Rossby Waves and the Polar Vortex
Rossby waves are the information-carriers of the atmosphere and ocean.
They transport energy and angular momentum, often driving the kind of
banded zonal jets seen on Jupiter and Earth. Here we illustrate their two
basic forms: oscillatory waves in a fluid nearly at rest, driven by a
small oscillating body (representing Green's function for Rossby waves);
and, Rossby
waves excited by westerly (aka eastward) mean flow past a mountain, both
on a polar ß-plane.
This is a polar ß-plane experiment, with rapidly rotating (Ω=3.2 sec-1) homogeneous fluid with a free surface. The paraboloidal shape of the free surface provides a potential vorticity gradient closely analogous to the effect of the spherical Earth. The 1m diameter cylinder has a wavemaker, in the form of a horizontal disk (10 cm diameter), driven by a long rod, oscillating slowly up and down. It produces Rossby waves, most visibly the short waves propagating east from the wavemaker. The green wavecrests have phase propagation westward toward the wavemaker, yet group velocity eastward, away from it. The longer waves propagate west, and are less visible. The sequence shows the development of the eastern wavetrain (with group velocity about 5 cm/sec, and the induction of systematic bands of zonal flow by the wave's potential vorticity flux. Westward mean flow occurs both north and south of the wavemaker (the deep blue and green bands), while an eastward jet (faint blue) develops at the latitudes encompassing the forcing. This jet was discovered in Whitehead's landmark experiments (Tellus, 1974). Because of bottom-frictional dissipation, fluid does traverse across latitude circles, for example connecting the eastward jet sucked into the wavemaker with a westward jet further north.
Yet, at this high rotation rate the potential vorticity 'barrier' to meridional flow is strong, as is the 'Rossby wave elasticity'. Orange fluid at the North Pole remains unmixed, like an 'ozone-hole' region in the stratosphere, for the duration of the experiment.
Stirring of potential vorticity leads to the predictable polar anticyclone, almost regardless of the detailed nature of the Rossby waves or geostrophic turbulence. Low pv from lower latitudes mixes poleward, replacing the high pv of the polar cap, and since the planetary vorticity is unchanged this must appear as anticyclonic (easterly, westward) zonal acceleration. Beautiful numerical experiments of Yoden and Yamada (J.Atmos.Sci. 1993, Phys.Fluids 1997) show this, and the general arguments are given by Rhines (The Sea, 1977, Dyn. Atmos. Oceans 1979, Ann.Revs. Fluid Mech. 1979, Geophys.Astrophys.Fluid Dyn. 1983). This particular wavemaker also generates small scale vortices, which flow with the induced circulation northwestward.
A 2 Mb animation of a fully developed Rossby wave field can be downloaded here;. The animation is time-lapsed, with a factor 12 faster than real time. {It is an .avi file with an MPEG-4 codec which plays on Windows Media Player. Use its contrast and brightness controls if needed.} A longer (9Mb) animation showing the wave-field start from rest, and sweeping across a range of frequencies, is shown here. There are several or many spatial modes at a given frequency in this 1 m. diameter cylinder, and the band of resonant, large amplitude motion (at about 9 sec. period) is quite narrow.
Notice once again that the orange polar cap is very unwilling to be mixed, despite tortuous straining by the waves. This is the essence of the ozone hole in the south polar stratosphere: captured by the potential vorticity barrier, high-altitude air loses its ozone in winter and it is not readily replenished by mixing (except in 2002!).
If you look at the dispersion relation connecting frequency and wavenumbers for simple Rossby waves, it is apparent that long waves with nearly zonal wave crests (and nearly zonal currents) reach westward from the source region, while short waves, with nearly meridional (north-south) wave-crests and currents reach eastward.
Here ψ is the wave-function;
it could be
the pressure, streamfunction, or the height of a layer of fluid with a
free
surface in which the wave is propagating. ω is
the wave frequency, (k,l) are the wavenumbers in
the east (x-) and
north
(y-) directions (together they are the 'wave-vector').
φ is the angle between the
wave-vector and the
eastward direction.This is
despite the phase propagation which is westward everywhere. Thus
in
the animation above the dominant waves are the short wavelengths
reaching
eastward from the forcing plunger. This is all summed up in the Green
function for linear waves on a ß-plane (Rhines, J.Fluid
Mechanics, 1969),
which is shown just below. Here ω is now the frequency of the
forcing, and H is the
Hankel function of the second kind, a complex Bessel function which by itself
has the spirit of a travelling wave (with spiral wavecrests).
Barotropic Rossby Waves on a mean westerly flow
The same basic waves exist when a zonal flow is forced to pass over a mountain, yet many new effects occur. The linear theory of waves generated by a moving forcing effect (equivalent to waves in a moving stream due to stationary forcing effect) have been described (Lighthill, Waves in Fluids, Cambridge University Press, 1978). The Green function for this problem exhibits semi-circular wave-crests downwind of the forcing region, centered upon it. These semi-circles fill a circular region which grows with time. They are nicely described by examining the locus of possible wavenumbers for a fixed mean flow, U, given by
which are just circles in wavenumber space, together with the vertical axis (k = 0). The k=0 solutions correspond to purely zonal velocity (and east-west wavecrests) which one might think are 'non-waves', judging by the dispersion relation above. However in the limit ω=>0 with the wavelength fixed we find the waves have finite (large!) group velocity and are capable of propagating zonal currents westward and eastward from a forcing function. These might be called Lighthill modes in honor of the late Prof. Lighthill. They are a linear representation of blocking of a zonal wind by a forcing region.
There is a
technicality about small mountains, which cannot excite these modes, yet
if you view the animations that follow you see that for small enough
U/βL2, experiments
from our GFD lab show strong upwind (westward) blocking of zonal flows.
The dominant lee Rossby waves have circular wave-crests, because
of the symmetry of the
wave equation with a uniform, steady westerly wind. Actually they are
semi-circular because the trail behind the mountain and cannot
propagate upwind. With time they fill out a region, also circular, that
expands at a rate 2U downwind. So, the region of trailing Rossby waves
expands downwind at twice the speed of the mean wind. This much we
learn from ray theory, for the wavefield far from the mountain. A
complete linear solution without this approximation is plotted just
below, for the case of a cylindrical mountain (from McCartney,
Journal of Fluid Mechanics 1976).
The first animation (2 Mb .avi
file) shows the
standing Rossby waves that develop in
the lee of a circular mountain that penetrates about 1/3 of the depth of
the fluid, on a polar
Where the K0 is now the imaginary Bessel function of the
second
kind. A plot (just below) of this solution shows a 'gyre' of
circulation driven by the
'tweak' of forcing at the origin.
The
animated experiment
shows some stationary waves developing, yet the wake is unstable,
generating time-dependent disturbances and Rossby waves with a wide
spectrum. This was anticipated by the numerical simulations of Polvani and
Plumb (J. Atmospheric Sciences, 1992).
The
second animation (2 Mb) shows the Rossby waves and wake
generated at a
larger value of U/βL2.
Here you will notice that the waves are much more stationary, the wake is
stable, and the upstream blocking is much weakened in comparison with the
first experiment. We still draw polar fluid into the wake of the
mountain, however, and there is still small-scale geostropic turbulence at
work.
Annular modes observed in both hemispheres of
the atmosphere have much in common with these zonal jets. They have
significant barotropic components, and time-variation of meridional and
vertical pv flux is known to be active in driving them (e.g., Hartmann
D.L.,
2000: The key role of lower-level meridional shear in baroclinic wave life
cycles J. Atm. Sci., 57, 389-401; DeWeaver, E., and S. Nigam, 2000: Do
stationary waves drive the zonal-mean jet anomalies of the Northern
Winter? J. Climate, 13, 2160-2176, also Stationary waves: structure and
forcing, Encyclopedia of Atmospheric
Sciences, Academic Press,2002). Vertical propagation of Rossby waves
in the stratified atmosphere occurs along ray paths that are sensitive to
the presence of the annular modes: this introduces the idea of a
'self-tuning wave-guide'. Dave Thompson has established a new website for annular
modes. An interesting recent meeting on stratosphere-troposphere
interaction, involving many aspects of vertically propagating Rossby
waves, is found
here.
The gyre is circular near the origin, at
small Cr, yet at larger distance becomes elongated along the negative x-axis,
that is in the westward direction. It is a sort of 'arrested' Rossby
wave, and is another expression of a Lighthill mode.
