Geoscience Reference
In-Depth Information
An importantimplicationof (2.19)isthat thesteady state volumetransport(i.e.,
thedepthintegralofvelocity)inthe boundarylayerrelativetothe geostrophicflow
isatrightangle(
cum sole
)tothesurfacestressandproportionaltoitsmagnitude.If
horizontalvelocityisexpressedasacomplexnumber
u
=
u
x
+
iu
y
thesteadyIOBL
momentumequationis
=
∂τ
∂
if
u
(2.20)
z
At some level near the far extentof the boundarylayer, the turbulentstress is zero,
sointegrating(2.20)fromthatlevelto thesurfaceprovides
0
if
u
dz
=
if
M
=
τ
0
(2.21)
z
bl
where
M
isthe vectorvolumetransportand
τ
0
isthe kinematicstress atthebound-
ary.Multiplyingahorizontalvectorby
i
rotatesitby90
◦
,thusvolumetransportwill
beapproximatelyperpendiculartosurfacestress,regardlessofdetailsofturbulence
intheIOBL.However,ashallowlayerwillrequirehighermeanvelocitythanadeep
layertoeffectthesametransport,whichplacesanimportantconstraintonboundary
layerscales.
2.5 Inertial Oscillations
Ekman (1905) in his classic paper, pointed out (with credit to Fredholm) the
possibility of oscillations in the upper ocean having the inertial period
2
f
.
Heuristically,
inertial oscillations
are easily demonstratedby consideringthe time-
dependentvolumetransportequationobtainedbyverticallyintegratingthehorizon-
tallyhomogeneousversionof(2.19):
π
/
∂
M
∂
t
+
if
M
=
τ
0
(2.22)
Suppose an upper ocean system initially at rest is subjected to an impulsive stress
in the
y
-direction,
i
=
τ
0
at time
t
0. It is easily verifiedthat the complexsolutionof
(2.22)is
=
τ
0
e
−
ift
M
f
(
1
−
)
The solution, sketched in Fig. 2.1, traces a circle in one inertial period about the
steady-state balance
M
ss
=
τ
0
/
f
, but because there is no friction in this system it
continues to oscillate with the inertial period, averaging
M
ss
, but never having the
steady-state value. Despite the seeming unreality of this example, it is instructive
to consider some numbers. A typical kinematic surface stress during a moderate
squallmightbe
10
−
4
m
2
s
−
2
,withamaximumvolumetransport(occurring
τ
0
=
2
×
75m
2
s
−
1
. If the summertime
mixedlayerwas25mthick,thedepthaveragedvelocityintheboundarylayerwould
bearound11cms
−
1
.
at
t
=
6, 18, 30h, etc., at the North Pole) of about 2
.
