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( 2.2 )-( 2.4 ) must balance each other. This layer is named from the Swedish physicist
and oceanographer W. Ekman (1874-1954), who for the first time derived mathe-
matically the influence of the Earth's rotation on marine and atmospheric flows.
A prominent wind feature which distinguishes the Ekman layer from the surface or
Prandtl layer below is the turning of wind direction with height. The Ekman layer
covers the major part of the ABL above the Prandtl layer (see Fig. 3.1 ). In the Ekman
layer the simplifying assumption is made that the height dependent growth of the
exchange coefficient K M = ju * z (see Sect. 3.1.1.1 ) stops at the top of the Prandtl
layer and that K M is vertically constant for the rest of the boundary layer.
3.2.1 Ekman Layer Equations
The balance of forces in the Ekman layer involves three forces. The Coriolis force
is relevant in addition to the pressure gradient force and the frictional forces.
Equating the three relevant terms III, V, and VII in ( 2.2 ) and ( 2.3 ) leads to:
K M ou
oz
ox o
fv þ 1
q
o p
¼ 0
ð 3 : 40 Þ
oz
K M o v
oz
oy o
fu þ 1
q
o p
¼ 0
ð 3 : 41 Þ
oz
Here, the right-most terms on the left-hand side of ( 3.40 ) and ( 3.41 ) are
substituted for the symbolic expressions for the frictional forces F x and F y in term
VII in ( 2.2 ) and ( 2.3 ). K M is the turbulent vertical exchange coefficient for
momentum, which has the physical dimension of a viscosity, i.e. m 2 /s. Using the
definition of geostrophic winds introduced in Eqs. ( 2.5 ) and ( 2.6 ) leads to:
K M o u
oz
fv þ fv g o
¼ 0
ð 3 : 42 Þ
oz
K M o v
oz
fu fu g o
¼ 0
ð 3 : 43 Þ
oz
The two left terms in ( 3.42 ) and ( 3.43 ) can be merged into one term containing
the so-called velocity deficits u g - u and v g - v. This yields the so-called defect
laws for the Ekman layer:
K M o u
oz
o
fv g v
¼ 0
ð 3 : 44 Þ
oz
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