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Internal Gravity Wave Generation and Hydrodynamic Instability

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  • 1 Department of Physics, University of Toronto, Toronto, Ontario, Canada
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Abstract

Two mechanisms are proposed whereby internal gravity waves (IGW) may radiate from a linearly unstable region of Boussinesq parallel flow that is characterized in the far field by constant horizontal velocity and Brunt-Väisälä frequency. Through what is herein referred to as “primary generation,” IGW may be directly excited by linear instability of the initial-state parallel shear flow. Characteristically, these waves propagate with horizontal phase speed and wavenumber equal to that of the most unstable mode of linear stability theory. Through the second mechanism, referred to as “secondary generation,” IGW may be excited via nonlinear modification of the initial instability into a form that couples strongly to a large amplitude outgoing internal wave field. The authors propose that the primary generation of IGW may occur provided a penetration condition, which is derived on the basis of linear theory, is satisfied. The penetration condition provides a limit on the growth rate of a disturbance of any particular frequency that is capable of propagating into the far field. This hypothesis is supported by a sequence of representative nonlinear numerical simulations in two spatial dimensions for both free mixing layer and jet flows with horizontal velocity profiles U(z) = tanh (z) and U(z) = sech2(z), respectively. For the purpose of these analyses, the fluid density is taken to be such that the square of the Brunt–Väisälä frequency is given by N2(z) = J tanh2(z/R). Such stratification allows both for the development of large-scale eddies in the region of low static stability and, in the far field where N2J is positive and approximately constant, for the radiation of a broad frequency spectrum of IGW.

Abstract

Two mechanisms are proposed whereby internal gravity waves (IGW) may radiate from a linearly unstable region of Boussinesq parallel flow that is characterized in the far field by constant horizontal velocity and Brunt-Väisälä frequency. Through what is herein referred to as “primary generation,” IGW may be directly excited by linear instability of the initial-state parallel shear flow. Characteristically, these waves propagate with horizontal phase speed and wavenumber equal to that of the most unstable mode of linear stability theory. Through the second mechanism, referred to as “secondary generation,” IGW may be excited via nonlinear modification of the initial instability into a form that couples strongly to a large amplitude outgoing internal wave field. The authors propose that the primary generation of IGW may occur provided a penetration condition, which is derived on the basis of linear theory, is satisfied. The penetration condition provides a limit on the growth rate of a disturbance of any particular frequency that is capable of propagating into the far field. This hypothesis is supported by a sequence of representative nonlinear numerical simulations in two spatial dimensions for both free mixing layer and jet flows with horizontal velocity profiles U(z) = tanh (z) and U(z) = sech2(z), respectively. For the purpose of these analyses, the fluid density is taken to be such that the square of the Brunt–Väisälä frequency is given by N2(z) = J tanh2(z/R). Such stratification allows both for the development of large-scale eddies in the region of low static stability and, in the far field where N2J is positive and approximately constant, for the radiation of a broad frequency spectrum of IGW.

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