1. 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 3. Depts. Physics and Earth & Atmospheric Sciences, University of Alberta EUROPEAN GEOSCIENCES UNION, APRIL 2014

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

3. Motivation: Periodic surface gravity waves From Wallet & Ruellan (1950) Solution to linearized problem for periodic waves: Circular motion to leading order O(α) and net horizontal drift at next order O(α 2 ), where α=ak (steepness).

4. Motivation: Surface gravity wave packets

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

6. What are internal gravity waves? Surface gravity wavesInternal 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-periodicx-periodic and z compact No Stokes driftHorizontal 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. Requirement: no indefinite growth in x. O(ε 4 ) O(ε 2 ) O(α 2 ε 3 ) Mean-flow response O(α 2 ) No induced flow in vertical: stratification inhibits all motion, Horizontal velocity cannot vary in horizontal direction: long response.

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!