or
at the surface. From eq 114 and 123, we see that
(U - Ug )
2
- f (V - Vg ) = K
K Ug
(116a)
τx / ρ =
z2
(125a)
δ
- K Ug sgnf
(V - Vg )
2
τy / ρ =
f (U - Ug ) = K
.
(125b)
(116b)
δ
2
z
That is, the surface stress is to the right of the geo-
since Ug and Vg are independent of height.
strophic wind in the Northern Hemisphere, as Nansen
If we define
observed; in the Southern Hemisphere, it is to the left.
Since by definition
S ≡ (U - Ug ) + i(V - Vg )
(117)
(
)
1/ 2
τ = τ2 + τ2
= ρ u*
2
(126)
we can combine eq 116 a and b into one equation in S
x
y
2S
ifS
= 2.
we also find from eq 125 another relation between the
(118)
K
z
constants
21 / 2 K Ug
The solution of this is easiest if we align the x axis with
=
2
u*
.
(127)
the geostrophic wind, that is, Vg = 0. Then the bound-
δ
ary conditions at z = 0 are
Using this in eq 124, we see that
U(0) = V(0) = 0
(119)
2
21 / 2 u*
δ=
.
(128)
f Ug
so
S(0) = U.
(120)
While solving the Ekman equations in the geo-
strophic wind frame is easiest, observers at the surface
As z approaches infinity, the boundary conditions are
know the direction of the surface stress (remember, in
an Ekman layer the surface wind vector is zero) better
lim U(z) = Ug
(121a)
than they know the direction of the geostrophic wind.
z→∞
Hence, for me at least, it is easier to visualize the
Ekman winds in a coordinate system aligned with the
lim V (z) = 0
(121b)
z→∞
surface stress vector than in a coordinate system aligned
with the geostrophic wind. The following equations
so
accomplish this transformation of velocity components
lim S(z) = 0.
(122)
in the geostrophic wind frame [denoted as (U,V)] to
z→∞
components in the surface stress frame [denoted as
(U′,V′)]:
With these boundary conditions, the solution of eq
U ′(z) = 2 -1 / 2 [U(z) + sgnf V (z)]
118 is fairly straightforward. It is (e.g., Businger 1982;
(129a)
Stull 1988, p. 211; Garratt 1992, p. 42)
V ′(z) = 2 -1 / 2 [-sgnf U(z) + V (z)].
(129b)
U(z) = Ug[1 - exp(-z / δ) cos(z / δ)]
(123a)
Thus, substituting eq 123 into eq 129 yields the
Ekman solution in a reference frame aligned with the
V (z) = sgnf Ug exp(-z / δ) sin(z / δ).
(123b)
surface stress
Here, sgnf is the sign of the Coriolis parameter: plus in
U ′(z) = 2-1 / 2 G
the Northern Hemisphere, minus in the Southern Hemi-
(130a)
sphere. And δ is a height scale
{1 - exp(-z / δ)[cos(z / δ) - sin(z / δ)]}
2
2K
δ=
.
(124)
V ′(z) = -2-1/ 2 G sgnf
f
(130b)
{
}
1 - exp(-z / δ)[cos(z / δ) + sin(z / δ)] .
Let us look at what eq 123 implies about processes
24
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