2. Linear gyrokinetic equation

Here, we concern ourselves with the electrostatic limit of the gyrokinetic equation in a flux tube geometry, where the fluctuations are periodic in the directions perpendicular to the magnetic field, which is given by $\boldsymbol{B} = \boldsymbol{\nabla }\psi \boldsymbol{\cdot} \boldsymbol{\nabla }\alpha$ . In these coordinates, $\psi$ acts as a ‘radial’ coordinate (in the sense that the plasma gradients will be in the coordinate $\psi$ ), and $\alpha$ is binormal to the magnetic field and the gradient direction. The coordinate along the magnetic field lines is denoted by $l$ . Because we wish to test the limits of the optimal modes’ ability to bound the growth of linear instabilities, we must consider a scenario where the linear eigenmodes are well understood.Footnote ¹ To this end, we consider the case of a uniform magnetic geometry (henceforth called the ‘slab’ geometry), in which a single Fourier mode along the magnetic field line can be considered, such that $\partial / \partial l \to i k_\parallel$ .

Moreover, we consider only a single kinetic species, focusing on the familiar limit of kinetic ions with adiabatic electrons (Kadomtsev & Pogutse Reference Kadomtsev, Pogutse and Leontovich1970; Biglari, Diamond & Rosenbluth Reference Biglari, Diamond and Rosenbluth1989; Plunk et al. Reference Plunk, Helander, Xanthopoulos and Connor2014). In this scenario, the gyrokinetic equation becomes

(2.1) \begin{align} \frac {\partial g}{\partial t} + i v_{\|} k_\| g = \frac {e F_{0}}{T_i}\left (\frac {\partial }{\partial t} + i\omega _{*}^T\right )\phi J_{0}. \end{align}

Here, $g = g(\mu , E,\boldsymbol{k}_\perp , k_\|)$ is the (Fourier transformed) non-adiabatic part of the perturbed ion distribution function. The phase space variables are the particle energy $E =m v^2/2$ and the magnetic moment $\mu = m v_\perp ^2/(2B)$ , where $v^2 = v_\perp ^2 + v_\parallel ^2$ , $m = m_i$ is the ion mass, $v_\perp$ and $v_\|$ are the components of the particle velocity in the plane perpendicular to, and parallel to $\boldsymbol{B}$ , respectively. The ions are assumed to have a Maxwellian background distribution $F_{0} = n(\psi )/(v_{T}^3\pi ^{3/2})\exp (-v^2/v_{T}^2)$ with a number density $n(\psi )$ , where the ion temperature $T_i$ enters via the thermal speed $v_{T} = \sqrt {2T_i/m}$ and $T_i = T_i(\psi )$ . The plasma gradients enter via the energy-dependent diamagnetic frequency $\omega _{*}^T = \omega _{*}(1 + \eta [v^2/v_{T}^2 - 3/2])$ where $\omega _{*} = ({k_\alpha T_i}/{e})({\mathrm{d}\ln n}/{\mathrm{d}\psi })$ and $\eta = \mathrm{d} \ln T_i/ \mathrm{d}\ln n$ , with $e$ the ion charge. The Bessel function of the first kind $J_0$ , which arises due to the gyro-average, has the argument $J_{0} = J_0(k_\perp v_\perp /\varOmega )$ where $\varOmega = e B/ m$ . The system is closed by quasi-neutrality

(2.2) \begin{align} \frac {e^2 n}{T_i}(1 + \tau ) \phi = e \int _{-\infty }^{\infty }\mathrm{d} v_\| \int _0^{\infty } \,\mathrm{d} v_\perp v_\perp 2 \pi g J_{0} = e \int _{-\infty }^{\infty } \bar g \mathrm{d}v_\|, \end{align}

where $\phi$ is the Fourier transformed electrostatic potential and $\tau$ is the ion–electron temperature ratio $\tau = T_i/T_e$ . Additionally, we have defined the reduced distribution function $\bar g(v_\|)= 2\pi \int _{0}^{\infty } v_\perp g J_0 \mathrm{d} v _\perp$ .

Because the wave–particle resonance is one-dimensional in phase space in a slab geometry, we may integrate (2.1) over $v_\perp$ to express the gyrokinetic equation in terms of $\bar g$ . This yields

(2.3) \begin{align} \frac {\partial \bar g}{\partial t} + i v_{\|} k_\| \bar g = \frac {e \bar F_{0}}{T_i}\left [G_{\perp 0}\left (\frac {\partial }{\partial t} + i\omega _{*}(1 + \eta (x_\|^2 - 3/2 ))\right ) + i \omega _* \eta G_{\perp 2}\right ]\phi , \end{align}

where $\bar F_0 = {n}/({v_T \sqrt {\pi }})e^{-v_\|^2/ v_T^2}$ and $x_\| = v_\|/v_T$ . The functions $G_{\perp 0}$ and $G_{\perp 2}$ can be written in terms of the familiar $\varGamma _n(b) = I_n(b)e^{-b}$ functions of gyrokinetic theory where $I_n$ is a modified Bessel function and $b = k_\perp ^2 \rho ^2$ with $\rho = v_T/(\varOmega\sqrt2)$ (see Plunk & Helander (Reference Plunk and Helander2023) for specifics)

(2.4) \begin{align} G_{\perp 0}(b) = \varGamma _0(b), \end{align}

(2.5) \begin{align} G_{\perp 2}(b) = \varGamma _0(b) - b[\varGamma _0(b) - \varGamma _1(b)]. \end{align}

The linear eigenmodes of (2.3) have been well studied (Kadomtsev & Pogutse Reference Kadomtsev, Pogutse and Leontovich1970; Plunk et al. Reference Plunk, Helander, Xanthopoulos and Connor2014) allowing clear comparison between the linear growth rates and the bounds derived here.

3. Tightest possible energetic bounds

Before attempting to tailor the optimal mode analysis to bounding the growth of linear instabilities, we will determine whether there is a fundamental limitation of the optimal modes when it comes to bounding linear eigenmode growth. In other words, how tightly can the growth of the linear modes be bounded by the instantaneous growth of an energetic norm? To answer this question, we will leverage techniques that are seldom used in gyrokinetics (Heninger & Morrison Reference Heninger and Morrison2018; Plunk Reference Plunk2013, Reference Plunk2015). We begin by defining $f = \bar g - e \bar F_0 G_{\perp 0} \phi /T_i$ , allowing us to write (2.3) as

(3.1) \begin{align} \frac {\partial f}{\partial t} + i v_\| k_\| f = - i k_\| S(v_\|, k_\|, b)\int f(v_\|') \mathrm{d} v_\|', \end{align}

where we have used quasi-neutrality to close the system in terms of an intergo-differential equation and defined

(3.2) \begin{align} S(v_\|) = \frac {\bar F_0}{n(1 + \tau - G_{\perp 0})}\left [v_\| G_{\perp 0} - \frac {\omega _*}{k_\|}\left (G_{\perp 0}(1 + \eta (x_\|^2 - 3/2)) + \eta \, G_{\perp 2}\right ) \right ], \end{align}

where we have suppressed the dependence on other parameters e.g. $k_\|$ and $k_\perp$ . Equation (3.1) is equivalent to the linear system studied by Plunk (Reference Plunk2013) with the additional inclusion of finite-Larmor-radius effects.

3.1. Case–Van Kampen energy

The linear eigenmodes of this equation satisfy

(3.3) \begin{align} (\omega - v_{\|}k_\parallel )f = k_\parallel S(v_\parallel ) \int f(v_\parallel ') \mathrm{d} v_\parallel ', \end{align}

where $\omega = \omega _r + i \gamma$ is the eigenvalue. We may also set the normalisation of the solutions to

(3.4) \begin{align} \int _{-\infty }^{\infty } f(v_\|) \mathrm{d}v_\| = 1. \end{align}

Equation (3.3) admits, depending on the choice of parameters (i.e. $k_\parallel$ , $\omega _*$ , $\eta$ , $b$ and $\tau$ ), a discrete set of damped and unstable modes (see Appendix A) which come in conjugate symmetric pairs (due to the time reversal symmetry of the kinetic equation without collisions) which we denote as

(3.5) \begin{align} f_n = S(v_\parallel )/(\omega _n/k_\parallel - v_\parallel ), \end{align}

and a continuum of singular undamped ‘Van Kampen’ modes with $\gamma = 0$ which we denote as

(3.6) \begin{align} f_\omega = P[{S(v_\parallel })/{(\omega /k_\parallel - v_\parallel )}] + \lambda (\omega )\delta (\omega /k_\parallel - v_\parallel ), \end{align}

where $P$ symbolically indicates that the Cauchy principal value should be taken upon integration. Integrating (3.3) for a discrete mode yields the linear dispersion relation for the slab ITG mode

(3.7) \begin{align} 1 - \int _{-\infty }^{\infty } \frac {k_\parallel S(v_\parallel )}{\omega _n - v_\parallel k_\parallel } \mathrm{d} v_\parallel = 0. \end{align}

In the case of the continuum modes, integrating (3.3) does not constitute a dispersion relation, due to the singularity at $v_\| = \omega /k_\|$ , and instead sets the normalisation of the singular modes that determines $\lambda (\omega )$ (see van Kampen & Felderhof Reference van Kampen and Felderhof1967)

(3.8) \begin{align} \lambda (\omega ) = 1 - P \int \frac {k_\| S(v_\|)}{\omega - v_\| k_\|} \mathrm{d}v_\|. \end{align}

This system is directly analogous to the system studied by Case and Van Kampen, where it was shown by Case (Reference Case1959) that this set of linear eigenmodes, comprising the discrete and continuum modes, is complete, in the sense that any distribution function $f(v_\parallel ,t )$ can be written as

(3.9) \begin{align} f(v_\parallel , t) = \sum _n a_n(t) f_n(v_\parallel ) + \int A(\omega , t) f_\omega (v_\parallel )\, \mathrm{d}\omega . \end{align}

The coefficients of this eigenmode decomposition can be computed by exploiting the orthogonality of eigenmodes with respect to the eigenmodes of the adjoint equation of (3.3) (given in Appendix B), which we denote as $\tilde {f}_n$ and $\tilde {f}_\omega$ for the discrete and continuum branches, respectively.

We now wish to use these features of the kinetic equation to construct an energetic norm which is ‘aware’ of the linear eigenmodes in which we are interested. To do this, we first project (3.1) onto the basis of discrete eigenmodes by applying

(3.10) \begin{align} \frac {1}{C_n}\int _{-\infty }^{\infty } \tilde f_n(v_\parallel)(\ldots ) \mathrm{d}v_\parallel , \end{align}

and noting that $\omega _n$ must satisfy (3.7). This yields the simple relation

(3.11) \begin{align} \frac {\partial a_n}{\partial t} = -i \omega _n a_n, \end{align}

which implies that

(3.12) \begin{align} \frac {\mathrm{d}}{\mathrm{d} t} \sum _n |a_n|^2 = 2 \sum _n \gamma _n |a_n|^2, \end{align}

where we have summed over all the discrete modes. Equation (3.12) states that the amplitudes of the projection coefficients all grow in time according to their growth rate $\gamma _n$ . Next, we turn our attention to the continuum modes. Performing the projection in a similar manner as for the discrete spectrum and recognising the normalisation condition of the Van Kampen modes gives $\partial A(\omega ,t) /{\partial t }= -i \omega A(\omega ,t)$ . Because $\omega$ is purely real for these modes, we have

(3.13) \begin{align} \frac { \mathrm{d}}{\mathrm{d} t }\int |A(\omega , t)|^2 \mathrm{d} \omega = 0. \end{align}

Now by combining (3.12) and (3.13) we can construct what we will refer to as the Case–Van Kampen energy, which we define as

(3.14) \begin{align} E = \sum _n |a_n(t)|^2 + \int |A(\omega , t)|^2 \mathrm{d} \omega . \end{align}

The energy balance equation for $E$ is now simply

(3.15) \begin{align} \frac {\mathrm{d} E}{\mathrm{d} t} = 2 \sum _n \gamma _n |a_n|^2. \end{align}

The energy $E$ constitutes a positive definite norm of any distribution function $f(v_\parallel )$ due to the completeness of the eigenmodes. For a given $f$ , E grows (or damps) at a rate determined by the magnitude of its projection onto the discrete eigenmodes.

3.2. Optimal modes and tightest possible bounds

The Case–Van Kampen energy is a natural choice as a norm for bounding the growth of linear instabilities as it is simply a sum of eigenmode amplitudes, thus it exhibits no ‘non-normal’ or ‘transient growth’ in the absence of linear instability (i.e. it is explicitly aware that the Van Kampen modes cannot sustain linear growth, whereas other norms may be made to grow transiently by a superposition of these modes). While such transient growth is important for understanding nonlinear instability growth e.g. in scenarios where subcritical turbulence is of interest (Krommes Reference Krommes1997; Barnes et al. Reference Barnes, Parra, Highcock, Schekochihin, Cowley and Roach2011; Landreman, Plunk & Dorland Reference Landreman, Plunk and Dorland2015; van Wyk et al. Reference van Wyk, Highcock, Schekochihin, Roach, Field and Dorland2016), it seems relatively unimportant in scenarios in which the relevant parameters are well above the linear instability threshold where robustly growing instabilities can be expected.

We can now derive the optimal modes of $E$ by writing the energy balance (3.15) as $\mathrm{d} E/ \mathrm{d} t = 2K$ and defining the instantaneous growth rate of $E$ as $\varLambda \equiv K/E$ . The optimal modes are the solutions $f$ which extremise $\varLambda$ , and can be constructed by writing $E \equiv (f, \mathcal{E} f)$ , and $K \equiv (f, \mathcal{K} f)$ with the familiar inner product

(3.16) \begin{align} (f_1, f_2) = \int \frac {f_1^* f_2}{\bar F_0} \mathrm{d} v_\parallel . \end{align}

The optimal modes then satisfy the generalised eigenvalue problem

(3.17) \begin{align} \varLambda \operatorname {\mathcal{E}} f = \operatorname {\mathcal{K}} f, \end{align}

where the operators $\mathcal{E}$ and $\mathcal{K}$ are given in Appendix B. To solve the optimal mode problem, we now consider the projection of (3.17) with a discrete eigenmode of the linear problem $f_m$ with the inner product $\varLambda (f_m, \mathcal{E} f) = (f_m, \mathcal{K} f)$ . Due to the orthogonality of the eigenmodes with the adjoint eigenmodes, we find (see Appendix B)

(3.18) \begin{align} \varLambda = \gamma _m. \end{align}

Similarly, considering the projection with a continuum mode, $\varLambda (f_\omega , \mathcal{E} f) = (f_\omega , \mathcal{K} f)$ , yields $\varLambda = 0$ . Thus there is a one-to-one correspondence between the linear growth spectrum and the optimal $\varLambda$ solutions. Moreover, the optimal modes maximising $E^{-1}\mathrm{d}E/ \mathrm{d} t$ are exactly the linear eigenmodes.

This may feel like a somewhat trivial consequence of the diagonalisation of the linear operator by the linear eigenmodes, but there are several takeaways from this result. Firstly, the growth of linear eigenmodes can be bounded with arbitrary tightness by the optimal modes of an energetic norm, with the tightest possible bound (i.e. equality of (1.1)) being given by the Case–Van Kampen energy. We note that this tightest bound may not be given uniquely by this energy norm. For instance, $E$ can be defined with arbitrary weights on each of the amplitude coefficients.

Secondly, any kinetic system for which the Case–Van Kampen energy is a nonlinear invariant is nonlinearly stable if it is linearly stable because $\varLambda$ will be identically zero for all $k_\perp$ , i.e. subcritical turbulence is impossible. An example of such a system is the two-dimensional drift-kinetic toroidal ITG discussed by Plunk (Reference Plunk2015). Moreover, as presented in DelSole (Reference DelSole2004), Landreman et al. Reference Landreman, Plunk and Dorland(2015) and Plunk & Helander (Reference Plunk and Helander2022), for statistically steady turbulence to exist in these systems where $E$ is a nonlinear invariant when summed over all wavenumbers, it must be the case that $\sum _k \langle \mathrm{d} E_k/ \mathrm{d} t \rangle = 0$ , where $\langle \ldots \rangle$ denotes a time average over the turbulent time scale, and the index $k$ has been added. In this case $\sum _{n,k} \gamma _{n,k} \langle |a_{n,k}|^2\rangle = 0$ , such that, in the turbulent phase, the distribution function must project equally on to growing and damped eigenmodes on average.

4. Constrained optimal modes

We have demonstrated that linear instability growth may be bounded with arbitrary tightness by the instantaneous optimals of an energetic norm. However, these tightest bounds require complete knowledge of the linear spectrum. In this section, we demonstrate that we can construct energetic bounds that have the same qualitative dependence on key parameters as the linear growth rate by including only some features of the linear eigenmodes in the analysis. This avoids the requirement of prior knowledge of the solution to the linear problem.

A key feature of the linear eigenmodes, which has been absent from the optimal mode analyses involving simple energetic norms, is the real frequency $\omega _r$ . This oscillation frequency determines the relative phases of the different degrees of freedom of the eigenmode (e.g. the various fluid moments) and is associated with some of the stabilising ‘resonant’ effects that are not captured by the instantaneous upper bounds, which do not have an associated real frequency. As such, the optimal modes considered thus far, with the exception of the Case–Van Kampen optimals discussed in the previous section, are largely unconstrained in the allowable phase relations between their different degrees of freedom.

Here, we seek a means to include phase information into the optimal mode analysis of a simple, positive definite energy norm – the Helmholtz free energy (Helander & Plunk Reference Helander and Plunk2022). For the reduced-dimensionality kinetic equation (2.3), the Helmholtz free-energy balance is

(4.1) \begin{align} \frac {\mathrm{d} H}{\mathrm{d} t} = 2 D, \end{align}

where

(4.2) \begin{align} H = T_i \int \frac {| \bar g|^2}{\bar F_0}\mathrm{d} v_\parallel - \frac {e^2 n}{T_i}(1 + \tau )G_{\perp 0} |\phi |^2, \end{align}

and

(4.3) \begin{align} D = \mathrm{Re}\left \{ i \omega _* \eta \, G_{\perp 0} e\phi \int \bar g^*x_\|^2 \,\mathrm{d} v_\| \right \}. \end{align}

To include the frequency information, we constrain the optimal mode analysis, such that the allowed distribution functions are those that share some key features with the linear modes. Thus, we seek distribution functions that maximise $\mathrm{d}H/ \mathrm{d} t$ subject to a set of constraint equations that are also satisfied by the linear eigenmodes.

Because the free-energy extraction $D$ depends only on two ‘fluid’ moments of $\bar g$ , natural choices for these constraints are the linear moment equations. To keep the analysis relatively simple, we consider only the two lowest-order moments as constraints. We denote these moments of $\bar g$ by

(4.4) \begin{align} \kappa _1 = \frac {1}{n}\int \bar g \mathrm{d} v_\|, \end{align}

(4.5) \begin{align} \kappa _2 = \frac {1}{n}\int \left (\frac {v_\|}{v_T }\right )\bar g \, \mathrm{d} v_\|, \end{align}

(4.6) \begin{align} \kappa _3 = \frac {1}{n}\int \left (\frac {v_\|^2}{v_T^2 }\right )\bar g \, \mathrm{d} v_\|. \end{align}

The moment equations for the density, $\kappa _1$ , and the parallel flow, $\kappa _2$ , obtained by taking the respective moments of (2.3), are given by

(4.7) \begin{align} \frac {\partial \kappa _1}{\partial t} + i v_T k_\| \kappa _2 = \frac {1}{1 + \tau }\left [G_{\perp 0}\left (\frac {\partial }{\partial t} + i\omega _{*}(1 - \eta )\right ) + i \omega _* \eta G_{\perp 2}\right ]\kappa _1, \end{align}

and

(4.8) \begin{align} \frac {\partial \kappa _2}{\partial t} + i v_T k_\| \kappa _3 = 0. \end{align}

We now require that these moments evolve in time like linear eigenmodes (i.e. they satisfy the same phase relation as the moments of a linear eigenmode), such that $\partial \kappa _{1,2}/\partial t = -i \omega ' \kappa _{1,2}$ , where $\omega ' = \omega _r' + i \gamma '$ . The linear moment equations then become

(4.9) \begin{align} \alpha \kappa _1 - v_T k_\| \kappa _2 = 0, \end{align}

and

(4.10) \begin{align} \omega ' \kappa _2 - v_T k_\| \kappa _3 = 0, \end{align}

where we have defined

(4.11) \begin{align} \alpha = \omega ' - \frac {1}{1 + \tau }(G_{\perp 0}\left [\omega ' - \omega _*(1 - \eta )\right ] - \omega _*\eta \, G_{\perp 2} ). \end{align}

In addition to these phase relations between moments, linear eigenmodes have the property that free energy increases at exactly twice the growth rate of the mode, which can be expressed in terms of the free-energy balance (4.1) as an additional constraint

(4.12) \begin{align} D = \gamma ' H. \end{align}

The distribution functions that satisfy (4.9), (4.10) and (4.12) make up a subspace of the larger space of all possible $\bar g$ and, importantly, the linear eigenmodes are themselves included in that subspace. In other words, $\omega '$ need not be an eigenvalue of (2.3), but the true eigenmodes also satisfy (4.9), (4.10) and (4.12) with $\omega ' \to \omega$ .

Therefore, distribution functions which extremise free-energy growth, while satisfying the constraints (4.9), (4.10) and (4.12) place a rigorous upper bound on the growth rate of true eigenmodes. To state this problem formally, we consider the Lagrangian

(4.13) \begin{align} L & \equiv D - \varLambda (H - H_0) - \lambda _1(\gamma ' H - D) \nonumber\\[3pt] & \quad - \lambda _2^*(\alpha ^* \kappa _1^* - v_T k_\| \kappa _2^*) - \lambda _2(\alpha \kappa _1 - v_T k_\| \kappa _2) \nonumber\\[3pt] & \quad - \lambda _3^*(\omega ^{\prime *} \kappa _2^* - v_T k_\| \kappa _3^*) - \lambda _3(\omega ' \kappa _2 - v_T k_\| \kappa _3). \end{align}

Here, to extremise $L$ is to extremise the free-energy drive $D$ , subject to several constraints, which are enforced by Lagrange multipliers denoted by $\varLambda$ (which will become the optimal growth) and $\lambda _n$ . Here, $\varLambda$ keeps the free energy fixed to some normalising value $H_0$ , the complex multipliers $\lambda _2$ and $\lambda _3$ enforce the moment constraints and $\lambda _1$ ensures that the value of $\gamma '$ from these moment constraints is consistent with free-energy balance. Each of these constraints are enforced at the extrema of $L$ , as is evident from taking $\delta L/ \delta \varLambda =0$ and $\delta L/ \delta \lambda _n =0$ .

Computing the optimal modes via $\delta L/ \delta \bar g =0$ and making use of the inner product

(4.14) \begin{align} (\bar g_1, \bar g_2 ) = T_i \int \frac {\bar g_1^* \bar g_2}{\bar F_0}\mathrm{d} v_\|, \end{align}

we arrive at the following kinetic problem for the optimal $\bar g$

(4.15) \begin{align} (\varLambda - \gamma ' \lambda _1)\left (\bar g - \frac {\bar F_0 G_{\perp 0}}{(1 + \tau )}\kappa _1 \right )& = (1 - \lambda _1) \frac {i \omega _* \eta \bar F_0 G_{\perp 0}}{2(1 +\tau )}\big(x_\|^2 \kappa _1 - \kappa _3\big) \nonumber\\ & \quad - \frac {\lambda _2^* \bar F_0}{n T_i}\big(\alpha ^* - x_\| v_T k_\|\big) - \frac {\lambda _3^* \bar F_0}{n T_i}\big(\omega ^{\prime *}x_\| - x_\|^2 v_T k_\|\big). \end{align}

This problem must be solved subject to constraint (4.9), (4.10) and (4.12).

Upon enforcing the constraint $\gamma ' H = D$ in the above expression, we find the expected result that $\varLambda = \gamma '$ , such that the Lagrange multiplier once again has the physical interpretation of the linear growth rate. We see that after enforcing this, the  $\lambda _1$ Lagrange multiplier is arbitrary (as long as it is non-zero and not unity) such that it amounts to a renormalisation of the other multipliers. The remaining unknowns can be eliminated by noting that the system is closed by projecting onto the $\kappa _n$ moments. The system of moment equations which results from this projection is given in Appendix C. After reducing the system of equations, we arrive at a quintic polynomial for $\tilde \varLambda = \varLambda /({\omega _*\eta })$ , for which $\tilde \varLambda = 0$ is a solution (corresponding to the null-space of $D$ ). The remaining solutions are given by the quartic equation

(4.16) \begin{align} P \tilde \varLambda ^4 + Q \tilde \varLambda ^2 + R = 0, \end{align}

where the coefficients $P$ , $Q$ and $R$ are given in Appendix D. We note that only solutions to (4.16) that yield a real value of $\varLambda$ are valid. Complex solutions, which are inconsistent with (4.12), are discarded. Solving this equation yields and optimal mode growth rate $\varLambda$ , with an explicit dependence on the parameter $\omega _r'$ . The upper bound on linear growth is given by

(4.17) \begin{align} \varLambda _{\mathrm{max}}\equiv \max \limits _{\omega _r'} \varLambda . \end{align}

The maximising value of $\omega _r'$ can, in principle, be found analytically. However, it is generally more convenient to evaluate numerically or with a computer algebra program such as Mathematica. The largest real solution to (4.16) at the optimal $\omega _r'$ gives a rigorous upper bound on the growth of the unstable eigenmodes of the slab ITG system. The bound is guaranteed to be at least as good as the unconstrained bound given by Helmholtz free-energy balance and the constraints serve to tighten this bound to the linear growth rate. Although the gyrofluid moment equations are an incomplete description of the linear eigenmodes, especially in the resonant limit, we emphasise that they are always valid constraints because they are always satisfied by the linear eigenmodes regardless of the specific parameters. While we have only considered the lowest-order moments here, the bound could be further tightened by considering a larger number of constraints, capturing more features of the linear modes.

4.1. Features of the constrained bound

To gain insight into the behaviour of the solutions of (4.16), we now consider some simple limits and explore the general features of the constrained optimal mode growth.

4.1.1. Resonant stabilisation

Firstly, to facilitate comparison with the results of Plunk & Helander (Reference Plunk and Helander2023) we consider the long-wavelength limit where $b \to 0$ and consider a scenario without a density gradient where $\eta \to \infty$ . In this limit, we have $G_{\perp 0,2} \to 1$ .

Figure 1.

Upper bound on linear growth (blue) alongside the linear growth rate (dashed black) from the linear dispersion relation (A.1), plotted as a function of the instability parameter $\kappa _\|$ in the low- $k_\perp$ limit with $\eta \to \infty$ and $\tau =1$ . Also shown is the fluid-limit dispersion relation (dot-dashed red) (Plunk et al. Reference Plunk, Helander, Xanthopoulos and Connor2014), which diverges as $\kappa _\| \to 0$ . The vertical light-grey dashed line is the critical gradient of Kadomtsev & Pogutse (Reference Kadomtsev, Pogutse and Leontovich1970).

In figure 1, we show the behaviour of the upper bound alongside the solution of the linear dispersion relation as a function of the instability parameter $\kappa _\| = \omega _*\eta /(v_T k_\|)$ . As one would expect, the bound agrees well with the linear growth rates for large $\kappa _\|$ , where the linear eigenmodes become more fluid-like as the lowest moments dominate the fluid hierarchy. In contrast to previous bounds (Plunk & Helander Reference Plunk and Helander2023), the constrained bound for the slab ITG exhibits a maximum growth rate at a finite $\kappa _\|$ , as is the case for linear eigenmodes. This also in contrast to the simple non-resonant dispersion relation for the slab ITG mode shown in figure 1, which actually diverges in this limit (Plunk et al. Reference Plunk, Helander, Xanthopoulos and Connor2014)

(4.18) \begin{align} \frac {\gamma }{\omega _*\eta } = (2 \tau \kappa _\|)^{-\frac {1}{3}}. \end{align}

Remarkably, the constrained bound also exhibits a critical gradient, a value of $\kappa _\|$ below which only $\varLambda = 0$ is a valid solution. The criterion for this critical value of $\kappa _\|$ can be found by setting $R = 0$ , such that $\varLambda = 0$ is a repeated root of (4.16) and seeking the lowest positive value of $\kappa _\|$ for which this can occur. In this limit, we have $\kappa _{\|, \mathrm{cr}} = \sqrt {2\tau (1 + \tau )}$ , which is precisely the critical gradient which can be derived from the linear slab ITG dispersion relation (Kadomtsev & Pogutse Reference Kadomtsev, Pogutse and Leontovich1970; Plunk et al. Reference Plunk, Helander, Xanthopoulos and Connor2014). Indeed, we see agreement between the critical gradient of the linear eigenmode and the bound in figure 1. This agreement in the ‘resonant’ limit of low- $\kappa _\|$ is notable given that the most unstable linear eigenmode becomes increasingly singular as the critical $\kappa _\|$ is approached. As a result, one might not expect to capture this behaviour with a finite-dimensional system of moment equations. On the other hand, the critical gradient expression derived by Kadomtsev & Pogutse (Reference Kadomtsev, Pogutse and Leontovich1970), which amounts to solving a two-dimensional system of equations, gives some indication that the resonant stabilisation of the slab ITG is not as complicated as one may expect and can be captured with relatively few degrees of freedom. The constrained optimal modes are, evidently, successful in capturing this resonant stabilisation due to the inclusion of information about the real frequency via $\omega _r'$ , which is responsible for the effect in the linear dispersion relation.

4.1.2. Density gradient stabilisation

We now consider a finite value of $\eta$ , such that the density gradient is non-zero. This is known to have a significant stabilising effect on linear ITG instabilities. The previous, nonlinear bounds of Helander & Plunk (Reference Helander and Plunk2022) and Plunk & Helander (Reference Plunk and Helander2023) do not capture this effect because the adiabaticity of the electrons precludes the density gradient from being a source of free energy. In figure 2, we show that the constrained optimal growth rate has the same qualitative dependence on $\eta$ as the linear instability growth. Once again, we see that the critical value of $\eta$ agrees with that of the linear dispersion relation. In the linear theory, this stabilisation by the density gradient is connected with the real frequency of the ITG, hence why the constrained upper bound captures this effect.

Figure 2.

Upper bound on linear growth (blue) alongside the linear growth rate (dashed black) from the linear dispersion relation (A.1), plotted as a function of $1/ \eta$ in the low- $k_\perp$ limit with $\tau = 1$ . Here, both the upper bound and the linear growth rate have been maximised over $k_\|$ .

4.1.3. Finite-Larmor-radius effects

Finally, we consider how the upper bound from the constrained optimal modes depends on the normalised perpendicular wavenumber $k_\perp \rho$ . In the case of the slab ITG, because the resonance does not involve the perpendicular velocity, the finite-Larmor-radius (FLR) effects manifest via the velocity-space-independent functions $G_{\perp 1,2}$ . As a result, the most consequential step for capturing the FLR dependence of the slab ITG is the integration over the $v_\perp$ coordinate when going from (2.1) to (2.3). In figure 3, we see that the constrained optimal growth has a very similar qualitative dependence on $k_\perp \rho$ to the linear growth rate, and is significantly lower than the pervious bound of Plunk & Helander (Reference Plunk and Helander2023).

Figure 3.

Upper bound on linear growth (blue) alongside the linear growth rate (dashed black) from the linear dispersion relation (A.1) and the nonlinear bound of Helander & Plunk (Reference Helander and Plunk2022) and Plunk & Helander (Reference Plunk and Helander2023) (dashed grey), plotted as a function of $k_\perp \rho$ in the limit of $\eta \to \infty$ for $\tau = 1$ , where both the constrained upper bound and the linear growth rate have been maximised over $k_\|$ .