omega }_*$) obtained by solving (4.8) as a function of $\hat {k}_{\parallel *}$ and $\hat {\nu }_*$. The effect of increasing $\hat {\nu }_*$ is primarily to shift the growth rate peak to larger values of $\hat {k}_\parallel ^{*}$ for all values of $\tau$. Additionally, while $g(\tau,\nu )$ increases with $\tau$ for all $\nu$, it is found that $f(\tau, \hat {\nu }_*)$ is an increasing function of $\tau$ at large values of $\hat {\nu }_*$.

Figure 4.

Estimates of the normalized sITG peak, $\hat {\gamma }_*$, obtained from (4.8), for increasing values of $\hat {\nu }_* = 0.015$ (from (a) to (b)), as a function of $\hat {k}_{\parallel *}$ and the temperature ratio $\tau$. The local maxima of the growth rate is indicated by the dashed black lines. The growth rates $\hat {\gamma }_*$ are normalized to their maximal values.

We remark that the collisionality dependence of the functions $f$ and $g$ is not present in previous fluid ITG models based on the drift-reduced fluid equations (Mosetto et al. Reference Mosetto, Halpern, Jolliet, Loizu and Ricci2015). This dependence arises here because the 4GM model (and more generally the gyromoment hierarchy equation in (2.14)) allows for a different evolution of the perpendicular and parallel temperature fluctuations assuming, in general, $T_\parallel \neq T_\perp$. On the other hand, the drift-reduced Braginskii fluid model in, e.g. Zeiler et al. (Reference Zeiler, Drake and Rogers1997), assumes that $T_\parallel = T_\perp$. In fact, the $\nu$ dependence in (4.8) enters in the dispersion relation, given in (4.8), through terms proportional to $\nu \alpha (T_\parallel - T_\perp )$ which arise from the gyromoment $\mathcal {C}^{20}$ and $\mathcal {C}^{01}$ components of the DK Coulomb collision operator.

To investigate the effect of anistotropic temperature fluctuations, we derive the evolution equation of the temperature, $T = (T_\parallel + 2 T_\perp ) /3 - N$. Using (4.1c) and (4.1d) and the continuity equation, (4.1a), this can be expressed as

(4.9)\begin{equation} \frac{\partial }{\partial t} T - {\rm i} k_\parallel \frac{ \sqrt{2 \tau}}{3}V + \frac{{\rm i}}{3} \left( \varGamma_2 - 2 \varGamma_3 - 2 \varGamma_1\right) \phi =0. \end{equation}

From (4.9) with (4.1a) and (4.1b), and considering the limits used to derive (4.8), we obtain the estimate of the maximum sITG growth rate,

(4.10)\begin{equation} \hat{\omega}_*^{3} - \hat{k}_{{\parallel} *}^{2} \left( 1 + \frac{2}{3}\tau \right) \hat{\omega}_* + \hat{k}_{{\parallel}*}^{2} \tau = 0. \end{equation}

From (4.10), one finds that the peak of the sITG growth rate is proportional to $\gamma \simeq g(\tau ) \eta \omega _*$ and occurs at $k_{\parallel } \simeq f(\tau ) \eta \omega _*$. It is found that $g(\tau )$ is an increasing function of $\tau$, while $f(\tau )$ decreases with $\tau$. We remark that (4.10) agrees with Mosetto et al. (Reference Mosetto, Halpern, Jolliet, Loizu and Ricci2015) by replacing the numerical factor $2/3$ by $5/3$, a difference due to the additional degree of freedom assumed in the 4GM model.

We now turn to the toroidal ITG mode, driven unstable by the magnetic drifts and by the presence of a finite temperature gradient. We derive a simple dispersion relation for the toroidal ITG from (4.6) indicative of its properties. Since, in contrast to the sITG, the toroidal ITG subsists at $k_\parallel = 0$, we set $k_\parallel =0$ thus removing the coupling with sound waves, and we adopt the long perpendicular wavelength limit, keeping terms up to $O(k_\perp ^{2})$, such that $\mathcal {K}_{0} \simeq 1 - \tau k_\perp ^{2}/2$, $\mathcal {K}_{1} \simeq \tau k_\perp ^{2}/2$, and $\mathcal {K}_{2} \simeq 0$. Since $\omega _B \ll \omega _*$ in the boundary region, we obtain

(4.11)\begin{equation} \omega^{3} + b_t \omega^{2} + c_t \omega + d_t = 0, \end{equation}

where

(4.12a)\begin{align} b_t & = \frac{3}{2} {\rm i} \alpha \nu + \omega_* - 2 \omega_B \left( 1 + 8 \tau \right) + k_\perp^{2} \left[ 2 \omega_B \left( 1 + \frac{3}{2} \tau \right) - \omega_* \left( 1 + \tau + \eta \tau \right) \right], \end{align}

(4.12b)\begin{align} c_t & = \frac{3}{2 } {\rm i} \alpha \nu \left[ \omega_* - 2 \omega_B - \frac{32}{3} \tau \omega_B + k_\perp^{2} \left( 2 \omega_B \left( 1 + \frac{19}{12} \tau \right) -\omega_* (1 + \tau + \tau \eta) \right) \right] \nonumber\\ & \quad + \omega_B \omega_* \left[ 2 \tau ( \eta -7) + k_\perp^{2} \left(10 \eta \tau \left(\tau - \frac{1}{5}\right) +\tau\left(13 \tau +14\right) \right) \right], \end{align}

(4.12c)\begin{align} d_t & ={\rm i} \alpha \nu \omega _{B } \omega _* \tau \left[ \left(3 \eta -13 \right) + k_\perp^{2} \left(3 \eta \left(2 \tau-1 \right) +11 \tau +13 \right)\right]. \end{align}

An estimate of the value of $k_\perp$ that yields the largest growing mode can be derived from (4.11) by neglecting terms proportional to $\nu$, for simplicity. Then, the dispersion relation in (4.11) becomes a second-order equation for $\omega$ presenting the largest solution $\omega$ when the coefficient $b_t$ is minimized. Hence, by minimizing $b_t$ and considering $\omega _B \ll \omega _*$, the growth rate is maximized when $k_\perp ^{2} \simeq 1 / [ 1 + \tau (1 + \eta )]$. Figure 5 shows the toroidal ITG growth rate obtained by solving (4.11), as a function of the perpendicular wavenumber $k_\perp$ and temperature ratio $\tau$. We observe that the decrease of the perpendicular wavenumber of the fastest growing mode with $\tau$ is present also at finite collisionality, $\nu$. We also remark that collisions tend to destabilize long wavelength modes.

Figure 5.

Toroidal ITG growth rate as a function of the perpendicular wavenumber, $k_\perp$ and the temperature ratio $\tau$ obtained from (4.11), for a low (a) and a high (b) value of collisionality. The local maxima of the growth rate are indicated by the dashed black lines. The growth rates are normalized to the maximum value. Here, $\eta = 7$ and $R_B = 0.1$.

6.1. Comparisons between the GK coulomb and collision operators models

We focus on the differences between the GK Coulomb collision operator, the GK/DK Sugama collision operator (Sugama et al. Reference Sugama, Watanabe and Nunami2009), the GK/DK momentum-conserving pitch-angle scattering (pitch) operator (Helander & Sigmar Reference Helander and Sigmar2002), the zeroth-order DK HSC collision operator (Hirshman & Sigmar Reference Hirshman and Sigmar1976a) and the GK/DK Dougherty collision operator (Dougherty Reference Dougherty1964). The gyromoment expansion of the Sugama collision operator is detailed in Frei et al. (Reference Frei, Ball, Hoffmann, Jorge, Ricci and Stenger2021), while the gyromoment expansions of the other collision operators are reported in appendices A–C.

Figure 8 shows the slab (figure 8a–c) and the toroidal (figure 8d–f) ITG linear growth rate at a relative temperature gradient strength of $\eta =3$ from low (figure 8a,d) to high (figure 8c, f) collisionality as a function of the perpendicular wavenumber $k_\perp$ using the GK collision operators. The collisionless and high-collisional limits are plotted for comparison. It is remarkable that the ITG growth rate is sensitive to both the collision operator models and to the presence of FLR related terms. In fact, the GK Dougherty collision operator produces the results that mostly differ with respect to the other GK operator models, yielding the strongest ITG damping. The effect of energy diffusion can be observed by comparing the GK pitch-angle and GK Sugama collision operators. While the former systematically overestimates the ITG growth rate (compared with the GK Coulomb operator) due to the absence of energy diffusion, the latter damps stronger the ITG mode with growth rates smaller (but closer) than the GK Coulomb. Finally, we remark that the GK Coulomb operator predictions approach the ones of the 6GM and 4GM models in the limit of long perpendicular wavelengths for all collisionalities, since the FLR collisional damping becomes negligible when $k_\perp \ll 1$.

Figure 8.

Comparisons between GK (solid lines) collision operator models in the case of the slab (a–c) and toroidal (d–f) ITG growth rate as a function of the perpendicular wavenumber $k_\perp$, from the low (a,d) to high (c, f) collisionality regime. The collisionless (dashed black) and 6GM and 4GM (red dotted and dashed–dotted lines, respectively) growth rates are plotted for comparisons. Here, $\eta = 3$, $k_\parallel = 0.1$.

By neglecting the FLR terms, the DK collision operators lead to a larger ITG growth rate than the GK operators as shown in figure 9. Contrary to the latter, the DK collision operators can eventually destabilize a SWITG peaking near $k_\perp \simeq 1.5$. The presence of this SWITG can be inferred from figure 9 from the increase of the ITG growth rate occurring for $k_\perp \gtrsim 1$ when $\nu \gtrsim 0.1$ (see § 6.2). This behaviour is particularly pronounced in the pitch angle and HSC collision operators at short wavelengths, and it is related to the absence of energy diffusion in these operators (Ricci, Rogers & Dorland Reference Ricci, Rogers and Dorland2006; Barnes et al. Reference Barnes, Abel, Dorland, Ernst, Hammett, Ricci, Rogers, Schekochihin and Tatsuno2009), which is retained in the Coulomb, Sugama and Dougherty collision operators. We also notice that the absence of energy diffusion in the pitch angle and HSC collision operators affects the high-collisional limit. While the deviations between the DK Coulomb, Sugama and Dougherty collision operators are reduced as the collisionality increases and, ultimately, becomes negligible when $\nu \gg 1$, converging to the 6GM and 4GM solutions, a difference subsists in the pitch-angle and HSC collision operators. Finally, among the DK collision operators considered, the DK Sugama approaches better the DK Coulomb collision operator.

Figure 9.

Same as figure 8 using the DK collision operator models.

The deviations of the collision operator models with respect to the GK Coulomb operator can be quantified by computing their signed relative difference of the growth rate, $\sigma _C = (\gamma _C - \gamma ) / \gamma _C$ where $\gamma _C$ is the growth rate using the GK Coulomb and $\gamma$ is the growth rate obtained using the other collision operator models considered in figures 8 and 9. The results are shown in figure 10 where we plot $\sigma _C$ as a function of collisionality, $\nu$, and of temperature gradient strength, $\eta$, for the toroidal branch ($R_B = 0.1$). The GK Sugama collision operator provides the results that best approach the GK Coulomb collision operator. Among the other GK operators considered, the largest deviation is produced by the GK Dougherty where the relative difference is larger than $30\,\%$ when $\nu \gtrsim 0.5$. Surprisingly, while the GK pitch-angle operator systematically overestimates the ITG growth rate (as observed from $\sigma _C< 0$), it yields an error smaller than (or of the order of) $30\,\%$ in almost all the parameter space considered. In particular, the relative signed difference of the GK pitch-angle collision operator is similar in absolute value to the one of the GK Sugama at high $\eta$ and all collisionalities, but with opposite sign. We remark also that the negative sign of $\sigma _C$ for the DK collision operators is the result of ignoring the FLR collisional damping. For all the considered operators, the relative deviation with GK Coulomb operator decreases at steeper temperature gradients while it increases with collisionality.

Figure 10.

Signed relative difference, $\sigma _C$ (defined in the main text), of the growth rate obtained using the collision operator models, used in figures 8 and 9, with respect to the GK Coulomb operator as a function of collisionality $\nu$ and temperature gradient strength $\eta$. The colourbars are saturated at a maximal relative deviation of $\sigma _C = \pm 0.4$. Here, $k_\parallel = 0.1$, $k_\perp = 0.5$ with $R_B = 0.1$ are considered, corresponding approximately to the peak ITG growth rate.

The gyromoment approach allows for the investigation of the ITG eigenvalue spectrum at finite collisionality. An example is shown in figure 11 for the toroidal case when $\nu = 0.1$ using the DK (figure 11a) and GK (figure 11b) pitch angle, Dougherty, Sugama and Coulomb collision operators near to the peak growth rate occurring at $k_\perp = 0.6$. First, focusing on the spectrum of the DK collision operators, we note that, while the DK Sugama reproduces qualitatively well the eigenvalues of the DK Coulomb, the DK Dougherty collision operator displays an eigenvalue spectrum with strongly damped subdominant modes characterized by $\gamma < 0$. Similar differences between the DK Dougherty and the DK Coulomb collision operators in the eigenvalue spectrum is reported in the study of electron plasma waves at arbitrary collisionality by Jorge et al. (Reference Jorge, Ricci, Brunner, Gamba, Konovets, Loureiro, Perrone and Teixeira2019b). In addition, despite the absence of energy diffusion, the DK pitch-angle operator features similar eigenvalues than the DK Coulomb. Focusing now on the spectrum of the GK collision operators, we remark that a large difference exists between the GK Dougherty operator and the other GK collision models. It is noticeable that the GK Coulomb collision operator produces subdominant modes that are less damped than the ones predicted by the GK Sugama and pitch-angle operators but, nevertheless, yields a similar eigenvalue spectrum. We remark that the eigenvalues have all a positive real frequency, $\omega _r > 0$, corresponding to the ion diamagnetic direction.

Figure 11.

Toroidal ITG eigenvalue spectrum when $\nu = 0.1$ and $k_\perp = 0.6$ using the DK (a) and GK (b) collision operator models. Here, $R_B = 0.1$ and $\eta = 3$.

6.2. Destabilization of the SWITG mode

An initial analysis of the effects of FLR terms in the collision operators can be carried out by observing, first, the discrepancies between the GK Coulomb and the 6GM and 4GM stability boundaries in the $\nu =1$ case (see figure 6) and, second, from the comparison between GK and DK collision operators (see figures 8 and 9). A clear deviation between GK and DK Coulomb operators occurs at sufficiently steep temperature gradients, $\eta \gtrsim 3$, and intermediate collisionality, i.e. when $\nu \gtrsim 0.1$. For a more detailed analysis, we plot in figure 12 the ITG growth rate evaluated with the DK Coulomb in the slab case for two different values of temperature gradient strength.

Figure 12.

Slab SWITG growth rate (unstable when the DK Coulomb model is used) when $\eta =3$ (a–c) and $\eta =5$ (d–f). The stability boundaries of the gyromoment hierarchy (dotted white line), 6GM and 4GM (red dotted and dot–dashed lines, respectively) are plotted for comparison. The main ITG branch is identified near $k_\perp \simeq 0.5$ and the SWITG near $k_\perp \simeq 2$, appearing as the collisionality increases when $\nu \gtrsim 0.5$. The colourbar is saturated at the maximum of $\gamma$ when $\nu = 0.05$.

The comparison between figures 6 and 12 reveals that a short wavelength branch of the ITG, referred to as the SWITG mode and typically peaking at $k_\perp \sim 2$, is present when $\nu \gtrsim 0.5$ if the DK Coulomb operator is used. We remark that the SWITG is also captured by the 6GM and 4GM models. This mode has been identified in the collisionless limit as a continuous extension of the main branch of the ITG mode in the $k_\perp > 1$ region. In fact, the collisionless ITG growth rate reveals a typical ‘double-humped’ behaviour when plotted as a function of $k_\perp$ (Smolyakov, Yagi & Kishimoto Reference Smolyakov, Yagi and Kishimoto2002; Gao et al. Reference Gao, Sanuki, Itoh and Dong2003, Reference Gao, Sanuki, Itoh and Dong2005). Additionally, figure 12 shows the presence of a stability region that increases with collisionality, isolating the conventional ITG and the SWITG modes. We note that, in the toroidal case, the SWITG is stabilized (or even suppressed) by increasing magnetic gradients, a similar feature observed also in the collisionless limit (Gao et al. Reference Gao, Sanuki, Itoh and Dong2005).

The SWITG mode is also predicted by all the other DK collision operators considered in this work, but it is suppressed by FLR collision damping in the GK collision operators. This questions the applicably of DK collision operator models to correctly predict the turbulent transport level in ITG driven turbulence in the boundary region, since the SWITG can produce an additional (yet spurious) contribution in addition to the main ITG mode peaking at larger $k_\perp$ (Gao et al. Reference Gao, Sanuki, Itoh and Dong2005). However, further investigations are required to go beyond the local approximation used in the present work.

8.1. Collisionality convergence

We first consider the convergence of the ITG linear growth mode as a function of the collisionality, $\nu$, using the GK and DK Coulomb collision operators. The collisionless and high-collisional limits are obtained by solving the collisionless GK dispersion relation, (3.3), and the 6GM and 4GM models (see § 4.1). The results are shown in figure 17(a) for the slab and figure 17(b) for the toroidal ITG modes. Different numbers of gyromoments, $(P,J)$, are used. The collisionless GK solution is retrieved for $(P,J) \simeq (18,6)$ with both the GK (dashed) and DK (solid) Coulomb collision operators. In the slab case, convergence is achieved with $P \lesssim 8$, when $\nu \simeq 0.1$, and with $P \gtrsim 18$, when $\nu \lesssim 10^{-3}$. This is in qualitative agreement with the estimate of $P$ that can be obtained from (7.7). In fact, solving the gyromoment hierarchy up to $P \gtrsim p_\nu$ yields $p_\nu \sim 3$ for $\nu = 0.1$ and $p_\nu \sim 20$ for $\nu = 10^{-3}$. Also consistent with the observations made in § 7 is the fact that only a few numbers of the Laguerre gyromoments are necessary to resolve the sITG, as illustrated by the overlap between the $(10,4)$ and $(10,6)$ lines in figure 17. This is because the sITG mode is mainly driven by the parallel dynamics, resolved by the Hermite part of the gyromoment spectrum (see figure 13), and the relatively small value of $k_\perp$ considered in this case (see figure 18). This remains also true for the toroidal ITG case considered in figure 17 because of the relatively small value of $R_B$ used. In fact, larger values of $R_B$ deteriorates the convergence as discussed below. While the GK Coulomb yields a strong FLR collisional damping as $\nu$ increases, the gyromoment hierarchy with DK Coulomb agrees with the 6GM and 4GM when $\nu \gg 1$. Nevertheless, we notice that, because of finite magnetic drifts (see § 3), agreements with the 6GM and the 4GM models occur at higher collisionality in the toroidal case. In fact, in the slab case, the high-collisional limits are retrieved when $\nu \gtrsim 1$, while for the toroidal case ($R_B = 0.1$) when $\nu \gtrsim 10^{2}$.

Figure 17.

Convergence of the ITG linear growth rate, $\gamma$, as a function of the collisionality $\nu$ for different values of $(P,J)$, using the GK (dashed lines) and DK (solid lines) Coulomb operators, in the case of slab (a) and toroidal ((b) with $R_B = 0.1$) ITG branches. The collisionless limit (dotted black lines) and the high-collisional limits, 6GM and 4GM models (red dotted and dash–dot lines, respectively) are shown for comparison. Similar plots are obtained for the other GK and DK operator models. The parameters are $k_\perp = 0.5$, $k_\parallel = 0.1$, $\eta = 3$.

Figure 18.

Slab ITG growth rate $\gamma$ as a function of $k_\perp$ for different values of $J$, in the low (a), intermediate (b) and high (c) collisionality regimes, with the GK (dashed lines) and DK (solid lines) Coulomb collision operators. Similar plots are obtained for the other GK and DK operator models.

8.2. Perpendicular wavenumber convergence

We now investigate the convergence associated with the FLR effects due to the gyroaverage of the electrostatic potential, i.e. with the number of Laguerre gyromoments. The coupling between the Laguerre gyromoments driven by magnetic gradients are neglected by focusing on the sITG. We scan the growth rate $\gamma$ as a function of the perpendicular wavenumber, $k_\perp$, for different values of $J$ and collisionality $\nu$. The number of the Hermite gyromoment is fixed at $P = 18$. The results are shown in figure 18. We observe that convergence occurs at lower $J$ at high collisionalities. Nevertheless, the value of $J$ necessary for convergence increases with $k_\perp$ at all collisionalities. This is in agreement with the unfavourable scaling of the FLR kernel with $b$, i.e. $\mathcal {K}_{j} \sim (b/2)^{2j}/j!$ (see § 7). We also notice that the ITG mode is strongly damped by the GK Coulomb collision operator that departs rapidly from the DK Coulomb as $k_\perp$ and $\nu$ increases, in agreement when comparing figure 8 with figure 9. The same qualitative observations hold in the toroidal case.

8.3. Magnetic gradient convergence

Finally, we consider the convergence of the gyromoment hierarchy as a function of the normalized magnetic gradient, $R_B$. As discussed in § 7, the presence of finite magnetic drifts broadens significantly the collisionless gyromoment spectrum. However, even at low collisionality, the gyromoment approach converges correctly to the collisionless solution $\gamma _{\text {GK}}$ obtained from (3.3). This is shown in figure 19 where the ratio of the ITG growth rate obtained using the gyromoment hierarchy to $\gamma _{\text {GK}}$ is plotted as a function of $P$ and $J$. It is remarkable that, consistently with § 7, a larger number of gyromoments is needed as $R_B$ increases.

Figure 19.

Ratio between the ITG growth rate $\gamma$ obtained using the gyromoment hierarchy and the collisionless solution $\gamma _{\textrm {GK}}$ as a function of $P$ with $J = 10$ (a) and as a function of $J$ with $P=32$ (b) for two different values of $R_B$. Here, $k_\perp =0.5$, $k_\parallel = 0.1$ and $\nu = 0.001$.

Collisions can compete against the strong kinetic drive of the magnetic drifts by damping higher-order gyromoments (see figure 16). To investigate the convergence of the gyromoment approach at finite $R_B$ in the presence of collisions, we focus on the toroidal ITG with $k_\parallel =0$ and, hence, neglecting the coupling between the Hermite gyromoments associated with the parallel streaming. We scan the toroidal ITG linear growth rate $\gamma$ as a function of $R_B$ and increase the collisionality. The results are reported in figure 20. We observe that, at all collisionalities, the convergence of the gyromoment approach deteriorates as $R_B$ increases. Nevertheless and consistently with § 7, the number of gyromoment to achieve convergence is reduced as the collisionality increases, demonstrating the competition between the kinetic effects driven by the magnetic gradient drifts and the collisional effects. We also remark that the 4GM limit is retrieved at higher collisionality when $R_B$ increases.

Figure 20.

Toroidal ($k_\parallel = 0$) ITG growth rate $\gamma$ as a function of the normalized magnetic gradient, $R_B$, for different $(P,J)$ and increasing collisionality (from (a) to (c)). Here, the GK (dashed lines) and DK (solid lines) Coulomb collision operators are shown, with the collisionless (black dotted) and high-collisional 4GM limit (red dotted–dashed lines).