Geoscience Reference
In-Depth Information
The influence of the pressure gradient forces can be derived in this manner:
given an infinitesimal volume
x
3
, the force exerted by the pressure on
the left side of the cube in
x
1
direction is given by
p
(
x
)
∆
V
=
∆
x
1
∆
x
2
∆
x
3
(pressure is force
per area). Similarly, the force exerted on the right side of the cube is
∆
x
2
∆
≅− −
∂
∂
p
x
−+
px
(
∆∆∆
x
)
x
x
px
( )
∆∆∆
xxx
1
2
3
1
2
3
1
x
1
). So the net force
on the volume in the
x
1
direction, after adding the two contributions, is
dV
,
and the force per unit volume is just . The same analysis can be carried out
for the other two directions, giving the total pressure gradient force per unit volume
of .
This result can now be used to insert the external forces into the general momen-
tum equation. Including the gravitational force in their form per unit volume
Here a Taylor series expansion has been applied to
p
(
x
+
∆
p
x
1
−
∂
∂
−
∂
∂
p
x
1
−
∂
∂
p
x
ρ
g
i
,
we have
=−
∂
∂
p
x
f
+ ρ
g
i
i
i
and, therefore,
du
dt
1
ρ
∂
∂
p
x
i
=−
+
g
(3.13)
i
i
In this form the equations are called
Euler equations,
or more precisely, Euler
equations in a nonrotating frame of reference. Note that the gravitational acceleration
is a vector with only the vertical component different from zero
g
i
=
(0, 0,
−
g
)
(pointing downward) and
g
is 9.81 m s
−2
.
3.4.1.2.2 The Euler Equations in a Rotating Frame of Reference
The above-derived equations are not suitable for their application to meso-scale or
basin-wide scale. This is because the Earth is not an inertial frame of reference but
is rotating. Although this has no impact on water bodies that are small in size, for
the larger applications it is very important to take account of this effect.
The influence of the Earth's rotation can be described with the introduction
of a new apparent volume force called the
Coriolis force
. In this case, the new
forces on the water parcel are composed not only of the pressure gradient, but
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