A perfectly balanced method for estimating the internal pressure gradients in σ-coordinate ocean models

Abstract The estimation of the internal pressure gradients (IPG) in σ -coordinate ocean models has been addressed both in text books and in many research papers. In this paper a perfectly balanced method for estimating internal density and pressure gradients is suggested. The method is perfect in the sense that for cases with ρ  =  ρ ( z ), where ρ is density and z the vertical coordinate, the numerical estimates of the density and pressure gradients are zero. The method has in addition another important property: for continuous stratification, the estimates of the internal pressure gradients vary continuously with changes in the stratification. The properties of the method are investigated using two very simple vertical column test cases, the seamount case, and two Nordic Seas test cases, one with ρ  =  ρ ( z ) and another more realistic case with ρ  =  ρ ( x , y , z ). For the seamount case and the simple Nordic Seas case, the errors are orders of magnitude smaller than the corresponding errors reported in earlier papers. For the simple vertical column case with non-zero density gradients, the estimates of the gradients produced with the new method converge quadratically towards the true values as the horizontal and vertical grid sizes both tend to zero. The new method may be regarded as a modified second order method calibrated such that the errors are zero for ρ  =  ρ ( z ).

[1]  L. Slørdal,et al.  THE PRESSURE GRADIENT FORCE IN SIGMA-CO-ORDINATE OCEAN MODELS , 1997 .

[2]  M. D. Sikirić,et al.  A new approach to bathymetry smoothing in sigma-coordinate ocean models , 2009 .

[3]  S. G. L. Smith,et al.  Numerical and Analytical Estimates of M2 Tidal Conversion at Steep Oceanic Ridges , 2006 .

[4]  James C. McWilliams,et al.  A method for computing horizontal pressure‐gradient force in an oceanic model with a nonaligned vertical coordinate , 2003 .

[5]  Changsheng Chen,et al.  An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Ocean Model: Application to Coastal Ocean and Estuaries , 2003 .

[6]  John M. Gary,et al.  Estimate of Truncation Error in Transformed Coordinate, Primitive Equation Atmospheric Models , 1973 .

[7]  M. Piggott,et al.  A Nonhydrostatic Finite-Element Model for Three-Dimensional Stratified Oceanic Flows. Part II: Model Validation , 2004 .

[8]  A. Ciappa.  A method for reducing pressure gradient errors improving the sigma coordinate stretching function: An idealized flow patterned after the Libyan near-shore region with the POM , 2008 .

[9]  Robert L. Haney,et al.  On the Pressure Gradient Force over Steep Topography in Sigma Coordinate Ocean Models , 1991 .

[10]  S. Østerhus,et al.  The Iceland–Faroe inflow of Atlantic water to the Nordic Seas , 2003 .

[11]  George L. Mellor,et al.  A Generalization of a Sigma Coordinate Ocean Model and an Intercomparison of Model Vertical Grids , 2002 .

[12]  Peter C. Chu,et al.  Sixth-order difference scheme for sigma coordinate ocean models , 1997 .

[13]  Julie D. Pietrzak,et al.  On the pressure gradient error in sigma coordinate ocean models: A comparison with a laboratory experiment , 1999 .

[14]  C. Estournel,et al.  Low-order pressure gradient schemes in sigma coordinate models: The seamount test revisited , 2009 .

[15]  John D. McCalpin.  A comparison of second‐order and fourth‐order pressure gradient algorithms in a σ‐co‐ordinate ocean model , 1994 .

[16]  Hernan G. Arango,et al.  Developments in terrain-following ocean models: intercomparisons of numerical aspects , 2002 .

[17]  Jarle Berntsen,et al.  Estimation of the internal pressure gradient in σ-coordinate ocean models: comparison of second-, fourth-, and sixth-order schemes , 2010 .

[18]  Y. Song,et al.  A General Pressure Gradient Formulation for Ocean Models. Part II: Energy, Momentum, and Bottom Torque Consistency , 1998 .

[19]  Dale B. Haidvogel,et al.  Numerical Simulation of Flow around a Tall Isolated Seamount. Part I: Problem Formulation and Model Accuracy , 1993 .

[20]  M. Batteen,et al.  On reducing the slope parameter in terrain-following numerical ocean models , 2006 .

[21]  A. J. Przekwas,et al.  A comparative study of advanced shock-capturing shcemes applied to Burgers' equation , 1992 .

[22]  Stephen M. Griffies,et al.  Fundamentals of Ocean Climate Models , 2004 .

[23]  Laurent White,et al.  High-order regridding-remapping schemes for continuous isopycnal and generalized coordinates in ocean models , 2009, J. Comput. Phys..

[24]  M. Levine,et al.  A Correction to the Baroclinic Pressure Gradient Term in the Princeton Ocean Model , 2001 .

[25]  George L. Mellor,et al.  The Pressure Gradient Conundrum of Sigma Coordinate Ocean Models , 1994 .

[26]  B. Ådlandsvik,et al.  Production of monthly mean climatological archives for the Nordic Seas , 1998 .

[27]  G. Mellor.  USERS GUIDE for A THREE-DIMENSIONAL, PRIMITIVE EQUATION, NUMERICAL OCEAN MODEL , 1998 .

[28]  A. Ciappa.  An operational comparative test of z-levels PGF schemes to reduce pressure gradient errors of the ocean model POM , 2006 .

[29]  J. Beckers,et al.  On the use of the σ-coordinate system in regions of large bathymetric variations , 1992 .

[30]  André B. Fortunato,et al.  Evaluation of horizontal gradients in sigma-coordinate shallow water models , 1996 .

[31]  G. Mellor,et al.  Development of a turbulence closure model for geophysical fluid problems , 1982 .

[32]  P. Chu,et al.  Hydrostatic correction for sigma coordinate ocean models , 2003 .

[33]  J. Berntsen,et al.  Development of eddies in an idealised shelf slope area due to an along slope barotropic jet , 2006 .

[34]  G. Stelling,et al.  On the approximation of horizontal gradients in sigma co‐ordinates for bathymetry with steep bottom slopes , 1994 .

[35]  H. Sundqvist.  On Truncation Errors in Sigma‐System Models , 1975 .

[36]  Nadia Pinardi,et al.  Ocean forecasting : conceptual basis and applications , 2002 .

[37]  A. Blumberg,et al.  A Description of a Three‐Dimensional Coastal Ocean Circulation Model , 2013 .

[38]  Y. Song,et al.  A General Pressure Gradient Formulation for Ocean Models. Part I: Scheme Design and Diagnostic Analysis , 1998 .

[39]  K. A. Orvik,et al.  Atlantic inflow to the Nordic Seas: current structure and volume fluxes from moored current meters, VM-ADCP and SeaSoar-CTD observations, 1995–1999 , 2001 .

[40]  Shian-Jiann Lin,et al.  A finite‐volume integration method for computing pressure gradient force in general vertical coordinates , 1997 .

[41]  L. Oey,et al.  Sigma Coordinate Pressure Gradient Errors and the Seamount Problem , 1998 .

[42]  J. Molines,et al.  A sigma-coordinate primitive equation model for studying the circulation in the South Atlantic. Part I: Model configuration with error estimates , 1998 .

[43]  Jarle Berntsen,et al.  Internal pressure errors in sigma-coordinate ocean models due to anisotropy , 2006 .

[44]  A. Arakawa,et al.  Numerical methods used in atmospheric models , 1976 .

[45]  U. Shaked,et al.  H∞ nonlinear filtering , 1996 .

[46]  D. Haidvogel,et al.  A semi-implicit ocean circulation model using a generalized topography-following coordinate system , 1994 .

[47]  J. Berntsen.  Internal Pressure Errors in Sigma-Coordinate Ocean Models , 2002 .

[48]  Robert N. Miller.  Numerical Modeling of Ocean Circulation , 2007 .

[49]  J. Berntsen,et al.  Estimating the internal pressure gradient errors in a σ-coordinate ocean model for the Nordic Seas , 2007 .

[50]  H. Berntsen,et al.  Efficient numerical simulation of ocean hydrodynamics by a splitting procedure , 1981 .

[51]  C. Estournel,et al.  Sigma Coordinate Pressure Gradient Errors: Evaluation and Reduction by an Inverse Method , 2000 .

[52]  R. Dmowska,et al.  International Geophysics Series , 1992 .

[53]  Jarle Berntsen,et al.  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 , 2005 .