Advection on the C-grid: one term, three ways to compute it

Abstract¶

The advection term moves a quantity along with the flow. In the continuous equations it can be written in three ways that are all equal. However, on a grid they stop being equal, as each one puts the approximation in a different place and keeps a different set of quantities exactly. In this page, we go from the PDE to the stencils, in finitevolX’s conventions, and run all three on the same test.

Keywords:advectionArakawa C-gridfinite volumeArakawa JacobianconservationfinitevolX¶

Open the full page

Where it sits. A refresher on the term the learned schemes replace, and what each discrete form conserves. The full page is a self-contained document with operator boxes (inputs, outputs, shapes and units), figures and code sketches, which opens on its own and links back here.

TL;DR: Same term, different approximation¶

Here $q(\vect{s}, t) \in \mathbb{R}$ is the advected quantity, $\vect{v} = (u, v) \in \mathbb{R}^2$ the velocity in m s⁻¹, $\psi(\vect{s}, t) \in \mathbb{R}$ the streamfunction, $J(\psi, q) \in \mathbb{R}$ the Jacobian of the two fields, and $\nabla := \nabla_{\vect{s}}$ is the gradient in the spatial coordinates $\vect{s} \in \mathbb{R}^2$ .

Flux form $\nabla \cdot (\vect{v} q)$ : move $q$ to the faces where $u$ and $v$ live, multiply, and take a difference. The difference is exact, so all of the error is in the face value of $q$ . It conserves $\sum q$ for any face rule, and this is where upwind and WENO live.
Advective form $\vect{v} \cdot \nabla q$ : take the gradient of $q$ at the centre and move $u$ and $v$ there. The error is in the gradient and in moving the velocity, and it conserves nothing by construction. This is the slot a spectral gradient fits.
Jacobian form $J(\psi, q)$ , for QG: combine $\psi$ and $q$ directly. Arakawa’s average of three versions conserves $\sum q$ , $\sum q^2$ and energy exactly, but it adds no dissipation, so fronts ring unless viscosity is added.
Centred schemes never damp, whereas upwind-biased ones always do. On a sharp front that is the whole difference: the centred schemes keep the variance and ring, upwind smears, and WENO5 does neither.

On the page¶

What the advection term says
Where the term appears
Three ways to write the same term
The grid
Form 1 · Flux form (finite volume)
Form 2 · Advective form
Form 3 · Jacobian form (Arakawa)
All three, on the same test
Momentum advection, briefly
Back to the question