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

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

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

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

6. Can Gravity Waves Influence Basin Scale Circulation? Climate lore: no Climate lore: no Data: not available Data: not available Lab: no experiments Lab: no experiments Basin scale circulation models do not incorporate this aspect 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  T  S)

9. The Conveyor Belt

10. Stommel’s 2-Box Model

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

12. Steady State Solutions: Haline Temp

13. Advective Effects f = (R x – y) dx/d = (1-x) - |f|[x(-s)-x()] dy/d = 1–y - |f|[y(-s)-y()] Kurtze, Restrepo, JPO, vol 31,’01

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

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

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

17. Air/Sea Interface Budgets

18. Energy Budget

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

20. Restrepo, Leaf, JPO, vol 32, ‘02

21. Quasi-Geostrophic Case

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

27. Empirical Estimates Planetary Geostrophic Balance

29. Wind-driven Spectra

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

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

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

36. 2 space scales, average over smaller ones 2 space scales, average over smaller ones 3 time scales, average over faster ones 3 time scales, average over faster ones Waves (amplitude equations) Waves (amplitude equations) Waves and Currents have depth and stratification dependence Waves and Currents have depth and stratification dependence Frequency/wavenumber evolution equations Frequency/wavenumber evolution equations Restrepo, Continental Shelf Res, 2001

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

41. Effect of CURRENTS WAVE Amplitude WAVE Phase NO CURRENTS

42. Wave Effects on Currents

43. NO WAVES WAVES

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

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

46. Dissipative Effects White capping Z t =  f(Z t,t)dt+s(Z t )dW with f(x,t) = a cos(k x -  t) =  (t-s) = 0 Yields dissipative coupling of the total rotation of the current and the Stokes drift velocity u S r £ [u S £  ] Dissipative effect… but how does it manifest itself?

47. BASIC DISSIPATION MODEL New particle motion: New particle motion: dZ t =  ( u,w) dt +  2 v dt + B(Z t,T) dW t dZ t =  ( u,w) dt +  2 v dt + B(Z t,T) dW t  = a cos (k x -  t – [ 2  ] 1/2 W t ) e -  t Sea Elevation:  = a cos (k x -  t – [ 2  ] 1/2 W t ) e -  t dx t =  u dt + [2 B(X,T)] 1/2 dW h t dz t =  w w dt

48. Stokes Drift with Dissipation V St = A 2  k/2 sinh 2 [kH] [cosh [2k(z+H)]+1/  2 (2  2 +[  -D 2 /2]) D [cosh [2k(z+H)]+1/  2 (2  2 +[  -D 2 /2]) D W st = - A 2  k/ 2 sinh 2 [kH] (16  /  ) D D = e -  T [1 + (  + D 2 /2) 2 /  2 ] -1

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

50. Effect of Dissipation No dissipation With dissipation Initial vorticity

51. NO DISSIPATION WITH DISSIPATION VELOCITY VELOCITY VELOCITY + DRIFT

52. Future Work Regional Ocean Model (ROMS) Regional Ocean Model (ROMS) Dissipative Mechanisms in Wave/Currents: wave breaking, bottom drag, surface pollution, stratification. Dissipative Mechanisms in Wave/Currents: wave breaking, bottom drag, surface pollution, stratification. Wind Forcing Wind Forcing

53. Further Information: Juan M Restrepo www.math.arizona.edu/~restrepo