Hydrodynamic Equations of the Sea
3.7 Tidal Forces
In principle, fluids respond to all gravitational fields exerted on them, but in practice, except for the earth’s own gravity, only the sun and the moon exert influences strong enough to induce detectable tidal excursions in the sea. The orbits of these two bodies carry them through a very complex se-
quence of motions that vary the frequency, phase, and amplitude of the tid- al response; the most important tidal forcings have frequencies ranging from approximately twice a day to twice a year.
Consider again the three-body configuration shown in Fig. 2.1, in which the sun, moon, and earth are all assumed to lie approximately in the plane of the ecliptic and to be rotating about their common center of mass. Be- cause of the enormous mass of the sun-approximately 1.99 x lo3' kg- the center of mass of the system lies only 0.06% of the solar radius from the center of the sun. On the other hand, for the earth-moon system, the system center of mass lies within the earth about 4660 km (some 7 3 % of its radius) from its center. It will be seen that not only gravitational but also centrifugal forces are important in the tidal problem. We shall delay the de- velopment of the tidal potential function until Chapter 5 , but for now it may be said that this function, expressed in spherical coordinates and evaluated on the surface of the earth, is a complicated one that reflects the astronomi- cal motions of the three bodies. The dominant periods fall in three groups- semidiurnal (half-day), diurnal (daily), and longperiod (14,28, and 180 days and longer). The 1 1 most important periods are listed in Table 3.1 along with their commonly accepted nomenclature: M 2 , K , , etc., with the subscript 2 denoting semidiurnal and 1 , diurnal periods.
TABLE 3.1 Major Tidal Modes
Period,
Tidal Mode T (h)
Semidiurnal Tides
M2 principal lunar 12.421
Nz elliptical lunar 12.658
K , declination luni-solar 11.967
Sz principal solar 12.000
Diurnal Tides
K , declination luni-solar 23.935
01 principal lunar 25.819
PI principal solar 24.066
QI elliptical lunar 26.868
Long-Period Tides
Mf fortnightly lunar 327.86
Mm monthly lunar 661.3 I
Ssa semiannual solar 4383.04
Fig. 3.7 Mean sea surface topography as derived from the radar altimeter on the Seasat satellite. Contours represent elevations and depressions as departures from an ellipsoid of revolution having an equatorial radius, Re, of 6378.137 km, and a flattening of 11298,257, and are spaced at 5 m intervals. Extremes of elevation are near 65 m above the ellipsoid; the extreme depression is near 100 m below it. [From Marsh, J. G., et al., J. Geophys. Res. (1986).]
Fig. 3.7. Anomaly range shown covers *60 mGal. A high correlation exists between bathymetry (Fig. 3.9) and gravity anomaly. [From Haxby, W. F., 1982-83 Lamont-Doherty Geological Observatory Yearbook.]
86 Principles of Ocean Physics
The periods of the diurnal constituents clearly support the rule of thumb that “the tides come one hour later each day” (i.e., have roughly a 25 h period) although this neglects the intervening semidiurnal cycle; however, the importance of this rule varies greatly from place to place. The maximum of the fortnightly modulation is said to be a “spring tide” (which, however, has nothing to do with the spring season), and occurs approximately at the time that the sun and moon are in either conjunction or anticonjunction.
The minimum tide over 14 days is termed a “neap tide” and happens ap- proximately when the two celestial objects are in quadrature. The graphs in Fig. 3.10 show tidal elevations at four near-shore locations that exhibit, respectively, semidiurnal, mixed/dominant semidiurnal, mixed/dominant di- urnal, and full diurnal types, with the mix at any location due in part to the
(a)
S P O E 01 A
(a)
S P 01 A
9 .o
6.0 3.0 0
(b)
1 . 2 0.6
0
0.3
0 -0.3
1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1 6 17
Time, t ( d a y )
Fig. 3.10 Time series of tidal amplitudes showing four characteristic types of tides:
(a) semidiurnal, (b) mixed, dominantly semidiurnal, (c) mixed, dominantly diurnal, (d) full diurnal. [Adapted from Oceanographic Atlas of the North Atlantic Ocean (1969).]
Fig. 3.9 Map showing bathymetry and land topography. [From U. S. Dtspartment
Hydrodynamic Equations of the Sea a7
resonant character, if any, of both the large and small bodies of water in the vicinity of the point of observation.
Figure 3.1 l a is a schematic version of how the gravitational and centrifu- gal forces add vectorially over the face of the earth t o give the net tide- generating forces when the moon is in the zenith. It is the horizontal compo- nent of the resultant vector that causes the tidal forcing; the vertical compo- nent makes miniscule contributions to g. When the moon has a declination (angle with respect to the equator) of about 28.5" (caused by a combination of the earth's axial tilt and the moon's own 5" inclination to the plane of the ecliptic), the forcing is as shown in Fig. 3.11;. The flows that are caused by these forces are conditioned by the presence of land masses that help to shape the response, by bottom friction, and by the variable depths of the ocean basins, the last because the tide is properly considered to be a shallow water wave that propagates at a speed c = (gw' in water of depth H , while being driven by a force at a speed of approximately R,Q sin 8.
An example of a theoretical tide derived using recent analytical methods is given in Fig. 3.12, which shows the M2 tidal response caused by the lu- nar forcing function over all of the oceans of the world. The lines shown are the co-tidal lines (i.e., the loci of constant phases with respect to the pas- sage of the moon over Greenwich) and co-range lines (i.e., the contours of constant elevations, or ranges, which approximately form an orthogonal coor- dinate system t o the co-tidal lines). From the variation in phase angle about the points of zero range, or arnphidromes, one can deduce the direction of the tidal advance, which can be considered as a combined rotation and tilt- ing of the surface, with the amphidrome as a stationary point. One such plot may be calculated for each tidal constituent of interest in Table 3.1. It is clear that the tidal responses-currents and heights-are exceedingly com- plex in both the open sea and near the shores, where bottom dissipation is strongest.
For purposes of completing our task of tabulating the fluid forces, now let us simply write the tidal potential,
a t ,
as a time-dependent function that implicitly varies with the polar coordinates, r, 0 and 4, and agree t o make its form more explicit later on. Thus we take(3.47) and define the body force per unit mass, f , , as the negative gradient of 9, :
f , = -v9,
.
(3.48)Fig. 3.11 (a) Force diagram showing tide-generating forces when the moon is 28.5" north of the equatorial plane. Open arrows:
gravitational force. Shaded arrows: centrifugal force. Solid arrows: tide-generating force. Only the horizontal component of the last of these is effective in producing tidal currents. (b) Distribution of horizontal components of tidal forces when the lunar declination is 28.5" from the equator (i.e., above the zenith point, Z). Approximately the same distribution holds on the opposite side of the earth and it is symmetrical about the nadir point, N. As the earth rotates, this force pattern remains oriented toward the moon, propagat- ing at 15" per lunar hour. The tidal responses to the forcing are complicated. [Adapted from Defant, A., Physical Oceanography(l961).]
Fig. 3.12 Chart of tidal response characteristics for the M , tide in the world’s oceans. Amphidromes are points of zero ampli- tude about which tidal currents rotate; co-range lines are contours of constant elevations (in meters); co-phase lines are contours of constant phase in degrees relative to passage of the moon over the Greenwich meridian. [From Schwerdski, E. W., Global Ocean Tides, Part I / (1979).]