https://www.researchgate.net/publication/347514163
The interaction of Stokes waves with log profiles of the wind in the atmosphere and the wind drift in the ocean is studied. Data assimilation is used to nudge the wavy portion of the base flow toward a Stokes wave and the vortical portion of the base flow toward log profiles. The data assimilation framework allows for free-surface vorticity and the turbulent diffusion of free-surface vorticity into the atmosphere and into the ocean for the vortical portion of the flow.
The results of numerical simulations show that coherent structures form on the free surface and diffuse upward into the atmosphere and downward into the ocean even in the absence of stratification. As the crests of the Stokes waves pass over the coherent structures, the amplitudes of the coherent structures abruptly increase and the phases of the coherent structures abruptly change. As the friction velocity in the water increases, the coherent structures surf the crests of steep Stokes waves. The resulting turbulent wakes that form behind the wave crests induce meandering flows with vertical structures that are helical in the atmosphere and the ocean. Cross sections of the meandering flows transverse to the wind form layers with a diagonal pattern that reflects the helical shedding into the wake. The successive passing of coherent structures beneath the wave crests gives rise to a multitude of spatial and temporal variations. In view of the important contributions that the coherent structures and meandering flows make to the mixing of the atmosphere and the ocean, the physics associated with these spatial and temporal variations are herein named the Ocean’s Heartbeat. As discussed in this paper, the Ocean’s Heartbeat can be heard using electromagnetic principles.
Windrows form at the interstices of the coherent structures even in the absence of breaking waves. As the steepnesses of the Stokes waves increase, the rate of lateral spreading of the windrows in- creases, and the windrows become more rectilinear, less sinuous, and less diffuse. The numerical simulations show good agreement for the following experimental observations: 1) Convergence zones form beneath windrows, 2) Divergence zones form between windrows, 3) Downwelling oc- curs beneath windrows; 4) Upwelling occurs between windrows; 5) Streamwise velocities are enhanced beneath windrows; 6) Streamwise velocities are diminished between windrows; and 7) Y-Junctions that point upwind form as windrows merge. Although Y-Junctions that point down- wind have been observed, their importance has not been previously recognized. The numerical results show that Y-Junctions that point downwind form as windrows split.
The kinetic energy of the coherent structures and meandering flows increases linearly with respect to time in correspondence with forced two-dimensional turbulence and the formation of an inverse energy cascade. Also, in correspondence with forced two-dimensional turbulence, the enstrophy of the vertical component of vorticity is constant on average in a surface-following coordinate system in planes that are parallel to the free surface in both the atmosphere and the ocean. The simulation with the longest duration shows evidence of energy condensation as the length scales of the coherent structures approach the size of the computational domain. The flux of energy into the vortical portion of the flow increases as the wave steepness increases. The flux of energy into the vortical portion of the flow also increases as the friction velocities in the atmosphere and ocean increase relative to the phase speed of the Stokes wave.
The linear growth rate of energy in the vortical portion of the flow is comparable to the initial exponential growth rate of wind-driven ocean waves for steep Stokes waves with intermediate age. The physical scales in this study correspond to a region where there is significant scatter in Plant (1982)’s wave-growth measurements for inverse wave ages less than u∗/c_o < 0.4, where u∗ is the friction velocity in the air and c_o is the phase speed of the Stokes wave. The inverse energy cascade can be so strong that modulation of the waves through a feedback mechanism occurs. As the waves are modulated by the vortical portion of the flow, the inverse energy cascade momentarily breaks down and then reestablishes itself. It is conjectured that growing seas jump back and forth between states of two and three-dimensional turbulence as is evident in the growth of energy and the oscillations in entrophy. During this phase, wave breaking occurs in such a manner that windrows do not break up, which supports Dommermuth (2020)’s conjecture that spilling breaking occurs in lanes. The spilling breaking waves and coherent structures work in concert to form windrows! Numerical simulations with larger domains are required to clarify the physics.
Preliminary results indicate that the growth rate of the kinetic energy in the vortical portion of the flow for fixed friction velocity in the water scales according to the turbulent diffusion of the initial free-surface vorticity, which is expressed in terms of a non-dimensional Ocean’s Heartbeat number, β_v ∼ R_H. R_H = ω^s/(k^2ν^s) ≈ 500, where k is the wavenumber of the Stokes wave, ν^s is the two-dimensional eddy viscosity evaluated on the free surface, and ω^s is the initial vorticity on the free surface at the crest of the Stokes wave in an irrotational flow. For a steady flow, the initial free-surface vorticity is expressed in terms of the surface curvature (κ) and the total tangential velocity (u_t) evaluated on the free surface: ω_s = −2κ u_t. ω_s quantifies both the initial enstrophy for the initial boundary value problems starting from rest and the ongoing production of turbulence through interactions with shear. For variations of the wave friction velocity with fixed wave steepness, it is conjectured that conservation of wave action should be considered.
The Ocean’s Heartbeat number R_H reflects scaling in accordance with forced two-dimensional turbulence. The forcing here is provided by continuously nudging the wavy portion of the flow to an irrotational Stokes wave while the vortical portion of the flow is nudged toward log profiles in the atmosphere and the ocean. The effects of the friction velocities in the atmosphere and the ocean are included indirectly in the eddy viscosity.
The formation of Langmuir circulations is associated with an inverse energy cascade. The forma- tion of windrows with large lateral spreading is the manifestation of this inverse energy cascade. Meandering flows form large coherent structures in the atmosphere and ocean due to this inverse energy cascade. Meandering flows are fundamental mechanisms for mixing in the ocean and atmosphere over large spatial and temporal scales. The Ocean’s Heartbeat is an extraordinary mechanism by which the wavy portion of the flow strongly forces the vortical portions of the flow in the atmosphere and the ocean.