# Tides

2-D Models First

First, imagine that there is no 3rd dimension, that everything exists in a flat space. Most of the important tide effects can be explained without considering the 3rd dimension, just by looking at a slice through the equator and imagining that none of the orbits are tilted.

2-D Earth Alone in Space

First we will consider just the Earth by itself with no Moon or Sun.

The Earth is a solid rigid ball of rock, turning once per day.

Everything on the Earth's surface is being pulled to the center by gravity. Because the Earth is solid and rigid, things on its surface cannot go below the surface so they just sit there. Because of friction, objects on the Earth's surface end up rotating once per day along with the whole Earth.

Objects on the Earth's surface travel in a circle with a radius of about 6400 kilometers once per day. This circular motion is called the centripetal motion. Because the Earth acts as a solid, rigid object, this motion is the same for every point on the Earth's surface.

The centripetal motion is at a constant speed but it is not in a straight line. Therefore, it is accelerated motion. The rate of acceleration is towards the center of the Earth, this is called the centripetal acceleration. The forces that cause the acceleration to occur are gravity and pressure.

Imagine a totally smooth Earth (no mountains or valleys) with water. The water would cover the whole Earth at a uniform depth.

For the purposes of this discussion, water is a non-compressible fluid. In other words, it can flow freely, but you cannot force two kilograms of water into 1 liter of space. If two liters of water want to flow in opposite directions, they have to go around each other somehow.

Because it is not compressible, any forces (like the gravity of another nearby planet) cannot make the water move towards the center of the Earth no matter how hard they try. Also, they cannot pull the water up directly away from the Earth unless the gravity were stronger than the Earth's gravity (and we will only be considering the relatively weak pull of the Moon, Sun, and possibly other planets). However, they are free to pull water from side to side. If a force is pulling at an angle, the horizontal part of the force will accomplish something while the vertical part will be offset by pressure against the hard surface.

If the same force were pulling at an angle on a large flat expanse of water (like an ocean), all the water would feel a slight pull to one side. Each bit of water would move a little bit in that direction, and the water at one end would rise slightly. The water at the end got lifted because all the water in the entire ocean is pushing together in the same direction.

(figure)

If the sideways force were about 5% as strong as the Earth's gravity, the resulting slope in the ocean's surface would be about 5%.

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Earth and Moon without Water

Now add the Moon, with 1/80 the mass of the Earth, orbiting in a perfect circular orbit once per 29.5 days at a distance of 300,000 km. Forget about the water for the time being.

In order for the system to balance, the Earth and moon must both orbit around a common center point called the barycenter. The barycenter is located 1/80 of the way from the Earth's center to the Moon's center, which turns out to be inside the Earth and about 2650 kilometers below the point on the surface where the Moon is directly overhead.

Because the Earth and Moon are orbiting the barycenter, they both have a centripetal motion, a circular motion that takes 29.5 days to complete one circle. There is also a corresponding centripetal acceleration. These are the orbital centripetal motion and orbital centripetal acceleration.

We are still assuming the Earth is rigid and solid, so every part of the Earth experiences the exact same amount of orbital centripetal acceleration.

However, the points on the Earth that are closer to the moon actually experience a bit greater gravitational pull from the Moon, and the points on the Earth farthest from the Moon experience less.

An object sitting on the Earth's surface will experience this lunar gravity, which changes slightly throughout the day.

If the force were always exactly enough to equal the centripetal acceleration, there would be no variation throughout the day. But instead, the lunar gravity never exactly balances the centripetal motion, so there is a little bit of force left over. This little bit left over is the tidal force.

The tidal force is a small fraction of the total gravitational pull of the Moon, which in turn is a small fraction of the force from the Earth's gravity. It only exists when viewed from a frame of reference that is sitting on the surface of the rotating Earth.

The strength of the Lunar tidal force is 1/45 of the Moon's gravity, which in turn is 1/180000 of the Earth's gravity. In other words, the Lunar tidal force is 1/8,000,000 the strength of the Earth's gravity.

(Want to add a diagram showing the tidal force at each of 8 times throughout a 24-hour day)

Earth and Moon with Water

Now consider the water. It is experiencing two forces: the pull of Earth's gravity, and the tiny tidal pull.

(Add a diagram showing the total force the water experiences throughout the day)

Notice the force is always down. But there is also a very small amount of horizontal pull. Because the water cannot be compressed, the downward part of the force will not show up in any way, but the small horizontal pull will encourage the water to flow east or west.

This diagram shows the horizontal pull at each point throughout the day.

Velocity and Position

As the water feels the horizontal force, it reacts by accelerating, that is increasing its velocity. Assuming it started at rest (no velocity) the water's velocity throughout the day will go like this:

As the water moves, it is changing its position. When it starts moving the other day, its position moves back. Here is a graph showing the position of one small bit of water as it moves throughout the day:

Opposite Velocities Cause Swells and Dips

Here we see that the water in different places, moving in opposite directions, causes a swells (local rise in water level) and corresponding dips.

How High?

Because the Lunar tidal force is 1/8,000,000 as strong as the Earth's gravity, we would expect the height of the ocean bulges to be about 1/8,000,000 their width (which is 1/4 of the circumference of the Earth). That works out to about 1.2 meters, so we expect the average high tide to be 1.2 meters above mean water level and the average low tide to be 1.2 meters below, or a total of 2.4 meters. That's actually a fair amount larger than the global average.

The Earth is Not Rigid

Part of the reason the tides are not as high as expected is because the Earth is not rigid, but is actually sort of fluid. Most of the Earth is molten metal and rock, and it deforms a little bit. It is well known that the Earth is flattened out a bit (by about 20 kilometers at the equator) because of the cetripetal acceleration of its spinning. The Moon's tidal forces also cause a distortion of about 1 meter — the land is rising and falling twice per day by a distance of 1 meter! These land tides have two bulges pointing towards and away from the Moon, just like the water bulges. There are also diurnal, solar, and perigee effects just as with the water tides (some of these will be described later). In most locations, 1 meter variation gets subtracted from the 2.4 meters computed above.

The tides are also a little lower than expected because much of the tidal energy is absorbed by the viscosity of the water and by friction (which causes smoothing and erosion of the ocean floor).

Semidiurnal Tide

Notice that there are two swells and dips. Because the Earth rotates once per day, an observer placed at some point on the Earth will see the water rise and fall twice per day. This is the tide most people are familiar with, and it is called the semidiurnal tide because it comes twice per day or once per half day (semi = half, diurnal = daily).

First Diurnal Effect

You may have noticed an unevenness in the lengths of the arrows and shapes of the curves. Gravity is an inverse-squared force, and as a result, the increase in the Moon's gravity on the closer side of the Earth is a little greater than the decrease on the farther side of the Earth. (One is about 2% bigger than the other, but the diagrams were drawn as if it were a 15% difference.) To balance this out, the region with higher-than-average Lunar gravity covers slightly less than half of the Earth's surface, and the lower-than-average gravity region covers the rest (slightly more than half). The result of this is that the high tide on the side towards the Moon is actually a little higher than the high tide on the opposite side of the Earth, because the water there is bunched up into a slightly smaller volume. This is a diurnal effect, because it produces a once-per-day pattern.

#Moon's Orbit is Not Circular*

A small variation in tides comes from the fact that the Moon is not orbiting in a circle, but rather an ellipse, with the closest point (perigee) about 5% closer than the further point. This causes a monthly variation in the height of both highs and both lows. When the Moon is closer, the tides are higher. This is the perigee tide.

The Sun's Tidal Pull

There is gravity and orbital centripetal motion associated with the Sun. Their strength is about 40% as great as the strength of the Moon's gravity and orbital centripetal motion. As a result, there are solar tidal forces with 40% of the strength of the Lunar tidal forces. (A curious fact: This is also the ratio between the density of the moon and that of the sun. This fact is related to the fact that the moon and sun have the same apparent size in the sky. If you know the formulas you can see why.)

Earth, Moon and Sun Together

When the Moon and Sun are at a 90-degree angle with respect to the Earth, the Sun's low tide coincides with the Moon's high tide and vice versa. The Sun's tide subtracts from the Moon's tide to create a combined tide about 60% of the height of the average tide. This is the neap tide. The Moon's phase is first quarter or third quarter.

When the Moon and Sun are in the same direction or in opposite directions with respect to the Earth, the Sun's high tide coincides with the Moon's high tide and likewise for the low tides. The Sun's tide adds to the Moon's tide to create a combined tide about 140% of the height of the average tide. This is the spring tide. The Moon's phase is new or full.

The Tidal Day

Because the Moon is orbiting the Earth, and the tides follow the Moon, the Moon and the tidal bulges move slowly throughout the day. They move in the same direction the Earth is turning but 1/29.5 as fast. As a result, it takes a little more than one day for you to get through an entire cycle of two high tides and two low tides. The actual amount of time is 24 hours 50 minutes; this period is called a tidal day.

The Solar Perigee Effect

There is also a distance variation between the Earth and Sun that varies throughout the year. The Earth is closest to the Sun on Jan 2nd. This creates a yearly variation in the strength of the Solar tidal force; making the tides a little more extreme in December and January and a little less extreme in June and July. This variation is much smaller than the Lunar perigee effect which in turn is smaller than the variation between spring and neap tides.

Earth and Moon in 3-D

Forget the Sun again for the moment. In 3 dimensions, the Earth is a sphere. The Moon's tidal effect on the water causes the water to bulge into a shape like an egg, with the point of the egg pointing towards the Moon.

Notice that you get the biggest tide if you on at the equator, and none at all if you are at the north or south pole. The tides at intermediate latitudes are smaller.

The Moon's Orbit is Tilted

The Moon does not orbit directly above the Earth's equator. Instead, its orbit is tilted by an amount that varies anywhere from 17 degrees to 28 degrees (the average tilt is 22.5). It crosses the equator twice per month. One week after crossing, it reaches its highest point (as much as 28 degrees north of the equator) and two weeks later it is at its lowest point.

The Second Diurnal Effect

Let's suppose the Moon is currently at 20 degrees north. The tidal bulges will also be oriented 20 degrees north.

When this happens, observers in the far north latitudes experience just one tide per day. Because of the tilt of the tidal bulge, the observer spends a greater part of the day in one of the high tides and doesn't go very far into the other high tide. If they are far enough north, they only get one high tide per day.

For most locations, this diurnal effect combines with the normal semidiurnal tides to make a mixed tide pattern. A mixed tide pattern has a higher high tide and a less-high high tide, and the two alternate for as long as the Moon stays above the equator, usually about 10 days. Then after a few days the alternation appears again, but the pattern has "skipped a beat". To an observer in the northern hemisphere, the moon is up longer and reaches a higher point in the sky during the part of the month when it is north of the equator. During this part of the month, the higher high tide will occur when the moon is up. During the other half of the month, the Moon is below the equator and will spend less time above the horizon, and will not reach as high a point in the sky (if you're far enough north, it doesn't rise at all). During this part of the month, the higher high tide will be the one that takes place while the Moon is below (or furthest below) the horizon.

The Moon's Nodes Do Not Follow the Phases

The point in the Moon's orbit when the Moon is 5 degrees north of the equator does not happen at the same time of the month. It varies by a couple days per month. This is the same cycle that causes eclipses to happen only at two times of the year. When the moon is crossing the equator during new moon or full moon, eclipses are possible, and the diurnal tide effect will take place during first and third quarter. Three months later, there are no eclipses and the diurnal tides are strongest during full and new moons.

Getting the Maximum Effect

If you want to see the biggest tides (greatest variation between high and low) you should wait for a time when the spring tide effect coincides with the diurnal effect. This happens when the new moon is at the same time of the month as the time when the moon reaches its maximum height above the equator (if you are in the northern hemisphere) or below the equator (if you are in the southern hemisphere). This happens on one or two days during one or two months at a particular time of year. The correct time of year for this is 3 months after eclipse season A (when the new moon coincides with the ascending node). In 2001, this occurs around the new moons on Sep 17th and Oct 17th and at that time the moon's height is about 22 degrees.

The Sun's Diurnal Effect

Of course, the Sun is not directly over the equator either, it goes as much as 22.5 degrees north or south. The Sun is furthest north on June 21st and furthest south on December 20th, and it is directly over the equator on March 21st and September 21st.

This causes a diurnal effect similar to the one just described, and it combines with the other effects to cause an annual variation in the pattern of the high and low tides.

The Moon's Orbit Precesses

From one year to the next, the point in the year when the new or full moon crosses the eauator shifts. There are two such periods each year called eclipse seasons. Each year the eclipse seasons take place about 3 weeks earlier than then they did the year before. It takes 19 years (actually 18 2/3) for the moon's orbit to come all the way around.

Getting the Maximum Effect

To see the biggest tides (greatest variation between high and low), in addition to the spring tide effect and lunar diurnal effect you should also worry about the solar diurnal effect. The desired alignment os for the new moon to be at its maximum height above the equator near June 21st (if you live in the northern hemisphere). This does not happen every year — it only happens during a few years out of the 19-year cycle just mentioned. The next time it will happen is in 2005 and 2006.

The Continents and Sea Floor

Of course, the continents and the varying depth of the sea floor cause great variations in tide time and height. The tidal bulges are like waves travelling around the Earth at up to 1600 kilometers per hour. These waves get slowed down by shallow water, diverted and amplified by coasts, resonate in seas and bays, firths and fjords, and so on. Certain coastline formations, most famously the Bay of Fundy in Nova Scotia, create very large resonant and amplifying effects.

The deformation of the Earth due to the land tides mentioned above does not get distorted in this way. The result is that, in some locations, the land tide actually runs out of phase with the water tides, making the water tides higher relative to the land.

References

http://home2.planetinternet.be/ballaux/

http://www1.pactide.noaa.gov/tide-explanation.htm