Rayleigh-Taylor instability

The Rayleigh-Taylor instability occurs when a denser fluid lies above a lighter fluid in a gravitational field, causing interface perturbations to grow.

The dimensionless density contrast $({\rho }_h-{\rho }_l)/({\rho }_h+{\rho }_l)$, conventionally using heavy and light densities. In the inviscid, zero-surface-tension heavy-over-light Rayleigh-Taylor instability, the growth rate is $s=\sqrt{\mathcal{A}gk}$. State the signed ordering if labels are not heavy/light.

For a deep inviscid liquid above a gas of negligible inertia, an inverted free surface has dispersion relation ${\omega }^2=q(\sigma q^2/\rho -g)$, $q=|k|$. The Rayleigh-Taylor instability affects $0<q<k_c=\sqrt{\rho g/\sigma }$. A bounded aperture can suppress it if its smallest allowed wavenumber is at least $k_c$. The capillary length is ${\ell }_c=k_c^{-1}$ and the marginal wavelength is $2\pi {\ell }_c$. A pinned one-dimensional span of width $L$ with a half-sine lowest mode instead has $k_{min}=\pi /L$, giving $L\le \pi {\ell }_c$. A numerical maximum container size therefore also requires aperture geometry and contact-line boundary conditions; the marginal wavelength alone is not a universal diameter.

Consider uniform fields $B_{-}\hat{\mathbf{y}}$ below and $B_{+}\hat{\mathbf{x}}$ above an interface, with local constant densities ${\rho }_{-},{\rho }_{+}$. For ideal magnetohydrodynamics and local incompressible potential flow, the interface displacement $h{e}^{st+ikx+imy}$ has displacement potentials ${\chi }_{-}=h{e}^{Kz}/K$ and ${\chi }_{+}=-h{e}^{-Kz}/K$, $K=\sqrt{k^2+m^2}$. Magnetic flux freezing gives ${\mathbf{b}}_i=({\mathbf{B}}_i\cdot \nabla )\nabla {\chi }_i$. The linear magnetohydrodynamic total pressure is $-[{\rho }_i{s}^2+({\mathbf{k}}_h\cdot {\mathbf{B}}_i{)}^2/{\mu }_0]{\chi }_i$. Matching Lagrangian total pressure gives the displayed dispersion relation. For ${\rho }_{+}>{\rho }_{-}$ the most unstable orientation has its wavevector along the weaker field.

// Target: continuum-mechanics.bigb

For the density jump of a magnetic pressure discontinuity in an isothermal atmosphere, with $0<|B_{+}|<|B_{-}|$, the fastest Rayleigh-Taylor instability has $m=0$ and $|k|=K_{\ast }=(B_{-}^2-B_{+}^2)/(4B_{+}^{2}H)$. Its vertical decay length is $K_{\ast }^{-1}$. Hence localization relative to both common scale heights requires $4B_{+}^2\ll B_{-}^2-B_{+}^2$. This also makes its growth speed small compared with the isothermal sound speed. If $B_{+}=0$, the ideal sharp-interface dispersion relation instead has unbounded growth as $|k|\to \infty$: there is no finite maximizing mode.

A dense viscous crust of thickness $a$ above lighter liquid can overturn on a Rayleigh-Taylor instability time ${\mu }_s/(\Delta \rho ga)$. If diffusive growth gives $a=A\sqrt{\kappa t}$, matching its age to that time gives ${\tau }_s\sim [{\mu }_s/(\Delta \rho gA\sqrt{\kappa }){]}^{2/3}$. A renewed crust then supplies average solid-volume flux $J_s\sim (\Delta \rho g/{\mu }_s{)}^{1/3}(A^2\kappa {)}^{2/3}$. This scale assumes that heat supplied by the interior does not arrest the crust before instability.

For two resting incompressible inviscid layers between impermeable horizontal walls, the upper layer has density ${\rho }_1$ and depth $L_1$, and the lower layer has ${\rho }_2<{\rho }_1$ and depth $L_2$. Solving the Laplace equation for the velocity potential and imposing the linearized kinematic and capillary pressure conditions gives the displayed dispersion relation. With positive surface tension $\gamma$, exponential Rayleigh-Taylor instability occurs for $0<k<k_c$, where $k_c=\sqrt{g({\rho }_1-{\rho }_2)/\gamma }$. The depths change the growth rate, but not this cutoff.

For an infinitely deep lower layer and $k_c{L}_1\ll 1$, write $A=g({\rho }_1-{\rho }_2)$. Uniformly over the unstable band, the finite-depth Rayleigh-Taylor dispersion relation gives ${\sigma }^2\sim (L_1/{\rho }_1)(A{k}^2-\gamma k^4)$. Its maximum occurs at $k_m\sim k_c/\sqrt{2}$, with the displayed growth rate. In two deep layers the maximum instead occurs at $k_c/\sqrt{3}$ and obeys ${\sigma }_m^2=2A{k}_c/[3\sqrt{3}({\rho }_1+{\rho }_2)]$. Thus confinement strongly reduces growth while leaving the cutoff unchanged. These thin-layer expressions are leading asymptotics, not exact finite-depth identities.