Global Modeling of Internal Tides Within an Eddying Ocean General Circulation Model 0602435 N 73-8677-025

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The new concurrent simulations demonstrate that the wavenumber spectrum of sea surface height-a measure of the energy contained in different length scales-is dominated in some locations by internal tides and in others by mesoscale eddies.
Tidal kinetic energies in the new concurrent simulations compare well with those in current-meter observations, as long as sufficient spatial averaging is performed.
The new concurrent simulations are being used in the planning of future-generation satellite altimeters, in the provision of boundary conditions for coastal ocean models, and in studies of ocean mixing.

TIdeS aNd GlOBal TIde MOdelING
Ocean tides result from the difference in the gravitational potential of the Moon and Sun across Earth's extent (Cartwright, 1999).The tidal potential, or equilibrium tide, has a simple shape first elucidated by Newton.Actual ocean tides represent a complex dynamical response, first elucidated in mathematical form by Laplace, to the equilibrium tidal forcing.Ocean tides are affected by several factors, including Earth's rotation, friction with the seafloor, water depth (which controls the speed of shallow-water gravity waves), and the presence of continents, as well as seemingly exotic effects such as tides in the solid Earth and perturbations in the gravitational self-attraction of mass in the ocean and solid Earth brought about by tidal motions (Hendershott, 1972).
The shallow-water equations used by modern scientists to model ocean tides are essentially updated versions of the Laplace tidal equations, in which the effects described above are included.An important assumption in the shallowwater equations is that motions have horizontal scales that are large compared to the water depth and therefore lie in hydrostatic balance (Pedlosky, 1987;Vallis, 2006).
As has been recognized for some time (e.g., Wunsch, 1975) It is therefore thought that tides exert important controls on the stratification and circulation of the deep ocean (Munk and Wunsch, 1998).
Internal tides have much smaller horizontal scales than barotropic tides, and by definition can only be numerically modeled in a system with more than one layer in the vertical direction.Because both horizontal and vertical resolution must be increased, the computational effort required to model internal tides is commensurately greater than the effort required to model barotropic tides.One approach to this challenge is to model internal tides on a regional scale, which permits higher spatial resolution than is possible in global models.Carter et al. (2012, in this issue) provide a review of regional internal tide models.Because of the great computational expense involved in global internal tide models, the first papers on global modeling of internal tides are less than a decade old, and the number of published papers that focus on, or at least include, global models of internal tides is, to the best of our knowledge, still quite small (Arbic et al., 2004;Simmons et al., 2004;Hibiya et al., 2006;Simmons, 2008;Arbic et al., 2010;Niwa and Hibiya, 2011) studies, astronomical tidal forcing was the only forcing present in the model.
Due to the lack of atmospheric forcing, which sets the oceanic stratification, the oceanic stratification was assumed to be horizontally uniform.Oceanic stratification in subtropical regions was assumed to hold everywhere in the ocean and, as a result, the internal tides in polar regions in early global internal tide models were noted to be quite inaccurate (Padman et al., 2006).
As in many subdisciplines of physical oceanography, the study of tides has been revolutionized by the advent of satellite altimetry (Fu and Cazenave, 2001).From an orbit of approximately 1,300 km, satellite altimeters can measure sea surface heights averaged over an area of several square kilometers with an accuracy of about 1 cm.This accuracy implies that any oceanographic phenomenon with a sea surface height (SSH) signature, including tides, can be examined in satellite altimeter data.Global barotropic tide models that are developed from satellite altimeter data (e.g., Egbert et al., 1994;Ray, 1999) compare extremely well to independent observations such as tide gauges and bottom pressure recorders (Shum et al., 1997).
Because tides contribute about 80% of the SSH variance measured by altimeters, examination of nontidal oceanic motions in altimeter data can only take place after tides have been accurately removed from altimeter records (Le Provost, 2001).This illustrates a fundamental truth about the place of tides in physical oceanography: they are "signal" in many applications, and "noise" in many other applications.Satellite altimeters have also been instrumental in the study of internal tides.Although internal tides take on their greatest amplitudes at depth, their signature in altimeter data is large enough to be detectable (e.g., Ray andMitchum, 1996, 1997).seen in Figure 1b as small-scale "wiggles" in the lines of constant phase (white lines in Figure 1a,b) and as small-scale "dimples" in the amplitude contours of

GlOBal GeNer al cIrcul aTION MOdelING
With the notable exception of gravitationally forced ocean tides, the atmosphere provides the primary forcing heat between the atmosphere and ocean (Yu and Weller, 2007).Salinities in the upper ocean are set by the difference between evaporation and precipitation at the ocean surface (Yu, 2007;Schmitt, 2008).Because the buoyancy (density) of seawater at the ocean surface is controlled by temperature and salinity, the forcing described above is known as buoyancy forcing.The atmosphere forces the ocean through pressure loading (Ponte, 1993) and wind stress (Pond and Pickard, 1983) in addition to the buoyancy forcing.Wind stress at the ocean surface drives near-inertial motions (Alford, 2003;Simmons and Alford, 2012, in this issue), Ekman flows (Pedlosky, 1996), and the large-scale general circulation (Pedlosky, 1996).
Winds blowing on the sea surface also drive oceanic motions on small scales (scales from about 1 cm to about 1 km), such as capillary waves (Cox, 1958), surface wind waves (Phillips, 1966), and Langmuir cells (Langmuir, 1938).These small-scale motions, however, are not resolved in models such as HYCOM, which make the hydrostatic approximation (Pedlosky, 1987;Vallis, 2006).
The oceanic general circulation consists of the well-known current systems around the globe, such as the Gulf Stream in the western North Atlantic Ocean, the Kuroshio in the western North Pacific Ocean, the Agulhas Current off of Southern Africa, and the Antarctic Circumpolar Current in the Southern Ocean (Schmitz, 1996a,b;Siedler et al., 2001).On timescales of about 10-200 days, these currents meander and generate highly energetic mesoscale eddies (Schmitz, 1996a,b;Stammer, 1997), the spinning oceanic dynamical counterparts of atmospheric weather systems generated through a hydrodynamical process known as baroclinic instability (Pedlosky, 1987;Vallis, 2006).
Because currents meander and mesoscale eddies are born and die over relatively long timescales (scales much longer than, for instance, tidal timescales), these currents and eddies are sometimes referred to as low-frequency motions.
A substantial percentage of the kinetic energy in the ocean resides in mesoscale eddies and other transient low-frequency features such as current meanders.
Because the horizontal length scales of strong currents and mesoscale eddies are relatively small (about 100 km), numerical models of the ocean must have high resolution in order for the modeled low-frequency kinetic energies to compare well to those recorded in observations from satellite altimeters and other instruments (Hurlburt and Hogan, 2000;Maltrud and McClean, 2005).The grid spacing in current state-of-the-art global high-resolution ocean models is about 1/10° to 1/12°, which is thought to be sufficient to capture most of the oceanic mesoscale.Indeed, ocean models having resolutions this high or higher are known as "eddying" models (Hecht and Hasumi, 2008).Note, however, that Thoppil et al.Using the new concurrent tide model, we have begun to examine a host of scientific and operational questions about internal tides.In the following sections, we give brief overviews of some of our preliminary results of this exploration.

GlOBal cOMparISON OF MOdeled INTerNal TIdeS WITh SaTellITe alTIMeTer eSTIMaTeS
We have begun to validate the internal tides in our model with a global comparison to satellite altimeter observations.
As in the previous section, we focus here on M 2 , the largest constituent of oceanic tides.We apply a 50-400 km band-pass filter to the M 2 amplitudes at the sea surface in order to remove both smallhorizontal-scale noise (in the altimeter to the surface M 2 elevations.Figure 3a displays results from along-track satellite altimeter data (Ray andMitchum, 1996, 1997;Richard Ray, NASA, pers. comm., 2011), while Figure 3b  TIdal aNd NONTIdal cONTrIBuTIONS TO The Sea SurFace heIGhT SpecTruM We have begun to use our model to distinguish between the high-frequency tidal and low-frequency nontidal contributions to quantities of interest such as the wavenumber spectrum of sea surface height.Because the satellite altimeter repeats tracks every 10 days, high-frequency motions such as the tides are aliased into longer periods (Parke et al., 1987).In satellite altimeter data, therefore, it is sometimes difficult to distinguish between tidal motions and lower-frequency motions.Because our model output is written at hourly intervals, we can easily separate low-from high-frequency signals in the model.
In Figure 4 et al., 2008;Xu and Fu, 2011).In regions where low-frequency motions are more energetic than tidal motions, such as the Kuroshio (see westernmost box in Figure 2b), low-frequency motions dominate the spectrum of total SSH (Figure 4a).However, in regions where internal tides are energetic, for instance near Hawai'i (see easternmost box in Figure 2b), high-frequency tidal motions dominate the high end of the wavenumber spectrum (Figure 4b).The latter observation implies that in data taken by the planned high-resolution wide-swath satellite altimeter (Fu and Ferrari, 2008), internal tides will have to be removed very accurately before low-frequency oceanic motions can be studied.Tides have always been an important source of noise in altimeter measurements of SSH.
However, internal tides will become a more important source of noise for the wide-swath altimeter mission than for previous altimeter missions.This issue is due to the much higher spatial resolution of the wide-swath mission, which will collect data at horizontal scales where the internal tides contain substantial energy.

GlOBal cOMparISON OF MOdeled TIdal curreNTS WITh curreNT MeTer OBSerVaTIONS
We have begun to compare the tidal cur-

IMplIcaTIONS aNd FuTure deVelOpMeNTS
The development of a model with barotropic and internal tides embedded within an eddying ocean general circulation has opened up a host of scientific questions and applications, some of which are described in earlier sections of this paper.As discussed, the model is expected to be useful in planning for the next-generation wide-swath satellite altimeter (Fu and Ferrari, 2008).expected to be useful for regional and coastal modeling.A number of improvements are needed for the new concurrent tide model.The self-attraction and loading term (Hendershott, 1972) is implemented in a rather crude way, using the so-called scalar approximation (Ray, 1998), and it needs to be improved upon in future versions of the model.The frequency dependence of topographic wave drag (Bell, 1975) is problematic because the model is run in the time domain, not the frequency domain.
Perhaps the greatest technical challenge remaining is how to perform data assimilation on the tidal as well as nontidal motions within the model.Successful c I a l I S S u e O N I N T e r N a l WaV e S B y B r I a N K .a r B I c , J a M e S G .r I c h M a N , J ay F. S h r I V e r , paT r I c K G .T I M K O , e .J O S e p h M e T z G e r , a N d a l a N J .Wa l l c r a F T aBSTr acT.Ocean tides, and the atmospherically forced oceanic general circulation and its associated mesoscale eddy field, have long been run separately in high-resolution global models.They are now being simulated concurrently in a high-resolution version of the HYbrid Coordinate Ocean Model (HYCOM).The incorporation of horizontally varying stratification with the addition of atmospheric forcing yields internal tides (internal waves of tidal frequency) in high-latitude, low-stratification regions that are qualitatively different from those in earlier global internal tide models, in which atmospheric forcing and horizontally variable stratification were absent.The internal tides in the new concurrent HYCOM simulations compare well with those measured in along-track satellite altimeter data.
were barotropic, meaning that the relatively small variations in ocean density from the top to the bottom of the water column were ignored.In a shallow-water model, the assumption of constant density implies that the horizontal velocities do not change with vertical position in the water column, though they do change with latitude and longitude.The spacing between adjacent horizontal grid points in early global tide models was about 6° (670 km).This relatively coarse grid spacing was dictated by the limited computer power available at the time.
, tidal flow over topographic features generates an internal tide.In a stratified fluid, the vertical motions induced by flow over topography take the form of internal waves (waves along interfaces between fluids of differing densities), which are the subject of this special issue of Oceanography.Internal tides are internal waves with tidal frequencies.Internal tides can have displacement amplitudes greater than 50 m and current speeds greater than 2 m s -1 .Internal tides are a topic of great current interest, for a number of reasons.The vertical motions (displacements) associated with internal tides affect the motions of gliders, floats, and other oceanographic instruments.They also impact the motions of submarines and acoustic (sound) waves in the ocean.Most importantly, the breaking of internal waves is a leading source of mixing, and internal tides are thought to be one of the largest energy sources powering mixing in the deep ocean.
Barotropic tides generate internal tides, and internal tides in turn feed back onto the barotropic tides.Inferences from altimetry-constrained barotropic tide models show that about one-third of global tidal energy dissipation occurs in regions of rough topography, where internal tides are generated (Egbert and Ray, 2000).Internal tide generation thus acts as a damping mechanism for the barotropic tides.

Figure 1a ,
Figure1a,b shows the surface elevation amplitude of the principal lunar semidiurnal tide M 2 , the largest tidal constituent in the ocean.Figure1adisplays the M 2 amplitude in TPXO(Egbert et al., 1994), a highly accurate altimetryconstrained barotropic tide model.

FigureFigure 1 .
Figure 1b displays the M 2 amplitude in the new concurrent HYbrid Coordinate Ocean Model (HYCOM) simulations described in this paper.Although the HYCOM M 2 amplitudes are clearly very similar to those in TPXO, they are less accurate due to the lack of constraints applied using satellite altimetry.(Thus far, our HYCOM simulations are "forward" simulations, that is, simulations unconstrained by data.)Another key difference is that the HYCOM simulations include internal tides, which can be

Figure 1b .
Figure 1b.The presence of internal tide perturbations to amplitude and phase in the new concurrent HYCOM simulation, and the lack of these perturbations in TPXO, is more easily seen by contrasting the insets to Figure 1a,b.The insets focus on the western Pacific, a region of strong internal tides.
of large-scale motions in the ocean (motions on scales from 1 km up to basin scales).For instance, exchanges between the ocean and atmosphere at the sea surface exert a strong control on ocean stratification.Seawater is stratified by density, and seawater density is a function of temperature, salinity, and pressure.Temperatures in the upper ocean (and lower atmosphere) are determined by fluxes of sensible and latent

(
figure displays the amplitude of the M 2 internal tide signature in the steric sea surface height (i.e., the sea surface height component having to do with stratification effects).Figure 2a, taken from a HYCOM simulation like those in earlier global internal tide studies-using a twolayer, horizontally uniform stratification typical of the subtropics-displays an artificially strong internal tide in highlatitude regions such as the Labrador Sea and Drake Passage.In the new concurrent HYCOM simulations with tides embedded within an atmospherically forced model (Figure 2b), this defect is removed.Figure 2b also demonstrates that the introduction of a more realistic 32-layer horizontally varying stratification leads to increased internal tide amplitudes at most of the major generation sites in the mid-and low-latitude oceans, when compared to the internal tides generated in a two-layer horizontally uniform stratification.See, for instance, the area around Hawai'i.
data) and large-scale barotropic tides (in the altimeter data and in the HYCOM simulation).The band passing therefore leaves only the low-mode internal tide remaining.Figure 3 displays the results of applying the band-pass filter Figure 3b, and found that the modeled rms values agree with the altimeter rms we show the wavenumber spectrum of SSH from the model.The wavenumber spectrum measures the energy content of a signal as a function of wavenumber, which is inversely proportional to the wavelength.The slope of the wavenumber spectrum is of great theoretical interest, as it is used to infer the dominant dynamics of low-frequency flows (e.g., Le Traon

Figure 4 .
Figure4.Wavenumber spectrum of total (black), high-frequency (red) and low-frequency (blue) sea surface height, where two days is the dividing point between low and high frequencies, for (a) Kuroshio region, a region of very strong low-frequency motions, and (b) region north of hawai'i, in which low-frequency motions are weaker than M 2 internal tidal motions.The two regions are delineated by the westernmost and easternmost boxes, respectively, in the North pacific in Figure2b.Spectra are computed from a hycOM simulation containing both tides and low-frequency atmospherically forced motions.extra green lines are drawn in at the best-fit slopes of -4.38 and -3.87 over lowfrequency motions in the same wavenumber band discussed inXu and Fu (2011).as discussed in leTraon et al. (2008) andXu and Fu (2011), -11/3 and -5 slopes imply surface quasi-geostrophic and quasigeostrophic dynamics, respectively.

Figure 5 .
Figure 5. (a) horizontal locations of current meters used to validate the tidal currents in the hycOM simulations.units of x-and y-axes are degrees.(b) Vertical distributions of the M 2 tidal kinetic energy, spatially averaged over depthbinned instruments, for current meter observations ("obs.")and hycOM simulations ("model").
Regional and coastal models often include tidal forcing, but this forcing is generally barotropic, such that internal tides are not included in the open boundary conditions.Kelly and Nash (2010) showed that the response of shelf tides to barotropic plus internal (i.e., fully three-dimensional) tidal forcing can be quite different from the response to forcing that includes only barotropic tides.The Australian North West Shelf is a region where a low vertical-mode internal tide is expected to propagate onto the shelf from the open ocean (Sam Kelly, University of Western Australia, pers.comm., 2012).Current meter observations (Holloway et al., 2001) show propagation of a mode-1 internal tide onto the shelf.The global model at 4 km resolution underestimates the strength of the internal tide compared to the observations, but shows onshore propagation from deep water onto the shelf consistent with the observations.However, a regional model forced by barotropic tidal boundary conditions shows onshore propagation of the internal tide only inshore of the critical slope, and offshore propagation over the outer continental slope, in disagreement with the observations.By including tidal forcing in a fully threedimensional global ocean circulation model, we will provide an internal tide capability everywhere, and allow nested models to include internal tides at their open boundaries.
data assimilation, once performed, will make the model much more useful for operational purposes of the United States Navy and other users.acKNOWledGeMeNTSWe thank two anonymous reviewers whose comments led to several improvements and clarifications in this manuscript.We thank Richard Ray for providing results from a global harmonic analysis of along-track satellite altimetry data, used in Figure 3.We thank Robert Scott for providing the current meter database used in Figure 5. BKA and 0487.JGR, JFS, EJM, and AJW were supported by the project "Eddy resolving global ocean prediction including tides" sponsored by the Office of Naval Research under program element number 0602435N.This is NRL contribution NRL/JA/7320-12-1135 and has been approved for public release.