Here's a late breaking question about the homework etc:
Prof. Rhines,
I am wondering if you could clarify something in #4. When you ask us to
"discuss -- in particular how do verylong waves differ from the rigid lid
case? What wave equation do they satisfy?", what do you mean exactly? When
you say the rigid lid case, do you mean #3? Do you mean the Rossby wave
dispersion example in class (was that rigid lid?)?
Thanks for clarifying Rossby waves this morning. By the way, Gill defines
Rossby (planetary) waves as, "a new form of wave whose frequencies are
considerably lower than those of gravity waves. THey represent an
important new class of motion that occurs because the potential vorticity
on the curved surface of the earth is not a constant, but varies with
latitude". I am trying to figure out examples of Rossby waves. Am I
correct in assuming that the jet stream is a Rossby wave? What other
examples would I be familiar with?
I am still grappling with the idea that a longer wave would have
a higher frequency than a shorter wave. This implies to me that
the longer waves will propogate much faster. Very strange. So, a Rossby
wave in the jet will move faster than a blip (small disturbance) on that
same wave?
Discussion:
The 'rigid lid' case in #4 refers to the constant-depth Rossby waves derived
and discussed in class (upper boundary condition being w=0 at z=0). Change this to allow
for a free upper surface (ocean model) with elevation eta(x,y,t), as outlined in the
problem. Note that several of the dependent variables are virtually the same:
geostrophic pressure (say, at depth z=0, which is a close approximation to exact pressure
there); streamfunction (multiplied by f ro), and, because of hydrostatic balance, eta
(multiplied by g ro). With just a single layer of constant density fluid of course we
have no vertical shear of the horizontal velocity.
So, you have the linearized pv equation in terms of the variable eta, which has one
additional term compared with the rigid lid equation for psi. Using scale analysis and the
dispersion relation, you can discuss the behavior of the waves when their wavelength is
very large (small wavenumber).
Note that a key property of Rossby waves is that they
are dispersive. Different wavelengths have different phase speeds. They are also
anisotropic so that wavevectors with different directions (at the same wavelength)
have different phase speeds. With non-dispersive waves, all waves have the same phase
speed, and hence arbitrarily shaped initial conditions propagate without exploding into
a sea of sin-waves; these occur in certain corners of our parameter space.
Rossby waves, surprisingly, are everywhere. Meandering of the jet-stream, and development
of cyclonic and anticyclonic weather systems below can be called 'generalized Rossby
waves'; that is, waves depending on a pre-existing pv gradient which may be beta, the
Earth curvature or it may be a north-south gradient of isopycnal layer thickness (as in
baroclinic instability). It can also be a gradient in relative vorticity of a jet, which
means that small-scale shear instability ('barotropic instability') is actually a form of
unstable Rossby wave.
Perhaps we should call all these 'vorticity waves'. There
is the neighboring topic of 'geostrophic turbulence' which is what happens to Rossby waves
when they are get strong and break; or, what happened in the lab when we introduced a
couple of baroclinic eddies by injecting dyed fluid in a 2-layer stratification; they
interacted, each winding up the pv field of the other.
As to the wave speed, try
the vorticity argument for Rossby waves shown in Pedlosky sec.3.16; as he remarks in
the final sentence, the longer wavelengths lead to stronger restoring velocities and
hence higher frequency and faster wavespeed. The beta effect is the Earth's curvature, and
short waves don't 'feel' that curvature very much. Thus a short blip on the jet stream
would indeed to move slowly relative to the mean flow, and a long-wave distortion to
propagate faster. But the relative vorticity of the jet stream (du/dy) and its baroclinic
profile (du/dz) both yield north-south pv gradients, so the jet-stream story is more
complex.
In the atmosphere and in ocean boundary currents, wavy meanders propagate relative to a
strong mean flow...so they may just stand still relative to the ground. Thus in fact we
have 'standing' or 'stationary' Rossby waves in the lee of mountains. This is really the
best simple model of the atmosphere's time-averaged flow. Note that this can happen only
with westerly (eastward) winds, because the Rossby waves have westward phase propagation,
and can stand still only on an eastward wind. The useful message for the jet stream is
that its meanders move downstream more slowly than fluid parcels move downstream, because
of the Rossby mechanism.
Handedout figs: For a glimpse look at the handout Xerox's from last week. On top
is a
numerical Rossby
wave in a north-south channel. Plotted is psi or pressure. The view is of the x-axis,
with rigid boundaries at left and right. The bizarre behavior is that the long wave at the
start (just one-half wavelength in the domain) bounces off the wall as shorter waves (see
homework prob #4). Superimposed on the psi dot-plot is a Hoevmueller plot (x vs. time)
showing zero crossings of psi. Time increases downward. The sloping curves show westward
phase propagation of the waves; note the fast propagation early on, and slower phase
propagation later. Connect this with the group velocity of the waves.
The second figure is 300mb v velocity showing something like a standing Rossby wave in the
South Atlantic, where the westerly wind flows over the Andes Mts.
THe third figure is a numerical model showing a Rossby wave pressure field
develop in the lee of a single mountain (stippled), with westerly winds. The flow from
west to east makes a standing wave (omega/k = -Uo) whose energy propagates downstream
because the group velocity is smaller than the Rossby wave's intrinsic westward phase
speed.
The fourth figure shows a simple one-layer model with an initial psi
field in the form of a Gaussian, and NO mean flow. On an f-plane this would be a single
eddy that would sit still. On a beta plane, with weak amplitude, the eddy 'bursts' into
Rossby waves. Notice their sort of parabolic wave crests, just as shown in lecture on
Monday. The model has periodic boundary conditions, so it's actually a 'row' of eddies
radiating Rossby waves.
================================
Have a look some time at animations of either satellite cloud/water vapor images or
numerical weather prediction model output and get a feel for the 'elasticity' of this
fluid (web refs are posted at the top of this page); and, specifically to look at
the propagation of jet-stream meanders:
http://www.weather.unisys.com/mrf/mfr_300_loop.html