Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra

This study uses the DNA code [24, 25], which solves a reduced gyrokinetic model in an unsheared-slab using a Hermite representation

$$\begin{eqnarray}&&f(v)=\displaystyle \sum _{m=0}^{\infty }{\hat{f}}_{m}{H}_{m}(v){{\rm{e}}}^{-{v}^{2}},\end{eqnarray} \tag{ 1 }$$

for parallel velocity space, where H_m(x) is the mth order Hermite polynomial. Hermite polynomials were shown empirically to be an efficient representation of velocity space in gyrokinetic simulations in [43]. Their use in this work was inspired, in particular, by earlier demonstrations of their utility in calculating linear eigenmode spectra [44, 50].

The gyrokinetic equations are integrated over perpendicular velocity space, retaining an approximation of finite-Larmour-radius effects by replacing gyroaverage operators with ${{\rm{e}}}^{-{k}_{\perp }^{2}/2}$ (valid for ${k}_{\perp }{\rho }_{i}\lt 1$ [45]). The model includes density and temperature gradients and collisions via a Lenard–Bernstein collision operator [46]. These ingredients produce the linear dynamics of interest for this work, namely temperature gradient driven instabilities, DWs, Landau damping and phase mixing, and collisional dissipation.

The evolution equation for the Hermite moments is

$$\begin{eqnarray}\displaystyle \frac{\partial {\hat{f}}_{{\bf{k}},m}}{\partial t} & = & \displaystyle \frac{{\omega }_{T}{\rm{i}}{k}_{y}}{{\pi }^{1/4}}\displaystyle \frac{{k}_{\perp }^{2}}{2}{\bar{\phi }}_{{\bf{k}}}{\delta }_{m,0}-\displaystyle \frac{{\omega }_{n}{\rm{i}}{k}_{y}}{{\pi }^{1/4}}{\bar{\phi }}_{{\bf{k}}}{\delta }_{m,0}-\displaystyle \frac{{\omega }_{T}{\rm{i}}{k}_{y}}{\sqrt{2}{\pi }^{1/4}}{\bar{\phi }}_{{\bf{k}}}{\delta }_{m,2}\\ & & -\displaystyle \frac{{\rm{i}}{k}_{z}}{{\pi }^{1/4}}{\bar{\phi }}_{{\bf{k}}}{\delta }_{m,1}-{\rm{i}}{k}_{z}(\sqrt{m}{\hat{f}}_{{\bf{k}},m-1}+\sqrt{m+1}{\hat{f}}_{{\bf{k}},m+1})\\ & & -\nu m{\hat{f}}_{{\bf{k}},m}+\displaystyle \sum _{{{\bf{k}}}^{\prime }}({k}_{x}^{\prime }{k}_{y}-{k}_{x}{k}_{y}^{\prime }){\bar{\phi }}_{{{\bf{k}}}^{\prime }}{\hat{f}}_{{\bf{k}}-{{\bf{k}}}^{\prime },m},\end{eqnarray} \tag{ 2 }$$

where ${\hat{f}}_{{\bf{k}},m}$ is the perturbed (i.e. deviation from Maxwellian) ion distribution function, n_i0 is the ion density, R is a macroscopic scale length, v_ti is the ion thermal velocity, m denotes the order of the Hermite polynomial, t is time, ${\omega }_{T}=R/{L}_{T}$ is the normalized ITG scale length, ${\omega }_{n}=R/{L}_{n}$ is the normalized inverse density gradient scale length, k_y is the Fourier wavenumber for the direction perpendicular to both the direction of the background gradients $[x\to {k}_{x}]$ and the coordinate aligned with the magnetic field $[z\to {k}_{z}]$. The perpendicular wavenumber is ${k}_{\perp }={({k}_{x}^{2}+{k}_{y}^{2})}^{1/2}$, ${\bar{\phi }}_{{\bf{k}}}$ is the gyro-averaged electrostatic potential, T_e0 is the background electron temperature, e is the elementary charge, and ν is the collision frequency. Hermite polynomials are, conveniently, eigenvectors of the Lenard–Bernstein collision operator: $\nu {\partial }_{v}$ $[(1/2){\partial }_{v}+v]\to \nu m$. Time quantities are normalized to $R/{v}_{{ti}}$, the perpendicular directions are normalized to the ion gyroradius ${\rho }_{i}$, the parallel direction is normalized to R, the potential is normalized to ${\rho }_{i}{T}_{e0}/{eR}$, and the distribution function is normalized to ${\rho }_{i}{n}_{i0}/{{Rv}}_{{ti}}^{3}$. We employ the adiabatic electron approximation, making electrostatic potential directly proportional to the zeroth-order Hermite polynomial as determined by

$$\begin{eqnarray}&&{\phi }_{{\bf{k}}}={\pi }^{1/4}{{\rm{e}}}^{-{k}_{\perp }^{2}/2}{\hat{f}}_{{\bf{k}},0}/[\tau +1-{{\rm{\Gamma }}}_{0}({k}_{\perp }^{2})],\end{eqnarray} \tag{ 3 }$$

where τ is the ratio of the ion to electron temperature, and ${{\rm{\Gamma }}}_{0}(x)={{\rm{e}}}^{-x}{I}_{0}(x)$, with I₀ the zeroth order modified Bessel function.

One advantage of the Hermite representation is the elegant form taken by the energy equations, as described in detail in [25]. Here we note simply that the free energy

$$\begin{eqnarray}&&{E}_{{\bf{k}},m}={E}_{{\bf{k}}}^{(\phi )}{\delta }_{m,0}+{E}_{{\bf{k}},m}^{(f)},\end{eqnarray} \tag{ 4 }$$

is composed of an electrostatic part ${E}_{{\bf{k}}}^{(\phi )}=\tfrac{1}{2}{(\tau +1-{{\rm{\Gamma }}}_{0}({k}_{\perp }^{2}))}^{-1}| {\phi }_{{\bf{k}}}{| }^{2}$ and an entropy part ${E}_{{\bf{k}},m}^{(f)}=\tfrac{1}{2}{\pi }^{1/2}| {\hat{f}}_{{\bf{k}},m}{| }^{2}$. In this system, Landau damping is a conservative process that transfers energy from the former to the latter (i.e. from waves to particles) via the term

$$\begin{eqnarray}&&{J}_{{\bf{k}}}^{(\phi )}={\mathfrak{R}}[-{\rm{i}}{k}_{z}{\pi }^{1/4}{\bar{\phi }}^{* }{\hat{f}}_{{\bf{k}},1}].\end{eqnarray} \tag{ 5 }$$

The net sources and sinks are, respectively, the gradient drive

$$\begin{eqnarray}&&{Q}_{{\bf{k}}}={\eta }_{i}{\mathfrak{R}}\left[-\displaystyle \frac{{\pi }^{1/4}}{{2}^{1/2}}{\rm{i}}{k}_{y}{\hat{f}}_{2}^{* }\bar{\phi }\right],\end{eqnarray} \tag{ 6 }$$

which is proportional the the heat flux, and the collisional dissipation

$$\begin{eqnarray}&&{C}_{{\bf{k}},m}=2\nu m{\varepsilon }_{{\bf{k}},m}^{(f)}.\end{eqnarray} \tag{ 7 }$$

which is proportional to the collision frequency ν and the Hermite number m, so that the total energy $E={\sum }_{{\bf{k}},m}{E}_{{\bf{k}},m}$ evolves according to

$$\begin{eqnarray}&&\displaystyle \frac{\partial E}{\partial t}=\displaystyle \sum _{{\bf{k}}}{Q}_{{\bf{k}}}+\displaystyle \sum _{{\bf{k}},m}{C}_{{\bf{k}},m}.\end{eqnarray} \tag{ 8 }$$

The quantities Q/E and C/E (having units of inverse time) represent the rate at which energy is injected or dissipated and will be called the drive and collisional growth rates, respectively. As an illustration, the sum ${\text{}}Q/{\text{}}E+{\text{}}C/{\text{}}E$ calculated from a linear eigenvector is equal to twice the linear growth rate (since the quantities are quadratic). These drive and collisional growth rates will be used extensively to characterize linear eigenmodes and SVD structures.

Pseudospectra have been popularized in fluid dynamics by the realization that nonlinear behavior often differs drastically from linear expectations when the linear operators are highly nonorthogonal. A prominent example is high-Reynolds-number Poiseuille flow, which becomes unstable far below the linear critical Reynolds number and does so in a manner that is consistent with the pseudospectra but very different from the linear spectrum [41]. In contrast, our motivation for this work is to find connections—which may be absent in the eigenvalue problem—between the linear operator and the nonlinearly excited structures.

Pseudospectra account for the fact that regions of the complex plane that are far removed from actual eigenvalues may be eigenvalues when the linear operator is subject to even very small perturbations. Definitions of the pseudospectrum rely on the resolvent of the operator $R={({zI}-A)}^{-1}$, where A is the operator, I is the identity matrix, and z is a complex number. If z is an eigenvalue then zI − A is singular and $| | R| |$ is infinite. We reproduce below three equivalent (with some restrictions [42]) definitions of the $\epsilon$-pseudospectrum from [42], each of which has a particular relevance for this work. The $\epsilon$-pseudospectrum ${{\rm{\Lambda }}}_{\epsilon }$ is defined as isocontours of $| | R| |$ in the complex plane such that the resolvent is larger than a given value:

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\epsilon }(A)=\{z\in {\mathbb{C}}\ :| | {({zI}-A)}^{-1}| | \geqslant {\epsilon }^{-1}\}.\end{eqnarray} \tag{ 13 }$$

The $\epsilon$-pseudospecturm can also be defined in terms of perturbations to the operator:

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\epsilon }(A)=\{z\in {\mathbb{C}}\ :z\in {\rm{\Lambda }}(A+{\rm{\Delta }}A),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\,\mathrm{for}\,\mathrm{some}\,{\rm{\Delta }}A\,\mathrm{with}\,| | {\rm{\Delta }}A| | \leqslant \epsilon \},\end{eqnarray} \tag{ 15 }$$

where ${\rm{\Delta }}A$ represents an arbritrary deviation from from A, and Λ is the eigenvalue spectrum. Such a deviation could represent, for example, the nonlinear term in the equations.

Another definition relates to the SVD of zI − A,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\epsilon }(A)=\{z\in {\mathbb{C}}\ :{\sigma }_{\min }({zI}-A)\leqslant \epsilon \},\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{\min }$ is the smallest singular value of the SVD of zI − A. This definition is useful because it defines a simple method for calculating pseudospectra numerically. It also identifies the right singular vector corresponding to ${\sigma }_{\min }$ as the most-resonant vector, i.e., the vector that is closest to being an eigenvector at the point z in the complex plane. This most-resonant vector will be called the pseudo-eigenvector below.

Pseudospectra of the kinetic slab ITG operator are shown in figure 3 for a wavenumber representative of the strongly unstable region of k-space. The contours are closely nested around the ITG mode, indicating that it is quite orthogonal to the remaining eigenvectors. The pseudospectra also distinguish (to a lesser degree) the DW ($\omega \approx -1$ and $\gamma \approx -0.2$) from the remaining eigenvalues. The region surrounding the Landau roots contains only pseudospectra corresponding to very small values of $\epsilon$. This is an indication that the eigenvectors corresponding to the Landau roots are highly nonorthogonal and any complex number in this region is close to being an eigenvalue. We would thus expect that structures arising in the nonlinear turbulence may not be easily identifiable with individual eigenvalues in this region of the complex plane.

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Since pseudospectra reflect the non-orthogonality of an operator, we note here that Landau damping represents a violation of Hermiticity. This is readily identified in the non-symmetric term in the matrix ${ \mathcal L }$ defined in equation (10), which produces the transfer of energy from waves to particles (i.e. Landau damping—quantified by ${J}_{{\bf{k}}}^{(\phi )}$ in equation (5)). We thus expect that as k_z increases and Landau damping becomes stronger, the pseudospectral contours will become increasingly distorted. An example is shown for a high k_z wavenumber in figure 4, where it is seen that the ITG mode (now stable) is embedded in elongated contours that no longer separate it from the lower region of the complex plane. Figure 5 shows the deviation of linear growth rates from the nearest point (vertically displaced—i.e., the nearest point with the same frequency) on the $\varepsilon =0.1$ pseudospectrum as a function of k_z for both the ITG mode and the DW. In both cases the deviation increases with k_z, indicating that less-damped regions of the complex plane become increasingly accessible. This suggests (1) that the structures that arise in the nonlinear turbulence may deviate strongly from linear expectations for such parameters where Landau damping is expected to be important, and (2) that the nonlinear structures may be substantially less damped than expectations from the linear growth rates.

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SVD is applied to the turbulence in the following manner: for a given ${\boldsymbol{k}}=({k}_{x},{k}_{y},{k}_{z})$, the distribution function ${\hat{f}}_{{\boldsymbol{k}},m}(t)$ is defined by its dependence on m and t, making it amenable to decomposition of the form ${\hat{f}}_{m}(t)={\sum }_{p=0,N}{\sigma }_{p}{\hat{f}}_{m}^{p}{h}^{p}(t)$, where all amplitude information is found in ${\sigma }_{p}$, m and t dependence has been separated into $\hat{f}$_m ^p and ${h}^{p}(t)$, respectively, and N is the number of elements in the SVD decomposition (minimum of the number of time points and the number of Hermite moments). An SVD produces these ingredients in the form of singular values ${\sigma }_{p}$, and orthonormal left and right singular vectors ($\hat{f}$_n ^p and ${h}^{p}(t)$, respectively). Moreover, the SVD produces an optimal decomposition in the sense that more energy ${\sum }_{m}| {\hat{f}}_{m}{| }^{2}$ is captured by a truncated SVD decomposition than by any other possible decomposition with the same truncation rank.

The left singular vectors ${\hat{f}}_{m}^{p}$ should then be considered the optimal structures for describing the turbulent phase space, and, having the same form (i.e., dependence only on m) as their linear counterparts (linear eigenvectors or pseudo-eigenvectors), can be compared directly. Using the turbulent distribution function, SVDs were constructed for each wave vector in the simulation. With this optimal decomposition in hand, the remaining challenge is extracting useful information about the turbulence.

We now have three different lenses—linear, pseudospectral, and SVD—through which we can extract and/or interpret nonlinear phase-space structures. Each lens has its distinctive advantages and disadvantages. The linear eigenvalue problem is intuitive and familiar from the historical literature. However, as the pseudospectral approach unveils, the linear eigenmodes may have little to do with what is actually observed in a realistic system. The pseudospectral techniques provide a more nuanced picture of the linear operator, but remain nebulous in other ways, making no certain prediction of the structures that will be excited in the nonlinear turbulence. The value of the SVD is clear—a delineation of the quantifiably most important structures and the manner in which they combine to represent the turbulent fluctuations. However, aspects of the SVD structures may remain as inscrutable as the underlying turbulence from which they are extracted unless further work is done to make connections with meaningful concepts with respect to the turbulence. In this section we describe just such a technique that links all three approaches, revealing patterns in the phase space structures that would be invisible to more conventional analyses.

We start with an SVD decomposition and, in the conceptual spirit of pseudospectra, construct NPEV for each SVD structure as follows. The proximity of an SVD structure to being an eigenvector for a prospective eigenvalue z is given by the quantity

$$\begin{eqnarray}&&{\delta }_{p}(z)=\displaystyle \frac{| | A{\hat{f}}_{n}^{p}-z{\hat{f}}_{n}^{p}| | }{| | {\hat{f}}_{n}^{p}| | },\end{eqnarray} \tag{ 17 }$$

where A is the linear operator. We define the NPEV for a given SVD structure to be the point in the complex plane where ${\delta }_{p}(z)$ is minimized. An authentic eigenvector would, for example, identify its eigenvalue as the location in the complex plane where ${\delta }_{p}=0$. We note the connections between this calculation and the calculation of pseudo-spectra; both treatments explore the complex plane for regions of special significance to the linear operator. Whereas the pseudo-spectral technique finds such regions for arbitrary vectors, the NPEV procedure finds points unique to a given vector that optimally represents the turbulent system. As will be outlined below, this procedure extracts some surprising and subtle features that link the linear, pseudo-linear, and SVD analyses.

Figures 6–8 show plots for a series of wave numbers representing the peak energy-containing scales (note that as k_y increases, the energy peaks at progressively larger values of k_z, roughly consistent with the concept of critical balance [61], which was identified in this system in [24, 25]). These figures encapsulate the main conclusions of this work, synthesizing all three analyses—eigenvalues, pseudospectra, and NPEV. In panel (A), the black x'(s) denote the linear eigenvalues and the contours denote the pseudospectra. The NPEVs are shown as colored symbols, whose color (log scale) denotes the singular value (i.e., average amplitude) of the corresponding SVD structure. The lower panels (B)–(D) show additional information about the SVD structures. The SVD modes are ordered by their singular values, while the symbol shapes denote this ordering and serve to make the connection with the NPEVs in panel (A). Panel (B) shows the drive (Q/E) and dissipation (C/E) growth rates of the structures, panel (C) shows the net contribution to energy balance for each mode (average Q + C), and panel (D) shows the minimized ${\delta }^{2}$ (the degree to which the SVD structure approximates an eigenvector).

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We now discuss several interesting trends that can be identified in this series of figures. We note that the trends identified are not universal—i.e., there is some minor variation to the observations described below. However, the main points are quite robust and the series of figures is representative of a broad range of wave numbers.

• (1)  
  We first note that one NPEV corresponds quite closely to the unstable ITG mode, as expected from the pseudo-contours that are closely nested about the linear eigenvalue. This ITG mode is invariably the most energetic SVD structure. We also note that the NPEV for the ITG mode tends to drift away from its linear eigenvalue as the wavenumbers (k_z and k_y) increase, which is consistent with the increasingly vertically elongated pseudo-contours. As discussed above, this is a result of the two symmetry-breaking terms in the linear operator being, respectively, proportional to k_z and k_y.
• (2)  
  As may be expected from the pseudospectra, a second NPEV closely corresponds to the marginally stable DW. Consistent with its linear counterpart, this mode has a relatively small Q and a smooth structure in velocity space, as evidenced by its small value of C/E. The DW linear eigenvalue is (aside from the ITG), the least stable eigenmode, which would suggest that it should achieve a penultimate degree of excitation in the nonlinear state. This is, surprisingly, often not the case, which brings us to perhaps the most puzzling element of the NPEV spectra.
• (3)  
  As noted in section 4, the pseudo-spectra are extremely sparse in the lower region of the complex plane, providing little if any prediction of potential nonlinear structures that may be found here. Interestingly, a distinct NPEV with relatively high amplitude consistently appears in this region. This NPEV corresponds to the structure with, often, the second most (to only the ITG mode) energy in the spectrum. The mode has a similar frequency to that of the ITG mode and is strongly stable, often due to a substantially negative value of Q/E. This combination of relatively high amplitude and strong damping rate makes this mode a robust energy sink, as can be seen in the (C) panels. The properties of this mode evoke those of the MITG mode, which is identified in the collisionless linear eigenvalue spectrum but completely vanished when collisions are included (as they are in the nonlinear simulation). This begs the question whether the MITG mode, which is not an eigenmode of the underlying linear operator, reappears in the nonlinear turbulent fluctuations. This will be discussed in detail in the next section.
• (4)  
  The remaining distinctive mode in the NPEV spectra is the fourth mode (diamond), which has a substantial value of C/E and appears consistently near $\omega =0$. This, as with the remaining modes, cannot be clearly identified with any feature of the linear spectrum. These structures, which appear to be purely nonlinear creations, describe the small velocity space scales of the distribution function and collectively represent a non-negligible energy sink.

Having summarized the NPEVs, we redirect the readers attention to table (1), to which we can now add a final column denoting the modes identified in the nonlinear turbulent state.

As discussed above, a mode with properties very similar to those of the MITG mode appears in the NPEV spectra even though the MITG mode is completely absent from the underlying linear eigenvalue spectra. As it turns out, a closer examination of the pseudo-spectra serves to bridge the gulf between the pure linear and nonlinear analyses, resolving this apparent paradox. The key piece of information lies in the pseudo-eigenvectors defined by the right singular vectors described in relation to equation (16). These pseudo-eigenvectors represent the most resonant structures for the linear operator at a given point in the complex plane. As with linear eigenvectors or SVD vectors, various mode properties can be calculated from pseudo-eigenvectors. The contours of Q/E for the pseudo-eigenvectors are plotted throughout the complex plane in figure 9. The thick black contour separates regions of positive Q from negative Q. Interestingly, the region of strongly negative Q is found opposite (across the real axis) from the ITG eigenvalue in exactly the region where the MITG mode is found in the (collisionless) linear spectrum and its counterpart in the NPEV spectrum. Apparently, the MITG mode, which disappeared from the linear eigenvalue spectrum, remains 'just under the surface' in the pseudospectra, poised for excitation in the nonlinear turbulent state.

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This identification of a remnant of the MITG mode in the pseudospectrum offers an explanation for the surprising reappearance of the mode in the nonlinear fluctuations—i.e., MITG-like structures represent some of the least-damped structures in the broad region of near-resonance with only very low-ε pseudospectral contours. However, the question remains as to why modes in this region of the complex plane are preferentially excited instead of, e.g., equally weakly damped structures with different frequencies. We propose here two factors that likely contribute to the preferential excitation of MITG modes.

First, the defining characteristic of the (M)ITG modes is the ${\phi }^{\ast }\,T$ cross-phase that determines the value of Q (i.e., the energy injected into the system by the instability). In the turbulent state, the nonlinearity will stochastically modulate this cross phase away from the value defined by the linear eigenmode, requiring, in combination with the positive-phase ITG mode, a negative-phase structure on which to project. The pseudospectra suggest that such a structure is most resonant in the MITG region of the complex plane.

Second, the dominant nonlinear coupling mechanism is likely to be strongest for resonant frequency triads that include a zonal (k_y = 0) component (note that zonal coupling remains important in these simulations even though we do not remove the flux-surface-averaged potential in the equation for ϕ since this strongly suppresses the turbulence in slab simulations [20, 24]). This would entail three wave coupling between wave numbers ${\boldsymbol{k}}+{{\boldsymbol{k}}}^{\prime }+{{\boldsymbol{k}}}^{\prime\prime }=0$, where, ${\boldsymbol{k}}=(0,{k}_{y},{k}_{z})$, ${{\boldsymbol{k}}}^{\prime }=({k}_{x},-{k}_{y},-{k}_{z})$, and ${{\boldsymbol{k}}}^{\prime\prime }=(-{k}_{x},0,0)$. The frequency of the zonal (k_y = 0) component ${\omega }_{\mathrm{zonal}}$ is (near) zero and the two finite k_y wave numbers have frequencies with near-opposite signs so that $| {\omega }_{\mathrm{zonal}}+{\omega }_{{k}_{y}}+{\omega }_{-{k}_{y}}| /| {\omega }_{{k}_{y}}| \ll 1$. The reciprocal of this frequency sum is the correlation time of three wave interactions. In closure theories, the nonlinear transfer term is inversely proportional to this frequency sum so that couplings that satisfy this frequency cancellation would be strongly amplified (this has been studied in the context of stable mode coupling in related systems in [32, 39]). Such a coupling mechanism, in combination with the pseudospectral considerations discussed above, provide a compelling explanation for the reemergence of MITG in the nonlinear spectrum. Moreover, the frequency sum is seen to be descriptive of the nonlinear self-organization that favors the MITG and access to dissipation. As such, it could be useful as a proxy in nonlinear confinement optimization schemes like those pursued for stellarators [65].

The NPEV analysis also has something interesting to say about Landau damping. Figures 10 and 11 show the NPEV spectrum for high-k_z wave numbers where the ITG and DW modes are stable due to Landau damping. The NPEVs follow the distortions in the pseudospectral contours and are clearly less damped than the linear eigenvalues that represent the Landau damping rates. This means that the Landau damping mechanism is much less effective than the pure linear solution would suggest. Figure 11 shows an NPEV that is a robust energy source even though there are no unstable linear eigenvalues. We generally observe that the nonlinear damping rate at high k_z is strongly reduced (or even reversed) from the linear expectation. This suggests that the damping rates in critically balanced cascades [61–63] may be less than expected from linear Landau damping. This is consistent with recent work [26, 64] demonstrating that Landau damping rates are strongly modified by random sources intended to roughly represent a turbulent nonlinearity. This work suggests that pseudospectral or related non-modal techniques may prove useful in formulating advanced gyrofluid models that faithful reproduce Landau damping in a turbulent systems.

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