FEBRUARY 2007 MUNK AND BILLS 135 Tides and the Climate: Some Speculations WALTER MUNK Scripps Institution of Oceanography, La Jolla, California BRUCE BILLS Scripps Institution of Oceanography, La Jolla, California, and NASA Goddard Space Flight Center, Greenbelt, Maryland (Manuscript received 18 May 2005, in final form 12 January 2006) ABSTRACT The important role of tides in the mixing of the pelagic oceans has been established by recent experiments and analyses. The tide potential is modulated by long-period orbital modulations. Previously, Loder and Garrett found evidence for the 18.6-yr lunar nodal cycle in the sea surface temperatures of shallow seas. In this paper, the possible role of the 41 000-yr variation of the obliquity of the ecliptic is considered. The obliquity modulation of tidal mixing by a few percent and the associated modulation in the meridional overturning circulation (MOC) may play a role comparable to the obliquity modulation of the incoming solar radiation (insolation), a cornerstone of the Milankovic´ theory of ice ages. This speculation involves even more than the usual number of uncertainties found in climate speculations. 1. Introduction et al. 2000). The Hawaii Ocean-Mixing Experiment (HOME), a major experiment along the Hawaiian Is- An early association of tides and climate was based land chain dedicated to tidal mixing, confirmed an en- on energetics. Cold, dense water formed in the North hanced mixing at spring tides and quantified the scat- Atlantic would fill up the global oceans in a few thou- tering of tidal energy from barotropic into baroclinic sand years were it not for downward mixing from the modes over suitable topography. A few specific sites warm surface layers. Mixing a stratified fluid takes en- account for most of the 20 GW of barotropic tidal en- ergy; the required rate of energy expenditure was esti- ergy dissipation along the Hawaiian Island chain, about mated at 2 TW (Munk and Wunsch 1998). Global tidal 2% of the 1 TW of global pelagic tidal dissipation (Eg- dissipation is 3.5 TW, two-thirds in marginal seas, one- bert and Ray 2001; Merrifield et al. 2001). third in the pelagic1 oceans, suggesting a tidal contri- The proposals for pelagic tidal mixing go back to the bution to the pelagic ocean mixing. Observational sup- 1960s and 1970s (Cox and Sandstrom 1962; Bell 1975; port comes from tidally induced fortnightly and Garrett 1979) but then fell out of fashion. The proposal monthly temperature variations in the Indonesian Seas for a long-period modulation goes back to Loder and (Ffield and Gordon 1996) and from measurements of Garrett (1978) who attributed an 18.6-yr cycle in ocean ocean microstructure in the deep Brazil Basin revealing surface temperature to shallow mixing associated with mixing over topography with enhanced intensity at the lunar nodal tide cycle. Otto Pettersson in 1910 dis- spring over neap tides (Polzin et al. 1995, 1997; Ledwell covered internal tides breaking over the bank that sepa- rates Gullmarfjord from the sea and spent much of his subsequent career trying in vain to convince his col- 1 “relating to . open oceans or seas rather than waters adja- cent to land or inland waters.” The American Heritage Dictionary leagues that tidal mixing is a factor in ocean climate of the English Language, 4th ed. 2004, Houghton Mifflin. (e.g., Pettersson 1930). A few decades ago the suggestion that the Moon played a role in determining global ocean properties Corresponding author address: Dr. Walter Munk, Scripps Insti- tution of Oceanography, 9500 Gilman Dr., Mail Code 0225, La was considered lunatic; now it is considered obvious Jolla, CA 92093-0225. (Wunsch and Munk were more comfortable working in E-mail: [email protected] the earlier times). There is wide agreement that pelagic DOI: 10.1175/JPO3002.1 © 2007 American Meteorological Society JPO3002 136 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 TABLE 1. Equilibrium coefficients. Amplitudes are given as per- Here, aE is the radius and mE the mass of Earth, and cent of the lunar equilibrium amplitude K ϭ 35.79 cm (Doodson mM, rM are the mass and mean distance of the Moon. 1921; Cartwright and Taylor 1971; Godin 1972). Sir George Dar- (In many ports the tide exceeds equilibrium values, and win (son of Charles) introduced the designations of constituents the reader might think of the tabulated percentages as within the long-period, diurnal, and semidiurnal species; Mn is a new notation. indicating centimeters of tidal range.) Frequency is expressed in terms of abbreviated Doodson Nos. Doodson numbers, giving cycles per (lunar) day, cpd cpm cpy cpn Amplitude (%) Darwin designations month, year, and nodal period, to be designated Long period (m ϭ 0) cpd, cpm, cpy, and cpn. [The Doodson columns for lu- Subannual nar apsidal (8.85 yr) and solar perigee (20 900 yr) are 0 0 0 0 73.6 Mean figure of Earth omitted.] 0 0 0 1 6.6 Mn The dominant constituent M2 is at exactly 2 cpd. Annual Ϫ ϩ There is monthly splitting, with N2, K2 at 1, 2 cpm 0 0 1 0 1.2 Sa 0 0 2 0 7.3 Ssa relative to M2, a yearly fine structure such as S2 at 2 Monthly cycle per solar day, ϩ2 cpy, and a nodal hyperfine struc- Ϫ 0101 8.3 Mm ture at 2 Ϫ 0.000 15 cpd. The spectral fine structure is 0 2 0 0 15.6 M f the equivalent in the frequency domain to the long- Diurnals (m ϭ 1) 1 Ϫ10Ϫ1 7.1 (Nodal splitting) term modulations in the time domain. Doodson’s he- Ϫ 1 1 0 0 37.7 O1 roic expansion was the last gasp in the Kelvin–Darwin– Ϫ 112 0 17.6 P1 Doodson development of tide prediction by adding si- 1 1 0 0 53.1 K1 (luni solar) nusoids. Ultimately, one is better off in the time domain ϩ 1101 7.2 (Nodal splitting) (Munk and Cartwright 1966; Wunsch 2000; Wunsch Semidiurnals (m ϭ 2) Ϫ 2002). 2 1 0 0 17.4 N2 200Ϫ1 3.4 (Nodal splitting) The nodal modulation is too large to be ignored, yet 2 0 0 0 90.8 M2 the available tide records are usually too short to re- Ϫ 222 0 42.4 S2 solve the hyperfine splitting. Darwin (1911) proposed 2 2 0 0 11.5 K 2 that any recorded spectral line A cos␻t be interpreted as a temporary manifestation of a modulated signal of amplitude tidal mixing must be taken into account in any realistic ␩ ϭ ␣͑t͒A cos͓␻t ϩ s͑t͔͒, ͑2͒ modeling of ocean properties. But we are a long way from understanding the underlying physics, and depend where ␣(t) is a slowly varying amplitude factor and s(t) heavily on a parameterization of the processes in- is a phase increment, both derived from equilibrium volved. theory. “Darwin factors” are related according to The demonstration that tides play a role in the ␣ coss ϭ 1 ϩ ␧ cosN and ␣ sins ϭϩ␧ sinN, present ocean climate has encouraged us, prematurely or, to first order in ␧, perhaps, to consider a possible role of tides in past climate changes. The discussion is centered on the ␣ ϭ 1 ϩ ␧ cosN and s ϭϩ␧ sinN, ͑3͒ 41 000-yr modulation of the inclination of Earth’s equa- with N increasing by 2␲ rad in 18.61 yr at a rate • torial plane relative to the ecliptic; the modulation is N ϭ dN/dt. Substituting in (2) yields caused by the gravitational pull from the other planets • in the solar system. Comparable effects from precession ␩ ϭ A cos␻t ϩ ␧A cos͑␻ ϩ N͒t. ͑4͒ and eccentricity are ignored for the sake of simplicity. The spectrum consists of a line at the principal fre- quency ␻, plus a single weak sideband at a slightly ␻ ϩ • • 2. Darwin factors lower frequency N (N is negative for the nodal regression). For M2, the fractional amplitude of the Table 1 shows the amplitude of 16 selected tidal con- split line is ␧ϭ3.4/90.8 ϭ 0.037. stituents [out of 400 listed by Doodson (1921)], ex- The nodal tide Mn (as well as other long-period tides) pressed as a percentage of the lunar equilibrium ampli- appear in two places in Table 1: (i) at the primary fre- ϭ tude quencies and their harmonics, for example, Mm, Mf 1, 2 cpm; S , S ϭ 1, 2 cpy; M ϭ 1 cpn; and (ii) in splitting ր a sa n mM mE K ϭ a ϭ 35.79 cm. ͑1͒ the principal short-period constituents. The mixing E ͑ ր ͒3 rM aE problem involves the split high-frequency constituents FEBRUARY 2007 MUNK AND BILLS 137 (ii), and not the primary lines (i) for the following rea- Ѩ␪ Ѩ2␪ son: mixing is associated with the tidal current u, and ϭ ␬ ͑10͒ Ѩt Ѩz2 not the tidal elevation ␩. Long-period tidal currents are very weak. For a basin of length L and depth H, the for a constant heat flow Q through the surface at z ϭ H horizontal particle2 velocity scales as and zero heat flow through the bottom z ϭ 0. The initial ␪ ϭ ␪ temperature (z,0) 0, a constant. Loder and Garrett L (1978) write the solution u ϭ ␩, ͑5͒ HT • 1 • 1 ␪͑z, t͒ ϭ ␪ ϩ ␪ t ϩ ͑␪ ր␬͒͑z2 Ϫ H2͒,tk ␶, where T is the characteristic period. For L ϭ 1000 km, 0 0 2 0 3 H ϭ 1 km, and T ϭ 18.6 yr, we have u ϭ 10Ϫ6 msϪ1. ͑11͒ ␪• ϵ ␳ where 0 Q/( CpH) is the warming of the entire wa- 3.