Dissipation range of the anisotropic plasma turbulence
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
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 . 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 . 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  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  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 .
Therefore, stationary Kolmogorov-type spectrum can be “driven” not only by energy cascade, but also by the cascade
of magnetic and generalized helicities . In Ref.  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  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.
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 .
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 . 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 
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
≪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 , 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 . 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  in the framework of
Hall magnetohydrodynamics  yields a = −1 and b1 = 0, g1 = 1. Therefore, Eq. (8) gives N(k) ∼ k−c1 , where
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  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.,  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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