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The RT instability occurs when a dense liquid is laid on top of a tenuous fluid. The more exact mathematical condition is that the density gradient is opposite to the gravitational force.

 \mathbf{g} \cdot \nabla \rho < 0

In the next sections we will develop the growth rate in the linear stage.

Equation of continuity

 \nabla \mathbf{v} = 0

Conservation of momentum

 \rho \frac{\partial \mathbf{v}}{\partial t} + \rho \mathbf{v} \cdot \nabla \mathbf{v} + \nabla P = \mathbf{g}

We can simplify the equation by replacing the velocity by a flow potential  \mathbf{v} = \nabla \varphi . Substituting in the equations of motion yields

 \nabla^2 \varphi = 0

 \nabla \left(\rho \frac{\partial \varphi}{\partial t} + \frac{1}{2} \rho \left(\nabla \varphi \right )^2 + P - \rho \mathbf{g} \cdot \mathbf{r} \right ) = 0

We assume an initial configuration where a heavy fluid of density  \rho_u is on top  y>0 of a lighter fluid of density  \rho_d (which occupies the region  y<0 .

Next, we introduce a perturbed flow potential

 \varphi = \delta \varphi_{u,d} \exp \left[ i \left(t \omega - k x + i q |y| \right) \right]

From the continuity of the normal velocity on the interface between the two fluids  \delta \varphi_u = - \delta \varphi_d . The Bernouli invariant should also be continuous across the interface

 \left[\rho \frac{\partial \varphi}{\partial t} + \frac{1}{2} \rho \left(\nabla \varphi \right )^2 + P - \rho g y \right] = 0

where the square brackets denote difference across the interface. The pressure is continuous across the interface, so it vanishes. The kinetic energy term also vanish because it is of a higher order in the perturbation. The position of the interface is

 y = \frac{1}{i \omega} \frac{\partial \varphi}{\partial y}

We are thus left with a dispersion equation

 \omega^2 = -k g \frac{\rho_u - \rho_d}{\rho_u + \rho_d}

Since the frequency is complex, the amplitude of this mode grows exponentially if a heavy fluid is on top, and decays if the lighter fluid is on top.

There is a more intuitive way to understand the this growth rate. Let us focus on a single cell of size  1/k near the interface between the two fluids. Let us denote by  z the position of the interface. The entire cell can be thought of as moving as a single body whose mass is  \left(\rho_u + \rho_d \right) k^{-2} L , where $L$ is in dimension along the z axis. The force driving this cells is given by Archimedes' buoyancy force  g \left(\rho_u - \rho_d\right) k^{-1} L z , so the equation of motion for the cell is  \left(\rho_u + \rho_d\right) k^{-2} L \ddot{z} = g \left(\rho_u-\rho_d\right) k^{-1} L z

and the growth rate is  \frac{z}{\ddot{z}} = \frac{g k \left(\rho_u - \rho_d\right)}{\rho_u+\rho_d}

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