Baroclinic and Barotropic Tides in the Ross Sea
Robin Robertson
Lamont-Doherty Earth Observatory
Columbia University
Palisades, New York 10964
USA
Office (845) 365- 8527
fax (845) 365- 8157
[email protected]
Revised and Resubmitted to Antarctic Science
July 16, 2003
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Abstract
The barotropic and baroclinic tides in the Ross Sea were simulated using a primitive equation, sigmacoordinate model, the Regional Ocean Model System (ROMS), for four tidal constituents, M2 , S2 , K1 , and
O1 . Small elevation amplitudes were predicted over most of the basin, with a combined standard
deviation of 30-50 cm. Larger amplitudes, with standard deviations ranging from 50-70 cm, occurred
deep within the ice shelf cavity, over the continental slope, and over Iselin Bank. Most of the elevation
response was associated with the diurnal constituents (K1 and O1 ), as was most of the depth-independent
(barotropic) velocity response. Baroclinic tides were generated at locations of steep topography for the
semidiurnal constituents, but not the diurnal. Diurnal continental shelf waves were generated by the
diurnal tides and found to amplify the semidiurnal elevations and baroclinic tidal velocities over the
continental slope. Comparisons with observations in both elevation and velocities showed very good
agreement for the semidiurnal constituents (M2 and S2 ) and moderate agreement for K1 and O1
constituents. The disagreement for the diurnal constituents was associated with diurnal frequency
continental shelf waves, which were overexcited along the shelf break. The baroclinic tides induced both
small-scale horizontal and vertical shear in the velocity fields in the Ross Sea.
Key Words: Tides, Ross Sea, Internal tides
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
1. Introduction
Recent studies indicate that much of the mixing energy in the deep oceans and along the continental
slopes originates from tides either through internal tides or boundary layer dynamics (Munk & Wunsch
1998; Garrett 2003). Tidal mixing over rough topography in the deep ocean is estimated to account for ~
1 TW (terawatt; 1 TW=1012 W) of energy due to the M2 constituent alone (Egbert & Ray 2001).
How do the tides influence the circulation and mixing? Tides affect the circulation directly through
increased boundary stress, which both retards the mean flow through additional friction and increases
mixing through additional shear in the benthic boundary layer. Tides influence circulation along the shelf
break and over rough topography through the excitation of continental shelf waves (CSW), which are also
referred to as topographicly trapped waves. Interactions between the barotropic tide and topography,
particularly the continental slope, ridges, and other rough topographic features, generate internal tides.
The internal tides affect the velocity structure and induce shear in the water column, which can promote
mixing through shear instabilities in low gradient Richardson number environments. Additionally, under
the sea ice in the polar regions, tides influence the dynamics through modifications in the sea ice structure
and through the additional friction in the surface boundary layer at the ice-ocean interface. The sea ice
structure is affected by tides, since tides induce lead formation in the ice due to convergence and
divergence of the velocities (Nansen 1898; Robertson et al., 1998).
These effects on circulation and mixing also affect the heat transfer both between the atmosphere and
the ocean and within the ocean. In the Antarctic seas, mixing has the potential to affect the heat transport
from the Warm Deep Water to the surface ice and influence the ice cover (McPhee et al., 1996; Muench
et al., 2002; Beckmann et al., 2001). A strong front exists at the continental shelf break in the Antarctic
Seas where Warm Deep Water and shelf waters meet and the deep waters are formed. The cross-slope
exchange in the Ross Sea from tidal mixing and other processes is being investigated by the Antarctic
Slope Initiative (AnSlope), an observational program, in order to determine the relative contributions of
the different mechanisms to the cross-slope exchange (A. Gordon, personal communication). Mixing
along the continental slope, including that induced by internal tides, can affect the formation of deep and
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
bottom waters and their ventilation. Tidally induced divergence in the surface velocities increases lead
formation and heat transfer to the atmosphere (Padman & Kottmeier 2000; Robertson 2001b).
Although it is known that tides affect the circulation and mixing, their exact contribution to the
processes is unknown. In order to understand and quantify the tidal contributions to mixing, lead
formation, and other processes in the Ross Sea, the structure of the tides must be known. Since only
sparse tidal observations are available in the Ross Sea (numbered sites in Figures 1b and 1c), models are
being used to fill in data gaps and provide estimates of the tidal structure. Barotropic tides in the Ross
Sea have been modeled for multiple tidal constituents by MacAyeal (1984) and Padman and associates
(Padman & Kottmeier 2000; Padman et al., 2002, Erofeeva et al. 2004).
The vertical structure of baroclinic tides, however, has only been simulated for one semidiurnal
constituent (M2 ) (Robertson et al. 2003). The goal of this project was to extend this modeling effort to
the four major constituents and develop a description of the three-dimensional structure of the tides in the
Ross Sea. For this study, the Regional Ocean Modeling System (ROMS), described in Sectio n 2, was
used. In Section 3, the model results are discussed and compared with existing observations. A summary
is given in Section 4.
2. Model Description
The ROMS model in this application is described elsewhere (Robertson et al., 2003; Robertson
2004). ROMS had been modified previously to include the presence of a floating ice shelf (Robertson et
al. 2003), including both mechanical and thermodynamic effects following Robertson (1999; 2001a) and
Hellmer & Olbers (1989), respectively. Here only the pertinent details and recent modifications will be
discussed. Recently, improvements have been made in the vertical mixing parameterization for primitive
equation models. Different vertical mixing parameterizations available in ROMS were evaluated using a
sensitivity study at a site with extensive hydrographic, velocity, and turbulence observations (Robertson
2004). For vertical mixing, the Generic Length Scale (GLS) parameterization of Umlauf & Burchard
(2003) was found to best replicate the observed velocities and vertical diffusivities and was used for these
simulations. Tidal forcing was implemented by setting elevations along all the open boundaries, with the
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
coefficients taken from a global, two-dimensional inverse model (TPXO6.2: Egbert and Erofeeva 2003).
Four major tidal constituents were used, two semi-diurnals, M2 and S2 and two diurnals, K1 and O1 .
The model domain (Figure 1) covered the Ross Sea with a grid spacing of 0.045o in latitude, 0.186o in
longitude (~5 km), and 24 vertical levels. The ice thickness and water column thickness, defined as the
distance between the bottom and the ice shelf base (or ocean surface), were taken from BEDMAP (Lythe
et al. 2000). In the northern region beyond the extent of BEDMAP, the water column thickness was
taken from Smith & Sandwell (1997). No smoothing was performed on either the water column or the
ice shelf thicknesses. The minimum water column thickness was set to 75 m, which entailed artificially
deepening a small portion of the region under the ice shelf, particularly near the eastern grounding line.
The barotropic and baroclinic mode time steps were 6 s and 180 s, respectively, and the simulations were
run for 30 days, with hourly data from the last 15 days saved for analysis. The kinetic and potential
energies stabilized after around 15 days, thus the first 15 days of simulated data was discarded.
Initial potential temperature, ?, and salinity, S, fields were assembled from the World Ocean
Circulation Experiment (WOCE) Hydrographic Programme Special Analysis Center (Gouretski & Janke
1999), except under the ice shelf. Here, the hydrography was essentially a four layer system based on a ?
and S profile through the ice shelf at a single location (J9 (Jacobs 1989)). The uppermost layer was 40 m
thick, with the potential temperature equal to the freezing point of water at the depth of the ice shelf base
and S =34.39 psu. The second layer was 100 m thick with the potential temperature 0.06o C warmer than
the freezing point of water for the depth. ? increased linearly between this value and that of the
lowermost layer within the 50 m thick third layer. The lowermost layer encompassed the remaining water
column (deeper than 190 m below the ice shelf base) with ? =-1.87o C and S =34.83 psu. S increased
linearly between values in the uppermost and the lowermost layers over the 150 m of the middle two
layers. The model elevation and velocities were initialized with the geostrophic velocities associated with
the initial hydrography as determined from a simulation without tidal forcing.
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Elevation, ? , and depth-independent velocities, U2 and V2 , are produced by the two-dimensional
(barotropic) mode of the model and ?, S, and depth-dependent velocities, U3 , V3 , and W3 , by the threedimensional (baroclinic ) mode. The depth-dependent velocities fully represent the velocity at each grid
cell and do not have the depth-independent velocity removed. Foreman’s tidal analysis routines were
used to analyze these fields (Foreman 1977; 1978).
3. Results
3.1 Elevations and Depth-Independent Velocities
3.1.1 Semidiurnal constituents (M2 and S2 )
The elevation amplitude response was similar for the two semidiurnal constituents, M2 and S2 . Their
prominent feature was an amphidromic point located in the western Ross Sea under the ice shelf at ~ 178o
W, 81o S (Figures 2a and 2b). These amphidromic points result from destructive interference of the tidal
Kelvin wave, which is ~180o out of phase on opposite sides of the ice shelf cavity. The locations are
dependent on the relative size of the basin, its water column thickness, and the tidal wavelength
(MacAyeal 1984). The semidiurnal tidal elevation amplitudes in the Ross Sea were generally small, ? 10
cm, although values > 30 cm were reached for both semidiurnal constituents in the tongue in the
southeastern portion of the ice shelf cavity (Figures 2a and 2b).
The model performance was evaluated by comparing the modeled elevation amplitudes and phases
with those of the existing observations. Unfortunately, the observations are not well distributed
throughout the region, but are concentrated at the ice shelf (locations are indicated by site numbers in
Figure 1b). With one exception for M2 and four exceptions for S2 , the model elevation amplitudes
matched the observations within the 2 cm observational uncertainty (Williams & Robinson 1980) (Table
1). The phase agreement was not as good, with only four of eleven locations agreeing within the
observational uncertainties for each semidiurnal constituent (Table 1). The phase changes rapidly in the
vicinity of the amphidromic point and small errors in the location of this point can result in large phase
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
errors. The errors in the phase are also larger when the amplitudes are small, since the phase calculation
becomes less accurate as the ratio of small numbers.
It is believed that most of the differences between the observations and the model estimates are a
result of topographic errors. The bathymetry and ice shelf thickness under the Ross Ice Shelf are not well
known and the model is quite sensitive to the topography. For an example of the flaws in the topography,
it was necessary to shift the position of two of the elevation observation points to the nearest submerged
grid cell, since the observation location was on land in the model topography. One of these locations was
site 1, with the largest differences between the modeled elevations and the observation. Similar
topographic dependence has been previously seen in model results (MacAyeal 1984; Robertson et al.
1998; (CATS2000) Padman et al. 2002).
The model estimates were compared to the results of two different two-dimensional tidal models
(MacAyeal 1984; (RossTIM) Erofeeva et al. 2004) (Tables 1 and 2). A description of the model used for
the CATS simulations is given in Robertson et al. (1998). The agreement between the observations and
ROMS for both semidiurnal constituents was equivalent to CATS02.01 (grid spacing: variable in
longitude and ~9 km in latitude) and better than MacAyeal’s model (grid spacing: ~10 km in longitude
and ~10 km in latitude). ROMS had lower rms differences (Table 2) and better phase agreement for S2
than the other two models (Table 2). The differences between the model responses are attributed to two
factors 1) ROMS is a three-dimensional model and the other two models are two-dimensional, and 2) the
models use different topographies.
The major axes of the tidal ellipses of the semidiurnal constituents for the depth-independent
velocities were quite small, generally < 5 cm s-1 . Major axes > 5 cm s-1 occurred at places along the
continental shelf break and at the entrance to the tongue in the southeast corner of the ice shelf cavity
(Figures 3a and 3b). The large major axes under the ice shelf are a result of the restricted topography in
that corner of the domain. Those along the shelf break correspond to the harmonics of continental shelf
waves (CSWs). At the shelf break, the diurnal constituents excite CSWs along the continental slope.
Since the semidiurnal constituent frequencies are at or near the first harmonic of the diurnal constituents,
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
interactions between the CSW and variations in topography transfer energy from the diurnal tides to their
first harmonic, which is interpreted as semidiurnal tides in the analysis. This can be seen by the local
regions of slightly larger major axes, >5 cm s-1 , along the continental slope and over Iselin Bank in
Figures 3a and 3b, which are present when four tidal constituents were used for forcing. These regions
are not present or are diminished (Iselin Bank) when forced by only the semidiurnal constituent M2
(Figure 3e). Again the topography was the primary controlling factor for the depth-independent
velocities, as has been previously noted for a two-dimensional tidal model (Robertson et al. 1998).
3.1.2 Diurnal Constituents (K1 and O1 )
The tidal elevation response of the diurnal constituents, K1 and O1 , was quite different than the
semidiurnal response. The diurnal constituent responses were westward propagating Kelvin waves
without amphidromic points (Figures 2c and 2d, respectively). Large elevation amplitudes, > 40 cm,
occurred in the ice shelf cavity, particularly in the narrow tongue in the southeastern corner of the ice
shelf cavity, over Iselin Bank, and along the continental slope. The large amplitudes in the slope regions
are a result of the generation of continental shelf waves (CSWs). Along the continental slope, there are
regions of alternating high and low elevation amplitudes and velocities, which correlate to the
predominant locations of crests and troughs of the CSWs.
Continental shelf waves are excited by interactions of currents with topographic irregularities
(Huthnance 1978; Brink 1991). Excellent summaries of their properties are given by Huthnance (1978)
and Brink (1991). They are coastally trapped waves with sub-inertial frequencies (Huthnance 1978;
Brink 1991). For the Ross Sea, CSWs are excited by the diurnal tides, but not the semi-diurnal tides. In
the southern hemisphere, CSW phase propagates westwards and the group velocity (energy) propagates
eastwards (Huthnance 1978; Middleton et al., 1982; Brink 1991). In weak stratification, they are
essentially barotropic (Huthnance 1978). Increasing stratification will increase the wave speed
(Huthnance 1989) and induce the frequency to increase. The CSWs decay primarily through friction
(Brink 1991). Realistic topography was found to concentrate the motion over the upper slope and shelf
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
(Huthnance 1978). In the presence of mean alongshore flow, the properties of CSWs change due to a
combination of Doppler shift, changes in the background vorticity, and the growth of instabilities (Brink
1991). The behavior of the mode one CSW present in the model results is consistent with this behavior,
propagating westward, and having a concentration of motion over the upper slope and shelf.
Comparisons of the model estimates of the elevation amplitudes with the observations for the diurnal
constituents, showed good agreement for O1 , but only moderate agreement for K1 with rms values of 3.4
and 5.6 cm for the O 1 and K1 constituents, respectively (Tables 1 and 2). ROMS outperformed the
MacAyeal (1984) and CATS02.01 (Padman et al., 2002) models in replicating tidal elevation amplitudes
for the K1 constituent (Table 2).
With the exception of over the continental slope, Iselin Bank and two regions under the ice shelf, the
diurnal major axes are small < 10 cm s-1 (Figures 3c and 3d). However, over the continental slope and
Iselin Bank, CSWs induced large velocities, > 50 cm s-1 , in the diurnal constituents (Figures 3c and 3d).
3.2 Depth-Dependent Response
3.2.1 Semidiurnal Constituents (M2 and S2 )
To further evaluate the model performance, the major axes of the tidal ellipses for depth-dependent
velocities were compared with observations. The observations consisted of seventy-two current meters
from twenty-six separate moorings, whose locations are indicated by their mooring numbers in Figure 1c.
(Although there are seventy-two current meter records, there are only sixty-nine separate sites on twentysix moorings, since three sites had repeat observations and the moorings had multiple instruments.) Most
of the observations are discussed in Robertson et al. (2003); however, eighteen new observations
(Johnson and Van Woert, 2004) have been included. Two different comparisons were done. The rms
differences between the model estimates and observations provide bulk quantitative values for evaluating
model performance. However, the rms difference reflects only the errors in the major axes. A more
comprehensive measure is the absolute error (defined later), which combines errors in the amplitude with
those in phase. Both of these provide quantitative estimates of the model performance, which can be used
to evaluate the model. And their estimates are only representative, if the observations are well spread and
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
numerous, which is not the case for the Ross Sea. Also they provide no information on the relative model
performance in different areas. Since this information is important to the potential uses of the tidal
information, the individual values are given in Table 4 and comparisons shown in Figures 4 and 7. The
comparisons in Figures 4 and 7 graphically show the performance of the model in replicating the vertical
structure of the observations.
The model estimates for the M2 major axes at forty-eight of the sixty-nine sites agreed with the
observed values (Figure 4 and Table 3) within the observational uncertainty of 1.7 cm s-1 (Pillsbury &
Jacobs 1985) and the estimates for the S2 major axes agreed for fifty of sixty-eight sites (Table 3). The
rms differences for the major axes were 4.7 and 3.0 cm s-1 for M2 and S2 , respectively (Table 4). The rms
differences were reduced to 0.9 and 0.6 cm s -1 for M2 and S2 , respectively (Table 4), when the bottom
record at mooring 18 was excluded. The bottom record at mooring 18 was cut short by 9 months (out of
13) indicating probable current meter malfunction. For one of the locations of disagreement (mooring 17
in Figure 4), the model overestimated the velocity for the upper current meter. This overestimation did
not occur in the simulation with only M2 tidal forcing, thus it results from amplification by the diurnal
CSW through a harmonic response. Two other disagreement locations are over Iselin Bank where the
observations reported extremely large tidal benthic velocities (bottom values at moorings 18 and 26 in
Figure 4). These large benthic values were not seen in the model results.
The absolute errors (EA ) for each constituent were calculated for both positive (anticlockwise) and
negative (clockwise) components according to E A ? L? 1 ? L D , with
D?
?
?
0.5 A2o ? A2m ? Ao Am cos?? o ? ? m? , where L is the number of observations and Ao and Am and ? o
and ? m are the amplitudes and phases, respectively, with the subscripts of o and m representing the
observations and model results, respectively (Cummins and Oey 1997). The absolute error is determined
for each component for each constituent. The absolute errors in the positive components for M2 and S2
were 0.69 and 0.47 cm s-1 , respectively (Table 4). Again, a significant portion of these errors was
associated with the benthic values of mooring 18 and the errors were reduced to 0.46 and 0.31 cm s-1 for
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
M2 and S2 , respectively, when this observation was excluded (Table 4). In the Southern Hemisphere,
most of the baroclinic response for the boundary layers should be associated with the positive rotating
component. The negative rotating component is barotropic in the Southern Hemisphere. The absolute
errors in the negative components for M2 and S2 were smaller, ~0.2 cm s-1 for both M2 and S2,, when the
bottom record of mooring 18 was excluded (Table 4). The larger absolute errors for the positive rotating
component indicate that more error is associated with the baroclinic response than the barotropic
response.
For the M2 semidiurnalconstituent, the critical latitude, 74o 28.8’ S, crosses through the domain
(Figure 1b and dashed line in Figure 5). At the critical latitude the inertial frequency equals the tidal
frequency, which affects both the generation and propagation of internal tides. Linear internal wave
theory predicts that internal waves will not be generated nor propagate poleward of the critical latitude.
Robertson (2001a) gives a more extensive explanation of the relevant linear internal wave theory and
Robertson (2001b) describes the response of the M2 constituent to critical latitude effects.
Generation and propagation of internal tides occurred for the semidiurnal constituents over the
continental slope and steep topography (for example, areas indicated with IT in Figures 5a-5e). The
semidiurnal internal tides had the expected theoretical wavelengths for mode 1 waves and propagated
along the expected wave characteristics according to linear internal wave theory (not shown). The
structure of the phases also supported propagation of internal tides (Figures 5f-5j). A previous study
showed that critical latitude effects increased M2 generation, especially in the western Ross Sea (Figures
5a-5c), where the continental slope is equatorward of the M2 critical latitude (Robertson et al, 2003). This
was also true with multiple constituents; however with multiple constituents, M2 critical latitude effects
were overwhelmed by amplification by CSWs. The locations of intense internal tide generation
(velocities > 5 cm s-1 ) over the continental slope and steep topographic features in the multiple constituent
simulation in Figure 5 have significantly smaller velocities, < 5 cm s-1 , in the simulation with only the M2
constituent (not shown). These areas coincide with the locations of high diurnal major axes (Figures 3c-d
and 6). Thus internal tide generation for the semidiurnal constituents is amplified by diurnal CSWs.
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Internal tides were also generated under and at the front of the ice shelf (Figures 5a-5e). Although the
existence of internal tides is not expected poleward of the critical latitude in a continuously stratified fluid
according to linear internal wave theory, it is quite possible for a baroclinic mode tide to exist in the
multiple layer system (Kundu 1990, p. 233), such as that of the hydrographic conditions in the ice cavity.
Comparison of the M2 major axes of the tidal ellipses for depth-independent and the surface depthdependent (Figure 3f) velocities show the depth-dependent surface velocities to have more horizontal
variability with many small-scale local features and velocities amplified over the depth-independent
values. This horizontal variability is induced by spatial differences in the baroclinic tides. The horizontal
variations, in turn, induce divergence and convergence in the surface velocity fields, which affects lead
formation and pack ice dynamics (Geiger et al. 1998).
3.2.2 Diurnal Constituents (K1 and O1 )
Agreement between the model estimates and the observations was not as good for the diurnal
constituents as for the semidiurnal constituents, particularly along the continental slope, with differences
at thirty-eight and eighteen of the fifty-one sites exceeding the observation uncertainty for K1 and O1 ,
respectively (Table 3). It should be noted that the observations represent average values for their
respective record lengths, which may not represent either the corresponding hydrographic conditions used
in the model or a multi-year average such as the model estimates. Observational tidal estimates have been
found to vary on seasonal and interannual time scales and on spatial scales < 150 km (Johnson and Van
Woert, 2004). The rms differences for the major axes were 8.4 and 11.0 cm s-1 for K1 and O1 ,
respectively. These were reduced to 8.3 and 5.8 cm s-1 for K1 and O1 , respectively, with the benthic
observation at mooring 18 excluded (Table 4). The absolute errors for the positive and negative rotating
components were 4.0 and 3.5 cm s-1 for K1 and 1.6 and 1.8 for O1 , respectively, with the benthic
observation at mooring 18 excluded (Table 4). Again more error was associated with the depth-varying
response than the barotropic response.
CSWs were overexcited in the model along the continental shelf break resulting in higher values than
the observations for those sites, with the exception of the large benthic value at mooring 18 (last plot in
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
third row of Figure 7). Disagreement with the observations also occurred at other locations (bottom row
in Figure 7) where the observations were depth-dependent, but the model estimates were not.
The overexcitement of CSW along the continental slope can account for much of the disagreement for
the diurnal constituents. Unfortunately, the cause for the overexcitation of CSWs by the model is
unknown, although there are several potential sources. It is possibly linked to the lack of wind-driven
mean flows in the model. Mean alongshore flows modify the CSW properties. Also stratification can
cause the frequency of the CSWs to shift. It is possible that in the natural conditions, a shift is caused by
the stratification and/or the presence of a mean alongshore flow. This shift transfers energy from the
diurnal components into other frequencies where they are not picked up in the tidal analysis. The model
conditions may not reproduce this shift due to the lack of a mean flow and/or different hydrographic
conditions. Resonance effects associated with errors inherent in the model are another potential source
for the discrepancy.
The diurnal response of the depth-dependent velocities was essentially barotropic (Figure 6). Large
major axes occurred at sites along the continental slope and over Iselin Bank and are associated with
CSWs (for example, areas marked CSW in Figure 6). The vertically variable response was slight, and
primarily surface and benthic boundary layer effects, which are most apparent with the largest velocities
(Figure 6c and 6e). The Ross Sea is 40o or more poleward of the diurnal critical latitudes (28o and 30o ),
so the diurnal response is expected to be barotropic without baroclinic tidal generation. Despite the
theory, the observations do show some variation in velocities in the vertical direction (diamonds in Figure
7 and Table 3), primarily along the ice shelf edge.
3.3 Combined Tidal Response
To this point, the discussion has considered the tidal response separately for the different constituents.
This is appropriate when evaluating the model performance; however, sea ice, water mass properties, and
tracers respond to the combined elevation and velocity fluctuations. The combined response will be
evaluated using the standard deviations of the elevations and velocities over the fifteen days of model
results.
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
The largest elevation standard deviations occurred in the tongue in the southeastern corner under the
ice shelf and over Iselin Bank (Figure 8a). The restrictive topography of the tongue in the southeastern
corner and the increase of Kelvin wave height in shallow water induce large standard deviations in that
region. The high variance over Iselin Bank is attributable to the generation of CSWs. The depthdependent surface velocity standard deviations are also highest along the continental slope and over Iselin
Bank (Figure 8b) again due to CSWs. It should be noted that this combined estimate of the variability is
slightly larger than it should be along the continental shelf break due to the overestimation of the diurnal
constituents.
Standard deviations of the depth-dependent velocity anomalies, defined as the difference between the
depth-dependent and depth-independent velocities, also are highest over regions of steep topography,
such as the continental slope and Iselin Bank (Figure 9). The combined standard deviations of the depthdependent velocity anomalies exhibit the typical pattern associated with propagating internal waves along
with boundary layer(s). Since the diurnal response was nearly barotropic except for boundary layers, this
mid-water column depth-dependent response was associated with the semidiurnal constituents.
Vertical shear in the horizontal velocities can induce mixing in the water column, through increased
bottom (and surface under the ice shelf) friction and shear instabilities. Inspection of the average of the
magnitudes of the two horizontal vertical shears in the depth-dependent velocities show that the tides
induce shear in the water column (Figure 10). Tides can induce vertical shear in two ways: 1) increasing
the velocities increases the benthic (and surface under the ice shelf) boundary layer(s) stress(es)
quadratically as the velocity increases and 2) through shear instabilities associated with baroclinic tides.
Both effects are present. The vertical shear in the benthic and surface boundary layers is apparent in
Figure 10, particularly 10a with the enlarged vertical scale. Vertical shear attributable to baroclinic tides
is also apparent throughout the water column, particularly north of the M2 critical latitude over steep
topographic features (Figure 10). The Richardson number (Ri) is often used to characterize water column
stabilities with Ri < 0.25 indicating an unstable water column capable of having shear instabilities. Ri
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
values less than 0.25 often occurred near the bottom and occasionally in the mid-water column in the
regions of high vertical shear (not shown).
4. Summary
Simulations of the barotropic and baroclinic tides in the Ross Sea basin and the cavity under Ross Ice
Shelf were performed both with the major semidiurnal constituent alone, M2 , and for four tidal
constituents, M2 , S2 , K1 , and O1 , combined using ROMS. Comparison of the model results with tidal
estimates from elevation and velocity observations showed very good agreement for the semidiurnal
constituents, M2 and S2 . The agreement for the diurnal constituents, K1 and O1 , were not as good,
particularly along the continental slope, where the model overestimated the tidal response in the form of
continental shelf waves.
The diurnal frequency continental shelf waves were found to amplify the semidiurnal response. Thus,
simulations with all the constituents combined showed more vertical variation in the velocities along the
continental slope at semi-diurnal frequencies than the single -constituent simulation.
The diurnal tides were essentially barotropic with vertical differences in their response associated
with the benthic and surface boundary layers. The mid-water column baroclinic response occurred at the
semidiurnal frequencies. Semidiurnal baroclin ic tides were generated along the continental shelf break
and over regions of steep topography, particularly over steep topography north of the critical latitude.
The semidiurnal baroclinic tides were significantly amplified by the diurnal continental shelf waves when
the shelf waves were present. The baroclinic tidal behavior was consistent with that for internal waves in
a continuously stratified fluid and propagating along wave rays with wavelengths corresponding to the
theoretical estimates for mode one internal waves.
Under the ice shelf, the hydrography was specified by a series of four layers. In the ice shelf cavity, a
multi-layer depth-dependent response was often present inducing shear in the water column in this region.
There is room for improvement in the model predictions. The model estimates will improve as
knowledge of topography and bathymetry improves. Presently, a topographic survey of the southeastern
portion of the Ross Ice Shelf cavity is being planned by Bell and Studinger (personal communication) at
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Lamont-Doherty Earth Observatory. The Anslope initiative is obtaining hydrographic and bathymetric
data for the shelf slope area of the Ross Sea. Data from these sources will be incorporated into the initial
conditions for the model in future efforts. Also, the problem of the overexcitation of diurnal continental
shelf waves needs to be addressed. The overexcitation may be linked to the lack of alongshore mean
currents in the model. Two improvements to the model that are presently in-progress are 1) addition of
the wind-induced mean circulation and 2) coupling to the dynamic sea ice model of Tremblay & Mysak
(1997). These improvements, along with improved hydrography, may alleviate the excessive continental
shelf wave excitation. Inclusion of more of the forcing factors and dynamics will improve the model
estimations. Additionally, through the Anslope program more tidal data is becoming available, which
will be incorporated into future comparisons. Furthermore, it is planned to delve into what the effects of
tides are on the surface pack ice, heat transfer to the atmosphere, and mixing in future efforts.
Acknowledgements:
This study was funded by (Office of Polar Programs) OPP grant OPP-00-3425 of the National
Science Foundation (NSF). I would like to thank Stan Jacobs for his input on the manuscript and Keith
Nicholls and Laurie Padman for their careful reviews. This is Lamont publication 6593. Any opinions,
findings, and conclusions or recommendations expressed in this materia l are those of the author and do
not necessarily reflect the views of the National Science Foundation.
16
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
References
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EGBERT, G. D., and R. RAY , 2001, Estimates of M2 tidal energy dissipation from TOPEX/Poseidon altimeter
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Robertson: Baroclinic and Barotropic Tides in the Ross Sea
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in the southern Weddell Sea, Journal of Physical Oceanography, 12, 618-634.
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18
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
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19
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
WILLIAMS, R. T., and E. S. ROBINSON , 1980, The ocean tide in the southern Ross Sea, Journal of
Geophysical Research, 85, 6689-6696.
20
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
List of Figures
Figure 1. a) The location of the model domain with respect to Antarctica. b) and c) The model domain
with the water column thickness contoured at 100, 500, 1000, 2000, and 3000 m. The approximate edge
of the Ross Ice Shelf is indicated by a dashed line in b) and c). The locations of tidal elevation
observations are indicated by numbers in b) and those of velocity observations by numbers in c).
Figure 2. The amplitude of the tidal elevation for the a) M2,, b) S2 , c) K1 , and d) O1 constituents in cm.
The overlaid heavy dashed lines indicate the phase for the elevation, with the zero phase line indicated in
a) and b). The approximate edge of the Ross Ice Shelf is indicated by a line with long dashes.
Figure 3. The major axes of the tidal ellipses for the depth-independent velocities for the M2,, b) S2 , c) K1 ,
d) O1 constituents, in cm s-1 . e) The same for M2 simulated with only M2 forcing and f) for M2 from the
depth-dependent surface velocities forced with four constituents. The approximate edge of the Ross Ice
Shelf is indicated by a dashed line.
Figure 4. Major axes of the M2 tidal ellipses from ROMS (crosses) compared with observational data
(diamonds). The error bars indicate the observational uncertainty (1.7 cm s-1 ).
Figure 5. The major axes of the M2 tidal ellipses from the depth-dependent velocities along N-S transects
at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W, and e) 160o W. The phases for the M2 tidal
ellipses from the depth-dependent velocities along N-S transects at f) 169o 30’ E, g) 175o E, h) 179o 30’
W, i) 175o W, and j) 160o W. The pairs of letters indicate points discussed in the text (IT-internal tides).
The location of the critical latitude is indicated by a dashed line.
21
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Figure 6. The major axes of the K1 tidal ellipses from the depth-dependent velocities along N-S transects
at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W, and e) 160o W, in cm s-1 . CSW indicates
continental shelf waves.
Figure 7. Major axes of the K 1 tidal ellipses from ROMS (crosses) compared with observational data
(diamonds). The error bars indicate the observational uncertainty (1.7 cm s-1 ).
Figure 8. The a) standard deviation of the elevations (cm) and the b) combined standard deviations of the
depth-dependent surface velocities (cm s-1 ). The locations of the front of the Ross Ice Shelf is indicated
by dashed line.
Figure 9. The combined standard deviations in the depth-dependent velocity anomalies (difference
between the depth-dependent and depth-independent velocities) over 15 days at a) 169o 30’ E, b) 175o E,
c) 179o 30’ W, d) 175o W, and e) 160o W. The M2 critical latitude is indicated by a dashed white line.
Figure 10. The average of the magnitudes of the two horizontal vertical shears in the depth-dependent
velocities along N-S transects at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W, and e) 160o W. The
M2 critical latitude is indicated by a dashed white line.
22
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
List of Tables
Table 1. Elevation amplitudes and phases from ROMS compared against observational data and other
two-dimensional models. Underlined values exceed the observational uncertainties, which were taken
from Williams and Robinson (1980). Note: Williams and Robinson were unable to estimate the
uncertainty at one site, which is reflected here by a ? for the value.
Table 2. Rms differences for the elevation amplitude and phases at the twelve observation locations for
each constituent for the ROMS simulations and for two-dimensional simulations of MacAyeal (1984) and
(CATS02.01) Padman et al. (2002).
Table 3. The major axes of the tidal ellipses from the observations and the corresponding model
estimates from ROMS. Underlined values exceed the observational uncertainty of 1.7 cm s-1 . An *
indicates a depth with multiple observations.
Table 4. Rms differences for the major axes of the tidal ellipses and absolute and relative errors for the
positive and negative rotating components from the depth-dependent velocities at the observation
locations for each constituent for ROMS.
23
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Location
Latitude
Longitude
M2
Amplitude
(cm)
Phase (o )
S2
Amplitude
(cm)
Phase (o )
K1
Amplitude
(cm)
Phase (o )
O1
Amplitude
(cm)
Phase (o )
Site
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
Observations
ROMS
MacAyeal1
CATS02.012
1
2
84o
15.0’
S
171o
19.8’
W
1
8? 2
14
16
9
258
?3
225
13
259
11?2
21
16
16
142
?4
146
160
214
41?2
50
57
56
206
?2
195
213
201
40?2
37
39
43
190
?2
180
193
187
82o
31.8’
S
166o
0.0’
W
2
8? 2
7
7
9
213
? 27
163
290
186
10?2
9
5
16
112
? 27
72
100
128
43?2
44
54
53
186
?4
178
193
181
35?2
34
39
41
174
?4
165
176
169
82o
22.2’
S
168o
37.8’
W
3
7? 2
6
4
8
205
? 14
162
282
185
8? 2
7
3
13
106
? 14
72
112
127
37?2
41
50
51
191
?4
180
194
181
37?2
33
36
40
172
?4
166
178
169
81o
11.4’
S
170o
30.0’
E
4
3? 2
2
6
9
310
? 16
292
117
285
2? 2
3
6
1
160
? 25
231
117
240
31?2
38
38
39
200
?2
194
216
199
27?2
29
29
32
190
?5
185
196
185
80o
11.4’
S
161o
33.6’
E
5
5? 2
7
15
9
130
? 27
89
209
103
10?2
9
8
17
26
? 27
349
333
37
44?2
46
55
48
160
?4
155
162
152
38?2
34
40
38
140
?4
143
145
141
79o
45.0’
S
169o
3.0’
W
6
3? 2
4
8
5
75
? 27
83
187
96
6? 2
5
5
8
25
? 27
339
316
33
37?2
45
45
44
160
?4
170
175
159
32?2
33
35
36
153
?4
149
158
148
79o
31.8’
S
163o
38.4’
E
7
4? 2
3
8
2
340
? 27
331
128
354
2? 2
5
8
3
190
? 27
260
271
300
31?2
38
37
38
208
?4
210
224
208
29?2
29
28
31
196
?4
194
202
192
79o
15.0’
S
170o
19.8’
E
8
3? 2
3
7
2
300
??
356
136
22
4? 2
4
7
3
130
??
280
280
330
30?2
36
35
35
200
??
208
219
200
34?2
26
27
29
190
??
178
198
186
78o
12.0’
S
162o
16.2’
W
9
3? 2
4
9
5
35
? 27
57
162
60
5? 2
5
6
9
342
? 27
305
299
349
34?2
39
41
40
154
?4
156
156
144
25?2
30
32
33
141
?4
136
141
133
77o
51.0’
S
166o
39.6’
E
10
4? 2
5
8
5
6
? 27
24
147
32
2? 2
6
8
6
268
? 27
307
307
346
26?2
30
28
26
196
?4
211
229
207
26?2
19
22
22
186
?4
193
204
192
77o
45.0’
S
166o
45.8’
E
11
4? 2
5
4
334
? 27
23
30
3? 2
6
6
292
? 27
306
344
25?2
30
27
208
?4
210
207
23?2
19
23
167
?4
193
192
MacAyeal (1984)
CATS02.01: Padman et al. (2002)
Table 1. Elevation amplitudes and phases from ROMS compared against observational data and other
two-dimensional models. Underlined values exceed the observational uncertainties, which were taken
from Williams and Robinson (1980). Note: Williams and Robinson were unable to estimate the
uncertainty at one site, which is reflected here by a ? for the value.
24
77o
1.9’
S
166o
9.8’
E
12
5? 2
6
5
342
? 27
25
32
5? 2
7
7
311
? 27
307
348
21?2
29
25
218
?4
213
212
21?2
19
21
178
?4
199
197
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Constituent
ROMS
MacAyeal
(1984)
CATS2000
(2002)
M2
Amplitude Phase
(cm)
(o )
1.4
32
5.4
124
1.4
31
S2
Amplitude
(cm)
2.2
4.4
Phase
(o )
45
65
3.2
58
K1
Amplitude Phase
(cm)
(o )
5.6
11
9.5
15
6.8
15
O1
Amplitude
(cm)
3.4
3.9
Phase
(o )
7
7
3.3
16
Table 2. Rms differences for the elevation amplitude and phases at the twelve observation locations for
each constituent for the ROMS simulations and for two-dimensional simulations of MacAyeal (1984) and
(CATS02.01) Padman et al. (2002).
25
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Latitude
Longitude
Mooring
78o
13.62’
S
78o
13.20’
S
78o
11.52’
S
78o
10.98’
S
78o
9.78’ S
1
2
443
3
436
4
530
5
558
6
567
7
558
8
550
9
595
10
602
77o
58.68’
S
77o
52.92’
S
77o
40.80’
S
77o
12.42’
S
76o
58.92’
S
172o
29.40’
W
172o
31.08’
W
172o
48.00’
W
174o
39.00’
W
170o
36.72’
W
174o
30.78’
W
175o
30.18’
W
175o
30.00’
W
176o
39.48’
W
177o
1.623’
W
175o
35.82’
W
178o
31.98’
W
160o
24.18’
W
175o
9.30’
E
171o
58.74’
E
Water
Depth
(m)
420
11
585
12
700
13
590
14
77o
40.98’
169o
91.02’
78o
6.48’ S
78o
5.88’ S
78o
5.52’ S
78o
4.68’ S
78o
0.00’ S
Instr.
Depth
(m)
211
283
383
215
291
395
240
315
390
237
310
492
228
305
M2
Observ. ROMS
(cm s-1)
(cm s-1)
0.7
0.8
0.5
0.4
1.0
0.4
0.2
1.0
0.2
0.5
0.1
0.5
0.5
0.4
210
285
S2
Observ. ROMS
(cm s-1)
(cm s-1)
0.7
0.4
0.4
0.7
0.4
0.8
1.0
0.5
0.4
0.3
0.6
0.8
0.6
0.4
0.3
0.8
0.5
0.5
0.4
1.0
1.8
0.3
0.4
0.2
0.3
1.3
0.5
0.4
0.4
0.4
0.4
125
225
300
253
327
508
220
425
0.4
0.4
0.4
0.9
0.3
0.3
0.6
0.4
244
390
579
210
285
515
340
425
650
255
540
685
15
671
16
810
K1
Observ. ROMS
(cm s-1)
(cm s-1)
0.3
0.5
0.7
0.3
0.5
0.7
0.4
0.5
0.6
0.4
0.4
0.6
0.5
0.6
2.1
5.6
4.6
5.4
6.7
5.2
0.2
4.0
0.1
4.0
2.0
3.8
2.9
3.0
0.5
0.5
0.5
0.5
0.4
0.4
0.6
0.5
0.7
0.5
0.6
0.7
0.5
0.4
0.4
0.5
0.4
0.1
0.2
0.4
0.5
0.5
0.5
0.5
0.4
0.3
0.3
0.4
0.4
0.9
0.7
0.5
0.8
1.1
0.3
0.4
0.6
0.4
0.3
0.4
1.2
0.7
175
350
0.1
0.3
107
210
497
657
40
220
0.6
0.5
0.6
0.4
0.4
0.5
O1
Observ. ROMS
(cm s-1)
(cm s-1)
6.1
5.9
6.1
6.0
5.9
6.1
6.3
6.3
6.3
5.8
5.9
5.7
6.0
6.0
1.0
6.1
2.8
3.3
3.2
0.2
4.3
0.2
2.7
0.9
2.7
2.7
2.3
3.3
3.2
3.1
3.4
3.2
3.1
3.1
3.2
3.1
3.4
3.4
3.4
2.6
2.6
4.9
4.7
6.0
6.0
3.3
3.2
3.4
3.4
0.5
0.5
0.4
0.4
0.3
0.3
0.7
0.2
4.6
4.5
4.2
4.3
3.7
2.5
2.0
3.9
5.8
5.8
5.9
5.9
5.8
5.8
5.6
5.5
3.1
3.0
2.9
3.1
2.7
1.8
1.5
2.7
3.5
3.5
3.5
3.5
3.5
3.6
3.7
3.6
0.5
0.4
0.5
0.4
0.5
0.3
0.4
0.3
0.4
1.2
1.2
0.6
0.3
0.6
0.3
0.5
0.4
0.3
0.3
0.3
1.5
1.1
4.2
4.9
4.3
4.5
4.4
4.2
3.5
3.3
3.8
1.4
1.4
5.5
5.6
5.7
6.0
6.0
6.0
5.2
5.1
4.8
4.5
4.3
2.9
3.2
2.9
2.9
2.8
2.7
2.6
2.5
2.7
1.5
1.4
3.7
3.7
3.6
3.6
3.6
3.6
3.7
3.6
3.2
0.6
0.6
0.8
0.5
0.1
0.4
0.5
0.5
0.4
4.1
4.9
5.0
0.3
2.6
5.6
5.7
1.3
1.0
0.6
0.9
1.2
0.5
0.6
0.6
0.6
0.5
0.6
0.3
0.9
0.7
0.7
0.8
0.3
0.3
4.4
4.1
3.9
4.0
2.8
3.1
2.6
2.6
2.6
2.6
2.0
2.0
2.7
2.8
2.5
2.8
2.1
2.3
3.5
3.5
3.5
3.5
3.1
3.1
26
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
S
76o
30.09’
S
76o
29.808’
S
76o
20.46’
S
76o
20.34’
S
E
167 o
29.97’
W
174o
59.595’
W
165o
1.8’ E
770
241*
740*
0.4
0.3; 0.1
0.4; 0.3
0.4
3.2
1.5
0.3
0.3; 0.1
0.1; 0.2
0.3
1.8
2.8
2.8
1.9; 0.6
0.4; 1.4
1.9
42.5
32.9
2.3
1.6; 0.5
0.4; 1.2
2.9
14.6
10.1
569
241*
534
0.8; 0.1
39.2
0.4
0.4
0.9; 0.1
24.6
0.6
0.3
9.2; 2.6
87.3
19.6
1.6
4.9; 1.6
2.4
11.1
2.2
19
828
174o
56.16’
E
20
625
177o
35.980’
W
177o
13.030’
E
164o
23.58’
E
175o
0.0’ E
175o
0.0’ E
172o
31.470’
E
21
625
22
912
23
879
36
210
813
24
196
427
604
293
473
624
122
565
766
40
780
0.3
0.4
0.4
0.8
0.5
0.5
0.5
0.7
0.8
0.7
0.4
0.3
0.3
0.6
0.3
0.5
0.5
0.3
1.5
0.6
0.6
0.6
0.8
1.1
1.6
0.1
0.1
0.2
0.2
0.2
0.3
0.6
0.4
0.6
0.4
0.4
0.3
0.7
0.7
0.6
0.4
0.3
0.3
0.7
0.2
0.5
0.3
0.3
0.4
0.3
0.5
0.3
0.3
0.6
1.2
0.1
0.1
0.2
0.2
0.1
0.3
0.4
0.2
4.1
5.0
4.0
3.8
9.8
9.8
6.7
1.8
1.2
1.8
1.8
1.7
1.6
1.6
1.5
3.1
3.1
3.0
2.9
13.5
13.5
11.0
0.5
0.4
0.8
0.4
0.5
0.3
0.4
0.4
1.4
1.4
1.1
1.2
5.1
5.0
3.1
1.4
1.5
1.7
1.3
1.6
2.2
2.2
2.1
4.3
4.3
4.3
4.0
11.3
11.3
9.0
0.8
0.4
1.1
0.8
1.6
75o
56.200’
S
75o
6.100’
S
74o
57.90’
S
74o
0.0’ S
72o
30.0’ S
72o
28.813’
S
24
588
25
456
26
533
220
555
230
425
241
498
0.8
0.6
2.7
3.1
2.7
6.9
1.4
1.0
1.5
1.5
1.5
1.9
0.6
0.9
1.0
1.6
0.6
2.1
0.7
0.3
0.9
0.7
1.1
0.4
11.4
9.8
41.9
24.6
18.2
15.4
11.0
9.5
10.0
10.0
18.9
13.3
3.6
3.3
13.6
6.1
18.9
13.6
15.2
12.9
21.9
22.1
47.6
35.7
17
775
18
Table 3. The major axes of the tidal ellipses from the observations and the corresponding model
estimates from ROMS. Underlined values exceed the observational uncertainty of 1.7 cm s-1 . The *
indicates a depth with multiple observations.
27
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Rms
Positive rotating component Absolute error
(anticlockwise)
Negative rotating component Absolute error
(clockwise)
No. of sites exceeding uncertainty
All Observations
76o 29.808’ S (18) Excluded
M2
S2
K1 O1
M2
S2
K1
O1
4.7 3.0 8.4 11.0
0.9
0.6
8.3 5.8
0.69 0.47 4.1 4.0 0.46
0.31
4.2 3.5
0.43 0.34 1.6
3
2
31
2.0
0.21
0.22
1.6
1.8
18
2
1
30
18
Table 4. Rms differences for the major axes of the tidal ellipses and absolute and relative errors for the
positive and negative rotating components from the depth-dependent velocities at the observation
locations for each constituent for ROMS.
28
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
a.
b.
Iselin Bank
500
12
500
11
10
-80 5
9
8
7
M critical latitude
Latitude
-75
30 0
2 000 0
1000
6
0
50
2
4
3 2
1
-85
c.
26 25
24
Latitude
-75
23
22
19
20
16 15
14
21
18
17
11
12496-9
11
84
7
61-3
31
2 510
13
-80
-85
170o E
180 o
170o W
Longitude
160o W
Figure 1. a) The location of the model domain with respect to Antarctica. b) and c)The model domain
with the water column thickness contoured at 100, 500, 1000, 2000, and 3000 m. The approximate edge
of the Ross Ice Shelf is indicated by a dashed line in b) and c). The locations of tidal elevation
observations are indicated by numbers in b) and those of velocity observations by numbers in c).
29
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
a.
M2
Latitude
-75
0o
-80
100
50
-85
170o E
180
o
170o W
160o W
30
Longitude
b.
20
10
S2
5
2
-75
Latitude
40
1
0
-80
0
Land
(cm)
o
-85
170o E
180o
170o W
160o W
Longitude
Figure 2. The amplitude of the tidal elevation for the a) M2,, b) S2 , c) K1 , and d) O1 constituents in cm.
The overlaid heavy dashed lines indicate the phase for the elevation, with the zero phase line indicated in
a) and b). The approximate edge of the Ross Ice Shelf is indicated by a line with long dashes.
30
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
c.
K1
Latitude
-75
-80
100
50
-85
170o E
180
o
170o W
160o W
30
Longitude
d.
20
10
O
1
5
2
-75
Latitude
40
1
0
Land
(cm)
-80
-85
o
170 E
o
180
o
170 W
Longitude
Figure 2. (cont.)
31
o
160 W
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
a.
M
2
Latitude
-75
-80
50
40
-85
170o E
180o
170o W
160o W
30
20
Longitude
10
S2
b.
2
-75
Latitude
5
1
0
Land
-80
-1
(cm s )
-85
o
170 E
180
o
o
170 W
o
160 W
Longitude
Figure 3. The major axes of the tidal ellipses for the depth-independent velocities for the M2,, b) S2 , c) K1 ,
d) O1 constituents, in cm s-1 . e) The same for M2 simulated with only M2 forcing and f) for M2 from the
depth-dependent surface velocities forced with four constituents. The approximate edge of the Ross Ice
Shelf is indicated by a dashed line
32
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
c.
K
1
Latitude
-75
-80
50
40
-85
170o E
180
o
170o W
160o W
30
20
Longitude
10
O
d.
1
2
-75
Latitude
5
1
0
Land
-1
(cm s )
-80
-85
170o E
180o
170o W
Longitude
Figure 3. (cont.)
33
160o W
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Figute 3. (cont.)
34
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Depth (m)
0
1
5
4
200
400
0
0
Depth (m)
3
2
2
4 0
2
10
7,8
4 0
2
4 0
2
12
11
4 0
2
4
2
4
20
40
4
8
13
200
400
600
0
Depth (m)
0
2
4 0
2
15
14
4 0
2
4 0
2
17
16
4 0
18
200
400
600
800
0
Depth (m)
0
19
2
4 0
20
2
4 0
2
21
4 0
2
24
4 0
26
400
800
0
2
4 0
2
4 0
2
4 0
-1
Major Axis (cm s
2
4 0
)
Figure 4. Major axes of the M2 tidal ellipses from ROMS (crosses) compared with observational data
(diamonds). The error bars indicate the observational uncertainty (1.7 cm s-1 ).
35
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
0
Depth (km)
a.
IT
o
-1 169 30' E
0
-2
175 o E
10.0
7.5
0
Depth (km)
c.
IT
-1
critical
latitude
Depth (km)
b.
-1
-2
0
Depth (km)
d.
Depth (km)
2.5
179o 30' W
IT
1.0
0.0
-2
Ice
Shelf
IT
IT
-4
0
e.
5.0
IT
175o W
Bottom
(cm s -1 )
IT
-2
-4 160o W
-84
IT
-82
-80
-78
Latitude
-76
-74
-72
Figure 5. The major axes of the M2 tidal ellipses from the depth-dependent velocities along N-S transects
at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W, and e) 160o W. The pairs of letters indicate points
discussed in the text (IT-interna l tides). The location of the critical latitude is indicated by a dashed line.
36
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
0
Depth (km)
f.
IT
o
-1 169 30' E
0
IT
-1
-2
critical
latitude
Depth (km)
g.
o
175 E
0
Depth (km)
h.
IT
-1
-2
0
Depth (km)
i.
Depth (km)
IT
-2
IT
IT
o
-4
0
j.
179o 30' W
175 W
IT
-2
-4 160o W
-84
IT
-82
-80
-78
Latitude
-76
-74
-72
Figure 5. (cont.) The phases for the M2 tidal ellipses from the depth-dependent velocities along N-S
transects at f) 169o 30’ E, g) 175o E, h) 179o 30’ W, i) 175o W, and j) 160o W.
37
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
0
Depth (km)
a.
o
-1 169 30' E
0
Depth (km)
b.
CSW
-1
-2
75
50
o
175 E
c.
40
Depth (km)
0
10
Depth (km)
Depth (km)
179o 30' W
5
2
1
CSW
-2
-4
0
e.
20
-1
-2
0
d.
30
CSW
0
Ice
Shelf
Bottom
175o W
(cm s -1 )
-2
CSW
-4 160o W
-84
-82
-80
-78
Latitude
-76
-74
-72
Figure 6. The major axes of the K1 tidal ellipses from the depth-dependent velocities along N-S transects
at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W, and e) 160o W, in cm s-1 . CSW indicates
continental shelf waves.
38
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
Depth (m)
0
1
5
4
200
400
0
0
Depth (m)
3
2
4
8 0
4
10
7,8
8 0
4
8 0
4
8 0
12
11
4
8
4
8
13
200
400
600
0
Depth (m)
0
4
8 0
4
15
14
8 0
4
8 0
8 0
4
17
16
18
200
400
600
800
0
Depth (m)
0
4
19
8 0
4
20
8 0
4
21
8 0
20
24
40
0
40
80
26
400
800
0
4
8 0
4
8 0
10
20 0
-1
Major Axis (cm s
10
20 0
10
)
Figure 7. Major axes of the K 1 tidal ellipses from ROMS (crosses) compared with observational data
(diamonds). The error bars indicate the observational uncertainty (1.7 cm s-1 ).
39
20
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
a.
100
50
Latitude
-75
40
30
20
10
-80
5
2
1
-85
0
170o E
o
180
o
170 W
o
160 W
Longitude
Land
(cm)
b.
50
40
Latitude
-75
30
20
10
-80
5
2
1
-85
170o E
180o
170o W
Longitude
160o W
0
Land
(cm s -1 )
Figure 8. The a) standard deviation of the elevations (cm) and the b) combined standard deviations of the
depth-dependent surface velocities (cm s-1 ). The locations of the front of the Ross Ice Shelf is indicated
by dashed line.
40
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
0
Depth (km)
a.
o
-1 169 30' E
0
Depth (km)
b.
-2
20
15
175o E
10
0
Depth (km)
c.
5
Depth (km)
3
179o 30' W
2
1
0
-2
-4
0
e.
4
-1
-2
0
Depth (km)
d.
-1
Ice
Shelf
175 o W
Bottom
(cm s -1 )
-2
-4 160o W
-84
-82
-80
-78
Latitude
-76
-74
-72
Figure 9. The combined standard deviations in the depth-dependent velocity anomalies (difference
between the depth-dependent and depth-independent velocities) over 15 days at a) 169o 30’ E, b) 175o E,
c) 179o 30’ W, d) 175o W, and e) 160o W. The M2 critical latitude is indicated by a dashed white line.
41
Robertson: Baroclinic and Barotropic Tides in the Ross Sea
0
Depth (km)
a.
o
-1 169 30' E
0
c.
2
-2
175o E
critical
latitude
-1
M
Depth (km)
b.
1E-001
5E-002
Depth (km)
0
Depth (km)
d.
1E-002
-1
o
-2 179 30' W
0
1E-003
-2
Ice
Shelf
-4
0
Depth (km)
e.
5E-003
175o W
Bottom
(s
-1
)
-2
-4 160o W
-84
-82
-80
-78
-76
Latitude
-74
-72
Figure 10. The ave rage of the magnitudes of the two horizontal vertical shears in the depthdependent velocities along N-S transects at a) 169o 30’ E, b) 175o E, c) 179o 30’ W, d) 175o W,
and e) 160o W. The M2 critical latitude is indicated by a dashed white line.
42