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u
w
+
v
w
)
∂τ
31
∂
x
3
e
1
+
∂τ
32
x
3
e
2
=
∂τ
z
=
−
∂
(
i
∂
∂
∂
z
where
τ
is a horizontal traction vector, expressed here as a complex number
τ
=
u
w
−
v
w
−
.Thishorizontaltractionvectorisoftenreferredtoasthe
Reynolds
stress
,butwiththeunderstandingthatthecompletedescriptionisatensor.Notethat
the trace of the Reynolds stress tensor (sum of the diagonal elements) is twice the
turbulentkineticenergyperunitmass.
i
2.3 Rotation: The Coriolis Force and Geostrophy
=(
)
Let
r
i, j, k
be a unit vector in a reference frame rotation with angular velocity
ω
.Inanunaccelerated(inertial)frame,thetimederivativesoftheunitvectorsare
d
i
dt
=
ω
×
d
j
dt
=
ω
×
d
k
dt
=
ω
×
i
j
k
Let
R
=
x
i
+
y
j
+
z
k
be a position vector in the rotating frame. Then differentiating
R
withtime
R
=
x
i
+
y
j
+
z
k
+
x
(
ω
×
i
)+
y
(
ω
×
j
)+
z
(
ω
×
k
)=
V
+
ω
×
R
(2.12)
where
V
is the velocity in the rotating frame. One further differentiation in time
expressesthe accelerationintherotatingframe:
R
V
=
+
ω
×
+
ω
×
(
ω
×
)
2
V
R
(2.13)
Thelasttermontherightiscentripetalacceleration,usuallyincorporatedintograv-
ityif consideredat all (it issmall at highlatitudes),whereasthesecondtermonthe
RHSof(2.13)istheCoriolisacceleration,andisofparamountimportanceformany
geophysicalflows.Theverticalcomponentoftherotationvectoractingonthehori-
zontal componentof flow is what we are commonlyinterested in, described by the
Coriolis parameter,
f
is latitude, with the convention
thatlatitudeispositiveinthenorthernhemisphereandnegativeinthesouth.Thein-
ertial periodis
2
=
2
ω
3
=
2
|
ω
|
sin
φ
where
φ
f
. At high latitudes, the Coriolis accelerationis strong(because
ω
3
is large)andthe effectsof rotationare morepronouncedthan atlowerlatitudes.
The angular rotation speed of the earth is 7
π
/
10
−
5
s
−
1
, so the inertial period,
.
292
×
f
,is12.23hat78
◦
N.
If the advective terms (including associated Reynolds stress) are ignored, the
horizontalpartoftheEulerequationin arotatingreferenceframebecomes
∂
2
π
/
H
p
ρ
u
t
+
f
k
×
u
=
−
∇
(2.14)
∂
Define
geostrophic current
,
u
g
, as the steady current that balances the horizontal
pressuregradient.Ifrapidvariationinairpressureisneglected,itmaybeexpressed
intermsofthe gradientofsea-surfaceelevation:
f
k
×
u
g
≡−
g
∇η
(2.15)