Encyclopedia of Environmental Science and Engineering, Volume I and II

OCEANOGRAPHY 795

With the simplifications and assumptions described

above, the Navier-Stokes equations governing the motion of

a homogeneous ocean in the northern hemisphere become:

r

r

fu A

dv

dz

fu A

du

dz

v

2

2

v

2

2



,

,

where f is the Coriolis parameter, and is equal to 2 v sin f,

where v is the angular velocity of the earth and f is the lati-

tude of interest.

The previous two equations can be combined to give two

fourth-order differential equations, one for u and one for v.

Solution of these equations requires knowledge of the boundary

conditions for u and v. We specify that u and v must decrease to

zero at infinite depth (i.e., as z goes to negative infinity). At the

water surface, z 
 0, we specify that the turbulent shear stress

must equal the applied surface wind stress. With these bound-

ary conditions, the solution of our equations becomes:

u

(A f)

cos((f/2A ) z 45)exp((f/2A ) z),

v

(A f

s

v

1/2 v

1/2

v

1/2

s

v



t

r

t

r ))

1/2sin((f/2A ) zv1/2 45)exp((f/2A ) z).v1/2

Examination of our result illustrates that the water velocity

decreases exponentially with depth from a maximum at the

surface, while at the same time experiencing a rotation in

direction. This vertical spiral, illustrated in Figure 5, is com-

monly referred to as the “Ekman Spiral”, after V.W. Ekman,

who first investigated the problem (Ekman, 1905). Note that

the surface water velocity (i.e., at z 
 0) is directed at an

angle of 45 to the right of the wind direction (in the northern

hemisphere). This theoretical result, obtained with several

simplifications and assumptions, is not far removed from

observations which indicate surface water motions directed

at approximately 10 to 40 to the right of the wind direction.

For a more complete theoretical treatment of this problem,

including the effects of unsteadiness, and a varying eddy vis-

cosity, the reader is referred to Madsen (1977).

A further interesting conclusion can be obtained by verti-

cally integrating our expressions for u and v from the surface

to the “bottom”, z 
 negative infinity. The quantities thus

obtained, U and V, represent the total volume flux of water in

the x and y direction, respectively. Performing the integration,

we obtain U 
 0.0 and V 
  t S (f r ). This result indicates that the

total water flux in the wind-driven water column (termed the

“Ekman Transport”) is directed 90 to the right (left) of the wind

stress in the northern (southern) hemisphere!

Our solution for the Ekman Transport offers an explanation

for the coastal phenomena known as “upwelling” and “down-

welling”. Clearly, if a wind acts along a coastline with the

coast to the left (in the northern hemisphere), the total Ekman

Transport will be directed in the offshore direction, with most

of the flux occurring in the surface region of the water column

(remember, the velocity decreases exponentially from the sur-

face). Conservation of water mass dictates that this volume of

water must be replaced, and this “replacement” water can only

come from the offshore, bottom region. This process, known

as upwelling, is actually much more complex than the general

Wind Surface current

FIGURE 5 Vertical variation of wind-driven water motion

(Gross, 1977).

z,w

y, v

x, u

τ

τ

τ

s

y

x

FIGURE 4 Coordinate system.

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