A data assimilation tutorial based on the Lorenz-95 system

where i = 1, 2 . . . 40, and cyclic boundary conditions are used x0 = x40, x−1 = x39, x41 = x1. The magnitude of the forcing is set to F = 8. For this forcing the system is chaotic, i.e. it has positive Lyapunov exponents. Lorenz (1995) concluded that similar error growth characteristics to operational NWP systems are obtained if a time unit in the L95-system is associated with 5 days. This scaling will be used here, too. Solutions of the system are obtained by numerical integration with a fourth-order Runge-Kutta scheme using a 3-hour time-step (∆t = 0.025). For the chosen forcing, the system has 13 positive Lyapunov exponents, the largest corresponds to a doubling time of 2.1 days. The dynamics is the 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 climatological standard deviation of σclim ≡ sigma clim = 3.6. A perturbation of the initial condition will grow with time and its leading edge propagates “eastward” (to higher indices) at a speed of about 25 degrees/day — this corresponds to a shift of 14 indices in a (nondimensional) time unit. See Lorenz (1995) and Lorenz and Emanuel (1998) for a more detailed discussion of the system.