202
DYNAMICAL OCEANOGRAPHY
where χ =0for the barotropic wave and χ = F 1 + F 2 for the baroclinic wave.
Ex. 9.2
We know that the phase speed of these Rossby waves is westwards. Now the
problem to be studied is what happens when these waves meet a continent. Consider an incoming Rossby wave (packet) with wavevector (k i ,l i ) and dispersion
relation
σ i =
−βk i
χ + k 2
i + l 2
i
,
(9.20)
at the western boundary. The streamfunction of this incoming wave can be written
as
ψ i = A i e
i(k i x+l i y−σ i t) .
(9.21)
For the energy of the wave (packet) to reach the western boundary, the group
velocity has to be negative In Fig. 9.2a, we see that the incoming wave has a
K i
K
r
C
r
C i
θ r
θ i
x
y
(a)
θ r
θ i
k
l
K r
K
i
(b)
Figure 9.2. (a) Reflection of Rossby waves, where k is the wavevector and c the group velocity.
(b) Geometrical view of the change in wavelength during reflection.
wavenumber which represents long Rossby waves. The wavenumber k i is explicitly given by
k i = −
β
2σ i
−
β 2
4σ 2
i
− (χ + l 2
i ).
(9.22)
The reflected wave is represented by
ψ r = A r e
i(krx+lry−σrt) .
(9.23)
The total streamfunction is hence given by ψ = ψ i +ψ r . Because u = −∂ψ/∂y =
0 at the wall (x =0), the relations
l i A i e
i(l i y−σ i t) = −l r A r e
i(lry−σrt) ,
(9.24)
must hold for every y and t which can only occur when
σ i = σ r ; l i = l r .
(9.25)
DYNAMICAL OCEANOGRAPHY
where χ =0for the barotropic wave and χ = F 1 + F 2 for the baroclinic wave.
Ex. 9.2
We know that the phase speed of these Rossby waves is westwards. Now the
problem to be studied is what happens when these waves meet a continent. Consider an incoming Rossby wave (packet) with wavevector (k i ,l i ) and dispersion
relation
σ i =
−βk i
χ + k 2
i + l 2
i
,
(9.20)
at the western boundary. The streamfunction of this incoming wave can be written
as
ψ i = A i e
i(k i x+l i y−σ i t) .
(9.21)
For the energy of the wave (packet) to reach the western boundary, the group
velocity has to be negative In Fig. 9.2a, we see that the incoming wave has a
K i
K
r
C
r
C i
θ r
θ i
x
y
(a)
θ r
θ i
k
l
K r
K
i
(b)
Figure 9.2. (a) Reflection of Rossby waves, where k is the wavevector and c the group velocity.
(b) Geometrical view of the change in wavelength during reflection.
wavenumber which represents long Rossby waves. The wavenumber k i is explicitly given by
k i = −
β
2σ i
−
β 2
4σ 2
i
− (χ + l 2
i ).
(9.22)
The reflected wave is represented by
ψ r = A r e
i(krx+lry−σrt) .
(9.23)
The total streamfunction is hence given by ψ = ψ i +ψ r . Because u = −∂ψ/∂y =
0 at the wall (x =0), the relations
l i A i e
i(l i y−σ i t) = −l r A r e
i(lry−σrt) ,
(9.24)
must hold for every y and t which can only occur when
σ i = σ r ; l i = l r .
(9.25)