# 3.7.5: Daily inequality

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Figure 3.23: The tidal bulges for a tide-generating body above the equator. Observers A and B both experience two high and low waters a day of equal height. The heights depend on the latitude.

So far, we have either explicitly or implicitly assumed that the sun and the moon are directly above the equator of the earth (or in other words that the orbits of the moon and the earth lie in the equatorial plane). Under that assumption a certain place on the earth experiences two high waters and low waters per day of equal height as can be seen from Fig. 3.23. The heights of those high waters and low waters depend on the latitude.

Figure 3.24: Daily inequality due to a non-zero declination angle 𝛿 between the equatorial plane and the line connecting the centres of the earth and the tide-generating body. For observer A the two high and low waters have the same height, but observer B experiences unequal heights in two successive high and low waters.

In reality, the orbits of the moon about the earth and of the earth about the sun are not in the equatorial plane, which complicates our simple picture of Fig. 3.23. While the moon’s orbit and the earth’s orbit are approximately in the same plane (5° difference), there is a time-varying declination angle between the equatorial plane and the earth-sun and earth-moon connection lines. As the tidal bulges tend to align themselves with the tide-generating body, the two high and low waters on a day are not equal (see Fig. 3.24). This phenomenon is referred to as daily inequality. Apparently, the declination of the sun has a diurnal (daily) effect on the tides.

According to Fig. 3.24, the daily inequality is zero at the equator and increases with latitude^{5}. At some higher latitudes, the daily inequality becomes so big that there is only one high and one low water (diurnal tide, see Sects. 3.7.6 and 4.4.1).

The earth’s axis is tilted by 23.5° with respect to a line perpendicular to the plane of the earth’s orbit around the sun. Due to the combination of this tilt and the orbiting of the earth around the sun, the declination of the sun varies seasonally. It is zero at the spring and autumn equinoxes (March 22 and September 22), positive during the Northern summer and negative during the Northern winter. The minimum declination of -23.5° (or 23.5° south) is reached on December 22 (Northern winter solstice) and the maximum of 23.5° on June 22 (Northern summer solstice). The daily inequality cycle of the solar tide therefore has a period of a year, with diurnal tides increasing with increasing declination south or north and semi-diurnal tides maximum at zero declination. An interesting consequence of the latter is that spring tidal ranges around the equinoxes are usually higher than average spring tidal ranges (equinoctial tides).

The lunar daily inequality cycle has a period of 27.3 days. The north-south difference of the declination of the moon during the month is twice 23.5° \(\pm\) 5°. The deviation of 5° has a cycle of 18.6 years and is a result of the 5° difference in moon’s and earth’s orbits. As with the sun, the largest daily inequalities correspond to the times that the moon is furthest south or north (minimum and maximum declination).

5. Some areas around the equator experience a large daily inequality. This is related to the presence of land masses and will be further explained in Sect. 4.4.