# The Rayleigh-Taylor instability#

This section was contributed by Cedric Thieulot.

This benchmark is carried out in Deubelbeiss and Kaus [2008], Gerya [2010], Thieulot [2011]) and is based on the analytical solution by Ramberg , which consists of a gravitationally unstable two-layer system. Free slip are imposed on the sides while no-slip boundary conditions are imposed on the top and the bottom of the box. Fluid 1 $$(\rho_1,\eta_1)$$ of thickness $$h_1$$ overlays fluid 2 $$(\rho_2,\eta_2)$$ of thickness $$h_2$$ (with $$h_1+h_2=L_y$$). An initial sinusoidal disturbance of the interface between these layers is introduced and is characterised by an amplitude $$\Delta$$ and a wavelength $$\lambda=L_x/2$$ as shown in Fig. 136.

Under this condition, the velocity of the diapiric growth $$v_y$$ is given by the relation

$\frac{v_y}{\Delta} = - K \frac{\rho_1-\rho_2}{2 \eta_2} h_2 g \qquad \qquad \text{with} \qquad \qquad K=\frac{-d_{12}}{c_{11}j_{22}-d_{12}i_{21}}$

where $$K$$ is the dimensionless growth factor and

$\begin{split}c_{11} &= \frac{\eta_1 2 \phi_1^2}{\eta_2(\cosh 2\phi_1 - 1 - 2\phi_1^2)} - \frac{2\phi_2^2}{\cosh 2\phi_2 - 1 - 2 \phi_2^2}\\ d_{12} &= \frac{\eta_1(\sinh 2\phi_1 -2\phi_1)}{\eta_2(\cosh 2\phi_1 -1 -2\phi_1^2)} + \frac{\sinh 2\phi_2 - 2\phi_2}{\cosh 2\phi_2 -1 -2\phi_2^2} \\ i_{21} &= \frac{\eta_1\phi_2 (\sinh 2 \phi_1 + 2 \phi_1)}{\eta_2(\cosh 2\phi_1 -1 -2\phi_1^2)} + \frac{\phi_2 (\sinh 2\phi_2 + 2\phi_2)}{\cosh 2\phi_2 -1 -2\phi_2^2} \\ j_{22} &= \frac{\eta_1 2 \phi_1^2 \phi_2}{\eta_2(\cosh 2\phi_1 -1-2\phi_1^2)} - \frac{2\phi_2^3}{ \cosh 2\phi_2 -1 -2\phi_2^2}\\ \phi_1&=\frac{2\pi h_1}{\lambda} \\ \phi_2&=\frac{2\pi h_2}{\lambda}\end{split}$

We set $$L_x=L_y=\text{ 512 km}$$, $$h_1=h_2=\text{ 256 km}$$, $$|\boldsymbol{g}|=\text{10 m/s^2}$$, $$\Delta=\text{3 km}$$, $$\rho_1=\text{3300 kg/m^3}$$, $$\rho_2=\text{3000 kg/m^3}$$, $$\eta_1=\text{1e21 Pa.s}$$. $$\eta_2$$ is varied between $$10^{20}$$ and $$10^{23}$$ and 3 values of $$\lambda$$ (64, 128, and 256km) are used. Adaptive mesh refinement based on density is used to capture the interface between the two fluids, as shown in Fig. 137 and Fig. 138. This translates as follows in the input file:

subsection Mesh refinement
set Initial global refinement = 4
set Initial adaptive refinement = 6
set Strategy = density
set Refinement fraction = 0.6
end


Description of benchmark files

The maximum vertical velocity is plotted against $$\phi_1$$ in Fig. 139 and is found to match analytical results.