^{1,a)}and John Methven

^{2,b)}

### Abstract

The optimal dynamics of conservative disturbances to plane parallel shear flows is interpreted in terms of the propagation and mutual interaction of components called counterpropagating Rossby waves (CRWs). Pairs of CRWs were originally used by Bretherton to provide a mechanistic explanation for unstable normal modes in the barotropic Rayleigh model and baroclinic two-layer model. One CRW has large amplitude in regions of positive mean cross-stream potential vorticity (PV) gradient, while the second CRW has large amplitude in regions of negative PV gradient. Each CRW propagates to the left of the mean PV gradient vector, parallel to the mean flow. If the mean flow is more positive where the PV gradient is positive, the intrinsic phase speeds of the two CRWs will be similar. The CRWs interact because the PV anomalies of one CRW induce cross-stream velocity at the location of the other CRW, thus advecting the mean PV. Although a single Rossby wave is neutral, their interaction can result in phase locking and mutual growth. Here the general initial value problem for disturbances to shear flow is analyzed in terms of CRWs. For the discrete spectrum (which could alternatively be described using normal modes), the singular value decomposition of the dynamical propagator can be obtained analytically in terms of the CRW interaction coefficient and the intrinsic CRW phase speeds. Using this formalism, optimal perturbations, the disturbances which grow fastest in a given norm over a specified time interval, can readily be found. The most natural norm for CRWs is related to air parcel displacements or enstrophy. However, if an energy norm is taken, it is shown to grow due to both mutual amplification of air parcel displacements and the untilting of PV structures (the Orr mechanism) associated with decreasing phase difference between the CRWs. A generalization of the CRW description to the optimal dynamics of the complete spectrum solution is outlined. Although the dynamics then involves the interaction between an infinite number of “CRW kernels,” the form of the simple interaction between any two CRW kernels is the same as in the discrete case.

E.H. wishes to thank Michael McIntyre and Orkan Mehmet Umurhan for illuminating discussions. J.M. is grateful for an Advanced Fellowship sponsored jointly by the Natural Environment Research Council and the Environment Agency.

I. INTRODUCTION

II. CRW INTERACTION ILLUSTRATED USING THE RAYLEIGH MODEL

III. GENERALIZED STABILITY THEORY IN TERMS OF CRWS

A. Formulation

B. Normal modes in terms of CRWs

C. Optimal growth in the enstrophy norm

1. Optimal growth in the unstable modal regime

2. Optimal growth in the neutral modal regime

3. Wavelength dependence of optimal growth

D. Optimal growth in the energy norm

IV. CONCLUDING REMARKS

### Key Topics

- Normal modes
- 33.0
- Vortex dynamics
- 27.0
- Eigenvalues
- 16.0
- Shear flows
- 12.0
- Rheology and fluid dynamics
- 6.0

## Figures

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.

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 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 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.

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.

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.

(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

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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