Weak turbulence in two-dimensional magnetohydrodynamics

Magneto-Hydro-Dynamics (MHD) is of great interest for modeling turbulence in magnetically confined and unconfined plasmas. In astrophysics its applications range from solar wind Marsch and Tu (1994), to the Sun Priest , to the interstellar medium Ng et al. (2003) and beyond Zweibel and Heiles (1997). At the same time, MHD is also relevant to large-scale motion in nuclear fusion devices such as tokamaks Strauss (1976). One of the pioneering results of incompressible MHD turbulence has been obtained by Iroshnikov Iroshnikov (1963) and Kraichnan Kraichnan (1965) (thereafter IK) who proposed an extension of the Kolmogorov phenomenology Kolmogorov (1941), originally derived for Navier-Stokes equations for Hydro-Dynamics (HD). For simplicity the assumptions of homogeneity and isotropy were made by Kolmogorov, and the energy cascade was supposed to be dominated by local (in scale) interactions between eddies of similar size. Then the Kolmogorov phenomenology leads to the well-known one-dimensional kinetic energy spectrum $E(k)\sim k^{-5/3}$, where $k$ is the wave vector. The associated cascade properties for its inviscid invariants differ for 3D and 2D turbulence: while in 3D the energy and the kinetic helicity exhibit direct cascades, in 2D the energy cascades inversely – still with a $-5/3$ scaling – whereas a direct cascade is found for the enstrophy which leads to spectrum $E(k)\sim k^{-3}$ at the small scales.

IK modified the Kolmogorov phenomenology by taking into account the magnetic field. They also assumed turbulence homogeneity, isotropy and locality of interactions. However, there exist fundamental differences between the Kolmogorov and the IK theories. First of all, in MHD the energy cascade is supposed to be dominated not by the interactions between eddies but between Alfvén wave packets propagating in opposite directions; this modification leads to the energy spectrum $E(k)\sim k^{-3/2}$. Furthermore, unlike hydrodynamics the cascades of the ideal MHD invariants exhibit some similarities in their behaviour in 2D and 3D Biskamp (2003): in both cases, the cascades have the same direction with a direct cascade for the energy and cross-helicity, and an inverse cascade for the magnetic helicity or the anastrophy.

The differences between MHD and HD turbulence go beyond these classical properties. In the IK theory the large-scale magnetic field is supposed to play the role of external field which is necessary for the existence of Alfvén waves but its main effect, i.e. anisotropy, is not taken into account. The importance of an external magnetic field has been discussed many times during the last two decades Montgomery and Turner (1981); Shebalin et al. (1983); Oughton et al. (1994); Ng and Bhattacharjee (1997); Bigot et al. (2008a); Verma (2004); Chandran (2005); Boldyrev (2006); Bigot and Galtier (2011) and the anisotropic behavior has been shown in direct numerical simulations for both 2D Shebalin et al. (1983) and 3D Oughton et al. (1994).

Despite some similarities – like the cascade directions – the question about the identification of differences between 2D and 3D MHD turbulence still represents an important issue. In early numerical studies Shebalin et al. (1983) mainly 2D simulations were performed because of the limited numerical resources available and the non-accessibility of 3D calculations at high Reynolds numbers. Nowadays, 3D MHD numerical simulations are commonly achieved Politano et al. (1998); Gomez et al. (1999); Biskamp and Mueller (2000); Haugen et al. (2004); Biskamp and Shwartz (2001); Biskamp and Mueller (2000); Merrifield et al. (2007, 2006); Bigot et al. (2008b). At the same time, 2D simulations are still used for the illustration of new numerical techniques Dritchel and Tobias (2012). There is also an interest for the investigation of freely decaying MHD turbulence because the Reynolds numbers can be higher in 2D than in 3D Biskamp and Shwartz (2001). When the magnetic field perturbations are small compare to a uniform background magnetic field, the 2D MHD equations are sometimes used to model turbulence. Such a situation is particularly relevant for solar coronal loops as well as for reduced models for fusion plasmas in slab geometry Strauss (1976).

Strong statements were made by some authors that 2D simulations can be safely used to model 3D situations because the properties of the 2D and the 3D MHD turbulence are essentially the same Biskamp and Shwartz (2001); Biskamp (1993). One of the motivations of the present paper is to test validity of this claim in a special case when the external magnetic field is strong.

In this paper, we consider 2D MHD in the presence of a strong background magnetic field which implies realization of the weak turbulence regime. One of the main advantages of this regime is the fact that it allows one to derive accurate analytical results for the spectrum. An explicit comparison will be made between the weak turbulence regimes in 2D and in 3D; the latter was analyzed rigorously in Galtier et al. (2000). The weak wave turbulence approach is widely familiar to the plasma physics community Achterberg (1979); Akhiezer et al. (1975); McIvor (1974); Zakharov (1974, 1984); Zakharov et al. (1992); Kuznetsov (1972a, b, b); Tsytovich (1970); Sagdeev and Galeev (1969); Vedenov (1968); Galtier et al. (2001); Galtier and Nazarenko (2008). It is a statistical description of a large ensemble of weakly interacting dispersive waves. The formalism leads to wave kinetic equations from which exact power law solutions can be found for the energy spectra. There were several reasons which postponed development of weak turbulence for Alfvén waves. The first one is their semi-dispersive nature. Typically, the wave-kinetic approach cannot be used for non-dispersive waves since such wavepackets propagate with the same group velocity even if their wavenumbers are different, the energy exchange between such waves may not be considered small, and may lead to possible energy accumulation over a long time of interaction. The Alfvén waves represent a unique exception to this rule because co-propagating waves do not interact and the nonlinear interaction is present only for counter-propagating waves. The latter pass through each other in some finite time and no long time cumulative effect occur. That is why the Alfvén waves represent a unique example of semi-dispersive waves for which the wave turbulence theory applies. The second reason which renders the weak turbulence theory for Alfvén waves very subtle is the fact that the domination of three-wave interactions – as assumed by IK – may be questionable. While in Shridar and Goldreich (1994) the three-wave interactions were declared absent, the IK argument has been re-established in Galtier et al. (2000); Ng and Bhattacharjee (1997); Montgomery and Matthaeus (1995). The weak turbulence theory for 3D incompressible MHD was developed in Galtier et al. (2000) (see also Galtier et al. (2002); Nazarenko (2011)) where three-wave kinetic equations were derived with their exact solutions via a systematic asymptotic expansion in powers of small nonlinearities.

The main goal of the present paper is to derive the weak turbulence equation for the 2D MHD, analyze it and make a comparison with the 3D case. The crucial technical step which allows a comprehensive theoretical analysis of the solutions consists in transforming the wave kinetic equation into an integrable form by Fourier transforming it and separating the transverse and the parallel dynamics by using a self-similar “effective time” variable.

This article is organized as follows. In section II we derive the weak turbulence kinetic equation through a general perturbative procedure. In sections III and IV we proceed with a detailed investigation of its properties in an anisotropic limit. The main goal of such a study is to verify whether or not the turbulence is local. First of all, we will consider the steady state behavior by looking for Kolmogorov-Zakharov type solutions and check their locality. Next, we will proceed with investigation of an unsteady spectrum evolution by considering two different cases with a gaussian-shaped source and different kinds of dissipation: a uniform friction and a viscosity. Due to integrability of the weak wave-kinetic equation in the first case it is possible to find an exact solution. In the second case, a qualitative analysis for the steady state is complemented by a numerical simulation of the spectrum evolution. The goal of section V is to develop some qualitative reasoning about the turbulent behavior of our system near the applicability margin of wave-kinetic formalism and beyond. Formation of steady state is also discussed. Finally, we present a summary of our results in section VI.

At the first step of our investigation of the wave-kinetic equation (21), we seek exact stationary power law solutions, $\eta(k_{y};\infty)^{\pm}\propto k_{y}^{\nu^{\pm}}$. For this, we will use so-called Zakharov transformation,

We split the integral into the r.h.s. of (21) in two equal parts and we perform the Zakharov transformation in the integrand of one of these parts. This leads to the following conditions on the exponents,

However, the resulting power law spectra represent true mathematical solutions if and only if the original (before the transformation) integral in the r.h.s. of (21) converges at these solutions. In Appendix B, we perform such a convergence study and show that this integral never converges and therefore, no Kolmogorov-Zakharov solutions are possible for 2D MHD system, contrary to the 3D case. We remark that the balanced turbulence spectrum with $\nu^{+}=\nu^{-}=-1$ has a logarithmic divergence. Thus, one could anticipate that such a marginal nonlocality could be “fixed” by logarithmic corrections. We will see later that this is false.

A remarkable property of the kinetic equation (19) is its simplicity. In this section we show that in some physical situations it can be solved analytically. Let us introduce into the equation (19) sources and sinks of waves:

The function ${\mathcal{F}}(k_{x},k_{y})$ may represent forcing or dissipation, depending on the choice of the sign before it, and the constant $\sigma_{d}$ introduces a uniform friction.

In order to use factorisation (20) and eliminate $\mu(k_{x})$ in both sides of the forced kinetic equation, we assume the following type of force/dissipation function, ${\mathcal{F}}(k_{y},k_{x})={\mathcal{F}}_{x}(k_{x}){\mathcal{F}}_{y}(k_{y})$. Then the parallel component of (20) must be chosen as $\mu(k_{x})={\mathcal{F}}_{x}(k_{x})$. Finally, we obtain the following forced/dissipated kinetic equation for the perpendicular component of the energy spectrum,

In this section we are providing, at a qualitative level of rigor, a description for the energy spectrum behaviour beyond the weak turbulence regime at the late stage of the evolution. For the wave-kinetic equation to be valid the nonlinearity has to remain weak, i.e. the nonlinear time scale should be much longer than the linear wave period. This results in the following condition,

Here we have used the estimate for the nonlinear evolution time taking the standard hydrodynamic non-linear time scale. In our dynamical equations, the nonlinearity related to the magnetic field is of the same order. Finally, the applicability condition can be rewritten as a limitation for the parallel wave number:

It means that for small values of parallel wave vector component near $k_{x}^{*}$ the kinetic equation becomes invalid. The question about applicability of the wave-kinetic equation near $k_{||}=0$ has been frequently discussed in the literature, eg. in the 3D MHD turbulence context Galtier et al. (2000). In particular it was speculated in Galtier et al. (2000) that a sufficient spectrum smoothness for wavenumbers near $k_{||}=0$ must be present for the wave-kinetic equation to be applicable. In the 2D case, the smoothness of spectrum near $k_{x}=0$ is asymptotically (in time) broken due to a special structure of the kinetic equation.

Indeed, as we mentioned before, the wave-kinetic equation for the 2D MHD system is formulated in terms of a self-similar ”time” variable $\tau=k_{x}^{2}t$. Therefore dependence on the parallel wavenumber is still present in the perpendicular part of the energy spectrum $\eta(k_{y},\tau)$ in an implicit way, via $\tau$. Such a self-similar dependence on $k_{x}$ is manifested, at each fixed $k_{y}$, in shrinking of the original $k_{x}$ profile along the $k_{x}$-axis as time grows (see Fig. 3). The spectrum is narrowing and its derivative is growing near small values of $k_{x}$, and when it is so steep that a significant variation occurs over the range $\sim k_{x}^{*}$, the kinetic equation breaks down. Time estimate for such a breakdown is $t\sim(k_{x}^{*})^{-2}$.

Therefore, the weak turbulence description will break down at the late evolution stages, and the wave-kinetic equation will no longer work. However, it is possible to amend this description to take into account the strongly nonlinear effects and develop a qualitative theory of subsequent evolution leading to a steady state. Below we will present a qualitative argument which will allow us to obtain such a theory.

First of all we note that the three-wave interaction is never exactly resonant: it involves all the quasi-resonant within a certain small distance from the exact resonant frequency – so called nonlinear resonance broadening $\Gamma\sim t_{nl}^{-1}$. In the other words, the delta function in kinetic equation $\delta(2k_{1x})=\delta\left(\omega_{k}-\omega_{1}-\omega_{2}\right)$ should be replaced by a peaked function $f(k_{1x})$ with a small but finite width $\Gamma$. For sufficiently smooth spectra the difference from the delta-function can be ignored, but for sharp and narrow spectra the integrand in the kinetic equation’s integral become itself peaked and the delta-function broadening becomes important. Its main effect is that of a filter $f(k_{x})$ in the $k_{x}$ variable which acts to smoothen any sharp changes over the range $\Delta k\sim k_{x}^{*}$. The energy is no longer conserved separately at each fixed $k_{x}$.

For large wavenumbers $k_{x}\gg k_{x}^{*}$ (where the spectrum remains slowly varying even when it is steep at $k_{x}\sim k_{x}^{*}$) the kinetic equation could be easily amended by replacing $n^{\mp}(0,k_{y},0)=\int n^{\mp}\left(k_{1x},k_{1y},\tau\right)\delta(k_{1x})dk_{1x}$ with $\langle n^{\mp}\rangle(k_{y},\tau)=\int n^{\mp}\left(k_{1x},k_{1y},\tau\right)f(k_{1x})dk_{1x}$. For small wavenumbers, $k_{x}\sim k_{x}^{*}$, the effect of the resonance broadening is not reduced to such a simple modification of just one function in the integral. It is clear that the spectrum at $k_{x}=0$, which was fixed in the wave-kinetic approximation, will suffer changes caused by the smoothing in the direction determined by spectral slope at small $k_{x}$: if the gradient is positive (negative) the value will increase (decrease), as illustrated in Fig. 4. The details of the evolution at the small wavenumbers are not important because the combined action of the self-similar shrinking and smoothing will lead to a very rapid wipeout of all the gradients in $k_{x}$ and formation of a steady state with $\eta$ independent of $k_{x}$. Correspondingly, the values of $\eta$ at $k_{x}=0$ will adjust themselves to the values at $k_{x}=\infty$. After this moment, when the rapid dependencies on $k_{x}$ disappear, the kinetic equation in its usual weak turbulence form is valid once again, and it can be used for finding the final steady state spectrum. Since $\eta$ is now independent of $k_{x}$, the steady state could be readily obtained from the formal condition $\eta(k_{y},0)=\eta(k_{y},\infty)$ which simply means our solution independent of $\tau$; it has nothing to do with the initial/final values of the spectrum in time or $k_{x}$.

Thus, the evolution can be summarized as follows. At the early stage, $t\ll(k_{x}^{*})^{-2}$, the evolution is described by the three-wave kinetic equation. Then at the advanced stage, with characteristic times scales $t\sim(k_{x}^{*})^{-2}$, the kinetic equation is broken down by its own evolution. Smoothing of strong gradients in $k_{x}$ occurs, which results in spectrum stabilisation and re-emergence of the kinetic equation description at the large time scales $t\gg(k_{x}^{*})^{-2}$. This kinetic equation describes spectrum evolution within the steady state regime. Let us now consider the properties of such a steady state.

We have shown that the three-wave interactions for the pseudo-Alfvén waves (PAW) in the 2D MHD system are non-empty and it is possible to obtain a three-wave kinetic equation within the weak turbulence approach. These interactions take place in the second order of the anisotropy parameter.

We found Kolmogorov-Zakharov power law spectra for PAW in 2D MHD system and showed that they are not realisable due to divergence of the collision integrals of the kinetic equation. In the balanced case this divergence is marginal. This is an indirect indication that the 2D PAW turbulence is nonlocal: it is dominated by interaction of waves with very different in size wavelengths. Our full analytical solution of the kinetic equation confirms such a nonlocality. It also dispels the myth that all marginally nonlocal spectra can be “fixed” by a logarithmic correction.

The crucial technique for our analysis is passing to pseudo-physical space via Fourier-transforming the kinetic equation and using a self-similar effective time variable. This has allowed us to dramatically simplify the kinetic equation, solve it analytically in some important cases and to fully analyse in the other important cases. The two main examples we analysed have a Gaussian-shaped forcing of low wavenumbers and a dissipation represented by either uniform friction or by viscosity. The first case is solvable analytically, and the second one is shown to possess a similar behaviour. Namely, the spectrum evolves independently at each $k_{x}$ and it tends to a flat steady-state spectrum in the inertial range (which is not a log-corrected Kolmogorov-Zakharov spectrum).

At each fixed $k_{y}$, the spectrum develops sharp gradients in $k_{x}$ at small $k_{x}$, which eventually leads to the breakdown of the weak turbulence description. We present a qualitative argument about what follows after this moment. We argue that the effect of strong turbulence is to smoothen the sharp gradients via the nonlinear resonance broadening effect. This will lead to a steady state with no gradient in $k_{x}$ for which the weak turbulence kinetic equation formally works once again, and we present an analytical solution for such a steady state.

On the practical side, one should derive from our work a warning that the 2D and the 3D MHD systems are dramatically different, and one should be careful when extrapolating the 2D results, eg. numerical ones, onto the 3D case. Indeed, in contrast to 2D, in 3D there is no gradient sharpening at small parallel wave numbers, and the Kolmogorov-Zakharov spectrum is a local and well behaved solution.

Sergey Nazarenko gratefully acknowledges support from the government of Russian Federation via grant No. 12.740.11.1430 for supporting research of teams working under supervision of invited scientists.