Harmonics of Forcing Term in Laplace Tidal Equations

Making use of Equations (12.25)-(12.27), (12.35), (12.44), (12.65), and (12.80), we can write the forcing term in the Laplace tidal equations in the form

$$\skew{5}\bar{\zeta}_2(\theta,\phi,t)={\rm Re}\sum_j \skew{5}\bar{\zeta}_j(\theta,\phi,t),$$ (12.138)

where

$$\skew{5}\bar{\zeta}_j(\theta,\phi,t) =\skew{5}\tilde{\zeta}_{j}\,P_2^{\,m_j}(\cos\theta)\,{\rm e}^{\,{\rm i}\,(m_j\,\phi+\sigma_j\,t)}.$$ (12.139)

The amplitudes, $\skew{5}\tilde{\zeta}_j$ , azimuthal mode numbers, $m_j$ , and frequencies, $\sigma_j$ , of the principal harmonics of the forcing term are specified in Table 12.1, where

$$\zeta_1 = \frac{{\mit\Omega}^{\,2}\,a^{\,2}}{3\,g},$$ (12.140)

$$\zeta_2 =\frac{1}{2}\,\frac{m'}{m}\,\frac{a^{\,4}}{R^{\,3}}.$$ (12.141)

Here, $m$ is the planetary mass, $m'$ the mass of the moon, and $R$ the moon's orbital major radius. As indicated in the table, the harmonics of the forcing term can be divided into four classes: an ``equilibrium'' term that is time independent; two ``long period'' terms that oscillate with periods much longer than the planet's diurnal (i.e., rotational) period; two ``diurnal'' terms that oscillate at periods close to the planet's diurnal period; and two ``semi-diurnal'' terms that oscillate at periods close to half the planet's diurnal period. Here, we have neglected a semi-diurnal term that is significantly smaller than the other two.

Table 12.2: The principal harmonics of the forcing term in the Laplace tidal equations for the Earth.

$j$	$\skew{5}\tilde{\zeta}_j$	$m_j$	$\sigma_j$	Classification	Symbol
0	$-(\zeta_{\mit\Omega} + \zeta_M+\zeta_E)$	0	0	Equilibrium	$S_0$ , $M_0$
$1$	$+(3/2)\,\sin^2\epsilon\,\,\zeta_E$	0	$2\,\omega_E$	Long Period	$S_{sa}$
$2$	$-3\,e_M\,\zeta_M$	0	$\omega_M-{\mit\Pi}_M$	Long Period	$M_m$
$3$	$+(3/2)\,\sin^2\epsilon\,\,\zeta_M$	0	$2\,\omega_M$	Long Period	$M_f$
$4$	$-\sin\epsilon\,\zeta_M$	$1$	${\mit\Omega}-2\,\omega_M$	Diurnal	$O_1$
$5$	$-\sin\epsilon\,\zeta_E$	$1$	${\mit\Omega}-2\,\omega_E$	Diurnal	$P_1$
$6$	$-\sin\epsilon\,(\zeta_M+\zeta_E)$	1	${\mit\Omega}$	Diurnal	$K_1$
$7$	$+(7\,e_M/4)\,\zeta_M$	$2$	$2\,{\mit\Omega}-3\,\omega_M+{\mit\Pi}_M$	Semi-diurnal	$N_2$
$8$	$+(1/2)\,\zeta_M$	$2$	$2\,{\mit\Omega}-2\,\omega_M$	Semi-diurnal	$M_2$
$9$	$+(1/2)\,\zeta_E$	$2$	$2\,{\mit\Omega}-2\,\omega_E$	Semi-diurnal	$S_2$

For the Earth-Moon system, $m=5.97\times 10^{24}\,{\rm kg}$ , $m'=7.35\times 10^{22}\,{\rm kg}$ , $a=6.37\times 10^6\,{\rm m}$ , $R= 3.84\times 10^8\,{\rm m}$ , ${\mit\Omega}=7.29\times 10^{-5}\,{\rm rad.\,s}^{-1}$ (Yoder 1995). It follows that $g=9.82\,{\rm m\,s}^{-2}$ , as well as $\zeta_1=\zeta_{\mit\Omega}$ , $\zeta_2=\zeta_M$ , where

$$\zeta_{\mit\Omega} = 7.32\times 10^3\,{\rm m},$$ (12.142)

$$\zeta_M = 1.79\times 10^{-1}\,{\rm m}.$$ (12.143)

Moreover, $\delta= \epsilon$ , $e=e_M$ , $\omega_1=\omega_M$ , and $\omega_2={\mit\Pi}_M$ , where $\epsilon=23.44^\circ$ , $e_M= 0.05488$ , $360^\circ/\omega_M=27.322$ (solar) days, $360^\circ/{\mit\Pi}_M=8.847\,{\rm years}$ (Yoder 1995). Here, $m$ , $a$ , ${\mit\Omega}$ , and $\epsilon$ are the Earth's mass, mean radius, (sidereal) rotational angular velocity, and inclination of the equatorial plane to the ecliptic, respectively. Furthermore, $m'$ , $R$ , $e_M$ , $\omega_M$ , and ${\mit\Pi}_M$ are the Moon's mass, orbital major radius, orbital eccentricity, mean orbital angular velocity, and rate of perigee precession, respectively. Here, we have neglected the small (i.e., about $5^\circ$ ) inclination of the Moon's orbital plane to the ecliptic.

For the Earth-Sun system, $m'=1.99\times 10^{30}\,{\rm kg}$ and $R=1.50\times 10^{11}\,{\rm m}$ (Yoder 1995). It follows that $\zeta_1=\zeta_{\mit\Omega}$ and $\zeta=\zeta_E$ , where

$$\zeta_E = 8.14\times 10^{-2}\,{\rm m}.$$ (12.144)

Moreover, $\delta= \epsilon$ , $e=e_E$ , $\omega_1=\omega_E$ , and $\omega_2={\mit\Pi}_E$ , where $e_E= 0.0167$ , $360^\circ/\omega_1= 365.24\,{\rm days}$ , and $360^\circ/{\mit\Pi}_E = 20940\,{\rm years}$ (Yoder 1995). Here, $m'$ is the solar mass. Furthermore, $R$ , $e_E$ , $\omega_E$ , and ${\mit\Pi}_E$ are the Earth's orbital major radius, orbital eccentricity, mean orbital angular velocity, and rate of perihelion precession, respectively.

Combining lunar and solar effects, we can write the forcing term in the Laplace tidal equations for the Earth in the form (12.142)-(12.143). The properties of the principal harmonics of the forcing term are specified in Tables 12.2 and 12.3. The symbols are due to G.H. Darwin (1845-1912).

Table 12.3: The principal harmonics of the forcing term in the Laplace tidal equations for the Earth, excluding the equilibrium harmonic.

$j$	$\skew{5}\tilde{\zeta}_j (\rm m)$	$m_j$	$f_j=\sigma_j/2\,{\mit\Omega}$	Period	Symbol
$1$	$+1.93\times 10^{-2}$	0	$2.73\times10^{-3}$	$182.62$ days	$S_{sa}$
$2$	$-2.95\times 10^{-2}$	0	$1.81\times 10^{-2}$	$27.555$ days	$M_m$
$3$	$+4.25\times 10^{-2}$	0	$3.65\times 10^{-2}$	$13.661$ days	$M_f$
$4$	$-7.12\times 10^{-2}$	1	$4.63\times 10^{-1}$	$25.819$ hours	$O_1$
$5$	$-3.24\times 10^{-2}$	1	$4.97\times 10^{-1}$	$24.066$ hours	$P_1$
$6$	$-10.36\times 10^{-2}$	1	$5.00\times 10^{-1}$	$23.934$ hours	$K_1$
$7$	$+1.72\times 10^{-2}$	2	$9.45\times 10^{-1}$	$12.658$ hours	$N_2$
$8$	$+8.95\times 10^{-2}$	2	$9.63\times 10^{-1}$	$12.421$ hours	$M_2$
$9$	$+4.07\times 10^{-2}$	2	$9.97\times 10^{-1}$	$12.000$ hours	$S_2$