Explicitly modelled deep-time tidal dissipation and its implication for Lunar history

Dissipation of tidal energy causes the Moon to recede from the Earth. The currently measured rate of recession implies that the age of the Lunar orbit is 1500 My old, but the Moon is known to be 4500 My old. Consequently, it has been proposed that tidal energy dissipation was weaker in the Earth's past, but explicit numerical calculations are missing for such long time intervals. Here, for the rst time, numerical tidal model simulations linked to climate model output are conducted for a range of paleogeographic con gurations over the last 252 My. We nd that the present is a poor guide to the past in terms of tidal dissipation: the total dissipation rates for most of the past 252 My were far below present levels. This allows us to quantify the reduced tidal dissipation rates over the most resent fraction of lunar history, and the lower dissipation allow re nement of orbitally-derived age models by inserting a complete additional precession cycle.


Introduction
Tidally induced energy dissipation in the earth and ocean gradually slows the Earth's rotation rate, changes Earth and lunar orbital parameters, and ! increases the Earth-Moon separation (Darwin, 1899;Munk, 1968). A long-" standing conundrum exists in the evolution of the Earth-Moon system relating # to the present recession rate of the moon and its age: if present day observed $ dissipation rates are representative of the past, the moon must be younger than % 1500 Ma (Hansen, 1982;Sonett, 1996). This does not t the age model of the & solar system, putting the age of the moon around 4500 Ma (Hansen, 1982;Sonett, ' 1996;Walker and Zahnle, 1986;Canup and Asphaug, 2001;Waltham, 2004), and the possibility that the tidal dissipation rates have changed signicantly over long time periods has been proposed (Hansen, 1982;Ooe, 1989;Poliakov, 2005;Green and Huber, 2013;Williams et al., 2014). A weaker tidal dissipation must ! be associated with a lower recession rate of the moon. Consequently, it can be " argued that prolonged periods of weak tidal dissipation must have existed in # the past (Webb, 1982;Bills and Ray, 1999;Williams, 2000). There is support $ for this in the literature using quite coarse resolution simulations driven by % highly stylized, rather than historically accurate, boundary conditions (Munk, & 1968;Kagan and Sundermann, 1996). However, with the present knowledge of ' the sensitivity of tidal models to resolution and boundary conditions, e.g., the oceans density structure (Egbert et al., 2004), the results of prior work should be revisited with state-of-the-art knowledge and numerical tools.
It was recently shown through numerical tidal model simulations with higher ! resolution than in previous studies that the tidal dissipation during the early " Eocene (50 Ma) was just under half of that at present (Green and Huber, 2013). # This is in stark contrast to the Last Glacial Maximum (LGM, around 20 ka) $ when simulated tidal dissipation rates were signicantly higher than at present % due to changes in the resonant properties of the ocean (Green, 2010;Wilmes & and Green, 2014;Schmittner et al., 2015). However, the surprisingly large tides ' during the LGM are due to a quite specic combination of continental scale ! bathymetry and low sea-level, in which the Atlantic is close to resonance when ! the continental shelf seas were exposed due to the formation of extensive conti-! nental ice sheets (Platzman et al., 1981;Egbert et al., 2004;Green, 2010). It is !! therefore reasonable to assume and proxies support this that the Earth has !" only experienced very large tides during the glacial cycles over the last 12 Ma !# and that the rates have been lower than at present during the Cenozoic (Palike !$ and Shackleton, 2000;Lourens and Brumsack, 2001;Lourens and et al., 2001 including the Meridional Overturning Circulation (Munk, 1966;Wunsch and " Ferrari, 2004). " The tidally induced lunar recession and increased day length also act to re-" duce the precession rate of Earth's axis and, as a result, produce falling rates of "! climatic precession and obliquity oscillation through time (Berger et al., 1992).
"" As a direct consequence, cyclostratigraphy may be severely compromised be- Here U = uH is the volume transport given by the velocity u multiplied by friction and is written as where C d is a drag coecient, and u is the total velocity vector for all the tidal &$ constituents. We used C d = 0.003 in the simulations described below, but for &% all time slices simulations were done where C d was increased or decreased by a && factor 3 to estimate the sensitivity of the model to bed roughness. This only &' introduced minor changes in the results (within a few percent of the control), ' and we opted to use the value which provided the best r to observations for ' the present. The second part of the dissipative term, F w = CU, is a vector ' describing energy losses due to tidal conversion. The conversion coecient C is '! here dened as (Green and Huber, 2013) '" in which γ = 100 is a scaling factor, N b is the buoyancy frequency at the sea- The tidal dissipation, D, is computed using (Egbert and Ray, 2001): in which W is the work done by the tide-producing force and P is the energy $ ux. They are dened as in which the angular brackets mark time-averages. When we discuss the accu-& racy and the energy dissipation rates we use a cuto between deep and shallow ' water at 1000 m depth.

Earth-moon separation
The tidal dissipation rate, D, should be (Murray and Dermott, 2010) Moon separation, Ω is the Earth's rotation rate and n is the lunar mean motion.

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The next step is to note that lunar recession is well approximated using (Lam-# beck, 1980; Bills and Ray, 1999; Waltham, 2015) where the tidal drag factor In which k 2 is Earth's Love number, Q is the tidal quality factor, R is Earth's & radius whilst, from Kepler's 3rd Law Note that the tidal dissipation rates calculated in Table 1   The integrated tidal dissipation rates (in TW) for the M 2 constituent for the global (total) and abyssal (deep, i.e., deeper than 1000 m) ocean. The relative rate for PD is normalised with the PD reconstructed rate, whereas the relative LGM rate is normalised with the PD rate (see Figure 1 and the text for a discussion  (2014), relative to PD and constituents representing the diurnal luni-solar and lunar declinations, re-" spectively). Here, we limit our discussion to M 2 as changes in the other con-" stituents are similar to those in M 2 but smaller in magnitude (see the discussion bathymetries (see Matthews et al., 2015). This simulation showed a total M 2 "' dissipation of some 4.5 TW, of which 1 TW dissipated in deep waters (Table 1 # and Figure 1). This is within a factor 2 of our values using present day observed # bathymetry (2.8 TW in total and 0.9 TW in the deep, respectively) and leads us # to conclude that we most likely overestimate the dissipation rates in our paleo- with respect to the modern degraded simulation. The one exception is the LGM $ study, which is normalized by the undegraded PD simulations since modern ob-$ served bathymetry was used in this simulation. In the following we refer the $ reader to Figure 1 and Table 1 (Figures 2b and 3b). This is due to sea-level being some 25m higher $& than at present during this period and is consistent with previously reported $' simulations with extreme sea level rise (SLR; Green and Huber, 2013). The dy-% namical explanation is that the large SLR cause global dissipation rates to drop % below present because the near-resonant North Atlantic experiences decreased % dissipation rates with SLR due to larger shelf seas (Green, 2010).
%! Simulated Miocene tides resemble the modeled degraded PD tides to some %" extent, but they are generally weaker than at present (Figures 2c and 3c). The We have carried out a set of climate model sensitivity runs to complement ' the earlier Eocene simulation (see Table 2). These used a tidally driven dif-'! fusivity parameterization (Green and Huber, 2013) Table 2), showed a 45% increase in the abyssal rates but a 9% reduction in total ! dissipation. This again puts us on the safe side with our conclusions because ! we probably overestimate the dissipation slightly in the PT control run.

!!
The horizontally integrated dissipation rates for the other constituents, S 2 , !" K 1 and O 1 , are shown in Figure 5. It is evident from Figure 5  The lower-than-modern tidal dissipation rates simulated through the Ceno-"" zoic and Mesozoic shows that the lunar recession rate was probably smaller than "# otherwise predicted in the past. The questions raised are i) by how much? and "$ ii) how did this impact on the lunar distance? Using the recession model in "% Section 2.2, we show that the relative tidal dissipations in Table 12 are also "& the relative tidal-drag ratios. It is notable that all but the most recent ratios this paper, are that present day tidal dissipation is anomalously high. Given #" the results in Table 2 assumed that tidal lag (which is closely related to tidal drag) did not vary from $ the present day value in the past. $ 4. Discussion $ It is obvious, especially from the sensitivity tidal simulations, that the lunar $! distance would have been changing more slowly in the past than would be pre-$" dicted assuming modern dissipation rates. It has been suggested that the aver- The solid grey line shows lunar-recession assuming that tidal-dissipation equalled the present day dissipation in the past, whereas the black dotted line shows the lunar-separation history predicted by the full numerical model from Laskar et al. (2004). Note that the Laskar model is virtually identical to our curve, assuming PD tidal drag, but that the lower mean-drag shown in this paper gives a reduced separation in the past. % the tidal-lag must have an uncertainty of a factor of 2 or more. This conrms, % using a very dierent approach, suggestions about uncertainty in Milankovitch %! periods and cyclostratigraphy (Waltham, 2015). Furthermore, sensitivity simu-%" lations (not shown) with sea-level being 80m higher or lower in each time slice %# did not signicantly change the results, except for PD, when large shelf seas are %$ present and allowed to dry out or ood further (see Green and Huber, 2013, %% for a discussion). From these results it also appears that Earth is near a tidal and much cruder representations of varying boundary conditions (Hansen, 1982;' Webb, 1982;Kagan and Sundermann, 1996;Poliakov, 2005) is also noteworthy. ' This similarity conrms that the physics of tidal dissipation and the bulk vari-' ables that cause it to vary are robust and constrainable. tend to be non-unique. It has been suggested that a tidal dissipation value of ! approximately half of the modern rate characterized the past 3 Ma well (Lourens ! and Brumsack, 2001). This is in agreement with our results, but that study did ! not explore sensitivity to dynamical ellipticity. Signicant uncertainty remains Moon's recession rate was slower in the deep past than predicted using PD !" dissipation rates, supporting the old-age Earth-Moon model. Furthermore, our