Mass conservation approximation#

First, we have to choose how to approximate the conservation of mass: \(\nabla \cdot (\rho \mathbf u) = 0\) (see Equation (2)). We provide the following options, which can be selected in the parameter file in the subsection Formulation/Mass conservation (see also Parameter name: Mass conservation):

  • “incompressible”:

\[\nabla \cdot \textbf{u} = 0,\]
  • “isothermal compression”:

    \[\nabla \cdot \textbf{u} = -\rho \beta \textbf{g} \cdot \textbf{u},\]

    where \(\beta = \frac{1}{\rho} \frac{\partial \rho}{\partial p}\) is the compressibility, and is defined in the material model. Despite the name, this approximation can be used either for isothermal compression (where \(\beta = \beta_T\)) or isentropic compression (where \(\beta = \beta_S\)). The material model determines which compressibility is used. This is an explicit compressible mass equation where the velocity \(\textbf{u}\) on the right-hand side is an extrapolated velocity from the last timesteps.

  • “hydrostatic compression”:

    \[\nabla \cdot \textbf{u} = - \left( \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_{T} \rho \textbf{g} + \frac{1}{\rho} \left( \frac{\partial \rho}{\partial T} \right)_{p} \nabla T \right) \cdot \textbf{u} = - \left( \beta_T \rho \textbf{g} - \alpha \nabla T \right) \cdot \textbf{u}\]

    where \(\beta_T = \frac{1}{\rho} \left(\frac{\partial \rho}{\partial p} \right)_{T}\) is the isothermal compressibility, \(\alpha = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T} \right)_{p}\) is the thermal expansion coefficient, and both are defined in the material model. The approximation made here is that \(\nabla p = \rho \textbf{g}\).

  • “reference density profile”:

    \[\nabla \cdot \textbf{u} = -\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u},\]

    where the reference profiles for the density \(\bar{\rho}\) and the density gradient \(\frac{\partial \bar{\rho}}{\partial z}\) provided by the adiabatic conditions model (Initial conditions and the adiabatic pressure/temperature) are used. Note that the gravity is assumed to point downwards in depth direction. This is the explicit mass equation where the velocity \(\textbf{u}\) on the right-hand side is an extrapolated velocity from the last timesteps.

  • “implicit reference density profile”:

    \[\nabla \cdot \textbf{u} + \frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u} = 0,\]

    which uses the same approximation for the density as “reference density profile,” but implements this term on the left-hand side instead of the right-hand side of the mass conservation equation. This effectively uses the current velocity \(\textbf{u}\) instead of an explicitly extrapolated velocity from the last timesteps.

  • “ask material model,” which uses “isothermal compression” if the material model reports that it is compressible and “incompressible” otherwise.

The stress tensor approximation

If a medium is incompressible, that is, if the mass conservation equation reads \(\nabla \cdot \textbf{u} = 0\), then the shear stress in the momentum and temperature equation simplifies from

\[\tau = 2\eta\left(\varepsilon\left(u\right) - \frac{1}{3}\left(\nabla\cdot\textbf{u}\right)1\right)\]

to

\[\tau = 2\eta\varepsilon\left(\textbf{u}\right)\]