Chapter 11 · Lorenz 96: a many-variable atmosphere analogue

Overview

Everything in Part III was measured on a three-variable system. A forecast model has $10^9$. Which of those conclusions survive the change of scale?

The question is not rhetorical, and chapter 16 already found a casualty: in Lorenz 63 the gradient of a forecast metric points almost exactly along the fastest-growing direction, and in a forty-variable system the two are nearly orthogonal. Something has to sit between three variables and a general circulation model.

Read it as a latitude circle: the quadratic terms conserve $\sum_k x_k^2$ and stand in for advection, $-x_k$ is dissipation, $F$ is the forcing. Nothing about it is derived from the equations of motion. It earns its place by being the smallest system that behaves like a field — errors have a wavelength, structures propagate, and the attractor grows with the domain.

The model

chaoslib.systems supplies the right-hand side and its Jacobian; chaoslib.spatial (new in this chapter, and used again in chapter 12) supplies the diagnostics that only make sense once a system has a space: power spectra, phase speeds, correlation lengths.

Two exact identities anchor the chapter. Every diagonal entry of the Jacobian is $-1$, so $\sum_i \lambda_i = \operatorname{tr}\mathbf{J} = -N$ for every $F$, every $N$ and every trajectory. And the Jacobian at the uniform state $x_k = F$ is circulant, so Fourier modes diagonalise it in closed form:

That is not an approximation — it reproduces all $N$ eigenvalues to $5\times10^{-14}$.

Four results

The wavelength is set by a linear instability, and comes out in closed form. Taking real parts, $\operatorname{Re}\sigma = -1 + F(\cos\theta - \cos2\theta)$; with $u = \cos\theta$ the bracket is $1 + u - 2u^2$, maximised at $u = 1/4$ with value $9/8$. So a long chain destabilises at $F = 8/9$ and a finite one at $F_{\rm crit} = [\max_m(\cos\theta_m - \cos2\theta_m)]^{-1}$ — exactly $2/\sqrt5 = 0.8944$ at $N = 40$, where the best available mode is $m = 8$. The nonlinear flow peaks one wavenumber away, at $m = 9$.

Linear theory gets the scale nearly right and the speed badly wrong. The measured phase speed of the dominant mode is $-2.1$ sites per time unit against $-7.3$ predicted — a factor of 3.5. The prediction is made about a state the system is nowhere near, and finite-amplitude waves propagate on a flow they have themselves modified. The same lesson as chapter 15’s window of validity, from a different direction: the linearisation is quantitatively reliable for the instability that creates a structure, not for that structure’s later life.

There are two thresholds, and they are far apart. The uniform state loses stability at $F = 0.894$, but $\lambda_1$ stays within $\pm0.025$ of zero until about $F = 4.25$ — periodic waves with no error growth at all — and only becomes robustly positive near $F = 4.5$. At $F = 8$ the spectrum has 13 positive exponents, $h_{KS} = 10.2$ nats per time unit and $D_{KY} = 27.1$, against Lorenz 63’s 1, 0.905 and 2.06. The forty exponents sum to $-40$ to within $2.5\times10^{-5}$, a residual that is RK4 truncation and not non-convergence: it is independent of the averaging time and falls as $\Delta t^4$, both asserted by tests.

The model is extensive, and that is the point. Because the dynamics are local and the correlation length is a couple of sites, doubling the ring does not create a faster instability — it creates more independent copies of the same one. Measured over a 6.7-fold range of domain size: $\lambda_1$ is flat above $N \approx 30$ (mean 1.711), the spectra collapse onto a single curve under $i \to i/N$, and $D_{KY} = 0.675\,N$ and $h_{KS} = 0.256\,N$ through the origin, with $D_{KY}/N$ varying by 2.4% across the whole range. Below $N \approx 30$ $\lambda_1$ is suppressed, for a visible reason: the preferred wavelength is 4.4 sites, so a 12-site ring holds fewer than three waves and the instability is cramped by its own periodicity.

The forecasting consequence is a ratio. Lead time is set by $\lambda_1$, an intensive quantity — a property of the dynamics that no amount of computing changes. The number of directions an ensemble must span is set by $D_{KY} \approx 0.68N$, an extensive one. At 0.34 growing directions per variable, a model with $10^7$ variables has of order $3\times10^6$ of them, so fifty members is a rounding error against what needs sampling. Every ensemble method in Part V is a strategy for living with that ratio, and chapter 19’s localisation is a direct exploitation of the locality that makes the system extensive in the first place.

Exercises

Analytic. Derive $\sigma(\theta)$ from the Jacobian at $x_k = F$, and show that the maximum of $\cos\theta - \cos2\theta$ is $9/8$ at $\cos\theta = 1/4$. Deduce $F_{\rm crit} \to 8/9$ in the continuum limit, and explain why a finite ring’s threshold is always above that value.

Computational. Verify the trace identity $\sum_i\lambda_i = -N$ at two values of $\Delta t$ and confirm that the residual falls by roughly 16 when $\Delta t$ is halved while being unchanged by doubling the averaging time. Explain what each of those two facts rules out.

Exploratory. Measure the correlation length at $N = 12$ and at $N = 40$, and use it to account for the suppression of $\lambda_1$ at small $N$. How many correlation lengths does a ring need before it counts as large?

Further reading

• Lorenz, E. N. (1996). Predictability: a problem partly solved. Proceedings of the ECMWF Seminar on Predictability, vol. 1, 1–18.
• Lorenz, E. N. and Emanuel, K. A. (1998). Optimal sites for supplementary weather observations. Journal of the Atmospheric Sciences, 55, 399–414.
• Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability, §5.5 [citation needed: confirm section].
• Grassberger, P. (1989). Information content and predictability of lumped and distributed dynamical systems [citation needed] — extensivity of the Lyapunov spectrum in spatially extended systems.