Plasma Waves and Rayleigh–Taylor Instability: Theory and Application

Abstract

The presence of plasma density gradient is one of the main sources of Rayleigh–Taylor instability (RTI). The Rayleigh–Taylor instability has application in meteorology to explain cloud formations and in astrophysics to explain finger formation. It has wide applications in the inertial confinement fusion to determine the yield of the reaction. The aim of the chapter is to discuss the current status of the research related to RTI. The current research related to RTI has been reviewed, and general dispersion relation has been derived under the thermal motion of electron. The perturbed densities of ions and electrons are determined using two fluid approach under the small amplitude of oscillations. The dispersion equation is derived with the help of Poisson’s equation and solved numerically to investigate the effect of various parameters on the growth rate and real frequency. It has been shown that the real frequency increases with plasma density gradient, electron temperature and the wavenumber, but magnetic field has opposite effect on it. On the other hand, the growth rate of instability increases with magnetic field and density gradient, but it decreases with electron temperature and wave number.

1. Introduction

In the fluid mechanics, there are mainly two types of flows, i.e., laminar and non-laminar. Streamline flow is the example of laminar flow, and turbulent flow is non-laminar. A laminar flow converted to non-laminar due to some gradient or due to some perturbation in pressure and temperature. During the process of converting a flow from laminar to non-laminar, it produces some patterns in between the two states which are unstable and caused due to instability in the fluid [1]. There are many instabilities occurred due to many reasons, for example, Rayleigh–Taylor instability, Kelvin–Helmholtz instability, electrostatic instability, drift instability, electromagnetic instability, etc. Chandrasekhar was the first to introduce viscosity while analyzing the RTI problem at spherical accelerated interface [1]. In general instability means redistribution of energy, and this redistribution can occur due to some perturbation (external small disturbance). In some cases, we need to compress/reduce the instability to make better use of the appliance or the system to improve the performance. As in case of Rayleigh–Taylor instability, a heavy fluid is loaded on top of the lighter fluid in presence of gravitation, which always acts in downward direction as shown in Figure 1. Rayleigh–Taylor instability is a type of buoyancy-driven instability [1, 2, 3]. One example that can be observed at home is, when cold milk is added to hot tea/water (lighter-less density), then there are spikes created in the tea/water due to milk, and this is basic example of Rayleigh–Taylor instability. Some more examples of Rayleigh–Taylor instability which occurs in nature and industrial process are the instability in cold and hot water of oceans and rivers, in ionosphere due to solar radiation, in nuclear explosion and gas bubbles formation under the water. Most of the time RTI was solved using linear equations due to small perturbation but nonlinear method also exists for the situation, when perturbed amplitude is more and nonlinear terms of the equation cannot be neglected due to their reasonable magnitude [1, 2, 3, 4, 5].

Figure 1.

Plasma species under a gravitational field.

Another type of instability is the Kelvin–Helmholtz instability is one another macro-instability, which is due to shearing effect in single fluid due to different velocities or having different velocities at the interface of two fluids relative to each other [5]. It can be observed, when clouds make spiral shape patterns and roll around each other, then there exists Kelvin–Helmholtz Instability. It is caused due to flow of bulk plasma and have considerably long wavelength than gyro-radii of any kind [3]. Some examples include the red spot on Jupiter and the Sun’s atmosphere [2]. Electrostatic instabilities K-H and R-T are caused by velocity space inhomogeneity [3]. As an electromagnetic wave can propagate in plasma, electrostatic wave cannot even leave and penetrate the plasma. The electromagnetic instability is also a micro-instability, and it is caused due to velocity-space inhomogeneity or deformation of phase-space distribution function. Both electrostatic and electromagnetic plasma are developed in an unmagnetized plasma with different specific conditions for both.

The another universal and linear instability known as drift instability is caused by destabilizing of drift waves (which is driven by pressure gradient) due to difference in motion of the two components of plasma, i.e., electrons and ions in a magnetized plasma. Generally, plasma are inhomogeneous and density gradient in plasma support drift wave, which trigger drift instability in microscopic scale. Maximum time this type of instability exists in astrophysical plasma [2, 3].

Talking about the Rayleigh–Taylor Instability, there are a lot of applications for Rayleigh–Taylor instability from space to laboratory, from plasma to any fluid. But there are certain applications that are more beneficial where Rayleigh–Taylor instability (RTI) is important to study to make good material, machine or to reduce the energy loss due to instability in some part of system [6, 7, 8, 9]. RTI is used in tokamak to confine the plasma and gases by giving the conditions using comparing time of instability and fusion. More different types of RTI were studied in the past with simulations in 2D and 3D and observed a RTI in case of magnetohydrodynamics in the supernova remnants. Recently, Rayleigh–Taylor instability is observed in the outer surface of the sun where plasma bubble with low density carries high density solar prominence. Rayleigh–Taylor instability occurs in interstellar gases containing magnetic field and a density gradient. In sonoluminescense phenomenon, the RTI limits driving pressure. Rayleigh–Taylor instability has a critical role in case of climate which leads to drastic drop in temperature [4]. RTI also exists in salt domes, which occurred over millions of years ago under Earth’s crust under density gradient caused by acceleration due to gravity in upward direction.

2. Review of the current status

The heat and mass transfer rates of different viscosity and density of liquids can be controlled through instabilities. These instabilities lead to unstable multilayer flows, which results in nonuniform film thickness of coating processes. Hydrodynamic stability is well-known problem of fluid dynamics, which tells that when and why laminar flow break down. Zanella et al. presented the Cahn–Hilliard/Navier–Stokes model at low Atwood number to simulate the development of RTI and low-intermediate Reynolds number in immiscible fluids. The diffuse-interface model is presented in 2 D and 3D simulations capture phenomenon [10]. Balestra et al. inquire the RTI inside a cylinder, coated with a thin liquid film and axis orthogonal to gravity to investigate the effects of geometry on the instability [11]. Zeng et al. analyzed the rotational part of the disturbance flow field at the cylindrical interface of RTI and found the most unstable mode for small radii interfaces [12]. Tamim and Bostwick performed an elastodynamic stability analysis in a cylindrical container for the viscoelastic layer and tally the dispersion relationship, which shows dependence on the elastogravity number, elastocapillary number, compressibility number and solid Deborah number. They calculated the associated growth rate and wavelength by stability diagram in solid RTI [13].

Gebhard et al. described the incompressible inhomogeneous Euler equations for two different fluids under the influence of gravity and derived a general criterion for the turbulent mixing of two different fluids. It is found that for ultra-high range Atwood number the mixing grows quadratically in time [14]. Zhu et al. presented a method to run an azimuthal electron current that utilizes a rotating magnetic field to decrease the charge separation due to RTI. The azimuthal electric field of the fluctuation approaches to zero [15]. The mathematical homotopy perturbation method is employed to solve complex nonlinear differential equations. Two superposed rotating infinite hydromagnetic Darcian flows under the influence of a tangential uniform magnetic field through porous media are solved [16]. Mondal and Korenaga presented a propagator matrix formulation of stokes flow and instability problem turned into small eigenvalue equation. They use Monte Carlo sensitivity analysis and derived formula to calculate the growth rate of RTI for the stability of early crust [17]. Rahman and San investigated Euler turbulence using a central seven-point stencil reconstruction scheme for the performance of a relaxation filtering approach. For single-mode and multi-mode, a high-resolution numerical experiments are performed for inviscid RTI problems in 2D canonical settings [18].

Song and Srinivasan investigate the RTI under the effect of resistivity, magnetic field, viscosity and thermal conduction [19]. Casner et al. reviewed different RTI experiments performed at classical embedded interface or at the ablation front. A bubble-merger, bubble- competition regime in 2D was evidenced for the first time in indirect-drive on the National Ignition Facility at the ablation front all due to an unprecedented x-ray drive [20]. Jha et al. studied volume of fluid method to see the stabilizing effects of RTI simulations. The effect of numerical stability of blending parameter has been investigated on the flow topology. The numerical stability comparison of blending scheme shows better result than the pure higher-order schemes [21]. Yilmaz uses an advanced and fully featured LES algorithm to simulated RTI at high Atwood numbers [22]. Ahuja and Girotra investigated the RTI of nanofluids having different densities with superimposed horizontal layers [23].

Zhao et al. given an analytical model at arbitrary Atwood number in a cylindrical geometry for the nonlinear evolution of 2D single-mode RTI [24]. Horne and Lawrie studied the turbulent mixing induced by gravitational force between two miscible fluids in a tightly confined domain [25]. Scase and Hill argued about the effect of rotation on the classical gravity-driven RTI, which resulted between the destabilization effect of gravity and stabilization effect of the rotation. A fourth-order Orr–Sommerfeld equation was derived via linear stability analysis [26]. Chen et al. studied the coexisting system and 2D Richtmyer–Meshkov instability system. The multiple-relaxation time discrete Boltzmann model was employed for Richtmyer–Meshkov instability system, and the correlation between non-uniformity of temperature and globally averaged non-organized energy flux is found one [27]. Elyanov et al. studied the condition for the appearance of various types of instabilities in the lean hydrogen–air mixture containing inhomogeneities at the flame front [28].

Alqatarl et al. created a clean initial stationary interface between two miscible fluids by preparing a density inversion inside the thin gas separating flat plates. As a result of these conditions, the instability is suppressed below a critical plate spacing [29]. Gopalakrishnan et al. combined the mechanism of double-diffused instability and RTI for porous media flows [30]. Scase and Sengupta carried out numerical simulations of the inviscid flow, and they also examine the special case of single-layer rotating viscous column [31]. Sauppe et al. found that the initial sinusoidal perturbation grows slowly at the time of implosion and develops the classical bubble in the experiment of RTI growth (laser-driven cylindrical implosion) [32]. Shimony et al. find a relation for growth parameter of the self-similar evolution and the density of bubbles. They examined the effect of density ratio on RTI [33]. Malamud et al. studied the evolution of RTI in the design of a laser-driven experiment and examined the re-acceleration phenomenon [34]. Sauppe et al. designed a cylindrical implosion experiment using the radiation-hydrodynamics code xRAGE for the OMEGA and NIF laser facilities [35]. Wu et al. concentrated on nonlinear dynamics and analyzed the 3D single-mode RTI at a higher Atwood number [36]. Liang et al. performed the numerical simulations using an improved phase field lattice Boltzmann method at low Atwood number for multimode immiscible RTI. The larger wavelength instability undergoes faster growth after analyzed the effect of the initial conditions in terms of the perturbed amplitude and wavelength [37]. Wolf studied the miscible liquids for dynamic stabilization of RTI and the related frozen waves [38]. Cohen and Oron studied the dynamics of a thin isothermal and non-isothermal liquid film under the influence of harmonic tangential force [39]. Nimmagadda et al. studied the dynamics of a gradually mixed fluid with pre-defined bubbles rising due to RTI [40]. Naveh et al. theoretically studied the late-time evolution inside a finite-sized spatial domain [41]. Panda et al. investigated the seasonal and Spatial-temporal effects of ionosphere irregularities on GNSS using various instances of ionospheric disturbances [42]. Hu et al. simulated the evolution in 2D single-mode RTI of the bubble vorticity and its velocity at different viscosities with different Reynold number and at low Atwood number [43].

Luo et al. numerically studied the effect of compressibility on RTI with different isothermal Atwood numbers and Mach number in the late-time evolution of 2-D single-mode [44]. Wieland et al. used wavelet-based adaptive mesh refinement to study the vorticity dynamics and the impact of isothermal stratification strength on it for compressible single-mode RTI [45]. Puillet et al. used rotating cylinder coated by liquid elastomer melt from outside, to produce an array of droplets on a thin layer of the melt to study the elastic amplification of RTI [46]. Baksht et al. considered the energy balance in the radiating metallic gas-puff Z pinch to study the energy balance in imploding Mg-shell-on-Bi-jet pinches and Mg-shell-on-Mg-jet. The power-law density distribution is observed in this type of Z pinch for promoting suppression of RTI that occurs during compression in the pinch plasma [47]. Mishra et al. analyze the observations of an eruption in the intermediate corona consisting of magnetic Rayleigh–Taylor unstable plasma segment using the data collected from Solar Terrestrial relations Observatory [48]. Hu et al. theoretically studied the radiation equivalent of RTI-Interface Discontinuous Acceleration phenomena under the incompressible limit of optically thin limit [49]. Ruiz obtained a theoretical model by asymptotically expanding an action principle, which leads to fully nonlinear magnetic Rayleigh–Taylor equations [50]. Jiang and Zhao investigated in a bounded domain of an inhomogeneous incompressible viscous fluid to study the existence of unstable classical solutions of the RT problem [51]. Chen et al. studied the linear RTI of stratified fluids and the effects of elasticity and magnetic fields on it [52]. Jiang et al. reported the phenomenon of inhibition of RTI in the inviscid, inhomogeneous and incompressible fluid by applying horizontal magnetic field along velocity damping [53].

Meshkov and Abarzhi theoretically and experimentally studied the RT flows by focusing of the effect of acceleration on disorder and order in RT mixing [54]. Wang and Zhao mathematically investigated the RTI in the presence of a bounded uniform gravitational field based on Oldroyd-B model inside a compressible viscoelastic fluid [55]. Su uses a sign-changing Taylor sign coefficients for proving the existence of water waves via the method that strong Taylor sign holds in starting time while breaks down at a later time and vice versa [56]. Kord and Capecelatro studied the role of interfacial perturbations to suppress and enhance the RTI growth rate through manipulation of the initial conditions of the evolution of a multi-mode RTI [57]. Li et al. presented a linear analytical model for RTI by considering the long wavelength to study the phase effects in initial confinement fusion caused by the phase difference of single-mode perturbation on two interfaces [58].

Zu et al. worked on the conservative form of Allen–Cahn (A-C) equation for phase field, and they proposed a Lattice Boltzmann (LB) model to track the interface of binary fluid systems [59]. Talat et al. numerically studied RTI problem in 2D by coping with moving boundary conditions and by enabling single domain fixed node approach with the procedure of meshless solution based on diffuse approximate method and phase-field formulation [60]. Liu et al. considered spike for the pure single mode cosine perturbation between interfaces of two classical fluid and employ a method of small parameter expansion for analytical exploration of history of evolution of bubble for potential function having up to third-order nonlinear corrections [61]. Morgan et al. simulate the RTI with large acceleration by studying RTI in gas phase experiments. To make a RTI a stably stratified system of two fluids having a diffused interface separation is accelerated using a rarefaction wave [62]. Wang et al. performed numerical simulations and analytical modeling for the interfacial instability of a water droplet induced by internal volume oscillation [63].

Gebhard and Kolumban considered the evolution of two fluids of homogeneous densities described by the inviscid Boussinesq equations and provides the explicit relaxation of the associated differential inclusion [64]. Bian et al. conducted a systematic analysis on RTI late-time growth for studying the effects of perturbed Atwood number and Reynold number. They show that at large Reynolds number, the bubble re-accelerates [65]. Srinivasan and Hakim work in the nonlinear phase of magneto-RTI and studied the role of electron inertia and Larmor radius effects [66]. Huneault et al. formed gas–liquid interface, which becomes RT unstable, and studied the effect on growth of spray like high mode number perturbation using the implosion of nominally smooth cavity [67]. Terrones and Heberling studied the most-unstable modes by performing a systematic analysis using computational method under the action of a radially directed acceleration over the entire parameter space [68].

Li et al. experimentally studied the evaporation behavior of a silicone-oil-seeded. The observation shows the formation of plumes of segregated fluid and as a result an instantaneous segregation of 1,2-hexanediol in the sessile droplet. [69]. Lherm et al. described the impact catering that crater deceleration after impact, which is responsible for a density-driven perturbation at the drop-pool interface, known as a spherical RTI [70]. Rigon et al. studied the case of young Supernovae Remnants, when interacted with interstellar medium [71]. Polavarapu et al. experimentally presented the result of 3D and 2D perturbation of RTI in the elastic–plastic materials. Results show that by decreasing the initial wavelength and amplitude a more stable interface is produced [72]. Zheng et al. used linear stability analysis to investigate a rigid cylindrical container for RTI confining an elastic soft gel. The gel is found quite sensitive to the aspect ratio and critical load on the surface of the gel [73].

Lyubimova et al. investigated two confined isothermal binary mixture of slowly miscible liquids in horizontal plane layer with the simultaneous convective and diffusive evolution [74]. Awasathi achieved a second-order polynomial in form of dispersion equation for the growth rate analysis of RTI under mass and heat transfer [75]. Chao et al. examined the Spatial-temporal and temporal stability of the system under uniformly cooled–heated inclined substrate and studied the RTI of gravity-driven viscous liquid films flowing [76]. To characterize the instability pattern in spherical geometry, Balestra et al. investigated the RTI of a thin viscous film coated inside a spherical substrate [77]. Livescu et al. used the results from direct numerical simulations of RTI to study the evolution of the flow after the gravity. In turbulent kinetic energy transport equation, the buoyancy term changes sign and becomes a destructive term as the gravity is reversed, which is given by the product between mean pressure gradient and the mass flux, leading to a rapid decay of the kinetic energy [78].

Aslangil et al. explore RTI-induced mixing for zero acceleration with variable periods. The results were compared with the case of classical constant gravity, when acceleration is removed for either some indefinite period or intermediate period [79]. Sabet et al. demonstrated that mixing properties of fluids influenced drastically due to viscosity contrast between two miscible fluids under the destabilization effect of RTI [80]. Chakrabarti et al. addressed the pattern selection of the elastic RTI. The dominant coupling mechanism takes place through three modes of resonance [81]. D’Ortona et al. interested in segregating dry granular flow and studied the self-induced RTI in it [82]. Zhang et al. show that mass ablation can reduce RTI nonlinear bubble growth and RTI is dominated by bubble competition with respect to the classical value for the same initial perturbation amplitude via numerical simulations in 2D and 3D multimode RTI [83].

Yang et al. used a phase-field model to investigate the side wall boundary effect on all boundaries under no-slip conditions. The coupled Cahn–Hillard and Navier–Stokes system equations are solved [84]. Khan and Shah numerically solve the coupled system of incompressible phase-field and Navier–Stokes equations by implementing Neumann and periodic boundary conditions and hence simulated successfully the long-time evolution of the RTI problem [85]. Ding et al. employed molecular dynamic simulation to obtain the growth behavior of RTI [86]. Chen et al. studied the effects of specific heat ratio on the compressible RTI by applying discrete Boltzmann method. They investigate two kind of non-equilibrium quantities related to heat flux and viscous stress [87]. Piriz and Piriz worked on a linear elastic perfect-plastic constitutive model to develop the 2D linear theory of the incompressible RTI [88].

Gancedoet al. studied the dynamics of an incompressible fluid in porous medium driven by capillarity and gravity forces [89]. Zhang et al. worked for spherical geometry and studied the 3D weakly nonlinear RTI by developing a theoretical model for ideal Euler equation and incompressible fluid, and third-order solutions are derived for interface perturbations of spherical harmonic modes [90]. Yang et al. studied the RTI in presence of a horizontal magnetic field along the film for an inviscid flow with finite film thickness. They analyzed the growth of small perturbations on free surfaces [91]. Hillier investigated the physics behind the formation of fragmentation of prominence eruption and plumes in prominences and found that the physical process behind these observations is the magnetic RTI [92]. Dolai andPrajapati investigated the linear RTI in the dust cloud, and hydrodynamics model was formulated by considering Boltzmann distributed electrons and dust particles under magnetic field [93]. Pruss et al. worked on the RTI without and with phase transition in the problem of Verigin [94]. Schulreich and Brietschwerdt demonstrated that powerful time-varying pressure gradient in homogeneous plasma has a sudden energy release effects [95]. Gibbon employed the variable density model and says that RT-turbulence is driven by Navier–Stokes turbulence via gradient of the buoyancy in the inhomogeneous and incompressible [96]. Li et al. theoretically analyzed RTI in gas stream on a spherical droplet [97]. Mahulikar et al. worked on the role of fluctuations that integrate the maximum entropy production and fluctuation theorem in entropy production by a thermodynamic fusion theorem [98].

Zhao. et al. modeled thin shell approximation for an ideal incompressible fluid by extension of thin layer model for nonlinear evolution and deformation of the RTI [99]. Zhou and Cabot utilized three big datasets with a goal of determining degree of departure, using direct numerical simulation with different Atwood numbers at moderate Reynolds number, of homogeneous and isotopic turbulence [100]. Braileanu et al. worked on the development of RTI to understand partial ionization effect between solar corona and magnetized prominence at a smooth realistic transition layer [101]. Zhao et al. generalize thin layer model and put forward thin shell model in 2D spherical geometry for RTI. They obtained equation of motion for the shell analytically, which can also be calculated numerically [102]. Sun et al. work on a unified model, where decomposition of unstable mode into a rotational and irrotational part was done. Non-conservative force and constitutive properties of matter determine the mode of decay with the velocity field [103].

Luttwak used staggered Mesh Godunov scheme to study RTI [104]. Zhang et al. compare the RTI in planar geometry and spherical geometry using the third-order WN solutions for RTI in their research [105]. Zhang et al. using a model established the multi-mode incompressible RTI for spherical geometry in 2D and derived the solutions within the second order. To study the difference between growth of perturbation and geometry effect at the equator and pole, they use the Gaussian-type and cosine-type perturbation [106]. Grouchy et al. experimentally investigated gas-puff Z-pinch implosion and studied Magneto RTI in the cylindrical shocks launched toward the symmetry axis in different plasma of argon and neon [107]. Li et al. developed a code based on Euler method called as massively parallel laser ablative RTI code using simulations by considering the laser energy deposition, hydrodynamics and the electronic thermal conductivity [108]. Kashkovsky et al. performed computation and study using the direct simulation Monte-Carlo method about the development of the RTI and show that it can be successfully simulated at the molecular-kinetic level [109]. The physics of Hall thrusters and existing instabilities are reported in the references [110, 111, 112, 113] the Hall thruster support both resistive and RTI. The electromagnetic and electromagnetic oscillation has been discussed in detail in the references [114, 115, 116, 117, 118, 119].

3. Introduction to Rayleigh: Taylor instability

Consider a gravitational field in the x direction. If we place a plasma in the field without a vessel, the plasma will fall. Now, consider a situation when a DC magnetic field is applied along $\widehat{z}$ (${\overrightarrow{B}}_s\vee \widehat{z}$. The gravitational force on the electrons of plasma can be balanced by the Lorentz force due to magnetic field (Figure 1).

This equilibrium, however, is not suitable to perturbation in the plasma boundary. Let the perturbation be denoted by dotted curves. The ion drifts along -Y-direction, and the electrons drift in the positive X-direction. Due to charge separation, an electric field set up in the -Y-direction.

This electric field produces a $\overrightarrow{E}\times \overrightarrow{B_s}/B_s^2$ drift velocity of electrons and ions in the minus X-direction, and both electrons and ions fall down, which is balanced by the gravitational field. Due to this, perturbation in the boundary grows. This is known as Rayleigh–Taylor instability.

4. Fluid equations and instability analysis

Under this situation, we write the continuity equation

$$\frac{\partial n}{\partial t}+\overrightarrow{\nabla }·\left(\overrightarrow{\upsilon }n\right)=0$$E1

The equation of motion for ion

$$\left(\frac{\partial }{\partial t}+\overrightarrow{\upsilon }·\overrightarrow{\nabla }\right)\overrightarrow{\upsilon }=\frac{e\left(\overrightarrow{E}+\overrightarrow{\upsilon }\times {\overrightarrow{B}}_0\right)}{M}$$E2

The equation of motion for electrons

$$\left(\frac{\partial }{\partial t}+{\overrightarrow{\upsilon }}_e·\overrightarrow{\nabla }\right){\overrightarrow{\upsilon }}_e=-\frac{e\left(\overrightarrow{E}+{\overrightarrow{\upsilon }}_e\times {\overrightarrow{B}}_0\right)}{m}-\frac{\overrightarrow{\nabla }{p}_e}{{\mathrm{mn}}_e}$$E3

4.1 First-order linearization of the equations

We use the linearized form of the above equations for small perturbations density, velocity and electric field. We represent perturbed density by $n_1$ and velocity by ${\overrightarrow{\upsilon }}_1$.The unperturbed density is taken as $n_0$.The unperturbed value of velocities of ions and electrons is taken as ${\upsilon }_0=-\frac{\mathrm{Mg}}{{\mathrm{eB}}_0}\widehat{y}=-\frac{g}{{\Omega }_{\mathrm{ci}}}\widehat{y}$ and ${\upsilon }_{0e}=\frac{g}{{\Omega }_{\mathrm{ce}}}\widehat{y}$, respectively. These relative drifts of the plasma species set up a perturbed electric field in the positive Y-direction. Let us write $n=n_0+n_1$, $\overrightarrow{\upsilon }={\overrightarrow{\upsilon }}_1+{\overrightarrow{\upsilon }}_0$ and $\overrightarrow{E}={\overrightarrow{E}}_1+{\overrightarrow{E}}_0$.The linearized form of the above equations thus read

$$\frac{\partial \left(n+n_1\right)}{\partial t}+\overrightarrow{\nabla }·\left[\left(n+n_1\right)\left({\overrightarrow{\upsilon }}_0+\overrightarrow{\upsilon }\right)\right]=0$$E4

$$\frac{\partial n_0}{\partial t}+\frac{\partial n_1}{\partial t}+\overrightarrow{\nabla }·\left[\left(n_0{\overrightarrow{\upsilon }}_0+{\overrightarrow{\upsilon }}_1{n}_1+n_1{\overrightarrow{\upsilon }}_0+n_0{\overrightarrow{\upsilon }}_1\right)\right]=0$$E5

$$\frac{\partial n_1}{\partial t}+\left({\overrightarrow{\upsilon }}_0·\overrightarrow{\nabla }\right)n_1+\left({\overrightarrow{\upsilon }}_1·\overrightarrow{\nabla }\right)n_0+n_0\left(\overrightarrow{\nabla }·{\overrightarrow{\upsilon }}_1\right)=0$$E6

The quadratic in perturbation terms $\left({\overrightarrow{\upsilon }}_1·\overrightarrow{\nabla }\right)n_1.$ and $n_1\left(\overrightarrow{\nabla }·{\overrightarrow{\upsilon }}_1\right)$ are neglected. The terms $n_0\left(\overrightarrow{\nabla }·{\overrightarrow{\upsilon }}_0\right)$ and $n_1\left(\overrightarrow{\nabla }·{\overrightarrow{\upsilon }}_0\right)$ reduce to zero for the constant velocity ${\overrightarrow{\upsilon }}_0$.

Since the mass of the electron is much less than from the mass of the ion, the initial gravitational drift of electrons may be ignored to simply the dispersion relation. Let us consider that $\omega$ is the frequency of oscillations and $k$ is the propagation constant of the oscillating quantities. We seek the solution of above equations for which we put $A=Aexp\left(\mathrm{iky}-\mathit{i\omega t}\right)$. Replacing $\frac{\partial }{\partial t}\to$ ($-\mathit{i\omega }$ and $\nabla \to$ ik. With the help of Eqs.(5) and (6), we obtain the following expression for the perturbed ion density

$$-{\mathit{i\omega n}}_{i1}+{\text{in}}_{i1}{\mathit{k\upsilon }}_0+{\upsilon }_{\mathrm{ix}}\frac{\partial n_0}{\partial x}+{\mathrm{ikn}}_0{\upsilon }_{\mathrm{iy}}=0$$E7

After simplifying, the perturbed density of ions

$$n_{i1}=\frac{{\mathrm{kn}}_0{\upsilon }_{\mathrm{iy}}-{\mathit{i\upsilon }}_{\mathrm{ix}}\frac{\partial n_0}{\partial x}}{\omega -{\mathit{k\upsilon }}_0}$$E8

From the equation of motion of ions, we get

$$\frac{\partial {\overrightarrow{\upsilon }}_1}{\partial t}+{\upsilon }_0\frac{\partial {\overrightarrow{\upsilon }}_1}{\partial x}=\frac{e}{M}{\overrightarrow{E}}_1$$E9

$$\frac{\partial n_{e1}}{\partial t}+u_0\frac{\partial n_{e1}}{\partial y}+n_0\left(\overrightarrow{\nabla }·{\overrightarrow{\upsilon }}_e\right)+{\upsilon }_{\mathrm{ex}}\frac{\partial n_0}{\partial x}=0$$E10

The linearized form of equations of electrons

$$\frac{\partial {\overrightarrow{\upsilon }}_{e1}}{\partial t}+{\overrightarrow{\upsilon }}_{0e}\frac{\partial {\overrightarrow{\upsilon }}_{e1}}{\partial y}=-\frac{e}{m}\left({\overrightarrow{E}}_1+{\overrightarrow{\upsilon }}_e\times {\overrightarrow{B}}_0\right)-\frac{\overrightarrow{\nabla }{p}_e}{{\mathrm{mn}}_0}$$E11

On the other hand, we obtain from Eq. (11)

$$\left(\omega -{\mathit{k\upsilon }}_{0e}\right){\upsilon }_{\mathrm{ex}}=-i{\Omega }_{\mathrm{ce}}{\upsilon }_{\mathrm{ey}}$$E12

$$\left(\omega -{\mathit{k\upsilon }}_{0e}\right){\upsilon }_{\mathrm{ey}}=-\frac{i{\Omega }_{\mathrm{ce}}{E}_y}{B_0}+i{\Omega }_{\mathrm{ce}}{\upsilon }_{\mathrm{ex}}+\frac{{\mathrm{kn}}_{e1}{T}_e}{n_{0}m}$$E13

Here, $\Omega =\frac{{\mathrm{eB}}_0}{m}$ is taken as the electrongyrofrequency. Above equations are solved for the X- and Y-components under the assumption $\Omega >>\omega -{\mathrm{ku}}_0$, and we obtain the components

$${\upsilon }_{\mathrm{ex}}=\frac{E_y}{B_0}-\frac{{\mathrm{iV}}_{{\mathrm{th}}^2}{\mathrm{kn}}_{e1}}{{\Omega }_{\mathrm{ce}}{n}_0}$$E14

$${\upsilon }_{\mathrm{ey}}=\frac{i\left(\omega -{\mathit{k\upsilon }}_{0e}\right)E_y}{{\Omega }_{\mathrm{ce}}{B}_0}-\frac{V_{{\mathrm{th}}^2}{\mathrm{kn}}_{e1}\left(\omega -{\mathit{k\upsilon }}_{0e}\right)}{{\Omega }_{{\mathrm{ce}}^2}{n}_0}$$E15

For ions, their thermal motions velocity has been ignored.

$${\upsilon }_{\mathrm{ix}}=\frac{E_y}{B_0}$$E16

And

$${\upsilon }_{\mathrm{iy}}=-\frac{i\left(\omega -{\mathit{k\upsilon }}_{0i}\right)E_y}{{\Omega }_{\mathrm{ci}}{B}_0}$$E17

The perturbed ion density

$$n_{i1}=-i\frac{{\mathrm{kn}}_0{E}_y}{{\Omega }_{\mathrm{ci}}{B}_0}-i\frac{E_y}{B_0\left(\omega -{\mathit{k\upsilon }}_0\right)}\frac{\partial n_0}{\partial x}$$E18

The perturbed electron density can be written as

$$n_{e1}=\frac{{\upsilon }_{\mathrm{ex}}\frac{\partial n_0}{\partial x}}{i\left(\omega -{\mathit{k\upsilon }}_{e0}\right)}$$E19

Substituting the value of electron velocity, we get

$$n_{e1}=-\frac{{\mathrm{iE}}_y\frac{\partial n_0}{\partial x}}{B_0\left(\omega -{\mathit{k\upsilon }}_{e0}\right)\left\{1+\frac{V_{{\mathrm{th}}^2}k\frac{\partial n_0}{\partial x}}{\left(\omega -{\mathit{k\upsilon }}_{e0}\right){\Omega }_{\mathrm{ce}}{n}_0}\right\}}$$E20

We use the Poisson’s equation with perturbed densities and get

$${\epsilon }_0\nabla .E=n_{i1}-n_{e1}$$E21

$${\epsilon }_0\mathit{ik}{E}_y=-i\frac{k{n}_0{E}_y}{{\Omega }_{\mathit{ci}}{B}_0}-i\frac{E_y}{B_0\left(\omega -k{\upsilon }_0\right)}\frac{\partial n_0}{\partial x}+\frac{i{E}_y\frac{\partial n_0}{\partial x}}{B_0\left(\omega -k{\upsilon }_{e0}\right)\left\{1+\frac{{V_t}^{2}k\frac{\partial n_0}{\partial x}}{\left(\omega -k{\upsilon }_{e0}\right){\Omega }_{\mathit{ce}}{n}_0}\right\}}$$E22

After solving the above expression, we get the quadratic dispersion equation under the effect of plasma density, wavenumber and gravitation drift of plasma species. To find out the unstable growing waves, we need to find out the unstable eigenvalues of the linearized problem.

$${\omega }^2-\left(a+b\right)\omega +\zeta =0$$E23

The various coefficients are given below

$$\begin{array}{l}a=\gamma -c=\gamma -k{u}_0,b=k{V}_{0i},\\ c=k{u}_0,\zeta =\mathit{ab}-\frac{\left(c-\gamma -b\right)\beta }{\mathit{k\alpha }},\\ \alpha ={ϵ}_0+\frac{n_0}{B_0{\Omega }_{\mathit{ci}}},\beta =\frac{n_{0x}}{B_0},n_{0x}=\frac{\partial n_0}{\partial x},\\ \gamma =\frac{k{V}_{\mathit{th}}^2{n}_{0x}}{n_0{\Omega }_{\mathit{ce}}}.\end{array}$$E24

5. Result and discussion

The above quadratic equation is solved numerically to find out the growth rate and real frequency of Rayleigh–Taylor instability under the effect of thermal motion of electron. It has been shown in Figure 2 that the growth rate and the real frequency increase linearly with density gradient.

Figure 2.

Variation of growth rate and real frequency with density gradient.

Figure 3 shows that the growth rate is maximum for the smaller value of wavenumber. In other words, the instability grows faster for the larger wavelength oscillations. In Figure 4, the effect of magnetic field on the growth and the real frequency is shown. The growth rate shows almost parabolic variation with magnetic field, and the real frequency decreases with higher value of magnetic field. The real frequency behavior with wavenumber and electron temperature shows that the phase velocity remains constant during the propagation. The growth rate increases with electron temperature up to certain limit, and then, it decreases with higher value of temperature as shown in Figure 5.

Figure 3.

Variation of growth rate and real frequency with wave number.

Figure 4.

Variation of growth rate and real frequency with electron temperature.

Figure 5.

Variation of growth rate and real frequency with wave number.

6. Conclusions

In Rayleigh–Taylor instability, heavy fluid supported by a light fluid in the presence of relative acceleration, and it measures the rate of mixing of turbulent layers of two fluids of different densities at the interface. The mixing rate and yields can be characterized by measuring the mass and heat exchange across the mixing layers. It is measured that short wavelength perturbations blow up exponentially in RTI. The gravitational force acts on an inverted density gradient and triggers the instability.

Acknowledgments

The University Grants Commission (UGC), New Delhi, India, is thankfully acknowledged for providing the startup Grant (No. F. 30-356/2017/BSR).