2.1 Electron Vlasov–Poisson plasma

Assuming cold ions, or, equivalently, restricting our consideration to perturbations with frequencies of the order of the electron plasma frequency,

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D714}\sim \unicode[STIX]{x1D714}_{\text{p}e}=\sqrt{\frac{4\unicode[STIX]{x03C0}e^{2}n_{0e}}{m_{e}}},\end{eqnarray}$$

where $-e\equiv q_{e}$ , we may set $\unicode[STIX]{x1D6FF}f_{i}=0$ . Further restricting our consideration to a single spatial dimension, $x$ , we have $\unicode[STIX]{x1D6FF}f_{e}=\unicode[STIX]{x1D6FF}f_{e}(x,\boldsymbol{v})$ . We may now introduce the following reduced, non-dimensionalised fields and variables:

(2.6) $$\begin{eqnarray}\displaystyle g(x,v)=\frac{v_{\text{th}e}}{n_{0e}}\int \text{d}v_{y}\,\text{d}v_{z}\unicode[STIX]{x1D6FF}f_{e},\quad F_{\text{M}}(v)=\frac{1}{\sqrt{\unicode[STIX]{x03C0}}}\,e^{-v^{2}},\quad v=\frac{v_{x}}{v_{\text{th}e}},\quad \unicode[STIX]{x1D711}=-\frac{e\unicode[STIX]{x1D719}}{T_{e}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We may also non-dimensionalise $t\unicode[STIX]{x1D714}_{\text{p}e}\rightarrow t$ and $x/\!\sqrt{2}\,\unicode[STIX]{x1D706}_{\text{D}e}\rightarrow x$ , where $\unicode[STIX]{x1D706}_{\text{D}e}=v_{\text{th}e}/\sqrt{2}\,\unicode[STIX]{x1D714}_{\text{p}e}$ is the electron Debye length. In this notation, the Vlasov–Poisson system becomes

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}t}+v\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}x}+vF_{\text{M}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}x}-\frac{1}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}v}=C[g], & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D711}=\unicode[STIX]{x1D6FC}\int \text{d}v\,g+\unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

where $-\unicode[STIX]{x1D6FC}$ is twice the inverse Laplacian operator, $\unicode[STIX]{x1D6FC}_{k}=2/k^{2}$ in Fourier space. We have added an ‘external’ potential $\unicode[STIX]{x1D712}$ to represent energy injection in an analytically convenient fashion (it will also acquire concrete physical meaning in §§ 2.3 and 2.4). Finally, we make a further simplification by using the Lenard & Bernstein (Reference Lenard and Bernstein1958) collision operator

(2.9) $$\begin{eqnarray}C[g]=\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}\left(\frac{1}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}+v\right)g,\end{eqnarray}$$

with the proviso that it must be adjusted to conserve momentum and energy. This will not be a problem as the collision frequency $\unicode[STIX]{x1D708}$ will always be assumed small and so will only matter for the part of $g$ that varies quickly with $v$ .

Besides being interesting in itself, the 1-D Vlasov–Poisson system (2.7)–(2.8) is an appealing minimal model that contains all the ingredients necessary for phase-space turbulence featuring a competition between phase mixing and nonlinearity.

2.2 Ion-acoustic turbulence

Another, mathematically similar, model describes electrostatic perturbations at low frequencies, where it is ion kinetics that matter, namely,

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D714}\sim kv_{\text{th}i}.\end{eqnarray}$$

Since $m_{e}\ll m_{i}$ and assuming $T_{i}\sim T_{e}$ , the electrons’ velocities are $(m_{i}/m_{e})^{1/2}$ larger than the ions’ and so the kinetic equation (2.1) for $s=e$ becomes, on ion time scales,

(2.11) $$\begin{eqnarray}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{e}+\frac{e}{m_{e}}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})\cdot \frac{\unicode[STIX]{x2202}f_{e}}{\unicode[STIX]{x2202}\boldsymbol{v}}=\left(\frac{\unicode[STIX]{x2202}f_{e}}{\unicode[STIX]{x2202}t}\right)_{\text{c}}.\end{eqnarray}$$

This is solved by the Maxwell–Boltzmann distribution

(2.12) $$\begin{eqnarray}f_{e}=\frac{n_{0e}}{(2\unicode[STIX]{x03C0}T_{e}/m_{e})^{3/2}}\exp \left[-\frac{1}{T_{e}}\left(\frac{m_{e}|\boldsymbol{v}|^{2}}{2}-e\unicode[STIX]{x1D719}\right)\right].\end{eqnarray}$$

The electrons, therefore, have a Boltzmann response:

(2.13) $$\begin{eqnarray}n_{e}=n_{0e}e^{e\unicode[STIX]{x1D719}/T_{e}}\approx n_{0e}\left(1+\frac{e\unicode[STIX]{x1D719}}{T_{e}}\right),\end{eqnarray}$$

assuming $e\unicode[STIX]{x1D719}/T_{e}\ll 1$ . If $k\unicode[STIX]{x1D706}_{\text{D}e}\ll 1$ , the Poisson equation (2.2) turns into the quasineutrality constraint:

(2.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n_{i}}{n_{0i}}=\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}=\frac{e\unicode[STIX]{x1D719}}{T_{e}},\end{eqnarray}$$

where the last equality follows from (2.13) and the ion density perturbation has to be calculated from the perturbed ion distribution function, $\unicode[STIX]{x1D6FF}n_{i}=\int \text{d}^{3}\boldsymbol{v}\,\unicode[STIX]{x1D6FF}f_{i}$ . Restricting consideration again to 1-D perturbations, $\unicode[STIX]{x1D6FF}f_{i}=\unicode[STIX]{x1D6FF}f_{i}(x,\boldsymbol{v})$ , and defining

(2.15) $$\begin{eqnarray}\displaystyle g(x,v)=\frac{v_{\text{th}i}}{n_{0i}}\int \text{d}v_{y}\,\text{d}v_{z}\unicode[STIX]{x1D6FF}f_{i},\quad F_{\text{M}}(v)=\frac{1}{\sqrt{\unicode[STIX]{x03C0}}}\,e^{-v^{2}},\quad v=\frac{v_{x}}{v_{\text{th}i}},\quad \unicode[STIX]{x1D711}=\frac{Ze\unicode[STIX]{x1D719}}{T_{i}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $Ze\equiv q_{i}$ , we find that $g$ again satisfies (2.7). Using (2.14) and again adding an external forcing $\unicode[STIX]{x1D712}$ , we have

(2.16) $$\begin{eqnarray}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D712}=\frac{ZT_{e}}{T_{i}}\frac{\unicode[STIX]{x1D6FF}n_{i}}{n_{0i}}=\frac{ZT_{e}}{T_{i}}\int \text{d}v\,g\equiv \unicode[STIX]{x1D6FC}\int \text{d}v\,g.\end{eqnarray}$$

This is the same as (2.8), except now $\unicode[STIX]{x1D6FC}=ZT_{e}/T_{i}$ is a constant rather than a differential operator.

Note that the spatial and temporal coordinates in (2.7) can now be normalised $x/L\rightarrow x$ and $tv_{\text{th}i}/L\rightarrow t$ with an entirely arbitrary scale $L$ because the fundamental dynamics described by the ion equations – (damped) sound waves – does not have a special length scale.

2.3 Zakharov turbulence

Models allowing perturbations only on electron or only on ion scales are, in fact, of limited relevance to real plasma turbulence because interactions of Langmuir waves will promote coupling to low-frequency modes at the ion scales while those low-frequency modes will locally alter the plasma frequency, giving rise to a ‘modulational’ nonlinearity in the electron-scale dynamics. Such a ‘two-scale’ system has been extensively studied, mostly using the so-called Zakharov (Reference Zakharov1972) equations, or, to be precise, a version of them in which both the electron and ion densities obey fluid-like equations (see reviews by Rudakov & Tsytovich Reference Rudakov and Tsytovich1978, Thornhill & ter Haar Reference Thornhill and ter Haar1978, Goldman Reference Goldman1984, Zakharov, Musher & Rubenchik Reference Zakharov, Musher and Rubenchik1985, Musher, Rubenchik & Zakharov Reference Musher, Rubenchik and Zakharov1995, Tsytovich Reference Tsytovich1995, Robinson Reference Robinson1997 and Kingsep Reference Kingsep2004, of which the first and the last are the most readable) For electrons, the fluid approximation requires $k\unicode[STIX]{x1D706}_{\text{D}e}\ll 1$ and for ions, $T_{i}\ll T_{e}$ , so neither electron nor ion Landau damping are then important (because the phase velocities of the Langmuir and sound waves are much greater than $v_{\text{th}e}$ and $v_{\text{th}i}$ , respectively).

In a traditional approach to plasma turbulence, turbulence is what occurs at the scales (and in parameter regimes) where the dominant interactions are between wave modes (e.g. Langmuir or sound), which conserve fluctuation energy and transfer it, via a ‘cascade’, to scales where waves can interact with particles – usually via Landau damping, linear and/or nonlinear. The latter processes are expected to lead to absorption of the wave energy by particles, i.e. its conversion into heat. Thus, regimes and scale ranges in which kinetic physics matters are viewed as dissipative (analogous to viscous scales in hydrodynamic turbulence).

If one is not committed to such a dismissive attitude to kinetics in the way that a turbulence theorist in search of a fluid model might be, one may wish to explore how Zakharov’s turbulence interfaces with the phase space. While the fluid approximation for electrons at long wavelengths ( $k\unicode[STIX]{x1D706}_{\text{D}e}\ll 1$ ) is sensible, the assumption of cold ions and hence unfettered sound propagation is fairly restrictive, so one may wish to remove it. It is then possible to derive a kinetic version of Zakharov’s equations (as Zakharov Reference Zakharov1972 in fact did), in which the ion kinetic equation stays intact [this is (2.1) with $s=i$ and $q_{s}=Ze$ ], but the potential $\unicode[STIX]{x1D719}$ in this equation is the ion-time-scale ( ${\sim}1/kv_{\text{th}i}$ ) averaged potential – a kind of mean field against the background of electron-time-scale Langmuir oscillations. This mean potential is determined from the Poisson equation, which, since $k\unicode[STIX]{x1D706}_{\text{D}e}\ll 1$ , again takes the form of the quasineutrality constraint (2.14), but the ion-time-scale electron-density perturbation now contains both the Boltzmann response and the so-called ponderomotive one – essentially an effective pressure due to the average energy density of the Langmuir oscillations:

(2.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n_{i}}{n_{0i}}=\frac{\unicode[STIX]{x1D6FF}\bar{n}_{e}}{n_{0e}}=\frac{e\bar{\unicode[STIX]{x1D719}}}{T_{e}}-\frac{\overline{|\mathfrak{E}|^{2}}}{8\unicode[STIX]{x03C0}n_{0e}T_{e}},\end{eqnarray}$$

where overbars denote averages over the electron time scales and $\mathfrak{E}$ is the electric field associated with the Langmuir waves. The resulting ion equations are the same as those derived in § 2.2, viz., the kinetic equation (2.7) with the definitions (2.15) (but $\unicode[STIX]{x1D719}\rightarrow \bar{\unicode[STIX]{x1D719}}$ ) and $\unicode[STIX]{x1D711}$ given by (2.16), or, equivalently, by (2.8), but with the ‘external’ forcing now having a concrete physical meaning:

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\frac{\overline{|\mathfrak{E}|^{2}}}{8\unicode[STIX]{x03C0}n_{0i}T_{i}}.\end{eqnarray}$$

This forcing is, in fact, not independent of either $\unicode[STIX]{x1D711}$ or $\unicode[STIX]{x1D6FF}f_{i}$ , as $\mathfrak{E}$ satisfies a ‘fluid’ equation for the Langmuir oscillations with the plasma frequency modulated by $\unicode[STIX]{x1D6FF}\bar{n}_{e}$ (see Zakharov Reference Zakharov1972 or any of the reviews cited above; a systematic derivation of Zakharov’s equations from kinetics, which is surprisingly difficult to locate in the literature, can be found in Schekochihin Reference Schekochihin2017).

Thus, yet again, we have a system of equations that is mathematically similar to the 1-D Vlasov–Poisson system (2.7) and (2.8).

2.4 Stochastic-acceleration problem

A simpler problem than the three preceding ones is to consider a population of ‘test particles’, embedded in an externally imposed, statistically stationary stochastic electric field $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ , and seek these particles’ distribution function. It satisfies Vlasov’s equation (2.1), with collision integral now omitted. It can be restricted to one dimension either by fiat or by considering particles in a magnetic field being accelerated by fast electric fluctuations parallel to it. In the limit of the field $E$ having a short correlation time compared to the characteristic time for particles to become trapped in the potential wells associated with $E$ , the particles’ spatially averaged distribution function is easily shown to satisfy a diffusion equation (Sturrock Reference Sturrock1966):Footnote ¹

(2.19) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}f_{0}}{\unicode[STIX]{x2202}t}=D\,\frac{\unicode[STIX]{x2202}^{2}f_{0}}{\unicode[STIX]{x2202}v^{2}},\quad D=\frac{e^{2}}{m^{2}}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\langle E(t)E(t-\unicode[STIX]{x1D70F})\rangle ,\end{eqnarray}$$

where the electric-field correlation function should generally speaking be taken along particles’ trajectories, but, in the limit of short correlation times, it is the same as the Eulerian correlation function.

The solution of (2.19) (assuming an initial $\unicode[STIX]{x1D6FF}$ -shaped particle distribution) is a 1-D Maxwellian with $v_{\text{th}}^{2}=4Dt$ , expressing gradual secular heating of the test-particle population. At long times, this evolution can be treated as slow compared to the evolution of the perturbation $\unicode[STIX]{x1D6FF}f$ and so the latter considered to evolve against the background of a quasi-constant Maxwellian equilibrium. With the same normalisations as in § 2.1, the perturbed distribution function again satisfies (2.7), but $\unicode[STIX]{x1D711}$ is now an external field with prescribed statistical properties, entirely decoupled from $g$ . This, of course, corresponds to setting $\unicode[STIX]{x1D6FC}=0$ in (2.8).

3.1 Hermite moments: waves and phase mixing

We will work in the Fourier–Hermite space, decomposing the perturbed distribution as follows

(3.1) $$\begin{eqnarray}\displaystyle g(x,v)=\mathop{\sum }_{k}\text{e}^{\text{i}kx}\mathop{\sum }_{m}\frac{H_{m}(v)F_{\text{M}}(v)}{\sqrt{2^{m}m!}}\,g_{k,m},\quad g_{k,m}=\int \frac{\text{d}x}{2\unicode[STIX]{x03C0}L}\,\text{e}^{-\text{i}kx}\int \text{d}v\,\frac{H_{m}(v)}{\sqrt{2^{m}m!}}\,g(x,v), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $L$ is the system size. The Hermite polynomials

(3.2) $$\begin{eqnarray}H_{m}(v)=(-1)^{m}\text{e}^{v^{2}}\frac{\text{d}^{m}}{\text{d}v^{m}}\text{e}^{-v^{2}},\quad \int \text{d}v\,H_{m}(v)H_{n}(v)F_{\text{M}}(v)=2^{m}m!\,\unicode[STIX]{x1D6FF}_{mn},\end{eqnarray}$$

form a convenient orthogonal basis for handling 1-D perturbations to a Maxwellian. It is in anticipation of the Hermite decomposition that the Lenard–Bernstein collision operator (2.9) was chosen, as the Hermite polynomials are its eigenfunctions:

(3.3) $$\begin{eqnarray}C[g_{k,m}]=-\unicode[STIX]{x1D708}mg_{k,m}.\end{eqnarray}$$

To enforce momentum and energy conservation, we overrule this with $C[g_{k,1}]=0$ and $C[g_{k,2}]=0$ .

Using the identities

(3.4) $$\begin{eqnarray}vH_{m}=\frac{1}{2}\,H_{m+1}+mH_{m-1},\quad \frac{\text{d}H_{m}}{\text{d}v}=2mH_{m-1},\end{eqnarray}$$

denoting the particle density, flow velocity and temperature by

(3.5) $$\begin{eqnarray}n_{k}=g_{k,0},\quad u_{k}=\int \text{d}v\,vg_{k}(v)=\frac{g_{k,1}}{\sqrt{2}},\quad T_{k}=\sqrt{2}\,g_{k,2},\end{eqnarray}$$

and noticing that (2.8) then amounts to

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{k}=\unicode[STIX]{x1D6FC}_{k}n_{k}+\unicode[STIX]{x1D712}_{k},\end{eqnarray}$$

we arrive at the following spectral representation of (2.7):

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}n_{k}}{\unicode[STIX]{x2202}t}+\text{i}ku_{k}=0, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{k}}{\unicode[STIX]{x2202}t}+\text{i}k\left(\frac{T_{k}}{2}+\frac{1+\unicode[STIX]{x1D6FC}_{k}}{2}\,n_{k}\right)+\frac{1}{2}\mathop{\sum }_{p}\text{i}p\unicode[STIX]{x1D711}_{p}n_{k-p}=-\frac{\text{i}k\unicode[STIX]{x1D712}_{k}}{2}, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{k}}{\unicode[STIX]{x2202}t}+\text{i}k\left(\sqrt{3}\,g_{k,3}+2u_{k}\right)+2\mathop{\sum }_{p}\text{i}p\unicode[STIX]{x1D711}_{p}u_{k-p}=0, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}g_{k,m}}{\unicode[STIX]{x2202}t}+\text{i}k\left(\sqrt{\frac{m+1}{2}}\,g_{k,m+1}+\sqrt{\frac{m}{2}}\,g_{k,m-1}\right)+\sqrt{\frac{m}{2}}\mathop{\sum }_{p}\text{i}p\unicode[STIX]{x1D711}_{p}g_{k-p,m-1}=-\unicode[STIX]{x1D708}mg_{k,m}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

the last equation describing all $m\geqslant 3$ .

In the absence of sources, nonlinearities and heat fluxes ( $g_{k,3}=0$ ), (3.7)–(3.9) describe 1-D hydrodynamics of plasma waves (Langmuir waves for the electron model and ion-acoustic waves for the ion one). This becomes particularly obvious if we work in terms of the linear eigenfunctions

(3.11) $$\begin{eqnarray}n_{k}^{\pm }=n_{k}\pm \frac{k}{\unicode[STIX]{x1D714}_{k}}\,u_{k},\quad \unicode[STIX]{x1D714}_{k}=k\sqrt{\frac{3+\unicode[STIX]{x1D6FC}_{k}}{2}},\end{eqnarray}$$

denote $\unicode[STIX]{x1D703}_{k}=T_{k}-2n_{k}$ (the non-adiabatic part of the temperature) and recast (3.7)–(3.9) as follows

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}n_{k}^{\pm }}{\unicode[STIX]{x2202}t}\pm \text{i}\unicode[STIX]{x1D714}_{k}n_{k}^{\pm }=\mp \frac{\text{i}k}{2\unicode[STIX]{x1D714}_{k}}\left[k(\unicode[STIX]{x1D712}_{k}+\unicode[STIX]{x1D703}_{k})+\mathop{\sum }_{p}p\unicode[STIX]{x1D711}_{p}n_{k-p}\right], & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{k}}{\unicode[STIX]{x2202}t}=-\text{i}k\sqrt{3}\,g_{k,3}-2\mathop{\sum }_{p}\text{i}p\unicode[STIX]{x1D711}_{p}u_{k-p}. & \displaystyle\end{eqnarray}$$

At any given $k$ , the fluctuating fields $n_{k}^{\pm }$ oscillate at frequency $\unicode[STIX]{x1D714}_{k}$ (the Langmuir frequency or the ion sound frequency), with their energy injected by the forcing $\unicode[STIX]{x1D712}_{k}$ . The nonlinear term, which, since $\unicode[STIX]{x1D711}_{p}=\unicode[STIX]{x1D6FC}_{p}n_{p}+\unicode[STIX]{x1D712}_{p}$ , includes both self-interaction and advection by the external potential $\unicode[STIX]{x1D712}_{p}$ , can, in general, transfer wave energy to different wavenumbers, drain it or inject it. The term containing $\unicode[STIX]{x1D703}_{k}$ connects the wave dynamics to the entire hierarchy of higher Hermite moments, which evolve according to (3.10).

In a linear system, the latter effect would give rise to Landau damping: the coupling of lower-order Hermite moments to higher-order ones that appears in the second term on the left-hand side of (3.10) ‘phase mixes’ perturbations to ever higher $m$ ’s which represents emergence of ever finer structure in velocity space (at large $m$ , the Hermite transform is effectively similar to a Fourier transform in $v$ , with ‘frequency’ $\sqrt{2m}$ , so moments of order $m$ represent velocity-space structures with scale $\unicode[STIX]{x1D6FF}v\sim \unicode[STIX]{x03C0}/\!\sqrt{m}$ ). Eventually this activates collisions, however small their frequency $\unicode[STIX]{x1D708}$ might be, and the dynamics becomes irreversible. In the presence of nonlinearity, the situation is more complicated, with the last term on the left-hand side of (3.10) causing a kind of advection of higher Hermite moments by the wave field  $\unicode[STIX]{x1D711}_{k}$ . This gives rise to filamentation of the distribution function not just in velocity but also in position space (O’Neil Reference O’Neil1965; Manheimer Reference Manheimer1971; Dupree Reference Dupree1972). The resulting coupling between different wavenumbers can trigger plasma echoes (Gould et al. Reference Gould, O’Neil and Malmberg1967), or anti-phase-mixing, leading to cancellation, on average, of the Landau damping (Parker & Dellar Reference Parker and Dellar2015; Parker et al. Reference Parker, Highcock, Schekochihin and Dellar2016; Schekochihin et al. Reference Schekochihin, Parker, Highcock, Dellar, Dorland and Hammett2016). It is with the latter phenomenon that we will be concerned in what follows, as we seek to characterise both spatial and velocity structure of the distribution function in terms of its $k$ and $m$ spectra.

3.2 Energy fluxes

We can define the energy spectrum of our waves to be

(3.14) $$\begin{eqnarray}W_{k}=\frac{3+\unicode[STIX]{x1D6FC}_{k}}{2}\left\langle |n_{k}|^{2}\right\rangle +\left\langle |u_{k}|^{2}\right\rangle =\frac{\unicode[STIX]{x1D714}_{k}^{2}}{2k^{2}}\left\langle |n_{k}^{+}|^{2}+|n_{k}^{-}|^{2}\right\rangle .\end{eqnarray}$$

Using (3.12), we find that it evolves according to

(3.15) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W_{k}}{\unicode[STIX]{x2202}t}=k\,\text{Im}\left\langle \unicode[STIX]{x1D712}_{k}u_{k}^{\ast }\right\rangle +k\,\text{Im}\left\langle \unicode[STIX]{x1D703}_{k}u_{k}^{\ast }\right\rangle +\mathop{\sum }_{p}p\,\text{Im}\left\langle \unicode[STIX]{x1D711}_{p}n_{k-p}u_{k}^{\ast }\right\rangle .\end{eqnarray}$$

The first term on the right-hand side is the energy injection, the last term involves interactions between waves (it does not in general integrate to zero because waves can exchange energy, nonlinearly, with particles), whereas the second term is responsible for energy removal via phase mixing. We can see how this is picked up by higher Hermite moments if we define the Fourier–Hermite spectrum and Hermite fluxFootnote ²

(3.16) $$\begin{eqnarray}C_{k,m}=\frac{1}{2}\left\langle |g_{k,m}|^{2}\right\rangle ,\quad \unicode[STIX]{x1D6E4}_{k,m}=k\sqrt{\frac{m+1}{2}}\,\text{Im}\left\langle g_{k,m+1}^{\ast }g_{k,m}\right\rangle\end{eqnarray}$$

and deduce from (3.10) the evolution equation for the spectrum:

(3.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}C_{k,m}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6E4}_{k,m}-\unicode[STIX]{x1D6E4}_{k,m-1}+2\unicode[STIX]{x1D708}mC_{k,m}=\sqrt{\frac{m}{2}}\mathop{\sum }_{p}p\,\text{Im}\left\langle \unicode[STIX]{x1D711}_{p}g_{k-p,m-1}g_{k,m}^{\ast }\right\rangle .\end{eqnarray}$$

The phase-mixing term in (3.15) is

(3.18) $$\begin{eqnarray}k\,\text{Im}\langle \unicode[STIX]{x1D703}_{k}u_{k}^{\ast }\rangle =k\left[\text{Im}\left\langle g_{k,2}g_{k,1}^{\ast }\right\rangle -2\left\langle n_{k}u_{k}^{\ast }\right\rangle \right]=-\unicode[STIX]{x1D6E4}_{k,1}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left\langle |n_{k}|^{2}\right\rangle\end{eqnarray}$$

(the sloshing about of the fluctuation energy associated with the wave motion, represented by the time derivative, averages out in the statistical steady state). The Hermite flux in (3.17) passes energy along to higher $m$ ’s until the collision term is large enough to erase it. Our strategy will be to work out a universal form for $\unicode[STIX]{x1D6E4}_{k,m}$ in a turbulent plasma at high $m$ . Ideally, one would deduce from that what $\unicode[STIX]{x1D6E4}_{k,1}$ is, on average. In practice, we shall be able to predict that if one keeps a certain unspecified order-unity number $m$ of Hermite moments (‘order-unity’ meaning finite and independent of collisionality, however small the latter is), the energy flux $\unicode[STIX]{x1D6E4}_{k,m}$ from/to these moments to/from the rest of phase space is zero in a certain ‘inertial’ range of wave numbers  $k$ .

3.3 High- $m$ dynamics

Let us focus on the dynamics at $m\gg 1$ , deep in phase space (one might think of this as the ‘inertial range’ of phase-space turbulence). If

(3.19) $$\begin{eqnarray}p\unicode[STIX]{x1D711}_{p}\ll k,\quad \frac{\unicode[STIX]{x1D714}_{k}}{k}=\sqrt{\frac{3+\unicode[STIX]{x1D6FC}_{k}}{2}}\ll \sqrt{m},\end{eqnarray}$$

then, to lowest order in $1/\!\sqrt{m}$ , (3.10) gives us simply

(3.20) $$\begin{eqnarray}g_{k,m+1}\approx -g_{k,m-1}.\end{eqnarray}$$

This implies

(3.21) $$\begin{eqnarray}g_{k,m+1}\approx \pm \text{i}g_{k,m},\end{eqnarray}$$

i.e. $\text{i}^{m}g_{k,m}$ is either continuous or sign alternating. When $k>0$ , these two possibilities correspond to phase-mixing and anti-phase-mixing modes, respectively, and vice versa for $k<0$ (Kanekar et al. Reference Kanekar, Schekochihin, Dorland and Loureiro2015; Parker & Dellar Reference Parker and Dellar2015; Schekochihin et al. Reference Schekochihin, Parker, Highcock, Dellar, Dorland and Hammett2016). Let us separate these two cases explicitly.

In view of (3.20), the function

(3.22) $$\begin{eqnarray}G_{k,m}=\frac{\text{i}^{m}g_{k,m}+\text{i}^{m+1}g_{k,m+1}}{2}\end{eqnarray}$$

will be approximately continuous in $m$ : indeed,

(3.23) $$\begin{eqnarray}G_{k,m}-G_{k,m-1}=\frac{\text{i}^{m+1}}{2}(g_{k,m+1}+g_{k,m-1})\approx 0.\end{eqnarray}$$

Therefore, it is legitimate to treat $m$ as a continuous variable and approximate

(3.24) $$\begin{eqnarray}G_{k,m+1}\approx G_{k,m}+\frac{\unicode[STIX]{x2202}G_{k,m}}{\unicode[STIX]{x2202}m},\quad G_{k,m-1}\approx G_{k,m}-\frac{\unicode[STIX]{x2202}G_{k,m}}{\unicode[STIX]{x2202}m},\quad \text{etc.}\end{eqnarray}$$

treating the derivative terms as small. Using this approximation, we can cast (3.10) in the following approximate form, valid to lowest order in $1/\sqrt{m}$ ,

(3.25) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}G_{k,m}}{\unicode[STIX]{x2202}t}+\sqrt{2}\,k\,m^{1/4}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}m}m^{1/4}G_{k,m}+\unicode[STIX]{x1D708}mG_{k,m}=\sqrt{\frac{m}{2}}\mathop{\sum }_{p}p\unicode[STIX]{x1D711}_{p}G_{k-p,m}.\end{eqnarray}$$

Finally, if we define

(3.26) $$\begin{eqnarray}\tilde{f}_{k}(s)=m^{1/4}G_{k,m},\quad s=\sqrt{m},\end{eqnarray}$$

(3.25) becomes

(3.27) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\tilde{f}_{k}}{\unicode[STIX]{x2202}t}+\frac{k}{\sqrt{2}}\frac{\unicode[STIX]{x2202}\tilde{f}_{k}}{\unicode[STIX]{x2202}s}+\unicode[STIX]{x1D708}s^{2}\tilde{f}_{k}=\frac{s}{\sqrt{2}}\mathop{\sum }_{p}p\unicode[STIX]{x1D711}_{p}\,\tilde{f}_{k-p}.\end{eqnarray}$$

This is very similar to (3.12) of Schekochihin et al. (Reference Schekochihin, Parker, Highcock, Dellar, Dorland and Hammett2016) and we have kept their notation for a faithful reader’s convenience.Footnote ³ It is manifest in (3.27) that when $k>0$ , $\tilde{f}_{k}$ propagates to higher $s$ (phase mixes) and when $k<0$ , it propagates to lower $s$ (anti-phase-mixes) and that the coupling between wavenumbers in the nonlinear term can turn phase-mixing perturbations into anti-phase-mixing ones and vice versa.

The distribution function itself can be reconstructed from $\tilde{f}_{k}$ , or, equivalently, from $G_{k,m}$ , as followsFootnote ⁴

(3.28) $$\begin{eqnarray}g_{k,m}=(-\text{i})^{m}G_{k,m}+\text{i}^{m}G_{-k,m}^{\ast }.\end{eqnarray}$$

The fact that, for any given $k$ , both $G_{k,m}$ and $G_{-k,m}$ are necessary to reconstruct $g_{k,m}$ reflects the presence of both phase-mixing and anti-phase-mixing modes in any distribution function. Maintaining a solution of (3.27) with no anti-phase-mixing, viz., $\tilde{f}_{k}=0$ for all $k<0$ , is clearly only possible in the absence of the nonlinearity.

Finally, if we define

(3.29) $$\begin{eqnarray}F_{k}=\langle |\,\tilde{f}_{k}|^{2}\rangle =\sqrt{m}\left\langle |G_{k,m}|^{2}\right\rangle ,\end{eqnarray}$$

both the Fourier–Hermite spectrum $C_{k,m}$ and the Hermite flux $\unicode[STIX]{x1D6E4}_{k,m}$ , defined in (3.16), can be reconstructed from $F_{k}$ :

(3.30) $$\begin{eqnarray}\displaystyle & \displaystyle C_{k,m}\approx \frac{C_{k,m}+C_{k,m+1}}{2}=\frac{\left\langle |G_{k,m}|^{2}+|G_{-k,m}|^{2}\right\rangle }{2}=\frac{F_{k}+F_{-k}}{2\sqrt{m}}, & \displaystyle\end{eqnarray}$$

(3.31) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{k,m}=k\sqrt{\frac{m+1}{2}}\left\langle |G_{k,m}|^{2}-|G_{-k,m}|^{2}\right\rangle \approx \frac{k}{\sqrt{2}}\left(F_{k}-F_{-k}\right). & \displaystyle\end{eqnarray}$$

The derivation of these relations relies on (3.28) and on noticing also that

(3.32) $$\begin{eqnarray}\text{i}g_{k,m+1}=(-\text{i})^{m}G_{k,m}-\text{i}^{m}G_{-k,m}^{\ast }.\end{eqnarray}$$

Thus, the Fourier–Hermite spectrum is the average of the spectra of the phase-mixing and anti-phase-mixing modes and the Hermite flux is their difference. It remains to solve for $F_{k}$ , which, using (3.27), is immediately found to satisfy

(3.33) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}t}+\frac{k}{\sqrt{2}}\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}s}+2\unicode[STIX]{x1D708}s^{2}F_{k}=s\sqrt{2}\,\text{Re}\mathop{\sum }_{p}p\langle \unicode[STIX]{x1D711}_{p}\,\tilde{f}_{k-p}\tilde{f}_{k}^{\ast }\rangle .\end{eqnarray}$$

This equation, which is an approximate continuous version of (3.17), is not closed and so we will need a plausible method for handling its right-hand side.

4.1 Kraichnan–Kazantsev model

We shall be brutal and obtain a closure for the right-hand side of (3.33) by modelling $\unicode[STIX]{x1D711}_{p}$ as a random Gaussian white-noise (short-time-correlated) field,Footnote ⁵

(4.1) $$\begin{eqnarray}\left\langle \unicode[STIX]{x1D711}_{p}(t)\unicode[STIX]{x1D711}_{p^{\prime }}(t^{\prime })\right\rangle =2\unicode[STIX]{x1D718}_{p}\unicode[STIX]{x1D6FF}_{p,-p^{\prime }}\unicode[STIX]{x1D6FF}(t-t^{\prime }).\end{eqnarray}$$

This assumption, pioneered by Kraichnan (Reference Kraichnan1968) (for the passive-scalar problem) and Kazantsev (Reference Kazantsev1968) (for the turbulent-dynamo problem) is of course quantitatively wrong, but there is a long and encouraging history in fluid dynamics and magneto-hydrodynamics of the resulting closure leading to results that are basically correct (e.g. Kraichnan Reference Kraichnan1974, Reference Kraichnan1994; Zeldovich, Ruzmaikin & Sokoloff Reference Zeldovich, Ruzmaikin and Sokoloff1990; Krommes Reference Krommes1997; Falkovich et al. Reference Falkovich, Gawȩdzki and Vergassola2001; Boldyrev & Cattaneo Reference Boldyrev and Cattaneo2004; Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004a ; Schekochihin, Haynes & Cowley Reference Schekochihin, Haynes and Cowley2004b ; Schekochihin et al. Reference Schekochihin, Iskakov, Cowley, McWilliams, Proctor and Yousef2007; Bhat & Subramanian Reference Bhat and Subramanian2015). In the present context, what we are doing formally amounts to ignoring the contribution of the density $n_{k}=g_{k,0}$ to $\unicode[STIX]{x1D711}_{k}$ in (3.6) and stipulating the statistics (4.1) for the external forcing $\unicode[STIX]{x1D712}_{k}$ . Physically, we are assuming that the advecting stochastic electric field can be treated as statistically independent of the phase-space structure of the distribution function. A reader unconvinced that this can ever be a valid approximation for any aspect of the Vlasov problem with a self-consistent electric field, might find comfort in considering the calculations that follow to apply solely to the stochastic-acceleration problem, where $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D712}$ (§ 2.4).

For a Gaussian field, by the theorem of Furutsu (Reference Furutsu1963) and Novikov (Reference Novikov1965),

(4.2) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}_{p}\tilde{f}_{k-p}\tilde{f}_{k}^{\ast }\rangle (t)=\int ^{t}\text{d}t^{\prime }\mathop{\sum }_{p^{\prime }}\left\langle \unicode[STIX]{x1D711}_{p}(t)\unicode[STIX]{x1D711}_{p^{\prime }}(t^{\prime })\right\rangle \left\langle \frac{\unicode[STIX]{x1D6FF}[\,\tilde{f}_{k-p}(t)\tilde{f}_{k}^{\ast }(t)]}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}_{p^{\prime }}(t^{\prime })}\right\rangle .\end{eqnarray}$$

Using (3.27) to write an evolution equation for $\tilde{f}_{k-p}(t)\tilde{f}_{k}^{\ast }(t)$ and then formally integrating it over time, we find

(4.3) $$\begin{eqnarray}\displaystyle \tilde{f}_{k-p}(t)\tilde{f}_{k}^{\ast }(t) & = & \displaystyle \int ^{t}\text{d}t^{\prime \prime }\left\{-\frac{k}{\sqrt{2}}\tilde{f}_{k-p}\frac{\unicode[STIX]{x2202}\tilde{f}_{k}^{\ast }}{\unicode[STIX]{x2202}s}-\frac{k-p}{\sqrt{2}}\tilde{f}_{k}^{\ast }\frac{\tilde{f}_{k-p}}{\unicode[STIX]{x2202}s}-2\unicode[STIX]{x1D708}s^{2}\tilde{f}_{k-p}\tilde{f}_{k}^{\ast }\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{s}{\sqrt{2}}\mathop{\sum }_{p^{\prime \prime }}p^{\prime \prime }\left[\unicode[STIX]{x1D711}_{p^{\prime \prime }}\tilde{f}_{k-p-p^{\prime \prime }}\tilde{f}_{k}^{\ast }+\unicode[STIX]{x1D711}_{-p^{\prime \prime }}\tilde{f}_{k-p^{\prime \prime }}^{\ast }\tilde{f}_{k-p}\right]\right\}(t^{\prime \prime }).\end{eqnarray}$$

Therefore, its functional derivative is

(4.4) $$\begin{eqnarray}\displaystyle \left\langle \frac{\unicode[STIX]{x1D6FF}[\tilde{f}_{k-p}(t)\tilde{f}_{k}^{\ast }(t)]}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}_{p^{\prime }}(t^{\prime })}\right\rangle =\frac{s}{\sqrt{2}}\,p^{\prime }\left[\left\langle \tilde{f}_{k-p-p^{\prime }}(t^{\prime })\tilde{f}_{k}^{\ast }(t^{\prime })\right\rangle -\left\langle \tilde{f}_{k+p^{\prime }}^{\ast }(t^{\prime })\tilde{f}_{k-p}(t^{\prime })\right\rangle +\cdots \,\right]H(t-t^{\prime }), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $H(t-t^{\prime })$ is the Heaviside function, expressing the fact that, by causality, $\tilde{f}(t)$ cannot depend on $\unicode[STIX]{x1D711}(t^{\prime })$ at a future time $t^{\prime }>t$ , and ‘ $\cdots \,$ ’ stands for terms that vanish when $t^{\prime }=t$ . Substituting (4.1) and (4.4) into (4.2) gives

(4.5) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}_{p}\tilde{f}_{k-p}\tilde{f}_{k}^{\ast }\rangle =-\frac{s}{\sqrt{2}}\,p\unicode[STIX]{x1D718}_{p}\left(F_{k}-F_{k-p}\right).\end{eqnarray}$$

Finally, using this in (3.33), we get

(4.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}t}+\frac{k}{\sqrt{2}}\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}s}+2\unicode[STIX]{x1D708}s^{2}F_{k}=s^{2}\mathop{\sum }_{p}p^{2}\unicode[STIX]{x1D718}_{p}\left(F_{k-p}-F_{k}\right).\end{eqnarray}$$

The $F_{k}$ term on the right-hand side is an additional, ‘turbulent’ collisionality (turbulent diffusion in velocity space); the $F_{k-p}$ term is the mode-coupling term responsible for moving energy around and for converting phase-mixing modes into anti-phase-mixing ones or vice versa.

In steady state, $\unicode[STIX]{x2202}F_{k}/\unicode[STIX]{x2202}t=0$ and (4.6) can be recast in an even simpler form: dividing through by $2s^{2}$ , we arrive at

(4.7) $$\begin{eqnarray}k\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D708}F_{k}=\frac{1}{2}\mathop{\sum }_{p}p^{2}\unicode[STIX]{x1D718}_{p}\left(F_{k-p}-F_{k}\right),\quad \unicode[STIX]{x1D70F}\equiv \frac{(\sqrt{2}\,s)^{3}}{3}=\frac{(2m)^{3/2}}{3}.\end{eqnarray}$$

4.2 Energy budget and collisions

It is an important property of the Kraichnan–Kazantsev model applied to our problem that, in (4.6), the nonlinear interactions disappear under summation over all $k$ and so the total ‘energy’ of the $\tilde{f}$ field has a conservation law:

(4.8) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int _{s_{0}}^{\infty }\text{d}s\mathop{\sum }_{k}F_{k}=\frac{1}{\sqrt{2}}\mathop{\sum }_{k}kF_{k}(s_{0})-2\unicode[STIX]{x1D708}\int _{s_{0}}^{\infty }\text{d}s\,s^{2}\mathop{\sum }_{k}F_{k},\end{eqnarray}$$

where $s_{0}$ is some suitably chosen lower cutoff and the energy balance is between the flux through that cutoff (from or towards the waves at low $m$ ) and collisional dissipation. Restating this in the steady state and with the $\unicode[STIX]{x1D70F}$ variable (4.7),

(4.9) $$\begin{eqnarray}\int _{-\infty }^{+\infty }\text{d}k\,kF_{k}(\unicode[STIX]{x1D70F}_{0})=\unicode[STIX]{x1D708}\int _{\unicode[STIX]{x1D70F}_{0}}^{\infty }\text{d}\unicode[STIX]{x1D70F}\int _{-\infty }^{+\infty }\text{d}kF_{k}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

This energy balance admits two distinct physical scenarios.

One is essentially similar to what happens in the absence of nonlinearity ( $\unicode[STIX]{x1D718}_{p}=0$ ): in the limit of small $\unicode[STIX]{x1D708}$ , the spectrum $F_{k}$ is independent of $\unicode[STIX]{x1D70F}$ (or of $s$ , or of $m$ ), giving us the shallow $C_{k,m}\propto 1/\sqrt{m}$ slope associated with a Landau-damped solution (Zocco & Schekochihin Reference Zocco and Schekochihin2011; Kanekar et al. Reference Kanekar, Schekochihin, Dorland and Loureiro2015). Collisions become important at $\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D708}^{-1}$ [see (4.7)] and so the dissipation term in the right-hand side of (4.9) is finite and independent of collisionality as $\unicode[STIX]{x1D708}\rightarrow +0$ . This in turn implies that the integral in left-hand side of (4.9) must be finite and non-zero (in the linear regime, $F_{k<0}=0$ , so the integral is always positive and will be finite as long as the wavenumber spectrum decays fast enough).

The second scenario arises from any Hermite spectral slope that makes $F_{k}(\unicode[STIX]{x1D70F})$ decay with $\unicode[STIX]{x1D70F}$ . Then the collisional dissipation vanishes as $\unicode[STIX]{x1D708}\rightarrow +0$ , i.e. the limit of vanishing collisionality is non-singular in this sense, giving us license simply to set $\unicode[STIX]{x1D708}=0$ in (4.7) and expect to find a legitimate solution. The solution is indeed a legitimate steady-state solution if the integral in the left-hand side of (4.9) vanishes for it, i.e. if the overall energy flux into Hermite space is zero:

(4.10) $$\begin{eqnarray}\int _{-\infty }^{+\infty }\text{d}k\,kF_{k}(\unicode[STIX]{x1D70F}_{0})=\int _{0}^{+\infty }\text{d}k\,k\left[F_{k}(\unicode[STIX]{x1D70F}_{0})-F_{-k}(\unicode[STIX]{x1D70F}_{0})\right]=\int _{0}^{+\infty }\text{d}k\sqrt{2}\,\unicode[STIX]{x1D6E4}_{k,m_{0}}=0,\end{eqnarray}$$

although there is no a priori requirement that the Hermite flux must vanish at every $k$ . We shall see that this is exactly the state that emerges in the nonlinear regime.

4.3 Batchelor limit

While (4.7) is a closed and compact equation, it is an integral one and not necessarily easily amenable to analytical solution. We are going to make a further simplification by assuming that $p^{2}\unicode[STIX]{x1D718}_{p}$ decays sufficiently steeply with $p$ that it is meaningful to consider $F_{k}$ at $|k|\gg p$ , an approach pioneered by Batchelor (Reference Batchelor1959) in the context of passive-scalar mixing (with the more quantitative theory due to Kraichnan Reference Kraichnan1974). We can then expand under the wavenumber sum in (4.7):

(4.11) $$\begin{eqnarray}F_{k-p}-F_{k}\approx -p\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}+\frac{1}{2}\,p^{2}\frac{\unicode[STIX]{x2202}^{2}F_{k}}{\unicode[STIX]{x2202}k^{2}}.\end{eqnarray}$$

Having noticed that the first term vanishes under summation because it is odd in $p$ while $\unicode[STIX]{x1D718}_{p}=\unicode[STIX]{x1D718}_{-p}$ , we obtain a rather simple differential equation:

(4.12) $$\begin{eqnarray}k\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D708}F_{k}=\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}F_{k}}{\unicode[STIX]{x2202}k^{2}},\quad \unicode[STIX]{x1D6FE}\equiv \frac{1}{4}\mathop{\sum }_{p}p^{4}\unicode[STIX]{x1D718}_{p}.\end{eqnarray}$$

The wavenumber diffusion rate $\unicode[STIX]{x1D6FE}$ can easily be scaled out.

It turns out (see appendix A) that $p^{2}\unicode[STIX]{x1D718}_{p}$ must decay more steeply than $p^{-2}$ at the very least, in order for this approximation to make sense, although it would have to be steeper than $p^{-3}$ in order for the $p$ integral that determines $\unicode[STIX]{x1D6FE}$ in (4.12) to converge without the need for a high- $p$ cutoff. In what follows, we shall effectively assume the advecting electric field to be single scale, with $\unicode[STIX]{x1D718}_{p}$ concentrated around some characteristic wavenumber  $p$ .

Equation (4.12) makes it clear how the nonlinearity causes anti-phase-mixing. At $k>0$ , the steady-state equation (4.12) can be thought of as a diffusion equation in $k$ (or, to be precise, in $|k|^{3/2}$ ; see § 5.1), with $\unicode[STIX]{x1D70F}$ playing the role of time. At $k<0$ , this ‘time’ reverses, i.e. diffusion turns into anti-diffusion. Whatever energy resides at any given $k>0$ and low $\unicode[STIX]{x1D70F}$ will, as $\unicode[STIX]{x1D70F}$ increases, spread over the $k$ space. Some of it can spread towards $k=0$ , where it crosses into the $k<0$ territory (the rules of this crossing are established in § 4.4) and diffuses back towards large (negative) $k$ and low $\unicode[STIX]{x1D70F}$ . This creates an anti-phase-mixing energy flux, which can (and will) cancel the phase-mixing flux. Collisions limit the values of $\unicode[STIX]{x1D70F}$ (and of $k$ ) available to these phase-space flows. We shall discuss their role more quantitatively in § 5.4, after we have the exact collisionless solution in hand.

4.4 Boundary conditions: continuity in $k$ space

We would like to be able to treat the solution of (4.7) in the region of low $|k|\sim p$ , where the Batchelor approximation is not valid, as a continuous extension of the solution of (4.12). In other words, we wish to prove that we can simply solve (4.12) with the boundary conditions

(4.13) $$\begin{eqnarray}F_{k\rightarrow +0}=F_{k\rightarrow -0},\quad {\left.\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}\right|_{k\rightarrow +0}=\left.\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}\right|}_{k\rightarrow -0},\end{eqnarray}$$

where $k\rightarrow \pm 0$ really means $k\rightarrow \pm p$ .

The continuity of $F_{k}$ across $k=0$ and all the way to $|k|\gg p$ (i.e. to the wavenumbers where the Batchelor approximation holds) can be inferred from (4.7) as follows. Let us ignore collisions and consider sufficiently small $k$ (and/or sufficiently large $\unicode[STIX]{x1D70F}$ ) so that the phase-mixing term is small compared to the nonlinear term: from (4.12), this is true for $|k|\ll (\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})^{1/3}$ , which can be satisfied already at $|k|\gg p$ , at least for $\unicode[STIX]{x1D70F}\gg p^{3}/\unicode[STIX]{x1D6FE}$ . Then (4.7) reduces to

(4.14) $$\begin{eqnarray}\mathop{\sum }_{p}p^{2}\unicode[STIX]{x1D718}_{p}F_{k-p}=\left(\mathop{\sum }_{p}p^{2}\unicode[STIX]{x1D718}_{p}\right)F_{k}.\end{eqnarray}$$

If we denote $p^{2}\unicode[STIX]{x1D718}_{p}=K_{p}$ , assume that the width of this function is smaller than the range of $k$ in which (4.14) is valid (unlike in appendix A, where the validity of this approach is probed), turn sums in (4.14) into integrals and Fourier transform (4.14), denoting the dual variable by $x$ , and endowing the transformed functions with hats, we get

(4.15) $$\begin{eqnarray}\hat{K}(x)\hat{F}(x)=\hat{K}(0)\hat{F}(x).\end{eqnarray}$$

The solution of this equation is $\hat{F}(x)\propto \unicode[STIX]{x1D6FF}(x)$ . Therefore, $F_{k}$ is independent of $k$ in a range of $k$ surrounding $0$ , with characteristic width ${\sim}p$ , the typical wavenumber of the advecting field  $\unicode[STIX]{x1D711}$ .

The second of the relations (4.13) (the continuity of the derivative) follows from the requirement that there can be no total energy flux in or out of $k=0$ (because $C_{k,m}$ must be even in $k$ by the reality condition). Therefore, from (3.30),

(4.16) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k}\left(F_{k}+F_{-k}\right)\right|_{k\rightarrow +0}=0\quad \Rightarrow \quad \left.\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}\right|_{k\rightarrow +0}=-\left.\frac{\unicode[STIX]{x2202}F_{-k}}{\unicode[STIX]{x2202}k}\right|_{k\rightarrow +0}=\left.\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}\right|_{k\rightarrow -0}.\end{eqnarray}$$

5.1 Plan of solution

If, as we promised in § 4.2, we are going to discover that phase mixing is substantially or fully suppressed, we must be able find such a solution from (4.12) with $\unicode[STIX]{x1D708}=0$ . Note that the $\unicode[STIX]{x1D70F}$ variable can now be shifted arbitrarily and so we can choose $\unicode[STIX]{x1D70F}=0$ to correspond to any true value of $m$ – so let $\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0}\rightarrow \unicode[STIX]{x1D70F}$ , where $\unicode[STIX]{x1D70F}_{0}$ is the lower cutoff introduced in § 4.2. Physically, the limit $\unicode[STIX]{x1D70F}\rightarrow 0$ corresponds to going back from the depths of phase space to low $m$ ’s, where the approximations that led to (3.27) break down.

With these further simplifications, equation (4.12) turns into a type of diffusion equation at $k>0$ and anti-diffusion at $k<0$ . Namely, letting

(5.1) $$\begin{eqnarray}F_{k}=\left\{\begin{array}{@{}l@{}}F^{+}(\unicode[STIX]{x1D709}),\quad k>0\\ F^{-}(\unicode[STIX]{x1D709}),\quad k<0,\end{array}\right.\quad \unicode[STIX]{x1D709}\equiv \frac{2}{3\sqrt{\unicode[STIX]{x1D6FE}}}|k|^{3/2},\end{eqnarray}$$

we find that $F^{\pm }$ satisfies

(5.2) $$\begin{eqnarray}\pm \frac{\unicode[STIX]{x2202}F^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}=\frac{1}{\unicode[STIX]{x1D709}^{1/3}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

The ‘ $+$ ’ version of this equation belongs to a class studied exhaustively by Sutton (Reference Sutton1943), who derived its Green’s functions for all standard initial and boundary-value problems. Armed with these, we are going to construct the full solution in the following way.

(i) First postulate an ‘initial’ condition and a boundary condition for $F^{+}$ and find the Green’s-function solution for $F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709})$ :

(5.3) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}F^{+}(\unicode[STIX]{x1D70F}\rightarrow 0,\unicode[STIX]{x1D709})=F_{0}^{+}(\unicode[STIX]{x1D709}),\\ F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}\rightarrow 0)=Y(\unicode[STIX]{x1D70F}),\end{array}\right\}\quad \Rightarrow \quad F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}),\end{eqnarray}$$

where $F_{0}^{+}(\unicode[STIX]{x1D709})$ and $Y(\unicode[STIX]{x1D70F})$ are some unknown functions. Obviously, as we are not interested in spectra that blow up at small scales, $F^{+}$ must vanish at $\unicode[STIX]{x1D709}\rightarrow \infty$ and the same will be required of $F^{-}$ .

(ii) Since we must have continuity, $F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}\rightarrow 0)=F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}\rightarrow 0)$ [see (4.13)], $F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709})$ is found as a Green’s-function solution with an unknown value $Y(\unicode[STIX]{x1D70F})$ on the boundary. Since the equation for $F^{-}$ is an anti-diffusion equation, the ‘initial’ condition for it must be set at large $\unicode[STIX]{x1D70F}$ and since there can be no energy at $\unicode[STIX]{x1D70F}\rightarrow \infty$ , this ‘initial’ condition is zero. Formally, this can be implemented by choosing some cutoff $\unicode[STIX]{x1D70F}_{\text{max}}$ , requiring the function to vanish there, solving, and then taking $\unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty$ . The validity of this operation will be confirmed by the finiteness of the result. Thus,

(5.4) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}\rightarrow 0)=Y(\unicode[STIX]{x1D70F}),\\ F^{-}(\unicode[STIX]{x1D70F}\rightarrow \unicode[STIX]{x1D70F}_{\text{max}},\unicode[STIX]{x1D709})=0,\end{array}\right\}\quad \Rightarrow \quad F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty .\end{eqnarray}$$

Operationally, the solution can be accomplished by changing variables to $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70F}$ , so the anti-diffusion equation turns into a diffusion one. Physically, $\unicode[STIX]{x1D70F}_{\text{max}}\sim 1/\unicode[STIX]{x1D708}$ .

(iii) The unknown function $Y(\unicode[STIX]{x1D70F})$ is now determined by the continuity of the energy flux across $k=0$ [see (4.13)]. Since $\unicode[STIX]{x2202}F_{k}/\unicode[STIX]{x2202}k=\pm \unicode[STIX]{x1D709}^{1/3}\unicode[STIX]{x2202}F^{\pm }/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ ,

(5.5) $$\begin{eqnarray}\left.\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}\rightarrow 0}=-\left.\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}\rightarrow 0}\quad \Rightarrow \quad Y(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

A key physical constraint is that $Y(\unicode[STIX]{x1D70F})$ should be a decreasing function, otherwise the assumption that the collisional dissipation is inessential would have to be abandoned.

(iv) At this point, we are in possession of the full solution, subject to the unknown function $F_{0}^{+}(\unicode[STIX]{x1D709})$ . We may now use this solution to determine

(5.6) $$\begin{eqnarray}F_{0}^{-}(\unicode[STIX]{x1D709})=F^{-}(\unicode[STIX]{x1D70F}\rightarrow 0,\unicode[STIX]{x1D709}).\end{eqnarray}$$

The net Hermite flux (3.31) at $\unicode[STIX]{x1D70F}\rightarrow 0$ , i.e. from low Hermite moments to high ones, is proportional to $F_{0}^{+}-F_{0}^{-}$ . We will show that there is a solution for which

(5.7) $$\begin{eqnarray}F_{0}^{+}-F_{0}^{-}=0\end{eqnarray}$$

and that this solution is the only physically sound one. Thus, the outcome of this procedure will be a universal structure of the Fourier–Hermite spectrum $F_{k}(\unicode[STIX]{x1D70F})$ in steady state. An impatient reader can skip what follows to find this spectrum in § 5.3.2 (xe will also find a shorter, more elementary, if perhaps less general, route to this solution in appendix B).

5.2 Green’s function solution

5.2.1 The ‘ $+$ ’ solution

The solution of the ‘ $+$ ’ equation (5.2) satisfying the initial and boundary conditions (5.3) is

(5.8) $$\begin{eqnarray}\displaystyle F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}) & = & \displaystyle \frac{\unicode[STIX]{x1D709}^{1/3}\text{e}^{-\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}}{2\unicode[STIX]{x1D70F}}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D702}\,\unicode[STIX]{x1D702}^{2/3}\text{e}^{-\unicode[STIX]{x1D702}^{2}/4\unicode[STIX]{x1D70F}}\text{I}_{1/3}\!\left(\frac{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}{2\unicode[STIX]{x1D70F}}\right)F_{0}^{+}(\unicode[STIX]{x1D702})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\left(\frac{1}{2}\unicode[STIX]{x1D709}\right)^{2/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}\unicode[STIX]{x1D70E}\,\frac{\text{e}^{-\unicode[STIX]{x1D709}^{2}/4(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E})}}{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E})^{4/3}}\,Y(\unicode[STIX]{x1D70E}),\end{eqnarray}$$

where $\text{I}_{1/3}$ is a modified Bessel function of the first kind. The first term in (5.8) is responsible for satisfying the initial condition and is zero at $\unicode[STIX]{x1D709}\rightarrow 0$ , the second term is equal to zero at $\unicode[STIX]{x1D70F}\rightarrow 0$ and to $Y(\unicode[STIX]{x1D70F})$ at $\unicode[STIX]{x1D709}\rightarrow 0$ .

The associated energy flux at $\unicode[STIX]{x1D709}\rightarrow 0$ is

(5.9) $$\begin{eqnarray}\left.\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}\rightarrow 0}(\unicode[STIX]{x1D70F})=\frac{2^{1/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\left[\frac{1}{\unicode[STIX]{x1D70F}^{1/3}}\int _{0}^{\infty }\text{d}z\,\text{e}^{-z}F_{0}^{+}(2\sqrt{\unicode[STIX]{x1D70F}z})-\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E})^{1/3}}\right].\end{eqnarray}$$

The first term is obtained by expanding

(5.10) $$\begin{eqnarray}\text{I}_{1/3}\!\left(\frac{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}{2\unicode[STIX]{x1D70F}}\right)=\frac{1}{\unicode[STIX]{x1D6E4}\!\left(\frac{4}{3}\right)}\left(\frac{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}{4\unicode[STIX]{x1D70F}}\right)^{1/3}+\cdots \,,\end{eqnarray}$$

under the integral, mopping up powers of $\unicode[STIX]{x1D709}$ , then taking $\unicode[STIX]{x1D709}\rightarrow 0$ , and, finally, changing the integration variable to $z=y^{2}/4\unicode[STIX]{x1D70F}$ . The second term in (5.9) takes some work – the derivation can be found in Sutton (Reference Sutton1943) (his equation 7.4).Footnote ⁶

5.2.2 The ‘ $-$ ’ solution

To obtain the solution of the ‘ $-$ ’ equation (5.2), let $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70F}$ and solve

(5.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}F^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}^{\prime }}=\frac{1}{\unicode[STIX]{x1D709}^{1/3}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\end{eqnarray}$$

subject to the boundary and initial conditions (5.4), which become, with the new variable,

(5.12) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}F^{-}(\unicode[STIX]{x1D70F}^{\prime },\unicode[STIX]{x1D709}\rightarrow 0)=Y(\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70F}^{\prime }),\\ F^{-}(\unicode[STIX]{x1D70F}^{\prime }\rightarrow 0,\unicode[STIX]{x1D709})=0.\end{array}\right\}\end{eqnarray}$$

The solution is the same as (5.8), but with $F_{0}^{+}$ replaced by $0$ , $\unicode[STIX]{x1D70F}$ by $\unicode[STIX]{x1D70F}^{\prime }$ and $Y(\unicode[STIX]{x1D70E})$ by $Y(\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70E})$ . Changing the integration variable $\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70E}\rightarrow \unicode[STIX]{x1D70E}$ and restoring $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}_{\text{max}}-\unicode[STIX]{x1D70F}$ , we get

(5.13) $$\begin{eqnarray}F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709})=\frac{\left(\frac{1}{2}\unicode[STIX]{x1D709}\right)^{2/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D70F}_{\text{max}}}\text{d}\unicode[STIX]{x1D70E}\,\frac{\text{e}^{-\unicode[STIX]{x1D709}^{2}/4(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})}}{(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})^{4/3}}\,Y(\unicode[STIX]{x1D70E}).\end{eqnarray}$$

Note that taking $\unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty$ produces no anomalies, assuming $Y(\unicode[STIX]{x1D70E})$ does not grow (which would not be physical anyway as even with linear Landau damping, $Y=\text{const}$ ).

The flux associated with this solution at $\unicode[STIX]{x1D709}\rightarrow 0$ is found from $F^{-}(\unicode[STIX]{x1D70F}^{\prime },\unicode[STIX]{x1D709})$ by the same procedure as the second term in (5.9), followed by the same changes of variables as described above. The result is

(5.14) $$\begin{eqnarray}\left.\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}\rightarrow 0}(\unicode[STIX]{x1D70F})=\frac{2^{1/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\int _{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D70F}_{\text{max}}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})^{1/3}}.\end{eqnarray}$$

Note that we should not rush into taking $\unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty$ here before we take the derivative as the integral may well (and indeed will) prove divergent.

5.2.3 Continuity of energy flux

Using (5.9) and (5.14), the condition (5.5) becomes

(5.15) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\left[\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E})^{1/3}}-\int _{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D70F}_{\text{max}}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})^{1/3}}\right]=\frac{1}{\unicode[STIX]{x1D70F}^{1/3}}\int _{0}^{\infty }\text{d}z\,\text{e}^{-z}F_{0}^{+}(2\sqrt{\unicode[STIX]{x1D70F}z}).\end{eqnarray}$$

This is an integral equation for $Y(\unicode[STIX]{x1D70F})$ in terms of the unknown function $F_{0}^{+}$ (or vice versa), but solving it directly is a thankless pursuit. We will cut to the chase and seek a particular solution for which the Hermite flux at $\unicode[STIX]{x1D70F}\rightarrow 0$ vanishes.

5.3 Zero-flux solution

From (5.13), we can deduce the spectrum of anti-phase-mixing modes at $\unicode[STIX]{x1D70F}\rightarrow 0$ :

(5.16) $$\begin{eqnarray}F_{0}^{-}(\unicode[STIX]{x1D709})=\frac{\left(\frac{1}{2}\unicode[STIX]{x1D709}\right)^{2/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D70F}_{\text{max}}}\text{d}\unicode[STIX]{x1D70E}\,\frac{\text{e}^{-\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x1D70E}^{4/3}}\,Y(\unicode[STIX]{x1D70E}).\end{eqnarray}$$

Let us explicitly look for such a solution that $F_{0}^{-}(\unicode[STIX]{x1D709})=F_{0}^{+}(\unicode[STIX]{x1D709})$ . We may then substitute the expression (5.16) for $F_{0}^{+}$ in (5.15) and thus obtain an equation for what $Y(\unicode[STIX]{x1D70F})$ would have to be in order for the zero-flux solution to be realised. If this $Y(\unicode[STIX]{x1D70F})$ is physically legitimate (decays with $\unicode[STIX]{x1D70F}$ ), we can put it back into (5.16) to determine $F_{0}^{\pm }(\unicode[STIX]{x1D709})$ and then use that and $Y(\unicode[STIX]{x1D70F})$ in (5.8) and (5.13) to determine the Fourier–Hermite spectrum everywhere.

The integral in the right-hand side of (5.15) with (5.16) for $F_{0}^{+}$ can easily be done after switching the order of $z$ and $\unicode[STIX]{x1D70E}$ integrations and changing the integration variable $z$ to $\unicode[STIX]{x1D701}=z(1+\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70E})$ . As a result, (5.15) becomes

(5.17) $$\begin{eqnarray}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\left[\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E})^{1/3}}-\int _{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D70F}_{\text{max}}}\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})^{1/3}}\right]=\frac{1}{3}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70E}\,\frac{Y(\unicode[STIX]{x1D70E})}{(\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D70E})^{4/3}}.\end{eqnarray}$$

The solution to (5.17) is, up to a multiplicative constant,

(5.18) $$\begin{eqnarray}Y(\unicode[STIX]{x1D70F})=\frac{1}{\unicode[STIX]{x1D70F}^{2/3}},\end{eqnarray}$$

which we will presently show by direct substitution. In appendix C, we show that this is indeed the only sensible solution. In appendix B, this scaling with $\unicode[STIX]{x1D70F}$ emerges via a much simpler (but less rigorous) argument arising from seeking a self-similar solution to our problem.

With (5.18) for $Y$ , all integrals in (5.17) are standard tabulated ones: the integral in the right-hand side of (5.17) is (after rescaling the integration variable $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70F}\rightarrow \unicode[STIX]{x1D70E}$ )

(5.19) $$\begin{eqnarray}\text{right-hand side of (5.17) }=\frac{1}{3\unicode[STIX]{x1D70F}}\int _{0}^{\infty }\frac{\text{d}\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70E}^{2/3}(1+\unicode[STIX]{x1D70E})^{4/3}}=\frac{1}{\unicode[STIX]{x1D70F}},\end{eqnarray}$$

the first integral in the left-hand side is a constant (because $\unicode[STIX]{x1D70F}$ can scaled out) and the second one is dominated by the upper limit, so it is ${\approx}\ln (\unicode[STIX]{x1D70F}_{\text{max}}/\unicode[STIX]{x1D70F})$ as $\unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty$ . It then follows immediately that their derivative in the left-hand side of (5.17) is

(5.20) $$\begin{eqnarray}\text{left-hand side of (5.17) }\approx \frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}}\left(\text{const}-\ln \frac{\unicode[STIX]{x1D70F}_{\text{max}}}{\unicode[STIX]{x1D70F}}\right)=\frac{1}{\unicode[STIX]{x1D70F}}\quad \text{as}\quad \unicode[STIX]{x1D70F}_{\text{max}}\rightarrow \infty .\end{eqnarray}$$

This is equal to (5.19) and so (5.18) is indeed a good solution.

For future reference, these results imply that the energy flux through $\unicode[STIX]{x1D709}=0$ associated with this solution is

(5.21) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}k}\right|_{k\rightarrow 0}=\pm \left(\frac{3}{2\unicode[STIX]{x1D6FE}}\right)^{1/3}\left.\unicode[STIX]{x1D709}^{1/3}\frac{\unicode[STIX]{x2202}F^{\pm }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}\rightarrow 0}(\unicode[STIX]{x1D70F})=\frac{3^{1/3}}{\unicode[STIX]{x1D6FE}^{1/3}\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\frac{1}{\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

5.3.1 Reconstruction of the full solution

Now let us describe the full solution for the Fourier–Hermite spectrum that follows from what we have just derived. The spectrum at $\unicode[STIX]{x1D709}\rightarrow 0$ ( $k\rightarrow 0$ ) is given by (5.18). In our original variables, this means

(5.22) $$\begin{eqnarray}F_{k\rightarrow 0}(\unicode[STIX]{x1D70F})=\frac{1}{\unicode[STIX]{x1D70F}^{2/3}}\quad \Rightarrow \quad C_{k\rightarrow 0,m}=\frac{\text{const}}{m^{3/2}}.\end{eqnarray}$$

The spectrum at $\unicode[STIX]{x1D70F}\rightarrow 0$ (low $m$ ) is, via (5.16) and (5.18) (changing integration variable to $z=\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70E}$ ),

(5.23) $$\begin{eqnarray}\displaystyle F_{0}^{\pm }(\unicode[STIX]{x1D709})=\frac{2^{4/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\frac{1}{\unicode[STIX]{x1D709}^{4/3}}\quad \Rightarrow \quad F_{k}(\unicode[STIX]{x1D70F}\rightarrow 0)=\frac{3^{4/3}}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\frac{\unicode[STIX]{x1D6FE}^{2/3}}{k^{2}}\quad \Rightarrow \quad C_{k,m\rightarrow m_{0}}=\frac{\text{const}}{k^{2}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

These two scaling laws, the Hermite spectrum (5.22) and the Fourier spectrum (5.23) also hold asymptotically across the entire phase space, at large enough $m$ and $k$ , as we shall see shortly.

Using (5.18) in (5.13) and changing the integration variable to $z=\unicode[STIX]{x1D709}^{2}/4(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70F})$ , we find

(5.24) $$\begin{eqnarray}F^{-}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709})=\frac{1}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{0}^{\infty }\text{d}z\,\frac{\text{e}^{-z}}{\left(\unicode[STIX]{x1D70F}z+\frac{1}{4}\unicode[STIX]{x1D709}^{2}\right)^{2/3}}=\frac{\text{e}^{\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D70F}^{2/3}}\frac{\unicode[STIX]{x1D6E4}\!\left(\displaystyle \frac{1}{3};\frac{\unicode[STIX]{x1D709}^{2}}{4\unicode[STIX]{x1D70F}}\right)}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}.\end{eqnarray}$$

In the first of these expressions, both asymptotics (5.22) and (5.23) are manifest. In the second expression, which is obtained by changing the integration variable $z+\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}\rightarrow z$ ,

(5.25) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}\!\left(\displaystyle \frac{1}{3};\frac{\unicode[STIX]{x1D709}^{2}}{4\unicode[STIX]{x1D70F}}\right)=\int _{\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}^{\infty }\text{d}z\,z^{-2/3}\text{e}^{-z}\end{eqnarray}$$

is an upper incomplete gamma function.

Finally, using (5.23) and (5.18) in (5.8), we find

(5.26) $$\begin{eqnarray}\displaystyle F^{+}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D709}) & = & \displaystyle \frac{\text{e}^{-\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D70F}^{2/3}}\left\{1+(-1)^{2/3}\left[\frac{\unicode[STIX]{x1D6E4}\!\left(\displaystyle \frac{1}{3};-\frac{\unicode[STIX]{x1D709}^{2}}{4\unicode[STIX]{x1D70F}}\right)}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}-1\right]\right\}\nonumber\\ \displaystyle & = & \displaystyle \frac{\text{e}^{-\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D70F}^{2/3}}\left[1+\frac{1}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{0}^{\unicode[STIX]{x1D709}^{2}/4\unicode[STIX]{x1D70F}}\text{d}z\,z^{-2/3}\text{e}^{z}\right].\end{eqnarray}$$

The first term inside the bracket comes from the second integral in (5.8) (the boundary-value term) and is obtained by the manipulations analogous to those that led to (5.24). The second term comes from the first integral in (5.8) (the initial-value term), which is turned into a tabulated integral by changing the integration variable to $z=\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}/2\unicode[STIX]{x1D70F}$ (and then changing $z\rightarrow -z$ to obtain the last integral representation, which is perhaps more transparent than the one in terms of the incomplete gamma function).

5.3.2 Self-similar solution

Assembling (5.24) and (5.26) together and returning them to the original variables, we arrive at the following solution:

(5.27) $$\begin{eqnarray}F_{k}(\unicode[STIX]{x1D70F})=\frac{\text{e}^{-k^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D70F}^{2/3}}\left\{\begin{array}{@{}l@{}}\displaystyle 1+\frac{1}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{0}^{k^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}}\text{d}z\,z^{-2/3}\text{e}^{z},\quad k>0,\\ \displaystyle \frac{1}{\unicode[STIX]{x1D6E4}\!\left(\frac{1}{3}\right)}\int _{|k|^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}}^{\infty }\text{d}z\,z^{-2/3}\text{e}^{-z},\quad k<0,\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}=(2m)^{3/2}/3-\unicode[STIX]{x1D70F}_{0}$ ( $\unicode[STIX]{x1D70F}_{0}$ is an order-unity offset). Hence the Fourier–Hermite spectrum $C_{k,m}$ and the Hermite flux $\unicode[STIX]{x1D6E4}_{k,m}$ can be calculated according to (3.30) and (3.31), respectively.

Figure 1. The function $F_{k}(\unicode[STIX]{x1D70F})$ given by (5.27) (with $\unicode[STIX]{x1D6FE}=1$ ). The phase-mixing part of the spectrum is on the right ( $k>0$ ), the anti-phase-mixing part on the left ( $k<0$ ). The total energy is the sum and the Hermite flux the difference of these two. The broader-range scaling behaviour is better represented on a log scale: see figures 2 and 3.

Figure 2. Typical cuts through the solution (5.27) (with $\unicode[STIX]{x1D6FE}=1$ ) at (a) constant $\unicode[STIX]{x1D70F}$ ( $\unicode[STIX]{x1D70F}=20$ ), (b) constant $k$ ( $k=7$ ). The spectra of phase-mixing modes, $F_{k}(\unicode[STIX]{x1D70F})$ , are shown in red and the spectra of anti-phase-mixing modes, $F_{-k}(\unicode[STIX]{x1D70F})$ , in black. The asymptotic scalings (5.23) and (5.22) are shown for reference and convergence to them is manifest in the limits $k\gg (9\unicode[STIX]{x1D70F})^{1/3}$ and $\unicode[STIX]{x1D70F}\gg k^{3}/9$ , respectively. Two-dimensional plots of $F_{k}(\unicode[STIX]{x1D70F})$ and of the resulting Hermite flux and total energy spectrum are in figure 3.

Figure 3. (a,b) Same as figure 1, but on a log scale and across a broader range of $k$ and $\unicode[STIX]{x1D70F}$ . Note the asymptotic features (zero-flux solution, power-law scalings) discussed in §§ 5.3.2 and 5.4. Cf. figure 4. (c) Normalised Hermite flux $\bar{\unicode[STIX]{x1D6E4}}$ defined by (5.28). Its contours are straight in the logarithmic coordinates because $\bar{\unicode[STIX]{x1D6E4}}=\bar{\unicode[STIX]{x1D6E4}}(k^{3}/9\unicode[STIX]{x1D70F})$ . (d) Total energy spectrum $[F_{k}(\unicode[STIX]{x1D70F})+F_{-k}(\unicode[STIX]{x1D70F})]/2$ [see (3.30)].

The solution (5.27) is plotted in figure 1, which shows a pleasingly non-trivial shape. The essential result is, however, extremely simple. The solution is self-similar and could, in fact, have been obtained as such, by a shorter, if marginally less general, route (see appendix B). The similarity variable $k^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}$ determines the demarcation of the phase space into two asymptotic regions: the asymptotic of the spectrum when $|k|\gg (9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})^{1/3}$ is (5.23) and the asymptotic when $\unicode[STIX]{x1D70F}\gg |k|^{3}/9\unicode[STIX]{x1D6FE}$ is (5.22) (this is particularly obvious in the second expression in (5.13)). The former describes fluctuations in the ‘wavenumber inertial range’ with a vanishing Hermite flux, the latter fluctuations in the ‘Hermite inertial range’, which also have zero Hermite flux. These scalings are illustrated in figure 2 and the normalised Hermite flux

(5.28) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D6E4}}=\frac{F_{k}(\unicode[STIX]{x1D70F})-F_{-k}(\unicode[STIX]{x1D70F})}{F_{k}(\unicode[STIX]{x1D70F})+F_{-k}(\unicode[STIX]{x1D70F})}=\frac{\unicode[STIX]{x1D6E4}_{k,m}}{\sqrt{2m}\,kC_{k,m}}\end{eqnarray}$$

is plotted in figure 3(c). $\bar{\unicode[STIX]{x1D6E4}}$ is a good measure of how different the nonlinear state is from the linear one: for linear Landau-damped perturbations, we would have had $\bar{\unicode[STIX]{x1D6E4}}=1$ everywhere (Kanekar et al. Reference Kanekar, Schekochihin, Dorland and Loureiro2015). Note that, as follows immediately from (5.27), $\bar{\unicode[STIX]{x1D6E4}}=\bar{\unicode[STIX]{x1D6E4}}(k^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})$ is a function of the similarity variable only.

Another useful result is the overall Hermite spectrum integrated over all wavenumbers. While this, of course, misses the relationship between structure in position and velocity space that we have focused on so closely, it is a good crude measure of how ‘phase mixed’ the distribution is (cf. Hatch et al. Reference Hatch, Jenko, Bratanov and Bañón Navarro2014; Servidio et al. Reference Servidio, Chasapis, Matthaeus, Perrone, Valentini, Parashar, Veltri, Gershman, Russell and Giles2017). So, from (3.30) and (5.27), after integrating out the self-similar functional dependence of $F_{k}(\unicode[STIX]{x1D70F})$ on $k^{3}/9\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}$ , we deduce

(5.29) $$\begin{eqnarray}\mathop{\sum }_{k}C_{k,m}=\mathop{\sum }_{k}\frac{F_{k}+F_{-k}}{2\sqrt{m}}\propto \frac{1}{m}.\end{eqnarray}$$

This scaling – or, equivalently, the $k$ -by- $k$ $C_{k,m}\propto m^{-3/2}$ scaling at large $m$ (see (5.22)), – being steeper than $m^{-1/2}$ , implies that our solution does indeed decay fast enough in $m$ in order for the collisional dissipation to vanish at vanishing collisionality and so treating the collisionless limit as non-singular was justified (see discussion in § 4.2).Footnote ⁷ Still, restoring finite $\unicode[STIX]{x1D708}$ leads to a kind of ‘Kolmogorov scale’ for our kinetic turbulence and to a quantitative measure of the applicability, or otherwise, of the linear approximation, so we are going to do this in § 5.4.

Finally, we may also calculate the overall wavenumber spectrum of the free energy: again integrating out the self-similar functional dependence of $F_{k}(\unicode[STIX]{x1D70F})$ , we get

(5.30) $$\begin{eqnarray}\mathop{\sum }_{m}C_{k,m}\propto \frac{1}{k}.\end{eqnarray}$$

Since the total variance of the perturbed distribution function is conserved in the Kraichnan–Kazantsev model (see § 4.2), the above result can be made sense of as the classical Batchelor (Reference Batchelor1959) scaling of a passive scalar advected by a single-scale stochastic field.

5.4 Phase-space energy flows and role of collisions

The structure of our self-similar solution of (4.12) is, in fact, easily understood already by means of a qualitative examination of the equation, which is also a useful approach in evaluating the role of collisions. Figure 4 is a cartoon of phase-space energy flows in aid of the discussion that follows (cf. figure 3 a,b).

Figure 4. Cartoon of energy flows in phase space, including collisional cutoff. Cf. figure 3(a,b). Reminder: $\unicode[STIX]{x1D70F}\sim m^{3/2}$ [see (4.7)] and $\unicode[STIX]{x1D6FE}$ is related to the stochastic electric field via (5.33).

5.4.1 Phase-mixing region

The phase-mixing term dominates over the nonlinearity when

(5.31) $$\begin{eqnarray}\unicode[STIX]{x1D70F}\ll \frac{k^{3}}{\unicode[STIX]{x1D6FE}}\quad \Rightarrow \quad \frac{\unicode[STIX]{x2202}F_{k}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}=0,\end{eqnarray}$$

so the solution is independent of $\unicode[STIX]{x1D70F}$ in this region (see figure 3 b). This is the linear phase-mixing solution, $C_{k,m}\propto 1/\sqrt{m}$ , obtained earlier by Zocco & Schekochihin (Reference Zocco and Schekochihin2011) and Kanekar et al. (Reference Kanekar, Schekochihin, Dorland and Loureiro2015). Its wavenumber scaling, $F_{k}\propto 1/k^{2}$ , is, however, a new feature, extracting which required matching with other regions.

A way of making sense of this solution is to go back to the time-dependent (3.27) [or (4.6)] and notice that, for $k>0$ , whatever solution $\tilde{f}_{k}$ (and, therefore, $F_{k}$ ) exists at low $s$ , it will propagate ‘upwards’ (to higher $s$ ) along the characteristic

(5.32) $$\begin{eqnarray}s=\frac{kt}{\sqrt{2}},\end{eqnarray}$$

and it will do so unimpeded by the nonlinearity as long as the time $t$ is shorter than the nonlinear time $t_{\text{nl}}$ associated with the mode-coupling term in the right-hand side of those equations. In the Kraichnan–Batchelor model,

(5.33) $$\begin{eqnarray}t_{\text{nl}}^{-1}\sim \frac{s^{2}\unicode[STIX]{x1D6FE}}{k^{2}},\quad \unicode[STIX]{x1D6FE}\sim p^{4}\unicode[STIX]{x1D718}_{p}\sim p^{4}\unicode[STIX]{x1D711}_{p}^{2}t_{\text{c}},\end{eqnarray}$$

where $t_{\text{c}}$ is the correlation time of the wave field $\unicode[STIX]{x1D711}_{p}$ (effectively assumed to be single scale). Requiring $t\ll t_{\text{nl}}$ in (5.32), we see that the phase-mixing region of the phase space extends to $s^{3}\ll k^{3}/\unicode[STIX]{x1D6FE}$ , which is the same as the condition in (5.31).

This gives us a way to estimate how small the wave amplitude must be in order for the nonlinearity and associated effects never to matter: indeed, if the collision time is short compared to the nonlinear time,

(5.34) $$\begin{eqnarray}t_{\unicode[STIX]{x1D708}}\sim \frac{1}{\unicode[STIX]{x1D708}s^{2}}\ll t_{\text{nl}}\quad \Leftrightarrow \quad k\gg \left(\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D708}}\right)^{1/2}\equiv k_{\unicode[STIX]{x1D708}},\end{eqnarray}$$

the phase-mixed distribution function will thermalise before the echo can bring any energy back from phase space. Putting $t_{\unicode[STIX]{x1D708}}$ into (5.32) tells us how far into phase space energy will travel:

(5.35) $$\begin{eqnarray}\unicode[STIX]{x1D70F}\sim s^{3}\ll \frac{k}{\unicode[STIX]{x1D708}}\equiv \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D708}},\quad k\gg k_{\unicode[STIX]{x1D708}}.\end{eqnarray}$$

Perhaps the most practically important conclusion from this is that, given the strength of the electric field and, therefore, via (5.33), the value of $\unicode[STIX]{x1D6FE}$ , we can predict the collisional cutoff wavenumber $k_{\unicode[STIX]{x1D708}}$ , given by (5.34), at which phase mixing (Landau damping) curtails the universal spectrum that we have derived above: we shall see in a moment that for $k\gg k_{\unicode[STIX]{x1D708}}$ , $\bar{\unicode[STIX]{x1D6E4}}=1$ , i.e. there is no echo flux from high to low Hermite moments.

5.4.2 Diffusion and echo regions

Considering the limit opposite to (5.31), i.e. the region of phase space where $s\gg kt_{\text{nl}}$ , we get

(5.36) $$\begin{eqnarray}\unicode[STIX]{x1D70F}\gg \frac{k^{3}}{\unicode[STIX]{x1D6FE}}\quad \Rightarrow \quad \frac{\unicode[STIX]{x2202}^{2}F_{k}}{\unicode[STIX]{x2202}k^{2}}=0,\end{eqnarray}$$

i.e. $k$ -space diffusion dominates. Our self-similar solution (§ 5.3.2) tells us that the appropriate solution in this region is one independent of $k$ (see figure 3 a,b) and $F_{k}(\unicode[STIX]{x1D70F})\propto 1/\unicode[STIX]{x1D70F}^{2/3}$ . This solution takes whatever values $F_{k}(\unicode[STIX]{x1D70F})$ has at $k\sim (\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})^{1/3}$ (figure 3 b) and transfers them across to $k\sim -(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F})^{1/3}$ (figure 3 a), where anti-phase-mixing picks them up and transfers them ‘downwards’ to low Hermite moments ( $\unicode[STIX]{x1D70F}\rightarrow 0$ ), over times that are again shorter than $t_{\text{nl}}$ (because $\unicode[STIX]{x1D70F}\ll |k^{3}|/\unicode[STIX]{x1D6FE}$ again), along the characteristic

(5.37) $$\begin{eqnarray}s=s_{0}-\frac{|k|t}{\sqrt{2}},\end{eqnarray}$$

where $s_{0}\sim \unicode[STIX]{x1D70F}_{0}^{1/3}\sim |k|/\unicode[STIX]{x1D6FE}^{1/3}$ .

This last piece of the solution is the echo flux. It cancels the phase-mixing flux exactly, provided the energy ( $F_{k}$ ) from $k>0$ has been successfully transferred by phase mixing from $\unicode[STIX]{x1D70F}=0$ to $\unicode[STIX]{x1D70F}\sim k^{3}/\unicode[STIX]{x1D6FE}$ to be picked up by diffusion and carried over to the anti-phase-mixing region $k<0$ . For $k\gg k_{\unicode[STIX]{x1D708}}$ , the energy gets intercepted at $\unicode[STIX]{x1D70F}\sim k/\unicode[STIX]{x1D708}$ [see (5.35)] and thermalised by collisions, so at $\unicode[STIX]{x1D70F}\sim k^{3}/\unicode[STIX]{x1D6FE}$ , $F_{k}(\unicode[STIX]{x1D70F})=0$ . This then gives $F_{-k}(0)=0$ , i.e. no echo flux. Thus, $k_{\unicode[STIX]{x1D708}}$ is indeed the wavenumber cutoff – a kind of ‘Kolmogorov scale’ for Vlasov-kinetic turbulence – beyond which Landau damping can act as an efficient route to (eventually collisional) dissipation.