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The Stokes Drift and Mean Flows Induced by Two-Dimensional Internal Gravity Wave Packets

The Stokes Drift and Mean Flows Induced by Two-Dimensional Internal Gravity Wave Packets. Ton van den Bremer 1,2 & Bruce Sutherland 2,3 1. Department of Engineering Science, University of Oxford 2. Woods Hole Oceanographic Institution

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The Stokes Drift and Mean Flows Induced by Two-Dimensional Internal Gravity Wave Packets

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  1. The Stokes Drift and Mean Flows Induced by Two-Dimensional Internal Gravity Wave Packets Ton van den Bremer1,2 & Bruce Sutherland2,3 1. Department of Engineering Science, University of Oxford 2. Woods Hole Oceanographic Institution 3. Depts. Physics and Earth & Atmospheric Sciences, University of Alberta EUROPEAN GEOSCIENCES UNION, APRIL 2014

  2. Motivation: Periodic surface gravity waves Orbit doesn’t close: net horizontal drift From Wallet & Ruellan (1950)

  3. Motivation: Periodic surface gravity waves Solution to linearized problem for periodic waves: From Wallet & Ruellan (1950) Circular motion to leading order O(α) and net horizontal drift at next order O(α2), where α=ak (steepness).

  4. Motivation: Surface gravity wave packets

  5. Motivation: Surface gravity wave packets • A deep return flow below surface waves ensures zero vertically-integrated momentum (at x=cgt). From Longuet-Higgins & Stewart (1962), see also McIntyre (1981).

  6. What are internal gravity waves? Surface gravity waves Internal gravity waves • Free surface • Gravity is restoring force • Fluid has uniform density • Stratification: layers of different density • Buoyancy is restoring force

  7. Stokes drift for periodic internal gravity waves (igws) x- and z-periodic x-periodic and z compact No Stokes drift Horizontal Stokes drift

  8. Problem definition • What is the structure of the Stokes drift / induced mean flow for a vertically and horizontally localized internal wave-packet? • Assumptions: • Uniformly stratified ambient, • Boussinesq approximation, • No dissipation, • Weakly non-linear: no feedback of induced flow on packet propagation, • Infinite two-dimensional domain (x,z), • Approach: • Separation of scales expansion of non-linear governing equations, • Fully non-linear numerical simulations.

  9. Governing equations Two-dimensional Boussinesq internal waves in linearly stratified fluid (stable): Momentum (x and z), internal energy and incompressibility: Vertical displacement

  10. Leading-order Induced mean flow

  11. Leading-order Induced mean flow Mean-flow response O(α2) O(ε4) O(ε2) O(α 2 ε3) • No induced flow in vertical: stratification inhibits all motion, • Horizontal velocity cannot vary in horizontal direction: long response. Requirement: no indefinite growth in x.

  12. The non-local response: wave-packet filtered • From numerical simulations: • α=0.01, ε=0.05, • Induced mean flow not included as initial condition.

  13. The non-local response: the long wave scale • From numerical simulations: • α=0.01, ε=0.05, • Induced mean flow not included as initial condition.

  14. The non-local response: the long wave scale

  15. Conclusions • A two-dimensional horizontally and vertically compact internal gravity wave packet: • No mean flow in the vertical, • Horizontally long (>>packet) waves are excited, • Contain finite amount of horizontal momentum predicted well by theory. • Theoretical and numerical validation of Bretherton’s (1969) anticipation.

  16. Thank you for your attention!

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