Vorticity
79
4.4. Shallow-water equations
In this section several concepts of vorticity are studied in more detail along with
the introduction of the shallow-water equations. As we will both use dimensional
and dimensionless quantities, we use a star subscript to indicate dimensional quantities.
Consider the flow in a shallow liquid layer having a constant density ρ on a
plane that is rotating with angular velocity Ω. We choose a Cartesian coordinate
system with the gravity vector parallel to the z ∗ -axis. Let the bottom be described
by z ∗ = −D 0 + h b∗ (x ∗ ,y ∗ ) and the liquid-gas interface by z ∗ = h ∗ (x ∗ ,y ∗ ,t ∗ ).
The equations (3.32), with f =2Ω,thenbecome
ρ(
Du ∗
dt ∗
− fv ∗ )=−
∂p ∗
∂x ∗
+ ρA H
∂ 2 u ∗
∂x 2
∗
+
∂ 2 u ∗
∂y 2
∗
+ ρA V
∂ 2 u ∗
∂z 2
∗
, (4.17a)
ρ(
Dv ∗
dt ∗
+ fu ∗ )=−
∂p ∗
∂y ∗
+ ρA H
∂ 2 v ∗
∂x 2
∗
+
∂ 2 v ∗
∂y 2
∗
+ ρA V
∂ 2 v ∗
∂z 2
∗
, (4.17b)
ρ
Dw ∗
dt ∗
= −gρ −
∂p ∗
∂z ∗
+ ρA H
∂ 2 w ∗
∂x 2
∗
+
∂ 2 w ∗
∂y 2
∗
+ ρA V
∂ 2 w ∗
∂z 2
∗
, (4.17c)
∂u ∗
∂x ∗
+
∂v ∗
∂y ∗
+
∂w ∗
∂z ∗
=0, (4.17d)
with
D
dt ∗
=
∂
∂t ∗
+ u ∗
∂
∂x ∗
+ v ∗
∂
∂y ∗
+ w ∗
∂
∂z ∗
.
(4.18)
4.4.1. Hydrostatic equilibrium
A characteristic vertical length scale of the problem is the average liquid layer
depth D, L is a characteristic horizontal length scale, U is a horizontal velocity
scale and τ is a characteristic time scale. Let W and P be a priori unknown scales
of vertical velocity and dynamic pressure, then we define dimensionless quantities
x =
x ∗
L
,y =
y ∗
L
,z =
z ∗
D
,t =
t ∗
τ
(4.19a)
u =
u ∗
U
,v =
v ∗
U
,w =
w ∗
W
,
(4.19b)
p ∗ = −gρz ∗ + pP.
(4.19c)
Because ∂u ∗ /∂x ∗ = O(U/L) and ∂v ∗ /∂y ∗ = O(U/L), it follows from (4.17d)
that W cannot be larger than
W =
D
L
U.
(4.20)
The estimate (4.20) is an upper boundary; W can be smaller than (4.20) if the
terms ∂u ∗ /∂x ∗ and ∂v ∗ /∂y ∗ partially cancel.