Starting from nonlinear equations on the f-plane containing frictional dissipation under the Boussinesq approximation, a new kind of generalized energy is proposed as the Lyapunov function, and averages are taken as any functions of (x, z) instead of the commonly-used means of bilinear functions of (x, z), thereby resulting in a new criterion of generalized nonlinear symmetric stability. It shows that not only must the dissipative coefficient be greater than a certain critical value but the initial disturbance amplitude must be synchronously smaller than another marginal value as well. It follows that the latter imposes a crucial constraint on the former, thus leading to the fact that when the amplitude is bigger compared to another critical value, generalized nonlinear subcritical symmetrical instability may occur. The new criterion contributes greatly to the improvement of the previous results of its kind.
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