Numerical weather prediction

working pillar 1 · Physical modeling and domain theory

In one minute

• An NWP model is a dynamical core plus a physics suite. The core is well-posed numerics; the physics is where the modelling judgement — and most of the error — lives.
• Predictability is finite. Deterministic skill runs out around 10–14 days regardless of model quality, which is why ensembles exist.
• The convective grey zone (roughly 1–10 km) is genuinely awkward: too fine for convective parameterization, too coarse to resolve convection.
• Terrain-following coordinates degrade over steep topography — a recurring problem in exactly the complex terrain people most want to forecast.
• Model output is a realisation, not a truth. Post-processing and calibration are part of the forecast, not an afterthought.

Key results

Anatomy of a model:

Component	What it does	Typical choices
Dynamical core	solves the resolved-scale equations	hydrostatic vs non-hydrostatic; grid; time scheme
Microphysics	cloud and precipitation	single/double moment
Radiation	SW and LW transfer	RRTMG
Boundary layer	vertical mixing	YSU, MYJ, MYNN
Surface layer	fluxes	MOST-based
Land surface	soil, vegetation	Noah-MP
Cumulus	subgrid convection	Kain–Fritsch, Grell — off below ~4 km

Resolution regimes:

\(\Delta x\)	Convection	Approach
\(> 10\) km	fully parameterized	global models
1–10 km	grey zone	scale-aware schemes, or neither option is right
0.1–1 km	convection-permitting	no cumulus scheme; updrafts marginally resolved
\(< 100\) m	LES regime	boundary-layer scheme must also be replaced

"Convection-permitting" is a carefully chosen word: at 3 km, storms form and organise realistically, but individual updrafts are 2–3 grid points across and are not converged.

Operational systems, as a rough orientation:

Model	\(\Delta x\)	Domain	Notes
ECMWF IFS	~9 km	global	spectral, generally the skill benchmark
GFS	~13 km	global	FV3 core
HRRR	3 km	CONUS	hourly, convection-permitting, radar assimilation
WRF	configurable	limited area	research workhorse
MPAS	variable-resolution	global	unstructured, avoids nesting

Predictability. Error growth is exponential at small scales and saturates:

\[ \frac{dE}{dt} = \alpha E\left(1 - \frac{E}{E_\infty}\right) \]

Doubling time is roughly 1.5–2 days at synoptic scales and hours at convective scales — so a convection-permitting forecast of individual storm placement has skill for hours, not days, however good the model.

Ensembles sample initial-condition uncertainty (perturbed analyses) and model uncertainty (stochastic physics, multi-physics, multi-model). The ensemble mean beats any member on average; the spread is the useful product, and a well-calibrated ensemble has spread matching RMSE.

Mental model

A forecast model is a coupled system in which the dynamics are largely settled and the physics is where competing approximations meet. The dynamical core can be verified against analytic solutions — it is a V&V problem with right answers. The physics cannot: every scheme is a different defensible approximation, and switching one changes the forecast in ways that are hard to attribute.

That asymmetry explains a great deal about how the field works — why physics suites are named and tracked, why ensembles perturb physics as well as initial conditions, and why model intercomparison is a discipline of its own.

Numerics / practice

• Spin-up matters. A cold-started limited-area run needs hours before its boundary layer and clouds are consistent with its own physics. Verification during spin-up measures the initialisation.
• Nest ratios of 3:1 or 5:1, and keep the parent domain large enough that the lateral boundary is far from the region of interest.
• Do not mix a cumulus scheme with a 2 km grid. Either resolve convection or parameterize it; doing both double-counts.
• Verify against the right thing. Point verification of a convection-permitting forecast penalises realistic storms in slightly the wrong place — neighbourhood and object-based methods (FSS, MODE) are the appropriate tools.

Failure modes

Running a cumulus scheme in the grey zone. At 3–4 km with Kain–Fritsch active, the scheme and the resolved dynamics both produce convection. Symptom: too much light precipitation, suppressed organised storms, and a damped diurnal cycle. Conversely, switching cumulus off at 8 km leaves convection entirely unrepresented.

Terrain-following coordinate errors over steep slopes. Horizontal pressure-gradient terms become the small difference of two large terms on sloping coordinate surfaces. Symptom: spurious circulations and temperature errors locked to topography that do not disappear with refinement. Hybrid or step-terrain coordinates mitigate it.

Verifying during spin-up. The first 3–6 hours of a cold-start run reflect the initialisation, not the model. Including them makes physics look worse (or better) than it is.

Lateral boundary contamination. Errors propagate inward from the boundary at the advective speed. A small domain at high resolution can be boundary-dominated within a few hours — the nest is then reproducing its parent, not adding value.

Ensemble spread mistaken for uncertainty. Under-dispersive ensembles are the norm: spread is often smaller than RMSE, so the ensemble is overconfident. Check the spread-skill relationship before using percentiles as probabilities.

Point verification of convection-permitting output. A forecast storm 20 km from the observed one scores worse than a smooth forecast with no storm at all — the "double penalty". It rewards blurry forecasts, which is exactly the wrong incentive.

Physics timestep coupling. Schemes are usually called sequentially and each sees the state left by the last. The ordering is a modelling choice with real consequences, and it differs between models.

Worked example

Why convective-scale forecasts have skill for hours, not days:

import numpy as np

E0, E_inf = 0.01, 1.0                 # initial error, saturation

for scale, doubling_h in (("synoptic (1000 km)", 40.0),
                          ("mesoscale (100 km)",  8.0),
                          ("convective (10 km)",  2.0)):
    alpha = np.log(2) / doubling_h
    # logistic growth solved for time to reach 50% of saturation
    t50 = np.log((0.5*E_inf/E0) * (E_inf - E0)/(E_inf - 0.5*E_inf)) / alpha
    print(f"{scale:22s} doubling={doubling_h:4.0f} h   "
          f"time to 50% saturation = {t50:6.1f} h ({t50/24:.1f} days)")

synoptic (1000 km)     doubling=  40 h   time to 50% saturation =  265.2 h (11.0 days)
mesoscale (100 km)     doubling=   8 h   time to 50% saturation =   53.0 h (2.2 days)
convective (10 km)     doubling=   2 h   time to 50% saturation =   13.3 h (0.6 days)

Roughly the observed limits: about ten days for synoptic patterns, half a day for individual storms. No improvement in model or observations moves the convective number much — it is set by the dynamics.

Connections

• Dynamics — what the dynamical core solves.
• Radiation and clouds · Boundary layer — the physics suite.
• Data assimilation — where initial conditions come from.
• Toolchains › Weather models — running WRF, MPAS and friends.

Sources

• Kalnay, Atmospheric Modeling, Data Assimilation and Predictability.
• Warner, Numerical Weather and Climate Prediction.
• Archive: see the course archive.