• Crutzen, P., M. Lawrence, and U. Pöschl, 1999: On the background photo chemistry of tropospheric ozone. Tellus, 51A , 123146.

  • Dunkerton, T., 1989: Body force circulations in a compressible atmosphere: Key concepts. Pure Appl. Geophys., 130 , 243262.

  • Durran, D., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, 465 pp.

  • Falcone, M., and R. Ferreti, 1998: Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal., 35 , 909940.

    • Search Google Scholar
    • Export Citation
  • Garcia, R., 1987: On the mean meridional circulation of the middle atmosphere. J. Atmos. Sci., 44 , 35993609.

  • Gregory, A., and V. West, 2002: The sensitivity of a model's stratospheric tape recorder to the choice of advection scheme. Quart. J. Roy. Meteor. Soc., 128 , 18271846.

    • Search Google Scholar
    • Export Citation
  • Hall, T., D. Waugh, K. Boering, and R. Plumb, 1999: Evaluation of transport in stratospheric models. J. Geophys. Res., 104 , 1881518839.

    • Search Google Scholar
    • Export Citation
  • Hill, J., 1976: Homogeneous turbulent mixing with chemical reaction. Rev. Fluid Mech., 8 , 135161.

  • Mesinger, F., and A. Arakawa, 1976: Numerical methods used in atmospheric models. GARP Publ. Series, No. 17, WHO, 65 pp. [Available from World Meteorological organisation. Case postale No. 5 CH-1211 Geneva, Switzerland.].

    • Search Google Scholar
    • Export Citation
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Advection Equation with Oscillating Forcing: Numerical Aspects

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  • 1 Meteorologisches Institut, Universität München, Munich, Germany
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Abstract

The numerical solution to the linear advection equation with oscillating forcing is derived in analytic form for two-level schemes and for the leapfrog scheme with an Asselin filter. The numerical solutions are compared to the analytic solution of the advection equation with emphasis on the forced part. A detailed analysis is presented for the trapeze, the backward, and the leapfrog scheme with Asselin filter. Large deviations are found in quasi-resonant situations where the period of forcing and advection are close. Damping schemes fail completely to capture the resonant case. As for amplitude errors, the backward scheme is generally better than the trapeze scheme outside the quasi-resonant domain. However, the backward scheme produces large phase errors while the trapeze solutions are free of such errors. The leapfrog scheme has a resonant solution but generates large-amplitude errors near resonance. On the other hand, phase errors are particularly small in that case. The amplitude of the numerical mode tends to be large if either the forcing period or the advective period are but coarsely resolved. Addition of the Asselin filter removes the numerical high-frequency oscillations but destroys the resonant solution.

Corresponding author address: Joseph Egger, Meteorologisches Institut der Universität München, Theresienstr. 37, 80333 München, Germany. Email: J.Egger@LRZ.uni-muenchen.de

Abstract

The numerical solution to the linear advection equation with oscillating forcing is derived in analytic form for two-level schemes and for the leapfrog scheme with an Asselin filter. The numerical solutions are compared to the analytic solution of the advection equation with emphasis on the forced part. A detailed analysis is presented for the trapeze, the backward, and the leapfrog scheme with Asselin filter. Large deviations are found in quasi-resonant situations where the period of forcing and advection are close. Damping schemes fail completely to capture the resonant case. As for amplitude errors, the backward scheme is generally better than the trapeze scheme outside the quasi-resonant domain. However, the backward scheme produces large phase errors while the trapeze solutions are free of such errors. The leapfrog scheme has a resonant solution but generates large-amplitude errors near resonance. On the other hand, phase errors are particularly small in that case. The amplitude of the numerical mode tends to be large if either the forcing period or the advective period are but coarsely resolved. Addition of the Asselin filter removes the numerical high-frequency oscillations but destroys the resonant solution.

Corresponding author address: Joseph Egger, Meteorologisches Institut der Universität München, Theresienstr. 37, 80333 München, Germany. Email: J.Egger@LRZ.uni-muenchen.de

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