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i.e., the depth
d
=
h
b
is very small compared to the horizontal length
scale of waves or other flow features, and in the next chapter we'll instead
consider the water very deep relative to this scale.
−
12.1 Deriving the Shallow Water Equations
12.1.1 Assumptions
The shallowness assumption means essentially that we can ignore vertical
variations in the velocity field: the fluid doesn't have “room” for vertical
features like vortices stacked on top of each other. We'll then just track
the
depth-averaged
horizontal velocities,
u
(
x, z
)and
w
(
x, z
), which are the
average of
u
and
w
for
y
varying along the depth of the water. For the
inviscid flow we're modeling in this chapter, you can think of
u
and
w
as
constant along
y
; for viscous flow a better model would be that
u
and
w
vary linearly from zero at the bottom to some maximum velocity at the
free surface.
Just as an interjection: the process of depth averaging is used in many
more contexts than the one here. For example, avalanches have been mod-
eled this way (the snow layer is thin compared to the extent of the moun-
tainside it flows along) as well as large-scale weather patterns (the atmo-
sphere and oceans are extremely thin compared to the circumference of
the Earth). The resulting systems of equations are still commonly called
“shallow water equations,” even if referring to fluids other than water.
The other fundamental simplification we'll make is assuming hydrostatic
pressure. That is, if we look at the vertical component of the momentum
equation
∂t
+
u
∂v
∂x
+
v
∂v
∂y
+
w
∂v
∂z
+
1
∂p
∂y
=
∂v
−
g,
ρ
(where
g
9
.
81m
/
s is the magnitude of acceleration due to gravity) we
will assume that the dominant terms are the pressure gradient and gravity,
with the rest much smaller. This is consistent with the requirement that
the water is shallow and relatively calm, with accelerations in the fluid
much smaller than
g
.
≈
Dropping the small terms gives the equation for
hydrostatic pressure:
1
ρ
∂p
∂y
=
−
g.
