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