A MODEL FOR THE NON-UNIVERSAL POWER LAW OF THE SOLAR WIND SUB-ION-SCALE MAGNETIC SPECTRUM

The magnetic field being stretched by electron velocity gradients and the dynamics dominantly transverse, we define, when the interactions are local, a stretching frequency (inverse of the nonlinear time ${\tau }_{\mathrm{NL}}$) ${\omega }_{\mathrm{NL}}\sim {k}_{\perp }{v}_{{ek}}$ where the transverse electron velocity v_ek at scale ${k}_{\perp }^{-1}$ is given by ${v}_{{ek}}=\bar{\alpha }{b}_{k}.$ Here, $\bar{\alpha }$ is a function of ${k}_{\perp }{\rho }_{i},$ equal to 1 in the MHD range and scaling like ${k}_{\perp }{\rho }_{i}$ in the far sub-ion range, with a smooth transition near the ion gyroscale. As in hydrodynamic turbulence, we write ${\omega }_{\mathrm{NL}}\sim {\left[{\bar{\alpha }}^{2}{k}_{\perp }^{3}{E}_{k}\right]}^{1/2}$ (Kovasznay 1948; Panchev 1969). Nevertheless, when E_k is decaying fast enough, the above expression does not necessarily ensure the expected monotonic growth of ${\omega }_{\mathrm{NL}}.$ In this case, nonlocal interactions cannot be neglected, and ${\omega }_{\mathrm{NL}}$ should rather be viewed as the stretching rate due to all of the scales larger than ${k}_{\perp }^{-1}.$ It is then taken equal to the rms value ${\omega }_{\mathrm{NL}}={\rm{\Lambda }}{\left[{\displaystyle \int }_{0}^{{k}_{\perp }}{\bar{\alpha }}^{2}\;{p}_{\perp }^{2}{E}_{p}{dp}\right]}^{1/2}$ (Elisson 1962; Panchev 1971; Lesieur 2008), where Λ is a constant. The local approximation is recovered when the integral diverges at large ${k}_{\perp },$ while the integral formula can be replaced by the equation $d{\omega }_{\mathrm{NL}}^{2}/{{dk}}_{\perp }={{\rm{\Lambda }}}^{2}{\bar{\alpha }}^{2}\;{k}_{\perp }^{2}{E}_{k}.$

Alfvén waves are characterized by a frequency of ${\omega }_{W}=\bar{\omega }{k}_{\parallel }{v}_{A}$ and a dissipation rate of $\gamma =\bar{\gamma }{k}_{\parallel }{v}_{A}.$ Here $\bar{\omega }$ (equal to 1 in the MHD range) and $\bar{\gamma }$ are functions of ${k}_{\perp }{\rho }_{i}$ provided by the kinetic theory. In a linear description, parallel and perpendicular wavenumbers are defined relatively to the ambient field (taken in the z direction), but distortions of the magnetic field lines should be retained in the nonlinear regime. We write ${k}_{\parallel }={k}_{z}+{v}_{A}^{-1}{[{\displaystyle \int }_{0}^{{k}_{\perp }}{p}_{\perp }^{2}{E}_{p}{dp}]}^{1/2},$ where k_z is a constant estimating the parallel wavenumber of the energy-containing Alfvén waves. It should not be confused with the inverse correlation length along the z direction. Alternatively, ${k}_{\parallel }^{-1}$ measures the correlation along the magnetic field lines (which is larger than along the z axis). A procedure to estimate ${k}_{\parallel }$ in numerical simulations is described in Cho & Lazarian (2004), while an approach using the frequency as a proxy for ${k}_{\parallel }$ is used in TenBarge & Howes (2012). We are led to write ${\omega }_{W}=\bar{\omega }{k}_{z}{v}_{A}+\delta {\omega }_{W}$ with a turbulent frequency shift of $\delta {\omega }_{W}={\left[{\displaystyle \int }_{0}^{{k}_{\perp }}{\bar{\omega }}^{2}{p}^{2}{E}_{p}{dp}\right]}^{1/2}$ or $d{(\delta {\omega }_{W})}^{2}/{{dk}}_{\perp }={\bar{\omega }}^{2}{k}^{2}{E}_{k}.$ We similarly write $\gamma =\bar{\gamma }{k}_{z}{v}_{A}+\delta \gamma$ with $d{(\delta \gamma )}^{2}/{{dk}}_{\perp }={\bar{\gamma }}^{2}{k}_{\perp }^{2}{E}_{k}.$ A multiplicative constant should also enter the definitions of $\delta {\omega }_{W}$ and γ, but is easily scaled out.

Another timescale originates from the compressible character of KAWs. The latter are subject to Landau damping, resulting in temperature homogenization along the magnetic field lines, on the correlation length ${k}_{\parallel }^{-1}$ in a time ${\tau }_{H\;r}\sim {({v}_{\mathrm{th}\;r}{k}_{\parallel })}^{-1}.$ Here the thermal velocity ${v}_{\mathrm{th}\;r}$ appears as the rms streaming velocity of the r-particles. This time scale, which arises explicitly in Landau fluid closures (Hammett et al. 1997; Snyder et al. 1997; Sulem & Passot 2015), is very short for the electrons, while for the ions it is comparable to that of the other relevant processes and can thus affect the dynamics. Due to magnetic field distortion, this process introduces additional nonlinear couplings characterized by the frequency ${\omega }_{H}=\mu {v}_{\mathrm{th}\;i}{k}_{\parallel }$ where μ is a numerical constant. We are thus led to write ${\omega }_{H}=\mu {v}_{\mathrm{th}\;i}{k}_{z}+\delta {\omega }_{H}$ with $d{(\delta {\omega }_{H})}^{2}/{{dk}}_{\perp }={\mu }^{2}\beta {k}_{\perp }^{2}{E}_{k}.$

Finally, we define the transfer time as ${\tau }_{\mathrm{tr}}={\tau }_{\mathrm{NL}}\left({\tau }_{\mathrm{NL}}/{\tau }_{W}+{\tau }_{\mathrm{NL}}/{\tau }_{H}\right),$ or, in frequency terms, ${\omega }_{\mathrm{tr}}\;={\omega }_{\mathrm{NL}}^{2}/({\omega }_{W}+{\omega }_{H}).$ When only one process competes with nonlinear stretching, the critical balance condition (Goldreich & Shridhar 1995; Schekochihin et al. 2009; Nazarenko & Schekochihin 2011) ensures the equality of the two associated timescales, and thus of the transfer and stretching times.

Retaining KAW linear Landau damping leads to the phenomenological equation (Howes et al. 2008, 2011a) ${\partial }_{t}{E}_{k}+{{\mathcal{T}}}_{k}=-2\gamma {E}_{k}+{S}_{k},$ where S_k is the driving term acting at large scales and ${{\mathcal{T}}}_{k}$ is the transfer term related to the energy flux by ${{\mathcal{T}}}_{k}=\partial \epsilon /\partial {k}_{\perp }.$ Due to Landau damping, energy is not transferred conservatively along the cascade, making scale-dependent. For a steady state and outside the injection range, one has $d\epsilon /{{dk}}_{\perp }=-2\gamma {E}_{k}.$ Note that the present setting differs from the asymptotic regime considered by Schekochihin et al. (2009), for which, under the simultaneous conditions ${k}_{\perp }{\rho }_{i}\gg 1$ and ${k}_{\perp }{\rho }_{e}\ll 1,$ KAWs are not subject to Landau damping because they transfer part of their energy via parallel phase mixing to the ion (electron) entropy cascades only at ion (electron) gyroscales, where it is cascaded both in physical and velocity spaces to collisional scales via perpendicular phase mixing.

The estimate of the energy flux relies on a basic turbulence description (overlooking intermittency). We write $\epsilon =C{\omega }_{\mathrm{tr}}{k}_{\perp }{E}_{k},$ where C is a negative power of the Kolmogorov constant. Arguing that the small-scale eddies cannot be sufficiently highly correlated with each other to contribute equally (Elisson 1962), here we use the local approximation ${k}_{\perp }{E}_{k}$ of the Reynolds stress rather than the original Obukhov's expression ${\displaystyle \int }_{{k}_{\perp }}^{\infty }{E}_{p}{dp}$, which leads to an unphysical behavior in the dissipation range of hydrodynamic turbulence (Panchev 1971).

Normalizing frequencies by ${{\rm{\Omega }}}_{i},$ wavenumbers by ${\rho }_{i}^{-1},$ energy spectra by ${v}_{A}^{2}{\rho }_{i},$ energy fluxes by ${v}_{A}^{2}{{\rm{\Omega }}}_{i},$ and denoting the ion beta by β, we obtain the non-dimensional model equations (keeping the same notations)

$$\begin{eqnarray}&&d{\omega }_{\mathrm{NL}}^{2}/{{dk}}_{\perp }={{\rm{\Lambda }}}^{2}{\beta }^{-1}{\bar{\alpha }}^{2}{k}_{\perp }^{2}{E}_{k}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&d{(\delta {\omega }_{W})}^{2}/{{dk}}_{\perp }={\beta }^{-1}{\bar{\omega }}^{2}{k}_{\perp }^{2}{E}_{k}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&d{(\delta \gamma )}^{2}/{{dk}}_{\perp }={\beta }^{-1}{\bar{\gamma }}^{2}{k}_{\perp }^{2}{E}_{k}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&d{(\delta {\omega }_{H})}^{2}/{{dk}}_{\perp }={\mu }^{2}{k}_{\perp }^{2}{E}_{k}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&d\epsilon /{{dk}}_{\perp }=-2[{\beta }^{-1/2}\bar{\gamma }{k}_{z}+(\delta \gamma )]{E}_{k}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{E}_{k}=[({\beta }^{-1/2}\bar{\omega }+\mu ){k}_{z}+\delta {\omega }_{W}+\delta {\omega }_{H}]\displaystyle \frac{{C}^{-1}\epsilon }{{k}_{\perp }{\omega }_{\mathrm{NL}}^{2}}.\end{eqnarray} \tag{ 6 }$$

Except possibly near ${k}_{\perp }=1,$ $\bar{\alpha }=\bar{\omega }.$ The nonlinearity parameter $\chi ={\omega }_{\mathrm{NL}}/{\omega }_{W}$ obeys

$$\begin{eqnarray}&&\displaystyle \frac{d\chi }{{{dk}}_{\perp }}=\displaystyle \frac{{\bar{\alpha }}^{2}{k}_{\perp }^{2}{E}_{k}}{2\beta {\omega }_{\mathrm{NL}}^{2}}\chi \left({{\rm{\Lambda }}}^{2}-\displaystyle \frac{{\bar{\omega }}^{2}}{{\bar{\alpha }}^{2}}{\chi }^{2}\right)-{k}_{z}\displaystyle \frac{{\chi }^{3}}{2{\omega }_{\mathrm{NL}}^{2}}f\end{eqnarray} \tag{ 7 }$$

where $f=\displaystyle \frac{2}{\sqrt{\beta }}\displaystyle \frac{d}{{{dk}}_{\perp }}\left(\bar{\omega }\delta {\omega }_{W}\right)+\displaystyle \frac{{k}_{z}}{\beta }\displaystyle \frac{d{\bar{\omega }}^{2}}{{{dk}}_{\perp }}$ is positive. At the (small) injection wavenumber k₀, turbulence is characterized by $A={k}_{z}/{[{k}_{0}^{3}{E}_{{k}_{0}}]}^{1/2}=({k}_{z}/{k}_{0})({B}_{0}/\delta {B}_{\perp 0})$ and, when taking ${\omega }_{\mathrm{NL}}^{(0)}={\rm{\Lambda }}{\beta }^{-1/2}{k}_{0}^{3/2}{E}_{{k}_{0}}^{1/2}$ and ${\omega }_{W}^{(0)}={\beta }^{-1/2}({k}_{z}+{k}_{0}^{3/2}{E}_{{k}_{0}}^{1/2}),$ ${\chi }_{0}={\rm{\Lambda }}/(1+A).$ In the strong turbulence regime with ${k}_{z}=0,$ $\chi ={\rm{\Lambda }}$ in the full spectral range, while for ${k}_{z}\ne 0,$χ starts growing near k_0, but cannot exceed Λ.

Assuming local interactions, we have

$$\begin{eqnarray}&&{E}_{k}^{2}\sim \displaystyle \frac{{\beta }^{1/2}{{\rm{\Lambda }}}^{-2}{C}^{-1}\epsilon }{{\bar{\alpha }}^{2}{k}_{\perp }^{4}}(\bar{\omega }+\mu {\beta }^{1/2})({k}_{z}+{k}_{\perp }^{3/2}{E}_{k}^{1/2}).\end{eqnarray} \tag{ 8 }$$

When neglecting dissipation (constant ), we recover the usual inertial energy spectra. For weak turbulence (${k}_{\perp }^{3/2}{E}_{k}^{1/2}\ll {k}_{z}$), ${E}_{k}\propto {k}_{\perp }^{-2}$ in the MHD range, while in the sub-ion range (as $\bar{\omega }\gg \mu {\beta }^{1/2}$), ${E}_{k}\propto {k}_{\perp }^{-5/2}.$ For strong turbulence (k_z negligible), ${E}_{k}\propto {k}_{\perp }^{-5/3}$ in the MHD range, while in the sub-ion range ${E}_{k}\propto {k}_{\perp }^{-7/3}.$ A ${k}_{\perp }^{-3}$ regime is also obtained in the sub-ion range when the effect of wave propagation is negligible, as observed in two-dimensional hybrid PIC simulations with an out-of-plane ambient magnetic field (Franci et al. 2015). Furthermore, as can easily be seen from Equation (6), an additional regime with ${E}_{k}\propto {k}_{\perp }^{-1}$ is possible at very large scales when, for small enough , turbulence is not yet developed and ${\omega }_{\mathrm{NL}}$ is almost constant. Such a spectral exponent is observed in the solar wind at scales larger than the ${k}_{\perp }^{-5/3}$ inertial range (Matthaeus & Goldstein 1986; Nicol et al. 2008; Wicks et al. 2010).

For strong turbulence, when assuming local interactions and neglecting k_z contributions,

$$\begin{eqnarray}&&{\epsilon }_{k}={\epsilon }_{0}\mathrm{exp}\left[-2{C}^{-1}{{\rm{\Lambda }}}^{-2}{\displaystyle \int }_{{k}_{0}}^{{k}_{\perp }}\displaystyle \frac{\bar{\gamma }}{\xi {\bar{\alpha }}^{2}}(\bar{\omega }+\mu {\beta }^{\frac{1}{2}})d\xi \right],\end{eqnarray} \tag{ 9 }$$

where the notation ${\epsilon }_{k}$ for the energy flux stresses its wavenumber dependence. This quantity is to be substituted in ${E}_{k}\sim {\beta }^{1/3}{{\rm{\Lambda }}}^{-4/3}{C}^{-2/3}{\epsilon }_{k}^{2/3}{k}_{\perp }^{-7/3},$ taking $\bar{\alpha }=\bar{\omega }$ and $\bar{\gamma }/{\bar{\omega }}^{2}\approx \delta (\beta )$ ($\approx 0.78{\rho }_{e}/{\rho }_{i}$ when using Equations (62) and (63) of Howes et al. (2006) with $\beta =1$). Furthermore, in this range $\bar{\omega }\approx a(\beta ){k}_{\perp }$ with $a{(\beta )=(1+\beta )}^{-\frac{1}{2}}.$ This leads to ${\epsilon }_{k}\sim {\epsilon }_{0}{k}_{\perp }^{-\zeta }$ $\mathrm{exp}(-2a(\beta ){C}^{-2}{{\rm{\Lambda }}}^{-2}\delta (\beta ){k}_{\perp }),$ with $\zeta =2\delta$ $(\beta ){C}^{-1}$ $\mu {{\rm{\Lambda }}}^{-2}{\beta }^{1/2}.$ This results in a steepening of the algebraic prefactor of the magnetic spectrum, which is now proportional to ${k}_{\perp }^{-(7/3+2\zeta /3)}\mathrm{exp}[-(4/3)a(\beta )\delta (\beta ){C}^{-1}{{\rm{\Lambda }}}^{-2}{k}_{\perp }].$ Compared with the spectrum obtained in Howes et al. (2008), the present model predicts that, in addition to an exponential cut-off, Landau damping leads to a correction of the power-law exponent. In contrast with intermittency corrections discussed in Boldyrev & Perez (2012), this correction is not universal, which is expected when dissipation and nonlinear transfer times display the same wavenumber dependence (Bratanov et al. 2013). This situation holds for the second term in the exponential arising in Equation (9), which leads to the correction of the $-7/3$ spectral index. We note that it defines a dissipation length of ${k}_{d}^{-1}=2{C}^{-1}{{\rm{\Lambda }}}^{-2}\mu {\beta }^{1/2}\bar{\gamma }/({k}_{\perp }{\bar{\alpha }}^{2})\propto {k}_{\perp }^{-1}$ and thus a dissipation rate of ${\omega }_{d}={v}_{e}{k}_{d}$ for ${\epsilon }_{k},$ which has the same ${k}_{\perp }$ dependency as ${\omega }_{\mathrm{NL}}$ that, by a critical balance argument, identifies with ${\omega }_{\mathrm{tr}}$ beyond the transition range. A similar power-law decay of is encountered in drift-kinetic plasma turbulence (see Equation (2.53) of Schekochihin et al. 2015).

Differently, for weak turbulence, we get

$$\begin{eqnarray}&&{\epsilon }_{k}={\left[{\epsilon }_{0}^{\frac{1}{2}}-{{\rm{\Lambda }}}^{-1}{C}^{-\frac{1}{2}}{\beta }^{-\frac{1}{4}}{k}_{z}^{\frac{3}{2}}{\displaystyle \int }_{{k}_{0}}^{{k}_{\perp }}\displaystyle \frac{\bar{\gamma }}{{\xi }^{2}{\bar{\alpha }}^{\frac{1}{2}}}d\xi \right]}^{2},\end{eqnarray} \tag{ 10 }$$

where the integral behaves like ${k}_{\perp }^{1/2}.$ Equation (10) predicts that ${\epsilon }_{k}$ and thus E_k vanish at a finite ${k}_{\perp },$ indicating the breaking of the analysis near the corresponding scale, an effect possibly related to the difficulty for weak turbulence to exist in the presence of a significant Landau damping.