1. Introduction
Tides occur throughout the oceans as periodic oscillations in currents and sea surface height, typically with diurnal or semidiurnal time scales. The amplitude of the surface oscillations is about 50 cm over much of the open ocean and about 1 m along many coastlines, but local resonances can lead to tides of over 5 m in special coastal locations (e.g., Garrett 1972; Arbic et al. 2007). However, tidal amplitudes are sensitive to the frequency of the lunar and solar forcing (i.e., the length of the day), the ocean depth and density stratification, and the coastal configuration. Considerable changes in each of these over the history of the earth must have led to corresponding changes in ocean tides. For instance, modeling studies of the Last Glacial Maximum (LGM) have shown that the implied sea level drop of about 120 m leads to considerable amplification of semidiurnal tides (Thomas and Sündermann 1999; Egbert et al. 2004; Arbic et al. 2004b; Uehara et al. 2006; Griffiths and Peltier 2008; Arbic et al. 2008). Such changes can impact the climate system in a variety of ways.
Globally, of interest is the history of tidal dissipation, which is mainly associated with energy loss to a turbulent bottom boundary layer in shallow seas (Taylor 1920) and conversion to internal tides in the deep ocean (e.g., Garrett and Kunze 2007). Tidal dissipation implies a slowing of the earth’s rotation rate and a corresponding recession of the moon, variations in which can be traced back over two billion years (Williams 2000). Furthermore, the portion of energy converted to internal tides is thought to play an important role in vertical mixing of the deep ocean (e.g., Wunsch 2000). Changes to this mixing rate could lead to changes in ocean circulation and climate, as discussed by Wunsch (2005), Munk and Bills (2007), and Montenegro et al. (2007). Tidal models, in which dissipation is parameterized, suggest that both internal tide dissipation and total dissipation were considerably higher at LGM than at present (Egbert et al. 2004; Uehara et al. 2006).
On a smaller scale, past variations in tidal amplitudes may have local significance. At high latitudes, recent attention has been focused on amplification of semidiurnal tides under glacial conditions in the Labrador Sea (Arbic et al. 2004b, 2008) and the Arctic Ocean (Griffiths and Peltier 2008), and how these large tides might have interacted with adjoining ice streams and shelves to cause rapid climate change events. On the northwest European shelf, postglacial changes in tidal amplitudes, currents, and mixing were examined by Uehara et al. (2006). They, and others (e.g., Shennan and Horton 2002), have shown that changes in tidal amplitudes over the Holocene need to be accounted for when interpreting sedimentary records used to construct relative sea level history. We note that a corresponding analysis of changes in tidal range is needed for the east coast of North America, where there is disagreement between predictions and observations of relative sea level in the interval 5000–10 000 yr before present (Peltier 1998, his Fig. 23).
Here we present new simulations of global tides at LGM. At this time, ice sheets covered much of northern Europe and Canada, and the changes to the ocean depth and coastal configuration were the most extreme over recent history. Using a range of datasets and numerical models, several studies have suggested that there were also significant changes to the tides. Although the likely pattern of the semidiurnal tides at LGM has now been established, the studies of Egbert et al. (2004) and Arbic et al. (2004b, 2008) did not resolve the Arctic Ocean (for numerical reasons), that of Uehara et al. (2006) did not include a full treatment of tidal self-attraction and loading, while that of Thomas and Sündermann (1999) was performed at a low resolution, which required the use of nonphysical damping. We avoid all of these shortfalls by using the modeling strategy applied in our recent study (Griffiths and Peltier 2008). As there, we use the most recent reconstruction of the LGM topography, the ICE-5G model of Peltier (2004), rather than the older ICE-4G model (Peltier 1996) used by most previous studies. Furthermore, our internal tide drag scheme uses a model-derived estimate of ocean stratification at LGM, rather than an ad hoc estimate. This is perhaps the first such estimate of LGM ocean stratification.
Although our solutions are global, we focus on changes in the polar tides. These changes are driven by the different polar coastal configuration at LGM, at which time ice sheets occupied the marginal seas of the present-day polar oceans. Polar tides have been neglected in recent high-resolution studies, some of which were unable to resolve the Arctic Ocean for numerical reasons, yet they are of special interest because of their interaction with adjoining ice shelves and ice streams. Furthermore, although the global sea level change at LGM is now well constrained by various data (Peltier and Fairbanks 2006), there remains uncertainty about the location and extent of marine ice sheets and shelves. Here, we shall explore the sensitivity of the polar tides to changes in these ice margins. Our simulations are ideally suited for this purpose, because they are performed on a global Mercator grid with enhanced resolution at high latitudes.
In the first half of our study we describe our modeling strategy in some detail. Our numerical global tide model is described in section 2. Key components are a new parameterization of internal tide drag, and a model of ocean stratification at LGM, which is required by this scheme. In section 3, we describe the global properties of the modeled tides, for present day and LGM, and examine their dependence upon model resolution and the internal tide drag scheme. In the second half of our study, we discuss polar tides. In section 4 we examine in detail the significant changes at LGM around the Antarctic coastline, and in section 5 we briefly discuss the changes around the Arctic coastline, extending the results of Griffiths and Peltier (2008). We conclude in section 6, by discussing the possibility of an interesting interplay between climate (which influences ice sheet extent) and polar tides (which influence ice sheet stability and the ocean circulation).
2. A global tide model
a. Numerical implementation
We use a spherical coordinate system with the model pole running through 75°N, 40°W (Greenland) and its antipode (Antarctica). We make a Mercator transformation within this rotated system, replacing latitude θ with a Mercacor latitude τ defined by tanh τ = sinθ. Solving on a regular Arakawa C grid in longitude and Mercator latitude τ, the effective latitudinal resolution increases poleward consistent with the convergence of meridians. At a typical resolution used here, with 720 points in each direction, the grid spacing smoothly varies from about 55 km in the tropics to 5 km around the coast of Greenland and in the Ross Sea.
Each simulation is performed for a single tidal constituent of frequency ω. To reduce the excitation of transients, which is particularly important when modeling diurnal tides, the tidal forcing is premultiplied by a term 1 − exp(−t2/t02), with t0 = 4 days. Over the nth tidal period, a harmonic analysis is performed, yielding a tide ζ(n)(x, t) = Re[ζ̂(n) (x) exp (−iωt)]. The integration is terminated when equilibrium is reached, typically after about 20 days, when the global average of ||ζ̂(n)| − |ζ̂(n−1)|| is a few millimeters.
b. Bathymetry
For the present-day simulations, H is constructed from a modified version of the 2-minute gridded elevations/bathymetry for the world (ETOPO2) dataset. For LGM, H is interpolated from the ICE-5G v. 1.3 dataset for 26 000 yr before present, which is based on the same 2-min topography but accounts for both the water taken up by the ice sheets, and the deformation of the earth beneath the ice sheets. This is an updated version of the widely used ICE-5G v 1.2 dataset (Peltier 2004). The locations of the continents and ice sheets for both present day and LGM are shown in Fig. 1. At LGM there are significant changes in the coastal configurations of the Arctic, North Atlantic, and western Pacific Oceans.
In our tidal modeling, no account is taken of floating ice shelves such as those that exist around much of present-day Antarctica. These imply a reduction in the ocean depth and an additional drag on the flow due to a turbulent boundary layer at the water–ice interface. The omission of these effects does not appear to degrade the present-day global solutions, perhaps because the extent of the ice shelves is small. At LGM, the extent of ice shelves was presumably much larger than at present, but to what degree is not well known. We thus solely examine the effects due to sea level drop and changes in coastal geometry, which are well constrained, and that are likely to be responsible for the largest changes in the global tides.
c. Internal tide drag
This parameterization is similar to that used by Carrère and Lyard (2003) and Lyard et al. (2006), in that (i) DIT is proportional to u · ∇H, so that there is no drag for along-slope flow, and (ii) DIT is directed normal to the slope, consistent with the most basic physical relationship in (3). However, in contrast to their scheme and others, such as that of Jayne and St. Laurent (2001), (5) has no free parameters. We will discuss the significance of this in section 3c.
d. Stratification
To implement (5), estimates of the vertically averaged buoyancy frequency
However, it is also possible to estimate ocean stratification from suitably sophisticated coupled atmosphere–ocean climate simulations. Here we analyze output from the simulations of Peltier and Solheim (2004), which were performed for both present-day and LGM conditions. They ran the Climate System Model version 1.4 (CSM 1.4) to equilibrium over a period of 2000 yr, resolving the ocean with 25 vertical levels and a 116 × 102 latitude–longitude grid. Time-averaged temperature, salinity, and buoyancy frequency were calculated from the final 50 yr of these simulations. Latitude–height cross sections of these quantities, running through the Atlantic and then into the Arctic Ocean, are shown in Fig. 2.
The present-day simulation captures many features of the real ocean. For instance, note the thermocline at a depth of about 50 m, the formation of Antarctic bottom water (seen most clearly in the temperature), and the formation of Antarctic intermediate water (seen most clearly in the salinity). The most serious deficiency is a surface freshwater bias in the tropics, caused by a well-known drift in CSM 1.4. This typically leads to buoyancy frequencies which are too high relative to those calculated from the World Ocean Atlas dataset (Antonov et al. 2006; Locarnini et al. 2006), as will be discussed shortly.
At LGM, generally the ocean is cooler and saltier, with an estimated average increase in salinity of 3.7% due to the 3.6% reduction in the volume of the ocean. Nevertheless, the buoyancy frequency remains largely unchanged over much of the ocean. This is consistent with the results of Adkins et al. (2002), where deep salinity increases of 2.7 ± 0.1%, 3.3 ± 0.3%, 4.2 ± 0.2% and 6.9 ± 0.5% were reported at LGM. These increases need not imply a generally fresher upper ocean and larger stratification, since only in the final case is the salinity increase anomalously large. Two regions of the CSM 1.4 results deserve special mention. First, around Antarctica, there is increased bottom-water formation, driven by increases in sea ice. The enhanced vertical mixing associated with this leads to a more homogeneous water column, and lower buoyancy frequencies. Second, in the Greenland Sea and Arctic Ocean, the buoyancy frequency is higher than at present. This again appears to be driven by enhanced sea ice formation, which leads to an accumulation of salt in the deep Arctic Ocean. Unlike the Southern Hemisphere, this salty water cannot readily escape into the Atlantic because of the relatively shallow water around Iceland.
The vertically averaged buoyancy frequency
Although there are certain to be errors in these model-derived buoyancy frequencies, we apply a consistent approach by using these values for both the present-day and LGM tidal simulations. This is in contrast to previous studies, in which LGM stratification was typically set to present-day values, or to a multiple of present-day values. With the exception of some simulations presented in section 3c to illustrate model sensitivity, we adopt these model-derived buoyancy frequencies exclusively.
3. Global tides
We focus on the largest semidiurnal tidal constituent (M2, of frequency ω = 1.405 × 10−4 s−1) and the largest diurnal constituent (K1, of frequency ω = 7.292 × 10−5 s−1), which account for 68% and 11% of the tidal energy in the present-day ocean (Egbert and Ray 2003). The observed amplitudes of these components are shown in Figs. 4a,b, as described by the complex amplitude ζ̂tp of the TPXO 6.2 dataset, an updated version of the data-constrained solutions described by Egbert et al. (1994). As is well known, semidiurnal tides are large in the North Atlantic, while diurnal tides are large in the North Pacific, due to near resonances with the ocean depth and coastal configuration (e.g., Platzman 1984).
Corresponding numerically determined solutions for LGM are shown in Figs. 4e,f. Although the pattern of the M2 tide remains similar, the amplitude is typically enhanced relative to the present-day solution (such as around New Zealand and off the east coast of Brazil). The amplification in the North Atlantic has been modeled previously (Egbert et al. 2004; Arbic et al. 2004b), and discussed in some detail by Uehara et al. (2006) and Arbic et al. (2008). Later, we shall discuss the amplified M2 tide in the Weddell Sea, visible at about 50°W on the Antarctic coastline. Evaluating (6) for the LGM M2 tide, taking the modeled present-day tide for ζ̂ref gives
a. Effects of model resolution
As is evident from Figs. 4a–d, the present-day modeled tides are too energetic, at least relative to the observationally constrained TPXO 6.2 solutions. This effect can largely be attributed to model resolution. To illustrate this, we have calculated M2 and K1 tidal solutions at a range of model resolutions, with the number of longitudinal grid points varying from M = 270 to M = 1080. The calculated energies are shown in Figs. 5a,b, along with the values given by Egbert and Ray (2003) for corresponding TPXO solutions. One can see that the energy decreases as the model resolution increases, for both present-day and LGM simulations. Indeed, the energies of modeled present-day tides appear to be approaching the observed values, but even at M = 1080 the energy is overestimated by 50% for M2 and 30% for K1. Egbert et al. (2004) demonstrated that better agreement can be obtained at yet higher resolution (see their Fig. 2b), whether or not one is using an internal-tide drag scheme. Some related results are given in appendix A of Simmons et al. (2004).
It is not surprising that the solutions should become more accurate as the resolution increases. For instance, the accuracy of the finite-difference approximations will improve leading to a better treatment of highly sensitive near-resonant modes, and the model grid will better resolve key coastal features and channels. Such effects can occasionally lead to large improvements in the solution for only a small increase in resolution, such as apparently occurs for the LGM M2 tide around M = 450. However, it is somewhat surprising that the energy should always be overestimated at finite resolution. This behavior is consistent with the presence of a dissipative process within (1a) and (1b), which acts more effectively at high model resolution. We discount internal tide drag as being the dominant such process, since E also falls with resolution when DIT is omitted from the formulation. Other possibilities are (i) small-scale bottom roughness and coastal roughness, which leads to a greater scattering of the tide as the resolution increases, reducing its coherency and energy; and (ii) bottom friction, which might become more effective as shallow regions and narrow channels with vigorous flow are better resolved. Either way, note that a higher effective drag need not lead to a higher dissipation D, for the reasons explained by Platzman (1984, his Fig. 1) and Egbert et al. (2004, their Fig. 11). Thus, even though D may increase with resolution in some cases, in others it may fall (e.g., Fig. 5c, modern) or level off (e.g., Fig. 5d, modern).
b. Energetics
For the present-day M2 tide, as the resolution increases D appears to be approaching the 2.44 TW quoted by Egbert and Ray (2003), as shown in Fig. 5c. At M = 1080, we have DBL = 1.90 TW and DIT = 1.38 TW, so that 42% of the dissipation is accounted for by internal tide drag. A slightly lower amount (36%) of the dissipation occurs in waters deeper than 1000 m, consistent with the 32% quoted by Egbert and Ray (2003). In contrast, for the K1 component, the present-day dissipation appears to have converged to about the correct value even at the low resolution of M = 360. At M = 1080, we have DBL = 0.20 TW and DIT = 0.16 TW, so that 44% of the dissipation is accounted for by internal tide drag. Here 39% of the dissipation occurs in the waters deeper than 1000 m, which is considerably larger than the 11% quoted by Egbert and Ray (2003).
At LGM, we find little change in the dissipation of the K1 constituent, consistent with the results of Egbert et al. (2004). However, the M2 tide appears to be approaching a state with D in excess of 4 TW, considerably higher than the 2.44 TW of the present-day ocean. This is consistent with the values estimated by Egbert et al. (2004) and Uehara et al. (2006). For both M2 and K1, the bulk of the dissipation is now accounted for by internal tide generation in the deep ocean. For M2, the proportion is 63%, while for K1 the proportion is 57%. In each case, this represents an approximate reversal of the dissipation split for the present-day modeled tides, consistent with the expectation that internal tide drag is the dominant dissipation mechanism at LGM. This energy is potentially available to drive enhanced vertical mixing in the deep ocean (e.g., Montenegro et al. 2007).
c. Internal tide drag: Sensitivity and tuning
As already noted in section 3a, at the resolutions typically used by global tide models, that is about 0.25°–0.5°, it seems that an untuned model will typically overestimate tidal energies. This problem is often “remedied” by increasing dissipation aphysically, for instance by adding and tuning a horizontal viscosity, as in Thomas and Sündermann (1999). Alternatively, Arbic et al. (2004a) demonstrated that errors in the present-day M2 tide can be minimized by increasing the drag coefficient cd in DBL. They found that cd = 0.64 was optimal for their model, and noted this as being unrealistically large. However, since horizontal viscosity has no physical basis here and cd is thought to be well-constrained, tidal solutions are instead often improved by increasing the parameterized internal tide drag DIT, perhaps simply by tuning a multiplicative constant (e.g., Jayne and St. Laurent 2001; Arbic et al. 2004a). It is then argued that such tuning merely corrects for deficiencies in the parameterization of internal tide drag, since the simple expressions used for DIT are crude and poorly constrained. Note that the optimal tuning is resolution dependent (Arbic et al. 2008).
We illustrate the consequences of this procedure by presenting additional simulations (at M = 720) with enhanced internal tide drag, for both present day and LGM. We first simply apply (5) globally, rather than in the restricted latitude band with | f| < ω. Physically, this might account for drag due to the generation of nonlinear internal waves or baroclinic edge waves, which can occur at high latitudes. As shown in Table 2, this only leads to changes in the K1 tide, with the M2 tide remaining almost unchanged for both present day and LGM. This is not unexpected, because the region of additional drag (where | f| > ω) is much larger for diurnal tides than for semidiurnal tides. For the present-day K1 tide, the additional drag does reduce the energy toward the observed value, and
In our second set of sensitivity simulations, we multiply the right-hand side of (5) by a factor of 4, which corresponds to increasing the parameter HIT in (4) from H/3 to 4H/3. Physically, this might account for an underestimate of the internal wave drag by our standard parameterization in (5). As shown in Table 2, this does lead to an improved present-day M2 solution, when measured in terms of energy and
Putting aside this problem of unrealistically large internal tide dissipation, such tuning is hard to justify in the context of the convergence demonstrated in Figs. 5a,b (and in Fig. 2b of Egbert et al. 2004). Furthermore, the amount of tuning required will clearly depend upon the tidal constituent, the configuration of the coastline, and the stratification of the ocean. Thus, it is dangerous to tune the internal tide drag to fit present-day tides and then to use the same parameterization to model tides at LGM. We prefer our modeled tides to be too energetic, and to interpret them in terms of the convergence shown in Figs. 5a,b. Furthermore, in this study we are most interested in tidal amplitudes, and a 30%–50% overestimate in energy implies that tidal amplitudes are only overestimated by 15%–20%.
The second set of sensitivity simulations, with DIT increased by a factor of 4, can alternatively be interpreted as results for an ocean with
4. Antarctic tides
a. The K1 tide
More detailed views of the modeled K1 tides around Antarctica (calculated using M = 720) are shown in Fig. 6. The modern-day amplitude (Fig. 6a) takes an approximately annular structure around the coastline, superposed with various other features around the shelf-break and in the semienclosed Ross and Weddell Seas. Along the coastline (Fig. 7), the annular structure has an amplitude of about 30 cm, while the Ross Sea tides reach an amplitude of about 75 cm. Snapshots of the tide separated by one-quarter of a tidal period (Figs. 6b,c) clearly show that the annular structure corresponds to a zonal wavenumber 1 Kelvin wave, propagating anticyclonically, as expected from standard theory (e.g., Gill 1982).
To apply this theory to Antarctica, we take H0 = 2000 m and θ = 70°, so that (10) implies ω1 = 6 × 10−5 s−1. Thus, the zonal wavenumber-1 Kelvin wave is close to resonance at diurnal frequencies, accounting for its strong signature in Figs. 6a–c. This same idea is implicit in the analysis of Platzman et al. (1981), who calculated that Antarctica has a normal mode of period 29 h, taking the form of a Kelvin wave with zonal wavenumber 1.
At LGM, the coastal margins of Antarctica were quite different, as ice sheets flowed out from the mainland to partially occupy the continental shelves. In the ICE–5G v 1.3 reconstruction, the ice sheets extend to approximately midshelf in both the Ross and Weddell Seas (cf. the black and blue areas shown in Fig. 8). As a result, the Antarctic coastline becomes considerably smoother. One would expect this to promote the free passage of Kelvin waves, since there will be less scattering from sharp coastal features, and shallow areas of strong dissipation are reduced in size. This expectation is confirmed by our simulations (Figs. 6d–f). At LGM the structure of the K1 tide is rather similar to that of the present-day tide, but there is an amplification around the entire continent (also see Fig. 7).
However, just how much of the continental shelves was occupied by grounded ice sheets at LGM is not precisely known. We have therefore explored other scenarios, as shown in Fig. 8, in which ice sheets extend farther out toward the continental shelf break. These configurations were generated by adding grounded ice to the ICE–5G v 1.3 topography within specified latitudinal ranges and for H < 700 m, with no isostatic adjustment. The most extreme configuration, denoted LGM 60 and with grounded ice poleward of 60°S wherever H < 700 m, resembles the maximum extent of grounded ice for which there is geological evidence (Anderson et al. 2002, their Fig. 13).
In this configuration the coastline is now rather smooth, with only the Antarctic Peninsula acting as a weak barrier to the Kelvin waves. As shown in Figs. 6g–i and 7, the K1 tidal amplitude is then quite uniform around the coastline, taking a value of about 50 cm. This represents a significant increase over the average amplitude of approximately 30 cm for the present-day configuration.
b. The M2 tide
In principle, the same ideas of Kelvin wave propagation around the Antarctic coastline also apply to semidiurnal tides. Taking H = 2000 m and θ0 = 70° as before, (10) implies ω2 = 1.3 × 10−4 s−1 and ω3 = 1.9 × 10−4 s−1, bounding the semidiurnal frequency. This is consistent with the results of Platzman et al. (1981), who reported Antarctic normal modes with zonal wavenumbers 2 and 3, and with periods of 15 and 10 h, respectively.
Shown in Fig. 9 are M2 solutions for the present-day, LGM, and the extended ice LGM 60 scenario. In all cases, the M2 tide certainly has a strong signature of zonal wavenumbers 2 and 3, but unlike the K1 tide, these do not take the simple form of propagating Kelvin waves. At these zonal wavelengths, the tide has the same scale as many of the coastal irregularities, such as the Weddell Sea. Kelvin waves of M2 frequency are therefore unable to propagate past such features without considerable interaction. The picture is further complicated by strong interactions with remote tides, including those of Patagonia and New Zealand. The latter are clear in the enhanced tidal amplitudes at about 170°W, extending all the way to 50°S.
A more detailed view of the present-day coastal tidal amplitude is shown in Fig. 10. There are two small regions of highly localized tides: one under the Amery ice shelf and another at the southern edge of the Ross Sea. The entire Weddell Sea also has amplified tides, taking the form of short wavelength Kelvin waves (Figs. 9b,c). The tidal amplitude there (shown in Fig. 9a) has the classic form for a rectangular gulf, originally calculated by Taylor (1921) and reproduced in Gill (1982).
In the standard LGM configuration, the M2 tide is amplified to over 4 m in both the Ross Sea and Weddell Sea. These amplifications appear to occur due to independent local resonances of the partially ice-filled seas. These local resonances lead to amplified tides along the rest of the coastline, where tidal amplitudes approach 1 m (Fig. 10). The amplification in the Ross Sea is highly sensitive to the coastal bathymetry, and even a small outward shift of the grounding line to the LGM 78 configuration (Fig. 8) considerably reduces this tide (Fig. 10). However, the tide in the Weddell Sea is more robust, and the grounding line must be advanced to the LGM 75 configuration for a corresponding reduction. In the extreme LGM 60 configuration, tidal amplitudes fall to less than 1 m along the entire coastline (Fig. 10), with remote interactions appearing to strongly influence the solution (Fig. 9g).
5. Arctic tides
Modeled K1 tides in the Arctic Ocean are small for both present day and LGM, as shown in Figs. 4b,d,f. Modeled present-day M2 tides are also small (Figs. 11a–c), although there are larger tides in adjoining seas. These results are consistent with existing models of Arctic tides (e.g., Kowalik and Proshutinsky 1994). In contrast, at LGM the Arctic Ocean has a strong M2 tide with an approximately annular structure (Fig. 11d). Snapshots of the tide separated by one-quarter of a tidal period (Figs. 11e,f) reveal this tide to be a zonal wavenumber-1 Kelvin wave, propagating cyclonically around the interior of the Arctic Basin. This tide is analagous to the K1 tide of Antarctica, which propagates anticyclonically around the continental exterior. Indeed, the Arctic tide can be described using the model of section 4a, by replacing the polar continent with a polar basin with depth of the form (8). Now, since the tide must decay poleward (away from the coastline), from (11) the admitted solution of (9) is proportional to A+(θ). As expected, this corresponds to an eastward-traveling wave, with the coast on the right (left) in the Northern (Southern) Hemisphere.
As explained in our recent study (Griffiths and Peltier 2008), this tide is made possible at LGM by the disappearance of the Barents Sea, the closure of the Bering Strait, and the elimination of the shallow seas around the Queen Elizabeth Islands, transforming the Arctic Ocean to a basinlike configuration. This alone does not guarantee a large-amplitude tide. However, the LGM Arctic Basin is near to resonance at semidiurnal frequencies: using (10) with H0 = 3000 m and θ0 = 80° implies ω1 = 1.5 × 10−4 s−1. Explicit calculation of normal modes of the LGM Arctic Ocean confirms that the zonal wavenumber-1 Kelvin wave is indeed close to resonance (Griffiths and Peltier 2008). Note that since the other Kelvin wave normal modes occur at higher frequencies ω2, ω3, … , the K1 tide cannot be near resonant in the same way.
As in Antarctica, the M2 tide is sensitive to the assumed shape of the ice margins. In the ICE-5G v. 1.3 reconstruction used thus far, the Innuitian ice sheet entirely covers the present-day Queen Elizabeth Islands (QEI), which lie between 75°–80°N and 90°–120°W. However, there is some doubt as to the exact position of the grounding line of this ice sheet at LGM, and at times some or all of the shallow water surrounding the present-day QEI would have been exposed (Dyke et al. 2002; England et al. 2006). Indeed, this is the configuration of the ICE-5G v 1.2 reconstruction, used in our previous study (Griffiths and Peltier 2008). The modeled M2 tides for this topographic reconstruction are shown in Figs. 11g–i. The character of a zonal wavenumber-1 Kelvin wave is still evident, but there is a second feature that is massively amplified around the QEI, with tides in excess of 8 m (Fig. 12), as shown in Griffiths and Peltier (2008).
We now examine the sensitivity of these Arctic tides in more detail, by considering configurations where the grounding line of the Innuitian ice sheet is taken to lie between the extremes of Figs. 11d,g. These configurations were generated by adding grounded ice to the ICE-5G v 1.2 topography between 90° and 120°W and for H < 700 m, with no isostatic adjustment. The grounding line was taken at either 76°, 77°, 78°, or 79°N, corresponding to a northward advance of approximately 100 km in each case. The coastal amplitudes of the modeled tides are shown in Fig. 12. Note how the amplitude of the tides around the QEI initially increases (to over 11 m) as the grounding line moves northward, but decreases dramatically when taken beyond 77°N. In the final configuration, with grounded ice to 79°N, the tide has an approximately annular structure, with an amplitude of about 1 m for much of the Arctic coastline. This is similar to the tide shown in Fig. 11d, obtained using the ICE-5G v 1.3 topography.
6. Discussion
Our results suggest that polar tides would have been considerably larger at LGM than during the present day. These changes are mainly driven by the sea level drop of about 120 m and the advance of ice sheets onto the continental shelves, leading to quite different coastal configurations at the poles (e.g., Figs. 8 and 11). In our standard model, using the new ICE-5G v 1.3 topographic reconstruction, Antarctic semidiurnal tides are amplified to more than 4 m in the Ross and Weddell Seas due to a near-resonance when the continental shelf is approximately half occupied with grounded ice. Both Antarctic diurnal tides and Arctic semidiurnal tides are amplified by a similar Kelvin wave resonance, which is strong because of the effective length of the coastline and coherent because the coastlines are relatively smooth during glacial conditions. This leads to amplified tides around the entire coastline, in excess of 50 cm for Antarctica and 1 m for the Arctic.
All of these amplitudes are sensitive to the assumed margins of the grounded polar ice sheets, the locations of which would have been changing continuously (e.g., Tarasov and Peltier 2004), and whose precise extent at any given time is not well known. We have illustrated how even small shifts of the grounding lines can lead to large changes in tidal amplitudes at both poles (Figs. 7, 10, and 12). When the Innuitian ice sheet is appropriately extended, our analysis predicts megatides with amplitudes of about 10 m in the Canadian Archipelago, consistent with our previous study (Griffiths and Peltier 2008). Because of great uncertainty in the extent of fringing ice shelves, we have not considered their possible impact. However, the implied reduction in water depth would alter wave propagation characteristics and impact tidal amplitudes to some degree.
The precise amplitude of these modeled tides must be treated with some caution. First, there are uncertainties in our assumptions about the LGM ocean, relating to the depth and coastal configuration, and the model-derived stratification (required for the internal tide drag scheme). Second, there are uncertainties associated with our modeling approach. For instance, nonlinear momentum advection (presently neglected) may play a moderating role where large amplitude tides are highly localized, while interactions between tidal constituents are neglected and the form of the prescribed internal tide drag is approximate. Finally, as discussed in section 3a, there are uncertainties associated with an overestimate of tidal amplitudes at finite model resolution. Although this can be remedied by artificially enhancing the internal tide drag, we rather anticipate a 10%–20% reduction in modeled tidal amplitudes as the resolution increases. Since the spatial pattern of the modeled tides is not resolution dependent, this does not influence our main conclusions that polar tides were (i) amplified at LGM and (ii) highly sensitive to the location of ice sheet grounding lines. Such behavior is expected not only at LGM, but also during other glacial periods of the Late Pleistocene.
a. Tides and polar climate
Tides impact polar climate in a variety of ways. The generation of internal waves, which can break locally leading to vertical mixing, modifies the temperature and salinity of the water column. Sea ice distribution is influenced by such temperature changes, and by direct fracturing and movement by horizontal tidal currents. Accounting for both of these processes, Holloway and Proshutinsky (2007) demonstrated a role for tides in the sea ice and heat budgets of the Arctic Ocean. Robertson et al. (1998) concluded that tidal currents and mixing strongly influence the circulation and heat flux in the Weddell Sea, and thus the formation of Antarctic Bottom Water, which in turn is coupled to the global ocean circulation. Our results suggest that large polar tides under glacial conditions would have led to greater vertical mixing of the water column than at present, impacting such processes. Since the strongest mixing would occur equatorward of 74° [see the comments leading to (5)], this conclusion does not apply in the high Arctic, or in the Ross and Weddell Seas.
Tides also influence grounded marine ice streams and floating ice shelves. Buckling and crevassing of ice shelves by vertical tidal motions causes them to weaken, perhaps directly leading to their collapse, or preconditions them for other destabilization mechanisms (e.g., Hulbe et al. 2004). The flow of ice sheets is modified by tidal processes close to the grounding line, with the effects being felt far inland, as recorded in present-day Antarctica (e.g., Bindschadler et al. 2003; Gudmundsson 2006). Such effects would have been stronger during glacial times, because of the greater extent of fringing ice and the larger tidal amplitudes predicted by our analysis.
Given the sensitivity of tides to the extent of marine ice streams (and shelves), there is the possibility of an interesting coupling between ice stream extent and tides. For instance, consider the advance of an ice stream onto a continental shelf, in a location where this causes tidal amplitudes to increase at the grounding line. (This would occur between Figs. 9a,d, for semidiurnal tides in the Ross and Weddell Seas.) Eventually, tidal amplitudes could become large enough to cause the breakup of a connected ice shelf, or to lead to enhanced calving at the grounding line. This might result in a steady state, with the tide effectively determining the ice shelf extent or grounding line location.
Alternatively, the large-amplitude tides might completely destabilize the marine ice sheet, leading to a coastward retreat of the grounding line and the discharge of ice into the ocean. Although the mechanism for such a tidally induced instability remains unknown, such a scenario is possible given the sensitivity of ice streams to sea level at the grounding line (e.g., Schoof 2007). This could naturally lead to a cyclical behavior of ice stream advance onto the continental shelf followed by a tidally induced collapse, with the time scale between events being simply that for the regeneration of a sufficiently large marine ice sheet.
Such behavior is reminiscent of the episodic periods of instability suffered by the Laurentide ice sheet during the late Pleistocene, known as Heinrich events (e.g., Hemming 2004). These events, during which large volumes of ice were discharged into the North Atlantic from close to the mouth of the present-day Hudson Strait, are associated with rapid climate change. The much smaller Innuitian ice sheet appears to have suffered a similar episodic instability, leading to ice discharge into the Arctic Ocean from the Queen Elizabeth Islands (Darby et al. 2002). In this scenario, the large-amplitude tides of the Labrador Sea are implicated in the collapse of the Laurentide ice sheet (Arbic et al. 2004b, 2008), while the megatides around the Queen Elizabeth Islands are implicated in the collapse of the Innuitian ice sheet (Griffiths and Peltier 2008).
Under glacial conditions, our tidal model predicts large tides around the Antarctic coastline, with particular amplifications in the Ross and Weddell Seas. If tides are implicated in the Northern Hemisphere instability events, it is therefore natural to ask whether the large marine ice sheets of Antarctica suffered similar episodic events of catastrophic instability. Unfortunately, data from around Antarctica and the Southern Ocean is much more sparse than that from the North Atlantic, so that relatively little can be inferred about Antarctic ice sheet stability. Although there is evidence of ice discharge events (e.g., Kanfoush et al. 2002) reaching as far away as New Zealand (Carter et al. 2002), it is not clear that these are a southern analog of Heinrich events (Ó Cofaigh et al. 2001). Nevertheless, Kanfoush et al. (2000) argued for the occurrence of episodic instability of grounded ice sheets in the Weddell Sea, with a similar time scale (of about 10 000 yr) to Heinrich events. Our analysis predicts large tides (with amplitudes in excess of 3 m) in the Weddell Sea under glacial conditions (Figs. 9d and 10).
An important aspect of these ice-discharge events is their apparent entrainment to particular phases of the Dansgaard–Oeschger oscillation (e.g., Sakai and Peltier 1998). Heinrich events were apparently entrained to cold phases (e.g., Hemming 2004), while Kanfoush et al. (2000) suggested that the Antarctic ice-discharge events were entrained to warm phases. Note that a period of relative warmth or cold could lead to a small migration of the grounding line, which, as shown here, could lead to a significant change in tidal amplitude for a sufficiently large ice sheet. Thus, a tidally induced destabilization mechanism would be sensitive to climate conditions and is thus consistent with the timing of past ice-discharge events.
Acknowledgments
This paper is a contribution to the work of the Polar Climate Stability Network, which is funded by the Canadian Foundation for Climate and Atmospheric Science, and by a consortium of Canadian universities. Additional support has been provided from NSERC Discovery Grant A9627. Thanks to Guido Vettoretti for preparing the CSM 1.4 dataset for our use and to Rosemarie Drummond for preparing the ICE-5G v 1.3 bathymetry.
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Surface height (m) and ice location of the ICE-5G v 1.3 reconstruction for (left) present day and (right) LGM.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Temperature, salinity, and buoyancy frequency N of the Atlantic and Arctic Oceans according to the results of Peltier and Solheim (2004). The longitudinal positioning of the cross section is shown by the dashed line in Fig. 3. Quantities are displayed on a nonuniform vertical and latitudinal grid, which is that used by the ocean model of Peltier and Solheim.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Vertically averaged buoyancy frequency
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Tidal amplitudes for the (left) M2 and (right) K1 constituents from (top) the TPXO 6.2 dataset, (middle) the present-day model with M = 1080, and (bottom) the LGM model with M = 1080.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
The effects of model resolution (measured by the number of longitudinal grid points M) on global energy E, total dissipation D, and response quality Q = ωE/D.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
The K1 tides (calculated at M = 720) for the (top) present-day model, (middle) LGM model, and (bottom) LGM 60 model with grounded ice occupying the entire continental shelf. (middle), (right) Instantaneous snapshots of the tide separated by one-quarter of a tidal period.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
The K1 tidal amplitude around the Antarctic coastline, modeled with M = 720. Each line gives the amplitude for a different ice margin with the colors corresponding to the configurations shown in Fig. 8.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Antarctic ice margins. Shown are the present-day model (black), the standard LGM model from ICE-5G v 1.3, and three models with extended ice margins.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
The M2 tides (calculated at M = 720) for the (top) present-day model, (middle) LGM model, and (bottom) LGM 60 model with grounded ice occupying the entire continental shelf. (middle), (right) Instantaneous snapshots of the tide separated by one-quarter of a tidal period.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Modeled M2 tidal amplitudes (at M = 720) around the Antarctic coastline. Each line gives the amplitude for a different ice margin with the colors corresponding to the configurations shown in Fig. 8.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
The M2 tides (calculated at M = 720) for the (top) present-day model, (middle) standard LGM model using ICE-5G v 1.3, and (bottom) LGM model using ICE-5G v 1.2. (middle), (right) Instantaneous snapshots of the tide separated by one-quarter of a tidal period.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Modeled M2 tidal amplitude (at M = 720) around the Arctic coastline, for simulations based on the ICE-5G v 1.2 bathymetry. The longitudes given are approximate, because the coastline winds back on itself in several places. The line color indicates the extent of grounded ice around the Queen Elizabeth Islands.
Citation: Journal of Climate 22, 11; 10.1175/2008JCLI2540.1
Horizontally averaged
Sensitivity of modeled tides to the internal tide drag scheme for both present-day (MOD) and LGM simulations at M = 720. Given are the global energy E, the parameterized boundary layer dissipation DBL, internal tide dissipation DIT, total dissipation D, and average global error