GFD-1 LAB #8 (2001); LAB #7 (2002) INTERNAL WAVES, MOUNTAIN WAVES, ESTUARIES
P.B. Rhines, E.G. Lindahl
Internal gravity waves without rotation
Visualized with 'shadowgraph', where a point light source illuminates a white
screen behind the wave tank, the waves are generated at a single frequency by a
cylinder
oscillated vertically. The stratification is closely linear, with buoyancy period,
2pi/N of
about 5 sec. Notice how the wave beam reflects off the base of the mixed layer, not
the top of the fluid (the linear stratification becomes mixed near the top by
evaporative cooling; the top of the water is at the base of the wide black line)).
The vertical purple dye line shows the horizontal fluid velocity, which is
concentrated in the 'beam' Energy moves along the beam, parallel with the wave
crests which propagate toward the horizontal; it is a little hard to connect this
pattern with a simple plane wave, because the oscillating cylinder is forcing just
about one wavelength, as it pushes the fluid. So you do not see many wavecrests,
really just a couple. However, recall that when the cylinder is suddenly started
moving from rest, a whole range of frequencies are impulsively forced, and you see a
'star-burst' pattern of many wavecrests, all moving toward the horizontal.
wavemake oscillating with period = 11 sec:
period = 8 sec:
period = 6 sec:
period = 5 sec:
period=4 sec:
Waves versus modes.
Here a 'paddle' is oscillated at the left
end of another stratified rectangular tank. You can see its 'X'-shaped motion in the
time-exposure image. Its shape tends to favor the generation
of the first vertical mode ('gravest' internal- or baroclinic- mode) with w, the
vertical velocity ,varying like sin(pi*z/H), H being the depth of the stratified
layer). The time exposure photos show the particle paths: a mixture of ellipses and
nearly straight lines (or small arcs of curves). Why? Think about the paths of
particles in a pure travelling wave (ellipses) and in a standing wave (lines or arcs
of curves; see the surface wave streak photos from earlier in the term). The waves
travel down the channel (rightward), reflect off the end wall and propagate back to
the left. But, if there is friction in the system they won't make it all the way
back. In, approximately, the left half of the channel, the wave field is dominated
by a rightward propagating mode and such a mode has a definite energy flux (average
value of p'u', the product of pressure and horizontal velocity for linear waves).
Very short
time-exposure images show streamlines, and these are also visible here. This wave
field can be described as a set of rays of the kind seen above, painting out the
entire space; or, it can be expressed more simply as a single vertical mode, the sum
of upward and downward propagating plane waves. Images by Jim Renwick
The mixed layer once again is a strong feature, reflecting the waves before
they reach the water surface. Notice the long particle paths near the base of the
mixed layer: a mean flow is induced there by the extra shear and dissipation at the
'turning point'.
Some vertical dye stripes show the horizontal velocity due to the wave and also the
strong mean streaming near the mixed layer base. Similarly, at the bottom, there is
mean flow generated.
Basic group velocity
Consider a pattern of bars drawn on a transparency. Your browser may not show a
simple set of parallel bars but it should. The wavelength (periodicity of bars)
changes from top to bottom but this is of no consequence for this demo.:
The figure below shows the interference pattern when two such bar-patterns are superimposed. You can
think of the individual bars as being wavecrests. Where they constructively interfere you have a region
several wavelengths long of high energy; where they destructively interfere, with the a wavecrest from
one pattern over a wave trough of the other pattern, you have little energy. Here the two waves have the
same wavelength yet different angle of propagation. Their interference pattern is
oriented perpendicular to wavecrests. As you move one pattern relative to the other the bands of energy
move along the wavecrests rather than in the'expected' direction, with the movement of the crests.
Actually it is a little more subtle because the direction of energy movement can't be uniquely seen from
this pattern. If a third bar pattern is added it can, since the interference then gives 'spots' of
energy rather than long ridges of energy.
Of course the wavecrests are a pattern in time as well as
space (here you simulate time by moving one sheet relative to the other). If we hear a pure note from
an idealized piano, all we have is the time-series at our ear. Two adjacent notes played together
should produce a 'beat' tone at much lower frequency, equal to the difference between the two primary
notes. This new note corresponds to the constructive interference bands as in the figure below yet for
the case with parallel wavecrests (so the bands are also parallel to them). You can't hear this
beat note easily unless you go to the bottom of the keyboard. There a subtle 'wow-wow'
sound does accompany the clashing two notes. The notes of a well-tempered instrument are separated by
the 12th root of 2 (=1.05946), because 12 of these notes gets you an octave, or a frequency doubling.
You can calculate the expected beat frequency (hint, the A above middle C is 440 Hertz (cycles per sec).
'Mountain' waves due to a towed cylinder
Here we tow the cylinder to the right after laying down vertical lines of dye. This
nearly linear wave field selects many different Fourier components (plane waves) all
of which have the same horizontal phase speed, omega/k which is equal to U, the
speed of the cylinder relative to the fluid. This is 'phase-speed matching' of the
wave source and the waves, and it follows from the linear nature of the wave
equation. From the dispersion relation with omega = Uk,
this set of waves obeys k*k + l*l = N/U, a constant. Thus all the waves
have the same wavelength. The result is a semi-circular pattern of wave crests in
the lee of the cylinder. In addition, low frequency waves, with nearly horizontal
wavecrests, race ahead of the obstacle (these are very subtle and
important: theycorrespond to another solution of the
dispersion, simply k = 0. Such waves have horizontal crests and zero frequency
relative to the fluid; yet they are still waves! They can be explored by taking the
limit of frequency => 0 and noticing that the group velocity remains very large ~
N/m in this limit, so the waves indeed can appear far ahead, and signal the presence
of the moving obstable.) You can see the low-frequency wavesarrive at the vertical
dye
lines, with gradually increasing vertical wavenumber. This is 'upstream blocking',
where the fluid cannot get out of the way of the moving mountain, and hence is
pushed ahead by it. Behind the obstacle, a there is also a set of such
low-frequency modes as well as the normal lee waves.
The lee waves are set up very quickly, yet the upstream blocking waves
take time to reach the end of the channel. In this experiment we excite the full
semicircle of lee waves (but for some distortion just behind the cylindrical
'mountain'; there is mixing and weakening of the stratification there but the main
effect is the mean velocity generated by the k=0 waves; it refracts the lee waves).
Generally the mix of waves appearing will depend on the projection of
the forcing effect on the wavenumber plane; the possible lee waves lie on a circle
in (m,n) space of radius sqrt(N/U). If the forcing is a wide mountain, compared
with sqrt(U/N), it will force primarily the waves near the vertical wavenumber axis,
where k << m. These are hydrostatic, and their approximate dispersion
relation is thus omega = Nk/m*m (here omega = Uk). Their wavecrests are nearly
horizontal and they correspond to the waves seen just above and below the cylinder,
which are so oriented. This is indeed the normal scaling for many atmospheric flows,
with extreme aspect ratio H/L << 1.
The group velocity seen by an observer at rest relative to the fluid is directed
along the wavecrests. However, an observer sitting on the 'mountain' watching
the wind blow over it, must add (-U,0) to that vector, and the resultant you will
find is directed radial outward from the mountain. This is just as it should be: the
mountain is the wave source and rays of energy move straight radially-outward.
In this last frame, the speed of the moving cylinder has been increased. Note above
that the predicted wavelength varies as sqrt(U/N), and you are seeing this effect in
the longer lee waves.
Waves and 'downslope winds' in a model of either an
ocean estuary or an atmospheric mountain range.
Here the flow is confined in a 2.5 m long channel, driven as an overturning
circulation
by buoyancy flux at one end (a fresh 'river' flowing into the saline estuary). The
general overturning flow may be seen in the initally vertical ink trace. A more
highly developed lab model of an estuary, built by Prof. MacCready, may be seen
within
the GFD Lab website at
www.ocean.washington.edu/research/gfd/Dec_96.html .
We add a
mountain-obstacle. What is
most striking is the plunging
downslope flow behind the mountain. In the atmosphere such winds are so famous as to
have names: Bora, Chinook, Santa Anna, Foehn. There is a premonition of this jet in
the basic linear lee-wave solution with downslope flow somewhat concentrated in the
lee of the mountian. But nonlinear effects make it more intense. A very
different idealization is a 1- or 2-layer hydrostatic flow. When the flow speed
approaches the free wavespeed sqrt(gh) 'hydraulic' effects become important and
a supercritical (fast) jet forms in the lee. There was in fact a
raging debate for years, between the basically linear-wave picture and the hydraulic
picture of this flow. See, for example, Peter Baines' book on Topographic Effects in
Stratified Flows, Sec 2.3.2 (Cambridge Univ Press).
One thing to think about: as you see the acceleration of the flow as it crosses the mountain, being
pinched into a small vertical height range, you can apply Bernoulli conservation along a streamline
(it's ok despite the stratified fluid, because you are sticking to one steady streamline along which
density is nearly constant). This tells you of the low pressure on the lee (downwind) side of the
mountain, and hence that there is a strong pressure drag on the mountain. This drag is the horizontal
force component given by pressure X slope of the topography. Thus the plunging downslope winds are
strong 'wave drag' events, and mountain drag can often exceed the predictions of linear wave theory that
don't include this concentration of flow. The drag force on the mountain takes momentum out of the
airstream (is this Newton's 3d Law?). Ironically, the slowing of the winds may occur high above the
mountain, where internal gravity waves deposit it. Such events can be simulated readily with a simple
numerical model.
Notice that upstream of the mountian there is a strongly blocked region with almost
no motion, and just above it a weak jet going upwind. Far above, the general
overturning circulation is of course moving to the right.
Internal waves with stratification and rotation
View from above, note both the mode shape traced by the short exposure
(first image), and the particle ellipses seen by longer exporsures. At
higher forcing amplitude and longer image exposure you see a mean
circulation as well as wave particle-orbits (third image). The Coriolis
force causes the particles to move in elliptical orbits, seen from above.
The frequency ratio N/f is here about 3.
Lee waves over Sicily
MM5 model over Olympic Peninsula
SLP during this event
below: PV Banners over the Alps
Cold air downburst over Georgia, flowing outward as a gravity current
(for this vivid animation from cims-Univ. of Wisconsin website go to
CIMSS, U Wisconsin).
