Analysis of Lower Hybrid Drift Waves in Kappa Distributions over Solar Atmosphere

Nonthermal particle distributions are ubiquitous at high altitudes in the solar wind and many space plasmas, their presence being widely confirmed by spacecraft measurements [4, 5, 6, 7, 8]. Such deviations from the Maxwellian distributions are also expected to exist in any low-density plasma in the Universe, where binary collisions of charges are sufficiently rare. The suprathermal populations are well described by the so-called Kappa ($\kappa$) or generalized Lorentzian velocity distributions functions (VDFs), as shown for the first time by Vasyliunas (1968) [9]. Such distributions have high energy tails deviated from a Maxwellian and decreasing as a power law in particle speed. Considering the suprathermal particles has important consequences for space plasmas. For instance, an isotropic Kappa distribution (instead of a Maxwellian) in a planetary or stellar exosphere leads to a number density n(r) decreasing as a power law (instead of exponentially) with the radial distance r and a temperature T increasing with the radial distance (instead of being constant).

Scudder (1992a,b) [10, 11] was pioneer to show the consequences of a postulated non thermal distribution in stellar atmospheres and especially the effect of the velocity filtration: the ratio of suprathermal particles over thermal ones increases as a function of altitude in an attraction field. The anti correlation between the temperature and the density of the plasma leads to this natural explanation of velocity filtration for the heating of the corona, without depositing wave or magnetic field energy. Scudder (Scudder 1992b) [11] also determined the value of the kappa parameter for different groups of stars. Scudder (1994) [12] showed that the excess of Doppler line widths can also be a consequence of non thermal distributions of absorbers and emitters. The excess brightness of the hotter lines can satisfactorily be accounted for by a two-Maxwellian electron distribution function (Ralchenko et al. 2007) [13] and should be also by a Kappa. The Kappa distribution is also consistent with mean electron spectra producing hard X-ray emission in some coronal sources (Kasparova and Karlicky 2009) [14]. Thus the stability analysis using kappa distribution is of particular interest for solar plasmas. We use a kappa distribution of the form given in Pierrard and Lazar, 2010 [15]

Out of the large number of waves undergoing quasilinear relaxation we pick out here one particularly interesting electrostatic mode in a magnetized plasma, the lower-hybrid drift mode. Since quasilinear theory describes the time evolution of the equilibrium distribution function, it is necessary to retain the dependence on the distribution function in the expression for the growth rate of the lower-hybrid instability,without assuming it to be a Maxwellian. For the sake of simplicity, we restrict ourselves to purely perpendicular propagation. Then the dispersion relation can be written as [1]

$$D(k_{\perp},\omega)=1+\chi_{e}(k_{\perp},\omega)+\chi_{i}(k_{\perp},\omega)$$

(1)

where $\,\chi_{e}(k_{\perp},\omega)=\frac{\omega_{pe}^{2}}{\omega_{ge}^{2}}(1-\frac{1}{k_{\perp}L_{R}}\frac{\omega_{ge}}{\omega-k_{\perp}v_{de}})\,$ and $\,\chi_{i}(k_{\perp},\omega)=\frac{\omega_{pe}^{2}}{k_{\perp}}\int\frac{dv_{\perp}}{\omega-k_{\perp}v_{de}}\frac{\partial{f_{0i}(v_{\perp},t)}}{\partial{v_{\perp}}}\,$ where $\,\omega_{pe}\,$ and $\,\omega_{ge}\,$ are the plasma and gyro frequency respectively. The growth rate in the low drift velocity regime $\,(v_{de}<v_{thi})\,$, is,

$$\gamma=-\frac{\pi v_{thi}^{2}k_{\perp}^{2}v_{de}}{2|k_{perp}|(1+\frac{k_{\perp}^{2}}{k_{max}^{2}})}\frac{\partial{f_{0i}}}{\partial{v_{\perp}}}\|_{v_{\perp}\rightarrow\omega/k_{\perp}}$$

(2)

Here $\,v_{thi}=\sqrt{\frac{2k_{B}T_{i}}{m_{i}}}\,$ is the thermal velocity of protons and $\,v_{de}\,$ is the drift velocity of electrons. It is assumed that $\,v_{de}=-v_{di}\,$ and $\,k_{max}\,$ defined as,

$$k_{max}^{2}\lambda_{Di}^{2}=2/(1+\omega_{pe}^{2}/\omega_{ge}^{2})$$

(3)

where $\,\omega_{pe}\,$ and $\,\omega_{ge}\,$ are the plasma and gyro frequencies respectively.

$$f=\frac{n}{2\pi\,(\kappa\,w_{\kappa i}^{2})^{3/2}}\frac{\Gamma(\kappa+1)}{\Gamma(\kappa-1/2)\,\Gamma(3/2)}\left(1+\frac{v_{\parallel}^{2}+v_{\perp}^{2}}{\kappa w_{\kappa i}^{2}}\right)^{-(\kappa+1)}$$

(4)

where $\,w_{\kappa i}=\sqrt{(2\,\kappa-3)\,k_{B}T_{i}/\kappa\,m_{i}}\,$ is the thermal velocity.

Substituting this in the general equation for the growth rate Eq.(2) we get the growth rate of lower hybrid drift wave corresponding to the kappa distribution as,

$$\gamma_{\kappa}=\frac{-\pi\,v_{thi}^{2}k_{\perp}v_{de}}{\left(1+\frac{k_{\perp}^{2}}{k_{max}^{2}}\right)^{2}}\frac{n_{i}\,\Gamma(\kappa+1)(-1-\kappa)}{\Gamma(\kappa-1/2)\Gamma(3/2)\,\kappa\,w_{\kappa i}^{2}}\frac{\omega}{k}\left[1+\frac{2}{\kappa}\frac{\omega^{2}}{k^{2}w_{\kappa i}^{2}}\right]^{-2-\kappa}$$

(5)

The instabilities are studied for large flares, medium flares and corona. Drift velocities $\,v_{de}=10^{-14}\,$cm/s, $10^{-13}\,$ cm/s and $10^{-10}\,$ cm/s taken for large flares, medium strength flares and corona respectively. .

A distribution function of this feature is described in Summers and Throne (1995) [16] and Singhal and Tripathi, 2006 [17]. It has the following form

$$\begin{array}[]{r}f=\frac{\Gamma(\kappa+l+1)}{\pi^{3/2}\theta_{\perp i}^{2}\,\theta_{\parallel i}\,\kappa^{l+3/2}\Gamma(l+1)\Gamma(\kappa-1/2)}(\frac{v_{\perp}}{\theta_{\perp i}})^{2l}\times\\ \\ \left[1+\frac{v_{\parallel}^{2}}{\kappa\theta_{\parallel i}^{2}}+\frac{v_{\perp}^{2}}{\kappa\theta_{\perp i}^{2}}\right]^{-(\kappa+l+1)}\end{array}$$

(6)

where, $\,\theta_{\perp i}=(\frac{2\kappa-3}{\kappa})^{1/2}(l+1)^{-1/2}(\frac{T_{\perp i}}{mi})^{1/2}\,$ and $\,\theta_{\parallel i}=(\frac{2\kappa-3}{\kappa})^{1/2}(\frac{T_{\parallel i}}{m_{i}})^{1/2}\,$.