Rayleigh-Taylor instabilities in stratified fluids

We study the growth of Rayleigh-Taylor instabilities at the interfaces of any number $N$ of stratified fluids forming an arbitrary density profile. Using the linear theory, we show that there are $2(N-1\mathrm{})\mathrm{}$ exponential growth rates which can be found by calculating the eigenvalues of an $\mathrm{}(N-1\mathrm{})\mathrm{}\times \mathrm{}(N-1\mathrm{})\mathrm{}$ band matrix. We illustrate analytically the case $N=3\mathrm{}$. For general $N$ we state and outline the proofs of two theorems: the first refers to the invariance of the spectrum of growth modes under inversion (${\rho }_i\to \frac{1}{{\rho }_{N+1-i}}$, $t_i\to t_{N+1-i}$), and the second relates the spectrum of any inversion-invariant density profile having free boundaries to the spectrum of the same profile between fixed boundaries. We compare and illustrate the results of our numerical code with the case of a continuous density profile $\rho ={\rho }_0{e}^{\beta y}$, and, setting $N=12\mathrm{}$, we apply our technique to solving a particular problem in designing a multishell target for intertial confinement fusion.