Diffusive Transport by Breaking Waves

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  • 1 Climate System Research Program, Department of Meteorology, Texas A&M University, College Station, Texas
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Abstract

A simple conceptual model of the relationship between advective transport by breaking waves and diffusive transport is derived. line model postulates that the displacement of fluid parcels by a breaking wave is analogous to molecular diffusion (in a manner similar to conventional mixing length theory). Unlike molecular diffusion, in which the fluctuations of nearby particles are independent, in fluid flow the motion of nearby parcels can be coherent. The effectively random phase of wavebreaking events with respect to an individual particle, however, results in a macroscopic “random walk” of the parcel with a step size related to the width of the breaking wave and a timescale related to the frequency of wave breaking events. As in the case of molecular diffusion, the motion of individual molecules or fluid parcels is unpredictable (chaotic, in fact), but averaging over a large ensemble of parcels results in an ensemble variance that increases linearly with time, formally equivalent to molecular diffusion of a gas.

Examples are shown of isentropic parcel dispersion in the stratosphere. Lagrangian trajectory calculations indicate that the latitudinal dispersion of air parcels increases linearly with time in wavebreaking regions, but first increases and then levels off in regions where little or no wavebreaking occurs. A simple model for barriers to mixing is also proposed.

These results suggest a new approach to parameterization of mixing in numerical models. The required parameters are the distributions of the frequency, width, and location of wavebreaking events. These in turn be related to the frequency, phase speed, and amplitude of the breaking waves, and the locations of critical lines in the fluid.

Abstract

A simple conceptual model of the relationship between advective transport by breaking waves and diffusive transport is derived. line model postulates that the displacement of fluid parcels by a breaking wave is analogous to molecular diffusion (in a manner similar to conventional mixing length theory). Unlike molecular diffusion, in which the fluctuations of nearby particles are independent, in fluid flow the motion of nearby parcels can be coherent. The effectively random phase of wavebreaking events with respect to an individual particle, however, results in a macroscopic “random walk” of the parcel with a step size related to the width of the breaking wave and a timescale related to the frequency of wave breaking events. As in the case of molecular diffusion, the motion of individual molecules or fluid parcels is unpredictable (chaotic, in fact), but averaging over a large ensemble of parcels results in an ensemble variance that increases linearly with time, formally equivalent to molecular diffusion of a gas.

Examples are shown of isentropic parcel dispersion in the stratosphere. Lagrangian trajectory calculations indicate that the latitudinal dispersion of air parcels increases linearly with time in wavebreaking regions, but first increases and then levels off in regions where little or no wavebreaking occurs. A simple model for barriers to mixing is also proposed.

These results suggest a new approach to parameterization of mixing in numerical models. The required parameters are the distributions of the frequency, width, and location of wavebreaking events. These in turn be related to the frequency, phase speed, and amplitude of the breaking waves, and the locations of critical lines in the fluid.

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