Recap Rossby’s Talk:
1) Tom talked about how it was difficult for columns of fluid to change there thickness in
a rotating frame:
Why was this?
Conservation of Potential Vorticity d/dt( (f+/h) =0
2) He also talked about the “radius of deformation” or c/f where c is the wave speed and
f the local Coriolis frequency. Most oceanographers refer to this as the Rossby Radius,
but Tom tries to remove his family’s name for this. As he does with the Rossby Wave—
which he calls planetary waves. However, apparently he can’t think of another name for
the Rossby Number.
Rossby radius of deformation is a fundamental length scale in the ocean—it’s the
distance that wave will travel before in essentially one inertial period. The wave speed
that is relevant is the fastest wave speed—which is the gravity wave. And the Gravity
wave is the topic of today’s lecture.
Before that—however—let’s look at the Rossby radius for ocean conditions:
For barotropic long waves (this will be defined more later) the phase speed is simply
sqrt(gh) where g is gravity and h is the water depth.
In the deep ocean h=4000m so sqrt(gh)=200 m/s. Much much faster than the Rossby
wave (which was cm/s!). This is the fastest wave in the ocean (outside of
So for f=10-4 1/s
c/f =200 m/s /10-4 1/s = 200*10-4 m=2000 km.
And this is the length scale of tidal motion in the deep ocean (we’ll see this later).
On the shelf—where say h=50 m then c=sqrt(10*50)=22 m/s
R=c/f=20 km.
However we also have an internal radius of deformation. This is the scale of ocean fronts.
This will form at c/f where c is the internal wave speed== or sqrt(g’h).
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So in the deep ocean if 3 and h=200 (upper layer thickness) then c=2.5 m/s and the
internal radius of deformation is 25 km. Thus to resolve such a feature model of the
global ocean much resolve variaility at 25 km—and even today this is not
computationally feasible.
In the coastal ocean—where the layer thickness might only be 10 meters and for a =3
kg/m3 and for f=0-4 1/s the internal Rossby Radius of deformation is around 5 km.
This is often the scale of upwelling fronts and coastal currents.
Chapter 9
Gravity Waves:
The class of waves are called Gravity waves because the restoring force is Gravity. This
is fundamentally different than the Rossby wave (or as they’re sometimes called
Planetary waves) where the restoring force is the Coriolis frequency.
Waves are characterized by there period (P) and wave length 

Just like the frequency  can be related to the wave period (P) we can define a
wavenumber k
k=
The wave number can simply be thought of how many waves you can see in a given
length. A high wave number means there will be a lot of waves in a given length—while
a low wave number means fewer waves. Think about this next time you fly over the
ocean. The more waves you see outside your little window—the bigger the wave number.
Thus the wave, which varies in both space and time, can be defined by
=Acos(t+kx)
Where a is the amplitude.
The character of gravity wave propagation depends fundamentally on the ratio of the
wave length to the water depth. If the wave length is long compared to the water depth
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this is referred to as a Long Wave or a Shallow water wave. Here the wave motion is
largely in the horizontal direction, with very little vertical motion. This results in a very
small vertical acceleration – and this means that we can use the hydrostatic assumption.
Aslo the wave motion extends throughout the entire water column
An example of a long wave would be the tide. Consider the tide. If the tidal range is 1
(=0.5) meter than the vertical velocity is:
n / t  A sin( t  kx)
This is the vertical velocity—so the vertical acceleration is going to be scaled with
squared times A
S~ 1.4 e-4
Thus s2*.5=1.e-8
So vertical acceleration is much smaller than gravity—and in developing a model for
these waves we can neglect vertical acceleration and make the hydrostatic assumption:
For the deep water waves (these are typically of much higher frequency) the vertical
acceleration can not be ignored—Here the vertical motion are large—and the wave has a
vertical structure and decays expontinally with its wave number.
More on this later:
Waves can also be progressive in nature or stading:
Define:
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Shallow water waves:
To discuss this dynamics of shallow water waves (long waves) of this lets return to the
momentum equation and look at propagating shallow water waves
2) If we assume a balance between acceleration and the pressure gradient—and a
vertically integrated continity equation—we obtain the classic wave equation
Question: Will these be deep water or shallow water waves.
u

 g
(1)
t
x

 ( Hu )

t
x
(2)
(2
Take d/dx of (1) and d/dt of (2) and Multiply 1 by H
and take difference
 2
 2
 gh 2
t 2
x
(3)
This is a form of the classic Wave-Equation that appears in many physical systems.
Substitute n=sin(kx+t) into (3) to yield
ghk

2
k2
 gh


k
 C  gh
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Note that for a given water depth all shallow (long) waves travel at the same speed. This
means that there are non-dispersion which means that a group of waves will stay
together and not dispersive. Thus the wave form will remain constant.
In contrast—you will see next lecture—that deep water waves are in fact dispersive. The
phase speed of wave increases with increasing wave length. Thus a packet of waves,
such as the one shown in class generated by a stone thrown in the water—will disperse
with the longer waves leading the way. This is why a far-away storm will be initially
apparent by long waves.
The frequency divided by the wave number is equal to the wave-length divided by the
period and this is the phase speed of the wave.
In a progressive tidal wave maximum tidal currents occur during high tide and low tide—
and since we’re only dealing with one wave we can easily relate the tidal range to the
tidal current speed with the momentum equation: (watch this animation and think about
the momentum and continuity equation written above # 1&2).
u

 g
t
x
n  A cos(kx  t )
u
  gAk sin( kx  t )
t
u

u
t 
t

gAk sin( kx  t ) 
gAk
sin( kx  t )

gAk gAT
gAT
g


A


H
gH T
So if A=1 meter and H = 10 m, tidal current speed is 1 m/s.
In contrast in the deep ocean H=4000, and A=0.25 m tidal current amplitudes are 1.25
cm/s.
Kelvin Wave
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As I mentioned in last lecture the tide in the ocean travels around the NOrth Atlantic
basin as a Kelvin wave. Lord Kelvin (yes—he was a Lord) noted that the flow normal to
the coast was zero (call this v) and thus v is everywhere zero (perhaps an act of faith—
but he was a Lord).
Neglecting the non-linear terms and friction and taking the hydrostatic assumption
(which is to say we are working with shallow water waves i.e. H) write the two
momentum equations as:
u

 fv   g
t
x
v

 fu   g
t
y
and the continuity equation
 u  v 

H
   
t
  x y 
Since v is zero (thank you Lord Kelvin!) this reduces to:
u

 g
t
x

fu  g
y
H
(4)
(5)
u


(6)
x
t
Equation 4and 6 are identical to 1 & 2 which were used to derive the wave equation.
In the other direction the flow is geostrophic. A solution to this equation is:
( x, y , t )  o e
fy
c cos(k ( x  ct ))
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Note that y=c/f is the efolding scale—and equal to a Rossby Radius
What this is a coastally trapped wave (it can also be an equatorially trapped wave) and
needs a coastline (or an equator) and it only propagates in one direction—because if c is
negative than the sea level grows exponentially moving off-shore
Deep water waves (Did not do this in 2006—Did El NiNO next—then Jim did Deep
water waves. Did show movies however)
Surface gravity waves—you are familiar with them
1)Begin with animations of Waves.
Note the difference between Deep water and shallow water waves—different dispersion
relations. Most interesting is the dispersion relationship for deep water waves—because
here the speed of the wave increases with the wave-length
C=sqrt(g/k)
Note that sin(kx + t) is a propagating wave—where  is the wave’s frequency and k is
the wave number = 2where is the wave length. The wave number is essentially the
number of waves you would see over a given length.
Note that a wave could be described
more generally as that sin(kx + ly +
mz \+ t) where k,l and m are the
east/west north/south and vertical
wave numbers
Even a casual observations of the
ocean would suggest that the wave
field is more complicated than one that
can be characterized by a single sin or
cosine curve—but rather composed of
a more complicated function.
Fortunately Fourier showed that any
feature can be characterized as the sum of an infinite number
of sin waves. For example, the figure to the left shows that
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the sum of 250 sin curves can nearly approximate a delta—or impulse function. The red
curve is an attempt to characterize the spike with only 5 sin curves, the green with 50 sin
curves and the blue with 250 sin curves. Clearly as the number of sin curves increases
the better we can represent the delta function.
A good example of a delta function occurs when a rock is tossed into calm water. The
photo to the left shows the disturbance of the sea-surface following the plunging of a rock
into the water and the subsequent evolution of that disturbance can be seen in this movie.
Therefore, if we take the 250 sin curves shown in the figure above and let the propagate
at a phase speed governed by the dispersion relationship for deep water waves—we see a
very similar evolution as the movie. Both the movie of the waves propagating away from
the disturbance and the animation of 250 sin curves propagating at speeds c=sqrt(g/k)
show two very important phenomena occurring in deep water waves. First we see that the
waves are dispersive. What this means is that over time the energy impulse—which
initially was focuses at a sigle point spreads out over time due dispersion. This occurs
because the impulse is composed of waves of many different wavenumbers and they
propagate at different phase speeds. It’s kinda like the New York City Marathon. 30,000
runners begin at the same spot at the same time—but 2 hours later some are near the
finish line, while others have not completed half the race. The other remarkable thing
about deep water waves, that can be seen in both the animation and the movie, is that the
wave crests move faster than the energy. Note that a wave crest moves though the packet
and disappears at the leading edge. This results in the weird phenomena that wave energy
travels at half the phase speed.
Animation of packet of deep water waves.
50 m
In the 1960’s an experiment lead by Walter
Munk tracked waves that were generated by
a storm in Antartica across the entire Pacific
250 m
ocean up to Alaska. One major conclusion
of the study was that there was very little
loss of energy due to friction. The passage
of waves past each observation point
500 m
showed the expected dispersion—with the
wave packet quickly passing stations that
are close to the storm while the further the
station from the storm the longer it takes the wave packet to pass. The longer it takes the
wave packet to pass—the slower the change in frequency of the observed wave (See
figure that shows results of calculation of wave passages past a point 50, 250 and 500
meters from a disturbance).
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Therefore by measuring the rate that the frequency changes you can determine how far
away a storm is. The distance will be Cg(t- t0) where Cg is the group velocity and t ist
the time of arrival and t0 is the time of generation. For short waves (deep water waves)
gk
C=k
Cg
Cg=C/2=g2
d= (g2(t- t0)
Or
gt-to)/d
Indicating that the frequency of the wave increases linearly with time—first a long swell
and later a shorter period wave. By plotting the frequency as a function of time you can
determine how away the storm was—emphasize this with figure.
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