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SPIDER: model overview

Here you can find a detailed overview of the SPIDER formulation.

Note

This model overview is taken from the notes and contains an extended description of the equations and derivations related to the SPIDER code. It is still work in progress.

Mixing length theory

Heat (and mass) transport for both the viscous (solid-state) and inviscid regime is parameterised using mixing length theory. The sensible heat flux is expressed as:

\[J_q = -\rho C_p \kappa \left( \frac{\partial T}{\partial r} \right)_S - \rho C_p \kappa_{\rm h}\Delta (\delta_r T)_S\]

where \(\kappa\) is thermal diffusivity (conduction) and \(\kappa_{\rm h}\) is eddy diffusivity (convection).

The heat-transport diffusivity is a piecewise function depending on the heat-transport regime:

\[\kappa_h= \begin{cases} \kappa & \text{if } \kappa_{\rm vis} \le \kappa \\ \kappa_{\rm vis} & \text{if } \kappa < \kappa_{\rm vis} < \nu \\ \kappa_{\rm invis} & \text{if } \nu \leq \kappa_{\rm vis} \end{cases}\]

In either regime, the appropriate diffusivity is:

\[\kappa_{\rm conv} \sim v_{\rm conv} l\]

where \(l\) is the convective mixing length.

To remain consistent with the entropy-pressure formulation, we rewrite in terms of entropy gradients:

\[\Delta (\delta_r T)_S = \frac{T}{C_p} \frac{dS}{dr}\]
\[J_q =-\rho C_p \kappa \left( \frac{\partial T}{\partial P}\right)_S \frac{dP}{dr} - \rho \kappa_{\rm h} T \frac{dS}{dr}\]

Velocity scalings

Convective velocities are given by:

\[v_{\rm vis} = \frac{\alpha |g| l^3}{18\nu} \Delta(\delta_z T)_S\]
\[v_{\rm invis} = \sqrt{\frac{\alpha |g| l^2}{16} \Delta(\delta_z T)_S}\]

Viscous scaling

For the viscous case, we balance the buoyancy force on each fluid parcel against the viscous drag:

\[U = \frac{\alpha g l^3}{18 \nu} \left( \frac{dT}{dz} - \left( \frac{dT}{dz} \right)_{\rm s} \right)\]

Inviscid scaling

For the inviscid case, kinetic energy of a fluid element is balanced by the work done by the buoyancy force:

\[v(x) = \sqrt{\alpha g x^2 \left( \frac{dT}{dz} - \left( \frac{dT}{dz} \right)_s \right)}\]

The average velocity is:

\[v_{\rm invis} = \sqrt{\frac{\alpha g l^2}{16} \left( \frac{dT}{dz} - \left( \frac{dT}{dz} \right)_s \right)}\]

Kamata and Wagner profile

According to 2 and 1, the classical mixing length profile is not able to reproduce realistic results with deviations up to 60% from 3D simulations. The profile can be characterized by two parameters: depth \(a\) and size \(b\), for mantle thickness \(D\).

Kamata's results

2 finds that coefficients \(a\) and \(b\) depend on relative mantle size \(f = R_{CMB}/R_{top}\) (about 0.55 for Earth) and viscosity contrast \(\gamma = \ln (\eta_{top}/\eta_{bottom})\):

\[a,b = a_2,b_2 f^2 + a_1,b_1f + a_0,b_0\]

Wagner et al.'s results

1 use a different parametrization in terms of \(\alpha\) and \(\beta\):

\[\alpha = \left( a_0 - a_1 \gamma - a_2 \log(\text{Ra}) \right) \tanh \left( a_3 \log (\text{Ra}/\text{Ra}_c) \right)\]
\[\beta = b_0 - b_1 \gamma - b_2 \log (\text{Ra})\]

for stagnant lid regime, or more complex expressions for mobile and sluggish lids. Values are given in their Table 4.

  1. Wang, Zaicong, Earth’s volatile-element jigsaw, Nat. Geosci., 2019. ↩↩

  2. Shunichi Kamata, One-dimensional convective thermal evolution calculation using a modified mixing length theory: Application to Saturnian icy satellites, J. Geophys. Res., 2018. ↩↩