Finite-Difference Schemes
Spatial derivatives are approximated on a staggered Arakawa-C grid. A second-order centred scheme for advection is:
\[\frac{\partial u}{\partial x} \approx \frac{u_{i+1} - u_{i-1}}{2\Delta x}\]Time integration uses a leapfrog or semi-implicit scheme. The Courant–Friedrichs–Lewy (CFL) stability criterion requires:
\[C = \frac{u\,\Delta t}{\Delta x} \leq 1\]Violating this condition causes numerical instability and exponential error growth.
Spectral Methods
Global models often expand horizontal fields in spherical harmonics $Y_n^m(\lambda,\phi)$:
\[T(\lambda,\phi,t) = \sum_{n=0}^{N}\sum_{m=-n}^{n} \hat{T}_n^m(t)\,Y_n^m(\lambda,\phi)\]Spectral truncation at triangular wavenumber $N$ gives an effective grid spacing of roughly $\pi a / N$ where $a$ is Earth’s radius. The semi-Lagrangian scheme relaxes the CFL constraint by tracing parcel trajectories back one time step and interpolating.
Error Growth and Predictability
Forecast errors grow roughly exponentially until they saturate at the climatological variance. The error doubling time for synoptic-scale features is approximately 2–3 days, setting a practical predictability limit near 2 weeks. Ensemble spread $\sigma_e(t)$ quantifies this growth:
\[\sigma_e^2(t) = \frac{1}{N-1}\sum_{i=1}^{N}\left[x_i(t) - \bar{x}(t)\right]^2\]Higher horizontal and vertical resolution consistently reduces systematic error in temperature, wind, and precipitation forecasts.