Stratification
195
All quantities describing the flow are independent of the coordinate y. Assume
that inertia and friction can be neglected, that |h|≪1 and that
¯
v ≡
h
−1
vdz =0
a. Derive the equation determining the slope of the sea surface in terms in the
density field.
b. Determine the velocity field of the flow and calculate the depth at which
v =0; this is the so-called ‘level of no motion’.
c. Make a sketch of the sea surface height and the velocity field. Provide a
physical interpretation of the result.
(8.4) Rossby waves in the Pacific
Rossby waves propagate westwards and can be observed at midlatitudes with
altimeters. In the figure below (source http://topex-www.jpl.nasa.gov/), measurements of sea surface height anomalies (in cm) are plotted as a function
of time along three latitudes in the Pacific (at 20 ◦ N, 32 ◦ N and 39 ◦ N, respectively).
a. Provide an estimate of the phase speed of the Rossby waves in these figures.
b. Calculate the dimensional phase speed of a Rossby wave with a zonal
wavelength of 5000 km and with an infinite meridional wavelength in a
constant density layer of water with a depth H 0 =3k ma t( i )1 0 ◦ N and (ii)
60 ◦ N.
c. Are the waves in the figures barotropic or baroclinic Rossby waves?
(8.5) Reflection of Rossby waves
As we have seen in section 8.4, for a baroclinic Rossby wave with certain
χ = χ n and with wavevector k =( k, 0), the dimensionless streamfunction ψ
is given by
ψ(x, y, t)=Ψ 0 exp [−i(kx − σt)]
with Ψ 0 being a complex amplitude and with the dispersion relation
σ =
−βk
χ + k 2
195
All quantities describing the flow are independent of the coordinate y. Assume
that inertia and friction can be neglected, that |h|≪1 and that
¯
v ≡
h
−1
vdz =0
a. Derive the equation determining the slope of the sea surface in terms in the
density field.
b. Determine the velocity field of the flow and calculate the depth at which
v =0; this is the so-called ‘level of no motion’.
c. Make a sketch of the sea surface height and the velocity field. Provide a
physical interpretation of the result.
(8.4) Rossby waves in the Pacific
Rossby waves propagate westwards and can be observed at midlatitudes with
altimeters. In the figure below (source http://topex-www.jpl.nasa.gov/), measurements of sea surface height anomalies (in cm) are plotted as a function
of time along three latitudes in the Pacific (at 20 ◦ N, 32 ◦ N and 39 ◦ N, respectively).
a. Provide an estimate of the phase speed of the Rossby waves in these figures.
b. Calculate the dimensional phase speed of a Rossby wave with a zonal
wavelength of 5000 km and with an infinite meridional wavelength in a
constant density layer of water with a depth H 0 =3k ma t( i )1 0 ◦ N and (ii)
60 ◦ N.
c. Are the waves in the figures barotropic or baroclinic Rossby waves?
(8.5) Reflection of Rossby waves
As we have seen in section 8.4, for a baroclinic Rossby wave with certain
χ = χ n and with wavevector k =( k, 0), the dimensionless streamfunction ψ
is given by
ψ(x, y, t)=Ψ 0 exp [−i(kx − σt)]
with Ψ 0 being a complex amplitude and with the dispersion relation
σ =
−βk
χ + k 2