1. Introduction

Various numerical errors arising from the baroclinic pressure gradient 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 baroclinic pressure gradient error in sigma coordinate models, considering specifically the Princeton Ocean Model (POM) (described in Blumberg and Mellor 1987; Mellor 1998; 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 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 baroclinic pressure gradient 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 M₂ 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 M₂ critical latitude ψ_crit, which is the latitude (ψ) at which the tidal frequency ω_tide equals the inertial frequency f. This newly identified error is associated with the incorrect treatment of time-varying surface elevation (η) 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 ψ_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 1998) and so are not repeated here. The topographic, hydrographic, and boundary conditions used are fully described in Robertson (1999) and Robertson (2001a,b). 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 M₂ tidal currents in the Weddell Sea (Robertson 1999), using a quasi-two-dimensional “transect” version of POM in a manner similar to that of Holloway (1996). The model was applied to a transect across the continental shelf and slope (Fig. 1), where ψ_crit(M₂) (≈74°28′S) is located near the shelf break. Wave dynamics are modified in the vicinity of ψ_crit, since it is the poleward limit for both internal (Kundu 1990) and Poincaré wave propagation (Middleton and Denniss 1993). Near ψ_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; Furevik and Foldvik 1996). Boundary effects are seen in both mooring data and analytical models, that is, 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, θ, and salinity, S). The model was forced by specifying the M₂ barotropic tidal elevations at the northern and southern ocean boundaries. In this configuration, an energetic depth-dependent oscillation developed over the continental slope. Both the magnitude and phase of the oscillation, which was initially interpreted as variation at the M₂ frequency, had a complex cross-slope structure (Figs. 2a and 3a). [For Fig. 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 ψ_crit to the south of the domain, the cross-slope velocity field was nearly constant with depth except, as expected, for the benthic boundary layer (Fig. 2b).

Is this depth-dependent oscillation in the critical-latitude case real? Velocity profiles with such a strong benthic signal, greater than 10 cm s⁻¹ in 1000-m water depth, at the M₂ 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 cross-slope depth-dependent velocity, V, expressed in sigma coordinates (x, y, σ). Here, x and y are the alongslope and cross-slope horizontal distances, respectively, and σ is the fractional vertical distance in the water column from the seabed to the free surface (−1 ≤ σ ≤ 0):

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In (1), U is the alongslope depth-dependent velocity, W′ is the vertical velocity, η is the surface elevation, and t is time. Depth D = H + η is the water column thickness, where H is the water depth for the system at rest. The perturbation density ρ′ 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 (m² s⁻¹), and the vertical viscosity coefficient K_M (m² s⁻¹). The momentum balance consists of terms for [in order of their appearance in (1)] acceleration, advection in the alongslope, cross-slope, 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, σ) coordinates, show that the BPG is at least 1–2 orders of magnitude smaller than the Coriolis term but varies significantly with y and σ, 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, near-resonant 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, C-grid 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:

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In (2), ρ′ is the perturbation density defined by

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In (3), ρ is the in situ density and ρ_MEAN is a horizontally averaged, time-independent initial in situ density field. To maximize numerical accuracy, ρ (and, thus, also ρ_MEAN), is normalized as

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where ρ^′_P,θ,S is the dimensional in situ density written as the sum of a function of θ and S plus a function of pressure, that is,

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In (4), ρ_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 meters per second, ρ_θ,S is the potential density, ρ_p is given by

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and the hydrostatic pressure is approximated by

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In (6) and (7), the factors 10⁴ and 10⁻⁴ convert between pressures in Pascals and in decibars. The sound speed c is approximated by

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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 ρ_MEAN does not include the time-varying pressure due to η(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 θ and S are constant everywhere. For simplicity, it was further assumed that c² is constant. Then it was expected that ρ′ = 0 everywhere. Indeed, the terms dependent on θ and S cancel, however,

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since the depth at which the pressure is calculated is slightly different in ρ and ρ_MEAN by ση = σ(D − H) meters. Note that ρ_o here represents the 1000 in (4). Now, defining two constants, α = gc⁻² and β = 2 × 10⁻⁵ρ_og²c⁻⁴, ρ′ becomes

ρασηβσ²D²H²(10)

Defining the integrand in (2) to be BC, it became

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Equation (11), after some arithmetic, reduces to

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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 η∂H/∂y will exceed H∂η/∂y over the continental slope, since the topographic scales of variability are generally smaller than the wavelength of the barotropic tide. Thus, neglecting H∂η/∂y and the second half of the first term, it was found that BC ∝ ησ∂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

_errorησ²Hy.(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 η 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 C grid is

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where j and k refer to the indices of the location being determined on the C grid in the cross-slope 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 C grid.

Since the BPG_error arises from η, 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 ψ_crit, BPG_error is nearly resonant with f and can, therefore, induce inertial oscillations whose amplitudes increase as the model time-stepping 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 would scale approximately with BPG_error (13), that is, U_e ∝ σ². In the high-viscosity run (Fig. 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 σ 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 (Fig. 6a). The difference between it and the stratified case (Fig. 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 ψ_crit, and even there, lateral diffusion of momentum due to horizontal viscosity will slowly detune the error inertial mode from the surface tidal forcing. Away from ψ_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 η(t). This beat period is simply τ_b = (ω_tide − |f|)⁻¹. Values of τ_b exceed 100 days for a band ∼3° wide around ψ_crit (Fig. 4). However, τ_b decreases to <10 days near ψ = 66°. Thus, at this latitude, during 5 days the phase between the forced inertial mode and the tidal forcing will shift by 180°, 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°N of ψ_crit so that the shelf break was near 65°S, supports this conclusion. In this case the effect of BPG_error was negligible (Fig. 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). Cross-slope variation arises from changes in bottom slope ∂H/∂y, the reduced efficiency of the resonance forcing as the meridional distance from ψ_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 ψ. Additional modifications to the basic expectations based on (13) occur due to vertical and lateral viscosity, and ignoring H∂η/∂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 ψ_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 η in the calculation of perturbation density ρ′ in (3). It was choosen to correct this error by using ρ directly, without subtracting ρ_MEAN. It is then possible to simplify further to ρ = ρ_θ,S (i.e., the potential density), since the contribution of ρ_P[p(σD)/c²)] 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 θ and S. For the pressure effects to exactly cancel, however, the pressure contributions of ρ_P must be determined at the location of the velocity point using derivatives and not calculated using differences on the C grid.

The primary justification for the present approach, subtracting ρ_MEAN from ρ, 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 ρ′ in (3) is already very close to being a normalized potential density, since the removal of ρ_MEAN also removes the pressure contribution to in situ density ρ.

The cancellation of the pressure contribution can be easily seen if the normalized in situ density, ρ, is split into two portions—one from the potential temperature and salinity, ρ_N, and one from the pressure, ρ_P:

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The BPG would then be determined from these two portions:

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Furthermore, the pressure portion of density, ρ_P, is determined at the velocity point being calculated using the pressure relation:

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Taking the derivatives in the alongsigma and cross-sigma directions, while treating c as a constant, yields

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Substituting these relations into BPG results in

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After canceling terms, this reduces to

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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 along sigma and cross-sigma 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 θ, S, p, and H.

A second possible method for eliminating BPG_error would be to set η = 0 in ρ_P used in the calculation of ρ, and then continue to use the perturbation density ρ′ (3) with ρ_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 Brunt–Väisälä 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, ρ, instead of the perturbation density, ρ′, was used for determining the BPG term in the baroclinic pressure gradient subroutine (BAROPG); in other words, ρ_MEAN was not removed from ρ 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 (ρ_o) based on the potential temperature and salinity biases at the ocean surface, normalizing the density by ρ_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 (4)–(8) to

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These corrections are, however, minor compared with removing the η-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 σ 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 re-run, for both a homogeneous and a realistically stratified ocean. In the homogeneous case (Fig. 5a), the velocity field now appears well behaved over the slope when compared with the uncorrected model run (Fig. 2a). Note that Fig. 2a shows the effect of diffusion of momentum upwards from the bottom-intensified 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 Fig. 6a) because the model-generated vertical viscosity K_M is very much smaller than in the homogeneous case. In a stratified case, depth-dependent velocity structure in the corrected model run (Fig. 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 M₂ tidal ellipse major axes with the uncorrected model (Fig. 6a) was not present when POM was modified (Fig. 6b). Internal tides also existed in the unmodified version of POM with stratification, but were obscured by the signal due to BPG_error (Fig. 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 15 of the 19 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 is given in another study (Robertson 2001a,b).

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 it is therefore recommended that the solutions be implemented even when the model is applied to regions well removed from critical latitude effects.