3.1. Preliminary closure discourse

The method of wave-turbulence closure refers to a set of various techniques by which the statistical closure of a hierarchy of moments of a random field is treated as a perturbation problem for a system of weakly interacting modes. The notion of a weak interaction refers to the existence of some small parameter $\varepsilon$ in terms of which either wave amplitudes or fluid moments may be perturbatively expanded to approximately solve the nonlinear differential equations characterizing these types of systems. Furthermore, $\varepsilon$ is representative of the ratio of time scales for linear to nonlinear processes ($\varepsilon \sim t_{\rm lin}/t_{\rm nl}$) and indicative of the strength of nonlinear interaction. In RMHD phenomenology, $\varepsilon$ weights the time for counter-propagating Alfvén waves to pass through each other versus the characteristic time required for nonlinear interaction during their spatial coincidence. This notion is quantified for a fluctuation with wavevector $\boldsymbol {k}=(k_\parallel,k_\perp )$ in the form

(3.1)\begin{equation} \varepsilon \sim \frac{t_{\rm lin}}{t_{\rm nl}} = \frac{k_\perp v_0}{k_\parallel v_{A}}. \end{equation}

This definition of $\varepsilon$ accounts for particular notable aspects of MHD turbulence phenomenology. For instance, the concept of critical balance (Sridhar & Goldreich Reference Sridhar and Goldreich1994; Goldreich & Sridhar Reference Goldreich and Sridhar1995, Reference Goldreich and Sridhar1997) for which a transition from weak to strong turbulence occurs is clearly demonstrated by the limit $\varepsilon \rightarrow 1$. In addition, the presence of strong turbulence in the neighbourhood of $k_\parallel =0$ (Galtier & Chandran Reference Galtier and Chandran2006) coincides with $\varepsilon$ becoming appreciably large; counter-propagating waves of sufficient parallel extent will interact for ample temporal intervals such that the nonlinearity would not be considered weak. Generally speaking, there exists a range of fluctuation length scales for which $\varepsilon \geq O(1)$ and the weak turbulence closure is not valid. The aforementioned fundamentally different linear and nonlinear dynamical processes occurring over disparate time scales are associated with the presence of a singular (as opposed to a regular) perturbation.

In a regular perturbation problem, the solution of the perturbed system (for $0 < \varepsilon \ll 1$) is qualitatively the same as the case for null-perturbation in which $\varepsilon = 0$. These solutions are characterized by a convergent expansion in $\varepsilon$, describing a combination of the $\varepsilon =0$ system and higher-order corrections in response to the regular perturbation. In contrast, a singular perturbation problem features marked differences between the perturbed and unperturbed systems. Additionally, solutions take the form of asymptotic and potentially divergent expansions in terms of $\varepsilon$ (Hunter Reference Hunter2004). For the RMHD model used in this work, the case of $\varepsilon = 0$ (as applied to (2.8)) corresponds to a trivially solvable ODE in the Fourier domain describing pure Alfvénic propagation and an absence of nonlinear interaction between counter-propagating waves; the time required for nonlinear interactions to occur is infinite, and the linear wave period is the only genuine dynamical time scale of interest in this situation. Non-zero, albeit small values of $\varepsilon$ describe a qualitatively different set of systems for which these waves may interact with one another over a time scale much longer than the linear wave period. These interactions then produce divergences in naïve perturbative expansions of dynamical expressions constructed from (2.8). Said divergences manifest as explicit polynomial time dependence found in solutions of the perturbed governing equations, resulting in higher-order terms eventually becoming larger than leading order $O(\varepsilon ^0)$ after the passage of sufficient amounts of time.

Methods for handling singular perturbations (see for instance Bender & Orszag Reference Bender and Orszag2009) can be classified according to whether the underlying differential equation is concerned with variations in space (e.g. the WKB approximation) or time (e.g. Poincaré–Lindstedt or the method of multiple scales). In this work, divergences associated with the temporally singular nature of (2.8) arising in asymptotic expansion of the two-time energy spectrum are dealt with using an AP method developed in Benney & Newell (Reference Benney and Newell1967) and Benney & Newell (Reference Benney and Newell1969). The AP method solves differential equations by naïvely assuming a regular perturbative solution to the desired order in $\varepsilon$ followed by renormalization of the zero-order term to remove any divergences arising due to the singular nature of the problem. The combination of asymptotic time-domain analysis and renormalization yields statistical closure as well as a dynamical description of the system, accurate through time scales of the order of the initial expansion.

The equivalence of this technique with the perhaps more commonly known method of multiple scales (among other methods for temporally singular perturbation problems) is trivial to demonstrate and may be found in a solution of the van der Pol equation in the introduction of Benney & Newell (Reference Benney and Newell1967). Application of the AP method to systems of weakly interacting dispersive waves may be found in the main body of the aforementioned work, while the equivalence of wave amplitude and cumulant-based expansions (we prefer the latter in this paper as the quantities of interest are functions in a statistical hierarchy) is demonstrated in the second and third sections of Benney & Newell (Reference Benney and Newell1969). In addition, it may be prudent (and is strongly recommended) for the uninitiated reader to peruse the demonstration of the AP method presented in Appendix B, which derives the WKE for the energy spectrum of incompressible MHD originally found by Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2000).

We admit that the level of rigor used in our analysis may seemingly verge on tedious mathematical verbosity, but argue that it is ultimately necessary for posterity to both understand the process by which one obtains these results, and how they may reapply these methods to their own nonlinear wave equations. The amount of detail presented at various points in for instance Benney & Newell (Reference Benney and Newell1967), Benney & Newell (Reference Benney and Newell1969), Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2000), Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2002), Galtier & Chandran (Reference Galtier and Chandran2006) and Newell & Rumpf (Reference Newell and Rumpf2011) may leave the mathematically curious, albeit wave-turbulence naïve, longing for a clearer and convincing explanation of the closure details. The reader who is more familiar (or not concerned) with this process may, in contrast, briefly glance over the details presented herein and proceed to the final spectral results found at the end of § 3.3.

If one were to momentarily forget any notion of plasma physics, the closures presented in this section as well as that of Appendix B amount to the solution of general singular perturbation problems characteristic of a first-order ODE whose linear homogeneous solution corresponds to a sinusoidal oscillation, which is then subject to an inhomogeneity in the form of a weak quadratic nonlinearity; the differences between the two derivations only arise in the physical interpretation of the desired functions and the amount of required analytical computation. The WKE supplies a description of a given Elsasser field Fourier mode's energetic tendencies according to integration over all nonlinear interactions in which it partakes at some instance in time. The two-time energy spectrum models the amount by which a given mode has become statistically dissimilar from itself (after some time $\tau$) due to being both transported through space by linear advection and distorted due to the aforementioned nonlinear interactions. While the naming convention for this function may somehow suggest that two different times are explicitly being evoked upon its evaluation, the assumption of stationary turbulence, that time-domain statistics are purely a function of the lag $\tau =t'-t$ (and not the ‘base-time’ $t$) between two instances of the Elsasser fields, results in the information associated with temporal decorrelation being encoded onto a single variable. In addition, it is easily observed that the application of the limit of $\tau =0$ to the two-time energy closure is approximately equivalent to performing half of the Appendix B derivation.

In this section, the multiple scales ($\tau _0 = \varepsilon ^0\tau$ and $\tau _2 = \varepsilon ^2 \tau$) arising in our implementation of the AP method are merely time lags of different order in $\varepsilon$. We exploit the creation of these artificial degrees of freedom (a process that is obviously not unique to the AP method, and rather ubiquitous in temporally singular perturbative techniques) to clearly isolate linear from nonlinear influence upon Fourier mode autocorrelation. At leading order, the autocorrelation oscillates with time lag $\tau =O(\varepsilon ^0)$ due to Alfvénic propagation, while it will be shown to exponentially decay with time lag $\tau = O(\varepsilon ^{-2})$ due to participation in resonant nonlinear energy transfer. The intuition is that a relatively large number of wave oscillations will have occurred before a time of $O(\varepsilon ^{-2})$ elapses and a single nonlinear interaction may transpire. The notion of the existence of these temporally disparate processes is not novel or original and is in fact identical to the foundation upon which the WKE (B64) is predicated. Nonetheless, the underlying mathematics is not concerned with the assignment of the appropriate physical meaning to the multiple scales. Reduced to their essence, the statistical closures presented in this work are but applications of rather elementary and commonplace singular perturbative methods for solving nonlinear differential equations. One could write down a generic form of the procedure lacking in any even remotely imaginable conceptualization attributed to the function or superficially distinct instances of the lone independent variable and the process would remain the same. However, such a presentation is outside of the intended scope of this work and is omitted; in this context, it would only serve as a redundant and repetitive abstraction of the analysis found in the subsequent subsections. The remainder of this section is dedicated to the development and closure of a two-time cumulant hierarchy, resulting in a functional description of a two-time energy spectrum accurate through time lag $\tau =O(\varepsilon ^{-2})$.

3.2. Development of two-time cumulant hierarchy

In this subsection, we develop a set of differential equations describing the evolution of a two-time cumulant hierarchy as a function of time lag. The set is developed with the intention of finding the leading-order behaviour of the two-time power spectrum $h_k^s(t,t')$ defined by (2.11). To facilitate analysis, an interaction representation

(3.2)\begin{equation} \phi^s_k(t) = \varepsilon \psi^s_k(t)\,{\mathrm e}^{\mathrm{i}\omega_k^s t}, \end{equation}

is introduced to separate linear and nonlinear effects on $h_k^s(t,t')$

(3.3)\begin{equation} h^s(\boldsymbol k,t,t') = \langle \phi^s_{k}(t)\phi^s_{k'}(t') \rangle = \langle \psi^s_{-k}(t) \psi^s_{k}(t') \rangle\,{\rm e}^{\mathrm i\omega_k^s (t'-t)}. \end{equation}

The slowly varying (i.e. the portion changing due to nonlinear interactions) two-time power spectrum is then defined as

(3.4)\begin{equation} \langle \psi^s_{-k}(t) \psi^s_{k}(t') \rangle \equiv \tilde h_k^s(t,t'). \end{equation}

In addition, shear-Alfvén Fourier amplitudes will then evolve nonlinearly according to a version of (2.8) recast as

(3.5)\begin{equation} \partial_t \psi^s_k= \varepsilon \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \psi^s_p\psi^{-s}_q \,{\mathrm e}^{-2\mathrm{i}\omega_q^st}\delta_{k,pq}.\end{equation}

We proceed to generate a hierarchy developed from (3.4) using (3.5) via wave turbulence in a manner comparable to the WKE derivation of Appendix B. A similar secularity associated with resonant interactions between counter-propagating Alfvén waves occurs for time lags of order $O(\varepsilon ^{-2})$ and subsequent reordering of the asymptotic expansion for $\tilde h_k^s(t,t')$ yields leading-order behaviour.

The process begins with the generation of a set of differential equations describing time-lagged evolution of the second- and third-order (in statistics) two-time cumulants. We begin with an approach comparable to § B.1 of this work by differentiating $\tilde h_k^s(t,t')$ of (3.4), but with respect to $t'$ instead of $t$:

(3.6)\begin{equation} \frac{\partial{}}{\partial{t'}} \tilde h^s_k(t,t') = \left\langle \psi^s_{-k}(t)\frac{\partial{}}{\partial{t'}}\psi^s_{k}(t')\right\rangle = \varepsilon \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \tilde Q^{(s,s,-s)(3)}_{-kp}(t,t')\,{\mathrm e}^{-2 {\mathrm i} \omega_q^s t'}\delta_{k,pq}.\end{equation}

A two-time third-order cumulant is encountered of the form

(3.7)\begin{equation} \langle \psi^s_{-k}(t)\psi^s_{p}(t')\psi^{-s}_{q}(t')\rangle \equiv \tilde Q^{(s,s,-s)(3)}_{-kp}(t,t')\delta_{k,pq}, \end{equation}

and we have found a closure problem for a hierarchy of two-time statistical functions. Thus, we seek an equation governing the lagged evolution of the third-order cumulant

(3.8)\begin{align} \frac{\partial{}}{\partial{t'}}\langle \psi_{-k}^s(t){\psi}_p^s(t'){\psi}_q^{-s}(t') \rangle & = \varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}}\langle \psi_{-k}^s(t){\psi}_l^s(t'){\psi}_n^{-s}(t'){\psi}_q^{-s}(t') \rangle\,{\rm e}^{-2 {\mathrm i} \omega_n^st'}\delta_{{p},{l}{n}} \end{align}

(3.9)\begin{align} & +\varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{q},{l}{n}}\langle \psi_{-k}^s(t){\psi'}_p^s(t'){\psi'}_l^{-s}(t'){\psi'}_n^s(t') \rangle\,{\rm e}^{2 {\mathrm i} \omega_n^s t'}\delta_{{q},{l}{n}}. \end{align}

A fourth-order moment decomposes into a fourth-order cumulant and all possible products of second-order cumulants. Noting the fast decorrelation of counter-propagating Alfvén waves (which may be demonstrated via the Riemann–Lebesgue lemma equation (B34)), the fourth-order moments on the right-hand side of (3.9) may be evaluated as

(3.10)\begin{equation} \langle \psi_{-k}^s(t){\psi'}_p^s(t'){\psi'}_l^{-s}(t'){\psi'}_n^s(t') \rangle =\tilde{Q}^{(s,s,s,-s)(4)}_{-kpn}(t,t')\delta_{k,pln}, \end{equation}

and

(3.11)\begin{align} \langle \psi_{-k}^s(t){\psi}_l^s(t'){\psi}_n^{-s}(t'){\psi}_q^{-s}(t') \rangle & =\tilde{Q}^{(s,s,-s,-s)(4)}_{-klq}(t,t')\delta_{k,lnq} +\tilde{h}^s(l,t,t')\delta_{k,l}e^{-s}_q(t')\delta_{qn} \end{align}

(3.12)\begin{align} & =\tilde{Q}^{(s,s,-s,-s)(4)}_{-klq}(t,t')\delta_{k,lnq}+ \tilde{h}^s(k,t,t')\delta_{k,l}e^{-s}_q(t')\delta_{qn}. \end{align}

The fourth-order moments on the right-hand side of (3.9) containing two-time statistics simplify analogously with (B4) and (B14) and the third-order cumulant denoted by (3.7) evolves as

(3.13)\begin{align} \frac{\partial{}}{\partial{t'}}\langle \psi_{-k}^s(t){\psi}_p^s(t'){\psi}_q^{-s}(t') \rangle & = \varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde{Q}^{(s,s,-s,-s)(4)}_{-klq}(t,t')\delta_{k,lnq}\, {\rm e}^{-2 {\mathrm i} \omega_n^s t'}\delta_{{p},{l}{n}} \nonumber\\ & \quad +\varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{q},{l}{n}} \tilde{Q}^{(s,s,s,-s)(4)}_{-kpn}(t,t')\delta_{k,pln}\,{\rm e}^{2 {\mathrm i} \omega_n^s t'}\delta_{{q},{l}{n}} \nonumber\\ & \quad +\varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde{h}^s_k(t,t')\delta_{k,l}e_q^{-s}(t')\delta_{qn}\, {\rm e}^{-2 {\mathrm i} \omega_n^st'}\delta_{{p},{l}{n}} . \end{align}

The third term on the right-hand side of (3.13) may be integrated over wave vectors from noting the delta function correlations

(3.14)\begin{equation} \int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde{h}^s_k(t,t')\delta_{k,l}e_q^{-s}(t')\delta_{qn}\, {\rm e}^{-2 {\mathrm i} \omega_n^st'}\delta_{{p},{l}{n}} =-M_{k,pq} \tilde{h}^s_k(t,t')e_q^{-s}(t')\, {\rm e}^{2 {\mathrm i} \omega_q^s t'}, \end{equation}

where we have used the fact that

(3.15)\begin{equation} M_{p,ln}\delta_{k,l}\delta_{qn}\delta_{p,ln} =-M_{k,pq}. \end{equation}

Equation (3.13) can then be expressed as

(3.16)\begin{align} & \frac{\partial{}}{\partial{t'}} \tilde Q^{(s,s,-s)(3)}_{-kp}(t,t') = \varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde Q^{(s,s,-s,-s)(4)}_{-klq}(t,t')\delta_{k,lnq} \,{\rm e}^{-2 {\mathrm i} \omega_n^s t'}\delta_{{p},{l}{n}} \nonumber\\ & \quad +\varepsilon\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{q},{l}{n}} \tilde Q^{(s,s,s,-s)(4)}_{-kpn}(t,t')\delta_{k,pln}\,{\rm e}^{2 {\mathrm i} \omega_n^s t'}\delta_{{q},{l}{n}} -\varepsilon M_{k,pq} \tilde{h}^s_k(t,t')e_q^{-s}(t')\, {\rm e}^{2 {\mathrm i} \omega_q^s t'}. \end{align}

Through the lens of wave-turbulence theory, (3.6) and (3.16) are sufficient to close the two-time statistical hierarchy through $O(\varepsilon ^2)$ in perturbation, and obtain an explicit expression for the two-time energy spectrum.

3.3. Weak turbulence closure for two-time energy spectrum

We naïvely perturb the hierarchy of time-lagged differential equations derived in the previous subsection and search for secular terms (associated with divergent expansions) by investigating the asymptotic behaviour of (3.6) and (3.16) for lag through order $\tau = O(\varepsilon ^{-2})$. Those who are unfamiliar with the process by which we seek out secularities may refer to the elementary functional analysis presented in Appendix B.1. Renormalization of the leading-order two-time energy spectrum using the AP method yields a solution of the said function describing exponential decay due to nonlinear interactions.

Equations (3.6) and (3.16) are subject to the asymptotic expansions

(3.17)\begin{gather} \tilde{h}^s_k(t,t') = \tilde{h}^s_{0,k}(t,t')+\varepsilon \tilde{h}^s_{1,k}(t,t')+\varepsilon^2 \tilde{h}^s_{2,k}(t,t')+\cdots, \end{gather}

(3.18)\begin{gather}\tilde Q^{(s,s,-s)(3)}_{-kp}(t,t') = \tilde Q^{(s,s,-s)(3)}_{0,-kp}(t,t')+\varepsilon \tilde Q^{(s,s,-s)(3)}_{1,-kp}(t,t')+\varepsilon^2 \tilde Q^{(s,s,-s)(3)}_{2,-kp}(t,t')+\cdots, \end{gather}

(3.19)\begin{gather}\tilde Q^{(s,s,-s,-s)(4)}_{-klq}(t,t') = \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}(t,t')+\varepsilon \tilde Q^{(s,s,-s,-s)(4)}_{1,-klq}(t,t')+\varepsilon^2 \tilde Q^{(s,s,-s,-s)(4)}_{2,-klq}(t,t')+\cdots. \end{gather}

After substitution and change of variables $t' = t+\tau$ under the assumption of statistically stationary turbulence, the resulting perturbed hierarchy of equations is of the form

(3.20)\begin{gather} \frac{\partial{}}{\partial{\tau}}\tilde{h}^s_{0,k}(\tau) = 0, \end{gather}

(3.21)\begin{gather}\frac{\partial{}}{\partial{\tau}}\tilde Q^{(s,s,-s)(3)}_{0,-kp}(\tau) = 0, \end{gather}

(3.22)\begin{gather}\frac{\partial{}}{\partial{\tau}}\tilde{h}^s_{1,k}(\tau) = \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \tilde Q^{(s,s,-s)(3)}_{0,-kp}\,{\mathrm e}^{-2 {\mathrm i} \omega_q^s (t+\tau)}\delta_{k,pq}, \end{gather}

(3.23)\begin{gather} \frac{\partial{}}{\partial{\tau}}\tilde Q^{(s,s,-s)(3)}_{1,-kp}(\tau) =\mathcal{P}_{sp,-sq}\int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}\delta_{k,lnq} \,{\rm e}^{-2 {\mathrm i} \omega_n^s (t+\tau)}\delta_{{p},{l}{n}} \nonumber\\ \quad - M_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s}\, {\rm e}^{2 { {\mathrm i} } \omega_q^s (t+\tau)}, \end{gather}

(3.24)\begin{gather} \frac{\partial{}}{\partial{\tau}}\tilde{h}^s_{2,k}(\tau) = \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \tilde Q^{(s,s,-s)(3)}_{1,-kp}(\tau)\,{\mathrm e}^{-2 { {\mathrm i} } \omega_q^s (t+\tau)}\delta_{k,pq}.\end{gather}

Before proceeding through solving the hierarchy, we observe the behaviour of the two-time cumulants in configuration space, where they correspond to space–time correlation functions. The two-time energy spectrum is related to the two-point two-time correlation function as

(3.25)\begin{equation} \langle \boldsymbol{z}^{s}(x,t)\boldsymbol{z}^{s}(x+\boldsymbol{r},t+\tau) \rangle = \int {\rm d}\boldsymbol{k} \, {\rm d} \boldsymbol{k}' \tilde h^s_k(\tau)\delta_{kk'}\,{\rm e}^{\mathrm i\omega_k^s \tau}\,{\rm e}^{\mathrm i \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{r}}. \end{equation}

A feature appears at this statistical order, which is absent from the energy spectrum transform expression (B37) in the form of an oscillation with respect to time lag. Terms in expansion (3.17) which do not identically cancel the complex phase ${\rm e}^{{\rm i}\omega _k^s\tau }$ present in (3.25) will decay in configuration space at large values of $\tau$ as $O(\tau ^{-1})$ in accordance with the Riemann–Lebesgue lemma and (B35). However, the secular terms in the expansion (3.17) will lead to exponential decay in the two-time energy spectrum due to resonant interactions, providing a faster and more stringent temporal decorrelation rate. The two-time third-order cumulant corresponds to a three-point two-time correlation function as

(3.26)\begin{align} & \langle \boldsymbol{z}^{-s}(\boldsymbol{x},t+\tau) \boldsymbol{z}^s(\boldsymbol{x}+\boldsymbol{r},t) \boldsymbol{z}^s(\boldsymbol{x}+\boldsymbol{r}',t+\tau) \rangle\nonumber\\ & \quad = \int {\rm d}\boldsymbol{k}\, {\rm d} \boldsymbol{p}\, {\rm d} \boldsymbol{q}\tilde Q^{(s,s,-s)(3)}_{-kp}(\tau) \delta_{k,pq}\,{\rm e}^{-2\mathrm i\omega_q^st}\,{\rm e}^{\mathrm i(\omega_k^s-2\omega_q^s) \tau}\,{\rm e}^{\mathrm i\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{r}}\,{\rm e}^{\mathrm i\boldsymbol{p}\boldsymbol{\cdot} \boldsymbol{r}'}. \end{align}

Oscillations are encountered involving both time and time lag. It will be shown that there is still a resonance present (analogous to that of one-time closure theory) for which the complex phase ${\rm e}^{-2\mathrm i\omega _q^st}$ is exactly cancelled at any given moment in time $t$, leading to a secularity. A similar form of configuration space correlation function may be shown for the two-time fourth-order cumulant.

Proceeding through the expansion, we observe $h^s_{0,k}(\tau )$ and $Q^{(s,s,-s)(3)}_{0,-kp}(\tau )$ are independent of the time lag to leading order. Direct integration of (3.22) results in

(3.27)\begin{align} \tilde{h}^s_{1,k}(\tau) & = \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \tilde Q^{(s,s,-s)(3)}_{0,-kp}\int_{0}^{\tau}\, {\mathrm d}\tau \,{\mathrm e}^{-2 { {\mathrm i} } \omega_q^s (t+\tau)} \delta_{k,pq} \nonumber\\ & = \int {\mathrm d}{\boldsymbol p}\,{\mathrm d}{\boldsymbol q}M_{{k},{p}{q}} \tilde Q^{(s,s,-s)(3)}_{0,-kp}\,{\mathrm e}^{-2 { {\mathrm i} } \omega_q^s t}\varDelta_\tau(-2\omega_q^s) \delta_{k,pq}. \end{align}

By a similar token, integration of (3.23) with respect to $\tau$ yields

(3.28)\begin{align} Q^{(s,s,-s)(3)}_{1,-kp}(\tau) & =\mathcal{P}_{sp,-sq} \int {\mathrm d}{\boldsymbol l}\,{\mathrm d}{\boldsymbol n}M_{{p},{l}{n}} \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}\delta_{k,lnq}\, {\rm e}^{-2 { {\mathrm i} } \omega_n^s t}\varDelta_\tau(-2\omega_n^s )\delta_{{p},{l}{n}} \nonumber\\ & \quad - M_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s}\, {\rm e}^{2 {\mathrm i} \omega_q^s t}\varDelta_\tau(2\omega_q^s ). \end{align}

Substitution of the second term of (3.28) into the Fourier transform equation (3.26)

(3.29)\begin{align} & \langle \boldsymbol{z}^{-s}(\boldsymbol{x},t+\tau) \boldsymbol{z}^s(\boldsymbol{x}+\boldsymbol{r},t) \boldsymbol{z}^s(\boldsymbol{x}+\boldsymbol{r}',t+\tau) \rangle_{res}\nonumber\\ & \quad =-\int {\rm d}\boldsymbol{k}\, {\rm d} \boldsymbol{p}\, {\rm d} \boldsymbol{q}\tilde M_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \varDelta_\tau(2\omega_q^s ) \delta_{k,pq}\,{\rm e}^{\mathrm i(\omega_k^s-2\omega_q^s)\tau}\,{\rm e}^{\mathrm i\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{r}}\,{\rm e}^{\mathrm i\boldsymbol{p}\boldsymbol{\cdot} \boldsymbol{r}'}, \end{align}

yields cancellation of the complex phase ${\rm e}^{-2\mathrm i\omega _q^st}$, implying that this particular three-wave interaction is still resonant at each particular instance in time $t$ in a manner similar to (B52). This term will lead to a secularity in the two-time energy spectrum at $O(\varepsilon ^2)$ in expansion (3.17). It should be noted that this correlation still decays as $O(\tau ^{-1})$ due to the Riemann–Lebesgue lemma as the modes that interacted at $\tau =0$ propagate away from each other as characterized by the remaining complex exponentials. Substitution of (3.28) into (3.24) results in

(3.30)\begin{align} \frac{\partial{}}{\partial{\tau}}\tilde{h}^s_{2,k}(\tau) & = \mathcal{P}_{sp,-sq} \int {\mathrm d}\boldsymbol{p}\,{\mathrm d}\boldsymbol{q}\,{\mathrm d}\boldsymbol{l}\,{\mathrm d}\boldsymbol{n}M_{k,pq}M_{p,ln} \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}\delta_{k,lnq} \notag\\ & \quad \times {\rm e}^{-2 {\mathrm i} (\omega_n^s+\omega_q^s) t}\varDelta_\tau(-2\omega_n^s )\delta_{{p},{l}{n}} \,{\mathrm e}^{-2 {\mathrm i} \omega_q^s \tau}\delta_{k,pq}\nonumber\\ & \quad - \int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \varDelta_\tau(2 \omega_q^s )\,{\mathrm e}^{-2 {\mathrm i} \omega_q^s \tau}\delta_{k,pq}. \end{align}

Noting the relationships

(3.31)\begin{equation} \varDelta_\tau(-2\omega_n^s )\,{\rm e}^{-2 {\mathrm i} \omega_q^s \tau} = \frac{{\mathrm e}^{-2 {\mathrm i} (\omega_n^s+\omega_q^s) \tau}-{\mathrm e}^{-2 {\mathrm i} \omega_q^s \tau}}{-2 {\mathrm i} \omega_n^s}, \end{equation}

and

(3.32)\begin{equation} \varDelta_\tau(2\omega_q^s )\,{\mathrm e}^{-2 {\mathrm i} \omega_q^s \tau} = \varDelta_\tau(-2 \omega_q^s ), \end{equation}

(3.30) is recast as

(3.33)\begin{align} \frac{\partial{}}{\partial{\tau}}\tilde{h}^s_{2,k}(\tau) & = \mathcal{P}_{sp,-sq} \int {\mathrm d}\boldsymbol{p}\,{\mathrm d}\boldsymbol{q}\,{\mathrm d}\boldsymbol{l}\,{\mathrm d}\boldsymbol{n}M_{k,pq}M_{p,ln} \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}\delta_{k,lnq}\, {\rm e}^{-2 {\mathrm i} (\omega_n^s+\omega_q^s) t}\nonumber\\ & \quad \times \frac{{\mathrm e}^{-2 {\mathrm i} (\omega_n^s+\omega_q^s) \tau}-{\mathrm e}^{-2 {\mathrm i} \omega_q^s \tau }}{-2 {\mathrm i} \omega_n^s}\delta_{p,ln}\delta_{k,pq} \nonumber\\ & \quad -{\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \varDelta_\tau(-2\omega_q^s )\delta_{k,pq}. \end{align}

Direct integration of (3.33) then results in

(3.34)\begin{align} \tilde{h}^s_{2,k}(\tau) & = \mathcal{P}_{sp,-sq} \int {\mathrm d}\boldsymbol{p}\,{\mathrm d}\boldsymbol{q}\,{\mathrm d}\boldsymbol{l}\,{\mathrm d}\boldsymbol{n}M_{k,pq}M_{p,ln} \tilde Q^{(s,s,-s,-s)(4)}_{0,-klq}\delta_{k,lnq}\, {\rm e}^{-2 {\mathrm i} (\omega_n^s+\omega_q^s) t} \nonumber\\ & \quad\times\frac{\varDelta_{\tau}(-2(\omega_n^s+\omega_q^s) ) -\varDelta_\tau(-2 \omega_q^s) }{-2 {\mathrm i} \omega_n^s}\delta_{p,ln}\delta_{k,pq} \nonumber\\ & \quad - \int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \int_{0}^{\tau}\, {\mathrm d}\tau \varDelta_\tau(-2\omega_q^s )\delta_{k,pq}. \end{align}

The procedure for evaluating the long $\tau$ behaviour is nearly identical to the one-time closure. Here $\tilde {h}^s_{1,k}(\tau )$ and $Q^{(s,s,-s)(3)}_{1,-kp}(\tau )$ only depend on functions of the form $\varDelta _\tau (x)$ and are thus well-behaved in the limit $\tau \rightarrow \infty$. The first term in (3.34) is of the same form (B59) and is thus non-secular. Meanwhile, the second term is of the form (B58), and in a manner analogous to the one-time case, is unbounded in time. In anticipation of subsequent calculations, we consider the possibility of time lag on the interval $-\infty < \tau < \infty$. It was demonstrated in the appendix of Benney & Newell (Reference Benney and Newell1969) that the case of $\tau < 0$ amounts to multiplying delta functions present in the asymptotic expressions such as (B58) by $-1$, and generalization of results from $\tau \in \mathbb {R}^+$ to $\tau \in \mathbb {R}$ leads to a secularity in the limit $|\tau | \rightarrow \infty$ of the form

(3.35)\begin{equation} [\tilde{h}^s_{2,k}(\tau)]_{\rm secular} =-\frac{{\rm \pi}|\tau|}{2 v_{A}}\int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \delta(q_\parallel)\delta_{k,pq}. \end{equation}

The resultant asymptotic expansion for $\tilde {h}^s(\boldsymbol {k},\tau )$ is therefore

(3.36)\begin{equation} \tilde{h}^s_k(\tau) =\tilde{h}^s_{0,k} - \frac{{\rm \pi} \varepsilon^2 |\tau|}{2 v_{A}}\int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \delta(q_\parallel)\delta_{k,pq} + \text{non-secular terms}. \end{equation}

In what follows, we expand the leading-order behaviour of the two-time energy spectrum using the AP method and allow $\tilde {h}^s_{0,k}$ to vary on the lagged time scale $\varepsilon ^2 |\tau | = \tau _2$ as

(3.37)\begin{equation} \tilde{h}^s_{0,k} \rightarrow \tilde{h}^s_{0,k} - \varepsilon^2 |\tau| \frac{\partial{\tilde{h}^s_{0,k}}}{\partial{\tau_2}}. \end{equation}

The choice of

(3.38)\begin{equation} \frac{\partial{\tilde{h}^s_{0,k}}}{\partial{\tau_2}} =-\frac{\rm \pi}{2 v_{A}}\int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \delta(q_\parallel)\delta_{k,pq} ,\end{equation}

removes the secularity and preserves the order of the asymptotic expansion for $\tilde {h}^s(\boldsymbol {k},\tau )$. Equation (3.38) then provides the leading-order behaviour of $\tilde {h}^s(\boldsymbol {k},\tau )$ through time lag $\tau =O(\varepsilon ^{-2})$ in the form

(3.39)\begin{equation} \frac{\partial{\tilde{h}^s_{0,k}}}{\partial{\tau}} =-\frac{\varepsilon^2 {\rm \pi}\, {\rm sgn}(\tau) }{2v_{A}}\int {\mathrm d}\boldsymbol{p}\, {\mathrm d}\boldsymbol{q} M^2_{k,pq} \tilde{h}^s_{0,k}e_{0,q}^{-s} \delta(q_\parallel)\delta_{k,pq}. \end{equation}

Expression (3.39) can be integrated over wavevector $\boldsymbol {p}$ as well as $q_\parallel$ utilizing the delta function constraints in combination with $M^2_{k,pq}$ (see Appendix C) to obtain

(3.40)\begin{equation} \frac{\partial{\tilde{h}^s_{0,k}}}{\partial{\tau}} =-\varepsilon^2 \left[\frac{{\rm \pi} k_\perp^2 \, {\rm sgn}(\tau)}{ v_{A}}g^{-s}(0)\int_0^\infty {\mathrm d}q_\perp q_\perp \mathcal{E}^{-s}(q_\perp)\beta(q_\perp{/}k_\perp)\right]\tilde{h}^s_{0,k},\end{equation}

where

(3.41)\begin{equation} \beta(\xi) = \int_0^{\rm \pi} {\mathrm d}\phi \frac{(1-\xi \cos^2 \phi)^2 \sin^2\phi}{1+\xi^2-2\xi \cos\phi}. \end{equation}

Equation (3.40) provides a first-order linear homogeneous ODE for the leading-order behaviour of the two-time energy spectrum in the interaction picture with the corresponding solution

(3.42)\begin{equation} \tilde{h}^s_{0,k}(\tau) = \tilde{h}^s_{0,k}(0)\,{\mathrm e}^{- \varepsilon^2 \gamma^s_k|\tau|}, \end{equation}

for which

(3.43)\begin{equation} \gamma^s_k = \frac{2{\rm \pi} k_\perp^2}{v_{A}}\lambda^{-s}_{0\|}\int_0^\infty {\mathrm d}q_\perp q_\perp \mathcal{E}^{-s}(q_\perp)\beta(q_\perp{/}k_\perp),\end{equation}

where we used the fact that the parallel correlation length can be defined as $2\lambda _{0\|}\equiv g^{-s}(0)$. This decorrelation rate can be further simplified to gain more intuition about the nature of the decorrelation of weakly interacting Alfvén-wave fluctuations by seeking the change of variables $x\equiv q_\perp /k_\perp$, and assuming the spectra of fluctuations follow the power-law $\mathcal {E}^s(k_\perp )\propto k_\perp ^{-\alpha _s}$, in which case it is straightforward to show that

(3.44)\begin{equation} \gamma^s_k = \frac{2{\rm \pi} k_\perp^4}{v_{A}}\lambda_{0\|}\mathcal{E}^{-s}(k_\perp)\int_0^\infty {{\rm d}\kern0.06em x} x^{1-\alpha_s}\beta(x). \end{equation}

From Galtier's weak turbulence closure it was found that the steady-state solutions of the WKE leads to $\alpha _+=\alpha _-=3$ in the balanced case, and it was argued by Boldyrev & Perez (Reference Boldyrev and Perez2009) that this is also the case in the imbalanced case. Based on these weak turbulence closures for the one-time spectra and dimensional arguments we assume that $\mathcal {E}^s(k_\perp )\sim v_0^2k_{0\perp }k_\perp ^{-3}$, in which case we have

(3.45)\begin{equation} \gamma^s_k \sim k_\perp v_0\chi_0, \end{equation}

where $\chi _0\equiv k_{0\perp } v_0/k_{0\|}v_{A}$ is the ratio of the nonlinear to the linear frequency at the outer scale, and indicative of the weakness of the turbulence which fluctuations of this size will experience. Hence for the regime of weak turbulence, $\chi _0 = O(\varepsilon )$. The decorrelation frequency equation (3.45) then exhibits similar scaling (albeit exponential instead of a Gaussian functional dependence in $\tau$) with the sweeping-based result obtained for strong turbulence by Bourouaine & Perez (Reference Bourouaine and Perez2019) and Perez & Bourouaine (Reference Perez and Bourouaine2020) in the form $k_\perp v_0$. In addition to exhibiting different functional dependences, our result suggests that in the regime of weak turbulence, the strength of sweeping by the outer scale is reduced in proportion to the weakness of the turbulence in which this scale participates. This is made evident purely through the definition of $\chi _0$. In the strong regime, $\chi _0 \geq O(1)$ ($\varepsilon$ is no longer a small parameter) and the strong/weak decorrelation rates become comparable in order of magnitude, although this limit would largely invalidate the results presented herein. In the analysis performed by Bourouaine & Perez (Reference Bourouaine and Perez2019), it is assumed from the outset of computation that random sweeping by the outer scale is the dominant means of temporal decorrelation, and occurs much faster than linear Alfvénic propagation or nonlinear energy transfer. In contrast, the weak turbulence assumptions employed in this work mandate that Alfvénic propagation happens much faster than sweeping or nonlinear interaction. Despite the differing preliminary assumptions made regarding the separation between time scales governing the aforementioned physical processes, our calculation remarkably suggests a decorrelation characterized by sweeping with strength that is reduced in proportion to its disparity (in the form of $\chi _0$) with the linear time scale. It is worth pointing out that this competition between sweeping and Alfvénic propagation at the outer scale requires additional investigation.

Finally, noting definitions (2.12) and (3.3), the leading-order behaviour of the two-time energy spectrum may be expressed as

(3.46)\begin{equation} h^s_{0,k}(\tau) = e^s_{0,k}\,{\mathrm e}^{ {\mathrm i} \omega_k^s\tau-\varepsilon^2 \gamma^s_k|\tau|}, \end{equation}

which implies a scale-dependent time correlation function of the form

(3.47)\begin{equation} \varGamma^s(\boldsymbol{k},\tau) = {\mathrm e}^{ {\mathrm i} \omega_k^s\tau-\varepsilon^2 \gamma^s_k|\tau|}. \end{equation}

Our solution (3.46) suggests oscillatory behaviour as well as exponential decay in the correlation between the temporally lagged states due to linear Alfvénic propagation and resonant nonlinear energy transfer, respectively. The former occurs on a time scale of order unity and is reminiscent of linear solutions to (2.7) while the latter becomes significant (keeping in mind $\gamma _k^s \sim k_\perp v_0 \chi _0 = O(\varepsilon t_{\rm nl}^{-1})$) at a temporal lag of $\tau =O(\varepsilon ^{-3}t_{\rm nl})$, an intermediary between $O(\varepsilon ^{-3})$ and $O(\varepsilon ^{-4})$.

The conclusion of exponential decay becoming appreciable at a lag of $\tau =O(\varepsilon ^{-3}t_{\rm nl})$ appears to instigate a most interesting predicament, bringing into question the legitimacy of the underlying closure process. Calculations were performed to ensure uniformity of the two-time energy spectrum expansion equation (3.17) accurate through $O(\varepsilon ^2)$ in perturbation and correspondingly a time lag of $\tau = O(\varepsilon ^{-2})$, yet a result for $h_k^s(\tau )$ is suggested herein which appears to exhibit perhaps the most significant aspect of its functional behaviour beyond the previously presumed domain of validity in $\tau$. This apparent contradiction may possibly be readily rectified by § 5 of Benney & Newell (Reference Benney and Newell1969) (hereafter referred to as BN69) on higher closures.

For systems whose wave amplitudes evolve nonlinearly in time, generally speaking according to the form of BN69 (2.5), or more specifically for instance our (3.5), it is clearly demonstrated in § 5 of BN69 that genuine (as opposed to spurious) secularities in cumulant expansions arise at $O(\varepsilon ^2)$ and $O(\varepsilon ^4)$ in perturbation, while none are present at $O(\varepsilon ^3)$ as shown by the analysis leading to BN69 (5.3). Hence, removal of these secularities using the AP method is only necessary in their work on time scales of $T_2 = \varepsilon ^2 t$ and $T_4 = \varepsilon ^4 t$. Correspondingly, the lower-order first-closure presented in our work only requires such corrections on lagged time scale $\tau _2 = \varepsilon ^2 \tau$ as we are not concerned with $O(\varepsilon ^4)$ behaviour. By direct mathematical analogy between the governing dynamical expressions of BN69 (2.5) and our own (3.5), a similar (albeit simpler, due to the quantity of interest being autocorrelation instead of energy) result is expected to follow, however, investigation of higher-order influence upon Fourier autocorrelation is well outside the scope of our own work.

Rigorous demonstration of the said absence of genuine secular growth in expansion (3.17) at $O(\varepsilon ^3)$ indeed requires a more complicated implementation of the AP method (see BN69 (3.3)) which contains a method for removing non-secular ‘free terms’ at lower orders in perturbation that otherwise lead to spurious closures (see BN69 (5.2)) at higher orders, including $O(\varepsilon ^{3})$. Being that our own calculation is only interested in a first closure with accuracy to $O(\varepsilon ^2)$ in perturbation, such a sophisticated analysis is unnecessary and not considered (Benney & Newell Reference Benney and Newell1969). Nonetheless, our calculation of (3.45) is likely rigorous through $O(\varepsilon ^3)$ in perturbation insomuch as accounting for terms present in our expansion equation (3.17) which will exhibit genuine secular growth. This is admittedly speculative in nature, but nonetheless likely true because the closure calculations presented in both our work and BN69 exhibit similar underlying mathematical structure, despite the differences in functions of interest as well as the pertinent independent variable. Rigorous investigation and potential proof of such a claim in a more general form is likely to be part of future work on weak turbulence closures for Fourier autocorrelation. Should this claim turn out to be false, there is still inherent value in the results of what would prove to constitute but a semiphenomenological analysis. This is demonstrated through the conceptuality suggested by the functional forms of the two-time energy and power spectra described by (3.46) and (3.56) as they nonetheless allude to the lowest-non-trivial-order in perturbation influence of nonlinear resonant energy transfer upon two-point two-time statistics for the model employed herein. In light of this, we remind the reader that this is the same resonance responsible for closure and obtention of the well-known WKE result of Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2000) (or (B64) herein). Such simultaneity of causality begs the question: should a comparable ordering characterize the time scale over which spectral energy fluctuates?

In fact, it is not difficult to show by means analogous to those which produced (3.44) and (3.45) an answer in the affirmative to this question. In Appendix D, it is demonstrated that the WKE

(3.48)\begin{equation} \frac{\partial{e^s_{0,k}(t)}}{\partial{t}} = \frac{{\rm \pi} \varepsilon^2}{2v_{A}}\int {\mathrm d} \boldsymbol{p}\, {\mathrm d} \boldsymbol{q} M^2_{k,pq} [e^{s}_{0,p}-e^s_{0,k}]e^{-s}_{0,q}\delta_{k,pq}\delta(q_\parallel), \end{equation}

may be approximately recast in the form

(3.49)\begin{equation} \frac{\partial{e^s_{0,k}(t)}}{\partial{t}} =2{\rm \pi} \varepsilon^2 k_\perp v_0 \chi_0 \left(\int_0^{\infty} x^{1-\alpha_s}[(1-x)^{-\alpha_s}-1]\beta(x)\,{{\rm d}\kern0.06em x}\right) e^s_{0,k}. \end{equation}

Defining

(3.50)\begin{equation} \varOmega_k^s =2{\rm \pi} k_\perp v_0 \chi_0 \left(\int_0^{\infty} x^{1-\alpha_s}[(1-x)^{-\alpha_s}-1]\beta(x)\,{{\rm d}\kern0.06em x}\right), \end{equation}

(3.49) becomes the differential equation

(3.51)\begin{equation} \frac{\partial{e^s_{0,k}(t)}}{\partial{t}} = \varepsilon^2 \varOmega_k^s e^s_{0,k}, \end{equation}

with corresponding solution

(3.52)\begin{equation} e^s_{0,k}(t) = e^s_{0,k}(0)\,{\rm e}^{\varepsilon^2 \varOmega_k^s t}. \end{equation}

Indeed, noting that $\varOmega _k^s \sim k_\perp v_0 \chi _0 = O(\varepsilon t_{\rm nl}^{-1})$, the solution described by (3.52) suggests that energy fluctuates over a time scale $(\varepsilon ^2 \varOmega _k^s)^{-1} = O(\varepsilon ^{-3}t_{\rm nl})$, as one may expect in the interest of phenomenological consistency (via common underlying causality) with the two-time energy spectrum equation (3.46). Again, we reference back to the concept demonstrated in BN69 that systems of this form only exhibit genuine secular behaviour on time scales of $O(\varepsilon ^{-2})$ and $O(\varepsilon ^{-4})$, and hence the WKE (3.48) as well as associated solution (3.52) are likely correct through the demonstrated relevant dynamical time scale $t=O(\varepsilon ^{-3}t_{\rm nl})$.

The property of exponential decay in correlation function equations (3.39) and (3.47) in the limit of large $\tau$ coincides with approximate solutions of a generic IDE for the SDTC derived and solved by Perez et al. (Reference Perez, Azelis and Bourouaine2020). The SDTC expression defined by (50) in that work is left in terms of an unspecified integral of the two-time energy spectrum, a function that is also not explicitly calculated by the authors. In contrast, our SDTC result (3.47) was extracted from a simple ODE and is dependent on parameters present in the RMHD model which are already fully determined at the outset of analytical computation. It is worth noting that the model of exponential decay has been proven applicable to the investigation of both simulation and spacecraft data (Matthaeus et al. Reference Matthaeus, Dasso, Weygand, Kivelson and Osman2010; Lugones et al. Reference Lugones, Dmitruk, Mininni, Wan and Matthaeus2016), however, the correct ${\mathrm e}$-folding time scale is obviously highly specific to the particular regime of turbulence being measured, assuming this functional form is even applicable to the relevant physical context under consideration.

We can acquire an intuition for the result (3.46) by comparing (3.39) with the one-time result (B64). Equation (B64) suggests that the nonlinear energy flux into or out of a given mode with wavevector $\boldsymbol {k}$ after ‘scattering’ off a $\boldsymbol {q}_\perp$ mode is proportional to the difference in energy between itself and all possible waves of wavevector $\boldsymbol {p}$ capable of taking part in the interaction according to the delta function constraints. In contrast, (3.39) suggests that the signature of a given mode will decorrelate from itself in proportion to its own energy as well as that of all $\boldsymbol {q}_\perp$ modes with which it interacts. On time scales of $t=O(\varepsilon ^{-3} t_{\rm nl})$, the energy $e^s_k$ of a mode will fluctuate according to (B64), but its statistical self-similarity $h^s_k$, which evolves according to (3.39), will exponentially decay when the time lag $\tau$ separating the two instances at which the Elsasser field is evaluated approaches $O(\varepsilon ^{-3} t_{\rm nl})$. Given a sufficiently long time, the vectors in the complex plane representing the system's set of constituent Fourier modes will both rotate and change their norms. The overall state of the system will become entirely dissimilar from itself and there will no longer be a memory of the $\tau =0$ field of turbulent fluctuations.

The quantity $\gamma _k^s$ defines the rate at which the autocorrelation of a Fourier Elsasser mode with wavevector $\boldsymbol k$ will exponentially decay via integration over all possible resonant nonlinear interactions in which it may partake during some temporal interval $\tau$. The various mathematical features present in expression (3.43) provide phenomenological insight into the dynamical means by which a given mode will decorrelate from itself. Firstly, the prefactor $k_\perp ^2 / v_{A}$ indicates that modes of smaller perpendicular spatial extent (higher $k_\perp$) will decorrelate faster than larger ones. Additionally, inverse proportionality with the Alfvén velocity simply follows the intuition that counter-propagating waves will pass through each other more rapidly, weakening the strength of interaction. The predominant feature present in the decorrelation rate is an integration over the energy stored in the set of all non-propagating modes $\boldsymbol q= (q_\parallel = 0, \boldsymbol q_\perp )$ with which $\boldsymbol k$ interacts. The presence of the spectrum $g^{-s}(0)\mathcal {E}^{-s}(q_\perp )$ merely suggests that the capacity for some turbulent grating $q_\perp$ to scatter $\boldsymbol k$ is directly proportional to the stationary structure's energy content. Recalling the spectrum's inverse dependence on wavenumber, it can be inferred that modes with lower $q_\perp$ scatter a given $\boldsymbol k$ with greater temporal efficiency. The spectral integral is modified by the presence of the function $\beta (q_\perp / k_\perp )$, which provides a geometric weighting to the modal decorrelation rate dependent on the ratio of perpendicular wavenumbers of interacting modes. Rudimentary numerical integration suggests that this function is strongly peaked at $\beta (0)={\rm \pi} /2$ and monotonically decays to an asymptotic value of $\beta \approx 0.2$ for $q_\perp / k_\perp > 2$. This functional behaviour reinforces the aforementioned concepts suggested by the numerator of the prefactor as well as the presence of the perpendicular spectrum that $q_\perp$ modes of larger spatial expanse scatter $\boldsymbol k$ modes with increased efficacy and the associated decorrelation is increasingly pronounced in proportion to $k_\perp$. This is to suggest that in a field of turbulent fluctuations, larger structures are both more resistant to being scattered and distorted while simultaneously performing these actions onto others more effectively than those of smaller scale. It is worth noting that this may suggest the presence of energy transfer that is somewhat non-local in wavenumber space, but further investigation of this possibility is required and is well beyond the scope of the present analysis.

When used in tandem with the Fourier transform expression (2.16), (3.46) has the potential to serve as an improvement over the TH in the regime of weakly turbulent MHD in the form of the space–time correlation function

(3.53)\begin{align} C^s(\boldsymbol r,\tau)= \int e^s_{0,k}\, {\mathrm e}^{ {\mathrm i} \omega_k^s\tau-\varepsilon^2 \gamma^s_k|\tau|}\, {\rm e}^{\mathrm i\boldsymbol k\boldsymbol{\cdot}\boldsymbol r} \, {\rm d} \boldsymbol k. = \int g^{s}(k_\|)\, {\rm e}^{\mathrm i k_\parallel ( s v_{A} \tau + z) } \, {\rm d} k_\parallel \int \mathcal{E}^{s}(k_\perp)\, {\rm e}^{\mathrm i k_\perp \boldsymbol{\cdot} \boldsymbol r_\perp- \varepsilon^2 \gamma_k^s |\tau| } \, {\rm d} \boldsymbol k_\perp, \end{align}

where we have used $\boldsymbol r = z \hat {\boldsymbol {b}} + \boldsymbol r_\perp$ to isolate parallel from perpendicular correlation lengths and separated parallel and perpendicular spectra according to $e_{0,k}^{s}=\mathcal {E}^{s}(k_\perp )g^{s}(k_\|)$. If one assumes the existence of $G^{s}$, the Fourier transform of $g^s$, integration with respect to $k_\parallel$ then yields the correlation

(3.54)\begin{equation} C^s(\boldsymbol r,\tau)= G^{s}(sv_{A} \tau +z) \int \mathcal{E}^{s}(k_\perp)\, {\rm e}^{\mathrm i k_\perp \boldsymbol{\cdot} \boldsymbol r_\perp- \varepsilon^2 \gamma_k^s |\tau| } \, {\rm d} \boldsymbol k_\perp. \end{equation}

The parallel structure of the correlation $G^{s}$ contains a TH-like argument for an advecting flow travelling at the Alfvén velocity $v_{A}$ in the $\pm \hat {\boldsymbol {b}}$ direction and appears to be unaffected by the presence of nonlinear interactions. This is in agreement with observations of MHD turbulence simulations first made by Shebalin, Matthaeus & Montgomery (Reference Shebalin, Matthaeus and Montgomery1983) and forged to rigorous mathematical form in the WKE of Galtier et al. (Reference Galtier, Nazarenko, Newell and Pouquet2000); both of which describe spectral energy transfer that is highly anisotropic and confined to planes of constant $k_\parallel$. The said anisotropy is manifest in the $\boldsymbol {k}_\perp$ integral in the form of resonant nonlinear energy transfer to higher $k_\perp$, lending to exponential decay in the correlation as the dynamics proceed in time. Elsasser field autocorrelation unsurprisingly approaches zero as the perpendicular structure evolves into a different and unrecognizable version of itself due to turbulence. Contrary to the TH, (3.54) makes no assumption of frozen fluctuations while also taking into account the state of the plasma at multiple instances in time. As a result, the formulation provides an authentic two-point two-time correlation function with space and time independent, as opposed to the two-point one-time resulting from the aforementioned simplifications allotted by the TH. Additionally, further manipulation of (3.46) leads to a prediction of temporal frequency broadening due to the presence of nonlinear interactions.

If one first considers the temporal Fourier transform of the two-time energy spectrum equation (3.46) for the limit $\varepsilon = 0$, such that there are no nonlinear interactions between fluctuations, the expression

(3.55)\begin{equation} h^s_{0,k}(\omega) = e^s_{0,k} \int_{-\infty}^\infty {\mathrm e}^{ {\mathrm i} \omega_k^s \tau}\, {\mathrm e}^{-{\rm i} \omega \tau} \, {\rm d} \tau = 2{\rm \pi} e^s_{0,k} \delta(\omega - s k_\parallel v_A) ,\end{equation}

is obtained. The above result describes a delta function peaked about the Alfvén frequency $\omega = \omega ^s_k$ and coincides with a TH-based assumption for which the advecting fluid flow travels at the Alfvén velocity in either $s=\pm 1$ direction along the background magnetic field. Alternatively, it may be inferred that in the absence of nonlinear interactions, the temporal structure of the system can be directly predicted from the shear-Alfvénic dispersion relation and the signature of a fluctuation $s k_\parallel$ will only be sampled at a single unique frequency. In contrast, inclusion of nonlinear interactions leads to a temporal Fourier transform of (3.46) of the form

(3.56)\begin{equation} h^s_{0,k}(\omega) = e^s_{0,k} \int_{-\infty}^\infty {\mathrm e}^{ {\mathrm i} \omega_k^s\tau-\varepsilon^2 \gamma^s_k|\tau|}\, {\mathrm e}^{-{\rm i} \omega \tau} \, {\rm d} \tau = \frac{2\varepsilon^2 \gamma^s_k }{(\varepsilon^2 \gamma^s_k)^2+(\omega-\omega_k^s)^2}e^s_{0,k}.\end{equation}

Equation (3.56) describes a power spectrum featuring Lorentzian peaks in the neighbourhood of $\omega = \omega _k^s$ that are both shifted away from pure linear Alfvénicity and appreciably broadened from infinitesimal width by the presence of the decorrelation frequency $\varepsilon ^2\gamma _k^s$. Exhibition of finite width indicates that interactions between fluctuations destroy the convenient, albeit superficial injective mapping between temporal and spatial structure allotted by the TH; a fluctuation with given $s k_\parallel$ will contribute to a continuum frequency band (as opposed to a single discrete delta function) in the system's power spectrum. While accurate interpretation of time-series measurements is made non-trivial by the presence of resonant nonlinear energy transfer, (3.56) attempts to make some prediction of expectation, at the very least in a qualitative manner.