Wind driven Ekman layers

• Momentum balance and mass transport

• Classical theory and first observation

• Modern observations

• Ekman layer recipe

• Latest developments

Momentum balance and mass transport

An upper Ocean Momentum balance

• Goal is to described the local response to wind stress:• no pressure gradients

• wind stress is steady : > several inertial periods• Deep ocean region: reference level at depth• Linear dynamics• Turbulence averaged solution• Some assumptions can be relaxed

!tu + u ·!u + fk" u = #1"!p +

1"!z#

tideseddieslarge scale pressure gradientsindirect effect of wind, pumping

!(z = 0) = !wind

!(z = !zr) = 0u = u! + up

! = !" < u!w! >

Wind driven mass transport

• After integration between depth z and surface:

• Total mass transport

• No turbulence parametrization, hence “robust” relationship. Observed, within 10-30%.

• Used as a diagnostic of the shear stress

fk! u =1!"z#

! 0

zu dz = !k" (!wind ! !(z))

"f

! 0

zr

u dz = !k" !wind

"f

Classical Ekman layer, simplest turbulence parametrization

Ekman 1905: interesting facts

• Suggested by Nansen, after observations of ice drift• First theory of the wind driven flow• Munk (who knew Ekman) said that he solved the problem in

one day ... (Munk 2002)• Ekman developed a current meter in order to observe the

spiral but was never successful• He knew that the constant eddy viscosity assumption was

unrealistic and considered K=K(|u|)• Also considered:

• Finite depth case - no slip at the bottom• Time dependance - spin up problem• Pressure gradient driven flow - bottom ekman layer• Influence of lands - combination of wind driven and

pressure driven flow

Classical Ekman theory: Ekman 1905

• Simplest turbulence parametrization:

• Let:

• Equation of momentum becomes:

• Boundary conditions become:

• Solution is:

K constant! = "K#zu

U = u + iv

ifU = !zK!zU

K!zU(z = 0) = "wind/# K!zU(z = !") = 0

U = !wind"E

#(1 + i)Ke(1+i)z/!E

zr = !"

!E = (2K/f)1/2

Classical Ekman theory: solution overview

• Overall look of the Ekman spiral:

• Ekman depth:

• Current scaling:

• Mass transport:

!E = (2K/f)1/2

|U | ! !wind/""Kf

! 0

!"Udz =

!i!wind

"f

Variations with K=constant

• finite:• : infinitely deep case, classical Ekman spiral

• : shallow case, friction overwhelms Coriolis, current and transport is in the direction of the wind stress

• No slip bottom boundary conditions for finite depth applications

• Time periodicity

• As long as K is constant, analytical solutions are easily found

zr

zr ! !E

zr ! !E

Hunkins (1966)(same Deep-Sea Research as Abyssal Recipes)

Current measured by lowering a drogue from an ice floe and

measuring the wire angle.

D=18 m

!=2.4!10-3 m2/s

First Observation: Hunkins 1966

• Difficulty of measuring currents in the surface layer of the ocean limited early observations of the wind driven flow

• Current measured by lowering a drogue from an ice floe and measuring the wire angle

• Fit to Ekman spiral:D = 18m, K=2.4 10^-3 m2/s

Modern observations

Modern observations

• Observational challenge is the signal-to-noise ratio problem:• Noise at low frequencies due to pressure gradient

driven flow - (Lenn 2009 : 40cm/s vs 1cm/s)Way to substract these other contributions

• Noise at high frequencies due to internal waves• Long time series with wind consistent enough

• Verification of Ekman dynamics made possible (starting around 1980) by the development of robust surface mooring and improved current meters:• Vector Measuring Current Meters: VMCM• Acoustic Doppler Current Profiler: ADCP -> best vertical

resolution

Modern observations

• MILE: Davis et al. 1981a/b

• FASINEX (Weller et al. 1991, Rudnick and Weller 1993)

(Price & Sundermeyer 1999:)• Long-Term Upper Ocean Study (LOTUS3)

Briscoe and Weller 1984 / Price et al. 1987• Eastern Boundary Current observations (EBC)

Chereskin 1995• 10N Wijffels et al. 1994

MILE

• Experimental setting:19 days with non persistent winds -> hence focus on higher frequencies49.5N, f=0.064cph (15.8h)2 stations with ~30 vector currents meters at depth ranging from 5 to 175m

• Findings:• Coherent and in phase inertial dominate current measurements within mixed layer

(0-30m) // slab like nature of inertial currents. Illustrate the time dependance• Considering geocentric acceleration “filters out” part of these inertial oscillations.• Vertically integrated momentum balances // mass transport:

• Pressure gradient term is estimated from current at depth (94m) and simplified momentum balance (ignoring vertical momentum mixing)

• Balances are good but estimate of pressure gradient leads to mixed results (p1446-1447)

MILE

! 0

zr

(!t + k!) u dz ="wind

#f

Pressure gradient uncorrected

Pressure gradient corrected

MILE

49.5N, f=0.064cph (15.8h)

U =1

i(! + f)µ(z)"wind/# µ =

i!E

!1 + "/f

(1 + i)Ke(1+i)z

!1+!/f/"E

MILE

!tu + u ·!u + fk" u = #1"!p +

1"!z#

Required whenRequires careful attention

|!| ! |f |

Price and Sundermeyer 1999

• Comparison between 3 experiments in fair weather conditions: tau<0.2Pa and significant solar heating

• Point is to that time dependance of the stratification can explain some features of the observed Ekman layers

• LOTUS3:• Briscoe and Weller 1984, Price 1987• Vector current meters, temperature profile• 170 days

• EBC:• Chereskin 1995• Similar conditions in LOTUS3 and EBC• Downward looking adcp ( good vertical resolution)• 4 months with persistent wind

• 10N • Wijffels et al 1994• cruise with downward looking ADCP• CTD casts• 68 days

Ekman layer recipe

1/ Separate Ekman currents from other contributions

• LOTUS3 / EBC /10N:• subtract current at a depth assumed to be below the

surface Ekman layer• assumes the pressure gradient driven flow is nearly

uniform with depth ( no geostrophic shear, check with XBT transects in Lenn and Chereskin 2009)

• Other methods:• sea surface height (Rio and Hernandez 2003, Elipot 2006)• CTD cast can leads to geostrophic shear (Chereskin and

Roemmich 1991)

2/ Relate current and wind stress

• “Simpler” case - persistent wind:• EBC/10N• Average

• Non persistent winds:• MILE: cross-spectral analysis• LOTUS3: currents are rotated along the wind direction

before being averaged - regression ?

3/ Compare observed and expected mass transports

• Average mass transports:• Agreement with Ekman

transport within 10-30%

• Uncertainties:

! 0

zr

u dz =k! !wind

"f

15-20%~<1cm/s

20%wind stress

LOTUS3EBC 10N

EBC

4/ Vertical structure: compare observed and expected vertical structure

EBC observations

Classical Ekman spiral• Similarities:• Spiral shape current: rotates and

decrease with depth

• Differences:• shallower spirals (smaller e-folding

depth)• decrease with depth more rapidly

than they rotate, hence observed spirals are flatter

K=274 cm2/s

5/ Estimate what a constant eddy viscosity should be

• Any kind of fit:• to current amplitude• to current direction

• Eddy viscosity estimates:

100-2000 cm2/s

U = !wind"E

#(1 + i)Ke(1+i)z/!E

Lenn and Chereskin 2009

6/ Diagnose the eddy viscosity

• After integration between depth z_r and z:

• Assuming:

• We get:

fk! u =1!"z#

! = "K#zu

K(z) =fk!

! zzr

u dz

!zu

! z

zr

k! u dz =!(z)"f

6/ Diagnose the eddy viscosity

• Complex Eddy viscosity !?

• Difficult to make sense physically

• Means that the shear and stress are not aligned:

stress is to the right

• Useful tool however for predictions but requires explanation

Lenn and Chereskin 2009

7/ Explain observations / Ekman layer models

• Price & Sundermeyer 1999:• Diurnal cycle in stratification leads to time variations of K• Integrated in time, leads to an Ekman layer similar to

observations

• Schudlich and Price 1998, LOTUS 4, winter Ekman spiral:• persistent downwind shear in the upper layer 15m at the

surface -> consistent with surface log layer (Madsen 1977)• Possible importance surface gravity waves / langmuir cell

(Lewis & Belcher 2004 / Polton 2005)

Latest developments

Recent contributions and undiscovered areas

• Cronin ????, deals with issue of getting rid of the geostrophic shear• Lenn and Chereskin 2009: Drake passage• Kim 2009

• Anistropy of the wind driven response close to the coast• Include issue of time dependance• Presence of pressure gradients

• Polton 2005:• includes Coriolis-Stokes(gravity wave) force contribution • theoretical model / LES / observations, “encouraging” agreement

• Numerical modelling, LES, Zikanov 2003:• Effect of horizontal components of the Coriolis force

• No measurements in the last meters away from the surface:• Moored ADCP, blanking time issue• Boats situation is even worst (dixit Teri - ?)

• Estimating tau(z)=<u’w’>• not done yet in the open ocean, technological challenge• shallow waters:

• ADCP moored at the bottom (Gargett 2007), pinging fast enough to measure turbulent stresses

Summary

• Observational difficulties are:• Good vertical resolution• Separating Ekman currents and background pressure

gradient driven flow• Long time averages to overcome noise to signal ratio• Time dependance ( close to inertial frequencies )

• Yes ! Ekman layers are observed in the ocean:• Relationship between mass transport and wind stress is

robust• Vertical structure is a spiral• Some discrepancies with classical theory

• Observations suggest the importance of:• diurnal variations of stratification/eddy viscosity• recent interest in the effect of surface gravity waves