Upwind‐weighted advection schemes for ocean tracer transport: An evaluation in a passive tracer context

Centered-in-space, centered-in-time integration has generally been used for the advection of scalars in ocean models. An assessment is made of the implications of centered leapfrog integration in the context of two-dimensional passive tracer advection within a Stommel (1948) gyre. Nonphysical ripples in the tracer field grow to alarming levels in purely advective integrations. Diffusive parameterizations of eddy mixing moderate these ripples, but it is found that Laplacian diffusion greatly reduces the peak amplitude of the tracer field, while biharmonic or weaker Laplacian diffusion allows ripples of large amplitude. Several forward-in-time, upwind-weighted schemes are found to provide better solutions. Smolarkiewicz's (1984) Multi-Dimensional Positive-Definite Advection and Transport Algorithm (MPDATA) scheme is slightly superior for an integration at moderate resolution within which the western boundary current is poorly resolved in typical fashion. Third-order, upwind-based schemes exhibit little sensitivity to the details of multidimensional treatment for this problem of passive tracer advection, with results nearly as good as for MPDATA.

[1]  W. Grabowski,et al.  The multidimensional positive definite advection transport algorithm: nonoscillatory option , 1990 .

[2]  David P. Stevens,et al.  A New Tracer Advection Scheme for Bryan and Cox Type Ocean General Circulation Models , 1995 .

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

[4]  Frank O. Bryan,et al.  An Overlooked Problem in Model Simulations of the Thermohaline Circulation and Heat Transport in the Atlantic Ocean , 1995 .

[5]  G. Strang On the Construction and Comparison of Difference Schemes , 1968 .

[6]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[7]  Willem Hundsdorfer,et al.  Method of lines and direct discretization: a comparison for linear advection , 1994 .

[8]  S. Zalesak Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[9]  J. Toole,et al.  Estimates of Diapycnal Mixing in the Abyssal Ocean , 1994, Science.

[10]  Andrew J. Watson,et al.  Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment , 1993, Nature.

[11]  P. Smolarkiewicz A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small Implicit Diffusion , 1984 .

[12]  K. Bryan A Numerical Method for the Study of the Circulation of the World Ocean , 1997 .

[13]  B. P. Leonard,et al.  Positivity-preserving numerical schemes for multidimensional advection , 1993 .

[14]  Richard B. Rood,et al.  Application of a Monotonic Upstream-biased Transport Scheme to Three-Dimensional Constituent Transport Calculations , 1991 .

[15]  Rüdiger Gerdes,et al.  The influence of numerical advection schemes on the results of ocean general circulation models , 1991 .

[16]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[17]  David L. Williamson,et al.  Review of Numerical Approaches for Modeling Global Transport , 1992 .

[18]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[19]  B. P. Leonard,et al.  The Flux-integral Method for Multidimensional Convection and Diffusion , 1994 .

[20]  P. Rasch Conservative Shape-Preserving Two-Dimensional Transport on a Spherical Reduced Grid , 1994 .

[21]  Piotr K. Smolarkiewicz,et al.  On Forward-in-Time Differencing for Fluids , 1991 .

[22]  Frank O. Bryan,et al.  Parameter sensitivity of primitive equation ocean general circulation models , 1987 .

[23]  B. P. Leonard,et al.  The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection , 1991 .