Problem Set #8
OCEAN 210: Ocean Circulation
Assigned: Friday, November 20, 2009
Due: Wednesday, November 25, 2009
Regional Wind Stress, Ekman Transport, Pumping, and Suction
1. Consider a box of surface coastal ocean in the northern hemisphere as illustrated in
Figure 1 below. The bottom of the box corresponds to the bottom of the Ekman layer, and
the box is bounded on the East by a North-South running coastline.
a) Sketch the wind direction and the direction of the resulting Ekman transport in the case
of a sustained northerly wind (left two panels) and southerly wind (right two panels).
b) Let's say our ocean box is part of an infinitely long coast, so that there is no net
north/south transport and no change in the dimensions of the box due to wind stress. In
each case, what additional flow must enter or leave the box to balance the volume
budget?
There must be an additional flow in (out) through the bottom of the Ekman layer to
balance the offshore (onshore) Ekman transport through the western side of the box in
Case 1 (Case 2).
Add these flows to frames c. and d. of Figure 1.
c) Which vertical flow from part b) will result in more nutrients being available in the
coastal photic zone? Consider the normal vertical gradient of nutrient concentration in the
world ocean.
Since nutrients are generally abundant in the aphotic zone and depleted in the photic layer
(roughly equivalent to the Ekman layer), upwelling which results from a sustained
northerly wind will lead to greater nutrient availability in the photic layer.
2. Now imagine the coastal box in Figure 1 is located at approximately 40N, is 100 km
wide (East-West) and has a 13 m s-1 northerly wind blowing across it. What is the
magnitude and direction of vertical velocity that will this produce? For this problem, take
the density of air (ρair) to be 1.3 kg m-3, the density of the coastal water (ρwater) to be 1027
kg m-3, and the kinematic viscosity (ν – pronounced “nu,” not to be confused with
velocity V) to be 0.06 m2 s-1. Additionally, the drag coefficient of the surface ocean (cD)
is 1*10-3, and the Coriolis parameter at this latitude (f) is 9.4*10-5 radians per second.
a) Calculate the wind stress (τ). τ = ρaircDU2
= (1.3 kg m-3)*(1x10-3)*(13 m s-1)2
= 0.22 kg m2 s-2 m-3
so τ = 0.22 N m-2 (since 1 N = 1 kg m s-2)
b) Calculate the depth of the Ekman layer, DE = (2*/f)0.5.
= (2*0.06 m2 s-1/ 9.4*10-5 s-1)0.5
= 35.7 m
c) Now find the resulting average horizontal transport velocity (VE) of the Ekman layer,
given by VE = (*(2*π)0.5) / (DE*water*f).
= 0.22 kg m2 s-2 m-3 * (2*π)0.5) / (35.7 m * 1027 kg m-3 * 9.4*10-5 s-1)
= 0.16 m s-1 or 16 cm s-1
d) Using a volume (or mass) balance (as you did conceptually in problem 1), calculate the
vertical velocity produced by this northerly wind. Hint: draw out a three-dimensional box
version of Figure 1, and keep in mind that the volume of the box is at steady state
(Δvolume/Δt = 0) and that there is no net North/South volume flux.
Δvolume/Δt = 0
0 = (volume transported into Ekman layer from below) –
(volume transported out of the Ekman layer to the west)
0 = x*y*w – y*z*VE
where x is the width of the theoretical box (in m), y is the length of the theoretical box (in
m), w is the vertical velocity (in m s-1), and VE is the horizontal velocity (in m s-1) in the
Ekman layer. Note that y drops out of the equation.
0 = x*w – z*VE
z*VE = x*w
w = z*VE/x
w = (35.7 m * 0.16 m s-1) / 100000 m
w = 5.7x10-5 m s-1