1. Response of the Upper Ocean to Winds
Chapter 9
Response of the Upper Ocean to Winds
Physical oceanography
Instructor: Dr. Cheng-Chien Liu
Department of Earth Sciences
National Cheng Kung University
Last updated: 10 November 2003

2. Introduction Question Possible answer
Same latitude but different climate?
Charleston v.s. San Diego
Norfolk v.s. San Francisco
Possible answer
Wind blows onshore
cool, moist, marine, boundary layer a few hundred meters thick capped by much warmer, dry air  San Diego
Charleston  warm, moist, marine, boundary layer that is much thicker
Convection, which produces rain, is much easier on the east coast than on the west coast.

3. Inertial motion The equation of motion
For a frictionless ocean at steady state (Eq. 9.1)
No horizontal pressure gradient & w << u, v  2W cosf << g (Eq. 9.2)
The Coriolis parameter f = 2W sinf
The equation for the harmonic oscillator & Solution (Eq. 9.4)
Inertia current or inertial oscillation (Fig 9.1)
The free motion of parcels of water on a rotating plane
The inertial period: Ti = (2p) / f = Tsd /(2 sinf ) (Table 9.1)
Rapid change of wind  inertia current
Incoherent, transient, anti-cyclonic (clockwise in the northern)

4. Ekman layer Steady wind  the Ekman layer
Few-hundred meters thick in the upper ocean
The doctoral thesis of Professor Walfrid Ekman
Table 9.2:
Contributions to the Theory of the Wind-Driven Circulation
Nansen’s qualitative arguments
Observation
Wind tended to blow ice at an angle of 200 – 400 to the right of the wind in the Arctic
Balance of three forces
W + F + C = 0

5. Ekman’s solution The equation of motion Steady state
A constant vertical eddy viscosity
Molecular viscosity n << turbulent eddy viscosity Az

6. Ekman’s solution Solution Equation The Ekman spiral (Fig 9.3)
Surface velocity
Exponentially decays with depth
Spiral
Ekman’s constants
Need two of three parameters (T, V0, Az)
Bulk formula: T = rair CDU102
V0 = V0(U10, f) (Eq. 9.14)

7. Ekman’s solution (cont.)
Ekman layer depth
V(DE) is opposite to V0
DE = p / a (Eq and 9.16)
Table 9.3: Typical Ekman Depth
The Ekman number
Ratio of the friction force and the Corilis force
Eq. 9.17: Ez
Eq. 9.18: at the Ekman depth DE, Ez = 1 / (2p2)

8. Bottom Ekman layer Solution Eq. 9.19
u = U [1 - exp(-az)] cos az
v = U exp(-az) sin az
Boundary conditions: u(zb) = v(zb) = 0
The Ekman spiral (Fig 9.4)
Direction: anti-cyclonic
Amplitude

9. Bottom Ekman layer (cont.)
Direction
U(z = 0-) is in the direction of U(z > zb) // p
U(z > zb)  p
U(z > zb) // p
U(z = 0+) is 450 to the left of U(z > zb)
U(z = 0-) is 450 to the right of U(z = 0+)
Fig 9.5

10. Examining Ekman’s assumptions
No boundaries
Valid away from coasts
Deep water
Valid if depth >> 200 m
f-plane
Valid
Steady state
Valid if wind blows for longer than a pendulum day
Time-dependent solution (Hasselmann 1970)

11. Examining Ekman’s assumptions (cont.)
Az = fn(U102)  fn(z)
Not a good assumption
If MLD < DE  Az = fn(z) from MLD to DE
Homogeneous density r  fn(z)
Probably good, except as it effects stability

12. Observations of flow near the sea surface
Measurements  Ekman's theory is remarkably good
Results indicate:
Inertial currents  the largest component of the flow
The flow is nearly independent of depth within the mixed layer for periods near the inertial period. Thus the mixed layer moves like a slab at the inertial period. Current shear is concentrated at the top of the thermocline
The flow averaged over many inertial periods is almost exactly that calculated from Ekman's theory. The shear of the Ekman currents extends through the averaged mixed layer and into the thermocline. The Ekman-layer depth DE is almost exactly that proposed by Ekman (9.16), but the surface current V0 is half his value (9.14)
The transport is 900 to the right of the wind in the northern hemisphere. The transport direction agrees well with Ekman's theory

13. Ekman mass transport Definition
The integral of the UE, VE from the surface to a depth d below the Ekman layer (Fig 9.6)

14. Ekman mass transport (cont.)
Volume transport
Definition:
The mass transport divided by the density of water and multiplied by the width perpendicular to the transport
Y: the north-south distance across which the eastward transport Qx is calculated
X: the east-west distance across which the northward transport Qy is calculated
Unit: Sv

15. Application of Ekman theory
Importance of upwelling
Enhance biological productivity, which feeds fisheries
Cold upwelled water alters local weather. Weather onshore of regions of upwelling tend to have fog, low stratus clouds, a stable stratified atmosphere, little convection, and little rain.
Spatial variability of transports in the open ocean leads to upwelling and downwelling, which leads to redistribution of mass in the ocean, which leads to wind-driven geostrophic currents via Ekman pumping

16. Application of Ekman theory (cont.)
Coastal upwelling
Along California coast  north wind  offshore Ekman transport (Fig 9.7 left)  upwelling (Fig 9.7 right)  region of cold water (Fig 10.16)
Upwelling  nutrient  productivity
Important regions: Peru, California, Somalia, Morocco, and Namibia

17. Application of Ekman theory (cont.)
Explain why the same latitude but different climate
region of cold water  cools the incoming air close to the sea  a thin, cool atmospheric boundary layer  cooling the city
Hadley circulation in the atmosphere (Fig 4.3)  downward velocity  the warmer air above the boundary layer  inhibits vertical convection  rain is rare
Rain forms only when winter storms coming ashore bring strong convection higher up in the atmosphere

18. Application of Ekman theory (cont.)
Explain why the same latitude but different climate (cont.)
Other reasons
MLD tends to be thin on the eastern side of oceans  upwelling can easily bring up cold water
Currents along the eastern side of oceans at mid-latitudes  bring colder water from higher latitudes
All these processes are reversed offshore of east coasts

19. Ekman pumping Conservation of mass
The spatial variability of the transports  vertical velocities at the top of the Ekman layer
ME: vector mass transport due to Ekman flow in the upper ocean
H: the horizontal divergence operator

20. Ekman pumping (cont.) Relate Ekman pumping to the wind stress
The spatial variability of the transports  vertical velocities at the top of the Ekman layer
Substitute
Into 
T: the vector wind stress
∵ no penetration ∴ w(0) = 0  need another vertical velocity to balance  the geostrophic velocity wG(0) at the top of the interior flow in the ocean

21. Important concepts
Changes in wind stress produce transient oscillations in the ocean called inertial currents
Inertial currents are very common in the ocean
The period of the current is (2p)/f
Steady winds produce a thin boundary layer, the Ekman layer, at the top of the ocean
Ekman boundary layers also exist at the bottom of the ocean and the atmosphere
The Ekman layer in the atmosphere above the sea surface is called the planetary boundary layer.
The Ekman layer at the sea surface has the following characteristics:
Direction: 450 to the right of the wind looking downwind in the Northern Hemisphere
Surface Speed: 1–2.5% of wind speed depending on latitude
Depth: approximately 40–300 m depending on latitude and wind velocity

22. Important concepts (cont.)
Careful measurements of currents near the sea surface show that:
Inertial oscillations are the largest component of the current in the mixed layer
The flow is nearly independent of depth within the mixed layer for periods near the inertial period. Thus the mixed layer moves like a slab at the inertial period
An Ekman layer exists in the atmosphere just above the sea (and land) surface
Surface drifters tend to drift parallel to lines of constant atmospheric pressure at the sea surface. This is consistent with Ekman’s theory.
The flow averaged over many inertial periods is almost exactly that calculated from Ekman’s theory
Transport is 900 to the right of the wind in the northern hemisphere

23. Important concepts (cont)
Spatial variability of Ekman transport, due to spatial variability of winds over distances of hundreds of kilometers and days, leads to convergence and divergence of the transport.
(a) Winds blowing toward the equator along west coasts of continents produces upwelling along the coast. This leads to cold, productive waters within about 100 km of the shore.
(b) Upwelled water along west coasts of continents modifies the weather along the west coasts.
Ekman pumping, which is driven by spatial variability of winds, drives a vertical current, which drives the interior geostrophic circulation of the ocean.

24. Momentum Equation

25. Equation 9.1

26. Equation 9.2

27. Equation 9.4

28. Ekman’s solution