Open access peer-reviewed chapter - ONLINE FIRST

Dynamics of Rayleigh-Taylor Instability in Plasma Fluids

By Sukhmander Singh

Submitted: November 8th 2019Reviewed: March 6th 2020Published: April 15th 2020

DOI: 10.5772/intechopen.92025

Downloaded: 43

Abstract

The chapter discusses the evolution of Rayleigh-Taylor instability (RTI) in ordinary fluids and in a plasma fluid. RT instability exits in many situations from overturn of the outer portion of the collapsed core of a massive star to laser implosion of deuterium-tritium fusion targets. In the mixture of fluids, the instability is triggered by the gravitational force acting on an inverted density gradient. The motivation behind the study of the instability has been explored by discussing the applications of RT instability. The basic magnetohydrodynamics equations are used to derive the dispersion relation (for an ordinary fluid and plasmas) for two fluids of unequal densities. The conditions of the growth rate of the instability and the propagating modes are obtained by linearizing the fluid equations. The perturbed potential is found to increase with the plasma parameters in a Hall thruster.

Keywords

  • instabilities
  • plasma
  • Navier-Stokes
  • growth rate
  • Hall thruster

1. Introduction

Flow instabilities are used to increase the heat and mass transfer rates as well as to fuse the fluids of dissimilar properties (viscosity, elasticity, density, etc.). In other technological applications, these instabilities are accountable to unstable the multilayer and free-surface flows. Multilayer flows are used in coating processes and lubricated pipeline transport. The presence of the instabilities in the system leads to nonuniform film thickness and defects, where good optical finishing and smooth edges are required by the industry, which further leads to poor product quality. Suppression of these instabilities has been a major task from a long time by the researchers to improve the product quality [1, 2]. Rayleigh-Taylor (RT) instability takes place when a lighter fluid supports a heavy fluid, then any perturbation of the interface grows and leads to spikes of the heavier fluid penetrating into the lighter one and the interface becomes unstable. The contact discontinuity between the two fluids is unstable to perturbations that grow by converting potential energy to kinetic energy, causing bubbles of the low-density fluid to rise, and spikes of the high-density fluid to sink. If the light fluid is above the heavy fluid, the interface is stable. In a magnetized plasma, the Rayleigh-Taylor instability can occur because the magnetic field acts as a light fluid supporting a heavy fluid (the plasma).

In curved magnetic fields, the centrifugal force on the plasma due to the charged particle motion along the curved field lines acts as an equivalent gravity force. When forces associated with the density gradient and gravity oppose each other, the RT instability sets in [3, 4]. The box of fluid shown in Figure 1 is now filled with two incompressible fluids of differing densities, separated by an interface with a perturbation imposed as shown in Figure 1 . Here, RTI is seen to play a wider role in many branches of science from astrophysical systems to industries.

Figure 1.

Two fluids inside of a large box.

2. Review of status of research

This instability occurs in many interesting physical situations, such as implosion of inertial confinement fusion capsules, core collapse of supernovae, or electromagnetic implosions of metal liners. The Rayleigh-Taylor problem was first studied by Lord Rayleigh in 1883 and Sir G.I. Taylor in 1950 [3]. Taylor used the theory of linearization for the small oscillations at the interface and obtained an exponential growth rate. Chandrasekhar, in 1961, studied the magnetic field case analytically for the fluids that are incompressible, inviscid, and have zero resistivity. Qin et al. [5] reported the synthesis of chains of metal nanoparticles with well-controlled particle sizes and spacing induced by the Rayleigh instability. Bychkov et al. [6] derived the dispersion relation for the internal waves and the RT instability in a nonuniform unmagnetized quantum plasma with a constant gravitational field. They have shown that the quantum effects always play a stabilizing role for the RT wave instability. Cao et al. [7] studied the RT instability incorporating the quantum magnetohydrodynamic equations and solved the second-order differential equation under different boundary conditions with quantum effects. Khomenko et al. [8] modeled the growth rate of the instability and the evolution of velocity and magnetic field vector in the prominence plasma (closer to Sun’s surface) under the presence of neutral atoms. Diaz et al. [9] derived the criterion for the growth rate of the RT instability in partially ionized plasma using single fluid theory. Ibrahim and Marshall theoretically investigated the impact of velocity profile on RTI within the jet to examine the effects of its relaxation on intact length [10]. Carlyle and Hillier experimentally verified that stronger magnetic fields can suppress the growth of the rising bubbles of the RTI [11]. Litvak and Fisch derived the necessary instability conditions of azimuthally propagating perturbations in a Hall thruster plasma [12]. Recently, investigators derived the dispersion for the Rayleigh-Taylor instabilities in a Hall thruster using the two - fluid theory [13, 14]. Shorbagy and Shukla investigated the RT instability in a nonuniform multi-ion plasma in a Hall thruster to obtain the growth rate of the instability [15]. Ali et al. [16] derived the modified dispersion relation for the Rayleigh-Taylor instability under the quantum corrections incorporating the terms of Fermi pressure and the Bohm potential force.

3. Basic fluid equations and Bernoulli’s theorem

First, we consider the two simple fluids separated by a smooth interface to derive the dispersion relation. Let us assume that in each separate region, the density is constant. The coordinate x is in the horizontal, z in the vertical, and y is going into the page. We consider a flow in the x-direction, which in the lower half-space z<0has density ρ1, whereas in the upper half-space z>0has density ρ2. In addition there can be a homogeneous gravitational field gpointing into the negative z-direction. We write the basic fluid equations for the ion and electron fluids as Navier-Stokes equations for an incompressible fluid are

ρt+ρυ=0E1
dυdt=υt+υυ=Pρ+g+η2υE2

Here, we have used total time derivative. Partial time derivative keeps an eye on a point and represents the rate of velocity change at that point. Total time derivative keeps an eye on fluid element and measures its velocities at tand t+Δt.

Let us consider the fluid is inviscid, so that we take viscosity η=0. We also assume that the fluid is irrotational, that is ×υ=0. Then the term υυreduces to 12υ2. The Stokes’ theorem permits us to express the velocity in terms of gradient of scalar function, that is υ=ϕ. The variable ϕis called the scalar velocity potential of fluid. We rewrite gravity acceleration into a gradient of gravity potential g=gz. Eq. (2) can be rewritten in terms of scalar function ϕunder the above assumptions:

ϕt+12υ2=PρgzE3

If the density remains constant in one region, we can write Eq. (3) as

ϕt+12ϕ2+gz=PρE4

Now integrating the above equation in horizontal and vertical directions, we get unsteady equation for the Bernoulli theorem.

ϕt+12ϕ2+gz+Pρ=ConstE5

That is, the total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure, and the kinetic energy of the fluid motion remains constant.

Let υ1and υ2be the velocities of the fluid in the lower half-space (z<0) and upper half-space (z>0) respectively. Now, it is convenient to write velocities of fluid in terms of scalar velocity potential ϕin both regions such that

υ1=ϕ1E6
υ2=ϕ2E7

For the incompressible fluid, Eq. (1) yields that υ=0. This also states that both fluids will satisfy the Laplace equation in both regions

2ϕ=0E8

The Bernoulli theorem state that quintiles ρϕt+12ρϕ2+ρgz+Pshould be constant across the fluid, so that we have

ρ1ϕ1t+12ρ1ϕ12+ρ1gz0+P1z=z0=ρ2ϕ2t+12ρ2ϕ22+ρ2gz0+P2z=z0E9

To understand the interface, we must impose boundary conditions. First of all the vertical velocities of the fluids must match with the interface, so we impose the kinematic boundary condition. Now we need to introduce the location of the interface by assigning variable z=z0t. Then dz0dtwill represent the velocity of the interface in the z-direction. In addition, at the interface, the velocity of both fluids must be continuous.

dz0dt=t+υz0=ϕ1zz=z0=ϕ2zz=z0E10

Let us say that the pressure is continuous along the interface, that is P1=P2. Then Eq. (9) leads to

ρ1ϕ1t+12ρ1ϕ12+ρ1gz0=ρ2ϕ2t+12ρ2ϕ22+ρ2gz0E11

4. Asymptotic boundary conditions at far field

We are looking for changes only on the interface at z=z0, therefore the velocity potentials and their derivatives must vanish at the boundaries, that is ϕ10as zand ϕ20as z.

5. Linear analysis

Eq. (9) contains a nonlinear term ρ1ϕ12of the second order. If the amplitude is chosen to be much smaller than the wavelength of the instability, the equations of motion can be linearized. We assume that all the perturbed quantities and various derivatives such as ϕand ϕare very small. In other words ϕ1zz=z0ϕ1zz=0. Hence, the difference in second order derivatives will be much smaller. Now we impose all boundary conditions at z=0, which yields the following set of equations,

2ϕ1=0E12
2ϕ2=0E13
dz0dt=ϕ1zz=0=ϕ2zz=0E14
ρ1ϕ1t+ρ1gz0z=0=ρ2ϕ2t+ρ2gz0z=0E15

6. Eigenvalue solution

Let us consider all the perturbed variables ϕand z0to have oscillating behavior such that ϕ=ϕ0expikxωtand should satisfy the Laplace Eq. (12). This implies

2ϕ1z2=k2ϕ1E16

The general solution of Eq. (16) is written as

ϕ1z=Aexpkz+Bexpkz,E17

The above two solutions must satisfy the boundary conditions such that ϕ10as zand ϕ20as z. So, we need to discard the unsatisfactory part of the solutions of Eq. (16) taking into account the boundary conditions. Therefore, zdependence goes as ϕ1zAexpkzand ϕ2zBexpkz. We note that the eigenfunction decreases exponentially on either side of the interface and the perturbation of wave number kpenetrates to a depth of order 1k=λ2π.

This solution further leads to the following form in Fourier mode,

ϕ1xzt=ϕ01expkzexpikxωtE18
ϕ2xzt=ϕ20expkzexpikxωtE19
z0xt=z00expikxωtE20

Here ϕ01, ϕ02, and z00are the amplitude of the modes. By substituting these solutions into Eq. (14), we obtain the boundary conditions at the interface.

kϕ01=kϕ02=z00atz=0E21

Then Eq. (20) changes into the form

z0xt=ikϕ01ωexpikxωtE22

Eq. (15) gives

iωρ1ϕ10+ρ1gikϕ01ω=iωρ2ϕ20+ρ2gikϕ01ωE23

Using Eq. (21) in Eq. (23) results in

iωρ1ϕ10+ρ1gikϕ01ω=iωρ2ϕ10+ρ2gikϕ01ωE24

Since the perturbed quantity ϕ010, the possible nontrivial solution of Eq. (25) gives the dispersion relation for small perturbations of the wave as below,

ω2=gρ1ρ2kρ1+ρ2=AtgkE25

Eq. (25) contains complete information about the linear stability of the two superposed fluid layers of different densities. The Atwood number At=ρ1ρ2ρ1+ρ2is a dimensionless number in fluid dynamics used to study the hydrodynamic instabilities in unequal density flows. Since the dispersion relation Eq. (25) is quadratic in ω, it has two real or complex conjugate roots depending on the values of the densities of the fluids. Here, we will discussion different cases.

6.1 First case: capillary-gravity waves (ρ2=0)

Henceω=gkandVph=gkE26

It is classical dispersion relation for gravity-capillary waves in deep water [17, 18]. These are also called short gravity waves. In this category the longer waves travel faster. Any initial disturbance may be regarded as the superposition of waves of a broad spectrum of lengths. The above relation then says that waves of different lengths will eventually separate, that is, disperse. This phenomenon is called dispersion, hence above relations are also known as the dispersion relation.

6.2 Second case: propagating modes (ρ1>ρ2)

If the lighter fluid is supported by heavier fluid, that is, ρ1>ρ2, then solutions of the equation leads to two waves with constant amplitude propagating in opposite directions with phase velocity ω±kwith ω±=gρ1ρ2kρ1+ρ2. Then the interface is stable and will only oscillate when perturbed. The phase velocity is given by

Vph=ω±k=gρ1ρ21kρ1ρ2+1E27

Figure 2 shows the variations of phase velocity of RT instability with (a) density ratio and (b) wave number respectively.

Figure 2.

Variation of phase velocity of RT instability with (a) density ratio and (b) wave number respectively.

6.3 Third case: Rayleigh-Taylor instability (ρ2>ρ1)

The frequency of oscillations will be negative imaginary and unstable if ρ2>ρ1, that is, when heavier fluid is supported by lighter fluid. Writing ω=where γis real and positive gives

γ=±gρ2ρ1kρ1+ρ2E28
γ=±gρ2ρ11k1+ρ2ρ1E29

Substituting the value of ω=into Eq. (18), the amplitude grows exponentially with the perturbation and is given by

ϕ1xzt=ϕ01expkzexpikxexp±γtE30

The term expγtincreases the amplitude of the oscillation exponentially as time progress. Figure 3(a) and (b) shows the variations of growth rate of RT instability with (a) density ratio and (b) wave number. The inverse of the growth rate γ1=tcharis called the linear characteristic timescale of the RTI. In other words, characteristic timescale has to be the order of the lifetime of the plasma oscillations to observe the RT instability. On the other hand, if the linear characteristic timescale is much larger than the oscillation lifetime, the plasma instability would not be observed.

Figure 3.

Variation of growth rate of RT instability with (a) density ratio and (b) wave number respectively.

7. Rayleigh-Taylor instability in plasma thruster

In the previous section, the general idea of RT instability has been explored. Here we have derived the RT equation for a plasma fluid using two fluid theory. In a Hall thruster, the propellant (plasma) is ionized and then accelerated by electrostatic forces. It has high thrust resolution, so it is best suited for the adjustment of the location of the satellite onboard [19, 20, 21, 22, 23, 24, 25, 26, 27]. Let us consider a plasma with nonuniform density confined under the crossed electric and magnetic fields.

Figure 4 shows the typical diagram of a Hall plasma thruster [26]. RT instability is common in Hall thrusters. Studies show that Rayleigh instability is driven by the presence of gradients in axial density, magnetic field, and velocity of the plasma species. Here we deduce a Rayleigh equation under the presence of ion temperature and check the variations of perturbed potential with plasma parameters.

Figure 4.

Typical diagram of a Hall plasma thruster.

7.1 Theoretical model for RTI in plasma

We consider plasma comprising of ions and electrons immersed in a magnetic field B=Bẑ. The magnetic field is strong enough so that only electrons get magnetized, but the ions remain unaffected due to their Larmor radius being much larger than the dimension of the thruster. These trapped electrons (due to crossed fields) drift in azimuthal direction along the annular channel [24]. The applied electric field Eis along the x-axis (axis of the thruster) and the magnetic field Bis taken along the z-axis (along the radius of the thruster). Hence, the azimuthal dimension is along the y-axis. We use Ωz=eBmas the electron gyro frequency and u0=E0Bŷas the initial drift of the electrons [14, 15, 16, 17] and write the continuity equation and equation of motion for plasma species as

nit+υini=0E31
t+υiυi=eEMpiMniE32
net+υene=0E33
t+υeυe=ϕmυe×ΩzE34

We use the linearized form of the above equations for small perturbations of the ion and electron densities, their velocities, and electric field. We write perturbed densities of ions (electrons) by ni1(ne1) velocities by υi1(υe1). The unperturbed electrons’ drift is u0in the y-direction. The unperturbed density (electric field) is taken as n0E0and the perturbed value of the electric field is taken as E(corresponding potential ϕ). Hence, the linearized form of Eqs. (31)(34) reads

ni1t+υix1n0x+n0υi1=0E35
υi1t=eMEpi1n0ME36
ne1t+u0ne1y+n0υe1+υex1n0x=0E37
υe1t+u0υe1y=ϕυe1×ΩzE38

The unperturbed ions’ velocity υ0is taken zero here for the case of simplification. We are looking for oscillating solution of the above equations that should vary as f=f0expiωtikyy. The ion thermal velocity can be written as V2thi=YiTiM. With the help of Eqs. (35) and (36) we obtain the following expression for the perturbed ion density in terms of the perturbed electric potential ϕ:

ni1=en0Mω2V2thiky2ky2ϕ2ϕx2YiTien02n0x2E39

Eq. (38) provides the velocity components of electron

iωkyu0υex1=emϕxΩzυey1E40
iωkyu0υey1+υex1u0x=ikyemϕ+Ωzυex1E41

In the above equations, the coordinate xlies in the interval 0<x<d, where dis the channel length. Let us define ωkyu0by ω̂in the above set of expressions. Further we readily obtain from the above equations

υex1=emiω̂ϕx+emikyΩzϕΩz2ω̂2Ωzu0xE42
υey1=emΩzϕx+emω̂2ϕx+emω̂kyΩzϕΩzΩz2ω̂2Ωzu0xE43

The electron cyclotron frequency is almost Ωz108/s (corresponding to 200 Gauss magnetic field). Generally, Ωzis much larger than the frequency of the oscillations. Therefore, under the condition Ωz>>ω,kyu0,u0x, the velocity components of electrons are reduced into the form

υx1=iemΩz2ω̂ϕx+Ωzkyϕ+u0xkyϕE44
υy1=emΩzϕx+kyω̂ϕΩz+kyω̂ϕΩz2u0xE45

The electron continuity equation gives the perturbed electron density newith the help of Eqs. (44) and (45)

ne1=en0mΩz2ky2ϕ2ϕx2+kyωkyu0ΩzxlnBn02u0x2ϕE46

The plasma frequency of oscillations for ion (electron) is defined as

ωiωe=n0e2Mmε0E47
The Poissonsequationε02ϕ=ene1ni1E48

Using Eqs. (39) and (46) in Eq. (48) gives the perturbed potential in the following form:

2ϕx2ky2ϕkyϕΩzxlnBn02u0x2ωkyu01+Ωz2/ωe2Ωz2ωi2ωe2ω2ky2V2thi=0E49

In the case of high frequency of oscillations and in the absence of ion thermal pressure, Eq. (49) turns into Rayleigh’s equation of fluid dynamics as below

2ϕx2ky2ϕ+ϕkyωkyVy2Vyx2=0E50

Here Vyis the flow velocity in the y-direction and ϕis called the flow function related to Vy=ϕ. The analytical eigenvalue solution of Eq. (49) is given in Ref. [12].

Resonance condition for the RT instability

From Eq. (49), it is clear that propagating mode may lead to instability if parameter ΩzxlnBn02u0x2=0at some point inside the Hall thruster.

7.2 Variations of perturbed potential

The RT Eq. (49) is solved numerically for the perturbed potential ϕalong with the boundary conditions such that ϕ0=ϕd=0. We plot perturbed potential of the instability with magnetic field B, initial drift of the electrons u0, channel length d, and ion temperature Ti. These parameters can have values as B=100250G, n0=5×10171018/m3, Ti=015eV, and u0106m/s [13, 15].

Figure 5 shows the variation of the perturbed potential with the magnetic field and it has been observed that the potential increases with the increasing magnetic field. These results are consistent with Keidar and Boyd model [28] and that other investigators [13, 14] for the potential of plasma plume. This situation is correspond to the plasma jet enters a transverse magnetic field with a high velocity under the condition that the magnetic field is relatively weak so that only the electrons are magnetized whereas the ions move out of the effect of magnetic field. However, ambipolar (both electrons and ions moving in opposite directions) plasma flow across the magnetic field may require an electric field to appear under the above conditions. Therefore, we can expect the potential to increase across the magnetic field.

Figure 5.

Effect of magnetic field on the perturbed potential ϕ .

The perturbed potential gets increased with the higher value of electron’s initial drift velocity (shown in Figure 6 ). Similar behavior of the potential was reported experimentally by King et al. [29] for the potential of plasma plume. Similar results are also reported in Refs. [13, 14]. The enhanced perturbed potential ϕwith the ion temperature is shown in Figure 7 which is consistent with an experiment [30].

Figure 6.

Dependence of perturbed potential ϕ on the drift velocity of the electrons.

Figure 7.

Variation of perturbed potential ϕ with the ion temperature.

8. Discussions and summary

In conclusion, we can say that short-wavelength perturbations blow up exponentially much more quickly in RTI. The primary source by which this instability is triggered is the gravitational force acting on an inverted density gradient (e.g., a heavy fluid supported by a light fluid). Stable and steady flows may become unstable depending on the ranges of the flow parameters. The instability takes free energy from the mean flow or externally supplied heat and the amplitude of waves grows exponentially. The instabilities exist in all natural and artificial phenomena (in smoke from chimneys, in rivers, in flickering flames) and their effects result in turbulence or random waves. The presence of plasma density and magnetic field gradients is one of the main sources for plasma instabilities in Hall thrusters. It is found that perturbed potential increases with the higher value of electrons’ drift velocity, magnetic field, and ion temperature.

Acknowledgments

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

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Sukhmander Singh (April 15th 2020). Dynamics of Rayleigh-Taylor Instability in Plasma Fluids [Online First], IntechOpen, DOI: 10.5772/intechopen.92025. Available from:

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