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Modal and Nonmodal Symmetric Perturbations. Part I: Completeness of Normal Modes and Constructions of Nonmodal Solutions

Qin XuNOAA/National Severe Storms Laboratory, Norman, Oklahoma

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Abstract

It is shown that the classic normal modes for symmetric perturbations in a layer of vertically sheared basic flow can be classified into three types: paired growing and decaying modes, paired slowly propagating modes, and paired fast propagating modes. In the limit of vanishing growth rate (or frequency), the paired growing and decaying (or slowly propagating) modes degenerate into paired stationary and linearly growing modes. Degeneracies occur only on a discrete set (which is infinite but has zero measure) in the wavenumber space when the Richardson number is smaller than one. A nonmodal solution is not affected by the degenerate modes unless the solution is horizontally periodic and contains wavenumbers at which the degeneracy occurs. The classic modes and degenerate modes form a complete set in the sense that they contain all the wavenumbers and thus can construct any admissible nonmodal solutions.

The mode structures are analyzed by considering the slopes of their slantwise circulations relative to the absolute-momentum surface and isentropic surface of the basic state in the vertical cross section perpendicular to the circulation bands. The cross-band streamfunction component modes are shown to be orthogonal between different pairs (measured by the subspace inner product associated with the cross-band kinetic energy). The full-component modes, however, are nonorthogonal (measured by the full-space inner product associated with the total perturbation energy), and two paired modes have exactly opposite polarization relationships between the cross-band motion and its driving inertial–buoyancy force (associated with the along-band velocity and buoyancy perturbations). These properties have important implications for the nonmodal growths examined in Part II.

In association with the complete set of normal modes, a complete set of adjoint modes is derived along with the biorthogonal relationships between the normal modes and adjoint modes. By using the biorthogonality, nonmodal solutions can be conveniently constructed from the normal modes for the initial value problem.

Corresponding author address: Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: qin.xu@noaa.gov

Abstract

It is shown that the classic normal modes for symmetric perturbations in a layer of vertically sheared basic flow can be classified into three types: paired growing and decaying modes, paired slowly propagating modes, and paired fast propagating modes. In the limit of vanishing growth rate (or frequency), the paired growing and decaying (or slowly propagating) modes degenerate into paired stationary and linearly growing modes. Degeneracies occur only on a discrete set (which is infinite but has zero measure) in the wavenumber space when the Richardson number is smaller than one. A nonmodal solution is not affected by the degenerate modes unless the solution is horizontally periodic and contains wavenumbers at which the degeneracy occurs. The classic modes and degenerate modes form a complete set in the sense that they contain all the wavenumbers and thus can construct any admissible nonmodal solutions.

The mode structures are analyzed by considering the slopes of their slantwise circulations relative to the absolute-momentum surface and isentropic surface of the basic state in the vertical cross section perpendicular to the circulation bands. The cross-band streamfunction component modes are shown to be orthogonal between different pairs (measured by the subspace inner product associated with the cross-band kinetic energy). The full-component modes, however, are nonorthogonal (measured by the full-space inner product associated with the total perturbation energy), and two paired modes have exactly opposite polarization relationships between the cross-band motion and its driving inertial–buoyancy force (associated with the along-band velocity and buoyancy perturbations). These properties have important implications for the nonmodal growths examined in Part II.

In association with the complete set of normal modes, a complete set of adjoint modes is derived along with the biorthogonal relationships between the normal modes and adjoint modes. By using the biorthogonality, nonmodal solutions can be conveniently constructed from the normal modes for the initial value problem.

Corresponding author address: Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: qin.xu@noaa.gov

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