Lection Notes Monday November 24, 2003
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 = 2where 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 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.
In the 1960’s an experiment lead by Walter Munk tracked waves that were generated by a storm in Antartica across the entire Pacific 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 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).
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
Cg
Cg=C/2=g2
d= (g2(t- t0)
Or
gt-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.
TIDES
Equilibrium tide—Meaning that the tidal bulge is in equilibrium (this is not the case as we will later see because of the geometry of the oceans).
The maximum equilibrium tide due to the moon would be 0.55 meters , and that due to the sun would be 0.24 meters. So during full and new moon the tidal range would be .79 meters, while at half moon
(See Figure 10.10 in Knauss and equations 10.26-10.31)
Earth-moon system (and earth-sun system) rotate around point.
Demonstrate two equal and opposite forces at points a and b whose force is proportional to the mass of the moon and inversely proportional to the third power of the distance.
Note that it really is more complicated than this—and it is the tangential forces that drive the tidal bulge—but this is the essence of the tidal forces.
The mass of the sun is 2.5 * 107 that of the moon bit it is also 400 times further away thus the tidal forces produced by the moon are more than double those forced by the sun.
Note that the period of the tide forced by the sun will be 12.00 hours—and that forced by the moon will be 12.42 hours. These are two “tidal constituents” known as the M2 ( Principle lunar) and S2 ( Principle Solar) tidal constituents. The beating of these two signals gives rise to the spring/neap variability—when they are in phase it is spring tide tide and when they are out of phase it is neap tide. The figure to the right shows spring neap variability at the battery.
In fact if we consider all the complicated motions of the earth, moon and sun system we find that there are many many tidal constituents. There are other semidiurnal constituents—most notably the N2 tide (the larger lunar elliptic) and the K1 and O1 ( the lunisolar and principle diurnal tide). These diurnal tides have approximately 1 cycle per day and give rise to diurnal inequalities in the tide—where one tide is larger than the other.
The diurnal tide is due to the tidal bulge not being on the equator—
In some regions of the world the tide is predominately diurnal—such as the plot showing sea-level in Hawaii.
The reason that tidal characteristics vary around around the world is because of the dynamic nature of the tide. The equilibrium tidal theory assumes an earth covered with water. However, as we know we have ocean basins, continental shelves, inland seas and estuaries—and the response of this system to the forcing of the moon gives rise to a complicated tidal structure.
To discuss this dynamics 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.
(2
Take d/dx of (1) and d/dt of (2) and Multiply 1 by H
and take difference
(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
Does this look familiar? 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.—i.e.
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).
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
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:
and the continuity equation
Since v is zero (thank you Lord Kelvin!) this reduces to:
(6)
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:
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
Tidal Potential
(From Stewart)
Tides are calculated from the hydrodynamic equations for a self-gravitating ocean on a rotating, elastic Earth. The driving force is the small change in gravity due to motion of the moon and sun relative to Earth. The small variations in gravity arise from two separate mechanisms. To see how they work, consider the rotation of moon about Earth.
- Moon and Earth rotate about the center of mass of the Earth-moon system. This gives rise to a centripetal acceleration at Earth's surface that drives water away from the center of mass and toward the side of Earth opposite moon.
- At the same time, mutual gravitational attraction of mass on Earth and the moon causes water to be attracted toward the moon.
If Earth were an ocean planet with no land, and if the ocean were very deep, the two processes would produce a pair of bulges of water on Earth, one on the side facing the moon, one on the side away from the moon. A clear derivation of the forces is given by Pugh (1987) and by Dietrich, Kalle, Krauss, and Siedler (1980). Here I follow the discussion in Pugh §3.2.
Figure 17.10 Sketch of coordinates for determining the tide-generating potential.To calculate the amplitude and phase of the tide on an ocean planet, we begin by calculating the forces. The tide-generating potential at Earth's surface is due to the Earth-moon system rotating about a common center of mass. Ignoring for now Earth's rotation, the rotation of moon about Earth produces a potential VM at any point on Earth's surface
/ (17.5)where the geometry is sketched in Figure 17.10, g is the gravitational constant, and M is moon's mass. From the triangle OPA in the figure,
/ (17.6)Using this in (17.5) gives
/ (17.7)r/R» 1/60, and (17.7) may be expanded in powers of r/ R using Legendre polynomials (Whittaker and Watson, 1963: §15.1):
/ (17.8)The tidal forces are calculated from the gradient of the potential, so the first term in (17.8) produces no force. The second term produces a constant force parallel to OA. This force keeps Earth in orbit about the center of mass of the Earth-moon system. The third term produces the tides, assuming the higher-order terms can be ignored. The tide-generating potential is therefore:
/ (17.9)The tide-generating force can be decomposed into components perpendicular P and parallel H to the sea surface. The vertical force produces very small changes in the weight of the oceans. It is very small compared to gravity, and it can be ignored. The horizontal component is shown in Figure 17.11. It is:
/ (17.10)where
/ (17.11)The tidal potential is symmetric about the Earth-moon line, and it produces symmetric bulges.
Figure 17.11 The horizontal component of the tidal force on Earth when the tide-generating body is above the Equator at Z. From Dietrich, et al. (1980).If we allow our ocean-covered Earth to rotate, an observer in space sees the two bulges fixed relative to the Earth-moon line as Earth rotates. To an observer on Earth, the two tidal bulges seems to rotate around Earth because moon appears to move around the sky at nearly one cycle per day. Moon produces high tides every 12 hours and 25.23 minutes on the equator if the moon is above the equator. Notice that high tides are not exactly twice per day because the moon is also rotating around Earth. Of course, the moon is above the equator only twice per lunar month, and this complicates our simple picture of the tides on an ideal ocean-covered Earth. Furthermore, moon's distance from Earth R varies because moon's orbit is elliptical and because the elliptical orbit is not fixed.
Clearly, the calculation of tides is getting more complicated than we might have thought. Before continuing on, we note that the solar tidal forces are derived in a similar way. The relative importance of the sun and moon are nearly the same. Although the sun is much more massive than moon, it is much further away.
/ (17.12)/ (17.13)
/ (17.14)
where R sun is the distance to the sun, S is the mass of the sun, R moon is the distance to the moon, and M is the mass of the moon.
Coordinates of Sun and Moon
Before we can proceed further we need to know the position of moon and sun relative to Earth. An accurate description of the positions in three dimensions is very difficult, and it involves learning arcane terms and concepts from celestial mechanics. Here, I paraphrase a simplified description from Pugh. See also Figure 4.1.
A natural reference system for an observer on Earth is the equatorial system described at the start of Chapter 3. In this system, declinationsd of a celestial body are measured north and south of a plane which cuts the Earth's equator.
Angular distances around the plane are measured relative to a point on this celestial equator which is fixed with respect to the stars. The point chosen for this system is the vernal equinox, also called the 'First Point of Aries'... The angle measured eastward, between Aries and the equatorial intersection of the meridian through a celestial object is called the right ascension of the object. The declination and the right ascension together define the position of the object on a celestial background...
[Another natural reference] system uses the plane of the Earth's revolution around the sun as a reference. The celestial extension of this plane, which is traced by the sun's annual apparent movement, is called the ecliptic. Conveniently, the point on this plane which is chosen for a zero reference is also the vernal equinox, at which the sun crosses the equatorial plane from south to north near 21 March each year. Celestial objects are located by their ecliptic latitude and ecliptic longitude. The angle between the two planes, of 23.45°, is called the obliquity of the ecliptic... Pugh (1987: 72).
Tidal Frequencies
Now, let's allow Earth to spin about its polar axis. The changing potential at a fixed geographic coordinate on Earth is:
where jp is latitude at which the tidal potential is calculated, d is declination of moon or sun north of the equator, and t1 is the hour angle of moon or sun. The hour angle is the longitude where the imaginary plane containing the sun or moon and Earth's rotation axis crosses the Equator.
The period of the solar hour angle is a solar day of 24hr 0min. The period of the lunar hour angle is a lunar day of 24hr 50.47min.
Earth's axis of rotation is inclined 23.45° with respect to the plane of Earth's orbit about the sun. This defines the ecliptic, and the suns declination varies between d = ± 23.45° with a period of one solar year. The orientation of Earth's rotation axis precesses with respect to the stars with a period of 26,000 years. The rotation of the ecliptic plane causes d and the vernal equinox to change slowly, and the movement called the precession of the equinoxes.
Earth's orbit about the sun is elliptical, with the sun in one focus. That point in the orbit where the distance between the sun and Earth is a minimum is called perigee. The orientation of the ellipse in the ecliptic plane changes slowly with time, causing perigee to rotate with a period of 20,900 years. Therefore Rsun varies with this period.