Mathematical Model
This part of the manual describes the mathematical model used in the Leeds Spherical Dynamo code. The following sections will describe various details of the model:
- Non-dimensionalisation information
- Governing Equations
- Boundary Conditions
- Control parameters
- Diagnostic Outputs
- Definitions of miscellaneous terms
Essential Features
Essential features of the model are summarised in the table below. Further details can be found by clicking on individual items, or in the sections above.
| Quantity | Definition |
|---|---|
| Timescale | \(D^2/\eta\) |
| Lengthscale | \(D = r_o-r_i\) |
| Temp scale | \(\Delta T~\), \(~~\beta/D\) |
| Composition scale | \(\Delta \xi~\), \(~~\beta_\xi/D\) |
| Magnetic scale | \(\left(2\Omega\rho\mu_0\eta \right)^{1/2}\) |
| Velocity scale | \(\eta/D\) |
| Ekman Number | \(\nu/\Omega D^2\) |
| Thermal Rayleigh | \(\alpha g\Delta T D^4/r_o \kappa \nu\), \(~\alpha g\beta D^3/r_o \kappa \nu\) |
| Compositional Rayleigh | \(\alpha_\xi g \Delta \xi D^4/r_o \kappa_\xi \nu\), \(~\alpha_\xi g \beta_\xi D^3/r_o \kappa_\xi \nu\) |
| Prandtl | \(\nu/\kappa\) |
| magnetic Prandtl | \(\nu/\eta\) |
| Schmidt Number | \(\nu/\kappa_\xi\) |