Weather and climate modelling

Level SpecializationDifficulty ★★★★★Application⌖ Open in the map

What is it?

Integrate the equations of the atmosphere and oceans on a global grid. Chaos limits deterministic forecasts to about two weeks, so services run ensembles; data assimilation fits the initial state to observations using adjoint (reverse-mode) gradients. Machine-learned forecasters now compete with physics models.

Formulas

$J(x_0) = \tfrac12\norm{x_0 - x_b}^2_{B^{-1}} + \tfrac12\sum_k\norm{y_k - H(x_k)}^2_{R^{-1}}$
4D-Var cost function, minimized with adjoint gradients

The mathematics behind it

Dynamical systems★★★★★fundamental
The atmosphere is a huge dynamical system; forecasting is integrating it forward.
Chaos and sensitivity to initial conditions★★★★★fundamental
Chaos limits deterministic forecasts to about two weeks; ensemble forecasting is the response.
Scalar fields★★★★★frequent
Temperature, pressure and humidity are scalar fields on a 3D grid of the atmosphere.
Runge–Kutta methods★★★★★frequent
Atmospheric models advance their discretized equations in time with Runge–Kutta-type schemes.
Attractors★★★★★frequent
Lorenz found his attractor in a toy convection model; climate can be seen as the statistics of the attractor, weather as a point on it.
Multivariable chain rule★★★★★advanced
4D-Var data assimilation uses adjoint (reverse-mode) models to fit the initial state of forecasts.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.