wave-current interactions and sediment dynamics juan m. restrepo mathematics department physics...
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Wave-Current Interactions and Sediment Dynamics
Juan M. RestrepoMathematics Department
Physics DepartmentUniversity of Arizona
Support provided by NSF, DOE, NASA
CollaboratorsCollaborators
• 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)
WAVESWAVES ANDAND MATHMATH Nonlinear and Dissipative WavesNonlinear and Dissipative Waves
dissipative Burgersdissipative Burgers Nonlinear and Dispersive WavesNonlinear and Dispersive Waves
Korteweg de VriesKorteweg de Vries Eikonal Equations/RaysEikonal Equations/Rays Amplitude EquationsAmplitude Equations
WHAT NEXT?
• Climate Dynamics Climate Dynamics (HEAT,TRANSPORT)(HEAT,TRANSPORT)
days-100 yrs, 1 Km-6 Kmdays-100 yrs, 1 Km-6 Km
• Shelf-Ocean DynamicsShelf-Ocean Dynamics
(TRANSPORT/WAVES-CURRENTS)(TRANSPORT/WAVES-CURRENTS)
10 sec-season, 10 m-100 Km10 sec-season, 10 m-100 Km
• Shoaling Zone Dynamics Shoaling Zone Dynamics
(RADIATION STRESSES,TRANSPORT)(RADIATION STRESSES,TRANSPORT)
5 sec-season, 1 m- 2 Km5 sec-season, 1 m- 2 Km
• Advection: waves, causal effects, Advection: waves, causal effects,
• Multiscale: resolving dynamicsMultiscale: resolving dynamics
• Stochasticity: turbulence, Stochasticity: turbulence, parametrizations, quantifying parametrizations, quantifying uncertainty, data assimilation.uncertainty, data assimilation.
ADVECTIVE/MULTISCALE ADVECTIVE/MULTISCALE STOCHASTIC FOCUSSTOCHASTIC FOCUS
Can Gravity Waves Can Gravity Waves Influence Basin Scale Influence Basin Scale Circulation?Circulation?• Climate lore: Climate lore: nono
• Data: Data: not availablenot available
• Lab: no Lab: no experimentsexperiments
• Basin scale Basin scale circulation models circulation models do not incorporate do not incorporate this aspectthis aspect
Ocean circulation is forced by radiation and surfacefluxes and results from balance of Earth’s rotation, viscous and buoyancy forces
Hemispheric, 2D Ocean Basin
TS)
The Conveyor BeltThe Conveyor Belt
Stommel’s 2-Box ModelStommel’s 2-Box Model
2-Box Steady Solutions2-Box Steady Solutions
f = (R x – y)dx/d = (1-x) - |f|xdy/d = 1–y - |f| y
Stommel’sEquations
Steady State Solutions:
densitydensity
temperature
salt
Steady State Solutions:
Haline
Temp
Advective EffectsAdvective 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
Conclusions?Conclusions?
• Advective effects potentially Advective effects potentially contribute to climate variabilitycontribute to climate variability
• Advective effects: important in THC?Advective effects: important in THC?
• Teleconnections in ENSO? (Tropical Teleconnections in ENSO? (Tropical Climate)Climate)
• Teleconnections in NAO? (North Teleconnections in NAO? (North Atlantic Oscillation)Atlantic Oscillation)
Wave Effects on ClimateWave Effects on Climate• Thermohaline Thermohaline
teleconnectionteleconnection• Residual flow due Residual flow due
to wavesto waves
McWilliams Restrepo, JPO, vol 32, ‘99
Air/Sea InterfaceAir/Sea Interface
• Momentum: waves, thermocline Momentum: waves, thermocline mixing, wind.mixing, wind.
• Mass: water evaporation and Mass: water evaporation and precipitation, river inflows, precipitation, river inflows, chemicals.chemicals.
• Energy: sun radiation, other thermal Energy: sun radiation, other thermal balances.balances.
Air/Sea Interface BudgetsAir/Sea Interface Budgets
Energy BudgetEnergy Budget
Transport Velocity due to Transport Velocity due to Oscillatory FlowsOscillatory Flows• Linear Waves: Linear Waves:
particle paths closeparticle paths close• Nonlinear Waves: Nonlinear Waves:
particle paths do particle paths do not closenot close
Restrepo, Leaf, JPO, vol 32, ‘02
Quasi-Geostrophic CaseQuasi-Geostrophic Case
Estimates on Wave/Driven Estimates on Wave/Driven FlowFlow
Wind driven transport:
Stokes transport:
Empirical EstimatesEmpirical Estimates
Planetary Geostrophic Balance
Wind-driven SpectraWind-driven Spectra
MathematicsMathematics• Vortex force representationVortex force representation UU¢r¢rU = 1/2U = 1/2rr|U|U22|+|+r£r£UU££ U U Radiation stress Radiation stress
representationrepresentation UU¢r¢rU = U = r¢r¢(UU)+U (UU)+U r¢r¢ U U
• Introduction of stochastic Introduction of stochastic componentcomponent
•Lagrangian/Eulerian mapping
Capturing multiscalebehavior of systemof hyperbolic pde’s
Shelf Wave/Current Shelf Wave/Current DynamicsDynamics
• 10 secs-months, 100m-100 10 secs-months, 100m-100 KmKm
• Speed: waves > currentsSpeed: waves > currents
• kH ~ 1kH ~ 1
• Applications: Applications:
erodible bed dynamicserodible bed dynamics
river plume evolutionriver plume evolution
algal/plankton blooms algal/plankton blooms
pollutionpollution
McWilliams, Restrepo, Lane, JFM 2004
Shelf Wave/Current ModelShelf Wave/Current Model
• Start with Shallow Water Equations Start with Shallow Water Equations (ignore dissipation, for now)(ignore dissipation, for now)
• Velocity field separation:Velocity field separation:
waveswaves
currentscurrents
long wave componentlong wave component
• 2 space scales, average over smaller ones2 space scales, average over smaller ones
• 3 time scales, average over faster ones3 time scales, average over faster ones
• Waves (amplitude equations)Waves (amplitude equations)
• Waves and Currents have depth and Waves and Currents have depth and stratification dependencestratification dependence
• Frequency/wavenumber evolution Frequency/wavenumber evolution equationsequations
Restrepo, Continental Shelf Res, 2001
Current Effects on WavesCurrent Effects on Waves
Current forcing:
Fixed bottom topography
Effect of CURRENTS
WAVEAmplitude
WAVEPhase
NO CURRENTS
Wave Effects on CurrentsWave Effects on Currents
NO WAVES WAVES
Inner Shelf/Shoaling RegionInner Shelf/Shoaling Region
• 5 seconds-6 hours, 5 seconds-6 hours, 1m-2Km1m-2Km
• Traditional Radiation Traditional Radiation Stress:Stress: wave-averaged wave-averaged effects on currents: effects on currents: divergence of a stress divergence of a stress tensortensor
• Vortex Force Vortex Force RepresentationRepresentation: wave-: wave-average effects: average effects: decomposed in terms decomposed in terms of a Bernoulli head of a Bernoulli head and a vortex force.and a vortex force.
Lane, Restrepo, McWilliams, JFM 2005
Radiation StressesRadiation Stresses
• Compared RS (Hasselmann), GML Compared RS (Hasselmann), GML (MacIntyre), VF (McWilliams, (MacIntyre), VF (McWilliams, Restrepo, Lane).Restrepo, Lane).
• Waves >> currents new Waves >> currents new interpretationinterpretation
• Revisit old problems: rip currents, Revisit old problems: rip currents, longshore currents.longshore currents.
Dissipative EffectsDissipative Effects
White capping
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 £ ]
Dissipative effect…but how does it manifest itself?
BASIC DISSIPATION MODELBASIC DISSIPATION MODEL
• New particle motion:New particle motion:
dZdZtt = = ( ( u,w) dt + u,w) dt + 22 v dt + B(Z v dt + B(Ztt,T) ,T) dWdWtt
Sea Elevation: = a cos (k x - = a cos (k x - t – [ 2 t – [ 2 ]]1/21/2 W Wtt) e) e-- t t
dxdxtt = = u dt + [2 B(X,T)] u dt + [2 B(X,T)]1/21/2 dW dWhhtt
dzdztt = = w www dt dt
Stokes Drift with DissipationStokes Drift with Dissipation
VVStSt = A = A22 k/2 sinh k/2 sinh22[kH] [kH]
[cosh [2k(z+H)]+1/[cosh [2k(z+H)]+1/22(2 (2 22+[+[--DD22/2])/2])DD
WWstst = - A = - A22 k/ 2 sinh k/ 2 sinh22[kH] (16 [kH] (16 //) ) DD
D D = e= e-- T T [1 + ( [1 + ( + D + D22/2)/2)22//22]]-1-1
Effect of DissipationEffect of Dissipation
DRIFT, NO DISSIPATION
Dissipation
DRIFT, DISSIPATION
Effect of DissipationEffect of Dissipation
No dissipation
With dissipationInitial vorticity
NO DISSIPATION WITH DISSIPATION
VELOCITY VELOCITY VELOCITY + DRIFTVELOCITY + DRIFT
Future WorkFuture Work
• Regional Ocean Model (ROMS)Regional Ocean Model (ROMS)
• Dissipative Mechanisms in Dissipative Mechanisms in Wave/Currents: wave breaking, Wave/Currents: wave breaking, bottom drag, surface pollution, bottom drag, surface pollution, stratification.stratification.
• Wind ForcingWind Forcing
Further Information:Further Information:
Juan M RestrepoJuan M Restrepo
www.math.arizona.edu/~restrepowww.math.arizona.edu/~restrepo