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Reflection of Nonlinear Baroclinic Rossby Waves and the Driving of Secondary Mean Flows

Federico GraefCentro de Investigación Científica y de Educación Superior de Ensenada, Ensenada, Baja California, Mexico

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Lorenz MagaardDepartment of Oceanography, University of Hawaii, Honolulu, Hawaii

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Abstract

The reflection of weakly nonlinear Rossby waves (RWs) from a vertical wall is examined analytically through perturbation methods, with the β-Rossby number (ε) as the small parameter. A uniformly valid solution up to O(ε3) is constructed using multiple scales. At O(ε), the nonlinear interaction between an incident and the reflected RW leads to 1) an Eulerian steady flow, us(1), parallel to the (nonzonal) wall and 2) a transient flow oscillating with a frequency twice (2ω) that of the RW pair. The steady forcing, whose response is us(1), can never be resonant, which implies, under the weak nonlinear regime, that us(1) is stable to the driving RWs. At the next order, the nonlinear interaction between the incident-reflected RW pair and us(1) plus the transient flow produces, in general, resonant forcing leading to a modification of the RWs’ phases: a shift in their offshore wavenumber. The steady flow that occurs at O(ε3) is driven by the modified RWs as well as through interactions of several components of the solution up to second order; it is the next correction to us(1). This correction can be significant for reasonable wave parameters that allow, at the same time, a meaningful perturbative solution. The entire steady circulation induced by the nonlinear dynamics, up to O(ε3), is immune to resonances for |sinα| > ⅓, where α is the angle between the wall and the circles of latitude [except for the resonance that occurs at O(ε2)]. Thus, the waves produce a mean current, the mean current affects the waves, which change the current, and so on. There is new observational evidence of the existence of the North Hawaiian Ridge Current (Mysak and Magaard), which has been hypothesized as an RW-driven current.

Abstract

The reflection of weakly nonlinear Rossby waves (RWs) from a vertical wall is examined analytically through perturbation methods, with the β-Rossby number (ε) as the small parameter. A uniformly valid solution up to O(ε3) is constructed using multiple scales. At O(ε), the nonlinear interaction between an incident and the reflected RW leads to 1) an Eulerian steady flow, us(1), parallel to the (nonzonal) wall and 2) a transient flow oscillating with a frequency twice (2ω) that of the RW pair. The steady forcing, whose response is us(1), can never be resonant, which implies, under the weak nonlinear regime, that us(1) is stable to the driving RWs. At the next order, the nonlinear interaction between the incident-reflected RW pair and us(1) plus the transient flow produces, in general, resonant forcing leading to a modification of the RWs’ phases: a shift in their offshore wavenumber. The steady flow that occurs at O(ε3) is driven by the modified RWs as well as through interactions of several components of the solution up to second order; it is the next correction to us(1). This correction can be significant for reasonable wave parameters that allow, at the same time, a meaningful perturbative solution. The entire steady circulation induced by the nonlinear dynamics, up to O(ε3), is immune to resonances for |sinα| > ⅓, where α is the angle between the wall and the circles of latitude [except for the resonance that occurs at O(ε2)]. Thus, the waves produce a mean current, the mean current affects the waves, which change the current, and so on. There is new observational evidence of the existence of the North Hawaiian Ridge Current (Mysak and Magaard), which has been hypothesized as an RW-driven current.

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