A Generalized Ekman Model for Frontal Regions

Meghan F. CroninWilliam S. Kessler

NOAA Pacific Marine Environmental Laboratory

Cronin, M.F. and W.S. Kessler. Near-surface shear flow in the tropical Pacific cold tongue front. Accepted pending minor revision J. Phys. Oceanogr., 2008.

Relevance to Climate Observing System

Ocean System: Upper 25 m is not always slab-like. Surface currents can have significant vertical shear.

Climate Observing System: The Climate Observing System is designed to monitor the 15-m currents. What part of the system monitors the near-surface shear?

Ocean Current Products: The generalized Ekman model can be used in gridding procedures that blend wind-forced current estimates with the geostrophic currents, e.g. OSCAR.

Heat and Freshwater Transport: Calculations of the heat and freshwater transport by the wind-forced Ekman flow depend upon understanding of Ekman flow in frontal regions.

Neither shipboard, nor moored upward-looking ADCPs measure currents above 20 m. Is there shear above 25m?

Extrapolated above 20m

Johnson et al. JPO (2001)

Sparse network of reference station moorings monitor near-surface shear. Within the TAO/TRITON array, near-surface

shear is monitored only at 2 equatorial sites.

• 5 Sonteks (acoustic Doppler current meters) were placed on a test mooring near the 2ºN,140ºW TAO mooring.

• Each Sontek had a thermistor.

5m

10m

15m

20m

25m

_2003_ _2004_ _2005_

SONTEK sensors TAO/EPIC 95W 2N140W Test Mooring KEO & IO

An opportunity to measure near-surface currents at 2 °N, 140 °W

Shear is very sensitive to stratification and mixing. Example: Diurnal Jet

Diurnal composite of wind, temperature, and u-u(25m)

• At 1600 local, currents at 5 m are 12 cm/s stronger than at 25m and are oriented in direction of wind. Nighttime shear is weak.

• Even weak daytime restratification can cause diurnal jet.

Local time (hours)

north is up

Mean near-surface currents at 2°N, 140°W

• Zonal flow is westward associated with SEC.

• Poleward flow is weaker than expected. Inferred transport is less than half that needed to balance expected equatorial upwelling transport.

• Zonal flow is surface intensified, but poleward current is not. Why? (Slab layer physics should work for both components).

Can near-surface shear be considered a combination of Classic Ekman Spiral and geostrophic thermal wind?

∆utot

∆ug

∆uag

(Ekman depth Dek= 25 m ?)

Mean for 24-May-2004 to 7-Oct-2004

Observed ageostrophic currents relative to 25m has Ekman-like spiral 70 to right of wind. But…

Can near-surface shear be considered a combination of Classic Ekman Spiral and geostrophic thermal wind? No!

(Ekman depth Dek= 25 m ?)

Mean for 24-May-2004 to 7-Oct-2004

Classic Ekman spiral has them aligned slightly to the right of the wind stress.

∆utot

∆ug

∆uag

∆uag

∆v re

lativ

e to

25

m (c

m s

-1)

∆u relative to 25 m (cm s-1)

(Ekman depth Dek= 80 m)

Observed ageostrophic currents relative to 25m has Ekman-like spiral 70 to right of wind. But…

€

ifu = − 1ρ∇P + ν ∂

2u∂z2 → ifua = ν ∂

2ua∂z2

€

at z = 0: ∂u∂z

= τ 0

ρν→ at z = 0: ∂ua

∂z= τ 0

ρνat z = −H: u = 0 → at z = −H: ua = 0

Equation of motion:

Boundary conditions:

“Classic Ekman Model”: Assume steady, linear flow; with uniform density and viscosity; subject to wind stress at surface, no drag at z=-H (-∞). Solves for ua(z).

€

note : u = ug + ua , where: ug = iρf

∇P and∂ug∂z

= igαρf

∇T = 0

€

ifu = − 1ρ∇P + ν ∂

2u∂z2 → ifua = ν ∂

2ua∂z2

€

at z = 0: ∂u∂z

= τ 0

ρν→ at z = 0: ∂ua

∂z= τ 0

ρνat z = −H: u = 0 → at z = −H: ua = 0

Equation of motion:

Boundary conditions:

“Classic Ekman Model”: Assume steady, linear flow; with uniform density and viscosity; subject to wind stress at surface, no drag at z=-H (-∞). Solves for ua(z).

€

note : u = ug + ua , where: ug = iρf

∇P and∂ug∂z

= igαρf

∇T = 0

Classic Ekman Model (Ekman 1905)Assumes steady, linear flow; with uniform density and viscosity; subject to wind stress at surface, no drag at z=-H (-∞).

€

Dek = 2νf

Figure from google imagehttp://www.eeb.ucla.edu/test/faculty/nezlin/PhysicalOceanography.htm

€

ifu = − 1ρ∇P + ν ∂

2u∂z2 → ifua = ν ∂

2ua∂z2

€

at z = 0: ∂u∂z

= τ 0

ρν→ at z = 0: ρν ∂ua

∂z= τ 0 − ρν

∂ug∂z

at z = −H: u = ug → at z = −H: ua = 0

Equation of motion:

Boundary conditions:

“Frontal Ekman Model”: Assume steady, linear flow; with uniform viscosity, subject to wind stress at surface, in a front that is uniform with depth; and with geostrophic flow at z=-H. Find ua(z).

€

u = ug + ua, where: ug = iρf

∇P and∂ug∂z

= igαρf

∇T ≡ vertically uniform

Ekman Spiral is forced by portion of wind stress that is out of balanced with geostrophic shear: eff = 0 - ug/z

€

ifu = − 1ρ∇P + ν ∂

2u∂z2 → ifua = ν ∂

2ua∂z2

€

at z = 0: ∂u∂z

= τ 0

ρν→ at z = 0: ρν ∂ua

∂z= τ 0 − ρν

∂ug∂z

at z = −H: u = ug → at z = −H: ua = 0

Equation of motion:

Boundary conditions:

“Frontal Ekman Model”: Assume steady, linear flow; with uniform viscosity, in a front that is uniform with depth; subject to wind stress at surface, and geostrophic flow at z=-H. Find ua(z).

u = ug + ua

∆u relative to 25 m (cm s-1)

In frontal region, the ageostrophic Ekman Spiral is forced by the portion of wind stress that is out of balance with surface geostrophic shear: eff = 0 - p

Mean for 24-May-2004 to 7-Oct-2004

∆utot

∆ug

∆uag

∆uag

∆v re

lativ

e to

25

m (c

m s

-1)

∆utot

∆ug

Ageostrophic Ekman response depends upon wind stress AND strength and orientation of the front relative to the wind. Ekman response is reduced when winds blow along a front.

Summary• Shear is very sensitive to both the horizontal and

vertical temperature distribution.

• Very weak daytime stratification (<0.2ºC/25m) resulted in a diurnal jet shear of 12 cm/s / 20m on average at 4 PM local!

• Wind stress balances the TOTAL surface shear (combined geostrophic and ageostrophic shears). The ageostrophic Ekman spiral is forced by the portion of the wind stress that is out of balance with the geostrophic shear.

• The effect of fronts on Ekman spiral is most pronounced at low latitudes.

• In frontal region, Ekman transport is not necessarily to right of the wind stress. Traditional Ekman heat transport implicitly assume that viscosity decays with depth.

• For 2ºN, 140ºW, southeasterly trades blowing across the cold-tongue front result in weaker than expected poleward currents. There may be a secondary circulation associated with the front, with downwelling on the cold side and upwelling on the warm side of the front.

Consequences

€

Mek = − i τ 0

ρ 0 f+ iν

f∂u g

∂z z=− H

Ekman Transport in Frontal Region

=