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Effects of horizontal and vertical magnetic field components on the Rayleigh-Taylor instability of stratified incompressible plasmas layer of variable density through Darcy porous medium are studied. The basic magnetohydrodynamic (MHD) set of equations has been constructed and linearized. Then the linear normalized growth rate is obtained analytically as a function of the physical parameters of the system considered. Numerical calculations have been performed to see the effects of various parameters on the normalized growth rate of Rayleigh-Taylor instability.(For more information,please refer to the PDF.)

The hydromagnetics stability of a magnetized plasma of varying density is of considerable importance in several astrophysical situations such as supernova explosions, in heating in solar corona, theories of sunspot magnetic fields, the formation and mixing of clouds and the stability of the stellar atmospheres in magnetic fields.

The classical study of the equilibrium of an incompressible, inviscid fluid of variable density was first undertaken by Rayleigh [

Under various physical effects, the Rayleigh-Taylor instability problem of a finite layer of a fluid has been studied by several authors in hydrodynamics and in magnetohydrodynamics domain; the stabilizing effect of magnetic field on RTI problem for an incompressible plasma has been demonstrated by Kruskal and Schwarzschild [

The RTI of magnetized plasma through porous medium problem has a great scientific interest, where this problem corresponds physically (in astrophysics) to the Rayleigh-Taylor instability of an equatorial section of a planetary magnetosphere or of a stellar atmosphere where the magnetic field is perpendicular or parallels to gravity. So, the RTI of a stratified plasma through porous medium in the presence or absence of magnetic field has been studied by a number of researchers (Chhajlani and Vaghela, Vyas and Chhajlani, Sharma and Bhardwaj, Sharma and Sharma, Sharma and Trilok, Sharma and Sunil, Shikha and Bhatia, Opara, Sharma and Sunil, Sunil and Sharma, Sharma and Thakur and Sharma and Rajput). In this case (Darcy’s model), the usual viscous term in the equation of motion is replaced by the resistive term

In all the above-mentioned studies, the behaviour of growth rates is considered with respect to the porosity of porous medium and the medium permeability in the presence of an variable magnetic field in

We consider the strata of incompressible and inviscous plasma as a fluid of electrons and immobile ions through Darcy porous medium in the presence of magnetic field

For incompressible flow the fluid elements move without changing density is to say that the Lagrangian total derivative of density is zero, that is (see reference [

where

One can see that the set of Equations (1)-(4) is complete for describing the magnetic field effects on the R-T instability of incompressible plasma, since its number of equations exactly equals its number of unknown quantities: Two unknown vector quantities

Now, we assume a small perturbation in the system of Equations (1)-(4), where the perturbations in the velocity

Now, let

If we assume that the perturbation in any physical quantity takes the form

where

Now, if we eliminate some of the variables from the system of Equations (16)-(21), we have a differential equation in

(23)

In this section we consider the case of incompressible continuously stratified plasma layer of thickness

where

where

Now, if we choose

Now, we define the dimensionless quantities

Then Equations (25) and (26), respectively, take the form

Now, we put

Now, if we rearrange the above two equations (Equations (30) and (31)), we will have

From Equations (32) and (33) maybe we can specialize the next special cases:

(i) In the case of

From Equation (33) we get

This case is considered by Goldston and Rutherford (see reference [

(ii) In the case of

A second time, from Equation (33) we get

This case studied in reference [

(iii) In the case of

A third time, from Equation (33) we get

Now, comparing between Equations (34) and (36), someone can observe that, the stabilizing role for the vertical magnetic field on the considerable system, where

(iv) In the case of

A fourth time, from Equation (33) we get

Then, the normalized growth rate becomes

From Equations (34) and (38) it is very clear that,

The stabilizing effects of the horizontal, vertical magnetic and resistive term, unaccompanied, (above cases (i)-(iv)) on the RTI have been numerically presented in

(v) For the general case (

In this case the normalized growth rate given as.

where

In fact, the square of normalized growth rate

The role of constant

rate

that is greater than

The general case (Equation (40)), that gives the effects of horizontal, vertical magnetic field and resistive term together on the instability of the considered system has been presented in

In the case

Then the maximum normalized growth rate gives by

Finally,

toward the point

In closing this paper, the Rayleigh-Taylor instability in stratified plasma in the presence of combined effect of horizontal and vertical magnetic field through Darcy porous medium is considered. The solution of the system leads to a dispersion relation where the physical parameters are put in the dimensionless form. Some special cases are particularized to explain the roles that play the variables of the problem and numerical solutions are made. Some stability diagrams are plotted and discussed. The results show that, as the growth rate depends on the horizontal and vertical components of magnetic field and resistive term (Darcy’s term) also depends on the parameter