Ensemble Weather Forecasting — Why Forecasts Lose Skill

Ensemble Setup

Initial perturbation ε₀ 0.06 hPa*

Ensemble members 5

Forecast speed 1.0×

Model turbulence ν 0.030

Forecast Readouts

Lead time

0.0 d

Ensemble spread

0.00 hPa*

Error e-folding time (predictability limit)

— d

Legend

Control run (member 0, "truth")

Perturbed ensemble members (wireframe)

Wind vectors (from control's pressure gradient)

How it works

Each ensemble member holds its own pressure-anomaly field p(x,y,t) on a periodic grid, evolved by the same simplified barotropic (geostrophic) model real NWP systems are built from:

u = -∂p/∂y,  v = ∂p/∂x        (rotated pressure gradient → wind)
∂p/∂t = -(u·∂p/∂x + v·∂p/∂y) + ν∇²p   (self-advection + diffusion)

The wind at every point is derived from the pressure field itself, and that wind then advects the very field that produced it — a nonlinear feedback loop. This is exactly the mechanism Edward Lorenz found in 1963 while running a toy weather model: two runs started ε₀ apart, however tiny, separate at a roughly exponential rate until they saturate at the climatological spread — the origin of "the butterfly effect" and the reason every real forecast center (ECMWF, GFS) runs 20-50 slightly perturbed members instead of one run.

ε₀ — how close the perturbed members start to the control run at t=0. Reseed to apply a new value.
Members — how many parallel forecasts run at once (real operational ensembles use dozens).
Forecast speed — simulated days advanced per second of wall-clock time.
Turbulence ν — diffusion strength; higher ν damps small scales faster, changing how quickly nonlinear advection can amplify the initial gap.
Spread is the RMS pressure difference between the perturbed members and the control, averaged over the whole grid; the e-folding time is the lead time it takes that spread to grow by a factor of e, a rough proxy for the model's predictability horizon.

* pressure units here are a dimensionless model anomaly, scaled for visual clarity rather than true hPa — the growth dynamics are what matters, not the absolute numbers.

Control (truth) Ensemble spaghetti

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