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Stable Linearized Waves in a Shear Flow

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Dynamics of the Equatorial Ocean

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Abstract

Instability is not the only trick of waves interacting with mean flows. Equatorial waves may be refracted by the currents and absorbed at critical surfaces where the phase speed matches the mean current. Atmospheric sciences pioneered such studies in the 60s; here, similar perturbative and arithmurgical methods are deployed in the ocean. Linearized about a mean flow that varies with both latitude and depth, the dynamics can be reduced to a single second order partial differential equation of mixed elliptic-hyperbolic type. This can be solved numerically by finite differences accompanied by sparse direct elimination methods. (Iterative methods are unreliable because the equation is of mixed elliptic-hyperbolic type.) The numerical computations are credible if care is taken near the critical surfaces where the mean flow matches the phase speed of the waves. The same wave-in-bivariate-mean-shear problems can also be attacked analytically by multiple scales perturbation theory. The horrendous algebra usual to this approach is enormously reduced by invocation of wave-mean flow conservations laws.

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Notes

  1. 1.

    The wave scale is not the wavelength, but rather the wavelength divided by \(2 \pi \); \(\cos (k z)\) varies on a length scale of 1 / k since each differentiation extracts a factor of k, i.e., \(u_{zz}=-k^{2} u\), but the wavelength is \(2 \pi /k\).

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Authors and Affiliations

  1. CLaSP Department, University of Michigan College of Engineering, Ann Arbor, MI, USA

    John P. Boyd

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  1. John P. Boyd
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Correspondence to John P. Boyd .

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Boyd, J.P. (2018). Stable Linearized Waves in a Shear Flow. In: Dynamics of the Equatorial Ocean. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55476-0_12

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