# Dissipation range of the anisotropic plasma turbulence

G. Gogoberidze

Georgian National Astrophysical Observatory, 2a Kazbegi ave.,

0160 Tbilisi, Georgia

Abstract. Dissipation range of the plasma turbulence in the presence of anisotropic kinetic dissipation is studied. In contrast

with viscosity, kinetic dissipation of plasma waves in the presence of backgroundmagnetic field is usually strongly anisotropic.

Here we show that in the presence of: (i) anisotropic kinetic dissipation; and (ii) if the nonlinear transfer is governed by the

scattering of the plasma waves by low frequency ones, then one should expect scale invariant power law spectrum of the

plasma turbulence in the dissipation range, in contrary to hydrodynamic turbulence. It should be emphasized that obtained

scale invariant spectrum is not associated with the constant flux of any physical quantity due to the presence of kinetic

dissipation. Application of the model to the high frequency part of the solar wind turbulence studied.

Keywords: Plasma turbulence, Dissipation, Solar wind

PACS: 52.35.Ra, 47.27.Eq, 96.50.Ci

INTRODUCTION

Various spacecraft observations show the presence of persistent magnetic fluctuations in the solar wind over a

broad range of frequencies [1, 2, 3]. For low frequencies ( f < 10−2 −10−3 Hz) the magnetic field spectrum vary

as approximately EM( f ) ∼ f −1. For higher frequencies, up to proton cyclotron frequency ( f ∼ 0.1 − 1 Hz) the

Kolmogorov spectrum f −5/3 is observed. This is believed to be the inertial interval of the solar wind turbulence.

The change of slope and rapid decrease in the intensity near the ion cyclotron frequency is usually considered to be

due to the absorption of Alfvén waves by ion cyclotron damping or Landau damping [3]. At the frequencies, higher

then ion cyclotron frequency, weak but persistent level of magnetic fluctuations, that can be well approximated by the

power law spectrum f −3 is observed up to the electron cyclotron frequency. These fluctuations are usually associated

with the whistler waves [2]. The nature of this high frequency part of the spectrum remains unexplained.

Whistler turbulence have been intensively studied by different authors both in strong [4, 5] and weak [6] turbulent

regimes. If one assumes the existence of the inertial interval of the whistler turbulence, then Kolmogorov-type

dimensional analysis yields for the magnetic spectrum [5] EM(k) ∼ k−7/3, that is incompatible with observations

(note that due to the relation EM( f )d f ∼ EM(k)dk, and taking into account Doppler shift and dispersion of whistler

waves f ∼ k2, observed f −3 spectrum corresponds to k−v with v ∼ 5−6 in the wave number space).

There exist several different directions of the research for explanation of high frequency solar wind spectrum.

The first approach [4, 7] is based on the fact that governing equations of Hall magnetohydrodynamics besides energy,

conserves two other second order (with respect to the field variables) quantities – magnetic and generalized helicity [8].

Therefore, stationary Kolmogorov-type spectrum can be “driven” not only by energy cascade, but also by the cascade

of magnetic and generalized helicities [4]. In Ref. [9] short wavelength dispersive properties of the magnetosonicwhistler

waves have been studied as a possibility reason of the spectrum steepening. Alternative approach to the

explanation of the high frequency magnetic fluctuations spectrum in the solar wind implies incorporation of the linear

kinetic effects, such as Landau and cyclotron damping. It has been shown [10] that simple incorporation of dissipation

term to the energy budget equation leads to sharp cut off of the energy spectrum. On the other hand, total ignore of

dissipation leads to much more smooth spectrum compared to the observed one.

THE MODEL

Dissipation range of incompressible hydrodynamic turbulence was extensively studied by different authors [11, 12, 13]

due to the fact that smallest scale perturbations display strong intermittency even at Reynolds numbers so low that there

is no basis for fractal cascade. The kinetic energy spectrum E(k) of the hydrodynamic turbulence in the far dissipation

range behaves as E(k) ∼ ka1 exp[−a2(k/kd)n], where kd is Kolmogorov dissipation wave number, a1 and a2 are some

constants and 1 ≤ n ≤ 2 [11].

In the case of plasma turbulence existence of various kinetic mechanisms of dissipation makes the situation much

more complicated. For instance, observations of the solar wind turbulence [1, 2, 3] strongly suggest steep power

law spectrum of the magnetic field fluctuations for the frequencies higher then ion cyclotron frequency for which

kinetic mechanisms of dissipation are dominant. In contrast with viscosity kinetic dissipation of plasma waves in

the presence of background magnetic field is usually strongly anisotropic. Here we show that in the presence of: (i)

anisotropic kinetic dissipation; and (ii) if the nonlinear transfer is governed by the scattering of the plasma waves

by low frequency ones, then one should expect scale invariant power law spectrum of the plasma turbulence [14]. It

should be emphasized that the scale invariant spectrum is not associated with the constant flux of any physical quantity

due to the presence of kinetic dissipation.

The general equation that governs the evolution of any averaged characteristic Z of the homogenous turbulence

which is conserved by nonlinear interactions (in the case of hydrodynamic turbulence Z is usually associated with

energy density) in the wave number space has the form

APPLICATION TO THE DISSIPATION RANGE OF SOLAR WIND TURBULENCE

The model considered in the presented paper could have important consequences for the explanation of the high

frequency part of the solar wind spectrum. Whistler waves propagating along the background magnetic field are

affected by neither Landau nor cyclotron damping. Based on the numerical solution of linear Vlasov equation [10]

the angle q0 at which kinetic dissipation becomes dominant can be estimated as q0 ∼p/6. The level of whistler wave

fluctuations is low in the sense that hb2

wi/B20

≪1, where hb2

wi is rms of the whistler wave magnetic field fluctuations

and B0 is the background magnetic field. Therefore the study can be held in the framework of the weak turbulence

theory. Possible nonlinear processes includes: (a) four wave resonant interactions of whistler waves [it can be shown

that if q0 <p/3 then three wave resonances of whistler waves are absent, i.e., with this restriction for all three waves

resonant conditions similar to (4) do not have nontrivial solutions]; (b) induced scattering of whistler waves by ions;

and (c) scattering of whistler waves by low frequency magnetohydrodynamic waves from the inertial range of the

turbulence, i.e., three wave interactions which involves two whistlers and one magnetohydrodynamic wave. Detailed

analysis of the nonlinear processes of the solar wind whistler waves will be presented elsewhere. Here we note that

characteristic time scales of these processes are respectively proportional to ta ∼ N−2, tb ∼ N−1, and tc ∼ 1, where

N(k)≡ (k)/w(k) is the number density of whistler waves. Consequently, the strongest nonlinear process that should

be responsible for the formation of the high frequency spectrum is the scattering of whistler waves by low frequency

magnetohydrodynamic ones. This process conserves the total number of whistler waves [17], and therefore Z ≡ N(k).

It can be shown that Alfvén waves do not interact with whistlers through three wave resonances, whereas kinetic Alfvén

waves do [21]. Another possibility is scattering of whistler waves by fast magnetosonic waves. Here we consider only

the second possibility.

Analytical calculations of diffusion coefficients are very complicated even in the incompressible limit. Analysis

of three wave interaction of whistler and fast magnetosonic waves performed in Ref [14] in the framework of

Hall magnetohydrodynamics [8] yields a = −1 and b1 = 0, g1 = 1. Therefore, Eq. (8) gives N(k) ∼ k−c1 , where

c1 =n1(d⊥/dk)1/2.

To obtain spectral index d of corresponding energy spectrum E( f ) ∼ f −d , we note that

Taking also into account that for whistler waves f ∼ k2, Eq. (9) yield d = (c1 −3)/2. As it was mentioned above,

according to observations d ≈ 3. Taking also q0 =p/6, we obtain [14] d⊥/dk ≈ 5. Obtained result seems reasonable,

due to the fact that in the magnetized media perpendicular cascade rate usually significantly exceeds parallel cascade

rate (see, e.g., [22] and references therein).

In the presented paper plasma turbulence in the presence of anisotropic kinetic dissipation is considered. It is

shown that if the nonlinear transfer is governed by the scattering of the plasma waves by low frequency ones, then

development of asymptotic scale invariant power law spectrum of the plasma turbulence is possible. Obtained scale

invariant spectrum is not associated with the constant flux of any physical quantity due to the presence of kinetic

dissipation. Corresponding spectral index is given by Eq. (7) with m = 1. Possible application of the presented model

to the high frequency part of the solar wind spectrum has been analyzed.

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