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Multiscale Momentum Flux and Diffusion due to Whitecapping in Wave–Current Interactions

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  • 1 Department of Mathematics, Department of Atmospheric Sciences, and Physics Department, The University of Arizona, Tucson, Arizona
  • | 2 Universidad Nacional de Colombia, Sede Medellin, Escuela de Matemáticas, Medellin, Colombia
  • | 3 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
  • | 4 School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia
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Abstract

Whitecapping affects the Reynolds stresses near the ocean surface. A model for the conservative dynamics of waves and currents is modified to include the averaged effect of multiple, short-lived, and random wave-breaking events on large spatiotemporal scales. In this study’s treatment, whitecapping is parameterized stochastically as an additive uncertainty in the fluid velocity. It is coupled to the Stokes drift as well as to the current velocity in the form of nonlinear momentum terms in the vortex force and the Bernoulli head. The effects of whitecapping on tracer dynamics, mass balances, and boundary conditions are also derived here. Whitecapping also modifies the dynamics and the size of the sea surface boundary layer. This study does not resolve the boundary layer, however, the authors appeal to traditional viscosity parameterizations to include these diffusive effects, modified for the context of wave–current interactions.

The parameterized breaking velocity field is endowed with empirical rules that link their generation in space and time to properties and dynamics of wave groups. The energy convergence rate of wave groups is used as an indicator for the onset of wave breaking. A methodology is proposed for evaluating this criterion over an evolving random Gaussian model for the ocean surface. The expected spatiotemporal statistics of the breaking events are not imposed, but rather computed, and are found to agree with the general expectation of its Poisson character. The authors also compute, rather than impose, the shear stress associated with the breaking events and find it to agree with theoretical expectations.

When the relative role played by waves and breaking events on currents is compared, this study finds that waves, via the vortex force, purely advect the vorticity of currents that are essentially only dependent on transverse coordinates. The authors show that currents will tend to get rougher in the direction of steady wind, when whitecapping is present. Breaking events can alter and even suppress the rate of advection in the vortex force. When comparing the rates of transport, the waves will tend to dominate the short term and the whitecapping of the long-term rate.

Corresponding author address: J. M. Restrepo, Dept. of Mathematics, The University of Arizona, Tuscon, AZ 85721. E-mail: restrepo@physics.arizona.edu

Abstract

Whitecapping affects the Reynolds stresses near the ocean surface. A model for the conservative dynamics of waves and currents is modified to include the averaged effect of multiple, short-lived, and random wave-breaking events on large spatiotemporal scales. In this study’s treatment, whitecapping is parameterized stochastically as an additive uncertainty in the fluid velocity. It is coupled to the Stokes drift as well as to the current velocity in the form of nonlinear momentum terms in the vortex force and the Bernoulli head. The effects of whitecapping on tracer dynamics, mass balances, and boundary conditions are also derived here. Whitecapping also modifies the dynamics and the size of the sea surface boundary layer. This study does not resolve the boundary layer, however, the authors appeal to traditional viscosity parameterizations to include these diffusive effects, modified for the context of wave–current interactions.

The parameterized breaking velocity field is endowed with empirical rules that link their generation in space and time to properties and dynamics of wave groups. The energy convergence rate of wave groups is used as an indicator for the onset of wave breaking. A methodology is proposed for evaluating this criterion over an evolving random Gaussian model for the ocean surface. The expected spatiotemporal statistics of the breaking events are not imposed, but rather computed, and are found to agree with the general expectation of its Poisson character. The authors also compute, rather than impose, the shear stress associated with the breaking events and find it to agree with theoretical expectations.

When the relative role played by waves and breaking events on currents is compared, this study finds that waves, via the vortex force, purely advect the vorticity of currents that are essentially only dependent on transverse coordinates. The authors show that currents will tend to get rougher in the direction of steady wind, when whitecapping is present. Breaking events can alter and even suppress the rate of advection in the vortex force. When comparing the rates of transport, the waves will tend to dominate the short term and the whitecapping of the long-term rate.

Corresponding author address: J. M. Restrepo, Dept. of Mathematics, The University of Arizona, Tuscon, AZ 85721. E-mail: restrepo@physics.arizona.edu
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