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Fourier Analysis of Ocean Tides I

3. The geometry of the solar tide

The solar and lunar tides are independent, and can be analyzed separately. The solar tide is considerably simpler.


The local strength of the solar "tide-producing force" depends on the angle between H, the direction of the Sun, and the local vertical P. (At this level of analysis, we will ignore the small yearly variation in the distance of the Sun due to the eccentricity of the Earth's orbit.) This image is drawn from the point of view of a stationary Earth with North N pointing up.
    During the day P rotates counter-clockwise around its circle of constant latitude. During the year the Sun-pointing vector H rotates, also counter-clockwise, in a plane inclined at a 23.452-degree angle to the plane of the Earth's equator.

To calculate the angle A(t) between H and P as a function of time, it is useful to draw cartesian coordinates with origin the center of the earth, with N on the positive z-axis, and with the x-axis pointing at the Spring equinox (the point where the Sun's trajectory crosses the equator on the way up).

If the Sun were traveling in the plane of the equator, its angular position would be (cos ht, sin ht, 0). Here h=0.04107 degrees/hour is the angular speed of the Earth in its orbit and therefore the angular speed of the Sun, as seen from Earth. To encode the inclination of the Earth's orbital plane (the ecliptic) with respect to the equator, we rotate this curve through an angle a=23.452o about the x-axis. The matrix for this rotation is

 1    0    0[0  cos a sin a], 0 -sin a cos a

and it transforms the path of H to (cos ht, cos a sin ht, -sin a sin ht).

The vector P rotates, at latitude L, at angular speed 15.04107 degrees/hour. In the 24 hours from noon to noon it turns 360 degrees plus the extra angle to catch up with the sun. We record this speed as T+h, where T=15 degrees/hour is the rate of change of the Sun's longitude. In our coordinates, this motion is encoded as (cos L cos[T+h]t, cos L sin[T+h]t, sin L).

Suppose that the spring equinox (H = (1,0,0)) occurred once exactly at noon (P = (cos L, 0, sin L)), so we can start the two t-clocks together at that moment.

Since H and P are unit vectors, their angle A(t)is the arc-cosine of their dot product. This dot product is a linear combination of time-varying functions, with coefficients coming from a and L. The actual value of the coefficients is not useful at this stage, so we record the dot product as LC{cos ht cos[T+h]t, sin ht sin[T+h]t, sin ht}, with LC{...} an abbreviation for ``linear combination of {...} with constant coefficients." With the aid of the ever-useful addition formula for cos(a+b) this can be simplified to:


A(t) = angle between H and P = arccos LC{cos Tt, cos[T+2h]t, sin ht}.