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Wave-Current Interactions and Sediment Dynamics

Wave-Current Interactions and Sediment Dynamics. Juan M. Restrepo Mathematics Department Physics Department University of Arizona. Support provided by NSF, DOE, NASA. Collaborators. Jim McWilliams (UCLA) Emily Lane (UCLA) Doug Kurtze (St. Johns) Paul Fischer (ANL)

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Wave-Current Interactions and Sediment Dynamics

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  1. Wave-Current Interactions and Sediment Dynamics Juan M. Restrepo Mathematics Department Physics Department University of Arizona Support provided by NSF, DOE, NASA

  2. Collaborators • Jim McWilliams (UCLA) • Emily Lane (UCLA) • Doug Kurtze (St. Johns) • Paul Fischer (ANL) • Gary Leaf (ANL) • Brad Weir (Arizona)

  3. WAVES AND MATH • Nonlinear and Dissipative Waves dissipative Burgers • Nonlinear and Dispersive Waves Korteweg de Vries • Eikonal Equations/Rays • Amplitude Equations WHAT NEXT?

  4. Climate Dynamics (HEAT,TRANSPORT) days-100 yrs, 1 Km-6 Km • Shelf-Ocean Dynamics (TRANSPORT/WAVES-CURRENTS) 10 sec-season, 10 m-100 Km • Shoaling Zone Dynamics (RADIATION STRESSES,TRANSPORT) 5 sec-season, 1 m- 2 Km

  5. ADVECTIVE/MULTISCALE STOCHASTIC FOCUS • Advection: waves, causal effects, • Multiscale: resolving dynamics • Stochasticity: turbulence, parametrizations, quantifying uncertainty, data assimilation.

  6. Can Gravity Waves Influence Basin Scale Circulation? • Climate lore: no • Data: not available • Lab: no experiments • Basin scale circulation models do not incorporate this aspect

  7. Ocean circulation is forced by radiation and surface fluxes and results from balance of Earth’s rotation, viscous and buoyancy forces

  8. Hemispheric, 2D Ocean Basin r=(1-aT+bS)

  9. The Conveyor Belt

  10. Stommel’s 2-Box Model

  11. 2-Box Steady Solutions Stommel’s Equations density density l f = (R x – y) dx/dt = d(1-x) - |f|x dy/dt = 1–y - |f| y temperature salt Steady State Solutions:

  12. Steady State Solutions: Temp Haline

  13. Advective Effects Kurtze, Restrepo, JPO, vol 31,’01 l f = (R x – y) dx/dt = d(1-x) - |f|[x(t-s)-x(t)] dy/dt = 1–y - |f|[y(t-s)-y(t)]

  14. Conclusions? • Advective effects potentially contribute to climate variability • Advective effects: important in THC? • Teleconnections in ENSO? (Tropical Climate) • Teleconnections in NAO? (North Atlantic Oscillation)

  15. Thermohaline teleconnection Residual flow due to waves Wave Effects on Climate McWilliams Restrepo, JPO, vol 32, ‘99

  16. Air/Sea Interface • Momentum: waves, thermocline mixing, wind. • Mass: water evaporation and precipitation, river inflows, chemicals. • Energy: sun radiation, other thermal balances.

  17. Air/Sea Interface Budgets

  18. Energy Budget

  19. Linear Waves: particle paths close Nonlinear Waves: particle paths do not close Transport Velocity due to Oscillatory Flows

  20. Restrepo, Leaf, JPO, vol 32, ‘02

  21. Quasi-Geostrophic Case

  22. Estimates on Wave/Driven Flow Wind driven transport: Stokes transport:

  23. Empirical Estimates Planetary Geostrophic Balance

  24. Wind-driven Spectra

  25. Mathematics Capturing multiscale behavior of system of hyperbolic pde’s • Vortex force representation U¢rU = 1/2r|U2|+r£U£ U Radiation stress representation U¢rU = r¢(UU)+U r¢ U • Introduction of stochastic component • Lagrangian/Eulerian mapping

  26. Shelf Wave/Current Dynamics • 10 secs-months, 100m-100 Km • Speed: waves > currents • kH ~ 1 • Applications: erodible bed dynamics river plume evolution algal/plankton blooms pollution McWilliams, Restrepo, Lane, JFM 2004

  27. Shelf Wave/Current Model • Start with Shallow Water Equations (ignore dissipation, for now) • Velocity field separation: waves currents long wave component

  28. 2 space scales, average over smaller ones • 3 time scales, average over faster ones • Waves (amplitude equations) • Waves and Currents have depth and stratification dependence • Frequency/wavenumber evolution equations Restrepo, Continental Shelf Res, 2001

  29. Current Effects on Waves Current forcing: Fixed bottom topography

  30. NO CURRENTS Effect of CURRENTS WAVE Amplitude WAVE Phase

  31. Wave Effects on Currents

  32. WAVES NO WAVES

  33. Inner Shelf/Shoaling Region • 5 seconds-6 hours, 1m-2Km • Traditional Radiation Stress: wave-averaged effects on currents: divergence of a stress tensor • Vortex Force Representation: wave-average effects: decomposed in terms of a Bernoulli head and a vortex force. Lane, Restrepo, McWilliams, JFM 2005

  34. Radiation Stresses • Compared RS (Hasselmann), GML (MacIntyre), VF (McWilliams, Restrepo, Lane). • Waves >> currents new interpretation • Revisit old problems: rip currents, longshore currents.

  35. Dissipative Effects Whitecapping Dissipative effect… but how does it manifest itself? Zt =f(Zt,t)dt+s(Zt)dW with f(x,t) = a cos(k x - t) <Wt Wx> =  (t-s) <Wt> = 0 Yields dissipative coupling of the total rotation of the current and the Stokes drift velocity uS r £ [uS£]

  36. BASIC DISSIPATION MODEL • New particle motion: dZt =  (u,w) dt + 2 v dt + B(Zt,T) dWt Sea Elevation:  = a cos (k x -  t – [ 2 ]1/2 Wt) e- t dxt =  u dt + [2 B(X,T)]1/2 dWht dzt =  ww dt

  37. Stokes Drift with Dissipation VSt = A2 k/2 sinh2[kH] [cosh [2k(z+H)]+1/2(2 2+[-D2/2])D Wst = - A2 k/ 2 sinh2[kH] (16 /) D D = e- T [1 + ( + D2/2)2/2]-1

  38. Effect of Dissipation DRIFT, DISSIPATION DRIFT, NO DISSIPATION Dissipation

  39. Effect of Dissipation No dissipation With dissipation Initial vorticity

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