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Use superscripts $T$ and $D$ for the thin-upper-layer and two-deep-layer cases. The cutoff is identical, whileThus confinement suppresses the Rayleigh-Taylor instability growth strongly, but moves the most amplified disturbance to a somewhat larger wavenumber. The cutoff depends on the gravity-capillary balance, whereas growth depends also on the inertia of the moving layers.

Let the perturbed interface be $z=\eta (x,y,t)$. For inviscid disturbances about rest, the initially irrotational velocities remain potential flows,in the upper layer $0<z<h$ and lower layer $-h<z<0$. The rigid walls impose no-penetration boundary conditionand zero normal derivative at $x,y=0,2h$. Linearizing the kinematic and pressure-continuity conditions at $z=0$ gives

The horizontal Neumann eigenfunctions arewhere $m,n$ are nonnegative integers, not both zero. For time dependence $e^{-i\omega t}$, the vertical factors satisfying the rigid lids areThe kinematic condition gives their interface valuesSubstitution into the dynamic condition yields the interfacial gravity-wave dispersion relationFor ${\rho }_2>{\rho }_1$, the lowest nontrivial wavenumber is $k=\pi /(2h)$, with the two degenerate modes $(m,n)=(1,0)$ and $(0,1)$. If ${\rho }_1\ll {\rho }_2$, the relation reduces to the finite-depth free-surface result ${\omega }^2\simeq gk\tanh (kh)$. If ${\rho }_1>{\rho }_2$, then ${\omega }^2<0$ and disturbances grow exponentially: this is the Rayleigh-Taylor instability.

Let the unperturbed interface rise at pore velocity $U/\varphi$, and write its displacement asIn each fluid, Darcy's law and incompressible flow implyThe decaying pressure perturbations are proportional to $e^{-\alpha |z|}$. Continuity of normal velocity, the kinematic boundary condition $\varphi {\eta }_t=u_n^{\prime }$, and continuity of pressure giveThus a less mobile displaced fluid, ${\mu }_2>{\mu }_1$, and a denser fluid above a lighter one both drive the Saffman–Taylor instability.

For immiscible fluids, the Young–Laplace equation adds the pressure jump $-\gamma {\eta }_{xx}$. The dispersion relation becomeswhereIf $A>0$, the unstable band is $0<\alpha <\sqrt{A/C}$ and differentiation gives

Define the signed characteristic buoyancy velocitySince ${\mu }_2/({\mu }_1+{\mu }_2)=(1+M)/2$ and $Ca={\mu }_{2}U/\gamma$,This expression applies when the quantity in the final parentheses is positive.

Every growth curve starts at the origin. For equal densities its initial slope is proportional to $M$; for ${\rho }_2>{\rho }_1$ buoyancy shifts that slope upward, while for ${\rho }_2<{\rho }_1$ it shifts it downward. When $A>0$, the curve rises to one positive maximum and then crosses zero before its stabilizing $-{\alpha }^3$ capillary tail. When $A\le 0$, every nonzero wavenumber decays.

For a prescribed nonzero wavenumber, neutral stability requiresorIn the quasistatic limit $U\to 0$, viscosity contrast disappears and this reduces to the capillary Rayleigh-Taylor instability threshold ${\rho }_2-{\rho }_1=\gamma {\alpha }^2/g$. Without surface tension, neutral stability in that limit simply requires equal densities.

Irrotationality gives ${\mathbf{u}}_i=\nabla {\varphi }_i$ and ${\nabla }^2{\varphi }_i=0$. At $z=\zeta$, the exact kinematic conditions areand pressure continuity gives $p_1=p_2$. Bernoulli equation in each fluid is

Linearize at $z=0$ and write $\zeta =Z{e}^{i(kx-\omega t)}$. Decay at infinity givesThe kinematic conditions give $-i\omega Z=-k{A}_1=k{A}_2$, while pressure continuity givesElimination yieldsStable waves require the lower fluid to be denser, ${\rho }_2>{\rho }_1$; otherwise this is the Rayleigh-Taylor instability.

A material interface obeys ${\eta }_t+{\varphi }_{j,x}{\eta }_x={\varphi }_{j,y}$ at its actual height. Keeping only first-order terms and evaluating at the undisturbed interface gives ${\varphi }_{1,y}={\varphi }_{2,y}={\eta }_t$ at $y=0$. Linearized unsteady Bernoulli gives the pressure perturbation $-{\rho }_j({\varphi }_{j,t}+g\eta )$, after absorbing time-only constants into the potentials. Pressure continuity therefore givesFor $\vartheta =kx-\omega t$, choose potentials decaying away from the interface:They are harmonic and their normal derivatives equal $A\omega \sin \vartheta ={\eta }_t$ at zero. The dynamic condition reduces toIf the denser fluid is above, the squared frequency is negative. The corresponding perturbations grow or decay exponentially instead of oscillating: this is Rayleigh-Taylor instability, with growing rate $\sqrt{gk({\rho }_2-{\rho }_1)/({\rho }_1+{\rho }_2)}$ in the ideal model without surface tension.

For potential flow, each velocity potential satisfies Laplace's equation in its fluid. The exact kinematic boundary condition on $y=\eta$ is ${\eta }_t+{\varphi }_{j,x}{\eta }_x={\varphi }_{j,y}$. Keeping only first-order terms givesThe linearized Bernoulli equation is $p_j=p_0-{\rho }_j({\varphi }_{j,t}+g\eta )$ at the interface, so pressure continuity givesFor a decaying disturbance away from the interface, put $\psi =kx-\omega t$ and takeThey satisfy Laplace's equation and the kinematic boundary conditions for $\eta =A\cos \psi$. Substituting in the dynamic condition gives the dispersion relationIf the heavier fluid is on top, ${\omega }^2<0$. Instead of oscillation, a Fourier disturbance has an exponentially growing component with growth rate $\sqrt{gk({\rho }_2-{\rho }_1)/({\rho }_1+{\rho }_2)}$. This is the Rayleigh-Taylor instability in the absence of surface tension.

For two inviscid incompressible half-spaces, lower density/velocity ${\rho }_1,U_1$, upper density/velocity ${\rho }_2,U_2$, downward gravity $g$ and surface tension $\sigma$, a mode $e^{i(kx-\omega t)}$ with $k>0$ obeysA negative right-hand side produces exponential growth. The gravitational term recovers Rayleigh-Taylor instability for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, viscosity, compressibility and magnetic tension lie outside this interface model.

A fluid has unstable density stratification when a vertical displacement amplifies itself. Denser fluid above lighter fluid can overturn by Rayleigh-Taylor instability or develop thermal convection when the density difference is thermal.

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.