Optimality of a nonlinear stability criterion of two-layer Phillips model

The optimality of a nonlinear stability criterion of motions of two_layer Phillips model is investigated. The result shows that for a given zonal periodical channel, there exist cases that there is no normal mode which grows exponentially even if the sufficient conditions of the criterion are destroyed. On the other hand, the above criterion is optimal in the following sense: when the sufficient conditions of the criterion are destroyed, except some critical cases, there exists a finite zonal periodical channel, in which there is a zonal normal mode growing exponentially.

The optimdity of a nonlinear stability criterion of motions of twelayer Phillips model is investigated. The result shows that for a given zonal periodical channel, there exist cases that there is no normal mode which grows exponentially even if the sufficient conditions of the criterion are destroyed. On the other hand, the above criterion is optimal in the following sense: when the sufficient conditions of the criterion are destroyed, except some critical cases, there exists a finite d periodical channel, in which there is a zonal normal mode growing exponentially.