## Nonlinear Model Storm Tracks

A number of numerical integrations of (2) are performed to investigate the relative intensity of the nonlinear model storm tracks under various parameter conditions. We use a time step of At = 0.05 in units of LU-1, corresponding to At = 0.8 x 103s « 14min. Each integration starts with an unbiased initial disturbance consisting of ten waves,

with a finite magnitude poo = 0.05. Each integration lasts for 1800 nondimensional time units (equivalent to ^354 days).

Let us first consider a control run in which we use uoo = 0.2 with a uniform friction coefficient for the whole domain, a = 0.03. The cos

Figure 4. Distribution of eigenvalues of normal modes under the influence of differential friction (aocean = 0.015, aland = 0.045). The unit of time is LU-1 =0.17 x 105 s.
Figure 5. Structure of a normal mode with ar = —0.0095, a = 0.22 under the influence of differential friction.

corresponding damping time scale is about 5 days. An overall measure of the intensity of an instantaneous disturbance field is the spatial variance of the departure stream function from that of the reference state. Figure 6 shows that this system has evolved to an equilibrated state for a sufficiently long time interval that robust statistical properties can be deduced

from the model output of the last 1000 time units. Even though none of the normal modes are unstable because a = 0.03 > (ar)miXscld = 0.023, an initial disturbance field is nevertheless able to intensify via transient growth (alternatively referred to as nonmodal instability) to a sufficiently significant magnitude that finite disturbances are repeatedly regenerated by continually extracting energy from the external forcing. The equilibrated state is established after about 100 days. The equilibrated flow is also maintained by continual self-sustained transient growth. This is compatible with the explicit assumption invoked in a linear stochastic model of storm tracks (e.g. Whitaker and Sardeshmukh, 1998). The output for the last 250 days is used to compute the statistics of the flow.

We will discuss the characteristics of the related model storm tracks in Subsec. 4.2.

### 4.1. Eddy feedback

Let us first examine the time mean equilibrated departure stream function from that of the reference state, { = ({¡total - Vref) (Fig. 7). The velocity vector field of the reference state is also plotted in this figure for comparison. This departure field, {¡, is induced by the mean feedback effect of the eddies. A dipole is induced in the region immediately downstream of each jet. It consists of an anticyclonic gyre to its north and a cyclonic gyre to its south. The dipole over the Atlantic sector is slightly stronger than the one over the Pacific sector (maximum

Figure 7. Velocity field of the reference state (vectors) and the time mean equilibrated departure stream function from that of the reference flow (contours) for the case of Uoo = 0.2 and a drag coefficient a = 0.03.
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