# The Rayleigh-Taylor instability *This section was contributed by Cedric Thieulot.* This benchmark is carried out in {cite:t}deubelbeiss:kaus:2008,gerya:2010,thieulot:2011) and is based on the analytical solution by Ramberg {cite}ramberg:1968, 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 {numref}fig:RTi_setup. {figure-md} fig:RTi_setup Setup of the Rayleigh-Taylor instability benchmark (taken from {cite:t}thieulot:2011)  Under this condition, the velocity of the diapiric growth $v_y$ is given by the relation {math} \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 {math} 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}  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 {numref}fig:RTi_grids_a and {numref}fig:RTi_grids_b. 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](../README.md)** {figure-md} fig:RTi_grids_a Grid with initial global refinement 4 and adaptive refinement 6.  {figure-md} fig:RTi_grids_b Density field with detail of the mesh.  The maximum vertical velocity is plotted against $\phi_1$ in {numref}fig:RTi_vels and is found to match analytical results. {figure-md} fig:RTi_vels Maximum velocity for three values of the \phi_1 parameter.  :::{toctree} ../README.md :::