Wave-Current Interactions and Sediment Dynamics

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

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

27. Empirical Estimates Planetary Geostrophic Balance

29. Wind-driven Spectra

33. 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

34. 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

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

36. 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

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

41. NO CURRENTS Effect of CURRENTS WAVE Amplitude WAVE Phase

42. Wave Effects on Currents

43. WAVES NO WAVES

44. 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

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

46. 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£]

47. 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

48. 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

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

50. Effect of Dissipation No dissipation With dissipation Initial vorticity