Skip to main content

Part of the book series: NATO ASI Series ((ASIC,volume 243))

Abstract

Some basic aspects of finite-difference methods in climate modeling are discussed. Emphasis is placed on comparisons of the dynamics of discrete systems with that of the original continuous system, including the problem of choosing vertical and horizontal grid structures and resolutions, and the problem of maintaining various integral constraints of physical importance.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Arakawa, A., 1966: ‘Computational design for long-term numerical integration of equations of fluid motion: Two dimensional incompressible flow. Part I.’ J. Comput. Phys., 1, 119–143.

    Article  Google Scholar 

  • Arakawa, A., 1972: ‘Design of the UCLA general circulation model.’ Numerical Simulation of Weather and Climate, UCLA Dept. of Meteor. Tech. Rep. No. 7, Los Angeles, CA, 90024.

    Google Scholar 

  • Arakawa, A., 1984: ‘Vertical differencing of filtered models.’ In ECMWF Seminar 1983, Numerical Methods for Weather Prediction, Vol. 1, pp. 183–206.

    Google Scholar 

  • Arakawa, A., and V. R. Lamb, 1977: ‘Computational design of the basic dynamical processes of the UCLA general circulation model.’ In Methods in Computational Physics, Vol. 17, ed. J. Chang, Academic Press, pp. 173–265.

    Google Scholar 

  • Arakawa, A., and V. R. Lamb, 1981: ‘A potential enstrophy and energy conserving scheme for the shallow water equations.’ Mon. Wea. Rev., 109, 18–36.

    Article  Google Scholar 

  • Arakawa, A., and M. J. Suarez, 1983: ‘Vertical differencing of the primitive equations in sigma coordinates.’ Mon. Wea. Rev., 111, 34–45.

    Article  Google Scholar 

  • Arakawa, A., and S. Moorthi, 1987: ‘Baroclinic instability in vertically discrete systems.’ J. Atmos. Sci. (submitted).

    Google Scholar 

  • Bates, J. R., 1984: ‘Non-linear advection: Lagrangian.’ In Short–and Medium-Range Weather Prediction Research Publication Series, No. 8, WMO, 157–168.

    Google Scholar 

  • Bleck, R., 1984: ‘Linear advection.’ In Short–and Medium-Range Weather Prediction Research Publication Series, No. 8, WMO, 105–116.

    Google Scholar 

  • Bretherton, F. P., 1966: ‘Critical layer instability in baroclinic flows.’ Quart. J. Roy, ileteor. Soc., 92, 325–334.

    Article  Google Scholar 

  • Burridge, D. M., and J. Haseler, 1977: ‘A model for medium range weather forecasting – adiabatic formulation.’ ECMWF Tech. Rep. No. 4, Bracknell, Berks., U.K., 46 pp.

    Google Scholar 

  • Cahn, A., Jr., 1945: ‘An investigation of the free oscillations of a simple current system.’ J. Meteor., 2, 113–119.

    Article  Google Scholar 

  • Chang, J., 1977: Methods in Computational Physics, Vol. 17, General Circulation Models of the Atmosphere. Academic Press, New York.

    Google Scholar 

  • Charney, J. G., 1955: ‘The use of primitive equations of motion in numerical prediction.’ Tellus, 7, 22–26.

    Article  Google Scholar 

  • Charney, J. G., 1971: ‘Geostrophic turbulence.’ J. Atmos. Sci., 28, 1087–1095.

    Article  Google Scholar 

  • Charney, J. G., and N. A. Phillips, 1953: ‘Numerical integration of the quasi-geostrophic equations for barotropic and simple baroclinic flows.’ J. Meteor., 10, 71–99.

    Article  Google Scholar 

  • Eady, E. T., 1949: ‘Long waves and cyclone waves.’ Tellus, 1, 33–52.

    Article  Google Scholar 

  • ECMWF, 1984: ECMWF Seminar 1983 on Numerical Methods for Weather Prediction, Vol. 1 and 2. European Centre for Medium Range Weather Forecasts, Reading, United Kingdom.

    Google Scholar 

  • Fjortoft, R., 1953: ‘On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow.’ Tellus, 5, 225–230.

    Article  Google Scholar 

  • GARP, 1979: ‘Numerical methods used in atmospheric models, Vol. II.’ GARP Publication Series No. 17, WMO/ICSU, Geneva.

    Google Scholar 

  • Grimmer, M., and D. B. Shaw, 1967: ‘Energy-preserving integrations of the primitive equations on the sphere.’ Quart. J. Roy. Meteor. Soc., 93, 337–349.

    Article  Google Scholar 

  • Haltiner, G. J., and R. T. Williams, 1980: Numerical prediction and dynamic meteorology. John Wiley and Sons, 477 pp.

    Google Scholar 

  • Hollingsworth, A., P. Kallberg, V. Renner and D. M. Burridge, 1983: ‘An internal symmetric computational instability.’ Quart. J. Roy.Meteor. Soc., 109, 417–428.

    Article  Google Scholar 

  • Janjic, Z. I., 1977: ‘Pressure gradient force and advection scheme used for forecasting with steep and small scale topography.’ Beitäge zur Physik der Atmosphäre, 50, 186–199.

    Google Scholar 

  • Janjic, Z. I., 1984a: ‚Nonlinear advection schemes and energy cascade on semi-staggered grids.’ Mon. Wea. Rev., 112, 1234–1245.

    Article  Google Scholar 

  • Janjic, Z. I., 1984b: ‘Non-linear advection Eulerian schemes.’ In Short–and Medium-Range Weather Prediction Research Publication Series, No. 8, WMO, 117–156.

    Google Scholar 

  • Janjic, Z. I., and F. Mesinger, 1984: ‘Finite-difference methods for the shallow water equations on various grids.’ In ECMWF Seminar 1983, Numerical Methods for Weather Prediction, Vol. 1, 29–102.

    Google Scholar 

  • Johnson, D. R., and M.-Y. Wei, 1985: ‘The planetary distribution of heat sources and sinks during FGGE.’ Proceedings of the First National Workshop on the Global Weather Experiment, Vol. Two, Part 1, National Academy Press, 299–315.

    Google Scholar 

  • Kasahara, A., 1974: ‘Various vertical coordinate systems used for numerical weather prediction.’ Mon. Wea. Rev.,102, 509–522. (See also Corrigendum in Mon. Wea. Rev., 1975, 103, 664.)

    Article  Google Scholar 

  • Kasahara, A., and W. M. Washington, 1967: ‘NCAR global general circulation model of the atmosphere.’ Mon. Wea. Rev., 95, 389–402.

    Article  Google Scholar 

  • Kasahara, A., and W. M. Washington, 1971: ‘General circulation experiments with a six-layer NCAR model, including orography, cloudiness and surface temperature calculations.’ J. Atmos. Sci., 28, 657–701.

    Article  Google Scholar 

  • Kirkwood, E., and J. F. Derome, 1977: ‘Some effects of the upper boundary condition and vertical resolution on modeling forced, stationary, planetary waves.’ Mon. Wea. Rev.,105, 1239–1312.

    Article  Google Scholar 

  • Kreiss, H., and J. Oliger, 1973: ‘Methods for the approximate solution of time dependent problems.’ GARP Publication Series No. 10, 107 pp.

    Google Scholar 

  • Kurihara, Y., 1965: ‘Numerical integration of the primitive equations on a spherical grid.’ Mon. Wea. Rev., 93, 399–415.

    Article  Google Scholar 

  • Lilly, D. K., 1965: ‘On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems.’ Mon. Wea. Rev., 93, 11–26.

    Article  Google Scholar 

  • Lindzen, R. S., E. S. Batten and J. W. Kim, 1968: ‘Oscillations in atmospheres with tops.’ Mon. Wea. Rev., 96, 133–140.

    Article  Google Scholar 

  • Lorenz, E. N., 1955: ‘Available potential energy and the maintenance of the general circulation.’ Tellus, 7, 157–167.

    Article  Google Scholar 

  • Lorenz, E. N., 1960: ‘Energy and numerical weather prediction.’ Tellus, 12, 364–373.

    Article  Google Scholar 

  • Mechoso, C. R., M. H. Suarez, K. Yamazaki, J. A. Spahr and A. Arakawa, 1982: ‘A study of the sensitivity of numerical forecasts to an upper boundary in the lower stratosphere.’ Mon. Wea. Rev., 110, 1984–1993.

    Article  Google Scholar 

  • Mesinger, F., 1984: ‘Finite-difference methods for the linear advection equation.’ In ECMWF Seminar 1983, Numerical Methods for Weather Prediction, Vol. 1, 1–28.

    Google Scholar 

  • Mesinger, F., and A. Arakawa, 1976: ‘Numerical methods used in atmospheric models.’ GARP Publication Series No. 17, Vol. I, 64 pp.

    Google Scholar 

  • Mesinger, F., and Z. I. Janjic, 1984a: ‘Finite-difference methods for the pressure gradient force and for the hydrostatic equation.’ In ECMWF Seminar 1983, Numerical Methods for Weather Prediction, Vol. 1, 103–158.

    Google Scholar 

  • Mesinger, F., and Z. I. Janjic, 1984b: ‘Pressure gradient force and hydrostatic equation.’ Short–and Medium-Range Weather Prediction Research Publication Series, No. 8, WMO, 175–234.

    Google Scholar 

  • Mihaljan, J. M., 1963: ‘The exact solution of the Rossby adjustment problem.’ Tellus, 15, 150–154.

    Article  Google Scholar 

  • Nakamura, H., 1978: ‘Dynamical effects of mountains on the general circulation of the atmosphere. I. Development of finite-difference schemes suitable for incorporating mountains.’ J. Meteor. Soc. Japan, 56, 317–340.

    Google Scholar 

  • Obukhov, A. M., 1949: ‘On the question of the geostrophic wind.’ Izv. Akad. Nauk SSR Ser. Geograf.-Geofiz., 13, 281–306.

    Google Scholar 

  • Phillips, N. A., 1957a: ‘A coordinate system having some special advantages for numerical forecasting.’ J. Meteor., 14, 184–185.

    Article  Google Scholar 

  • Phillips, N. A., 1957b: ‘A map projection system suitable for large-scale numerical weather prediction.’ J. Meteor. Soc. Japan, 75th Anniversary Volume, 262–267.

    Google Scholar 

  • Phillips, N. A., 1959: ‘An example of non-linear computational instability.’ The Atmosphere and the Sea in Motion. Rossby Memorial Volume, Rockefeller Institute Press, 501–504.

    Google Scholar 

  • Phillips, N. A., W. Blumen and O. Cote, 1960: ‘Numerical weather prediction in the Soviet Union.’ Bull. Amer. Met. Soc., 41, 599–617.

    Google Scholar 

  • Richardson, L. F., 1922: Weather Prediction by Numerical Process. Cambridge University Press, reprinted, 1965, 236 pp.

    Google Scholar 

  • Rossby, C.-G., 1937: ‘On the mutual adjustment of pressure and velocity distributions in certain simple current systems, I.’ J. Marine Res., 1, 15–28.

    Google Scholar 

  • Rossby, C.-G., 1938: ‘On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II.’ J. Marine Res., 1, 239–263.

    Google Scholar 

  • Sadourny, R., 1975: ‘The dynamics of finite-difference models of the shallow-water equations.’ J. Atmos. Sci., 32, 680–689.

    Article  Google Scholar 

  • Sadourny, R., 1984: ‘Entropy coordinate, quasi-geostrophic turbulence and the design of lateral diffusion in general circulation models.’ In ECMWF Seminar 1983, Numerical Methods for Weather Prediction, Vol. 1, 255–290.

    Google Scholar 

  • Sadourny, R., A. Arakawa and Y. Mintz, 1968: ‘Integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere.’ Mon. Wea. Rev., 96, 351–356.

    Article  Google Scholar 

  • Simmons, A. J., and D. M. Burridge, 1981: ‘An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates.’ Mon. Wea. Rev., 109, 758–766.

    Article  Google Scholar 

  • Sundqvist, H., 1979: ‘Vertical coordinates and related discretization.’ In Numerical Methods Used in Atmospheric Models. GARP Publication Series No. 17, Vol. II, 1–50.

    Google Scholar 

  • Tatsumi, Y., 1984: ‘Time integration methods used in atmospheric models.’ Short–and Medium-Range Weather Prediction Research Publication Series, No. 8, WMO, 43–104.

    Google Scholar 

  • Tokioka, T., 1978: ‘Some considerations on vertical differencing.’ J. Meteor. Soc. Japan, 56, 98–111.

    Google Scholar 

  • Wallace, J. M., S. Tibaldi and A. J. Simmons, 1983: ‘Reduction of systematic forecast errors in the ECMWF model through the introduction of an envelope orography.’ Quart. J. Roy. Meteor. Soc., 109, 683–717.

    Article  Google Scholar 

  • Washington, W. M., and D. L. Williamson, 1977: ‘A description of the NCAR global circulation models.’ In Methods in Computational Physics, Vol. 17, General circulation models of the atmosphere, ed. J. Chang, Academic Press, New York, 111–172.

    Google Scholar 

  • Williamson, D. L., 1968: ‘Integration of the barotropic vorticity equation on a spherical geodesic grid.’ Tellus, 20, 642–653.

    Article  Google Scholar 

  • Williamson, D. L., 1979: ‘Difference approximations for fluid on a sphere.’ In Numerical Methods Used in Atmospheric Models, GARP Publication Series No. 17, 51–120.

    Google Scholar 

  • WMO, 1984a: ‘Workshop on Limited-Area Numerical Weather Prediction Models for Computers of Limited Power, Part I.’ WMO Short–and Medium-Range Weather Prediction Research Publication, Series 8.

    Google Scholar 

  • WMO, 1984b: ‘Workshop on Limited-Area Numerical Weather Prediction Models for Computers of Limited Power, Part II.’ WMO Short–and Medium-Range Weather Prediction Research Publication, Series 11.

    Google Scholar 

  • WMO, 1984c: ‘Workshop on Limited-Area Numerical Weather Prediction Models for Computers of Limited Power, Part III.’ WMO Short–and Medium-Range Weather Prediction Research Publication, Series 13.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Kluwer Academic Publishers

About this chapter

Cite this chapter

Arakawa, A. (1988). Finite-Difference Methods in Climate Modeling. In: Schlesinger, M.E. (eds) Physically-Based Modelling and Simulation of Climate and Climatic Change. NATO ASI Series, vol 243. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3041-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-3041-4_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7867-2

  • Online ISBN: 978-94-009-3041-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics