Compressibility

MILET has two ways to treat the compressibility of rock:

• Boussinesq (formulation = "boussinesq", the default). The flow is incompressible, $\mathrm{\nabla }\cdot \mathbf{u}=0$, and compressibility enters only through the buoyancy. Each marker's density follows the rock's equation of state at lithostatic pressure,

  $$\rho ={\rho }_0(1-\alpha (T-T_{\mathrm{ref}}))(1+\beta (P-P_{\mathrm{ref}})),$$

  with the per-rock keys reference_density (${\rho }_0$), thermal_expansion ($\alpha$) and compressibility ($\beta$, 1/Pa). The energy equation uses the constant rho_ref_heat, and there is no adiabatic heating unless do_adiabatic is set.

• TALA (formulation = "tala"). One adiabatic reference state is used in the mass, momentum and energy equations, as described below. This is the consistent compressible option for convection with a significant dissipation number.

ALA, the full anelastic approximation with a dynamic-pressure density term, is not implemented. formulation = "ala" stops with an error.

TALA

formulation = "tala" (in [rheology]) replaces the Boussinesq equations with the truncated anelastic liquid approximation of King et al. (2010). One adiabatic reference state (src/physics/ReferenceState.jl), with $y$ the depth,

$$\overline{\rho }(y)={\rho }_0\exp (\frac{{\alpha }_{r}gy}{{\gamma }_r{c}_p}),\overline{T}(y)=T_0\exp (\frac{{\alpha }_{r}gy}{c_p}),$$

from the keys ref_rho0, ref_T0, ref_alpha, ref_cp, ref_gamma, is used in all three equations:

$$\mathrm{\nabla }\cdot (\overline{\rho }\mathbf{u})=0,\mathrm{\nabla }\cdot (2\eta \dot{\epsilon }-\frac{2}{3}\eta (\mathrm{\nabla }\cdot \mathbf{u})\mathbf{I})-\mathrm{\nabla }p=\overline{\rho }\alpha (T-\overline{T})\mathbf{g},$$$$\overline{\rho }{c}_p\frac{DT}{Dt}=\mathrm{\nabla }\cdot (k\mathrm{\nabla }T)+\overline{\rho }\alpha g{u}_{y}T+\tau :\dot{\epsilon }+H.$$

• Mass: build_stokes_forms tests $-q(\overline{\rho }\mathrm{\nabla }\cdot \mathbf{u}+\mathbf{u}\cdot \mathrm{\nabla }\overline{\rho })$, with $\mathrm{\nabla }\overline{\rho }$ exact.

• Buoyancy: the marker density is taken about the reference state, $\rho =(\overline{\rho }/{\rho }_0){\rho }_{\mathrm{rock}}(T;T_{\mathrm{ref}}=\overline{T}(y),P=0)$. A reference-rock marker on the adiabat therefore has exactly $\overline{\rho }(y)$, and the dynamic-pressure contribution to density is dropped (TALA).

• Energy: $\rho c_p$ uses $\overline{\rho }(y)$ in place of rho_ref_heat. Adiabatic heating $\overline{\rho }\alpha g{u}_{y}T$ belongs to the formulation and is always on. Shear heating is the full deviatoric $2\eta (\epsilon :\epsilon -\frac{1}{3}(\mathrm{\nabla }\cdot \mathbf{u}{)}^2)$ at the quadrature points.

The dissipation number of a box of depth $H$ is $Di={\alpha }_{r}gH/c_p$. Verification against King et al. (2010) is in Benchmarks. For planetary models, set ref_T0 to the mantle potential temperature and T_bot to $\overline{T}$ at the box floor. The [[init_T]] profiles hsc_adiabat and steady_lid build lids on that adiabat (see Parameter file).

Reference-state keys

Key	Default	Meaning
formulation	"boussinesq"	"tala" switches on everything on this page
ref_T0	T_top	$T_0$, reference temperature at $y=0$ (K)
ref_rho0	rock 1 reference_density	${\rho }_0$, reference density at $y=0$ (kg/m³)
ref_alpha	rock 1 thermal_expansion	${\alpha }_r$ (1/K)
ref_cp	rock 1 specific_heat	$c_p$ (J/kg/K)
ref_gamma	1.0	Grüneisen parameter ${\gamma }_r$

The keys can go in any section; the bundled cases put them under [physics]. A planetary model typically looks like this:

[rheology]
formulation = "tala"

[physics]
ref_T0    = 1600.0     # mantle potential temperature (K)
ref_rho0  = 3300.0
ref_alpha = 3e-5
ref_cp    = 1250.0
ref_gamma = 1.2

[temperature]
T_bot = 1869.0         # T0 exp(alpha g H / cp) at the floor (g = 9.81, H = 660 km)

Source

src/physics/ReferenceState.jl (reference state, marker density, $\rho$ in the energy equation), src/physics/Stokes.jl (mass and compressible stress), src/physics/Gravity.jl (buoyancy) and src/physics/HeatSources.jl (adiabatic and shear heating).

Reference

King, S. D., Lee, C., van Keken, P. E., Leng, W., Zhong, S., Tan, E., Tosi, N., & Kameyama, M. C. (2010). A community benchmark for 2-D Cartesian compressible convection in the Earth's mantle. Geophys. J. Int., 180, 73–87.