On reduced modeling of the modulational dynamics in magnetohydrodynamics

Coherent-structure formation from turbulence is ubiquitous in nature, intrinsically compelling, and one of the precious few aspects of turbulence that are relatively yielding to analytical efforts (Hussain, 1986; Karimabadi et al., 2013; Smolyakov et al., 2000; Krashennikov et al., 2008). For magnetohydrodynamic (MHD) turbulence (Biskamp, 2003; Beresnyak, 2019; Schekochihin, 2022) and the turbulent-dynamo problem (Brandenburg et al., 2012; Tobias, 2021; Rincon et al., 2016), this has spawned a vast body of work known as mean-field electrodynamics, or mean-field dynamo theory (Rädler, 2007; Krause & Rädler, 1980; Moffatt, 1978; Brandenburg, 2018; Gruzinov & Diamond, 1994). In such approaches, the velocity and magnetic fields are split into fluctuations and mean, or average, fields (the definition of average depends on the problem); then, various closures are implemented to obtain the mean-field evolution in response to the fluctuations (Blackman & Field, 2002; Brandenburg & Subramanian, 2005b; Pouquet et al., 1976; Kraichnan, 1977; Nicklaus & Stix, 1988).

These closures ultimately rely on a truncation of a formally infinite chain of correlation functions of increasingly high order.¹¹1Here, we use the term ‘truncation’ to refer to low-order closures in general, as opposed to precisely the closure choice of setting all correlations to zero after some cutoff. The latter, which we will refer to as a simple truncation, is done in the quasilinear approximation discussed below but not, for example, in the various $\smash{\tau}$ approximations (Brandenburg & Subramanian, 2005b; Blackman & Field, 2002). This is a common approach to reduced modeling of nonlinear systems, and one can expect it to work when the fluctuations are small compared to the mean field. However, in turbulent structure formation, the usual roles of perturbation and background are switched, that is, mean fields emerge as small modulations of order-one correlation functions that characterize turbulent fluctuations. Hence, the question of the validity of low-order closures becomes nontrivial.

In this paper, we are concerned with the popular closure called the first-order smoothing (Krause & Rädler, 1980) or second-order correlation approximation (Rädler, 1982). It implies that the mean-field evolution is determined only by second-order correlations of the fluctuations, which, in a broader context, is also known as the quasilinear approximation (Dodin, 2022). On the one hand, one can expect its validity domain to be extremely limited. It can be rigorously justified only for small magnetic Reynolds numbers or short correlation times (Brandenburg & Subramanian, 2005a; Rincon, 2019), and comparisons with direct numerical simulations often corroborate this expectation (Pipin & Proctor, 2008; Schrinner et al., 2005). On the other hand, the quasilinear approximation endures in its popularity as a workhorse that is too convenient to cast aside in analytical calculations (Squire & Bhattacharjee, 2015; Lingam & Bhattacharjee, 2016; Masada & Sano, 2014; Gopalakrishnan & Singh, 2023). Furthermore, in some cases, quasilinear calculations have been found to produce good agreement with direct numerical simulations (DNS) even outside of their formal validity domain (Käpylä et al., 2006).²²2Confusingly, various analytical closures and DNS can conflict and agree in various combinations (Zhou & Blackman, 2021). But this issue is beyond the scope of the present paper. It is therefore important to understand when, and how exactly, the quasilinear approximation fails. Answering this question requires exploring a mean-field model that retains high-order correlations. That is what the present paper is about.

Here, we explore structure formation in MHD turbulence as a modulational instability (Zakharov & Ostrovsky, 2009) (MI) of the flow-velocity and magnetic-field fluctuations comprising the background turbulence. (For a recent overview of this approach in application to drift-wave and hydrodynamic turbulence, see Zhu & Dodin (2021); Tsiolis et al. (2020).) For simplicity, we restrict our consideration to two-dimensional (2-D) turbulence. Although bounded planar flows cannot sustain long-lasting magnetic fields in two dimensions (an anti-dynamo theorem by Zeldovich (1957)), 2-D turbulence nevertheless hosts rich physics and has long served as popular testing grounds for various aspects of MHD turbulence theory due to its (relative) tractability (Pouquet, 1978; Biskamp & Schwarz, 2001; Nazarenko, 2007; Giorgio et al., 2017; Biskamp et al., 1996). The goal of this paper is to complement those works with a detailed study of MIs using both analytical calculations and DNS.

Specifically, we study how energy is transferred from primary fluctuations to mean magnetic field and mean velocity during the early stages of structure formation, when these mean fields are weak and the primary structure can be treated as fixed. To analyze this process in detail, we assume a maximally reduced model of the turbulent fluctuations, namely, a spatially monochromatic primary structure. The simplicity of this model has been detracting attention from it in the literature in favor of more realistic turbulence settings, but there is more to it that meets the eye. Even with a monochromatic primary structure, the modulational dynamics remains remarkably intricate and, depending on the system parameters, can change drastically (figure 1) in ways that are not captured by the common closures or obvious from the standard eigenmode analysis (Friedberg, 1987). However, the simplicity of our model helps unravel this mystery.

We find that, when the primary structure truly experiences a MI, this MI can be described well with typical truncated models that neglect high-order correlations. However, already in adjacent regimes, such truncated models can fail spectacularly and produce false positives for instability. To study this process in detail, we propose an ‘extended’ quasilinear theory (XQLT) that treats the primary structure as fixed but includes the entire spectrum of modulational harmonics (as opposed to just the low-order harmonics, as usual). From XQLT, also corroborated by DNS, we find that the difference between said regimes is due to a fundamental difference in the structure of the modulational spectrum. For unstable modes, the spectrum is exponentially localised at low harmonic numbers, so truncated models are justified. But this localization does not always occur. At other parameters, modulational modes turn into constant-amplitude waves propagating down the spectrum, unimpeded until dissipative scales. These spectral waves are self-maintained as global modes with real frequencies and cause ballistic energy transport along the spectrum, breaking the feedback loops that could otherwise sustain MI. The ballistic transport drains energy from the primary structure at a constant rate until the primary structure is depleted. Notably, these effects are overlooked if one assumes from the start that the modulation spectrum is confined and that, accordingly, dissipation entirely vanishes in the limit of zero viscosity and resistivity.

A comment is due here regarding how the established truncated MHD closures fit into this picture. Given the indispensability of such simple closures, a higher-resolution understanding of their limitations can help guide and interpret their inevitable application. For example, we show that both conservative and dissipative corrections to ideal MHD tend to restore the validity of the quasilinear approximation. In this sense, ideal incompressible MHD is a ‘singular’ theory that is particularly sensitive to the accuracy of the closure within mean-field models. That said, it is an important conclusion of our work that, unless deviations from ideal incompressible MHD are substantial, changing the turbulence parameters even slightly can produce qualitative inaccuracies in an otherwise workable reduced model. Therefore, efforts to extrapolate closures or infer them from general considerations should be supplemented with first-principle calculations based on the underlying modulational dynamics. As a step in this direction, we propose a spectral closure that captures both the propagating spectral waves and MIs within our model. (But, of course, more work remains to be done for generic MHD turbulence.)

The paper is organised as follows. In section 2, we present the basic equations and derive the XQLT. In section 3, we examine the ability of truncated models of low-order to predict the MI rate. In section 4, we introduce spectral waves and describe their properties. In section 5, we propose a closure that can simultaneously capture MIs and spectral waves. In section 6, we demonstrate the impact of corrections to ideal MHD on the modulational dynamics. In section 7, we summarise our main results. Auxiliary calculations are presented in appendices.

A common way to truncate a problem like (29) is to postulate that all $\smash{\psi_{n}}$ with $\smash{|n|>N}$ are negligible. One can also understand this as imposing reflective boundary conditions in the oscillator chain $\smash{\{\boldsymbol{\psi}_{n}^{\sigma}\}}$:

Retained in this case are harmonics with wavevectors $\smash{\boldsymbol{p}}$ and $\smash{\boldsymbol{q}+n\boldsymbol{p}}$, with $n=0,\pm 1,\ldots,\pm N$. This means that the resulting system has $\smash{Q=1+(1+2N)}$ degrees of freedom. We call this procedure $\smash{Q}$-mode truncation ($\smash{Q}$MT). The corresponding truncation of $\smash{\boldsymbol{\psi}}$ will be denoted as $\smash{\boldsymbol{\psi}_{Q\text{MT}}}$, and the corresponding truncation of $\smash{\boldsymbol{M}}$ will be denoted as $\smash{\boldsymbol{M}_{Q\text{MT}}}$, so (29) becomes

Because of (31), eigenmodes of this system can be understood as standing waves in the oscillator chain $\smash{\{\boldsymbol{\psi}_{n}^{\sigma}\}}$ and have the form $\smash{\boldsymbol{\psi}_{Q\text{MT}}=\boldsymbol{Y}\exp(-\mathrm{i}\omega\tau)}$, where $\smash{\omega}$ are global frequencies and $\smash{\boldsymbol{Y}}$ are constant ‘polarization vectors’. According to (29), these vectors satisfy $\smash{\boldsymbol{D}\boldsymbol{Y}=0}$, where $\smash{\boldsymbol{D}=\mathrm{i}\omega\mathbb{1}+\boldsymbol{M}_{Q\text{MT}}}$ and $\smash{\mathbb{1}}$ is a unit $\smash{Q\times Q}$ matrix. Then, $\smash{\omega}$ can be found from

and MIs correspond to modes with $\smash{\operatorname{Im}\omega>0}$.

The success of this approach hinges on the assumption that higher harmonics truly remain negligible. In other words, a $\smash{Q}$MT should be able to adequately describe the evolution of a modulational mode if its eigenvector $\smash{\boldsymbol{\psi}_{Q\text{MT}}}$ is heavily weighted towards the low-order harmonics. Below, we explore to what extent this assumption is satisfied in various regimes.

The lowest-order $\smash{Q}$MT is 4MT, which corresponds to $\smash{N=1}$. Within this model, the primary harmonic with $\smash{\boldsymbol{k}=\boldsymbol{p}}$ is assumed to interact with the modulational mode at $\smash{\boldsymbol{k}=\boldsymbol{q}}$ only through two sidebands $\smash{\boldsymbol{k}=\boldsymbol{q}\pm\boldsymbol{p}}$; then, $\smash{\boldsymbol{\psi}_{\text{4MT}}=(\boldsymbol{\psi}_{-1},\boldsymbol{\psi}_{0},\boldsymbol{\psi}_{1})^{\intercal}}$ and

(This model can be understood as quasilinear, because, although the modulational dynamics remains nonlinear, the second- and higher-order harmonics of the perturbation are neglected.) The 4MT equations yield

[Note that these are normalised frequencies corresponding to the normalised time $\smash{\tau}$. To obtain the actual physical frequencies, one must multiply the right-hand side of (35) by $\smash{\mathcal{A}}$.] Assuming $\smash{\phi=\upi/4}$, one finds that two out of four branches of this dispersion relation are unstable at $\smash{|r|<1}$. The corresponding growth rate $\smash{\Gamma\doteq\operatorname{Im}\omega}$ is maximised at $\smash{\theta=0}$ and $\smash{\theta=\upi/2}$ at the value

(from now on, we assume $\smash{r>0}$ for clarity of notation), and the corresponding polarization vectors are as follows:

The case $\smash{\theta=0}$ can be recognised as a purely hydrodynamic Kelvin–Helmholtz instability (KHI), i.e. one that does not involve magnetic field. The case $\smash{0<\theta<\upi/4}$ can be understood as the MHD generalization of the KHI.

Potentially more accurate is a model with $\smash{N=2}$, or 6MT, which accounts for harmonics with $\smash{\boldsymbol{k}=\boldsymbol{q}\pm 2\boldsymbol{p}}$; then, $\smash{\boldsymbol{\psi}_{\text{4MT}}=(\boldsymbol{\psi}_{-2},\boldsymbol{\psi}_{-1},\boldsymbol{\psi}_{0},\boldsymbol{\psi}_{1},\boldsymbol{\psi}_{2})^{\intercal}}$ and

One can derive an analytical expression for $\smash{\omega}$ from here just like we derived (35) from (34). However, this expression is cumbersome and not particularly instructive, so it is not presented explicitly. Truncations with larger $\smash{Q}$, albeit also explored by us, will not be presented either for the same reason.

Figure 3 shows a comparison of the 4MT and 6MT predictions and the MI growth rates inferred numerically from nonlinear DNS of (9). Note that the truncated models quantitatively agree with each other and with the DNS at some parameters but drastically disagree at other parameters. In particular, the 4MT predicts that $\smash{\Gamma}$ is an even function of $\smash{\upi/2-\theta}$, while the 6MT predicts that this is not the case. Furthermore, the DNS predicts that, for example, at $\smash{r=0.5}$, the system is stable at all $\smash{\theta>\upi/2}$ [figure 3(b)]. Let us discuss this in detail.

As expected from the analytical formula (36) based on the 4MT, and as we have also found numerically, MI is typically suppressed at $\smash{r\gtrsim 1}$; hence, below we focus on the regime $\smash{r\in(0,1)}$. A comparison of the growth rate predicted by nonlinear DNS and truncated models in this range is presented figure 5. The corresponding primary modes with $\smash{\theta\lesssim\upi/4}$ are modulationally unstable, and the growth rates predicted by 4MT and 6MT are in good agreement with DNS. Such primary structures are dominated by the velocity field, $\smash{\mathcal{E}_{\text{pri},v}>\mathcal{E}_{\text{pri},b}}$ [see (27)], and thus will be called $\smash{\boldsymbol{v}}$-dominated primary modes (VDPMs). At $\smash{\theta\gtrsim\upi/4}$, primary structures are dominated by the magnetic field, $\smash{\mathcal{E}_{\text{pri},b}>\mathcal{E}_{\text{pri},v}}$, and thus will be called $\smash{\boldsymbol{b}}$-dominated primary modes (BDPMs). As seen in figure 5, BDPMs are modulationally stable, but truncated models predict otherwise. Figures 4(a) and (b) show the discrepancy between the predictions of nonlinear DNS, 4MT, and 6MT for representative parameters, specifically, $\smash{\theta\in(0,\upi/2)}$ and $\smash{\phi=\upi/4}$. (Stability is primarily determined by $\smash{\theta}$ and $\smash{r}$. Deviations of $\smash{\phi}$ from $\smash{\upi/4}$ result only in quantitative adjustments, unless $\smash{\phi}$ is close to $\smash{0}$ or $\smash{\upi/2}$.) In other words, truncated modes tend to overestimate the growth rate and systematically produce false positives for instability. Figure 5 shows that, although the magnitude of disagreement decreases with $\smash{N}$, the systematic overprediction of instability persists.

The reason for this overprediction can be understood by analyzing the eigenmode structure. (See also appendix A for an alternative explanation.) As to be discussed in section 4, unstable modes are evanescent spectral waves, whose $\smash{|Y_{n}|}$ exponentially decrease with $\smash{|n|}$ [figure 6(a)]:

(We assume $\smash{\phi=\upi/4}$ for simplicity, so $\smash{|Y_{n}^{+}|=|Y_{n}^{-}|}$ and the upper index in $\smash{|Y_{n}^{\sigma}|}$ can be omitted.) That is why truncated models correctly predict MI for modes that are actually unstable. The spectral scale $\smash{l}$ depends on the system parameters, and the smaller $\smash{l}$ the more accurate $\smash{Q}$MT is. At some parameters, though, particularly when $\smash{\theta\rightarrow\upi/4}$, $\smash{l}$ becomes large or even infinite, such that $\smash{|Y_{n}|}$ asymptotes to a nonzero constant at $\smash{|n|\to\infty}$ [figure 6(b)]. In this case, $\smash{Q}$MT is bound to fail. The corresponding modes are propagating spectral waves (PSWs). While they also receive energy from the primary structure at low $\smash{n}$, PSWs transport this energy along the spectrum to $\smash{|n|\to\infty}$. There, the energy is unavoidably dissipated by viscosity or resistivity, however small those are. This dissipation makes PSWs more stable than any $\smash{Q}$MT would predict, because $\smash{Q}$MTs assume that the mode energy forever resides at small $\smash{n}$, where dissipation is small or zero. Below, we discuss these effects in more detail.

First, let us consider spectral waves at $\smash{|n|\gg 1}$. In this limit, one has $\smash{\alpha_{n}^{\pm}\rightarrow\mp r}$ and $\smash{\beta_{n}^{\pm}\rightarrow 0}$, so (29) yields two decoupled wave equations for $\smash{\psi_{n}^{+}}$ and $\smash{\psi_{n}^{-}}$:

These equations allow solutions in the form of monochromatic waves:

with constant amplitude $\smash{\Psi^{\sigma}}$, frequency $\smash{\omega}$, and ‘wavenumber’ $\smash{K^{\sigma}}$. The quotation marks are added as a reminder that the corresponding waves propagate in the spectral space rather than in the physical space. Also note that $\smash{K^{\pm}}$ are defined unambiguously only within the first Brillouin zone, $\smash{K^{\pm}\in(-\upi,\upi)}$.

From (40), one readily finds that spectral waves obey the following dispersion relations:

where $\smash{\omega_{0}^{+}=2r\sin\phi}$ and $\smash{\omega_{0}^{-}=2r\cos\phi}$. Accordingly,spectral waves in the $\smash{n}$ space along the $\smash{\psi^{\sigma}}$ channel have the group velocity

and can propagate either up or down the spectrum (figure 7). The dependence of $\smash{\omega}$ and $\smash{v_{g}^{\pm}}$ on $\smash{\theta}$ can be removed by a gauge transformation. Specifically, $\smash{\kappa^{\pm}}$ becomes the true wavenumber if one adopts the variables $\smash{\bar{\psi}_{n}^{\pm}\doteq\psi_{n}^{\pm}\mathrm{e}^{\mp\mathrm{i}n\theta}}$ instead of $\smash{\psi_{n}^{\pm}}$. Also notably, to the extent that the (large) index $\smash{n}$ can be approximately considered a continuous variable, (40) can be written as usual nondispersive-wave equations for $\smash{\zeta^{\pm}(\tau,n)\doteq\bar{\psi}_{n}^{\pm}(\tau)}$:

This model corresponds to the small-$\smash{\kappa^{\sigma}}$ limit of (42), that is, $\smash{\omega\approx\omega_{0}^{\sigma}\kappa^{\sigma}}$.

The above equations can be used to describe PSW packets localised at large $\smash{|n|}$ but not the global eigenmodes (because the latter involve dynamics at small $\smash{|n|}$). In particular, (42) are not enough to find the global-mode frequencies. Yet they allow one to determine the mode structure at $\smash{|n|\gg 1}$ if one knows the mode frequency from other considerations (or, vice versa, to find the mode frequency if $\smash{\kappa^{\pm}}$ are known). In particular, unstable modes have complex $\smash{\omega}$, so, according to (42), their asymptotic wavenumber cannot be real. This explains why unstable modes are evanescent. In contrast, for real $\smash{\omega}$, (42) allows for real wavenumbers (provided that $\smash{|\omega|\leq\omega_{0}^{\pm}}$), i.e. predicts PSWs. These results are corroborated by DNS of (29). For example, figure 6 shows the mode structures inferred from the asymptotic dynamics of $\smash{\psi_{n}^{\sigma}(\tau)}$ at large enough $\smash{\tau}$ for the initial conditions

Here, $\smash{\epsilon}$ is a constant small amplitude (within the linear approximation, one can always rescale $\smash{\psi_{n}^{\sigma}}$ such that $\smash{\epsilon=1}$), and $\smash{\xi_{n}}$ are parameters that change the polarization of the initial conditions while keeping the initial modulation energy fixed. (The specific values of $\smash{\xi_{n}^{\sigma}}$ are given in the captions of figures in which initial condition dependent results of DNS are presented.) This form of the initial conditions is chosen to emulate a simple modulation of the primary mode.

The fact that the properties of spectral waves are determined only by the asymptotic properties of $\smash{\alpha_{n}^{\pm}}$ and $\smash{\beta_{n}^{\pm}}$ makes these waves particularly robust. In particular, note that the asymptotic form of $\smash{\alpha_{n}^{\pm}}$ and $\smash{\beta_{n}^{\pm}}$ at $\smash{|n|\to\infty}$ remains the same even when $\smash{\boldsymbol{p}}$ and $\smash{\boldsymbol{q}}$ are not orthogonal, so the above results readily extend to general $\smash{\boldsymbol{q}}$. Likewise, a similar picture holds also for three-dimensional interactions, although the mode polarization can be more complicated in this case. Notably, PSWs provide ballistic rather than diffusive (or super-diffusive) energy transport along the spectrum. This distinguishes them from the cascades that are commonly discussed in turbulence theories and result from eddy–eddy interactions, or wave–wave collisions (Galtier et al., 2000).

In realistic settings, spectral waves are excited at finite $\smash{|n|}$ and propagate down the spectrum. Evanescent waves reach $\smash{|n|\sim l}$ [see (39)] in finite time $\smash{\tau_{0}}$, get reflected, and eventually (asymptotically) settle as stationary standing waves on time scales of the order of a few $\smash{\tau_{0}}$. Such waves can be adequately described by truncated models. In contrast, PSWs continue to propagate toward infinity indefinitely (until they dissipate at scales where viscosity or resistivity is no longer negligible). Thus, they never develop components propagating up the spectrum at large $\smash{|n|}$ and so they never become quite like the standing-wave eigenmodes predicted by truncated models (section 3.1). This means that actual PSWs cannot be adequately described by analyzing $\smash{Q}$MT in principle. Instead, they should be considered in the context of the initial-value problem, much like Langmuir waves appear in plasma kinetic theory within Landau’s initial-value approach as opposed to the Case–van Kampen true-eigenmode approach (Stix, 1992).

Consequently, even though (29) has infinitely many degrees of freedom, the number of relevant global modes, i.e. those that propagate towards infinity, is finite. We find that there are only two of them, which can be attributed to the fact that $\smash{\dim\boldsymbol{\psi}_{n}=2}$. (Likewise, at a given wavenumber, only one, Langmuir, branch of relevant electrostatic waves in electron plasma remains relevant after transients are gone.) For example, for $\smash{\phi=\upi/4}$, we find from DNS that

This dependence and the corresponding mode structures are illustrated in figure 8. Each of the two modes has two wavenumbers associated with it: one determines $\smash{\psi^{\sigma}_{n}}$ at $\smash{n\to+\infty}$, and the other one determines $\smash{\psi^{-\sigma}_{n}}$ at $\smash{n\to-\infty}$ (figure 9). Thus, there are four $\smash{\kappa^{\sigma}_{d}}$ overall, where $\smash{\sigma}$ denotes the corresponding component of $\smash{\boldsymbol{\psi}_{n}}$, and $\smash{d}$ denotes the direction of propagation ($\smash{d=\pm 1}$ is the sign of the group velocity at $\smash{|n|\to\infty}$). They can be found by combining (46) with (42) and applying $\smash{\operatorname{sgn}v_{g}=d}$, i.e. $\smash{\omega_{0}^{\sigma}d\cos\kappa^{\sigma}_{d}>0}$. This yields

where $\smash{\kappa^{\sigma}_{d}=K^{\sigma}_{d}+\sigma\theta}$ and the values of $\smash{K^{\sigma}_{d}}$ are defined modulo the Brillouin zone (i.e. $\smash{K^{-}_{-}=-3\upi/2}$ should be understood as $\smash{\upi/2}$ and so on). Remember, though, that this applies only to global eigenmodes. Transient waves propagating in the region $\smash{|n|\gg 1}$ can have any $\smash{\kappa^{\sigma}}$ (figure 7).

The fact that the frequencies of these modes are real indicates that the modes are self-sustained notwithstanding the dissipation that they unavoidably experience at $\smash{|n|\to\infty}$. The corresponding evolution of the wave spectrum is illustrated in figure 9(a) and the corresponding evolution in the physical space is shown in figure 10. This remarkable dynamics is understood from the fact that the energy drain via this dissipation is balanced by the energy injection from the primary mode at small $\smash{|n|}$. As seen from (19) and (16), the dimensionless modulational energy (21b), or

is governed by

where $\smash{I_{n}}$ and $\smash{F_{n}}$ are given by

Because $\smash{\sum_{n}F_{n}^{\sigma}=0}$, the terms $\smash{F_{n}^{\sigma}}$ represent the energy flux that is carried along the modulation spectrum and is conserved within each sub-channel $\smash{\sigma}$. In contrast, $\smash{I_{n}}$ can be understood as injection terms in that they also appear, with the opposite signs, in the equation for the primary wave,

which follows from (20).

The linear spectral waves discussed so far correspond to the regime when the change of $\smash{\mathcal{E}_{\text{pri}}}$ is negligible. Eventually, though, as higher harmonics are populated and thus absorb more energy, the primary wave will be depleted. This means, in particular, that no primary wave can be truly stationary. Even an arbitrarily small perturbation will generally launch a PSW, and this PSW will eventually deplete the primary wave.

Since a PSW mode carries energy towards $\smash{|n|\rightarrow\infty}$, eventually, a large enough $\smash{|n|}$ is reached where the energy is dissipated regardless of how small the viscosity and resistivity are. Thus, a PSW exhibits an effective, or ‘anomalous’, dissipation rate $\smash{\gamma}$ that is independent of $\smash{\nu}$ and $\smash{\eta}$ in the limit $\smash{\nu,\eta\to 0}$. This effect is different from the anomalous transport caused by eddy–eddy collisions in turbulence (for example, see Donzis et al. (2005)) in that the energy transport caused by a PSW is ballistic. As a result, $\smash{\gamma}$ is straightforward to calculate, which is done as follows.

As discussed in section 4.2, a global PSW at large $\smash{|n|}$ has a flat mode structure, i.e. a structure with $\smash{\mathcal{E}_{n}}$ independent of $\smash{n}$. This structure establishes itself as an expanding ‘shelf’ whose edges (wave fronts) propagate across the modulational spectrum to $\smash{n\to\pm\infty}$ at the group velocity $\smash{v_{g}}$ (figure 11). Since the height of the shelf, $\smash{\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{E}_{n}\mkern-1.5mu}\mkern 1.5mu}$ (the overbar here denotes averaging over the PSW temporal period and over all $\smash{n}$ within the shelf), remains constant, this process drains energy from the primary mode linearly in time:

where we consider the dynamics average over the PSW temporal period. Hence,

(The rate relative to the physical time $\smash{t}$, as opposed to $\smash{\tau}$, is $\smash{\mathcal{A}}$ times this $\smash{\gamma}$.)

Let us again assume the initial conditions (45) and $\smash{\phi=\upi/4}$. Through DNS of (29), we find that the global PSW-mode amplitude is maximised when $\smash{|\arg(\xi^{\sigma}_{\pm 1}/\xi^{-\sigma}_{\pm 1})|=|\arg(\xi^{\sigma}_{\pm 1}/\xi^{\sigma}_{\mp 1})|=\upi}$, irrespective of $\smash{\theta}$. (Any particular choice of $\smash{\xi^{\sigma}_{n}}$ that satisfies this condition affects only phases of the modulational dynamics and does not impact averaged quantities that determine $\smash{\gamma}$.) This corresponds to the case when all modulational-seed energy is in the magnetic field; i.e. $\smash{\mathcal{E}_{b}=\mathcal{E}}$ and $\smash{\mathcal{E}_{v}=0}$. Figure 11 shows the corresponding anomalous dissipation rate normalised to the seed energy, along with its determining factors – the system-dependent PSW group velocity $\smash{v_{g}}$ and the average spectral energy density $\smash{\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{E}_{n}\mkern-1.5mu}\mkern 1.5mu}$, which is determined by the initial conditions. It can be seen that, for the MI unstable VDPMs ($\smash{\theta<\upi/4}$), spectral waves are relatively slow and have a small amplitude. The transition to modulational stability occurs at $\smash{|v_{g}|\sim\Gamma}$. This condition can be understood as a threshold beyond which (i.e. at $\smash{|v_{g}|\gtrsim\Gamma}$) PSWs provide a sufficiently fast escape route for the energy injected at small $\smash{|n|}$, such that the positive feedback loops [see (50)] supporting MIs can no longer be sustained. In the context of $\smash{\mathcal{PT}}$ symmetry (section 2.2.2), the spectral group velocity can be understood as the effective coupling between the links in the oscillator chain, connecting the energy injection from the primary mode at small $\smash{n}$ to the energy sink at $\smash{n\to\infty}$. As $\smash{|v_{g}|}$ increases with $\smash{\theta}$, the system transitions from broken to unbroken $\smash{\mathcal{PT}}$ symmetry.

Also, notice the following. If modulations at multiple $\smash{\boldsymbol{q}}$s are present, they launch independent cascades along $\smash{\boldsymbol{k}=\boldsymbol{q}+n\boldsymbol{p}}$ and thus the drain on the shared primary mode will be additive. This regime is illustrated by figures 1(d)-(f), which show the modulational dynamics for a BDPM ($\smash{\theta=\upi/3}$) seeded with noise. The rather nondescript modulational dynamics seen in figures 1(d) and (e) can be understood as the superposition of the many structures like that in figure 10 for various $\smash{\boldsymbol{q}}$. Also, the linear-in-time depletion of the primary mode seen in figure 1(f) can be understood as the sum of the PSW-driven energy fluxes along the constituent modulational channels.