A Correction to the Baroclinic Pressure Gradient Term in the Princeton Ocean Model
Robin Robertson^{1}, Laurie Padman^{2}, and Murray D. Levine
College of Oceanic and Atmospheric Sciences, Oregon State University
Corvallis, Oregon ^{1} Now at: LamontDoherty Earth Observatory, Columbia University
Palisades, New York
^{2}Now at: Earth and Space Research, Seattle, Washington
For submission to J. Atmos. Ocean Technology
Revised November 27, 2000
ABSTRACT
An error in the calculation of the baroclinic pressure gradient term in the Princeton Ocean Model (POM) was identified while modeling the M_{2} tidal currents near its critical latitude in the southern Weddell Sea. The error arises from the present calculation of density, which involves the subtraction of a background density profile from the density field calculated at each internal time step. The small displacement of sigma surface depths relative to the surface, as surface elevation changes, causes a slight error in the calculation of the vertical and horizontal gradients of potential density. The error is largest at the seabed over rapidly changing bathymetry such as the continental slope. The baroclinic pressure gradient error is typically much smaller than the Coriolis term in the momentum equations, and therefore usually unimportant. Close to the critical latitude, however, nearresonance between the error and Coriolis terms can cause an energetic and spatially complex spurious inertial mode to develop. The error is significant when modeling tides near their critical latitudes, and will contribute to the error in the baroclinic pressure gradient in other simulations. Two methods were suggested for fixing this problem. The preferred method was tested by applying the new form of POM to the southern Weddell Sea. The new results are consistent with both current meter data and predictions of linear internal wave theory.
1. INTRODUCTION
Various numerical errors arising from the baroclinic pressure gradient (“BPG”) term in sigma coordinate models have been previously discussed [e.g., Kleim and Pietrzak, 1999; Mellor et al., 1994, 1998]. Kleim and Pietrzak [1999] evaluate the BPG error in sigma coordinate models, considering specifically the Princeton Ocean Model (POM) [described in Blumberg and Mellor, 1989; Mellor, 1996; Robertson, 1999]. For cases where the ocean surface is treated as a rigid lid, they state that “The pressure gradient error arises owing to truncation error in the discretization of (1) [the baroclinic pressure gradient equation].” The standard version of POM, however, uses a free surface boundary condition, which leads to an additional source of error in the BPG term. This error, which is the topic of this paper, can produce a spatially complex inertial oscillation and was identified while modeling the vertical structure of M_{2} tidal currents in the Weddell Sea [Robertson, 1999]. Baroclinic tides are generally understood to be generated through interactions of the barotropic tide and the continental slope. The continental slope in this region is close to the M_{2} critical latitude _{crit}, which is the latitude () at which the tidal frequency _{tide} equals the inertial frequency f. This newly identified error is associated with the incorrect treatment of timevarying surface elevation () in the calculation of in situ density. The error term is orders of magnitude smaller than the dominant term (Coriolis) in the momentum equations. Close to _{crit}, however, the error can resonantly force an inertial oscillation that can grow to exceed the original energy in the barotropic tide.
First, the error and its source are described (section 2). Details of POM appear elsewhere [e.g., Mellor, 1996] and so are not repeated here. The topographic, hydrographic and boundary conditions used are fully described in Robertson [1999] and Robertson [2000a; 2000b]. The modifications that were made to POM to eliminate the error are described in section 3, and comparisons of model results before and after the modification are presented in section 4.
2. THE PROBLEM
The research that led to identification of this error was a study of the depth variability of M_{2} tidal currents in the Weddell Sea [Robertson, 1999], using a quasitwodimensional “transect” version of POM in a similar manner as that used by Holloway [1996]. The model was applied to a transect across the continental shelf and slope (Figure 1), where _{crit}(M_{2}) (74^{o} 28’ S) is located near the shelf break. Wave dynamics are modified in the vicinity of _{crit}, since it is the poleward limit for both internal [Kundu, 1990] and Poincaré wave propagation [Middleton and Denniss, 1993]. Near _{crit}, inertial oscillations can be induced by the tide, and the momentum boundary layer for the tidal constituent can become thick and complex, even without stratification effects [Foldvik et al., 1990; Nøst, 1994; Furevik and Foldvik, 1996]. Boundary effects are seen in both mooring data and analytical models, i.e., they are not simply numerical model artifacts of the type reported herein.
As part of the diagnosis of suspected errors in the baroclinic version of POM, the model was applied to a homogeneous ocean (uniform potential temperature, , and salinity, S). The model was forced by specifying the M_{2} barotropic tidal elevations at the northern and southern ocean boundaries. In this configuration, an energetic depthdependent oscillation developed over the continental slope. Both the magnitude and phase of the oscillation, which was initially interpreted as variation at the M_{2} frequency, had a complex crossslope structure (Figures 2a and 3a). (For Figure 3a, the major axes of the tidal ellipses were determined using Foreman’s routines [Foreman, 1978] applied only to the last 2 days of the model run.) When POM was used with the same topography and forcing but with _{crit} to the south of the domain, the crossslope velocity field was nearly constant with depth except, as expected, for the benthic boundary layer (Figure 2b).
Is this depthdependent oscillation in the criticallatitude case real? Velocity profiles with such a strong benthic signal, greater than 10 cm s^{1} in 1000 m of water depth, at the M_{2} frequency have not been observed in current meter measurements made in this region [Middleton and Foster, 1977; Middleton et al., 1982; Foldvik et al., 1990; Woodgate et al., 1998]. These observations, combined with the strong dependence of the amplitude of the final velocity field on model run length, persuaded us that the oscillation was somehow an artifact of the numerical scheme.
What then, is the apparently numerical cause of these oscillations? This problem was approached by examining the “baroclinic mode” momentum equation for the crossslope depthdependent velocity, V, expressed in sigma coordinates (x, y, ). Here, x and y are the alongslope and crossslope horizontal distances, respectively, and is the fractional vertical distance in the water column from the seabed to the free surface (1 0).
(1)
In (1), U is the alongslope depthdependent velocity, W’ is the vertical velocity, is the surface elevation, and t is time. Depth D= H+ is the water column thickness, where H is the water depth for the system at rest. The perturbation density ’ will be described more fully later. Other variables are the Coriolis parameter f, gravitational acceleration g (9.8 m s^{2}), horizontal viscosity coefficient A_{M} (m^{2} s^{1}), and the vertical viscosity coefficient K_{M} (m^{2} s^{1}). The momentum balance consists of terms for (in order of their appearance in (1)): acceleration, advection in the alongslope, crossslope, and vertical directions, Coriolis force, surface elevation pressure gradient, baroclinic pressure gradient (BPG), vertical diffusion, and horizontal diffusion in that order. Only the Coriolis force, BPG, and diffusion terms varied with depth in the POM simulation for a homogeneous ocean. The BPG in this case, however, should be zero, indicating that the BPG is a probable source of error.
Time series (not shown) of the various terms in (1) for the homogeneous case and for various (y, ) coordinates, show that the BPG is at least 1 to 2 orders of magnitude smaller than the Coriolis term but varies significantly with y and , being greatest at the seabed over the continental slope. The effect of this error acceleration term in the momentum equations would normally be small since the term itself is small relative to other terms (in this case the Coriolis term). When the frequency of the error BPG is close to f, however, nearresonant forcing may cause the growth of a significant inertial mode during a sufficiently long model run.
Plausible candidates for the specific feature of the BPG calculation that produces the error include truncation errors, Cgrid errors, and a systematic numerical error in the calculation of p/y. Kleim and Pietrzak [1999] reported truncation errors in POM, which result from the differencing of two large nearly equal values used in the present BPG calculation. Due to the use of the sigma coordinate system, the BPG term in (1) is the difference of two terms:
. (2)
In (2), ’ is the perturbation density defined by
(3)
In (3), is the in situ density and _{MEAN} is a horizontallyaveraged, timeindependent initial in situ density field. To maximize numerical accuracy, (and thus also _{MEAN}), is normalized as
(4)
where _{P,}_{}_{,S} is the dimensional in situ density written as the sum of a function of and S plus a function of pressure, i.e.,
_{P,}_{}_{,S}( x, y, , t) = _{}_{,S} ((x, y, , t), S(x, y, , t)) + _{p} (p/c^{2}) (5)
In (4), _{o} is a general mean density, which was set to 1025 (or 1000) in the original POM.
In (5), c is the sound speed in m s^{1}, _{}_{,S} is the potential density, and _{p} is given by
(6)
and the hydrostatic pressure is approximated by
(7)
In (6) and (7), the factors 10^{4} and 10^{4} convert between pressures in Pascals and in decibars. The sound speed c is approximated by
c = 1449.1+0.00821p+4.550.45^{2}+1.34(S35) (8)
Note that the constant in (8), 1449.1, is as in the original version of the POM code. This form of the equation follows Mellor [1991]; however, the constant in (8) in Mellor [1991] is quoted as 1449.2 whereas it was 1449.1 in the original POM code, which was used. This difference is not expected to contribute significantly to this error. A full discussion of the errors in the equation of state is given in Mellor [1991].
The focus of this paper is the error that arises in the BPG because _{MEAN} does not include the timevarying pressure due to (t) that is included in the calculation of p in (7). While this error is very small, it will be shown to be significant under certain conditions that are experienced in the real ocean. To demonstrate, consider the case of an unstratified ocean where and S are constant everywhere. For simplicity, it was further assumed that c^{2} is constant. Then it was expected that ’ = 0 everywhere. Indeed, the terms dependent on and S cancel, however,
(9)
since the depth at which the pressure is calculated is slightly different in and _{MEAN} by = (DH) meters. Note that _{o} here represents the 1000 in (4). Now, defining two constants, = gc^{2} and = 2x10^{5}_{o}g^{2}c^{4}, ’ becomes
’ =   ^{2}(D ^{2} H^{2}) (10)
Defining the integrand in (2) to be BC, it became
(11)
Equation (11), after some arithmetic, reduces to
(12)
In (12), the second half of the first term is much smaller than the first half due to the c^{4} factor; furthermore, it is expected that H/y will exceed H/y over the continental slope, since the topographic scales of variability are generally smaller than the wavelength of the barotropic tide. Thus, neglecting H/y and the second half of the first term, it was found that BC H/y. Since the error BPG (denoted BPG_{error}) is the downward integral (from the sea surface) of BC, it is given to leading order by
BPG_{error} ^{2}H/y. (13)
As expected from the model simulations with the uncorrected version of POM, (13) predicts that the strongest error forcing is at the seabed over steep topography. The value of BPG_{error} is proportional to so that a rigid lid upper boundary condition, as in the cases studied by Kleim and Pietrzak [1999], removes this particular source of error. The exact BPG_{error} in POM as determined on the Cgrid is
(14)
where j and k refer to the indices of the location being determined on the Cgrid in the crossslope and vertical directions, respectively. Note that this term is of the same form as (12), however, with additional factors resulting from the use of a Cgrid.
Since the BPG_{error} arises from , which in this problem is primarily due to the barotropic tide that is forced by the northern and southern boundary conditions, it fluctuates at the tidal frequency. Near _{crit}, BPG_{error} is nearly resonant with f and can, therefore, induce inertial oscillations whose amplitudes increase as the model timestepping progresses. This occurs even though the magnitude of BPG_{error} is much less than the Coriolis term.
Two factors act to ultimately stabilize the spurious inertial oscillations forced by BPG_{error}. First, as velocities increase, the increased benthic friction and momentum diffusion slow further growth of the inertial oscillations. The effect of friction was tested with a comparison of two homogeneous model runs, one using the values of K_{M} generated by POM, and the other with friction eliminated by removing the vertical diffusion of momentum. Without friction, it was expected that the magnitude U_{e} of the error inertial mode for fixed y to scale approximately with BPG_{error} (13), i.e., U_{e}^{2}. In the highviscosity run (Figure 2a), vertical profiles of V over the slope are quite smooth whereas, for the no friction case (not shown), U_{e} is much higher near the seabed as expected from the quadratic dependence of BPG_{error} on in (13). For this case, the spatial structure of U_{e} is qualitatively similar to that shown in the major axes for the stratified case (Figure 6a). The difference between it and the stratified case (Figure 6a) is a reduced signal in the upper water column, which was due to the generation of a slight internal tidal signal in the stratified case. The effects of stratification are twofold: it may generate internal tides and it reduces the vertical viscosity coefficient, correspondingly adjusting the benthic boundary layer thickness.
Second, true resonance only exists precisely at _{crit}: even there, lateral diffusion of momentum due to horizontal viscosity will slowly detune the error inertial mode from the surface tidal forcing. Away from _{crit}, the period during which BPG_{error} will amplify the energy of the spurious inertial mode is some fraction of the “beat period” of the inertial mode and the tidal periodicity of (t). This beat period is simply _{b}=(_{tide}f)^{1}. Values of _{b} exceed 100 days for a band ~3^{o} wide around _{crit} (Figure 4). However, _{b} decreases to <10 days near =66^{o}. Thus at this latitude, during 5 days the phase between the forced inertial mode and the tidal forcing will shift by 180^{o}, and the maximum amplification of the inertial mode by BPG_{error} will be limited to the growth of the mode during a fraction of this time interval. A model run with a homogeneous ocean, and the domain shifted 10^{o} N of _{crit} so that the shelf break was near 65^{o}S, supports this conclusion. In this case the effect of BPG_{error} was negligible (Figure 2b).
It is then seen how a complex structure can develop in a velocity field, when the velocity time series are analyzed for just the end of the model run. Although spectral analysis indicates that the variability occurred near the forced tidal frequency, it is unable to resolve the inertial and tidal frequencies, which results in the inertial oscillations being attributed to tidal motion. Depth dependence arises from the quadratic dependence of BPG_{error} on depth (13). Crossslope variation arises from changes in bottom slope H/y, the reduced efficiency of the resonance forcing as the meridional distance from _{crit} increases, and because, at any given time, the phase relationship between the true tidal current and the forced inertial mode varies as a function of . Additional modifications to the basic expectations based on (13) occur due to vertical and lateral viscosity, and ignoring H/y and the second term in (12). Since most previous internal tide investigations, such as those of Holloway [1996] on the Australian slope or Chen and Beardsley [1998] over Georges Bank, have not been located near _{crit} for the tidal constituent being studied, these studies have not encountered this resonant error inertial response. This error also exists for cases with stronger stratification; however, the error is masked by the more intense internal tidal field, which develops with the stronger stratification.
3. MODEL MODIFICATIONS
The error in the BPG calculation arises through the incorrect handling of in the calculation of perturbation density ’ in (3). It was choosen to correct this error by using directly without subtracting _{MEAN}. It is then possible to simplify further to _{ = } _{}_{,S} (i.e, the potential density), since the contribution of _{P}(p(D)/ c^{2}) to BC is identically zero that is, the pressure effects are exactly removed when the horizontal gradient of density is calculated for the BPG, leaving only the contribution from variations of and S. For the pressure effects to exactly cancel, however, the pressure contributions of _{P} must be determined at the location of the velocity point using derivatives and not calculated using differences on the Cgrid.
The primary justification for the present approach, subtracting _{MEAN} from , is to reduce the dynamic range of density in the calculations of the BPG and the buoyancy frequency N. The results, shown in the following section, indicate that the perturbation approach is not necessary, at least with weak stratification when working in double precision. It should be noted that ’ in (3) is already very close to being a normalized potential density, since the removal of _{MEAN} also removes the pressure contribution to in situ density .
The cancellation of the pressure contribution can be easily seen if the normalized insitu density, , is split into two portions; one from the potential temperature and salinity, _{N}, and one from the pressure, _{P}.
(16)
The BPG would then be determined from these two portions:
. (17)
Furthermore, the pressure portion of density, _{P}, is determined at the velocity point being calculated using the pressure relation:
. (18)
Taking the derivatives in the alongsigma and crosssigma directions while treating c as a constant yields
and (19)
, respectively. (20)
Substituting these relations into BPG results in
. (21)
After canceling terms, this reduces to
. (22)
Since the gradients of the pressure portion cancel each other, only the potential temperature and salinity portion remains. It is easy to see that with a homogeneous ocean, the gradients of the potential temperature and salinity portion would also be zero, as would BPG. The dependence of c in the alongsigma and crosssigma directions has been ignored in these derivatives. An error analysis has determined that it is negligible, <10^{10} % given realistic values for the maximum gradients of , S, p, and H.
A second possible method for eliminating BPG_{error} would be to set = 0 in _{p} used in the calculation of , and then continue to use the perturbation density ’ (3) with _{MEAN} removed. If this is done, the pressure adjustments for the density and background density exactly cancel each other in (3) and the result is equivalent. However, this second solution method unnecessarily increases the number of computations performed and can also increase the truncation error.
To implement the above modification in POM, the pressure correction to density was removed in the density subroutine (DENS) as was the pressure correction to the BruntVäisälä frequency in the subroutine which determines the vertical viscosity (PROFQ). Additionally, since stratification is weak in the region of interest and the majority of the change in the background density is due to pressure, the normalized density, , instead of the perturbation density,_{ }’, was used for determining the BPG term in the baroclinic pressure gradient subroutine (BAROPG), in other words, _{MEAN,} was not removed from before the calculation.
Other modifications were made to POM to increase the accuracy of the density calculation. These modifications include using double precision for all variables, using a mean density (_{o}) based on the potential temperature and salinity biases at the ocean surface, normalizing the density by _{o} instead of 1000, and using 1449.2 instead of 1449.1 in the calculation of the speed of sound in the density pressure corrections. These modifications effectively change the set of density calculations from (48) to
(23)
These corrections are, however, minor compared with removing the dependent error in the BPG calculation (i.e. removing the pressure contribution to the density).
For these studies of tides in the southern Weddell Sea, POM was also modified to accommodate the existence of an ice shelf. This is unnecessary for most other locations. A term was added for friction at the ice shelf base as a surface stress for the ice shelf portion of the domain. In addition, the pressure head of the ice shelf was accounted for in pressure calculations at depth, with scaled by the water column thickness (depth from seabed to ice base) instead of the bottom depth. The ice shelf was assumed to float freely.
4. TEST OF MODEL AMENDMENTS
Having made the previously described modifications to POM, the Weddell Sea cases were rerun, for both a homogeneous and a realistically stratified ocean. In the homogeneous case (Figure 5a), the velocity field now appears well behaved over the slope when compared with the uncorrected model run (Figure 2a). Note that Figure 2a shows the effect of diffusion of momentum upwards from the bottomintensified oscillation due to BPG_{error}. In a stratified case with the uncorrected POM code, the vertical variation of V is more intensified towards the seabed (see Figure 6a) because the modelgenerated vertical viscosity K_{M} is very much smaller than in the homogeneous case. In a stratified case, depthdependent velocity structure in the corrected model run (Figure 5b) is constrained to the region where linear wave theory predicts internal tide generation, over the upper slope. The large benthic oscillation that was evident in the M_{2} tidal ellipse major axes with the uncorrected model (Figure 6a) was not present when POM was modified (Figure 6b). Internal tides also existed in the unmodified version of POM with stratification, but were obscured by the signal due to BPG_{error} (Figure 6a).Some tidal observations have been made in this region and are fully described in Robertson et al. [1998] and Robertson [1999]. The major axes of the tidal ellipses predicted by the modified version of POM agree within the approximate measurement uncertainty (2 cm s^{1}) for fifteen of the nineteen observations (Table 1). This agreement is encouraging, especially since many of the observations are not exactly along the transect and often have different water depths than at their location mapped onto the transect line. A more complete discussion of the observational data will be given in a future study in preparation [Robertson, 2000a; 2000b].
It was concluded that these modifications to the calculation of density in POM eliminated a small but sometimes significant error in the baroclinic pressure gradient term. Removing this error was found to be crucial to obtaining a satisfactory model solution for baroclinic tide generation in the vicinity of the tidal constituent’s critical latitude. It was not investigated whether this error is primarily a problem only when forcing is purely sinusoidal at a single frequency near f: it is possible that additional energy sources prevent the resonant growth of the inertial mode. However, no known problems were introduced to POM by these proposed solutions, and therefore recommend that the solutions be implemented even when the model is applied to regions well removed from critical latitude effects.
Acknowledgements: This study was supported by NSF, contract OPP9896041. We would like to thank Drs. Andrew Bennett, John Allen, and Ricardo Matano for their help in locating the problem in POM.
REFERENCES
Blumberg, A. F. and G. L. Mellor, 1987: A description of a threedimensional coastal ocean circulation model, in ThreeDimensional Coastal Ocean Models, 4, edited by N. Heaps, pp. 208, AGU, Washington, D. C.
Chen, C., and R. C. Beardsley, 1998: Tidal mixing and crossfrontal particle exchange over a finite amplitude asymmetric bank: A model study of Georges Bank, J. Mar. Res., 56, 11631201.
Foldvik, A., J. H. Middleton, and T. D. Foster, 1990: The tides of the southern Weddell Sea, DeepSea Res., 37, 13451362.
Foreman, M. G. G., 1978, Manual for tidal current analysis and prediction, Pacific Marine Science Report No. 786, Institute of Ocean Sciences, Patricia Bay, Sidney, B.C., 70 pp.
Holloway, P. E., 1996: A numerical model of internal tides with application to the Australian north west shelf, J. Phys. Oceanogr., 26, 2137.
Kleim, N. and J. D. Pietrzak, 1999: On the pressure gradient error in sigma coordinate ocean models: A comparison with a laboratory experiment, J. Geophys. Res, 104, 29,78129,799.
Kundu, P., 1990: Fluid Mechanics, Academic Press, Inc., San Diego, CA, pp. 668.
Mellor, G. L., 1991, An equation of state for numerical modeling of oceans and estuaries, J. Atmos. and Ocean. Tech., 8, 609611.
Mellor, G. L., T. Ezer, and L.Y. Oey, 1994: The pressure gradient conundrum of sigma coordinate models, J. Atmos. Ocean. Tech., 11, 11261134.
Mellor, G. L., 1996: User's guide for a threedimensional, primitive equation, numerical ocean model. Unpublished report, Atmospheric and Ocean Sciences Program, pp. 35, Princeton University, Princeton, NJ.
Mellor, G. L., L.Y. Oey, and T. Ezer, 1998: Sigma coordinate pressure gradient errors and the seamount problem, J. Atmos. Ocean. Tech., 15, 11221131.
Middleton, J. H., and T. Denniss, 1993: The propagation of tides near the critical latitude, Geophys. Astrophys. Fluid Dynam., 68, 113.
Middleton, J. H., and T. D. Foster, 1977: Tidal currents in the central Weddell Sea, DeepSea Res., 24, 11951202.
Middleton, J. H., T. D. Foster, and A. Foldvik, 1982: Lowfrequency currents and continental shelf waves in the southern Weddell Sea, J. Phys. Oceanogr., 12, 618634.
Robertson, R. A., L. Padman, and G. D. Egbert, 1998: Tides in the Weddell Sea, in Ocean, Ice and Atmosphere: Interactions at the Antarctic Continental Margin, Antarc. Res. Series, 75, edited by S. S. Jacobs and R. F. Weiss, 341369 pp., AGU, Washington, D. C..
Robertson, R., 1999: Mixing and heat flux mechanisms in the upper ocean in the Weddell Sea, Ph.D. thesis, 173 pp., Oregon State University, Corvallis, OR.
Robertson, R., 2000a: Internal tides and baroclinicity in the southern Weddell Sea. Part I: Model description and comparison of model results to observations, submitted to J. Geophys. Res.
Robertson, R., 2000b: Internal tides and baroclinicity in the southern Weddell Sea. Part II: Effects of the critical latitude and stratification, submitted to J. Geophys. Res.
Woodgate, R. A., M. Schröder, and S Østerhus, 1998: Moorings from the Filchner Trough and the Ronne Ice Shelf Front: Preliminary Results, FilchnerRonne Ice Shelf Program, Report No. 12, AlfredWegener Institute.
______________
M. D. Levine (College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR, 973315503; ph. 5417373047; email: levine@oce.orst.edu)
L. Padman (Earth and Space Research, 1910 Fairview Ave. E., Suite 102, Seattle, WA, 981023620; ph. 5417536695; email: padman@esr.org)
R. Robertson (LamontDoherty Earth Observatory, Rt. 9W, Palisades, NY 10964; ph. 9143658576; email: rroberts@ldeo.columbia.edu)
List of Figures:
Figure 1. The water column thickness for the Weddell Sea, contoured at 200, 500 (thick line), 1000 (dashed line), and 3000 m. The FilchnerRonne Ice Shelf area is indicated by hatching. The location of the transect used for the model domain is shown as a dashed line.
Figure 2. Crossslope velocity profiles at a time near the peak onshore flow over the continental slope from simulations using the original version of POM without stratification both (a) with and (b) without the critical latitude in the domain. The location of _{crit} is indicated in (a).
Figure 3. The major axes of the tidal ellipses of the depthdependent velocities (cm s^{1}) from simulations with the critical latitude in the domain and a homogeneous ocean from (a) the original version of POM and (b) the modified version of POM.
Figure 4. The beat period (in days) between _{tide} and f as a function of latitude.
Figure 5. Crossslope velocity profiles at a time near the peak onshore flow over the continental slope from simulations using the modified version of POM with the critical latitude in the domain and (a) without and (b) with stratification. The location of _{crit} is indicated in (a). The region where linear internal wave theory predicts internal tides with this location of _{crit} is indicated in (b).
Figure 6. The major axes of the tidal ellipses of the depthdependent velocities (cm s^{1}) from simulations with the critical latitude in the domain and a stratified ocean from (a) the original version of POM and (b) the modified version of POM.
List of Tables:
Table 1. Comparison between POM and RPE predictions and the observations for the major axis of the tidal ellipses (cm s^{1}) at the observation locations. The location and water depth are given for each of the observations along with the principal author and year of the reference. The differences are shown in parentheses. A dash indicates a difference less than the observational uncertainty.
Figure 1. The water column thickness for the Weddell Sea, contoured at 200, 500 (thick line), 1000 (dashed line), and 3000 m. The FilchnerRonne Ice Shelf area is indicated by hatching. The location of the transect used for the model domain is shown as a dashed line.
Figure 2. Crossslope velocity profiles at a time near the peak onshore flow over the continental slope from simulations using the original version of POM without stratification both (a) with and (b) without the critical latitude in the domain. The location of _{crit} is indicated in (a).
Figure 3. The major axes of the tidal ellipses of the depthdependent velocities (cm s^{1}) from simulations with the critical latitude in the domain and a homogeneous ocean from (a) the original version of POM and (b) the modified version of POM.
Figure 4. The beat period (in days) between _{tide} and f as a function of latitude.
Figure 5. Crossslope velocity profiles at a time near the peak onshore flow over the continental slope from simulations using the modified version of POM with the critical latitude in the domain and (a) without and (b) with stratification. The location of _{crit} is indicated in (a). The region where linear internal wave theory predicts internal tides with this location of _{crit} is indicated in (b).
Figure 6. The major axes of the tidal ellipses of the depthdependent velocities from simulations with the critical latitude in the domain and a stratified ocean from (a) the original version of POM and (b) the modified version of POM.
Location

Reference

On or Off

Water Depth (m)

M_{2} Major Axis for Tidal Velocity Ellipse



Transect

Observation

Model

Instrument

Observation

POM






257

5.7

5.3 ()

75^{o} 1’ S

Woodgate

Off

610

362

378

6.3

7.3 ()

31^{o} 46’ W

[1998]



(601)^{}

484

6.4

6.2 ()






590

5.3

2.5 (3)






191

7.1

7.0 ()

75^{o} 2’ S

Woodgate

Off

574

365

342

6.3

7.2 ()

33^{o} 33’ W

[1998]



(582)^{}

448

5.5

5.2 ()






554

3.3

0.9 (2)

74^{o} 40’ S

Middleton^{+}

Off

475

379

375

6.8

7.2 ()

33^{o} 56’ W

[1982]




450

4.5

3.6 ()

74^{o} 26’ S
 Middleton^{+} 
On

475

483

375

6.8

7.6 ()

39^{o} 24’ W

[1982]




450

3.4

2.5 ()

74^{o} 24’ S

Middleton^{+}

On

465

507

400

5.0

7.4 (+2)

39^{o} 6’ W

[1982]




450

2.6

1.2 ()

74^{o} 23’ S
37^{o }39’ W

Foldvik [1990]

Off

475

518

450

2.2

1.3 ()

74^{o} 8’ S
39^{o }19’ W

Foldvik [1990]

On

650

828

627

3.0

0.3 (3)

74^{o} 6’ S
39^{o }22’ W

Middleton^{+} [1982]

On

720

868

620

2.9

2.2 ()

73^{o} 43’ S

Middleton^{+}

On

1915

1906

1815

1.3

1.7 ()

38^{o} 36’ W

[1982]




1890

0.9

1.5 ()

^{+} Foldvik et. al [1990] is a more useful reference for these observations
Differences shown in parentheses
 for difference indicates the value agreed within the uncertainty range
^{} depth for the altered position
Table 1. Comparison between POM and RPE predictions and the observations for the major axis of the tidal ellipses (cm s^{1}) at the observation locations. The location and length of the record are given for each of the observations along with the principal author and year of the reference. The differences are shown in parentheses. A dash indicates a difference less than the observational uncertainty.
