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In-Depth Information
the interior geostrophic region. The Ekman condition is
expressed by the vertical velocity on top of the Ekman
layer, which is proportional to the relative vorticity of the
interior low:
w
Inserting these expressions together with the horizontal
divergence (7.15) in the evolution equation (7.9) yields:
∂ω
∂t
+
J(q
,
ψ)
δ
E
2
h
∇
δ
E
2
h
ω(ω
+
f )
(7.17)
|
z
=
δ
E
=
δ
E
ω/
2. Using this condition as the
bottom boundary condition when the continuity equa-
tion is vertically integrated (together with the rigid-lid
approximation), it is easily verified that
2
ω
−
ψ
·∇
q
=
ν
∇
−
where
J
is the Jacobian operator. The relative vorticity is
written in terms of the transport function:
∂u
∂x
+
∂v
∂y
=
1
2
E
2
ω
,
(7.12)
1
h
∇
2
ψ
+
1
ψ
+
δ
E
−
ω
=
h
2
∇
h
·∇
h
3
J(h
,
ψ)
.
(7.18)
with
E
2
=
δ
E
/H
. Thus, the horizontal divergence is pro-
duced by the entrainment or detrainment of fluid from
the Ekman layer and is proportional to
ω
. When the flow
has positive relative vorticity, the Ekman layer pumps
fluid into the geostrophic interior domain, whereas fluid
columns with negative vorticity imply a flow into the
Ekman layer. In both cases the relative vorticity
ω
in
the interior decays by stretching and squeezing effects,
respectively. By using this result in the vorticity equation
(7.9), and neglecting all nonlinear terms and lateral fric-
tion (in order to isolate the bottom damping effects), it is
found that
This model is essentially a shallow-water formulation
with a rigid lid. The inviscid version (omitting all vis-
cous terms) and its properties are clearly explained by
Grimshaw et al.
[1994]. A more conventional formula-
tion consists of considering topographic variations much
smaller than the total fluid depth. This is the quasi-
geostrophic approximation model, which can be derived
by writing the fluid depth as
h(x
,
y)
=
H
h(x
,
y)
,where
H
is the mean depth, and small deviations are such that
|
−
h(x
,
y)
|
H
. The vorticity equation has the form
∂ω
∂t
+
J(q
qg
,
ψ
qg
)
=
ν
1
2
E
1
/
2
f ω
,
2
ω
∇
−
(7.19)
∂ω
∂t
=
1
2
E
2
f ω
.
−
(7.13)
where now the potential vorticity is defined as
q
qg
=
ω
+
f f h/H
and the stream function as
ψ
qg
=
ψ/H
. Note
that units of these two fields are different from their coun-
terparts in the shallow-water model. Another difference
is that the horizontal velocity has zero divergence, and
therefore the velocity components are
u
=
∂ψ
qg
/∂y
and
v
=
Thus the relative vorticity decay induced by the Ekman
layer is exponential,
ω
t/T
E
)
, where the decay
rate defines the characteristic Ekman time scale
∝
exp
(
−
2
fE
2
≡
H
(ν)
2
T
E
=
.
(7.14)
∂ψ
qg
/∂x
. The corresponding expression of the rela-
tive vorticity in terms of the stream function is the Poisson
equation
ω
=
−
7.2.3. Quasi-Two-Dimensional Models
2
ψ
qg
. In most studies, only linear Ekman
terms are considered.
We make a short digression here: For a geophysical
flow, all models above apply under the so-called
f
-plane
approximation, which consists of a plane tangent to
Earth's surface centered at a reference midlatitude
φ
0
.The
difference is that the geophysical Coriolis parameter is
now given in terms of the angular velocity component per-
pendicular to the plane,
f
−∇
Inviscid and viscous topography effects can be incorpo-
rated in a single formulation as derived by
Zavala Sansón
and van Heijst
[2002]. Considering both effects, the
z
inte-
gration of the continuity equation gives the horizontal
divergence as the sum of their contributions:
∂u
∂x
+
∂v
1
h
Dh
Dt
+
δ
E
∂y
=
−
2
h
ω
,
(7.15)
f
0
=2
e
sin
φ
0
, with
e
the
angular speed of the planet. The approximation is valid
for low motions up to order
L
≡
where the kinematic boundary condition is used at the
free surface and the Ekman pumping-suction condition
is used at the lower boundary. The key point is to use
the rigid-lid approximation
∂h/∂t
∼
100 km. For larger scales,
of order
L
1000 km, but keeping the plane approxi-
mation, corrections due to the curvature of Earth's sur-
face must be included. Such corrections are of the form
f
=
f
0
+
βy
,where
y
is the latitudinal direction and the
corresponding variation of
f
is given by the parameter
β
=2
e
cos
φ
0
/R
e
, with
R
e
the mean radius of Earth. The
so-called
β
-effect implies profound consequences on the
evolution of geophysical flows, and it can be simulated
in laboratory experiments with topography, as shall be
explained below.
∼
∼
0, which implies that
now
h
h(x
,
y)
. This allows the definition of a transport
function
ψ
from (7.15) such that, up to
O
≡
(δ
E
/h)
2
[
]
,the
horizontal velocity components are
∂ψ
∂y
−
,
.
u
=
1
h
δ
E
2
h
∂ψ
∂x
v
=
1
h
∂ψ
∂x
−
δ
E
2
h
∂ψ
∂y
−
(7.16)
