GEF4610 – OBLIGATORY EXERCISE 2
All symbols are defined in the GEF4610 Lecture Notes
Problem 1
Shallow-water waves in a non-rotating ocean with a slowly-varying bottom
Consider propagating surface waves in two dimensions on the form
η = Aei ( kx+ly −ωt ) .
(1.1)
The bottom is slowly varying, and can be expressed as z = − H ( x) = − H 0 (1 + β x) where β is
small; see Fig.1. We will in this problem study the propagation of such waves.
z
η
x
0
-H0
z = -H(x)
Figure 1. Model sketch.
a) The equations and surface boundary conditions governing the motion are:
p
ρ0
= −φt − gz
∇ 2φ = 0
(1.2)
p ( z = η ) = 0.
Show that the velocity potential can be written
φ=−
iωA cosh κ ( z + H ) i ( kx+ly −ωt )
e
,
κ sinh(κH )
(1.3)
where κ 2 = k 2 + l 2 . What is the necessary assumption regarding the bottom boundary for this
to hold?
b) Using the shallow water assumption, show that the frequency may be expressed as
1
ω = κ gH (x) .
(1.4)
c) The wave field is assumed stationary, and there is no variation in the y-direction in this
problem. In addition, the wave number vector is irrotational (can you show this?). Use this to
show that the wave number component l = l0 is constant in space.
d) We have arrived at a relation on the form
∂ω dk ∂ω dH
+
=0.
∂k dx ∂H dx
(1.5)
Assume that k ( H = H 0 ) = k 0 and show that
H0
κ
(k 02 + l02 ) − l02 ≈ k 0 (1 − 0 2 β x)
H ( x)
2k 0
2
k ( x) =
(1.6)
where κ 02 = k 02 + l02 .
e) Energy propagation is along a path (“ray”) defined as dr × c g = 0 where c g is the group
velocity and dr = dxi + dyj . Show that
dy =
l0
dx
k ( x)
(1.7)
and derive the equation for the ray y(x). Sketch and comment your findings.
Problem 2
Continental shelf waves
We now consider waves under a rigid lid and variable topography H = H 0 exp(2by ) as shown
in Fig. 2.
2
z
x
Coast
B
y
0
H = H0exp(2by)
Figure 2. Model sketch.
a) From the linearized continuity equation and conservation of potential vorticity, show that
the equation controlling the motion is
∇ 2ψ t −
H'
(ψ yt + fψ x ) = 0 ,
H
(2.1)
where H ' = dH / dy , Hu = −ψ y and Hv = ψ x . The Coriolis parameter f is assumed constant
and positive.
b) We assume wave motion on the form
ψ = F ( y )e i ( kx−ωt )
F ( y) = ϕ ( y) H .
(2.2)
Show that the equation for ϕ is
d 2ϕ
2kf b
+ ( −b 2 − k 2 +
)ϕ = 0
2
dy
ω
(2.3)
c) Assume now that v( y = 0) = v( y = B) = 0 . Determine the dispersion relation and make a
plot for the lowest nonzero mode. Show that ω < f for all ω .
d) Determine the group and phase velocity for this case. What can you say about the wave and
energy propagation for very long versus very short waves? How does this relate to the
frequency?
3