Sigma Coordinate Ocean Model Research Articles

A high-resolution modelling study of the circulation along the Campania coastal system, with a special focus on the Gulf of Naples

A high-resolution modelling study of the circulation along the Campania coastal system (CCS) in the southern Tyrrhenian Sea is presented. The area includes three adjacent, semi-enclosed shallow basins (the gulfs of Gaeta, Naples and Salerno) facing the open deep sea, and represents an ideal site for investigating relevant dynamical processes of general interest in coastal oceanography. A Campania Regional sigma-coordinate Ocean Model (CROM) has been implemented in this region, with a 1/144° resolution; nesting with an operational circulation model covering the whole Tyrrhenian Sea with a 1/48° resolution is performed. Forcing is provided by surface momentum, heat, and freshwater fluxes computed from the non-hydrostatic SKIRON/Eta atmospheric modelling system outputs. A winter and a summer period of the reference year 2009 are analyzed in detail. The relative importance of the flow induced by remote large-scale currents through topographic interactions and of that induced locally by the wind are found to vary, even over a weekly time scale, in all of the three gulfs; the Gulf of Salerno appears to be the location where remote and local forcing are more often competing. An analysis of the high frequency variability shows that, besides the current changes induced by the typical mid-latitude atmospheric synoptic variability, inertial currents in the open sea and sea breeze-induced currents in the gulfs are present. Model-data comparison is performed in the CCS with altimeter data and satellite-derived turbidity distributions, and in the Gulf of Naples with the latter and with measurements performed with a high-frequency radar system. Significant model-data agreement is generally found.

Intrinsic variability of the Antarctic Circumpolar Current system: low- and high-frequency fluctuations of the Argentine Basin flow

Abstract. In this paper, the variability of the Antarctic Circumpolar Current system produced by purely intrinsic nonlinear oceanic mechanisms is studied through a sigma-coordinate ocean model, implemented in a large portion of the Southern Ocean at an eddy-permitting resolution under steady surface heat and momentum fluxes. The mean transport through the Drake Passage and the structure of the main Antarctic Circumpolar Current fronts are well reproduced by the model. Intrinsic variability is found to be particularly intense in the Subantarctic Front and in the Argentine Basin, on which further analysis is focused. The low-frequency variability at interannual timescales is related to bimodal behavior of the Zapiola Anticyclone, with transitions between a strong and collapsed anticyclonic circulation in substantial agreement with altimeter observations. Variability on smaller timescales shows clear evidence of topographic Rossby-wave propagation along the eastern and southern flanks of the Zapiola Rise and of mesoscale eddies, also in agreement with altimeter observations. The analysis of the relationship between the low- and high-frequency variability suggests possible mechanisms of mutual interaction.

An accurate implementation of the compressibility terms in the equation of state in a low order pressure gradient scheme for sigma coordinate ocean models

In a previous study, the authors studied a low order pressure gradient force (PGF) scheme referred to as the Primitive-Modified scheme, that appears to be equivalent to the Pressure-Jacobian PGF ( Lin, 1997). The scheme was successfully tested on the seamount experiment using a simplified equation of state (EOS). Yet, a complete equation of state, including compressibility effect, can raise a serious problem of accuracy. A new implementation is thus proposed in the present paper. The scheme is rewritten using a (numerically equivalent) geopotential formulation. The PGF truncation errors are removed by computing the EOS compressibility terms with potential temperature and salinity interpolated on a suitable geopotential level. The so-called Equivalent Geopotential Formulation (EGF) method is compared to the Finite-Volume approach proposed by Adcroft et al. (2008).

Estimation of the internal pressure gradient in σ-coordinate ocean models: comparison of second-, fourth-, and sixth-order schemes

Sigma-coordinate ocean models are attractive because of their abilities to resolve bottom and surface boundary layers. However, these models can have large internal pressure gradient (IPG) errors. In this paper, two classes of methods for the estimation of the IPGs are assessed. The first is based on the integral approach used in the Princeton Ocean Model (POM). The second is suggested by Shchepetkin and McWilliams (2003) based on Green’s theorem; thus, area integrals of the pressure forces are transformed into line integrals. Numerical tests on the seamount problem, as well as on a northwestern Atlantic grid using both classes of methods, are presented. For each class, second-, fourth-, and sixth-order approximations are tested. Results produced with a fourth-order compact method and with cubic spline methods are also given. The results show that the methods based on the POM approach in general give smaller errors than the corresponding methods given in Shchepetkin and McWilliams (2003). The POM approach also is more robust when noise is added to the topography. In particular, the IPG errors may be substantially reduced by using the computationally simple fourth-order method from McCalpin (1994).

A new approach to bathymetry smoothing in sigma-coordinate ocean models

σ -coordinate ocean models are currently used in large basin and coastal seas simulations. When there is a strongly varying bathymetry, σ -coordinate system solutions exhibit an intrinsic error in the horizontal pressure gradient (HPG) term. In this paper, a new, linear programming (LP) procedure has been proposed to deal with this problem. The success of the procedure in alleviating HPG error is gauged in comparison with two implementations of the Shapiro filter, the Mellor–Ezer–Oey (1994) algorithm (MEO), and the recently (2006) proposed procedure of Martinho and Batteen (MB). All are applied to three idealised cases and then to the Adriatic Sea, as modelled with the Rutgers version of ROMS. The comparisons reveal that even the improved Shapiro filter creates very large modifications to the bathymetry and should be avoided. The MB procedure performance is comparable to that of LP when the slope factor is larger than approximately 0.2, and inferior to it at smaller slope factor values. In cases where volume conservation is required, the MB method is inferior to both MEO and LP. The difference between the above-mentioned solutions is more pronounced for coarser grids.

On reducing the slope parameter in terrain-following numerical ocean models

Sigma coordinate ocean models, such as the Princeton Ocean Model, are a type of terrain-following model, which are currently being used in regions with large topographic variability such as entire ocean basins, shelf breaks, continental shelves, estuaries and bays. The main concern when using a terrain-following ocean model is to reduce the pressure gradient force error (PGFE). Regardless of the method of calculation of the pressure gradient, the PGFE will not be reduced to an acceptable value without first reducing the slope parameter, defined by the absolute value of the ratio of the difference between two adjacent cell depths and their mean depth. Here two methods for reducing the slope parameter are compared: a traditional two-dimensional smoothing with Gaussian filters and an alternative one-dimensional robust direct iterative technique. While both methods efficiently smooth the bottom topography so that the pressure gradient errors are reduced to acceptable levels, the alternative method is shown to have a unique advantage of maintaining coastline irregularities, continental shelves, and relative maxima such as seamounts and islands.

Internal pressure errors in sigma-coordinate ocean models due to anisotropy

Sigma-coordinate ocean models, or models based on more generalized topography following coordinate systems, are presently widely used in oceanographic studies. Terrain following models are attractive because of their abilities to resolve the surface and bottom layers. However, the internal pressure gradient estimation is problematic in such models, and artificial pressure gradients may create artificial flow. In fine resolution studies where topography and stratification is well resolved, the artificial flow due to erroneous pressure gradients is small because the errors typically converge to zero with the square of the grid size. In coarse resolution studies, however, there may still be large erroneous pressure gradients that may create strong artificial flow. To avoid this, larger horizontal viscosities than one would like to apply from physical considerations, are often applied. This means that smaller scale phenomena in the flow will be poorly represented. The response of the flow to the pressure errors in long time simulations with low horizontal viscosity is very important and different ways to reduce these pressure errors have been suggested. The present study focus on how the grid orientation affect the internal pressure errors in sigma-coordinate ocean models. It is shown that if the internal pressure gradients are computed both along the original axes and along coordinate axes that are rotated 45° to the original axes, the erroneous velocities are substantially reduced if the two internal pressure gradient estimates are averaged properly.

An operational comparative test of z-levels PGF schemes to reduce pressure gradient errors of the ocean model POM

Numerical schemes for computing pressure gradient force (PGF) in sigma-coordinate ocean models are usually tested in absence of external forcing and with a horizontally uniform stratification, a condition where the ocean should be at rest. In this study PGF schemes based on z-coordinate levels are compared with the standard density scheme of Princeton Ocean Model (POM) in tests of this kind with different topographies and stratifications (including the idealized seamount problem and a realistic case in a basin-wide domain) to investigate if idealized results are verified in realistic cases. Results of the performed tests show that all the tested schemes produce stronger spurious currents in the realistic case respect to the seamount problem and the behavior of z-levels schemes, less robust than the standard POM scheme, is more stable over a realistic topography than in the seamount case. In the test realized with realistic topography and stratification spurious barotropic currents produced by z-levels schemes are strongly reduced respect to the POM standard scheme and the ‘spline interpolation’ scheme performs the best among the z-coordinate schemes, giving spurious barotropic currents of a few cm/s. Considering that pre-operational tests of this kind (unforced and with a flat stratification) can give useful indications on the effects of PGF errors in realistic cases (baroclinic, externally forced) on condition that bathymetry, stratification and distribution of grid cells is the same of the operational case, the reduced level of errors indicates the possibility to use z-levels PGF schemes in practical applications.

Internal pressure errors in sigma-coordinate ocean models—sensitivity of the growth of the flow to the time stepping method and possible non-hydrostatic effects

Sigma coordinate ocean models, or models based on more generalised topography following coordinate systems, are presently widely used in oceanographic studies. Terrain following models are attractive because of their abilities to resolve the surface and bottom layers. However, the internal pressure gradient estimation is problematic in such models, and artificial pressure gradients may create artificial flow. In fine resolution studies where topography and stratification is well resolved, the artificial flow due to erroneous pressure gradients is small because the errors typically converge to zero with the square of the grid size. In coarse resolution studies, however, there may still be large erroneous pressure gradients that may create strong artificial flow. To avoid this, larger horizontal viscosities than one would like to apply from physical considerations, are applied in most studies so far. This means that smaller scale phenomena in the flow will be poorly represented. The response of the flow to the pressure errors in long time simulations with low horizontal viscosity is very important. In this study it is demonstrated that the choice of the time stepping method may affect the growth of the erroneous flow considerably. Increasing the implicit nature of the time stepping method, will generally reduce the growth. Recently many oceanic model studies have been performed with models that allow non-hydrostatic pressure effects. In the present study it is shown that when allowing the flow to adjust to non-hydrostatic pressure, the growth of the flow due to internal pressure errors may be strongly affected.

Simulating Upwelling in a Large Lake Using Slippery Sacks*

A Lagrangian numerical model is used to simulate upwelling in an idealized large lake. This simulation is carried out to test the model’s potential for simulating lake and ocean circulations. The model is based on the slippery sack (SS) method that was recently developed by the authors. It represents the lake as a pile of conforming sacks. The motions of the sacks are determined using Newtonian dynamics. The model uses gravity wave retardation to allow for long time steps and has pseudo-Eulerian vertical mixing. The lake is exposed to northerly winds for 29 h. Upwelling develops in the eastern edge of the basin, and after the winds shut off, upwelling fronts propagate around the lake. This case was previously simulated using a height- and sigma-coordinate ocean model. The SS model produces circulations that are similar to those produced by the other models, but the SS simulation exhibits less mixing.

Hydrostatic correction for sigma coordinate ocean models

How to reduce the horizontal pressure gradient error is a key issue in terrain‐following coastal models. The horizontal pressure gradient splits into two parts, and incomplete cancellation of the truncation errors of those parts cause the error. Use of the finite volume discretization leads to a conserved scheme for pressure gradient computation that has better truncation properties with high accuracy. The analytical coastal topography and seamount test cases are used to evaluate the new scheme. The accuracy of the new scheme is comparable to the sixth‐order combined compact scheme (with an error reduction by a factor of 70 comparing to the second‐order scheme) with mild topography and much better than the sixth‐order combined compact scheme with steep topography. The computational efficiency of the new scheme is comparable to the second‐order difference scheme. The two characteristics, high accuracy and computational efficiency, make this scheme useful for the sigma coordinate ocean models.

Numerical simulation of the Gulf Stream System: The Loop Current and the deep circulation

The Loop Current and the deep circulation in the Gulf of Mexico are numerically investigated by a primitive equation, sigma coordinate ocean model with realistic surface fluxes obtained from an atmospheric forecast model. A deep cyclonic circulation, bounded by the deep basin in the eastern Gulf of Mexico, is spun up by the Loop Current; the deep cyclonic circulation is coincident with a southward current of the Loop Current eastern limb and weakens after Loop Current ring separation and cessation of the southward current. The anticyclonic, semienclosed Loop Current also induces anticyclonic lower layer columnar eddies in the eastern gulf. These lower layer eddies decouple from the upper layer Loop Current. The westward translation speed of a Loop Current ring is about 2.16–5.18 km d−1; the lower layer eddies have a higher speed and lead the rings into the central gulf. The time‐averaged surface circulation of the Gulf of Mexico basin is anticyclonic, mainly because of the transport of anticyclonic vorticity by Loop Current rings in the surface layer an average lower layer cyclonic circulation occurs along the continental slope of the basin.

Internal Pressure Errors in Sigma-Coordinate Ocean Models

Abstract Sigma-coordinate ocean models, or models based on more generalized topography following coordinate systems, are presently widely used in oceanographic studies. The controversy over internal pressure errors in sigma-coordinate ocean models is, however, still worrisome to at least some of the users. In the present study, experiments with the seamount case using both constant and large horizontal viscosities and a Smagorinsky-type viscosity are performed. For the constant viscosity (2000 m2 s−1) case, the errors do not grow prognostically. For the more realistic case with Smagorinsky viscosity and a value of the viscosity parameter in the range usually recommended, 0.2, the errors grow very strongly prognostically. Large vertical transports associated with the eight cyclones and anticyclones around the seamount create strong and real internal pressures that add on to the initial erroneous internal pressure. The growth may be balanced by multiplying the viscosity parameter by approximately a factor o...

On the seasonal mixed layer simulated by a basin‐scale ocean model and the Mellor‐Yamada turbulence scheme

Seasonal changes and vertical mixing processes in the upper layers of the North Atlantic Ocean are simulated with a basin‐scale sigma coordinate ocean model that uses the Mellor‐Yamada turbulence closure scheme. The cause of insufficient surface mixing and a too shallow summertime thermocline, common problems of ocean models of this type, is investigated in detail by performing a series of sensitivity experiments with different surface forcing conditions and different turbulence parameterizations. A recent improvement in the parameterization of the dissipation term in the Mellor‐Yamada turbulence scheme, which has shown a significant improvement in one‐dimensional calculations, had a positive but relatively small influence on the three‐dimensional calculations. The results quantify the improvement in the model upper ocean thermal structure as surface forcing becomes more realistic from one experiment to another, for example, when monthly mean winds are replaced by 6 hour variable winds. The inclusion of shortwave radiation penetration is especially important to prevent overly shallow model mixed layers during the summer and seems to affect not only the surface layer but also the thermal structure of the upper 200 m of the ocean. The difficulty of evaluating turbulent mixing processes in three‐dimensional models due to errors in surface fluxes, spatial changes, and three‐dimensional effects, as shown here, points to the important role still left for one‐dimensional turbulence models in improving parameterizations used in three‐dimensional realistic models.

Horizontal Pressure Gradient Errors of the Monterey Bay Sigma Coordinate Ocean Model with Various Grids

A coastal ocean σ-coordinate model of Monterey Bay (MOB) with realistic bottom topography and coastlines is developed using the Princeton Ocean Model (POM) and grid generation technique (GGT) to study the horizontal pressure gradient errors associated with the MOB steep topography. The submarine canyon in MOB features some of the steepest topography encountered anywhere in the world oceans. The MOB grids are designed using the EAGEAL View and GENIE++ grid generation systems. A grid package developed by Ly and Luong (1993) is used in this study to couple grids to the model. The MOB model is tested with both orthogonal and curvilinear nearly-orthogonal (CNO) grids. The CNO grid has horizontal resolution which varies from 300 m to 2 km, while the resolution of the orthogonal grid is uniform with δx = 1.25 km and δy = 1.38 km. These grids cover a domain of 180 × 160 km with the same number of grid points of 131 × 131. Vertical resolutions of 25, 35 and 45 vertical sigma levels are tested. The error in the MOB are evaluated in terms of mean kinetic energy and velocity against various grids, vertical, horizontal resolution and σdistributions, and bottom topography smoothing. Simulations with various grids show that GGT can be used as another tool in reducing σ-coordinate errors in coastal ocean modeling besides increasing resolution and smoothing bottom topography. Topographical smoothing not only reduces topographic slope, but changes realistic topography. A CNO grid with a high grid density packed along steep slopes and Monterey Submarine Canyon reduces the errors by 40% compared to a rectangular grid with the same number of grid points. The CNO grid is more efficient than the rectangular grid, since it has most of its grids over water. The simulations show that the presented MOB σ-coordinate model can be used with a confidence regarding horizontal pressure gradient error.

Simulations of the Atlantic Ocean with a free surface sigma coordinate ocean model

A sigma coordinate, free surface numerical model with turbulence dynamics has been implemented for the Atlantic Ocean and the Greenland Sea, from 80°S to 80°N. It is driven at the surface by monthly mean sea surface temperature and wind stress climatologies and is executed for 30 years. This is the first time that a model of this type, previously used mostly for coastal and regional simulations, has been implemented for the entire Atlantic Ocean and run for a long period of time. The model horizontal circulation, thermohaline overturning circulation, and meridional heat fluxes are described; the results are compared with observations and the results of other models. The model produces intense deep western boundary currents and complicated gyre structures associated with small‐scale topographic variations. The meridional overturning circulation consists of about 14 Sv (1 Sv=106 m3 s−1) of southward flowing deep water mass crossing the equator and a northward heat flux with a maximum value of more than 1 PW (1015 W). Although the maximum meridional heat flux is comparable to estimates obtained from observations, the amplitude of the seasonal variations of northward heat flux across 26°N is underestimated in comparison with observations; it is similar to that obtained by other models.

Sea level variations induced by heating and cooling: An evaluation of the Boussinesq approximation in ocean models

In this paper a sigma coordinate ocean model is modified to remove the commonly used Boussinesq approximation so that the effect of thermal expansion is exactly included in the basic equations in order to cope with the seasonal heating cycle and the detection of climate change through variation in sea level height. Tests are performed to evaluate the differences between Boussinesq and non‐Boussinesq calculations under different heating and cooling conditions and different model domains. For an idealized case of a flat bottom, shallow ocean basin without wind forcing, simulations of a warm eddy show that the non‐Boussinesq dynamics have only a minor effect on the baroclinic current field. However, vertically averaged velocities, though small compared with the baroclinic velocities, are cyclonic for the Boussinesq calculation and anticyclonic for the non‐Boussinesq calculation. The results indicate that global or closed basin Boussinesq models should be able to simulate most of the observed steric sea level changes on seasonal or climate timescales, when corrected by a spatially uniform, time‐dependent factor calculated from the volume‐averaged density change. The seasonal variation of the globally averaged sea level calculated from climatological data is small, about 1 cm. Variations in steric sea level in regional models, both Boussinesq and non‐Boussinesq, may differ from those of global models owing to the unknown transport across their boundaries associated with the local heating and cooling. A spatially uniform, time‐dependent correction, similar to that associated with thermal expansion, is proposed to account for transport across open boundaries of regional models. Variations of sea level obtained from a Boussinesq model of the Atlantic Ocean approximate the seasonal signal due to the heating/cooling cycle of each hemisphere as observed by satellite altimeter data.

The Pressure Gradient Conundrum of Sigma Coordinate Ocean Models

Abstract Much has been written of the error in computing the horizontal pressure gradient associated with sigma coordinates in ocean or atmospheric numerical models. There also exists the concept of “hydrostatic inconsistency” whereby, for a given horizontal resolution, increasing the vertical resolution may not be numerically convergent. In this paper, it is shown that the differencing scheme cited here, though conventional, is not hydrostatically inconsistent; the sigma coordinate, pressure gradient error decreases with the square of the vertical and horizontal grid size. Furthermore, it is shown that the pressure gradient error is advectively eliminated after a long time integration. At the other extreme, it is shown that diagnostic calculations of the North Atlantic Ocean using rather coarse resolution, and where the temperature and salinity and the pressure gradient error are held constant, do not exhibit significant differences when compared to a calculation where horizontal pressure gradients are c...

Diagnostic and prognostic calculations of the North Atlantic circulation and sea level using a sigma coordinate ocean model

The North Atlantic circulation and sea surface height, determined from hydrographic and wind stress data are calculated with a free surface, primitive equation ocean model. The model grid, a vertical sigma ‐ coordinate and horizontal curvilinear orthogonal system, makes it possible to resolve coastal regions with complicated topography that were unresolved in previous calculations. When running in a diagnostic mode, in which the temperature and the salinity fields are fixed and equal to the Levitus annual mean fields, the vertically integrated North Atlantic circulation is very similar to though more detailed than that obtained from previous calculations using simpler diagnostic models. On the other hand, the meridional, zonally averaged flows and the poleward heat transport from the purely diagnostic calculations are noisy and unrealistic. However, short prognostic calculations of only 30 days following the diagnostic run allow intensification of the western boundary current and removal of noise due to inconsistencies between the hydrographic data and bottom topography and produce a more realistic meridional circulation and poleward heat transport, with a maximum value of 1.2×1015 W which is, comparable to estimates based on observations. The calculated sea level along the North America continent agrees with previous estimates but provides more spatial detail. Analysis of the dynamic adjustment process shows that this process is dominated by the effect of bottom topography through the action of the bottom pressure torque. This study is a first step in applying a realistic numerical model, previously used primarily for estuaries and coastal regions, to basin‐scale ocean problems.

On the Pressure Gradient Force over Steep Topography in Sigma Coordinate Ocean Models

Abstract The error in computing the pressure gradient force near steep topography using terms following (σ) coordinates is investigated in an ocean model using the family of vertical differencing schemes proposed by Arakawa and Suarez. The truncation error is estimated by substituting known buoyancy profiles into the finite difference hydrostatic and pressure gradient terms. The error due to “hydrostatic inconsistency,” which is not simply a space truncation error, is also documented. The results show that the pressure gradient error is spread throughout the water column, and it is sensitive to the vertical resolution and to the placement of the grid points relative to the vertical structure of the buoyancy field being modeled. Removing a reference state, as suggested for the atmosphere by Gary, reduces the truncation error associated with the two lowest vertical modes by a factor of 2 to 3. As an example, the error in computing the pressure gradient using a standard 10-level primitive equation model appl...