Tides and the Climate: Some Speculations

FEBRUARY 2007 MUNK AND BILLS 135

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 energy; 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 separates 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

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 frequency ␻, 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. Loder and Garrett on nodal mixing ter column, and ␶ ϭ H2/␬ is the characteristic diffusion k ␶ Loder and Garrett (1978, hereinafter LG) base their time. The assumption t limits the validity to turbulent coastal waters. The top-to-bottom temperature dif- analysis on the published values (Godin 1972) of the • ␪ H t Ϫ ␪ t ϭ 1 2␪ ␶ split constituents with fractional amplitude modulations ference is then ( , ) (0, ) ⁄ 0 . A fractional ␧ ␬ ␬ ϩ increase in the tidal potential increases to 0(1 7.2 7.1 3.4 2␧) and decreases the surface temperature by a fraction ϭ 11%͑K ͒, ϭ 19%͑O ͒, and 53.4 1 37.7 1 90.8 4⁄3␧ of the top-to-bottom temperature difference: ϭ ͑ ͒ ͑ ͒ 3.7% M2 . 6 4 ␦␪ ϭ ␧͓␪͑H͒ Ϫ ␪͑0͔͒. ͑12͒ 3 The diurnals K1, O1 have roughly half the amplitude of the principal semidiurnal constituent M2 but a much Typically, this is on the order of a few tenths of degree. larger fractional modulation. There is an important In regions with dominant semidiurnal tides, SST consideration with regard to the phase of the modula- minima are expected around 1886, 1904, 1923, 1941, tion. Whereas the semidiurnals are strongest when the 1960, 1978, 1997, 2015, . . . (when the Moon’s orbit is Moon is closest to the equatorial plane (at 18.6-yr in- most nearly equatorial). In regions dominated by the tervals), the diurnals are then weakest. We emphasize diurnal species, SST maxima are to be expected at these that although the modulation is of very low frequency, times. we are here concerned with the relatively high fre- These expectations are fulfilled. Loder and Garrett quency semidiurnal and diurnal spectral bands. (1978) examined 12 stations with records extending Next LG refer to certain relations for turbulent mix- over 30–70 yr; six stations dominated by diurnal tidal ing in unstratified water: currents and six stations by semidiurnal tidal currents. The nodal cycle contributes only 20% to the variance of u*2 fH ␬ ϭ , u*2 ϭ ␥u2, provided Ͼ 0.1, ͑7͒ annual mean SSTs, not adequate to show conclusive 200f u* evidence for a spectral peak at the nodal frequency. Even so, “The coincidence of the phase of the fitted where ␬ is the turbulent diffusivity, f is the Coriolis Ϫ nodal cycle with the phase of the tidal variation is re- parameter, H is depth, and ␥ ϭ 2.5 ϫ 10 3 is the bottom markable, and seems to be good evidence....”(Loder drag coefficient. Thus, ␬ is quadratically dependent on and Garrett 1978). the tidal current u. For a modulation at frequency ␴ of the ␻ tide 4. Obliquity versus nodal modulation ͑␩, u͒ ϭ ͑A, U͒͑1 ϩ ␧ cos␴t͒ cos␻t, ͑8͒ In estimating a possible role for the obliquity modu- we have lations, we need to discuss similarities and differences with the nodal modulations. Unlike the LG analysis, we ␬ ϭ ␬ ͑ ϩ ␧ ␴ ͒2 ϭ ␬ ͑ ϩ ␧ ␴ ϩ ͒ ͑ ͒ 0 1 cos t 0 1 2 cos t · · · . 9 have no recorded data for testing the speculation. The reviewer has suggested that quantities such as the pa- To interpret the nodal variation in diffusivity, LG solve leostratification are probably more readily interpreted the temperature diffusion equation as related to tidal mixing than to insolation. The long chain of arguments is outlined below.

2 The usual terminology “orbital” (as distinct from “phase”) 1) The tidal potential induces a spheroidal distortion in velocity here leads to a confusion with orbital planetary motion. the figure of Earth, with elongations toward and 138 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37

away from the tide-producing body; for a fixed po- mate the variability in the poleward flux of heat as a sition of the tide-producing body, this would appear consequence of the temperature perturbation, using as a zero-frequency term in the tidal expansion a simple model of the meridional overturning circu- (Table 1). Monthly, yearly, and other long-period lation (MOC). These are two independent steps, variations in the positions of the Moon, Earth, and each subject to great uncertainty. Pelagic turbulence the Sun shift some of the zero-frequency variance involves nearly all of physical oceanography and is into the long-period species. The diurnal rotation of not well understood. Earth generates semidiurnal and diurnal frequen- 6) Nodal and obliquity variabilities in the tidal current cies, akin to a conversion from DC to AC electric u (as well as height ␩) are proportional to the vari- current. The semidiurnal and diurnal spectra in tide ability in the tidal potential V. For the nodal modu- potential are complex spectral clusters, with cpm, lation, V is proportional to the inclination of the cpy, and cpn fine structure related to the nonlinear- Moon’s orbit relative to the ecliptic by 5.1°; for the ity of Newton–Kepler (NK) orbital dynamics (tidal obliquity modulation, the determining factor is the potential ϳ1/r3). variation 23.5° Ϯ 1.0° of the obliquity itself. The 2) The long-period ocean forcing is not attributed to squared potential ͗V2(t)͘ of the SD species varies by the primary long-period tidal species (18.6 yr, 41 000 7.8% and 1.5% for nodal and obliquity modulations, yr) but to the spectral fine structure in the semidi- respectively. urnal and diurnal species. The reason is that mixing 7) The fractional modulation of 7.8% by the nodal is associated with the particle velocity u not the el- tides is associated with changes in surface tempera- evation ␩, and for the long-period species u is ture by only a few tenths of degrees and is difficult extremely small, of order 1 ␮msϪ1 [Eq. (5)]. The to detect. The fractional variation in diffusivity of long-period ocean response is the result of the non- 1.5% by the obliquity tide is even smaller. Yet the linearities in Navier–Stokes (NS) fluid dynamics re- ratio in periods is of order 2000:1. Given the appro- sponding to the difference frequencies in the com- priate long time constants of global ocean overturn- plex SD and D tide potential clusters. ing, the weak obliquity tides might play a role in 3) So far, the discussion holds for both the nodal and ocean climate; it is a matter of joules versus watts. obliquity variability. Loder and Garrett (1978) re- The magnitude of the MOC variability may be com- lated the nodal tide to an 18.6-yr variability of SST parable to that of the incoming solar radiation (in- in coastal waters, using a simple diffusion equation solation). Both depend on the obliquity time scale, for guidance. They chose a quadratic nonlinearity, but the detailed time histories differ, with the tides ␬ ϳ͗u2͘, based on boundary layer turbulence in depending on the lunar and solar orbits, but insola- nonstratified fluids. These processes are not appli- tion depending only on the solar orbit perturbations. cable to tidally produced turbulence in the pelagic oceans. (But we end up using the same ␬ ϳ͗u2͘ 5. Tide potential proxy: it is traditional and simple, and does not of- We need to sketch the derivation of the nodal and fend any presently known evidence.) obliquity perturbations. The height of the “equilibrium 4) We associate the pelagic stirring and mixing to Ri- tide” is given by chardson-type instabilities provided by internal V r 3 m a 3 tides. The instability is associated with an increase of ␩ ϵ ϭ ͩ ͪ ͑␮͒ ϭ M ͩ ͪ ϭ eq K P2 , K a 35.8 cm, the (inverted) ambient Richardson number (du/ g r mE r 2 2 ␮ ϭ dz) /N to some critical value. Internal tides contrib- where P2( ) is the Legendre polynomial of degree n ute by an increase in the numerator and a decrease 2 (the case of interest) and ␮ ϭ cos␥, where ␥(t)isthe in the denominator; by an enhancement in the am- angular distance between source and station coordi- bient shear, and by a divergence of neighboring nates: the sublunar (subsolar) latitude ␪(t) and longi- isopycnals. These effects of shearing and straining in tude ␭(t), and the fixed station coordinates ␪*, ␭*. ␮ the internal tide field are of comparable magnitude Simple geometry relates P2( ) to the associated Leg- m ␮ and are expected to lead to enhanced turbulent mix- endre function P2 ( ): ing. 2 ͑ ␥͒ ϳ m͑␪͒ im␭ m͑␪ ͒ im␭* 5) Loder and Garrett (1978) considered a perturbation P2 cos ͚ P2 e P2 * e . in the near-surface temperature gradient in a diffu- mϭ0 sive model. Here, we consider the variation of tem- Using the normalization of the Legendre functions by perature in the entire water column, using a diffu- Munk and Cartwright (1966) and Cartwright and Tay- sion–advection model for guidance. Next, we esti- lor (1971), FEBRUARY 2007 MUNK AND BILLS 139

3 1 P0͑␪͒ ϭ sin2␪ Ϫ , P1 ϭ 3 sin␪ cos␪, and 2 2 2 2 2 ϭ 2␪ ͑ ͒ P2 3 cos , 13 corresponding to the long-period, diurnal, and semidiurnal species. For fixed station coordinates ␪*, ␭*, Eq. (13) relates the tidal potential to the latitude ␪(t) and longitude ␭(t) of the sublunar (subsolar) point. The diffusivity at a given station is taken to vary as ␬ ϳ m 2 (p2 ) . There are daily variations between the times of flood tide and slack tide, and monthly variations between spring and neap tides. Suppose these have been duly averaged (see later), yielding ␬, which varies slowly in the course of a year and at longer time scales. The solar inclination relative to the equator varies from ␧ϭϩ23.5° in summer to ␧ϭϪ23.5° in winter, according to FIG. 1. Mean-square tide potential for the semidiurnal, diurnal, and long-period tide species. Semidiurnal and long-period tides sin␪͑L͒ ϭ sin␧ sinL,0Յ L Յ 2␲, are largest if the tide-producing body is in the equatorial plane ϭ where L is the mean solar longitude. The required an- (obliquity 0). Diurnal tides vanish for zero obliquity. The curves 1 ␧ are nomalized to correspond to ⁄9 fSD, fD, and 4fLP, where f( ) are nual mean-square values the functions in Eq. (14). 2␲ ϵ ͗͑ m͒2͑͘ ␲͒Ϫ1͵ ͑ m͒2 ϩ ϩ f P2 2 P2 dL are as compared with the 22% (K1), 38% (O1), and 0 Ϫ 7.4% (M2) used by LG [2 times the values in Eq. (6) 1 3 27 1 3 to allow for squaring]. Here, K1 depends on both lunar f ϭ Ϫ sin2␧ ϩ sin4␧, f ϭ 9ͩ Ϫ sin2␧ͪ sin2␧, LP 4 4 32 D 2 8 and solar forcing and this accounts for the lower value (22%) as compared with O1 (38%). and The obliquity modulations are easily estimated. Tak- 3 ing averages over the nodal period, further variations f ϭ 9ͩ1 Ϫ sin2␧ ϩ sin4␧ͪ ͑14͒ SD 8 are associated with the variable inclination of the equatorial plane relative to the ecliptic due to the pull on for the long-period, diurnal, and semidiurnal species, Earth by other planets, principally Venus (because it is respectively (Fig. 1). With increasing obliquity ␧, the f so close) and Jupiter (because it is so massive). The functions decrease for the semidiurnal species [which obliquity varies between 22.5° and 24.5° and is now very are largest for the Moon (Sun) at the equator], but near the central value. The result is increase for the diurnal species (which vanish at zero obliquity); this accounts for the 180° phase difference 1 ⌬f between the diurnal and semidiurnal species as ob- 2 ϭϪ4%͑LP͒, ϩ6.9%͑D͒, and Ϫ1.3%͑SD͒. served by LG. f͑23.5Њ͒ Relative to the ecliptic pole the lunar orbit pole is ͑16͒ inclined at a nearly constant angle of 5.1° and Earth’s spin pole remains near 23.5°. The plane of the lunar Perhaps the traditional separation into the long-period, orbit precesses with a period of 18.6 yr in response to diurnal, and semidiurnal species is not called for, and the solar attraction. Lunar tides on the planet Earth one should have a single discussion in the (variable) depend on the angle of the lunar pole relative to plane of the tide-producing forces. Earth’s spin pole and are highly variable, with extreme ␧ ϭ Ϯ ⌬ ϭ values M 23.5° 5.1°. The f function varies by f 6. Quadratic nonlinearities f(28.6°) Ϫ f(18.4°), with a fractional variation We now consider various sources of generating long- 1 period modulations (Fig. 2). The simplest case is that of ⌬f 2 the Sun and Moon pulling in the same and opposite ϭϪ20%͑LP͒, ϩ35%͑D͒,andϪ6.7%͑SD͒, directions at fortnightly intervals, thus producing the f͑23.5Њ͒ strong fortnightly modulation. The trigonometric iden- ͑15͒ tity 140 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37

FIG. 2. Cartoons illustrating low-frequency forcing (red) resulting from nonlinear interactions between high-frequency tidal constituents (black) in the (left) frequency and (right) time domains. The tide potentials associated with (e) a Kepler elliptic orbit and (f) the regression of the lunar nodes are compared with (a)–(d) various elementary models.

Fig 2 live 4/C FEBRUARY 2007 MUNK AND BILLS 141

1 1 south to north of the ecliptic plane) travels westward, cosx ϩ cosx ϭ 2 cos ͑x Ϫ x ͒ cos ͑x ϩ x ͒, 1 2 2 1 2 2 1 2 completing an orbit in 18.61 yr. The result is the single ϭ ͑ ͒ sideband of the Darwin factors [Eq. (4)]. Obliquity is xi wit, 17 more akin to an amplitude modulation, and gives a split renders the two-body forcing indistinguishable from an triplet. Monthly, yearly, nodal, and obliquity modula- excitation at the mean frequency whose amplitude is tions are quite distinct, yet the results of squaring and modulated at half the difference frequency. Since New- averaging are similar. Similar comments apply to non- ton–Kepler dynamics is linear with regard to the super- linearities other than squaring. position of two tide-producing bodies (no gravitational interaction), there is no lunar excitation at the fort- 7. Internal tides and shear instability nightly frequency, only the interference pattern (as from two independent turning forks). Low-frequency Consider first, the long surface waves (kH K 1) of forcing depends entirely on the Navier–Stokes nonlin- small amplitude (A/H K 1), with earities, as portrayed by a squaring devise (case a). ␩ ϭ A cos͑kx Ϫ ␻t͒ and u ϭ U cos͑kx Ϫ ␻t͒ For a near-circular orbit with eccentricity e there will be both amplitude and phase modulation. Consider the ͑20͒ simple cases of amplitude modulation only (case b) and designating vertical displacement and horizontal par- ␩ ϭ ␻ ϩ x ϭץ/uץ phase modulation (case c), according to A cos( t ticle velocity. Conservation of mass requires H t,orץ/␩ץs), with Ϫ A͑t͒ ϭ 1 ϩ ␧ cos␴t and s͑t͒ ϭ ␧ sin␴t, U A ␻ ϭ , C ϭ ϭ ͌gH. ͑21͒ where ␧ is a small parameter and ␴ K ␻. For the am- C H k plitude modulation, the linear high-frequency triplet at For a horizontal displacement ␰ ϭ B sin(kx Ϫ ␻t), we ␻ Ϫ ␴ ␻ ␻ ϩ ␴ , , and , upon squaring, produces low have ϪB␻ ϭ U and from (21) frequencies 0, ␴ (order 1, ␧) (in addition to high fre- A quencies, which we ignore). For pure phase modula- ͯ ͯ ϭ kH K 1. ͑22͒ tion, the original spectrum extends beyond the triplet, B but the low frequencies of the squared spectrum are For H ϭ 4 km, T ϭ 12.4h, C ϭ 200 m sϪ1, ␭ ϭ CT ϭ very weak. 8920 km, kH ϭ 0.0028, and A ϭ 0.5 m, we have B ϭ 177 m. Case d for two independent (but unequal) lines (in For a sloping bottom, m ϭ 0.1 say, the horizontal the ratio 0.458 of solar to lunar forcing) is not funda- displacement B is associated with a vertical displace- mentally different from case a. For the Kepler orbit ment (case e), to first order mA ϭ ͑ ϩ ␧ ͒ ϭ Ϫ ␧ ϭ ␴ mB ϭ ϭ 18 m. ͑23͒ r r0 1 coss0 , s s0 2 sins0, and s0 t kh ͑18͒ Under suitable bottom conditions, surface tides will scatter into internal tides with amplitudes of order so that the “swept area” ϭ AIT mB (18 m), larger that the amplitude of the 1 1 1 surface tide (0.5 m). r rds ϭ r2͑1 ϩ 2␧ cos ͒͑ds Ϫ 2␧ coss ds ͒ Ϸ r2ds 2 2 0 0 0 0 0 2 0 0 Simmons et al. (2004b) parameterizes ocean mixing by a diffusivity remains constant. The equilibrium tide equals 1 Ϫ Ϫ ␩ ϭ ͑ ր ͒Ϫ3 ͓␻ ϩ ͑ ͔͒ ␬ ϭ ␬ ϩ ⌫␧րN2, ␧ ϭ NL 1H2L 1F ͑z͒͗u2͘, A r r0 cos t s t 0 2 B T 3 ϭ A cos␻t Ϫ ␧A͓cos͑␻ ϩ ␴͒t ϩ cos͑␻ Ϫ ␴͒t͔ ͑ ͒ 2 24 ␧ ϩ ␧ ͓ ͑␻ ϩ ␴͒ ϩ ͑␻ Ϫ ␴͒ ͔ ͑ ͒ where is the tidal dissipation per unit mass (dimension A sin t sin t . 19 2 Ϫ3 ⌫ m s ). Here, is the mixing efficiency, LB and H are Amplitude and phase modulation are in the ratio 1.5:1, the length and height scales of bottom roughness, and and in quadrature. Squaring leads to 0, ␴,2␴ frequen- F(z) is a vertical turbulence structure function with

cies. The contribution of the phase modulation to the scale LT. Note the dependence on the squared tidal squared record is negligible. particle velocity u2, the pelagic equivalent to Eq. (7). Kepler splitting differs from nodal splitting. The “as- All these parameters can be tuned to provide the cending node” (the point where the Moon crosses from expected magnitudes. They tell us nothing about the 142 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 physical processes involved in turning tidal energy into mixing, and the assumed dependence on the squared turbulent mixing. We can gain some insight from recent potential is a matter of simplicity, not understanding. measurements on (Alford and Pinkel 2000). Topo- graphic scattering produces multiple internal tide 8. Ocean time scales modes. The intensive modes 1 and 2 radiate away from the island chain and can be traced over 1000 km by We have considered tidal modulations on various satellite altimetry. There appear to be strong resonant time scales. Take a quasi diffusivity that varies as the interactions accompanied by turbulent mixing at lati- square of the tidal potential. The spring–neap modula- tude 28.9° where the Coriolis parameter f equals half tion is of order 1, but the duration is too short to have the M2 frequency. The radiating low modes may con- any climate implications. Other perturbations are of vert into the internal wave continuum and power some order of a few percent, but with vastly different time of the pelagic mixing. scales: month, year, 18.6 yr, and 41 000 yr. The high modes, 3, 4, ..., are locally dissipated. For an ocean of depth H ϭ 4000 m, with diffusivity HOME has found evidence for separate instabilities ␬ ϭ 10Ϫ4 m2 sϪ1, and upwelling velocity w ϭ 10Ϫ7 msϪ1 associated with the shear and strain of internal tides at [corresponding to 25 Sv (Sv ϵ 106 m3 sϪ1) of deep- the flank of Kaena Ridge (Pinkel et al. 2000). The strain water formation], we can define three time scales: instabilities occur 6–9 h later than the shear instabili- ␬ր 2 ϭ ր ϭ 2ր␬ ϭ ties. w 300y, H w 1300y, and H 5, 000y. Instabilities occur for large root-reciprocal Richard- All three scales are long relative to the nodal modula- son numbers: tion, and accordingly the nodally modulated tidal en- ͗ 2͘ |duրdz| g d␳ ergy V is a good measure of the instantaneous ocean RiϪ1ր2 ϭ , N2 ϭϪ , N ␳ Ѩz mixing. The obliquity forcing has a period long compared to the ocean time scales. The equator-to-Pole where N is the buoyancy (Brunt–Väisälä) frequency. heat flux is a time-integrated measure (over 10 000 yr, Ϫ1/2 Large Ri is associated with large shear (du/dz) and say) of obliquity mixing variability. weak stratification (low d␳/dz). Increased positive strain separates the distance between adjoining isopycnals and so reduces N2 to N2(l Ϫ d␩/dz). 9. Poleward heat flux Ϫ1/2 ϭ Let Ri0 (dU0/dz)/N0 represent the ambient con- An increase in flux lowers the equatorial tempera- dition, with ture and raises the polar temperature. For a simple 3 u ϳ coskx coskz cos␻t and ␩ ϳ sinkx sinmz sin␻t, model of a meridional overturning circulation (MOC), set ͑25͒ ␷͑ ͒ ϭ ͑ Ϫ ր ͒ ͓ ͑ ͒ ͔ ͑ ͒ ϭ ͑ Ϫ ր ͒ z V 1 2z d sin k z z , k z k0 1 z d , designating a system of superimposed internal waves. Then, ͑28͒ ͑ Ј ϩ Ј͒2 Ј for the volume flux across 30°N (say). The constants U0 u 2u RiϪ1 ϭ Ϸ RiϪ1ͩ1 ϩ ϩ ␩Јͪ, ͑26͒ 2 0 Ј Ϫ1 N ͑ Ϫ ␩Ј͒ U0 ϭ ϭ ϭ 0 1 d 10.5 km, k0 2.44 km , and V 1.22 w/ are so chosen that the flow is northward above a depthץx ϭϪץ/uץ where uЈϭdu/dz, and so on. Then using 2 y and the internal wave dispersion ␻/N ϭ h ϭ 1.5 km, southward beneath h, and vanishes at theץtץ/␩ץy ϭϪץ k/m, we end up with surface z ϭ 0 and the bottom z ϭ H ϭ 4.5 km (Fig. 3). Ј Ј Here, V is a scaling factor such that the integrated vol- Ϫ Ϫ 2u Ϫ ր u Ri 1 ϭ Ri 1ͩ1 ϩ ϩ iRi 1 2 ͪ. ͑27͒ ume flux (dimensionless) between 0 and h is ϩ1 and 0 UЈ 0 UЈ 0 0 between h and H is Ϫ1. Ϸ 1 For a near-critical ambient Ri0 ⁄4, the strain and We take an exponential temperature profile shear instabilities are of equal magnitude and in ͑ ͒ ϭ ͑Ϫ ͒ ϭ ր␬ ϭ Ϫ1 quadrature. T z; p T0 exp pz , p w 1km , In summary, the conversion of pelagic tidal energy ͑29͒ into mixing energy is a manifold process that is not understood (St. Laurent and Garrett 2002; Garrett 2003; Simmons et al. 2004a,b). We simply do not have a 3 The model is adopted for computational convenience. The good rule for the conversion of tidal energy into pelagic actual situation differs greatly from ocean to ocean. FEBRUARY 2007 MUNK AND BILLS 143

Simmons et al. (2004b) have examined two models of tidally driven mixing in a numerical model of the gen- eral ocean circulation. At 30°N the two models produce 1.2 and 0.8 ϫ 1015 W, respectively, of poleward heat transport. For orientation, consider a perturbation T(z), leaving ␷(z) unchanged (this is highly unrealistic). The change in poleward heat transport can be positive or negative, depending on the value of p. For our choice of p ϭ 1 kmϪ1,a1%increase in diffusivity ␬ is accompanied by a1%decrease in p, and a 0.3% increase in poleward heat flux. This differs in sign from the result of Sim- mons et al. (2004b), and compounds the ambiguity associated with the opposite sign in the response of the semidiurnal and diurnal tides to the obliquity modulation [Eq. 16)]. We end up with an effect of (allegedly) significant magnitude but unknown sign, hardly a sat- isfactory situation.

10. Insolation We compare the perturbation in the tide-producing forces to the perturbation in the incoming solar radiation. They have quite similar spectra, both dominated by obliquity, but they differ in three important aspects: (i) radiation is entirely of solar origin, whereas both the Moon and Sun contribute to the tides; (ii) Earth is opaque to radiation and transparent to gravitation; “clipping” associated with the opacity renders the radiation spectrum more complex; and (iii) solar radiation directly delivers energy to Earth’s surface, whereas FIG. 3. Assumed current profile V(z) of the meridional overturning circulation, so normalized that the integrated volume tidal mixing modifies the poleward transport of heat by transport between the surface and the node at h ϭ 1.5 km ϭϩ1, the MOC, with large (and unknown) phase lag. and between h and the bottom, is 1. The top-to-bottom heat trans- On an annual- and diurnal-averaged basis, insolation ϭ Ϫ port for a temperature profile is T(z) exp( pz) and has a varies with the terrestrial latitude ␪* according to maximum near p ϭ 1. S 5 3 ␫ ϭ ͫ Ϫ ͩ Ϫ 2␧ͪ 0͑␪ ͒ͬ 1 1 sin P2 * ͌ Ϫ e2 2 2 for a simple model of diffusive–convective balance, 4 1 ϭ Ϫ7 Ϫ1 ϭ with w 10 ms (consistent with Q 25 Sv), ϵ ␫ ͑e͒ ϩ ⌬␫͑e, ␧͒P0͑␪*͒. ͑32͒ ␬ ϭ Ϫ4 2 Ϫ1 ␳ ϭ Ϫ3 ϭ Ϫ1 0 2 10 m s , 1026 kg m , Cp 3994 J kg Ϫ1 ϭ Eccentricity e and obliquity ␧ influence the pattern °C , and T0 20°C. A scale factor for the integrated heat flux is quite differently. Eccentricity changes the global mean, but does not influence the spatial pattern. Obliquity H does not change the global mean but does influence the F ͑p͒ ϭ ͵ dz ␷͑z͒T͑z; p͒. ͑30͒ ␫ Pole-to-equator gradient. The first term 0(e) is bal- 0 anced by back radiation and determines the mean tem- Ϫ⌬␫ 0 ␪ ϭ Then, F(0) → 0 (uniform temperature) because there is perature. The second term varies from P2( *) equal volume flux northward and southward; F(ϱ) → 0 ϩ1⁄2⌬␫ at the equator, to Ϫ⌬␫ at the Poles, and vanishes ␪ ϭϮ ͌ ϭϮ corresponding to a infinitely thin warm surface layer at *0 arc sin(1/ 3) 36.25°. The total area- ␪ where the flow vanishes. For p ϭ 1, F(1) ϭ 0.46, the weighted fluxes between the equator and 0*, and be- ␪ Ϯ ͌ ⌬␫ total poleward heat flow is tween 0* and the Poles are ( 3/9) , respectively. [The reviewer has pointed out that higher-order terms ϭ ␳ ͑ ͒ ϭ ϫ 15 ͑ ͒ PHF CpT0QF 1 0.95 10 W. 31 in Eq. (32) (Rubincam 1994) change the hinge latitude 144 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37

go back to Croll (1890) and Milankovic´ (1941). There has always been the problem of reconciling the 40-kyr obliquity period with the 100-kyr scale of ice ages. A simple remedy is to assign to the obliquity the role of a pacemaker that terminates the ice sheets every second or third obliquity cycle at times of high obliquity. Huy- bers and Wunsch (2005) have examined the last seven terminations (the best-defined features of the paleotemperature records) and find that “the null hypoth- esis that glacial terminations are independent of obliquity is rejected at the 5% significance level.” Accordingly, the terminations are timed by obliquity, whatever the manifestation: orbital perturbations in the FIG. 4. (top) Sketch of proposed radiative balance. (bottom) incoming solar radiation, the traditional view, or tidal The assumed radiative flux density relative to a mean radiation, with a zero crossing at 36° latitude; in a steady state the integrated modulations in the meridional ocean heat flux. Figure 5 tropical excess equals the polar deficit, and the poleward trans- shows the time histories of insolation and semidiurnal ϭ ϭ port of heat across the zero latitude: FTropics Fpolar F.A tidal equilibrium energy plotted side by side (for better ␦ variable obliquity modulates the incoming radiation by FTropics, comparison the sign for the tidal energy is reversed ␦F and independently modulates the poleward transport polar since high obliquity results in low tidal energy). We by ␦F. have ignored eccentricity; it is now included and accounts for a high variability in the obliquity cycles of from 36° to 43° and significantly alter the area-weighted insolation and tidal energy. The obliquity perturbations fluxes.] take into account the gravitational pull from all of the Equilibrium is maintained by a poleward flux across planets in the solar system. ␪ ͌ ⌬␫ ␧ 0* of ( 3/9) (e, 0) W (neglecting the small differ- Fractional changes are roughly by Ϯ2% for insola- ence in tropical and polar back radiation). The fraction and by Ϯ1% for tidal energy. But here the com- tional departure from equilibrium associated with the parison ends. Whereas insolation has a direct interpre- obliquity variation from 22.5° to 24.5° is given by tation for climate change, the tidally related heat flux 3 F(t) is related to the tide potential V(t) in a most com- ͓sin2͑24.5Њ͒ Ϫ sin2͑22.5Њ͔͒ 4 plex way, as we have seen. We express this symbolically ϭ 2.6%. ͑33͒ by the convolution integral 3 1 Ϫ sin2͑23.5Њ͒ 2 ␦F ␦V2 Increased obliquity increases the polar radiation and ϭ ⌽* , F 2 decreases the tropical radiation (Fig. 4). Equation (33) V can be compared to the fractional perturbation in the squared semidiurnal tidal potential [Eq. (14) to order ␧2], where ⌽ includes nearly all that is unknown in oceanography. In the most recent termination, global sea 1 sin2͑24.5Њ͒ Ϫ sin2͑22.5Њ͒ level started rising at Ϫ20 kyr, and had reached 100 m 2 Ϫ ϭ 1.5%. ͑34͒ (out of 120 m total) by the time ( 10 kyr) the insolation 1 (obliquity) was at a maximum (Munk 2002). One would 1 Ϫ sin2͑23.5Њ͒ 2 feel more comfortable if the obliquity forcing were ad- Fractional perturbation in radiation and the mean- vanced by 10 000 yr, and there are many ways to do so square tide potential are of the same order (not sur- with the additional degrees of freedom afforded by in- prisingly). Increasing obliquity increases incoming po- cluding tidal mixing (we have withstood the tempta- lar radiation, decreases the semidiurnal ͗V2͘, but may tion). There are other complexities. The rising sea level increase or decrease the poleward ocean heat flux. significantly alters the depth and dissipation in the shallow seas; Egbert et al. (2004) estimate that the North Atlantic tides during glacial times were 2 times as high 11. Ice ages and the pelagic dissipation almost three times the The role of obliquity in marine ␦18O records was first present rate. These feedbacks dwarf the astronomic shown by Hays et al. (1976), but the original associa- forcing. tions of ice ages with orbital perturbations of insolation But the numbers will not go away. It takes 1025 Jto FEBRUARY 2007 MUNK AND BILLS 145

FIG. 5. Obliquity and eccentricity on (a) 1 ϫ 106 yr and (c) 140 000-yr time scales. Associated fractional departures in the average polar insolation and semidiurnal tidal energy [drawn positive downward for ready comparison in (c) and (d)]. The departures are relative to a mean obliquity of ␧ϭ23.50 and eccentricity ␧ϭ0.006 (present values are 23.5 and 0.018). In a linear world with meridional heat flux proportional to equilibrium tidal energy, the curves would be additive. Curves in (d) also show paleotemperature (green) and sea level (dotted) for the last two “terminations.” Maximum obliquity and minimum equilibrium tidal energy occur well after the start of termination at Ϫ20 kyr. 146 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 melt enough ice to raise the sea level by 120 m. This Cox, C., and H. Sandstrom, 1962: Coupling of surface and internal corresponds to only 300 yr of the 1015 W flux. A 3% waves in water of variable depth. J. Oceanogr. Soc. Japan, 20, increase in the MOC heat flux could account for the 499–513. Croll, J., 1890: Climate and Time in their Geological Relation: A entire melting. Theory of Secular Changes of the Earth’s Climate. 2d ed. Appleton, 577 pp. Acknowledgments. We are grateful to the reviewer Darwin, G. H., 1911: The Tides and Kindred Phenomena in the (Peter Huybers) for his critical review and for many Solar System. 3d ed. Murray. (Republished in 1962 by W. H. positive recommendations. We thank Chris Garrett, Freeman, 378 pp.) Rob Pinkel, and Harper Simmons for having put us Doodson, A. T., 1921: The harmonic development of the tide- straight on many issues; so has Carl Wunsch, unaware generating potential. Philos. Trans. Roy. Soc. London, A100, 305–329. that he was to be the victim of this missile. Egbert, G. D., and R. D. Ray, 2001: Estimates of M2 tidal energy One of us (WM) was the recipient of a 65th birthday dissipation from TOPEX/POSEIDON altimeter data. J. Geo- celebration on 19 October 1982, organized by Chris and phys. Res., 106, 22 475–22 502. Carl. On that morning, I telephoned Carl at the Mas- ——, ——, and B. G. Bills, 2004: Numerical modeling of the glob- sachusetts Institute of Technology (MIT) for advice on al semidiurnal tide in the present day and in the last glacial some manuscript, unaware that he had spent the night maximum. J. Geophys. Res., 109, CO3003, doi:10.1029/ in our basement guest room. Carl’s secretary professed 2003JC001973. ignorance of his whereabouts, and when I called Mar- Ffield, A. L., and A. Gordon, 1996: Tidal mixing signatures in the Indonesian Seas. J. Phys. Oceanogr., 26, 1924–1937. jory she said, “Carl does not always tell me where he is Garrett, C., 1979: Mixing in the ocean interior. Dyn. Atmos. going.” Oceans, 3, 239–265. Carl’s contribution to the birthday volume was ——, 2003: Internal tides and ocean mixing. Science, 301, 1858– Acoustic Tomography and Other Answers (Wunsch 1859. 1984). The “other answers” in this title refer to “9-W” Godin, G., 1972: The Analysis of Tides. University of Toronto and two other answers from which one was to infer the Press, 264 pp. question by “inverse theory” (IT). Carl had recently Hays, J. D., J. Imbrie, and N. J. Shackleton, 1976: Variations in embarked on his lifelong obsession with IT. If you can- the earth’s orbit: Pacemaker of the ice ages. Science, 194, 1121–1132. not infer the question corresponding to “9-W,” you can Huybers, J., and C. Wunsch, 2005: Obliquity pacing of the late blame it on a defect of IT. Pleistocene glacial terminations. Nature, 434, 491–494. The other part of the title, acoustic tomography, re- Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. St. Laurent, fers to our joint efforts to rely on the intrinsic proper- R. W. Schmitt, and J. M. Toole, 2000: Evidence for enhanced ties of ocean acoustics for monitoring ocean variability, mixing over rough topography in the abyssal ocean. Nature, and was to result in eight joint papers and a book. At 403, 179–182. one time we thought that the combined application of Loder, J. W., and C. Garrett, 1978: The 18.6-year cycle of sea surface temperature in shallow seas due to variations in tidal satellite altimetry (Carl’s major contribution to ocean mixing. J. Geophys. Res., 83, 1967–1970. observation) and acoustic tomography would become Merrifield, M. A., D. E. Holloway, and T. M. S. Johnston, 2001: the cornerstones for large-scale ocean monitoring, but The generation of internal tides at the Hawaiian Ridge. Geo- this has not been the case, not yet. phys. Res. Lett., 28, 559–562. We have shared so many interests since Carl, then a Milankovic´, M., 1941: Kanon der Erdbestrahlung und Seine An- Stommel graduate student, first showed up at my door- wendung auf das Eiszeitenproblem (Canon of Insolation of the Earth and Its Application to the Ice-Age Problem). Royal step in 1966 to have a look at tides. 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