simplified.md

Simplified system

Equations

$$ \dot{n} + \nabla \cdot \left ( n \left ( v_{\parallel} + v_D \right )\right ) = n_0 $$

$$ \dot{(n v_{\parallel})} + \nabla \cdot \left ( n v_{\parallel} \left ( v_{\parallel} + v_D \right )\right ) = - T \nabla_{\parallel} n $$

$$ \dot{\omega} + \nabla \cdot \left ( \omega \left ( v_{\parallel} + v_D \right )\right ) = - \nabla_{\parallel} (n v_{\parallel}) $$

$$ \nabla_{\perp}^2 \phi = \omega $$

$$ v_D = \left ( \frac{\partial \phi}{\partial y}, -\frac{\partial \phi}{\partial x}, 0 \right ) $$

is the gradient RHS for omega needed?

Reduction to form used in Firedrake

The first two eqs are

$$ \dot{n} + \nabla \cdot \left ( n \left ( v_{\parallel} + v_D \right )\right ) = n_0 $$

$$ \dot{(n v_{\parallel})} + \nabla \cdot \left ( n v_{\parallel} \left ( v_{\parallel} + v_D \right )\right ) = - T \nabla_{\parallel} n. $$

By writing $\dot{(n v_{\parallel})} \equiv n \dot{v_{\parallel}} + \dot{n} v_{\parallel}$ and substituting from the first equation one obtains for $v_{\parallel}$

$$ n \dot{v_{\parallel}} + v n_0 + n \nabla{v_{\parallel}} \cdot (v_{\parallel}+v_D) = -T \nabla_{\parallel} n. $$

(If unsure about the vector calculus see the vector calculus identities Wiki page, that's what I do.)

CG version

Script: (LAPD-like_simplified_CG.py)

Implements the equations above, plus an attempt at streamline-upwind correction to add artificial viscosity.

DG version

Work in progress