Schematic illustration of the counterpropagating Rossby waves (CRWs) in the Rayleigh model in the configuration of the most unstable normal mode (wavenumber ). Blank arrows indicate the basic state velocity on the edges . Bold horizontal arrows represent the CRWs’ propagation direction. The edge positive and negative vorticity anomalies are indicated by and the circulation they induce is illustrated by the circled arrows and by the cross-stream solid arrows located out of phase of . The undulating solid lines illustrate the CRWs’ cross-stream displacement, where the basic state vorticity at the vicinity of the edges is given by (1b). The two CRWs are phase-locked with phase difference . The dashed arrows indicate the velocity induced by each CRW on the opposite edge [attenuated by , (3b)]. Since the phase difference each CRW advects the basic state vorticity on the opposite edge in a way that makes other CRW grow and hinders its natural propagation.
The CRW parameters which determine the Rayleigh normal mode dispersion relation as functions of the normalized horizontal wavenumber . The modal growth rate , (11), normalized by the shear , is indicated by asterisks. The positive branch of the modal phase speed, (14), normalized by is shown by the plus symbols. The cosine of the CRW phase difference in the growing mode configuration, , (8), is indicated by circles. The interaction coefficient , (7a), normalized by , and the intrinsic upper CRW phase speed , (7c), normalized by , are indicated, respectively, by the solid and the dashed lines.
The relative CRW phase angle in the growing regime of . corresponds to the CRW phase difference in the growing normal mode configuration, while corresponds to its biorthogonal configuration. The maximal instantaneous growth occurs at . For finite target time, (19) indicates that the initial and final optimal phases are symmetric with respect to . Hence for all target times, when is in the hindering regime, increases during the optimal evolution (filled arrows), and is positive. For target time infinity .
The ratio between the optimal growth and the normal mode growth as a function of and , in the regime where each CRW must help the counterpropagation of the other to enable phase-locking. are the CRW phase difference of the normal mode and the final phase difference of the optimal perturbation.
Optimal evolution in the modal stable regime. In this example , (12b), and the consecutive normalized target times . (a) The optimal growth (29b) is indicated by the solid line. The global optimal (30a) is indicated by the dashed line and is achieved first at the normalized time . (b) The initial and final CRW optimal phase differences and vs target time are indicated, respectively, by the dashed and the solid lines. Note that the phase difference always decreases with time during optimal evolution.
(a) The normalized effective optimal growth rate , (31), in the Rayleigh model, as a function of wavenumber, for the normalized target times , indicated, respectively, by the dashed, dashed-dot, and solid lines. These are compared with the normalized instantaneous growth rate , (7a), (bold dots), and the normal mode growth rate (asterisks). (b) The optimal change in phase difference between CRWs, , as a function of , for the three target times. The change in phase difference required for maximal growth is and is achieved as target time tends to infinity.
The maximal instantaneous growth rate in the energy norm (dashed line) vs the maximal instantaneous growth rate in the enstrophy norm (dashed dot line) and the normal mode growth rate (solid line), as functions of wavenumber
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