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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.

Notes specific to derivations in 1.

Phase Separation

Under the assumption of no melting/solidification, melt-solid separation is treated as a mass-transfer process. Average density of mixture:

\[\frac{1}{\rho} = \frac{1}{\rho_s}(1-\phi)+\frac{1}{\rho_m}\phi\]

where \(\phi\) is mass fraction of melt.

The masses of solid and melt phases per unit volume are:

\[\rho_s^\ast \equiv (1-\phi) \rho = \frac{\rho_s \rho_m (1-\phi)}{\rho_s \phi + \rho_m(1-\phi)}\]
\[\rho_m^\ast \equiv \phi \rho = \frac{\rho_s \rho_m \phi}{\rho_s \phi + \rho_m(1-\phi)}\]

Define the velocity of the local barycenter:

\[v \equiv \phi v_m + (1-\phi) v_s\]

And the vertical mass flux of melt relative to the barycenter:

\[J_m \equiv \rho \phi (1-\phi)(v_m-v_s)\]

The phase separation equation becomes:

\[\frac{\partial \phi}{\partial t} + v \frac{\partial \phi}{\partial z} = \frac{\rho_m \rho_s}{\rho (\rho_s-\rho_m)} \frac{\partial v}{\partial z} = - \frac{1}{\rho} \frac{\partial J_m}{\partial z}\]

Time Scale

Characteristic time scale \(\tau\) of melt-solid separation for a partially molten layer of thickness \(L\):

\[\tau = \frac{\rho L}{2 J_m} \min (\phi_0, 1-\phi_0 )\]

Impact Stirring

During planetary accretion, planetesimal impacts stirred the mantle. Assuming a roughly linear accretion rate:

\[\frac{dm}{dt} \sim \frac{M_E}{\tau_{acc}}\]

The impactor mass distribution is:

\[\frac{dN}{dm} = {K_{max}\left(\frac{m}{M_{max}}\right)^{-q}}\]

with \(q=1.5\) and \(M_{max}=0.1 M_E\).

The depth-dependent impact stirring timescale is:

\[\tau_{s}(d) = \frac{\tau_{acc}}{N_{s}(d)}\]

where \(N_{s}\) is the effective number of complete stirring events at depth \(d\).

Thermal Evolution of a Magma Ocean

The energy (enthalpy) balance equation:

\[\frac{\partial H}{\partial t} = - \frac{1}{\rho}\nabla \cdot \vec{J_{tot}} + \Delta V_m|g|\vec{J_m} \cdot \hat{r} + q_{heat}\]

The total heat flux is a combination of sensible and latent heat:

\[J_{tot} = J_q + T\Delta S_m J_m\]

The melt fraction can be approximated as:

\[\phi(H) = \frac{H - H_{sol}}{T\Delta S_m(P)}\]

for \(H_{sol} < H < H_{liq}\).

Key assumptions for single-component modeling: - Use a single component melting curve roughly corresponding to the 50% solidus - Entropy of melting adjusted to ensure bounds match realistic mantle - Use realistic heat capacity and density values for both phases - Chemical differentiation negligible (second order effect) - Thermal range of partially molten region (~200 K) small compared to mantle temperature difference (~2500 K)

Gravitational potential energy

The gravitational potential energy per unit mass is:

\[E_{\rm grav} = - \frac{G M(r)}{r} = - |g(r)|r\]

Melting influences gravitational potential energy through its effect on \(g(r)\). However, since \(g(r)\) is an integrated quantity it is not sensitive to small changes in density distribution. We therefore neglect changes in \(E_{\mathrm{grav}}\) for silicate melting in a magma ocean.

  1. Yutaka Abe, Thermal Evolution and Chemical Differentiation of the Terrestrial Magma Ocean, Evolution of the Earth and Planets, 1993. ↩