TY - JOUR

T1 - Relating optimal growth to counterpropagating Rossby waves in shear instability

AU - Heifetz, Eyal

AU - Methven, John

N1 - Funding Information:
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.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21244475975&partnerID=8YFLogxK

U2 - 10.1063/1.1937064

DO - 10.1063/1.1937064

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AN - SCOPUS:21244475975

SN - 1070-6631

VL - 17

SP - 1

EP - 14

JO - Physics of Fluids

JF - Physics of Fluids

IS - 6

M1 - 064107

ER -