Inert Trace Constituent Transport in Sigma and Hybrid Isentropic–Sigma Models. Part I: Nine Advection Algorithms

Fred M. Reames Space Science and Engineering Center, University of Wisconsin—Madison, Madison, Wisconsin

Search for other papers by Fred M. Reames in
Current site
Google Scholar
PubMed
Close
and
Tom H. Zapotocny Space Science and Engineering Center, University of Wisconsin—Madison, Madison, Wisconsin

Search for other papers by Tom H. Zapotocny in
Current site
Google Scholar
PubMed
Close
Restricted access

We are aware of a technical issue preventing figures and tables from showing in some newly published articles in the full-text HTML view.
While we are resolving the problem, please use the online PDF version of these articles to view figures and tables.

Abstract

The University of Wisconsin hybrid isentropic–sigma (θ–σ) coordinate channel model and the nominally identical sigma (σ) model are used to test the relative capabilities of nine trace constituent transport algorithms. The nine are “standard” second-order finite differencing, the standard with two local “borrow and fill” fixers, the standard with a global fixer, four conservative flux-integrated approaches, and the conservation of second-order moments (CSOM). Transport of two analytically specified initial trace constituent distributions is simulated within a common initial atmosphere, which includes a baroclinically amplifying synoptic-scale wave. Two different vertical resolution θ–σ models and four vertical resolution σ models provide excellent test beds for comparison of the transport algorithms because their 48-h predictions of standard synoptic fields are virtually identical.

Although no analytic solution exists against which detailed comparisons can be made, the constraint of adiabatic conditions for a continuum provides that the maximum of a trace constituent within explicit or implicit isentropic layers of a model should be conserved throughout the simulations, and that the area between any two trace constituent contours on an isentropic surface should remain constant. With these conditions as bases of comparison, several results are unambiguous. First, in the σ models the standard with fixers is better than the other schemes except for the CSOM at the highest resolution. Second, in the θ–σ models, the piecewise parabolic and CSOM schemes produce results approximately as accurate as the standard with fixers. Third, when comparing all algorithms, models, resolutions, and distributions, the CSOM scheme produces the most consistent results. Finally, for a large majority of the cases, the θ–σ models perform more accurately than the σ models with respect to the conservation of constituent extrema.

* Current affiliation: Department of Mechanical Engineering, University of Wisconsin—Madison, Madison, Wisconsin.

Corresponding author address: Dr. Fred M. Reames, SSEC, University of Wisconsin—Madison, 1225 West Dayton Street, Madison, WI 53706-1695.

Abstract

The University of Wisconsin hybrid isentropic–sigma (θ–σ) coordinate channel model and the nominally identical sigma (σ) model are used to test the relative capabilities of nine trace constituent transport algorithms. The nine are “standard” second-order finite differencing, the standard with two local “borrow and fill” fixers, the standard with a global fixer, four conservative flux-integrated approaches, and the conservation of second-order moments (CSOM). Transport of two analytically specified initial trace constituent distributions is simulated within a common initial atmosphere, which includes a baroclinically amplifying synoptic-scale wave. Two different vertical resolution θ–σ models and four vertical resolution σ models provide excellent test beds for comparison of the transport algorithms because their 48-h predictions of standard synoptic fields are virtually identical.

Although no analytic solution exists against which detailed comparisons can be made, the constraint of adiabatic conditions for a continuum provides that the maximum of a trace constituent within explicit or implicit isentropic layers of a model should be conserved throughout the simulations, and that the area between any two trace constituent contours on an isentropic surface should remain constant. With these conditions as bases of comparison, several results are unambiguous. First, in the σ models the standard with fixers is better than the other schemes except for the CSOM at the highest resolution. Second, in the θ–σ models, the piecewise parabolic and CSOM schemes produce results approximately as accurate as the standard with fixers. Third, when comparing all algorithms, models, resolutions, and distributions, the CSOM scheme produces the most consistent results. Finally, for a large majority of the cases, the θ–σ models perform more accurately than the σ models with respect to the conservation of constituent extrema.

* Current affiliation: Department of Mechanical Engineering, University of Wisconsin—Madison, Madison, Wisconsin.

Corresponding author address: Dr. Fred M. Reames, SSEC, University of Wisconsin—Madison, 1225 West Dayton Street, Madison, WI 53706-1695.

Save
  • Allen, D. J., A. R. Douglass, R. B. Rood, and P. D. Guthrie, 1991: Application of a monotonic upstream-biased transport scheme to three-dimensional constituent transport calculations. Mon. Wea. Rev.,119, 2456–2464.

  • Carpenter, R. L., Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 1990: Application of the piecewise parabolic method (PPM) to meteorological modeling. Mon. Wea. Rev.,118, 586–612.

  • Chahine, M. T., 1992: The hydrological cycle and its influence on climate. Nature,359, 373–380.

  • Denning, A. C., I. Y. Fung, and D. Randall, 1995: Latitudinal gradient of atmospheric CO2 due to seasonal exchange with land biota. Nature,376, 240–243.

  • Johnson, D. R., T. H. Zapotocny, F. M. Reames, B. J. Wolf, and R. B. Pierce, 1993: A comparison of simulated precipitation by hybrid isentropic–sigma and sigma models. Mon. Wea. Rev.,121, 2088–2114.

  • Kurihara, V., and J. L. Holloway, 1967: Numerical integration of a nine level global primitive equation model formulated by the box method. Mon. Wea. Rev.,95, 509–529.

  • Lin, S.-J., W. C. Chao, Y. C. Sud, and G. K. Walker, 1994: A class of the Van Leer-type transport schemes and its application to the moisture transport in a general circulation model. Mon. Wea. Rev.,122, 1575–1593.

  • Liu, G., J. A. Curry, and R.-S. Sheu, 1995: Classification of clouds over the western equatorial Pacific Ocean using combined infrared and microwave satellite data. J. Geophys. Res.,100, 13 811–13 826.

  • Mahlman, J. D., and R. W. Sinclair, 1977: Tests of various numerical algorithms applied to a simpler trace constituent air transport problem. Fate of Pollutants in the Air and Water Environments, I. H. Suffet, Ed., Vol. 8, John Wiley, 223–252.

  • Matsuno, T., 1966: Numerical integrations of the primitive equations by a simulated backward difference method. J. Meteor. Soc. Japan,44, 768–784.

  • Navarra, A., W. F. Stern, and K. Miyakoda, 1994: Reduction of Gibbs oscillation in spectral model simulations. J. Climate,7, 1169–1183.

  • Pielke, R. A., and Coauthors, 1995: Standardized test to evaluate numerical weather prediction algorithms. Bull. Amer. Meteor. Soc.,76, 46–48.

  • Pierce, R. B., D. R. Johnson, F. M. Reames, T. H. Zapotocny, and B. J. Wolf, 1991: Numerical investigations with a hybrid isentropic–sigma model. Part I: Normal-mode characteristics. J. Atmos. Sci.,48, 2005–2024.

  • Prather, M. J., 1986: Numerical advection by conservation of second-order moments. J. Geophys. Res.,91, 6671–6681.

  • Reames, F. M., and T. H. Zapotocny, 1999: Inert trace constituent transport in sigma and hybrid isentropic–sigma models. Part II:Twelve semi-Lagrangian algorithms. Mon. Wea. Rev.,127, 188–200.

  • Rood, R. B., 1987: Numerical advection algorithms and their role in atmospheric transport and chemistry models. Rev. Geophys.,25, 71–100.

  • Temperton, C., 1976: Dynamic initialization for barotropic and multi-level models. Quart. J. Roy. Meteor. Soc.,102, 297–311.

  • Thuburn, J., 1993: Use of a flux-limited scheme for vertical advection in a GCM. Quart. J. Roy. Meteor. Soc.,119, 469–487.

  • Van Leer, B., 1974: Towards the ultimate conservative difference scheme. Part II: Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys.,14, 361–370.

  • ——, 1977a: Toward the ultimate conservative difference scheme. Part III: Upstream-centered finite-difference schemes for ideal incompressible flow. J. Comput. Phys.,23, 263–275.

  • ——, 1977b: Toward the ultimate conservative difference scheme. Part IV: A new approach to numerical convection. J. Comput. Phys.,23, 276–299.

  • ——, 1979: Toward the ultimate conservative difference scheme. Part V: A second order sequel to Godunov’s method. J. Comput. Phys.,32, 101–136.

  • Zalesak, S. T., 1979: Fully multidimensional flux-corrected transport for fluids. J. Comput. Phys.,31, 335–362.

  • Zapotocny, T. H., D. R. Johnson, F. M. Reames, R. B. Pierce, and B. J. Wolf, 1991: Numerical investigations with a hybrid isentropic–sigma model. Part II: The inclusion of moist processes. J. Atmos. Sci.,48, 2025–2043.

  • ——, ——, and ——, 1993: A comparison of regional isentropic–sigma and sigma model simulations of the January 1979 Chicago blizzard. Mon. Wea. Rev.,121, 2115–2135.

  • ——, A. J. Lenzen, D. R. Johnson, T. K. Schaack, and F. M. Reames, 1997: A comparison of inert trace constituent transport between the University of Wisconsin isentropic–sigma model and the NCAR Community Climate Model. Mon. Wea. Rev.,125, 120–142.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 637 551 260
PDF Downloads 33 19 1