# 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(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 Figure 1.

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{aligned} 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{aligned}$$$We set$$L_x=L_y=\SI{512}{km}$$,$$h_1=h_2=\SI{256}{km}$$,$$|\boldsymbol{g}|=\SI{10}{m/s^2}$$,$$\Delta=\SI{3}{km}$$,$$\rho_1=\SI{3300}{kg/m^3}$$,$$\rho_2=\SI{3000}{kg/m^3}$$,$$\eta_1=\SI{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 Figure 3. 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

The maximum vertical velocity is plotted against $$\phi_1$$ in Figure 4 and is found to match analytical results.

Deubelbeiss, Y., and B. J. P. Kaus. 2008. “Comparison of Eulerian and Lagrangian Numerical Techniques for the Stokes Equations in the Presence of Strongly Varying Viscosity.” Physics of the Earth and Planetary Interiors 171: 92–111.

Gerya, T. 2010. Introduction to Numerical Geodynamic Modelling. Cambridge University Press.

Ramberg, Hans. 1968. “Instability of Layered Systems in the Field of Gravity.” Phys. Earth Planet. Interiors 1: 427–47.

Thieulot, C. 2011. “FANTOM: two- and three-dimensional numerical modelling of creeping flows for the solution of geological problems.” Phys. Earth. Planet. Inter. 188: 47–68.