‍Incomplete. This page gives an overview of the core equations. Several physically important terms are not yet included, and this list is not exhaustive:

Energy equation: adiabatic heating/cooling ( \(\propto D_i\,\alpha\,T\,u_r\)), viscous (shear) dissipation ( \(\propto D_i/\mathrm{Ra}\;\boldsymbol{\tau}:\boldsymbol{\varepsilon}\)), latent heat, and compositional transport.

Stokes equation: full compressibility model (anelastic liquid approximation), non-linear and non-Newtonian rheology, inertia.

Coupling: the buoyancy forcing is currently purely thermal; chemical buoyancy and phase transitions are absent.

Non-dimensionalisation: the choice of scales and the resulting dimensionless parameters are not documented here.

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The mantle convection model

Mantle convection is modelled as the slow creep of a highly viscous fluid in a thick spherical shell. The governing equations couple an instantaneous force balance (Stokes) to a time-dependent heat transport equation (energy / advection-diffusion).

Stokes equations

‍See the Stokes Discretization and Solver chapter for the full discretization, solver, and parameter guidance.

The momentum balance and mass conservation read

\[ \begin{aligned} -\nabla \cdot \boldsymbol{\tau}(\mathbf{u}) + \nabla p &= \mathbf{f}, \\ -\nabla \cdot \mathbf{u} &= g, \end{aligned} \]

where

\[ \boldsymbol{\tau}(\mathbf{u}) = 2\eta\!\left(\boldsymbol{\varepsilon}(\mathbf{u}) - \tfrac{1}{3}(\nabla\cdot\mathbf{u})\,\mathbf{I}\right), \qquad \boldsymbol{\varepsilon}(\mathbf{u}) = \tfrac{1}{2}\!\left(\nabla\mathbf{u} + (\nabla\mathbf{u})^T\right). \]

Variables and parameters:

\(\mathbf{u}\) — velocity,
\(p\) — dynamic pressure,
\(\eta > 0\) — dynamic viscosity (spatially varying),
\(\mathbf{f}\) — body force. In the Boussinesq approximation this is the thermal buoyancy \(\mathbf{f} = \mathrm{Ra}\,T\,\hat{r}\), with Rayleigh number \(\mathrm{Ra}\) and outward radial unit vector \(\hat{r}\),
\(g\) — mass-conservation source term ( \(g = 0\) for incompressible / Boussinesq flow; \(g \neq 0\) for anelastic extensions — see Stokes: compressible extensions).

Because inertia is negligible in mantle flow ( \(\mathrm{Re} \ll 1\)), no time derivative of velocity appears: the Stokes system is solved quasi-statically at every time step given the current temperature field.

Energy equation

‍See the Advection-Diffusion chapter for the full discretization, compressibility corrections, and boundary condition details.

The temperature \(T\) evolves according to the advection-diffusion equation

\[ \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T - \nabla \cdot (\kappa\,\nabla T) = f_T, \]

where \(\kappa\) is the thermal diffusivity and \(f_T\) collects volumetric heat sources (e.g. radiogenic heating). The advective form ( \(\mathbf{u}\cdot\nabla T\), not \(\nabla\cdot(\mathbf{u}T)\)) is appropriate for a passive scalar such as temperature; in the incompressible case the two are equivalent.

Coupling and time integration

The two systems are coupled through the buoyancy force and are advanced in time by operator splitting:

Stokes solve — given \(T^n\), solve for \((\mathbf{u}^n, p^n)\).
Energy update — advance \(T^{n+1}\) using \(\mathbf{u}^n\) as the advecting velocity.

This decoupled, sequential approach is first-order accurate in time. The Stokes solve is computationally dominant; the energy step is typically cheaper due to its scalar nature and the finite-volume discretization used.

Non-dimensionalisation

‍Not yet documented. The key dimensionless group is the Rayleigh number \(\mathrm{Ra} = \rho_0\,g_0\,\alpha\,\Delta T\,D^3 / (\eta_0\,\kappa_0)\), which controls the vigour of convection. A full description of the non-dimensionalisation and the resulting parameter choices will be added here.