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

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The Stokes Drift and Mean Flows Induced by Two-Dimensional Internal Gravity Wave PacketsTon van den Bremer1,2 & Bruce Sutherland2,31. Department of Engineering Science, University of Oxford2. Woods Hole Oceanographic Institution 3. Depts. Physics and Earth & Atmospheric Sciences, University of Alberta

EUROPEAN GEOSCIENCES UNION, APRIL 2014Motivation: Periodic surface gravity waves

From Wallet & Ruellan (1950)

Orbit doesnt close: net horizontal driftMotivation: Periodic surface gravity waves

From Wallet & Ruellan (1950)

Solution to linearized problem for periodic waves:

Circular motion to leading order O() and net horizontal drift at next order O(2), where =ak (steepness).

Motivation: Surface gravity wave packets

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=cgt).What are internal gravity waves?

Surface gravity wavesInternal gravity wavesFree surfaceGravity is restoring forceFluid has uniform densityStratification: layers of different densityBuoyancy is restoring forceStokes drift for periodic internal gravity waves (igws)x- and z-periodicx-periodic and z compactNo Stokes drift

Horizontal Stokes drift

Problem definitionWhat 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.

Governing equationsTwo-dimensional Boussinesq internal waves in linearly stratified fluid (stable): Momentum (x and z), internal energy and incompressibility:

Vertical displacement9Leading-order Induced mean flow

Leading-order Induced mean flow

Requirement: no indefinite growth in x.

O(4)O(2)O( 2 3 )Mean-flow response O(2)

No induced flow in vertical: stratification inhibits all motion,Horizontal velocity cannot vary in horizontal direction: long response.

The non-local response: wave-packet filtered

From numerical simulations: =0.01, =0.05,Induced mean flow not included as initial condition.

The non-local response: the long wave scale

From numerical simulations: =0.01, =0.05,Induced mean flow not included as initial condition.

The non-local response: the long wave scale

ConclusionsA 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 Brethertons (1969) anticipation.

Thank you for your attention!