Tutorial

A data assimilation tutorial based on the Lorenz-95 systemMartin Leutbecher, ECMWF, May 5, 20041. Introductiona. The Lorenz-95 systemWe will explore different data assimilation schemes using a low-dimensional dynamicalsystem introduced by Edward Lorenz in 1995. The system is given bydxi=−x x + x x − x + F, (1)i−2 i−1 i−1 i+1 idtwhere i = 1,2...40, and cyclic boundary conditions are used x = x , x = x , x =0 40 −1 39 41x . Themagnitudeoftheforcingissetto F = 8. Forthisforcingthesystemischaotic,i.e.1it has positive Lyapunov exponents. Lorenz (1995) concluded that similar error growthcharacteristics to operational NWP systems are obtained if a time unit in the L95-systemis associated with 5 days. This scaling will be used here, too. Solutions of the system areobtainedbynumericalintegrationwithafourth-orderRunge-Kuttaschemeusinga3-hourtime-step (Δt = 0.025). For the chosen forcing, the system has 13 positive Lyapunovexponents, the largest corresponds to a doubling time of 2.1 days. The dynamics isthe same for each variable as eqn. (1) is invariant under the transformation i → i +1. Variables fluctuate about the mean in a non-periodic manner with a climatologicalstandard deviation of σ ≡ sigma clim = 3.6. A perturbation of the initial conditionclimwill grow with time and its leading edge propagates “eastward” (to higher indices) ata speed of about 25 degrees/day — this corresponds to a shift of 14 indices in a (non-dimensional) time unit. See Lorenz (1995) ...