# Discretisation of flow on a sphere

These are some quick notes on the discretisation of flow on a sphere presented in LN-JFM-2022, written with a view to coding it up at some point.

## Coordinate system

 $$a$$ radius $$z$$ polar axis $$\mathbf{r}$$ vector from the sphere's centre $$\theta$$ colatitude (zero at zenith) $$\phi$$ longitude $$\psi(\theta, \phi)$$ a real-valued scalar function

## Expansions

Expansion of $$\phi$$, $\psi(\theta, \phi) = \sum_{l=0}^\infty \psi_l(\theta, \phi),$ $\psi_l(\theta, \phi) = \sum^l_{m=-1} a_{lm}Y^m_l(\theta, \phi).$

Spherical harmonics $Y_l^m(\theta, \phi) = \sqrt{ (2l + 1) \frac{(l-m)!}{(l+m)!} } P_l^m(\cos \theta ) e^{i\phi},$ where $$P_l^m$$ are the Legendre functions.

## Properties

Because $$\psi$$ is real, $$a_{l(-m)} = (-1)^m (a_{lm})^*$$ (so the $$\psi_l$$ are also real).

$$Y_l^m$$ are eigenfunctions of the Laplacian, $\nabla^2 Y_l^m = - \frac{l (l+1)}{a^2} Y_l^m.$

They are orthogonal wrt to integration over the sphere (as you would expect of eigenfunctions of a Laplacian), $\frac{1}{4\pi} = \int_\Omega Y_l^m Y_s^{p*} d\Omega = \delta_{ls} \delta_{mp}$

and have symmetry $Y_l^{-m} = (-1)^m Y_l^{m*}.$

The $$\psi_l$$ are invariant under change of coordinate system.

## Streamfunction definition of the velocity field

Define $$\nabla$$ to give the surface gradient along a vector tangent to the sphere. Introduce the Hodge star operator, $$\star$$. In this case the $$\star$$ rotates the gradient anti-clockwise by $$\pi/2$$ in the tangent plane (around $$\mathbf{r}$$).

Then, $\mathbf{u} := \star \nabla \psi.$

The velocity field is then divergence-free, according to $\nabla \cdot \mathbf{u} = \nabla \cdot \star \nabla \psi = 0.$

The vorticity is defined as $\omega := \nabla \cdot \star \mathbf{u} = - \nabla^2 \psi.$

Then $\mathbf{u} = \sum_{l=1}^\infty \mathbf{u}_l, \quad \mathbf{u}_l = \star \nabla \psi_l,$

$\omega = \sum_{l=1}^\infty \omega_l, \quad \omega_l = \frac{l (l+1)}{a^2} \psi_l.$

$\partial_t \mathbf{u} = - \mathbf{u}\cdot\nabla\mathbf{u} - \nabla\left(\frac{p}{\rho}\right) - (2\Omega \cdot \mathbf{e}_r) \star \mathbf{u} + \nu \left( \Delta \mathbf{u} + \frac{\mathbf{u}}{a^2} \right),$

$\Delta \mathbf{u} = - \star \nabla (\nabla \cdot \star \mathbf{u}) + \frac{\mathbf{u}}{a^2}.$

### To do

• time evolution of expansion coefficients
• efficient transformation between physical and harmonic coordinates