Tides

Ocean tides on Earth are due to the gravitational pull of the Moon, with a smaller contribution of the Sun. In 1695, Halley concluded that the Moon's orbital velocity around the Earth was increasing with time. To conserve the total angular momentum of the Earth-Moon system, Halley concluded that the distance between the Moon and the Earth was decreasing. The most precise measure of this distance comes from laser ranging observations made available after the installation of reflectors on the lunar surface by the Apollo mission in 1969 (Dickey et al., 1994). The Earth-Moon distance is in fact increasing, because the angular momentum of the system decreases due to tidal friction. Frictional dissipation occurs at a rate of $2.4 TW$ for the lunar semidiurnal constituent $M_2$ alone, and $3.7 TW$ for all constituents. Most of this energy loss ($95\%$ ) is associated with ocean tides. It was first thought that nearly all the energy was lost by friction in the shallow seas where tidal currents and associated bottom drag are strong (Taylor, 1919). Using satellite altimetry and an inverse model of the tides, Egbert (1997) and later Egbert and Ray (2001) estimated that only $1.8 TW$ of $M_2$ energy is dissipated in shallow seas (out of $2.4 TW$ ), leaving $0.6 TW$ of energy to be dissipated in the deep ocean. Several numerical modeling experiments have since shown that most of the energy lost by the surface tide in the open ocean occurs in regions of rough topography such as the Hawaiian Ridge or the Mid-Atlantic Ridge (Simmons et al., 2004; Merrifield and Holloway, 2002; Niwa and Hibiya, 2001). At these sites, the surface or barotropic tide is converted into internal or baroclinic tides. The generated internal tide is either dissipated locally, or radiated away and subsequently dissipated. St Laurent and Garrett (2002) estimate that less than $30\%$ of the internal tide energy is generated at small enough vertical scales suitable for local dissipation, while the rest is in the form of long-range propagating, low mode internal tides. At the Hawaiian Ridge in particular, Merrifield and Holloway (2002) found that $9.7\times10^{9} W$ or $9.7 GW$ of tidal energy is radiated away from the ridge as internal tides while Niwa and Hibiya (2001) and Simmons et al. (2004) found an $M_2$ energy conversion from barotropic to baroclinic of $15 GW$ and $30 GW$ respectively. Egbert and Ray (2000) estimated from altimetry that $20 GW$ of barotropic energy is lost over the same area.