Conservation of properties in a free-surface model

In height coordinate ocean models, natural conservation of tracers (temperature, salinity or any passive tracer) requires that the thickness of the surface cell varies with the free-surface displacement, leading to a non-linear free-surface formulation (NLFS). However, NLFS does not guarantee exact conservation unless special care is taken in the implementation, and in particular the time stepping scheme, as pointed out by Griffies et al. (Monthly Weather Rev. 129 (2001) 1081). This paper presents a general method to implement a NLFS in a conservative way, using an implicit free surface formulation. Details are provided for two tracer time stepping schemes, both second order in time and space: a two time-level scheme, such as Lax–Wendroff scheme, guarantees exact tracer conservation;a three time-level scheme such as the Adams–Bashforth II requires further adaptations to achieve exact local conservation and accurate global conservation preventing long term drift of the model tracer content. No compromise is required between local and global conservation since the method accurately conserves any tracer. In addition to the commonly used backward time stepping, the implicit free surface formulation also offers the option of a Crank–Nickelson time stepping which conserves the energy. The methods are tested in idealized experiments designed to emphasize problems of tracer and energy conservation. The tests show the ability of the NLFS method to conserve tracers, in contrast to the linear free-surface formulation. At test of energy conservation reveals that free-surface backward time-stepping strongly damps the solution. In contrast, Crank–Nickelson time stepping exactly conserves energy in the pure linear case and confirms the NLFS improvement relative to the linear free-surface when momentum advection is included. � 2003 Elsevier Ltd. All rights reserved.

[1]  R. Huang,et al.  Real Freshwater Flux as a Natural Boundary Condition for the Salinity Balance and Thermohaline Circulation Forced by Evaporation and Precipitation , 1993 .

[2]  L. Perelman,et al.  A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers , 1997 .

[3]  E. Maier‐Reimer,et al.  The Hamburg Ocean primitive equation model - HOPE , 1996 .

[4]  S. Panchev.  Introduction to Dynamic Meteorology (Kinematics of the Atmospheric Motions) , 1985 .

[5]  W. Krauss.  The Warmwatersphere of the North Atlantic Ocean , 1996 .

[6]  Detlef Stammer,et al.  Generation and distribution of mesoscale eddies in the North Atlantic Ocean , 1996 .

[7]  J. Dukowicz,et al.  Implicit free‐surface method for the Bryan‐Cox‐Semtner ocean model , 1994 .

[8]  A. Adcroft,et al.  Representation of Topography by Shaved Cells in a Height Coordinate Ocean Model , 1997 .

[9]  H. Hasumi,et al.  Developments in ocean climate modelling , 2000 .

[10]  S. Griffies,et al.  Tracer Conservation with an Explicit Free Surface Method for z-Coordinate Ocean Models , 2001 .

[11]  H. Stommel,et al.  The westward intensification of wind‐driven ocean currents , 1948 .

[12]  K. Ruddick,et al.  A coastal ocean model intercomparison study for a three-dimensional idealised test case , 1998 .

[13]  D. Webb,et al.  The Development of a Free-Surface Bryan–Cox–Semtner Ocean Model , 1991 .

[14]  L. Perelman,et al.  Hydrostatic, quasi‐hydrostatic, and nonhydrostatic ocean modeling , 1997 .

[15]  Gurvan Madec,et al.  Salt conservation, free surface, and varying levels: A new formulation for ocean general circulation models , 2000 .

[16]  A. Arakawa.  Computational design for long-term numerical integration of the equations of fluid motion: two-dimen , 1997 .