Tag: Coherent Structures

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The Ocean’s Heartbeat: An Inverse Energy Cascade that Mixes the Lower Atmosphere and Upper Ocean

Oblique-wave instabilities affect spilling breaking waves when the local wave steepness exceeds H/λ_o > 0.08, where H is the height and λ_o is the wavelength. Close to the ocean surface, spilling breaking waves resonate with Langmuir circulations in both the atmosphere and ocean. Away from the ocean surface, larger Langmuir circulations form under the action of inverse energy cascades. Surface currents due to Langmuir circulations and surfing effects due to spilling breaking waves affect the formation of windrows. The formation of meandering flows with broad spatial and temporal scales affects mixing in the atmosphere and ocean. Meandering drift currents have spatial and temporal scales conducive to the formation of internal waves in the ocean.

https://www.researchgate.net/publication/361643897

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Modeling the Ocean’s Heartbeat

Wave breaking affects turbulent fluctuations close to the free surface in the atmosphere and ocean. Away from the free surface, the large coherent structures that form through an inverse energy cascade in the atmosphere and ocean are not very sensitive to wave breaking. The formation of windrows is affected by the surfing and scrubbing actions of spilling breaking waves (Dommermuth, 2020f,g) and the surface currents that are induced by coherent structures transverse to the wind (Langmuir, 1938). Recent videos of windrows are suggestive of the complex interplay that can occur between wave breaking and surface currents (Tunli, 2021a,b,c,d,e).

https://www.researchgate.net/publication/352410045

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An Annotated Bibliography of the Ocean’s Heartbeat

Dommermuth (2020) shows that Langmuir circulations form under the action of an inverse energy cascade. The Ocean’s Heartbeat (OH) is an inverse energy cascade that occurs through interactions between surface gravity waves and organized vortical structures in the atmosphere and the ocean. Windrows are visible manifestations of the inverse energy cascade. The vortical portion of the flow and spilling breaking waves work in combination to generate windrows. Breaking waves generate meandering currents and winds in the oceanic and atmospheric boundary layers. This report provides an annotated bibliography of research on the Ocean’s Heartbeat.

https://www.researchgate.net/publication/351234250

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The Ocean’s Heartbeat

The effects of the Ocean’s Heartbeat on the ocean and atmosphere are profound. My ultimate aim is to help establish a research group to study the Ocean’s Heartbeat using petascale supercomputing resources. Data assimilation will be used to nudge iLES of two-phase flows with VOF interface capturing to improve our understanding of the Ocean’s Heartbeat. I also feel that it is important to lay the foundation for graduate studies in this research area. I welcome opportunities to work with research groups from other countries. If you share my vision, contact me at [email protected].

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Whitecaps, Inverse Energy Cascades, and Energy Budgets

https://www.researchgate.net/publication/350874498

Ocean waves induce vortical flows in the oceanic and atmospheric boundary through an inverse energy cascade. Depending on the growth rate, the energy density in the vortical portion of the flow is about 6-10% of the wave kinetic energy. I call the mechanics of the energy transfer the Ocean’s Heartbeat (Dommermuth, 2020g). The depths and heights of the computational domain are currently 1/2 a wavelength, but the vortical portion of the flow will diffuse higher into the atmosphere and deeper into the ocean. Over 50-100 wave periods, large-scale coherent structures fill the width of the computational domain, which at present is no greater than one wavelength wide due to the limits of my computational resources. The length of the computational domain should be greater than two wavelengths to permit interactions between successive whitecaps. There are strong interactions between the vortical and wavy portions of the flow that induce whitecaps. The shedding of vorticity out the back of whitecaps feeds the vortical portion of the flow. This study of the energy density of the vortical portion of the flow is driven by atmospheric forcing. Another study of the energy density is underway whereby log profiles of the wind and wind drift are imposed using data assimilation.

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Frequency Downshifting and Inverse Energy Cascades

https://www.researchgate.net/publication/348419601

The occurrence of frequency downshifting and angular spreading under the influence of Benjamin-Feir instabilities and spilling breaking waves is confirmed. Knots and upwellings form on the free surface in the wake of spilling breaking waves. Upwellings are due to vortex rings. Knots are due to intense monopoles, dipoles, tripoles, and quadrupoles. Coherent structures in the ocean and atmosphere resonate with oblique wave modes. Whitecaps indicate that frequency downshifting and angular spreading are occurring-just as windrows and knots indicate the presence of an inverse energy cascade 

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The Ocean’s Heartbeat

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.

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Research Papers and Reports

Langmuir Circulations and Meandering Flows

Windrows

The Ocean’s Heartbeat

Electric and Magnetic Fields of Meandering Flows

  • Dommermuth, D. G., “Magnetic Induction due to the Effects of Breaking Ocean Waves,” ResearchGate preprint, Oct 2020.
    https://www.researchgate.net/publication/344482876
  • Dommermuth, D. G., “The Generation of Electric Fields by Meandering Flows,” ResearchGate preprint, Oct 2020. https://www.researchgate.net/publication/344787449 (Please see more recent technical reports on the electric field that is induced by the transport of space charge density by the meandering wind.)
  • Dommermuth, D. G., “The Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate preprint, Sep 2021. https://www.researchgate.net/publication/354665883 (Please see more recent technical reports on the electric field that is induced by the transport of space charge density by the meandering wind.)
  • Dommermuth, D.G., “The Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate preprint, Sep 2021. https://www.researchgate.net/publication/354935485
  • Dommermuth, D.G., “The Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate preprint, Oct 2021. https://www.researchgate.net/publication/355215804
  • Dommermuth, D.G., “The Magnetic Fields due to Magnetic Induction by Meandering Drift Currents,” ResearchGate preprint, Oct 2021. https://www.researchgate.net/publication/355651651
  • Dommermuth, D.G., “A Parametric Study of the Electric Field in the Atmosphere due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate preprint, Nov 2021. https://www.researchgate.net/publication/356002487

Maxima Scripts for Meandering Flows

  • Dommermuth, D.G., “Maxima Coding for Solving the Electric and Magnetic Fields due to the Transport of Space Charge Density over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate code, Sep 2021. https://www.researchgate.net/publication/354935522
  • Dommermuth, D.G., “A Maxima Script for Solving the Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355209396

Fortran Codes for Meandering Flows

  • Dommermuth, D.G., “A Fortran Code for Calculating Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate code, Sep 2021. https://www.researchgate.net/publication/354935467
  • Dommermuth, D.G., “A Fortran Code for Calculating the Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355209298
  • Dommermuth, D.G., “F90 Coding for Calculating the Magnetic Fields due to Magnetic Induction by Meandering Drift Currents,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355651566
  • Dommermuth, D.G., “F90 Coding for a Parametric Study of the Electric Field in the Atmosphere due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate code, Nov 2021. https://www.researchgate.net/publication/356002311

The Effect of the Wind Drift on Wave Growth, Wave Breaking, and the Production of Turbulence

The Effect of Standing Waves on the Wave Energy Cascade

Helmholtz Decompositions into Wavy and Vortical Portions

  • Dommermuth, D. G., “The laminar interactions of a pair of vortex tubes with a free surface,” J. Fluid Mech., Vol. 246, 1993, pp. 91–115. https://doi.org/10.1017/S0022112093000059
  • Mui, R. C. and Dommermuth, D. G., “The vortical structure of a near-breaking gravity-capillary wave,” Journal of Fluids Engineering, Vol. 117, 1994,355–361. https://doi.org/10.1115/1.2817269
  • Dommermuth, D. G., Novikov, E.A., and Mui, C.Y., “The Interaction of Surface Waves with Turbulence,” The Proceedings of the Symposium on Free-Surface Turbulence, ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, California, USA, 1994. https://www.researchgate.net/publication/271527603

Entrainment and Mixing due to Plunging Breaking Waves

Numerical Methods