Rossby waves

[Ari Epstein <awe@pimms.mit.edu>]

James McLean wrote:

> Ari Epstein says:
> > Rossby waves are
> > among the most important ways in which changes in oceanic and/or
> > atmospheric conditions can propagate for long distances,
>
> Can you give a few details on how?  When I think of waves, I imagine the
> medium (ocean water) not moving very much.  In other words, the waves would
> propagate the water hieght, but not any of it's other properties
> (e.g. temperature).  So I don't understand how this can affect atmospheric
> conditions.
>
>  What am I missing?


It's kind of tricky, since in a Rossby wave the thing doing the
"waving" is not necessarily the water's surface height (which is what
most people imagine when they think of water waves). What propagates
might be some inhomogeneity in, for example, horizontal velocity, or
in some other property.

Let me give one example, however: 

Consider a stratified ocean, in which water at depth is colder than
water at the surface. Suppose some event occurs (say an atmospheric
disturbance) that causes the cold water to rise up toward the surface
in a certain region. A vertical cross-section of that region would
show a "bump" in the isotherms (lines of constant temperature) there:
an isotherm normally found at, say, 500 meters depth might come within
a few meters of the surface in that one region. The surface water
there would then be colder than the surface water elsewhere (since
some isotherms would actually "shoal," or intersect the surface of the
water).  Rossby waves are one mechanism by which such inhomogeneities
can propagate on global scales. In the mid-latitude ocean far from
land, they are probably the most important. 

What does this mean for climate? Well, if some localized event makes
the surface water in a certain region colder, Rossby waves will act to
propagate that "signal" (the displacement of the isotherms), causing
the surface water elsewhere to become colder some time later. And the
signal remains strong over long distances, in part because Rossby
waves are directional: a given Rossby wave will propagate only in
certain directions. Rossby waves are one of the ways in which
temperature changes propagate during the so-called El Nino-Southern
Oscillation events. 


I can't quit here without explaining a little bit about how Rossby
waves work, because it's just so interesting. As I mentioned in an
earlier posting, the "restoring force" in a Rossby wave really comes
from the conservation of angular momentum on a spinning Earth. In
addition, the phase speed (but not necessarily the group velocity)
always has a westward component. Here's a (necessarily oversimplified)
description of the mechanism at work:

Consider a column of water on the surface of a spinning planet. In the
planet's frame of reference (a *very* noninertial frame), a quantity
called the water's "potential vorticity" will always be conserved if
friction and other torque-producing forces are absent. Potential
vorticity is kind of a measure of how fast the column is "spinning"
relative to an inertial frame. Roughly speaking, it is equal to the
water's "relative vorticity" (how fast the column is spinning relative
to the earth's surface--a tornado, for example, has extremely high
relative vorticity) plus its "planetary vorticity," all divided by the
column's height. "Planetary vorticity" is a measure of how fast the
water is "spinning" (relative to the local vertical) simply by being
at a certain latitude. For example, a column of water at the North
Pole has high planetary vorticity because a "stationary" object at the
North Pole is actually spinning counter-clockwise once per day;
conversely, a column of water at the South Pole would have negative
planetary vorticity, because (relative to the local vertical) it spins
clockwise once per day. A column of water at the equator has zero
planetary vorticity.  Planetary vorticity therefore increases as you
move northward. (It is proportional to the sine of the local
latitude.)

To see how this can make waves, set a few coins in a horizontal row
flat on the desk in front of you. Pretend that they are columns of
water somewhere in the ocean, and that they begin with zero relative
vorticity (i.e.  they are not spinning relative to the
desktop). Suppose a disturbance of some sort pushes one column of
water slightly "northward" (away from you); for the sake of this
discussion, make it the easternmost column (the coin farthest to your
right). As that column moves northward, its planetary vorticity will
increase. To conserve potential vorticity, it must therefore take on
some negative relative vorticity: it must begin spinning in a
clockwise direction. The resulting flow field will tend to pull the
next column of water (the coin second from the right) northward as
well. That column will then also begin spinning in a clockwise
direction, pushing the first column back toward its initial position,
and also pulling the next column (the third coin from the right)
northward. The disturbance will continue to propagate in this way,
always moving westward. This is an analogue of a Rossby wave.



Ari Epstein