Laurence Padman, ESR  
email: padman@esr.org  
 
Lana Erofeeva, OSU  
email: serofeev@coas.oregonstate.edu  

This modeling effort is a collaboration between ESR (L. Padman) and Oregon State University (G. Egbert and S. Erofeeva). We have created two models: the Arctic Ocean Dynamicsbased Tide Model (AODTM5) and the Arctic Ocean Tidal Inverse Model (AOTIM5). Each is described below. The models are coded on a 5km cartesian grid, and contain tide height and velocity information for 8 constituents: 4 major tides (M_{2}, S_{2}, K_{1}, O_{1}); and 4 lesser tides (N_{2}, K_{2}, P_{1}, Q_{1}).
The models are available to the public. Two access packages are provided: a Fortranbased package; and a Matlabbased GUI (Graphical User Interface) which also includes batch processing scripts ("Mfiles") for many common processes such as extracting amplitude and phase fields for a specific tidal harmonic, and making predictions. For most users we expect that the inverse model AOTIM5 will be the more useful model, since its accuracy is improved through assimilation of coastal and benthic tide gauges, plus TOPEX/Poseidon and ERS satellite radar altimetry. Thus, at this time the AOTIM5 model is the only one available at this site. However, users interested in AODTM5 can obtain the model by contacting L. Padman (ESR).
A PDF version of a paper describing these models is also available.
This work was funded by the National Science Foundation grant OPP0125252 (ESR) and NASA JASON1 grant NCC5711 (OSU). We appreciate the assistance of A. Proshutinsky and G. Kivman in developing the data base of Arctic tide gauge harmonic constants used in this study.
If you experience problems with this page, or need further information, contact Laurie Padman (ESR).
It is now widely recognized that tidal currents contribute significantly to general patterns of hydrography and circulation in the world ocean through their effect on mixing in the ocean interior [e.g., Munk and Wunsch, 1998; Wunsch, 2000; Egbert and Ray, 2000]. This also holds true in highlatitude seas. Padman [1995] summarizes many of the processes by which tides contribute to the largerscale structure of the Arctic Ocean. These include elevated mixing in the benthic layer and also in the ocean interior through baroclinic tide generation, and ocean influences on the concentration and properties of sea ice leading to modifications to oceanatmosphere exchanges of heat and fresh water. Kowalik and Proshutinsky [1994] (hereinafter denoted KP94) used a barotropic tidal model (14km grid) with simple coupling of an ice cover to demonstrate that the increased openwater ("leads") fraction associated with tides provided a significant component of the total oceantoatmosphere heat loss in winter and a corresponding increase in net ice growth, locally between 10 and 100 cm per year due to tides alone. This process has been further discussed (for the Antarctic) by Padman and Kottmeier [2000]. Another consequence of tides was suggested by Robertson et al. [1998], who noted that increased benthic friction associated with tidal flow could significantly lower the net windforced transport of the Weddell Gyre. Strong tidal flows may also, however, contribute to "mean flow" through tidal rectification effects; see, e.g., Loder [1980], Padman et al. [1992], and Padman [1995].
The model domain (Figure 1) uses the International Bathymetric Chart of the Arctic Ocean ("IBCAO") [Jakobsson et al., 2000] digitized on a uniform 5km grid. The domain includes all of the central Arctic Ocean, the Greenland Sea, the Labrador Sea and Baffin Bay, and the Canadian Arctic Archipelago ("CAA"). There are two features of this domain that will be relevant to further discussions. First, the Arctic Ocean is dominated by the broad continental shelf seas of the eastern (or Eurasian) Arctic. Second, the passages of the CAA are narrow, and the flow of tidal energy through these passages (including Nares Strait at the northern end of Baffin Bay) critically depends on adequate resolution of these passages. As with other tide modeling efforts, the quality of dynamicsbased models depends to a large extent on the accuracy of the bathymetry grid.
From previous studies we have identified 310 coastal and benthic tide gauge records in the model domain that provide tidal coefficients for at least a few of the most energetic tidal harmonics [KP94; Tidal tables, 1941, 1958]; for locations, see Figure 1. Additional data are available from the T/P and ERS satellite radar altimeters: T/P measures sea surface height (SSH) for icefree ocean to ~66^{o}N, while ERS measures SSH for icefree ocean to ~82^{o}N. The T/P orbit was specifically chosen to allow tides to be accurately measured [Parke et al., 1987]; however, individual ERS height measurements are of lower accuracy, the orbit is unfavorable for resolving solar constituents (e.g., S_{2} and K_{1}), and the higherlatitude data must frequently be discarded because of ice cover within the radar's footprint. Nevertheless, for some constituents the ERS data set provides useful information about ocean tides above the T/P turning latitude.
The tide gauge data set ("TG") has been divided into 7 regions (see Figure 1) in order for us to present general comparisons between various tide models and data. This division is roughly based on the amplitudes of semidiurnal and diurnal tides.
The 5km Arctic Ocean Dynamicsbased Tide Model (AODTM5) is the numerical solution to the shallow water equations (SWE), which are essentially linear. Following Egbert and Erofeeva [2002] (hereinafter denoted EE), we solve the SWE by direct matrix factorization for 8 tidal constituents: M_{2}, S_{2}, N_{2}, K_{2}, K_{1}, O_{1}, P_{1}, and Q_{1}. Potentially significant simplifications to the SWE dynamics include our use of: tidal loading and self attraction computed from a global model ( TPXO6.2 Global Tide Model); and the use of linear benthic friction to parameterize the dissipation term. We use a constant friction velocity value of 0.5 m s^{1} for semidiurnal constituents and 2 m s^{1} for diurnals. The larger value for diurnal constituents is required because their currents are frequently strongest in relatively deep water along the shelf break. We assume that the errors introduced by the simplified dynamics can be corrected by the data assimilation. The AODTM5 uses elevations taken from the latest Ό degree global solution ( TPXO6.2 as open ocean boundary conditions. Additional forcing is provided by the specified astronomical tide potentials.
The 5km Arctic Ocean Tidal Inverse Model (AOTIM5) was created following the data assimilation scheme described by Egbert et al. [1994], hereinafter denoted EBF], Egbert [1997], and EE. Only the 4 most energetic tides (M_{2}, S_{2}, O_{1}, and K_{1}) were simulated with AOTIM5: for prediction purposes we use N_{2}, K_{2}, P_{1}, and Q_{1} from AODTM5. Assimilated data consists of coastal and benthic tide gauges (between 250 and 310 gauges per constituent), and 364 cycles of T/P and 108 cycles of ERS altimeter data from a modified version of the "Pathfinder" database [Koblinsky et al., 1999] with no tidal corrections applied [B. Beckeley, personal communication, 2003]. We used T/P altimeter data for 11178 data sites with a spacing of ~7 km, and ERS altimeter data for 18224 data sites, illustrated in Figure 1. Data assimilation was done for 4 major constituents only: TG+T/P+ERS for M_{2} and O_{1}, TG+T/P for K_{1} and TG only for S_{2}. These choices are based on the ability of the satellite data to resolve specific constituents depending on orbit characteristics; see, e.g., Parke et al., 1987, Smith [1999], and Smith et al. [2000].
The dynamical error covariance was defined following the considerations outlined in EBF, using the "prior" solution (AODTM5) to estimate the spatially varying magnitudes of errors in the momentum equations. The correlation length scale for the dynamical errors was set to 50 km (10 grid cells). The continuity equation was assumed to be exact.
To compute the inverse solution we used the singleconstituent reduced basis approach (EBF) to calculate the representer coefficients. The efficient calculation scheme described by EE was applied. Figure 2 shows elevations for the inverse solution for the M_{2}, S_{2}, K_{1} and O_{1} constituents. The most significant changes from the prior solution for the semidiurnals were in the Barents Sea near the entrance into the White Sea (amplitude changes >60 cm for M_{2} and >30 cm for S_{2}), in the White Sea (amplitude changes >40 cm for M_{2} and >10 cm for S_{2}) and in the northern part of Baffin Bay (amplitude changes >20 cm for M_{2} and >10 cm for S_{2}). The most significant changes from the prior solution for the diurnals were in the Baffin Bay and the Gulf of Boothia in the CCA (maximum amplitude changes ~20 cm for K_{1} and ~5 cm for O_{1}), in the Barents Sea near the entrance to the White Sea (maximum amplitude changes ~10 cm for both K_{1} and O_{1}), and in the Greenland Sea (maximum amplitude changes ~10 cm for K_{1} and ~5 cm for O_{1}).
Maps of tide height amplitude and phase for the 4 most energetic tides (Figure 2) are qualitatively similar to previously published maps (e.g., figures 25 in KP94). The M_{2} amplitude exceeds 1 m in the southern Barents Sea near the entrance to the White Sea, in the Labrador Sea, and at the northern end of Baffin Bay. The distribution of S_{2} amplitude is similar to M_{2}, but about a factor of 3 smaller. Diurnal tide amplitudes are largest in Baffin Bay and the Labrador Sea ("BBLS") and in the Gulf of Boothia in the southern CAA. Maximum K_{1} amplitudes are ~0.4 m, and about 0.2 m for O_{1}.
We calculate errors for each location averaged over the "inphase" and "quadrature" components for each constituent, i.e.,
(1) 
where are the measured harmonic constants and are the modeled harmonic constants for the constituent l at location x_{i}, and the sum is over the N data sites. For conciseness, we present averaged error values for the 7 subdomains described in section 2.2 (see Figure 1). Table 1 lists the value of RMS_{RI} for the comparison of TG data with 4 models; KP94, AODTM5, TPXO6.2, and AOTIM5. Note that both TPXO6.2 and AOTIM5 have assimilated these tide gauges, hence the fits in these cases represent the assigned uncertainty in data coefficients and the effect of other assumptions in the inverse calculation.
Averaged over the entire domain, AODTM5 performs significantly better than KP94 for the M_{2} tide, which contains most of the total tide height signal (see Figure 2). The values of RMS_{RI} averaged over the entire domain for S_{2}, K_{1}, and O_{1} are similar for both models. The improvement from KP94 to AODTM5 is most pronounced for M_{2} in the CAA and Nares Strait: we attribute this result primarily to the higher resolution (5 km for AODTM5 vs. 14 km for KP94) of the passages through the CAA and Nares Strait. The diurnal constituents are slightly worse in AODTM5 than in KP94 for the North Atlantic and Eastern Arctic seas, but are significantly better for the CAA, BBLS, and Nares Strait. Our studies indicate that dynamicsbased models of diurnal tides are sensitive to the choice of the boundary location south of Davis Strait: the location in AODTM/AOTIM is chosen to minimize RMS_{RI} in AODTM5 for the BBLS region.
As expected, the errors for the 2 inverse models TPXO6.2 and AOTIM5 are smaller than for either dynamicsbased model. For semidiurnal tides, AOTIM5 is a significantly better fit to the tide gauge data than is TPXO6.2. This occurs because the coastline of the Barents and White Seas are less well resolved in the Ό degree global model, semidiurnal tide elevations there approach 2 m, and more than 40% of all tide gauges are located in the Barents Sea. Furthermore, the decorrelation length scale for TPXO6.2 is 250 km, which does not allow as accurate a fit to closely spaced tidal data as can be achieved with the 50 km decorrelation length scale in AOTIM5. For diurnal tides, errors for the 2 inverse models are comparable throughout the domain.
The best way to look at the output from AOTIM5 is to download the model and play with it. Two software bases are available: Fortranbased (OSU) and Matlabbased (ESR). However, maps of amplitude and phase of sea surface elevation for each tidal harmonic are available for a quick look.
While the new models have been validated against tide height data (see Table 1), most practical oceanographic interest in tides is related to the strength and gradients of tidal currents rather than height variations. We assume that the highresolution data assimilation model AOTIM5, which best fits the available tide height data (Table 1), also best represents tidal currents. This is a reasonable assumption since the inverse model is consistent with the shallowwater wave equations to within the assumed accuracy of the bathymetrybased and dissipation terms. For each grid node in the model domain we calculate the timeaveraged speed based on all 8 modeled constituents:
(2) 
In (2), i=1,2,...T is the summation is of hourly currents over a time interval of 14 days, which encompasses the beat periods of the 4 major constituents. The components of the current vector
are calculated as:
(3) 
and α ^{l}(t) gives the slowly modulated periodic ("nodal": ~18.6 y period) time variations for constituent l (for details see, e.g., EE).
The map of ū (Figure 3) is comparable to the map of "maximum tidal current" u_{max} plotted by KP94 (their Plate 1): in general, u_{max}≈ ū . The largest currents are over the broad and shallow Eurasian shelf seas, with typical values for ū are ~515 cm s^{1}. Values of ū >80 cm s^{1} are found in the western Barents Sea south of Bear Island [Kowalik and Proshutinsky, 1995] and in the southern Barents Sea near the entrance to the White Sea. Strong currents are also found in Davis Strait in the Labrador Sea, and in Nares Strait and various locations within the CAA. Currents are weak over the deep basins and along the northern coast of Alaska.
We have ignored the possible effects of seaice cover in the present models. KP94 note that sea ice may change tidal amplitudes by up to 10% and phases by 12 h, presumably leading to quasiseasonal variability in tidal coefficients.
The inverse model (AOTIM5) is the most accurate Arctic tide model available at this time, as judged by comparisons with tide gauge data and satellite altimetry. Nevertheless, the longterm goal will be to develop dynamicsonly models with comparable accuracy, through further improvements in resolution, addition of seaice interactions, and more sophisticated dissipation parameterizations including benthic friction and the conversion of barotropic tidal energy to internal tides.
Instructions for accessing the model are provided here.
M_{2}  KP94  AODTM  TPXO6.2  AOTIM  S_{2}  KP94  AODTM  TPXO6.2  AOTIM 
All  25.4  19.2  19.6  8.5  All  9.4  9.2  6.8  2.3 
1  14.5  13.9  11.4  9.2  1  7.5  5.7  7.8  2.4 
2  34.8  26.9  28.0  10.4  2  11.6  13.2  8.9  2.7 
3  9.1  8.9  6.8  4.0  3  4.4  4.6  3.9  1.7 
4  9.0  9.5  10.9  8.8  4  3.4  3.1  4.1  1.2 
5  20.3  6.2  5.6  3.7  5  7.1  2.6  3.2  1.8 
6  17.7  14.4  13.1  11.6  6  14.8  6.4  5.0  2.7 
7  9.2  4.1  4.6  3.4  7  9.7  2.2  3.2  2.1 
K_{1}  KP94  AODTM  TPXO6.2  AOTIM  O_{1}  KP94  AODTM  TPXO6.2  AOTIM 
All  6.3  5.7  3.9  2.5  All  3.0  3.0  1.8  1.7 
1  2.1  4.5  2.3  2.8  1  1.5  2.2  1.0  1.9 
2  4.9  6.9  3.7  3.8  2  2.2  4.5  1.8  1.6 
3  3.4  3.8  2.2  2.3  3  1.6  2.2  1.2  2.1 
4  2.5  4.2  3.3  1.5  4  8.2  3.1  1.4  1.7 
5  9.5  5.4  6.0  2.1  5  4.2  3.4  2.6  1.3 
6  13.7  4.4  3.6  2.9  6  6.7  4.4  1.7  1.6 
7  4.5  1.5  3.2  0.9  7  1.6  1.5  1.7  0.5 
Figure 1: Model domain, showing locations of tide gauge data (red squares), and ERS and TOPEX/Poseidon radar altimetry (magenta and yellow dots, respectively). The domain is partitioned into 7 regions for modeldata comparisons (see Table 1). These regions are: (1) North Atlantic; (2) Barents Sea (including the White Sea); (3) the Russian shelf seas (Kara, Laptev, East Siberian, Chukchi); (4) the northern coast of Alaska and the Canadian Northwest Territories; (5) the Canadian Arctic Archipelago ("CAA"); (6) Baffin Bay and the Labrador Sea ("BBLS"); and (7) Nares Strait. Click on the figure for a clearer (*.png) image.
Figure 2: Amplitude (color shading, in m) and phase (white contours) for the 4 most energetic tidal harmonics, M_{2}, S_{2}, K_{1}, and O_{1}, for the Arctic Ocean Tidal Inversion Model (AOTIM5). Peak M_{2} amplitudes are close to 1.8 m in the southern Barents Sea near the entrance to the White Sea. Click on the figure for a clearer (*.png) image.
Figure 3: Mean tidal current speed (cm s^{1}) based on simulating 14 days of hourly total tidal speed from the 8constituent inverse solution AOTIM5. The largest values exceed 80 cm s^{1} in the southern Barents Sea near the entrance to the White Sea and around Bear Island in the western Barents Sea south of Svalbard. Click on the figure for a clearer (*.png) image.