Math 2.0

Geophysics Definition

Numerical weather prediction

Builds on Initial value problemPrimitive equationsData assimilationNumerical analysisButterfly effect

Definition

Let MM be a numerical model of the atmosphere. It is a discretisation in space and time, by the methods of numerical analysis, of partial differential equations for the atmospheric state. Typical equations are the primitive equations, together with parametrisations of unresolved processes. The model state is a vector x∈ℝnx \in \R^n of grid-point or spectral values, and one step of the model maps xkx_k to xk+1=M(xk)x_{k+1} = M(x_k). Numerical weather prediction is the forecasting of the weather by the following procedure.

  1. An analysis x0ax_0^a of the current state is computed from observations and a previous forecast by data assimilation.
  2. The initial value problem is solved forward: xk+1=M(xk)x_{k+1} = M(x_k) with x0=x0ax_0 = x_0^a, up to the forecast range.
  3. The forecast fields xkx_k are converted into weather variables at the times and places of interest.

Remarks

The procedure is repeated in cycles, typically every 66 or 1212 hours, and the short-range forecast of one cycle is the background for the next analysis. In ensemble prediction, step 2 is repeated from several perturbed analyses, and the spread of the forecasts estimates their uncertainty.

Example: the first successful numerical forecast, in 1950, used the barotropic vorticity equation on a grid over North America and was computed on the ENIAC computer.

Non-example: a statistical forecast that predicts tomorrow's temperature by regression on past observations, without integrating equations of motion, is not numerical weather prediction in this sense.

Because the atmosphere is chaotic, errors in x0ax_0^a grow with time, the butterfly effect, and this limits the useful range of deterministic forecasts to about two weeks.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Butterfly effect, concept, no evidence levelButterfly effectconceptData assimilation, concept, no evidence levelData assimilationconceptInitial value problem, concept, no evidence levelInitial value problemconceptNumerical analysis, concept, no evidence levelNumerical analysisconceptPrimitive equations, concept, no evidence levelPrimitive equationsconceptNumerical weather prediction, definition, no evidence levelNumerical weather pre…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Butterfly effect concept
  2. Data assimilation concept
  3. Initial value problem concept
  4. Numerical analysis concept
  5. Primitive equations concept
  6. Numerical weather prediction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Numerical_weather_prediction
About this page
TagED6Q47PKTG
URIm2://ED6Q47PKTG
Kinddefinition
WritingW0 stub
MSC86A10, 65M06, 35Q86
DefinesNumerical weather prediction
UsesInitial value problem, Primitive equations, Data assimilation, Numerical analysis, Butterfly effect
Changed3 November 2026

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