Geoscience Reference
In-Depth Information
2.3.1 Geostrophic Shear
Differentiating steady, geostrophic velocity (2.15) with respect to
z
, provides an
equationforverticalshearin termsofhorizontaldensitygradients:
∂
×
∂
u
g
∂
z
=
−
∇
p
+
∇
H
p
ρ
2
∂ρ
H
ρ
f
k
(2.16)
∂
z
∂
z
In nearly all practical applications the change in density with depth is small com-
paredtoitsmeanvalue,andthesecondtermontherightisnegligible.Thuswiththe
hydrostaticequation,
∂
p
/
∂
z
=
−
g
ρ
,an expressionforshear in geostrophiccurrent
isgiventogoodapproximationby
∂
u
g
∂
g
ρ
=
−
f
k
×
∇
ρ
(2.17)
H
z
Bythisrelation,verticalcurrentshearmaybepresentevenintheabsenceoffriction,
potentiallyasignificantfactorasicedriftsacrosswaterwithlargehorizontalsalinity
gradients. In the atmosphere, where density is controlled mainly by temperature,
(2.17) defines the
thermal wind
relation, i.e., vertical wind shear and differential
advectionassociatedwithtemperaturefronts.
2.4 Boundary-Layer Equations
Withrotation,theturbulentBoussinesqequation(2.11)becomes
p
ρ
−
g
ρ
ρ
∂
u
t
+
u
·
∇
u
+
f
k
×
u
=
−
∇
k
+
∇
·
∼
(2.18)
∂
where the impact of the Coriolis force on the deviatory velocities is considered
negligible in most IOBL applications, especially at high latitudes. In general, the
horizontalcomponentsof (2.18) are of most interest. In the absence of wind stress
or forcing from internal stress gradients, ice will respond to a horizontal pressure
gradient from tilt of the sea; i.e., shear between it and the underlying ocean will
vanishso that in a steady state the ice velocityis just
u
g
. Itis thusoftenconvenient
to consider boundary layer flow in a frame of reference translating with the mean
geostrophic current, so that the pressure gradient can be eliminated from (2.18),
yieldinganequationforthehorizontalcomponentsofrelativevelocity
∂
u
r
∂
t
+
u
r
·
∇
u
r
+
f
k
×
u
r
=
∇
·
∼
(2.19)
where
u
r
=
u
g
. Unless otherwise noted,in what follows we will tacitly assume
that IOBL velocity is velocity with respect to the undisturbed ocean flow and drop
fromthe
r
subscriptfrom(2.19).
u
−
