From equations describing the atmosphere to probabilities describing tomorrow.
Weather forecasting begins with a surprisingly simple idea: the atmosphere can be represented mathematically as a system whose state changes through time.
At any particular moment the atmosphere has a state containing quantities such as temperature, pressure, humidity and wind speed at millions of locations throughout the atmosphere.
We can represent the entire atmospheric state by a vector x. A simplified mathematical description is:
The function F represents the physical laws governing how the atmosphere evolves. If we knew the atmospheric state perfectly and could solve the equations perfectly, we could theoretically calculate the future.
Modern numerical weather prediction is based on physical conservation laws. The atmosphere is treated as a fluid and its evolution is governed by a system of coupled differential equations.
Air cannot simply appear or disappear. Changes in density are related to the movement of air.
Wind changes because of pressure gradients, gravity, rotation and friction.
Heating, cooling, compression, expansion and phase changes affect the energy of the atmosphere.
Temperature, pressure and density are related through the properties of air.
The momentum equation can be viewed as a form of the Navier–Stokes equations used to describe fluid motion. A simplified representation is:
Every term has a physical meaning. The challenge is that these equations are coupled, nonlinear and must be solved across an enormous three-dimensional grid covering the atmosphere.
A computer cannot simply solve the atmosphere as one enormous continuous equation. Instead, the atmosphere is divided into a numerical grid.
One of the deepest mathematical problems in weather forecasting is that the atmosphere is a chaotic dynamical system.
Two atmospheric states that are almost identical can eventually evolve into dramatically different states.
A simplified description of this behaviour is:
where d0 is the initial difference between two states and λ represents the rate at which that difference grows.
Because the initial atmospheric state is uncertain, modern forecasting does not rely on one single future.
Instead, a numerical model is run many times with slightly different initial conditions and sometimes different model assumptions. These are called ensemble members.
Suppose an ensemble contains 50 forecasts and 30 of them predict rain at a particular location. A simple estimate of the probability of rain is:
This is an important conceptual difference between a deterministic and probabilistic forecast.
“Rain will occur at 3pm.”
“There is a 70% probability of measurable rain around 3pm.”
Weather models constantly receive new observations from satellites, weather stations, aircraft, ships, weather balloons, radar and other sources.
The mathematical challenge is to combine these imperfect observations with an existing model estimate.
A simplified Bayesian formulation is:
Here:
An ensemble can be interpreted as a probability distribution. For example, instead of predicting one temperature, the model can describe a range of plausible temperatures.
The standard deviation gives a measure of forecast spread. A narrow distribution represents greater confidence; a wide distribution represents greater uncertainty.
Rainfall is the result of several interacting physical processes. The simplified chain is:
The actual process is considerably more complicated. Cloud microphysics, convection, humidity, temperature and atmospheric stability all influence the eventual precipitation.
A forecast is not useful simply because it looks plausible. Forecast systems must be tested against observations.
Measures the average absolute difference between forecast and observation.
Penalises large forecast errors more heavily.
Shows whether a forecast systematically over-predicts or under-predicts.
Used to evaluate probabilistic forecasts such as the probability of rain.
Weather forecasting is ultimately a problem of predicting the future state of a chaotic physical system from incomplete and uncertain information.
The fascinating part is that the mathematics of weather forecasting connects many of the mathematical ideas explored elsewhere on KerryTech:
Forecasts describe distributions of possible futures rather than a single certain outcome.
Observations are noisy and forecasts must be calibrated and evaluated.
The evolution of atmospheric variables is described using differential equations.
Small differences in the initial state can grow into major differences in the forecast.
The equations must be approximated and solved on enormous computer grids.
Ultimately the forecast is about quantifying uncertainty about future outcomes.