Rayleigh-Taylor of Gravitational instability

The Rayleigh-Taylor instability is a fluid instability that occurs at the interface between two fluids of different densities when the heavier fluid is supported by the lighter fluid. This instability can also occur in the context of gravitational forces.

In the case of gravitational instability, the Rayleigh-Taylor instability arises when there is a gravitational field acting on the interface between two fluids with different densities. The heavier fluid is located above the lighter fluid, and the gravitational field exerts a downward force on the heavier fluid and an upward force on the lighter fluid.

As a result of these opposing forces, small perturbations or disturbances at the interface can grow and lead to the mixing of the two fluids. The heavier fluid tends to sink into the lighter fluid, and the lighter fluid rises through the heavier fluid. This process can lead to the formation of complex structures and patterns, such as bubbles or spikes, depending on the initial conditions.

The Rayleigh-Taylor instability due to gravitational forces has applications in various fields, including astrophysics, geophysics, and engineering. It plays a role in the dynamics of supernovae explosions, where the explosion shockwave interacts with the stellar material, leading to the mixing of different layers. It is also relevant in the study of oceanic and atmospheric phenomena, such as the formation of clouds and the mixing of water masses.

Understanding and characterizing the Rayleigh-Taylor instability in the context of gravitational forces is essential for predicting and modeling various natural and engineered systems where fluid interfaces are subject to gravitational effects.

In hydrodynamics, Rayleigh Taylor instability occurs when heavy fluid is supported by light fluid. In plasma, the magnetic field acts as light fluid supporting heavy fluid (the plasma). In curved magnetic fields, the centrifugal force may arises die to particles motion along the curved lines of force. The centrifugal force is balanced by gravitational force.

Consider a plasma boundary lying in the y z plane. let KTi = KTe =0 and Bo uniform. In equilibrium, the ion obey the equation

For constant vo, taking cross product with Bo,

The electron have an opposite drift which can be neglected in the limit m/M -> 0

If a ripple should develop in the interface as the result of random thermal function the drift vo will cause the ripple to grow.

The ripple grows as a result of properly phased E1 x Bo drifts.

To find the growth rate, we can perform the usual linearized wave analysis for wave propagating in the y direction k = ky. There is no diamagnetic drift since KT = 0

The equation of motion for ions is

For the electrons (in the limit m/M ->0)

Vex = Ey/Bo and Vey = 0 ......10

The perturbed equation of continuity for ion is

Using 8, 9 in 11 we get

Using 10 in 12

Using 14 in 13

This is the equation for w.

There is instability if

k²Vo²/ 4 + gno'/no < 0

=> - g no'/no > 1/4 k²Vo² .......18

Hence, the instability requires g and no'/no to have opposite sign. The left-hand side of the equation represents the gravitational restoring force, which tends to stabilize the interface. The right-hand side represents the destabilizing effect caused by the inertia associated with the perturbation.

If the inequality holds, i.e., the left-hand side is greater than the right-hand side, the Rayleigh-Taylor instability can occur, and the interface between the two fluids becomes unstable, leading to the growth of perturbations. If the inequality is not satisfied, the interface remains stable, and the perturbations do not grow significantly.

It's important to note that this is a simplified criterion, assuming idealized conditions, and may not capture all the complexities of the Rayleigh-Taylor instability. Real-world situations may involve additional factors, such as surface tension, viscosity, and compressibility, which can influence the stability and modify the criterion accordingly.

=> The light fluid supporting heavy fluid otherwise w is real and plasma is stable

Re (w) = 1/2 kVo ....19

= Im (w) = √ - g no'/ no .....20

This note is a part of the Physics Repository.