As seen before, (detailed mathy description here) the divergence of gravitational field lines from the moon creates a tidal force, which gives a two-sided bulge. The moon's orbit is west to east and takes it from full-moon to full-moon once every 29.5 days. This is called the synodic month, and is roughly the origin of the 30 to 31 days one finds in the calendar. The moon moves east at about 12 degrees per hour, so the timing between moon rises is a bit longer than a day - 24 hours 49 minutes on average, but varies, as the orbit of the moon is a bit wobbly. The period of 24 hours 49 minutes is sometimes called a tidal day.
Since there are two lunar bulges, that means that the earth rotates underneath these bulges with a period of 12 hours 24 minutes.
The plot below shows an example of a 24 hour cycle of a lunar tide.
One day (24 hour) cycle of lunar tide.
I've drawn this to illustrate a few features before we get too complicated. I chose a time cycle so that high tide occurs at midnight in the start of the 24-hour cycle. You can see that you effectively get two high tides in the tidal day. If you look at the end of the day - close to midnight at the end of the 24-hour cycle, you'll see that you don't quite come back to a high tide at midnight. This is because the lunar cycle is 24 hours and 50 minutes long. Every so often, you'll only get one lunar high-tide per day, because of this slippage.
Another feature of the plot is the vertical axis, which is called "Height above datum". In nautical charts, depths are represented in relation to something called a datum. This is some reference sea-level that has to be defined. In United States nautical charts, the datum is taken to be something called the "Mean Lower Low Water" or MLLW. This is basically an average over a month of the daily low water level. The term "lower" comes into play when there is something called a "mixed tide", which we'll get to.
The lunar tide that showed two high tides in a 24 hour cycle (most of the time) is called a semi-dirunal tide. The figure below shows the occasional exception to the rule. Again, I started with a high tide at midnight on the first day. If you look closely, you'll see that on day 7, there's only one low tide per day, as the difference between the 24 hour 50 minute lunar cycle and the 24 daily cycle sweep the phase around.
Several days of tide. Usually two low tides and two high tides occur in a 24 hour cycle of a semi-diurnal tide, but if you look closely at day 7, you see only one low tide in the 24 hour cycle.
The moon isn't the only celestial object driving the tides. The sun plays a significant role. It is much farther away than the moon, so its gravitational lines of force are much more parallel than the moon's. However, the sun's gravitational force is so large that there is still a significant tidal field from the sun. In fact, it's about 1/2 the lunar tidal field, so contributes significantly.
Since the earth's rotation is 24 hours, you get two solar bulges per day, 12 hours apart. Twice every synodic month, the lunar and solar bulges line up and will generate higher tides. These are spring tides. At other times, e.g. when there's a half-moon, the bulges of the moon and sun are out of phase, and the tides are smaller. These are called neap tides. The alignment of the sun and moon for these tides are shown below.
Spring and neap tides.
The alignment for spring tides is when the moon is either on the same side of the earth as the sun (conjunction) or when the moon is on the opposite side of the sun (opposition). These two configurations are called syzygy (great Scrabble word, except you need a wild-card). The alignment for neap tides is when the sun and moon are at ninety degrees to each other, a configuration called quadrature.
In the figure below, you can see ten days in a tidal cycle that has both the moon and sun acting on the earth. It starts out at midnight at the beginning of day 1, where the moon and sun are in syzygy - let's say it's a full moon. At day one the combined tidal forces of the moon and the sun team up to give a larger than normal tide, a spring tide. But, by day five or day six, you can see that the moon and sun are now approaching quadrature, and the tidal range gets smaller - approaching a neap tide.
Ten days in a cycle with both the sun and the moon.
You'll note in the above figure that during the spring tide part of the cycle, the low tide goes below the MLLW datum. In this case, you need to understand that you'll be closer to underwater obstructions than the depths quoted on charts. For this reason, other countries often use a datum called "Lowest Astronomical Tide" or LAT. The moon's orbit has a 19 year cycle where effects like the distance of closest approach to the earth (perigee) and other orbital parameters can conspire to give very large tides. One example is a "super moon" when the moon is full and has its closest approach to earth. The LAT datum takes the lowest possible tide that can occur in the 19 year cycle, and hence is a bit safer to quote depths based on this datum. It's considered the international standard, although the US persists in using MLLW.
Here's a minor digression about astrology. In some explanations that try to back-up the efficacy of astrology, authors will often lead off with the seasons being caused by the position of the sun against a fixed background of stars. This is an allusion to the angle of the sun's rays causing winter and summer. Then the next discussion is related to the tides - first the lunar cycle, and then the relative positions of the sun and the moon. Clearly these are astronomical phenomena that have influences on things on the earth. A great example of this line of reasoning can be found in Ptolemy's treatise on astrology, Tetrabiblos.
Back to tides. Typically the bulges cause the earth to be displaced by about half a meter, but this isn't the real effect that causes water levels to rise in oceans. An instructive example is the LEP accelerator tunnel in Geneva, Switzerland. LEP was the "Large Electron-Positron accelerator". It had a high precision for its energy and was constantly monitored. At some point, there was a strange shift in the energy that the physicists couldn't figure out. Finally they figured out that it had to do with the tides. Now, earth moves up and down by about half a meter, and you think this would give a significant effect, but it's not really the full movement that makes a difference, it's the differential motion. The difference in tidal forces from one spot to another creates a differential force that creates measurable differences. In the case of the LEP accelerator, the differential tidal forces distorted the shape of the tunnel by a few millimeters. This was the shift from one side of the ring to the other, which is about 27 kilometers in circumference. Here is a link to an article on LEP and tides. Be warned, they invoke the "centrifugal force" explanation, my bête noire.
The distortion of the earth's crust is relatively small because it's fairly rigid and cannot flow from one place to another. On the other hand, water is an incompressible fluid, and can flow over long distances. In this case the differential tidal forces can create a substantial flow from one place to another.
One issue that makes the tides inscrutable is the response of large embayments, like the Bay of Fundy, which is known for its large tides. There's a phenomenon known as resonance. The best analogy to describe resonance is to think of a child on a swing. The length of the swing determines a natural frequency for the swing. This is to say: if you give it one push, it'll naturally swing back and forth with a period that is determined by the length of the swing. All well, and good, right?
Now, let's say someone is giving a push to the child on the swing. They can give it a push at any frequency they like. If they push it faster than the swings natural frequency, it won't go terribly high. If they push it slower than the spring's natural frequency, it won't go terribly high, either. BUT, if you push it at just the same frequency as the natural frequency, the swing will go very high, as anyone who has been on a playground knows.
The passing of the lunar (and solar) bulges are like the pushing on the swing, and embayments open to the ocean have a natural frequency associated with them. If the embayment has a natural frequency close to the frequency of the lunar bulge passing, it has a large response.
Resonant response and natural frequency of embayments.
The figure above shows you an example of different embayments. Typically, the larger the embayment, the longer the response time and the lower the natural frequency, like a swing with a long rope. So, the Caribbean, being relatively large, has a long response time, like a couple of days, while Nantucket sound has a natural response time of a few hours. On the other hand, the Bay of Fundy has a natural response time of about 12.42 hours, meaning that the lunar period of 12 hours, 24 minutes is extremely close to its natural frequency. As a result, it exhibits a very large response and has the largest tides on the planet, along with Ungava Bay, in northern Quebec.
I should note that the natural resonance frequencies of embayments are due to their size and bathymetry. In modeling the Bay of Fundy, researchers have found that the natural resonance frequency is indeed nearly identical to the lunar tidal frequency. A link to the study can be found here. Similarly, Ungava Bay has a natural resonance frequency of 12.7 hours, leading to extraordinarily high tides. A link to the study can be found here.
Other factors can come into play with tides. The amount of water that can flow in and out of the oceans, for example. The Mediterranean is very close to be land-locked, except for the narrow Straits of Gibraltar and the Dardanelles, which will not permit much water flow in and out. Partly as a result if this and its size, the Mediterranean has a low tidal range.
From the static theory of the tides, one might expect the highest tides to occur when the moon is directly overhead, or on the opposite side of the earth. This is true for some areas, but not for others. Why not? Mainly water has to flow from one place to another to create tides. In some cases, the flow is quite simple, as in Boston Harbor, where there's a direct opening to the open ocean. On the other hand, there is a complicated tidal system that included Long Island Sound, Newport RI, and Nantucket Sound. The tidal flood starts around Newport, then floods west into Long Island Sound and east into Nantucket Sound. There's quite a difference between the time the moon is directly overhead at Newport and the timing of the high tide there.
There are different forcing frequencies associated with tides. The orbit of the moon about the earth is inclined at an angle. When the moon in its orbit is above a point in the northern hemisphere, the near-side bulge is larger in the northern hemisphere than the away side bulge. This creates a difference in the forcing function that 'pings' the earth with a frequency of once a day - well actually once every 24 hours and 50 minutes. When one takes the different frequencies - twice a day for the sun, once a day for the sun, twice a day for the moon, once a day for the moon, etc, it's called harmonic analysis. By tracking the tides in an observing station for some time, one can get an idea which frequencies affect the local tides the strongest and then can use this to create predictions for the tides in different locations.
For a given location, there are a set of harmonic constituents. NOAA uses 37 of these in total for predicting tides. Familiar ones are M2, the principal lunar diurnal constituent, or S2, the principal solar diurnal constituent, or O1, the lunar diurnal component. If you want to look at all the constituents for a location, you can look at the NOAA website. Here is a link to the constituents for Bar Harbor, Maine.
Those of us on the east-coast of the US (excluding the Gulf of Mexico) are used to mostly semi-diurnal tides with high tides happening twice daily. The Gulf of Tonkin off of Vietnam, on the other hand, experiences more of a diurnal tide, with a high tide happening once a day. The size and bathymetry of the Gulf of Tonkin give it a natural resonant frequency of about 29 hours, much closer to the diurnal (once daily) harmonic components of the forcing function. A link to this analysis can be found here.
Throughout a lot of the west coast, and in Puget Sound in particular, there is a mixed semi-diurnal tide. An example of that is shown below. In this kind of tidal cycle, there are two low tides and two high tides in a tidal day, but the height of each of the pair are quite different. This is do to some combination of diurnal and semi-diurnal components contributing.
Mixed semi-diurnal tides with two highs per tidal day, but of unequal height.
As before with Ungava Bay and the Bay of Fundy, the response of an embayment or section of the coast is largely due to to its size and bathymetry (underwater topography), and even the friction on the bottom. The complex of the strait of Juan de Fuca, Puget Sound and the Georgia Strait have a resonant frequency between 17 and 21 hours, somewhere between the main lunar tidal period and double that. Also there's a fair amount of friction along the bottom of the straits as water ebbs and floods Puget sound. This gives rise to the mixed semi-diurnal tide. A link to a study of the resonant properties of this complex can be found here.
In the last post, I described something called the static theory of tides. In the latter half of the 18th century, the physicist Pierre-Simon LaPlace advanced the dynamic theory of tides. This included the resonant properties of embayments, the Coriolis effect in water transported over substantial distances, and the friction in motion over the ocean floor. His work gave rise to a fairly complicated set of equations that I won't display here, but have some curious consequences. One of these is the phenomenon of rotary tides.
A rotary tide has a node with zero tidal range at the center. There is a high tide line that circulates around this node, called an amphidromic point. The farther away you are from the amphidromic point, the higher the tidal range. Some rotary tides are in a fairly restricted area, like the North Sea.
Rotary tides in the North Sea. The nodes or amphidromic points are where the white lines converge. The white lines represent the high tide at a given moment on the tidal cycle. The tide range gets larger as one moves away from the amphidromic point.
There is also a global system of rotary tides. It is not a priori obvious where the amphidromic points show up, but is rather due to the complexities of the ocean and shorelines.
Global rotary tide systems. The color code represents the tidal range in centimeters.
In the next post, I take up where to go for information on the tides and figuring tidal currents and how to take them into account in trip planning.
No comments:
Post a Comment