Influence of the beta-effect on non-modal baroclinic instability

We focus on the influence of the beta -effect on the singular modes of baro clinic instability in the Phillips model. An analytical intercomparison bet ween normal and singular modes for the Eady problem on an f-plane has alrea dy been performed, showing that the amplification rate of the singular mode for the kinetic-energy norm after a dimensional time of three days is abou t 1.2 times larger than for the normal mode. We show here that, in the pres ence of the beta -effect, the maximum amplification rate of the singular mo de can be 1.5 times larger than for the normal-mode case. This difference i s due to the lesser stabilizing effect of beta on the singular modes compar ed with the normal modes. This result is obtained for the case of equal lay er depths which is relevant for the atmospheric situation. For the oceanic situation with unequal layer depths the amplification due to beta is furthe r increased leading to a factor of 1.8 in amplification rates between the t wo types of modes after a dimensional time of about 20 days. This behaviour is explained by the fact that beta only intervenes in the anti-Hermitian p art of the linear dynamical operator, the main effect of which is to alter the vertical phase tilt of the perturbations. As a consequence, the structu re of the bi-orthogonal mode becomes more inclined to the vertical in the p resence of beta while, on the contrary, the structure of the unstable norma l mode becomes less inclined.