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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)
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