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 Dynamics-based Tide Model (AODTM-5) and the Arctic Ocean Tidal
Inverse Model (AOTIM-5). Each is described below. The models are coded on a
5-km 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 Fortran-based package; and a Matlab-based GUI (Graphical User Interface) which also includes batch processing scripts ("M-files") 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 AOTIM-5 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 AOTIM-5 model is the only one available at this site. However, users interested in AODTM-5 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 OPP-0125252 (ESR) and NASA JASON-1 grant NCC5-711 (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 high-latitude seas.
*Padman* [1995] summarizes many of the processes by
which tides contribute to the larger-scale 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
ocean-atmosphere exchanges of heat and fresh water.
*Kowalik and Proshutinsky*
[1994] (hereinafter denoted KP-94) used a
barotropic tidal model (14-km grid) with
simple coupling of an ice cover to demonstrate that the increased open-water
("leads") fraction associated with tides provided a significant component of the total
ocean-to-atmosphere 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 wind-forced 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 5-km 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 dynamics-based 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 [KP-94;
*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 ice-free ocean to
~66^{o}N, while ERS measures SSH for ice-free 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 higher-latitude 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 5-km Arctic Ocean Dynamics-based Tide Model (AODTM-5) 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 semi-diurnal 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
AODTM-5 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 5-km Arctic Ocean Tidal Inverse Model (AOTIM-5) 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 AOTIM-5: for prediction purposes we
use N_{2}, K_{2}, P_{1}, and Q_{1} from AODTM-5.
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 (AODTM-5) 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 single-constituent 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 semi-diurnals 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 2-5 in KP-94). 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 "in-phase" 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; KP-94, AODTM-5, TPXO6.2, and AOTIM-5. Note that both
TPXO6.2 and AOTIM-5 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, AODTM-5 performs significantly better
than KP-94 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 KP-94 to AODTM-5 is most pronounced
for M_{2} in the CAA and Nares Strait: we attribute this result
primarily to the higher resolution (5 km for AODTM-5 *vs*. 14 km for
KP-94) of the passages through the CAA and Nares Strait.
The diurnal constituents are slightly worse in AODTM-5 than in KP-94 for the
North Atlantic and Eastern Arctic seas, but are significantly better for the
CAA, BBLS, and Nares Strait. Our studies indicate that dynamics-based 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 AODTM-5 for the BBLS region.

As expected, the errors for the 2 inverse models TPXO6.2 and AOTIM-5 are smaller than for either dynamics-based model. For semidiurnal tides, AOTIM-5 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 AOTIM-5. For diurnal tides, errors for the 2 inverse models are comparable throughout the domain.

The best way to look at the output from AOTIM-5 is to download the model and play with it. Two software bases are available: Fortran-based (OSU) and Matlab-based (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 high-resolution data assimilation model AOTIM-5, 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 shallow-water wave equations to within the assumed accuracy of the bathymetry-based and dissipation terms. For each grid node in the model domain we calculate the time-averaged 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 KP-94
(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 ~5-15 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 sea-ice cover in the present models. KP-94 note that sea ice may change tidal amplitudes by up to 10% and phases by 1-2 h, presumably leading to quasi-seasonal variability in tidal coefficients.

The inverse model (AOTIM-5) is the most accurate Arctic tide model available at this time, as judged by comparisons with tide gauge data and satellite altimetry. Nevertheless, the long-term goal will be to develop dynamics-only models with comparable accuracy, through further improvements in resolution, addition of sea-ice 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.

- Egbert, G.D., and S.Y. Erofeeva. Efficient inverse modeling of barotropic ocean tides, J. Atmos. Ocean. Tech., 19, 2, 183-204, 2002.
- Egbert, G.D., Tidal data inversion: Interpolation and Inference. Progress in Oceanography, 40, 53-80, 1997.
- Egbert, G.D., Bennett, A.F., and M.G.G. Foreman, TOPEX/POSEIDON tides estimated using a global inverse model, J. Geophys. Res., 99, 24821-24852, 1994.
- Egbert, G. D., and R. D. Ray, Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data, Nature, 405, 775-778, 2000.
- Jakobsson, M., N. Cherkis, J. Woodward, B. Coakley, and R. Macnab, A new grid of Arctic bathymetry: A significant resource for scientists and mapmakers, EOS Transactions, American Geophysical Union, v. 81, no. 9, p. 89, 93, 96, 2000.
- Koblinsky, C.J., Ray, R.D., Beckeley, B.D., Wang, Y.-M., Tsaoussi, L., Brenner, A., and R. Williamson, NASA ocean altimeter Pathfinder project report 1: Data processing handbook, NASA/TM-1998-208605, 1999.
- Kowalik, Z., and A.Y. Proshutinsky, The Arctic Ocean Tides, in The Polar Oceans and Their Role in Shaping the Global Environment, Geophysical Monograph 85, edited by O. M. Johannessen, R. D. Muench, and J. E. Overland, AGU, Washington, D. C., pp. 137-158, 1994.
- Kowalik Z. and A. Yu. Proshutinsky. Topographic enhancement of tidal motion in the western Barents Sea. J. Geophys. Res., 100, 2613-2637, 1995.
- Loder, J. W., Topographic rectification of tidal currents on the sides of Georges Bank, J. Phys. Oceanogr., 10, 1399-1416, 1980.
- Munk, W., and C. Wunsch, Abyssal recipes II: Energetics of tidal and wind mixing, Deep-Sea Res., 45, 1977-2010, 1998.
- Padman, L., Small-scale physical processes in the Arctic Ocean, In: Arctic Oceanography: Marginal Ice Zones and Continental Shelves, Coastal and Estuarine Studies, 49, edited by W. O. Smith and J. M. Grebmeier, pp. 97-129, AGU, Washington, D.C., 1995.
- Padman, L., and C. Kottmeier, High-frequency ice motion and divergence in the Weddell Sea, J. Geophys. Res., 105, 3379-3400, 2000.
- Padman, L., A. J. Plueddemann, R. D. Muench, and R. Pinkel, Diurnal tides near the Yermak Plateau, J. Geophys. Res., 97, 12,639-12,652, 1992.
- Parke, M. E., R. H. Stewart, D. L. Farless, and D. E. Cartwright, On the choice of orbits for an altimetric satellite to study ocean circulation and tides, J. Geophys. Res., 92, 11,693-11,707, 1987.
- Robertson, R.A., L. Padman, and G.D. Egbert, Tides in the Weddell Sea, in Ocean, Ice, and Atmosphere: Interactions at the Antarctic Continental Margin, Antarctic Research Series 75, edited by S. S. Jacobs and R. F. Weiss, AGU, Washington DC, 341-369, 1998.
- Smith, A. J. E., Application of Satellite Altimetry for Global Antarctic Tide Modeling, PhD Thesis, Delft University press, Delft, The Netherlands, 182pp, 1999.
- Smith, A. J. E., B. A. C. Ambrosius, and K. F. Wakker, Ocean tides from T/P, ERS-1, and GEOSAT altimetry, J. Geodesy, 74, 399-413, 2000.
- Tide Tables: Gidrographic Board of VMF of USSR, Tidal tables (in Russian), vol. 2, Harmonic constants for tide prediction, Leningrad, 1941.
- Tidal Tables, Vol. 1: Waters of European part of the USSR, part 2: Tidal harmonic constants, (in Russian), Leningrad, Gidrometizdat, 1958.
- Wunsch, C., Moon, tides and climate, Nature, 405, 743-744, 2000.

_{2},
S_{2}, K_{1} and O_{1}, evaluated with (eq. 1).
The 4 models are dynamics-only models by *Kowalik and
Proshutinsky* [1994] (KP94) and the Arctic Ocean Dynamics-based Tide Model
(AODTM: this paper), and inverse models TPXO6.2 (global solution) and the
Arctic Ocean Tidal Inverse Model (AODTM). Results are presented for the
entire domain (All), and for the 7 subdomains shown in Figure 1.

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 model-data 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.

**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
(AOTIM-5). Peak M_{2} amplitudes are close to 1.8 m in the
southern Barents Sea near the entrance to the White Sea.

**Figure 3:** Mean tidal current speed (cm s^{-1}) based on
simulating 14 days of hourly total tidal speed from the 8-constituent
inverse solution AOTIM-5. 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.