4. Flow over mountains: making waves
To understand flow over mountains (as well as just about any other dynamical
phenomenon in the atmosphere or ocean that is not zonally uniform), one must think in
terms of waves. Fluids (e.g. the atmosphere) experience a rich variety of them, some
more familiar than others.
The most familiar fluid wave is a gravity wave on a fluid surface. This wave involves
interaction between inertia, which tries to carry upward-moving fluid on in the same
direction, and a restoring force due to gravity which tries to flatten the surface height
anomalies back to zero. As you know from throwing rocks into ponds, waves radiate
from the point where they are excited, with some propagation velocity c. They are also
described by a wavelength l and frequency n. The dominant l and n that occur are
determined by the horizontal size and/or oscillation frequency of the source (and possibly
by its magnitude, if large enough). Large sources are less able to produce short waves,
but localized sources can produce all wavelengths, depending on how the source varies
with time.
The surface gravity wave involves water motions that extend down into the fluid below
(and up into that above, if there is one), decaying exponentially with depth. If there is a
shallow bottom (compared to this natural decay distance) then the fluid motions will be
about the same at all depths and the wave is called a shallow water wave. For these
waves, the wave speed c=÷(gh), h the depth and g the gravitational constant. For waves
where h is large (say, ocean swells far offshore), the decay depth of fluid motions will be
l/2p. Then c=÷(gh'), where h' is the equivalent depth---for surface gravity waves, just the
decay depth.
This means, in deep water, longer waves propagate faster (this is the main reason why
longer boats can go faster). When c depends on l we call the wave dispersive. Sound
and light waves are nondispersive, but most fluid waves are dispersive. Nonetheless,
there is a tendency for c to vary less than l or n such that, by and large, the most slowly
varying waves have the longest wavelengths.
Internal waves
In general, waves can occur within a fluid anywhere density decreases with height, but
these will propagate much more slowly than surface waves since the restoring force is
much weaker. These are called internal waves. If the density change is gradual (as it
nearly always is within a real fluid) rather than sudden (e.g. the interface between water
and air), then waves can propagate vertically as well as horizontally. This is important in
the atmosphere.
Waves where n >>1/day are all of the gravity wave type. At longer periods, the rotation
of the earth enters and significantly complicates things. Very slow waves are
geostrophically balanced; the longest-wavelength of these are planetary waves.
Mountain impacts on climate occur largely through the generation of stationary planetary
waves. Stationary means the phase of the wave does not change with time at a particular
location on the planet; a wave can remain stationary only if it propagates upstream and
c+u=0, where u is the background flow speed. Doppler shifting is obviously important
to keep track of in fluid waves.
The easiest way to understand a planetary (also called Rossby) wave is to consider
potential vorticity. Recall that most of an air mass's vorticity is due to the planet, with a
small contribution coming from the rotation of the fluid relative to the ground. If such an
air mass is moving eastward, and is somehow deflected poleward (equatorward), it will
soon find itself rotating anticyclonically (cyclonically) with respect to the surface. This
will cause the flow to turn equatorward (poleward), returning the air mass to its previous
latitude. This restoring tendency operates analogously to buoyancy in a gravity wave, but
the resulting fluid displacements are mostly meridional rather than vertical (though there
is a vertical component too, since thickness varies with latitude). If the initial velocity is
westward, on the other hand, the initial perturbation will amplify rather than restore, and
the flow will disintegrate (hence there are no geostrophic easterly jets). Mathematical
analysis reveals that Rossby waves can propagate only westward, and that longer ones
propagate faster. That means there will be a specific ls that is stationary, depending on u,
for positive u.
Planetary waves are usually described in terms of their wavenumber n, rather than
wavelength. A wave that fits just once around the planet (the longest possible stationary
wave) is “wavenumber 1”, two full cycles around a longitude line is wavenumber 2, etc.
The parts of the wave that dip down toward the equator are called troughs, the upwardreaching parts ridges.
Wave causes and effects
Waves affect weather and climate patterns by altering the winds that advect energy and
water vapor. The waves can be driven either by flow over topography, or by “diabatic
forcing,” which means zonally uneven heating of the atmosphere. For example, major
mountain ranges excite stationary planetary waves that cause undulations in the storm
tracks, while uneven distributions of tropical latent heating of the atmosphere also excite
Rossby waves that have similar effects. These causes can be coupled, as upslope flow
can cause latent heat release---a sticky problem for the theorist. Note that the wave
excited by the source is likely to express itself remotely, especially since the longer
wavelengths are most easily excited by stationary sources, and for mountain waves this
expression can be upstream, though most effects are downstream or in the lee of the
mountain (see Fig. 1).
Another important role for waves is in carrying energy and momentum. Waves exert a
drag force on the surface, extracting momentum from the planet. This momentum is
deposited in the atmosphere as the waves propagate upward, at levels and latitudes that
depend on details of the atmospheric structure. This momentum alters the strength and
position of jets. It also indirectly drives the ventilation of the stratosphere and
mesosphere, by interfering with the thermal wind balance and allowing a weak residual,
overturning circulation connecting these to the layers below. A good deal of the wave
effects result from the first two wavenumbers, so we are talking about long waves!
Mathematical treatment
There is obviously a lot of math that underlies much of the ideas discussed above, which
we will mostly avoid here. However, a few theoretical concepts are helpful for
understanding some of the papers.
Most wave analysis is based on linear theory. This means that the flow field is
decomposed into the “background” and “perturbation” components, e.g. u = <u>+u’,
and we assume u’ << <u>. Then if two perturbed fields are multiplied together, only the
first order terms usually need to be retained (e.g., uv ª <u><v> + <u>v’ + <v>u’ and we
forget about u’v’), so the problem remains linear and we say it is “linearized”. Linear
theory works until the wave amplitudes get too big. Very large waves start to distort, and
eventually break.
Variables used to describe winds are customarily u,v, and w for the westerly, southerly,
and upward components, respectively. Vertical velocities are denoted w in the pressure
coordinate system and have units of, e.g., hPa/day, with positive being toward the
surface. A convenient variable for describing geostrophic or 2-D wind fields is the
streamfunction y, defined such that {u,v} = k ¥ —y. Lines of constant y are parallel to
the flow. One quantity (y) is enough to fully specify the flow, since mass conservation
provides a second constraint on how u and v are related to each other. y is positive in the
middle of anticyclones, negative in the middle of cyclones.
A key parameter determining wave behavior is the local atmospheric static stability,
usually quantified by the Brunt-Väisälä or buoyancy frequency N:
N2 = g
†
d ln q
dz
This is the frequency with which a parcel of air would oscillate up and down in the
absence of any forces other than buoyancy and inertia; a typical period of oscillation is 8
minutes in the troposphere. Note that in a well-mixed atmosphere, N=0 since there is no
restoring force on a displaced parcel. The behavior of buoyancy waves typically depends
on how the intrinsic frequency n of the wave (which for stationary waves must be near
u/L, L the scale of the obstacle) compares with N. The behavior also depends, however,
on how this frequency compares with the rotation rate of the Earth. These issues will be
explored a bit in some of the papers.
Fig. 1. Streamlines of two-dimensional flow over a wide mountain.