Modal instability with the influence of friction a

The effect of linear drag would be trivially simple if the friction coefficient was spatially uniform. There would simply be a reduction of the growth rate by an amount equal to the fric-tional coefficient a. In that case, the distribution of eigenvalues on a (ar, ai) plane would be identical to Fig. 2(a), except that the (ar) axis is to be relabeled as (ar + a). There would be no change in the structure of the modes. According to Fig. 2(a), there would be no unstable mode if the uniform friction coefficient is a = 0.03 > (ar )Zifid = 0.023.

To determine the impacts of spatially nonuniform friction on the instability, we need to make additional computations. The frictional coefficient for generic land surfaces may be several times larger than that for water surfaces at moderate wind speeds (Arya, 1988). Thus, we present the result for aocean = 0.015 and aiand = 0.045. The corresponding damping time scales are about 10 days and 3 days respectively. It is seen that the distribution of eigenvalues on the (ar,ai) plane is substantially different when there is differential friction (Fig. 4 vs Fig. 2(a)). Since we have aocean = 0.015 ^ aland = 0.045, it is not surprising to find only one remaining weakly unstable mode which has a small growth rate of about ar « 0.003. That weakly unstable mode is a single-jet mode solely associated with the model Pacific jet [similar to Fig. 3(a)]. Figure 5 is an example of the structure of one synoptic-frequency mode under the influence of differential friction. It has a synoptic length scale. It should be emphasized that this is a weakly stable mode with ar = —0.0095 and has a frequency of ai = 0.22. Its appearance is broadly similar to that of Fig. 3(c). One might then ask: "Would there still be a storm track(s) in this model under a parameter condition where there is no relevant unstable normal mode?" It is possible, if nonmodal growth of transient disturbances is sufficiently strong. The answer to this question can only be ascertained by performing nonlinear experiments with this model.

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