KINETIC EXTENSION OF CRITICAL BALANCE TO WHISTLER TURBULENCE

We build the kinetic critical balance by comparing the eddy turnover time with the whistler wave scattering time on the assumption that an electron fluid develops into strong turbulence with eddies in the directions around the mean magnetic field and that whistler waves propagate in highly oblique (or quasi-perpendicular) directions with wave–wave interactions.

The phenomenology developed here is an algebraic model, and should not be mixed up with a closure scheme based on the fundamental nonlinear equations for the plasma and the electromagnetic field. That is, the phenomenological model does not start with dynamical equations but with balancing the timescale under dynamical equilibrium (i.e., the energy transfer rate is assumed to be constant) as presented in the textbook by Biskamp (2003, p. 95). The phenomenological model offers a reference to evaluate the use of the closure scheme for kinetic plasma turbulence. The construction of an a phenomenological model is inspired by the success of the Kolmogorov phenomenological model for hydrodynamic turbulence (Kolmogorov 1941), which was later confirmed and veriied by the closure scheme or the renormalization method of Lagrangian-history direct interaction approximation (Kraichnan 1965a).

The method of phenomenological energy cascade modeling is applied to the inertial-range spectra for MHD turbulence (Iroshnikov 1964; Kraichnan 1965b), electron MHD turbulence (Biskamp 1996), and whistler wave turbulence (Narita & Gary 2010; Saito et al. 2010). The novel approach in this paper is that the phenomenological model of whistler wave interaction is combined with the concept of critical balance.

The phenomenological model is more suited to low-beta plasmas because the breakup or the frequency gap of the whistler mode into Bernstein mode branches is sufficiently small. In that case, the whistler dispersion is regarded as nearly continuous even in the perpendicular wavevector limit. The model is valid for frequencies below the lower-hybrid frequency because an analytic form for the whistler dispersion relation is used. The corresponding wavenumber range is up to 10–20 times higher than the ion inertial wavenumber (Gary 1993).

The underlying plasma model is the electron MHD with the following momentum equation:

$$\begin{eqnarray}{m}_{{\rm{e}}}{n}_{{\rm{e}}}\left(\displaystyle \frac{\partial {{\boldsymbol{u}}}_{{\rm{e}}}}{\partial t}+{{\boldsymbol{u}}}_{{\rm{e}}}\cdot {\rm{\nabla }}\right){{\boldsymbol{u}}}_{{\rm{e}}} & = & -\,{\rm{\nabla }}{p}_{\mathrm{tot}}+\displaystyle \frac{1}{{\mu }_{0}}{\boldsymbol{B}}\cdot {\rm{\nabla }}{\boldsymbol{B}}\\ & & +\,{m}_{{\rm{e}}}{n}_{{\rm{e}}}\nu {{\rm{\nabla }}}^{2}{{\boldsymbol{u}}}_{{\rm{e}}}\end{eqnarray} \tag{ 1 }$$

where m_e denotes the electron mass, n_e the electron number density, ${{\boldsymbol{u}}}_{{\rm{e}}}$ the electron fluid velocity, ${p}_{\mathrm{tot}}={p}_{\mathrm{th}}+\tfrac{{B}^{2}}{2{\mu }_{0}}$ the total pressure consisting of the thermal pressure p_th and the magnetic pressure $\tfrac{{B}^{2}}{2{\mu }_{0}}$, ${\boldsymbol{B}}$ the magnetic field, and ν the viscosity. In the present model, we focus on the eddies and the whistler waves (the R mode) originating in the advection term and the magnetic tension term, respectively. We neglect the pressure gradient and the viscosity terms on the assumption of cold, incompressible, nearly inviscid electron fluid. The L mode and the electrostatic components are also neglected. The timescales for the eddies and the whistler waves are estimated as the eddy turnover time $\tfrac{{\ell }}{{{\boldsymbol{u}}}_{e}}$ and the whistler wave interaction time $\tfrac{{\ell }}{{v}_{\mathrm{gr}}}$, where ℓ is the length scale and v_gr the group speed of the whistler waves (Narita & Gary 2010; Saito et al. 2010).

It is assumed that MHD scale fluctuations already reach a developed turbulence state with strong anisotropy (${k}_{\perp }\gg {k}_{\parallel }$) and two fluctuation modes, linear mode waves (which are whistler waves) and zero-frequency mode (identified as eddies here), are the primary constituents. Linear mode waves in the wavenumber range from kc/ω_pp (wavenumber for the proton inertial length) to kc/ω_pe (wavenumber for the electron inertial length) are whistler, kinetic Alfvén, kinetic slow, and ion Bernstein modes. The whistler mode is assumed to the leading component among the linear modes for the reasons described in Section 1. In the model, wave nonlinearities are considered to transport the fluctuation energy onto smaller scales. For example, modulational-type wave coupling can excite waves at higher frequencies and higher wavenumbers as schematically presented in Figure 1 in Hoshino & Goldstein (1989) for parallel-propagating MHD waves.

The geometrical setup is displayed in Figure 1. Magnetized plasma has the mean magnetized field in the vertical direction (parallel direction). Electrons are treated as a fluid and develop strong turbulence with eddies in the plane perpendicular to the mean magnetic field. Whistler waves propagate highly obliquely to the mean magnetic field. The energy transfer rate , the ion density (equal to the electron density), and the number of whistler wave scattering in the kinetic critical balance N are assumed to be constant. Wavevectors are quasi-perpendicular such that the relation $k=| {\boldsymbol{k}}| \simeq {k}_{\perp }\gg {k}_{\parallel }$ holds.

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The theoretical building blocks are as follows.

• 1.  
  Kinetic critical balance comparing the eddy turnover time τ_ed with N-time whistler wave scattering time τ_w:

  $$\begin{eqnarray}&&{\tau }_{\mathrm{ed}}=N{\tau }_{{\rm{w}}},\end{eqnarray} \tag{ 2 }$$

  or in normalized units with tilde,

  $$\begin{eqnarray}&&{\tilde{\tau }}_{\mathrm{ed}}=N{\tilde{\tau }}_{{\rm{w}}}.\end{eqnarray} \tag{ 3 }$$

  Details of the normalization are listed in Table 1.
• 2.  
  Eddy turnover time for the electron fluid in the perpendicular plane

  $$\begin{eqnarray}&&{\tau }_{\mathrm{ed}}=\displaystyle \frac{{{\ell }}_{\perp }}{{v}_{\mathrm{ed}}},\end{eqnarray} \tag{ 4 }$$

  or after normalization,

  $$\begin{eqnarray}&&{\tilde{\tau }}_{\mathrm{ed}}={({\tilde{k}}_{\perp }{\tilde{v}}_{\mathrm{ed}})}^{-1},\end{eqnarray} \tag{ 5 }$$

  where the wavenumber expression is used, ${\tilde{k}}_{\perp }={\tilde{{\ell }}}_{\perp }^{-1}$.
• 3.  
  Kolmogorov inertial-range scaling

  $$\begin{eqnarray}&&{v}_{\mathrm{ed}}={(\epsilon {{\ell }}_{\perp })}^{1/3},\end{eqnarray} \tag{ 6 }$$

  or

  $$\begin{eqnarray}&&{\tilde{v}}_{\mathrm{ed}}={\tilde{\epsilon }}^{1/3}{\tilde{k}}_{\perp }^{-1/3}.\end{eqnarray} \tag{ 7 }$$

  Here the energy transfer rate for electron fluid turbulence is assumed to be constant and the pre-factor or coefficient in the Kolmogorov scaling is set to unity. From Equations (5) and (7) we obtain the eddy turnover time as a function of the length scale:

  $$\begin{eqnarray}&&{\tilde{\tau }}_{\mathrm{ed}}={\tilde{\epsilon }}^{-1/3}{\tilde{k}}_{\perp }^{-2/3}.\end{eqnarray} \tag{ 8 }$$
• 4.  
  Whistler scattering time

  $$\begin{eqnarray}&&{\tau }_{{\rm{w}}}=\displaystyle \frac{{\ell }}{{v}_{\mathrm{gr}}},\end{eqnarray} \tag{ 9 }$$

  or

  $$\begin{eqnarray}&&{\tilde{\tau }}_{{\rm{w}}}={(\tilde{k}{\tilde{v}}_{\mathrm{gr}})}^{-1}.\end{eqnarray} \tag{ 10 }$$

  Four different combinations are possible to estimate the whistler scattering time in choosing the length scale either in the parallel or perpendicular directions(${{\ell }}_{\parallel }$ and ℓ_⊥, respectively) and directions of the group velocity for whistler wave scattering (${v}_{\mathrm{gr}\parallel }$ and ${v}_{\mathrm{gr}\perp }$, respectively). This point is discussed in the next subsection.
• 5.  
  Whistler dispersion relation up to the lower hybrid frequency

  $$\begin{eqnarray}&&\displaystyle \frac{\omega }{{{\rm{\Omega }}}_{{\rm{p}}}}=\displaystyle \frac{| {\boldsymbol{k}}| {V}_{{\rm{A}}}}{{{\rm{\Omega }}}_{{\rm{p}}}}+\displaystyle \frac{| {\boldsymbol{k}}| {k}_{\parallel }{V}_{{\rm{A}}}^{2}}{{{\rm{\Omega }}}_{{\rm{p}}}^{2}},\end{eqnarray} \tag{ 11 }$$

  or

  $$\begin{eqnarray}&&\tilde{\omega }=\tilde{k}+\tilde{k}{\tilde{k}}_{\parallel }.\end{eqnarray} \tag{ 12 }$$

  For highly oblique whistler waves, one may set $\tilde{k}\simeq {\tilde{k}}_{\perp }$.
• 6.  
  Ampère's law

  $$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \frac{1}{{\mu }_{0}}{\rm{\nabla }}\times {\boldsymbol{B}},\end{eqnarray} \tag{ 13 }$$

  or by taking a perpendicular component to the mean magnetic field and by normalizing the variables,

  $$\begin{eqnarray}&&{\tilde{v}}_{\mathrm{ed}}={\tilde{k}}_{\perp }\tilde{b}.\end{eqnarray} \tag{ 14 }$$

  Here we used the expression where the current is carried by the electron fluid (which is in turbulent motion):

  $$\begin{eqnarray}&&{\boldsymbol{j}}={{en}}_{{\rm{e}}}{{\boldsymbol{v}}}_{\mathrm{ed}}\end{eqnarray} \tag{ 15 }$$

  and the conversion rule for the proton inertial length from the Alfvén speed V_A and the proton gyro-frequency Ω_p into the speed of light c and the proton plasma frequency ω_pi:

  $$\begin{eqnarray}&&\displaystyle \frac{{V}_{{\rm{A}}}}{{{\rm{\Omega }}}_{{\rm{p}}}}=\displaystyle \frac{c}{{\omega }_{\mathrm{pi}}}.\end{eqnarray} \tag{ 16 }$$

  It is worth noting that Equation (14) makes a difference from MHD turbulence because the scaling law can be the same between the flow velocity and the magnetic field in MHD if one assumes highly Alfvénic fluctuations such that the relation ${\tilde{v}}_{\mathrm{ed}}\simeq \tilde{b}$ holds.
• 7.  
  Faraday's law

  $$\begin{eqnarray}&&{\partial }_{t}{\boldsymbol{B}}=-{\rm{\nabla }}\times {\boldsymbol{E}},\end{eqnarray} \tag{ 17 }$$

  or in normalized units,

  $$\begin{eqnarray}&&\tilde{\omega }\tilde{b}={\tilde{k}}_{\perp }\tilde{e}.\end{eqnarray} \tag{ 18 }$$

  Equation (18) is used to derive the electric field spectra.

Table 1.  Normalization of Physical Quantities

Variable	Normalization
time scale	$\tilde{\tau }=\tau {{\rm{\Omega }}}_{{\rm{p}}}$
frequency	$\tilde{\omega }=\tfrac{\omega }{{{\rm{\Omega }}}_{{\rm{p}}}}$
wavenumber	$\tilde{k}=\tfrac{{{kV}}_{{\rm{A}}}}{{{\rm{\Omega }}}_{{\rm{p}}}}$
velocity scale	$\tilde{v}=\tfrac{v}{{V}_{{\rm{A}}}}$
magnetic field	$\tilde{b}=\tfrac{\delta B}{{V}_{{\rm{A}}}\sqrt{{\mu }_{0}\rho }}$
electric field	$\tilde{e}=\tfrac{\delta E}{{V}_{{\rm{A}}}{B}_{0}}$
energy transfer rate	$\tilde{\epsilon }=\tfrac{\epsilon }{{V}_{{\rm{A}}}^{2}{{\rm{\Omega }}}_{{\rm{p}}}}$

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An interesting property of the kinetic critical balance theory is that the discussion or self-consistency of wavevector anisotropy uniquely imposes how whistler waves scatter. To see this, we first compute the group velocities of the whistler waves in the parallel and perpendicular directions:

$$\begin{eqnarray}&&{\tilde{v}}_{\mathrm{gr}\parallel }=\displaystyle \frac{\partial \tilde{\omega }}{\partial {\tilde{k}}_{\parallel }}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&=\tilde{k}+\displaystyle \frac{{\tilde{k}}_{\parallel }}{\tilde{k}}+\displaystyle \frac{{k}_{\parallel }^{2}}{\tilde{k}}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\simeq {\tilde{k}}_{\perp }\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&{\tilde{v}}_{\mathrm{gr}\perp }=\displaystyle \frac{\partial \tilde{\omega }}{\partial {\tilde{k}}_{\perp }}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{{\tilde{k}}_{\perp }}{\tilde{k}}+\displaystyle \frac{{\tilde{k}}_{\parallel }{\tilde{k}}_{\perp }}{\tilde{k}}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\simeq 1+{\tilde{k}}_{\parallel }.\end{eqnarray} \tag{ 24 }$$

The parallel–parallel and perpendicular–perpendicular combinations for the whistler interaction time, ${{\ell }}_{\perp }/{v}_{\mathrm{gr}\parallel }$ and ${{\ell }}_{\perp }/{v}_{\mathrm{gr}\perp }$, respectively, turn out to be an unphysical choice because the kinetic critical balance (Equation (3)) leads us to the relation ${\tilde{k}}_{\parallel }{\tilde{k}}_{\perp }^{1/3}={\tilde{\epsilon }}^{1/3}N$ and consequently to a positive slope of the energy spectrum in the parallel wavenumber domain. Another choice, perpendicular–parallel combination for the interaction time, ${{\ell }}_{\perp }/{v}_{{\mathrm{gr}}_{\parallel }}$ is not meaningful either because the parallel wavenumber ${\tilde{k}}_{\parallel }$ vanishes out of the anisotropy scaling and the relation becomes ${\tilde{k}}_{\perp }^{4/3}={\tilde{\epsilon }}^{1/3}N$. The only physically meaningful estimate of the whistler interaction time is the parallel–perpendicular combination, ${{\ell }}_{\parallel }/{v}_{\mathrm{gr}\perp }$, such that

$$\begin{eqnarray}&&{\tilde{\tau }}_{{\rm{w}}}={({\tilde{k}}_{\parallel }{\tilde{v}}_{\mathrm{gr}\perp })}^{-1}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{1}{{\tilde{k}}_{\parallel }(1+{\tilde{k}}_{\parallel })}\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}=\left\{\begin{array}{cc}{\tilde{k}}_{\parallel }^{-1} & ({\tilde{k}}_{\parallel }\lt 1)\\ {\tilde{k}}_{\parallel }^{-2} & ({\tilde{k}}_{\parallel }\geqslant 1)\end{array}\right.\end{eqnarray} \tag{ 27 }$$

In other words, the wave interaction length scale is longer than the wavelength of the oblique whistler waves. The situation is illustrated in Figure 2.

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Now we are ready to derive the anisotropy scaling for the kinetic critical balance (Equation (3)). For high wavenumbers ${\tilde{k}}_{\parallel }\geqslant 1$ (i.e., dispersive range of the whistler waves) we obtain the scaling

$$\begin{eqnarray}&&\displaystyle \frac{{k}_{\parallel }^{2}}{{k}_{\perp }^{2/3}}={\epsilon }^{1/3}N,\end{eqnarray} \tag{ 28 }$$

and for low wavenumbers ${\tilde{k}}_{\parallel }\lt 1$,

$$\begin{eqnarray}&&\displaystyle \frac{{k}_{\parallel }}{{k}_{\perp }^{2/3}}={\epsilon }^{1/3}N.\end{eqnarray} \tag{ 29 }$$

Interestingly, the MHD critical balance is restored as a low-wavenumber limit of the kinetic critical balance for whistler turbulence because the scaling ${\tilde{\tau }}_{{\rm{w}}}={\tilde{k}}_{\parallel }^{-1}$ holds, which is the same as the Alfvén wave scattering time.

The energy spectra for the flow velocity (kinetic energy), the magnetic field, and the electric field are derived as follows. The energy spectra have the dimensions of squared amplitude per wavenumber. Using the flow velocity (Equation (8)) and the wavenumber scaling ${\rm{\Delta }}{\tilde{k}}_{\perp }\simeq {\tilde{k}}_{\perp }$ (i.e., equi-distant gridding in the logarithmic space), we obtain the kinetic energy spectrum in the perpendicular wavenumber domain as

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}({\tilde{k}}_{\perp })={\tilde{\epsilon }}^{2/3}{\tilde{k}}_{\perp }^{-5/3}.\end{eqnarray} \tag{ 30 }$$

Using the anisotropy scaling (Equation (28)) and the scaling ${\rm{\Delta }}{\tilde{k}}_{\parallel }\simeq {\tilde{k}}_{\parallel }$, we obtain the spectrum in the perpendicular wavenumber domain as

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}({\tilde{k}}_{\parallel })=\tilde{\epsilon }N{\tilde{k}}_{\parallel }^{-3}.\end{eqnarray} \tag{ 31 }$$

From Ampère's law (Equation (14)) we obtain an estimate of the squared magnetic field amplitude as

$$\begin{eqnarray}&&{\tilde{b}}^{2}={\tilde{\epsilon }}^{2/3}{\tilde{k}}_{\perp }^{-8/3},\end{eqnarray} \tag{ 32 }$$

and the magnetic energy spectra are, again using the wavenumber scaling and Equation (28),

$$\begin{eqnarray}&&{E}_{\mathrm{mag}}({\tilde{k}}_{\perp })={\tilde{\epsilon }}^{2/3}{\tilde{k}}_{\perp }^{-11/3}\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{E}_{\mathrm{mag}}({\tilde{k}}_{\parallel })={\tilde{\epsilon }}^{2}{N}^{4}{\tilde{k}}_{\parallel }^{-9}.\end{eqnarray} \tag{ 34 }$$

Finally, the energy spectra for the electric field in the high-wavenumber range are estimated from the induction equation (Equation (18)) and the whistler dispersion relation (Equation (12)) as

$$\begin{eqnarray}&&{E}_{\mathrm{elc}}({\tilde{k}}_{\perp })=\tilde{\epsilon }N{\tilde{k}}_{\perp }^{-3}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{E}_{\mathrm{elc}}({\tilde{k}}_{\parallel })={\tilde{\epsilon }}^{2}{N}^{4}{\tilde{k}}_{\parallel }^{-7}.\end{eqnarray} \tag{ 36 }$$

Figure 3 shows schematically the energy spectra.

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