The last two chapters treated the analysis as a static problem: a Gaussian prior, a Gaussian observation, one posterior. Real data assimilation lives in a harder setting — the model is a chaotic dynamical system, the forecast spreads, and after long enough the error distribution is not Gaussian at all. This chapter looks at where those difficulties come from, using two toy dynamical systems: the 40-variable Lorenz-96 model for chaos and predictability, and the vort3d tropical-cyclone model for an unpredictable process (rapid intensification) and for the non-Gaussianity that position error produces. The analysis schemes in the next part are, at bottom, answers to exactly these challenges.
Lorenz-96 is the standard toy for predictability studies: 40 variables on a ring, $\dot x_i = (x_{i+1}-x_{i-2})x_{i-1} - x_i + F$ with $F=8$. It is deterministic and chaotic: two states that differ by anything — even $10^{-12}$ — separate exponentially, until the difference saturates at the system's natural variability. A weather model does the same thing with its own initial-condition errors, which is why data assimilation exists: the analysis exists to shrink those errors before they have time to grow.
Try it: slide the initial perturbation $\sigma$ down and the fan in panel (a) tightens while the growth curves in panel (b) shift right; slide it up and the ensemble starts already-spread.
Three features of panel (b) carry the whole lesson. First, the exponential phase: at short lead times the error doubles on a characteristic timescale (here ≈ 2 days — the famous Lorenz-96 value), so the readout's error-doubling time is the model's intrinsic error-doubling time. Second, the saturation: error and spread both level off at the climatological std ≈ 3.5, the limit of what is predictable about this system from initial conditions alone. Third, spread tracks error — the ensemble is built from the same perturbation statistics that generated the truth error, so its spread is a faithful forecast of the error. The vertical marker is the predictability limit: past it, the forecast carries no more information than a random draw from the climatology.
The vort3d model is a minimal 3-D tropical-cyclone model (Zhu, Smith & Ulrich 2001, in NEDAS). The demo below launches 20 forecasts of the same storm from slightly perturbed initial states, and tracks each one for 11 days. Two challenges stand out that the Gaussian picture of Chapters 1–2 cannot capture:
Rapid intensification is a regime change. Some members intensify quickly to Vmax > 60 m/s while others stall or decay — watch the intensity panel in row 1: the 20 realizations separate into distinct stories. The storm's tendency is what is uncertain, not just a small additive error, and the ensemble mean of a bimodal set of intensities is not a storm anyone would see. This is the practical form of the "less predictable process" problem: predictability depends on the flow, and a fast-developing regime is where it drops most.
Position error is non-Gaussian. Hover over a realization and look at the wind-speed maps in row 2: members that agree on intensity can disagree wildly on where the vortex is. An ensemble whose members carry position uncertainty produces, at any fixed grid point, a wind error distribution that is bimodal (storm here or there), skew, and generally far from Gaussian — the raw material that the linear-Gaussian analysis of the next chapters cannot represent. You met the germ of this idea in Chapter 2 (position uncertainty produces strong negative correlations); here it is in a real model. This is what motivates the nonlinear filters of Chapter 7.
Row 1 — vortex track, intensity (maximum boundary-layer wind) and size (radius of the 15 m/s wind ring) for 20 of 200 free-running pool realizations (Ying, Anderson & Bertino 2023, MWR, Fig. 1 layout); dashed line: ensemble mean. Row 2 — the selected realization's structure at the chosen time: boundary-layer wind speed map (1280 km window centred on the vortex: dashed track = ensemble mean, coloured track = the selected realization), and a radius–height cross-section of the azimuthally-averaged radial wind (shading, inflow negative) with tangential-wind contours. The white ring on the wind map is the 15 m/s isotach — the same definition as the "size" timeseries of row 1.
Next: Chapter 4 — From best linear unbiased estimate to the Kalman filter — the first analysis scheme, built on exactly the Gaussian assumptions these examples will stress.