Analysis of Lower Hybrid Drift Waves in Kappa Distributions over Solar Atmosphere
AAnalysis of Lower Hybrid Drift Waves inKappa Distributions over Solar Atmosphere
Antony Soosaleon & Blession Jose
School of Pure & Applies PhysicsMahatma Gandhi University, Kottayam, Kerala, INDIA [email protected]
February 3, 2021
Abstract
Kappa distributions and with loss cone features have been fre-quently observed with flares emissions with the signatures of Lowerhybrid waves. We have analysed the plasma with Kappa distributionsand with loss cone features for the drift wave instabilities in perpen-dicular propagation for Large flare and Normal flare and Coronal con-dition . While analysing the growth/damping rate, we understandthat the growth of propagation of EM waves increases with kappa dis-tribution index for all the three cases. In comparing the propagationlarge flare shows lesser growth in compared with the normal and thecoronal plasmas. When added the loss cone features to Kappa dis-tributions, we find that the damping of EM wave propagation takesplace. The damping rate EM waves is increases with T ⊥ T (cid:107) and loss coneindex l, in all the three cases but damping is very high for large flareand then normal in comparision with coronal condition. This showsthat the lower hybrid damping may be the source of coronal heating. Drift instabilities are sometimes also called universal instabilities, a term in-dicating that drift instabilities are the most general linear instabilities which1 a r X i v : . [ a s t r o - ph . S R ] F e b ppear in almost everyplace and at all occasions. The reason is that plas-mas are always inhomogeneous at least on the microscopic scales. One maytherefore be sure that drift instabilities will be met under all realistic con-ditions. One of the most important instabilities is the lower- hybrid driftinstability. The reason for its importance is that it excites waves near thelower-hybrid frequency which is a natural resonance. Hence, the instabilitycan reach large growth rates. The energy needed to excite the instabilityis taken from the diamagnetic drift of the plasma in a density gradient [1].This is similar to the modified two stream instability insofar that the dia-magnetic drift gives rise to a transverse current in the plasma which acts ina way corresponding to the current drift velocity of the modified two streaminstability. In general, the lower-hybrid drift instability is an electromagneticinstability causing whistler waves near the resonance cone to grow [1]. Nonthermal particle distributions are ubiquitous at high altitudes in the so-lar wind and many space plasmas, their presence being widely confirmed byspacecraft measurements [4, 5, 6, 7, 8]. Such deviations from the Maxwelliandistributions are also expected to exist in any low-density plasma in the Uni-verse, where binary collisions of charges are sufficiently rare. The suprather-mal populations are well described by the so-called Kappa ( κ ) or general-ized Lorentzian velocity distributions functions (VDFs), as shown for thefirst time by Vasyliunas (1968) [9]. Such distributions have high energytails deviated from a Maxwellian and decreasing as a power law in particlespeed. Considering the suprathermal particles has important consequencesfor space plasmas. For instance, an isotropic Kappa distribution (instead ofa Maxwellian) in a planetary or stellar exosphere leads to a number densityn(r) decreasing as a power law (instead of exponentially) with the radial dis-tance r and a temperature T increasing with the radial distance (instead ofbeing constant).Scudder (1992a,b) [10, 11] was pioneer to show the consequences of apostulated non thermal distribution in stellar atmospheres and especiallythe effect of the velocity filtration: the ratio of suprathermal particles overthermal ones increases as a function of altitude in an attraction field. Theanti correlation between the temperature and the density of the plasma leadsto this natural explanation of velocity filtration for the heating of the corona,without depositing wave or magnetic field energy. Scudder (Scudder 1992b)2igure 1: Figure shows plot of growth rate ( γ ) versus wave vector (k) forvarious values of κ index , for large flares. The growth rate electromagneticwave in Kappa distribution with loss cone feature shows significant propaga-tion, also the propagation rate increases with kappa index: v de = 10 cm/s T i = 10 K, T ⊥ = 1 . × K, T (cid:107) = 10 K, T m = 10 K, T t = 2 × K, n i = n e = 10 cm − , B=500 Gauss, ω = 10 rad/s, v de = 10 − cm/s. [2, 3][11] also determined the value of the kappa parameter for different groups ofstars. Scudder (1994) [12] showed that the excess of Doppler line widths canalso be a consequence of non thermal distributions of absorbers and emitters.The excess brightness of the hotter lines can satisfactorily be accounted forby a two-Maxwellian electron distribution function (Ralchenko et al. 2007)[13] and should be also by a Kappa. The Kappa distribution is also consistentwith mean electron spectra producing hard X-ray emission in some coronalsources (Kasparova and Karlicky 2009) [14]. Thus the stability analysis usingkappa distribution is of particular interest for solar plasmas. We use a kappadistribution of the form given in Pierrard and Lazar, 2010 [15]3igure 2: Figure shows plot of growth rate ( γ ) versus wave vector (k) forvarious values of κ index , for normal flares. The growth rate electromagneticwave shows significant propagation, also the propagation rate increases withkappa index. The growth rate has a wide range of frequencies and higherin comparision with larger flare and has wide,: v de = 10 cm/s T i = 10 K, T ⊥ = 1 . × K, T (cid:107) = 10 K, T m = 10 K, T t = 2 × K, n i = n e =10 cm − , B=300 Gauss, ω = 10 rad/s, v de = 10 − cm/s.[2, 3]4igure 3: Figure shows plot of growth rate ( γ ) versus wave vector (k) forvarious values of κ index , for coronal condition. The growth rate is higher incomparision with larger but less than normal flare. Corona: v de = 10 cm/s T i = 10 K, T ⊥ = 1 . × K, T (cid:107) = 10 K, T m = 10 K, T t = 2 × K, n i = n e = 10 cm − , B=10 Gauss, ω = 10 rad/s, v de = 10 − cm/s. [2, 3].5 .2 Theory Out of the large number of waves undergoing quasilinear relaxation we pickout here one particularly interesting electrostatic mode in a magnetizedplasma, the lower-hybrid drift mode. Since quasilinear theory describesthe time evolution of the equilibrium distribution function, it is necessaryto retain the dependence on the distribution function in the expression forthe growth rate of the lower-hybrid instability,without assuming it to be aMaxwellian. For the sake of simplicity, we restrict ourselves to purely per-pendicular propagation. Then the dispersion relation can be written as [1] D ( k ⊥ , ω ) = 1 + χ e ( k ⊥ , ω ) + χ i ( k ⊥ , ω ) (1)where χ e ( k ⊥ , ω ) = ω pe ω ge (1 − k ⊥ L R ω ge ω − k ⊥ v de ) and χ i ( k ⊥ , ω ) = ω pe k ⊥ (cid:82) dv ⊥ ω − k ⊥ v de ∂f i ( v ⊥ ,t ) ∂v ⊥ where ω pe and ω ge are the plasma and gyro frequency respectively. Thegrowth rate in the low drift velocity regime ( v de < v thi ) , is, γ = − πv thi k ⊥ v de | k perp | (1 + k ⊥ k max ) ∂f i ∂v ⊥ (cid:107) v ⊥ → ω/k ⊥ (2)Here v thi = (cid:113) k B T i m i is the thermal velocity of protons and v de is the driftvelocity of electrons. It is assumed that v de = − v di and k max defined as, k max λ Di = 2 / (1 + ω pe /ω ge ) (3)where ω pe and ω ge are the plasma and gyro frequencies respectively. f = n π ( κ w κi ) / Γ( κ + 1)Γ( κ − /