The Adjoint of the Semi-Lagrangian Treatment of the Passive Tracer Equation

M. Tanguay Meteorological Research Branch, Atmospheric Environment Service, Dorval, Quebec, Canada

Search for other papers by M. Tanguay in
Current site
Google Scholar
PubMed
Close
and
S. Polavarapu Meteorological Research Branch, Atmospheric Environment Service, Downsview, Ontario, Canada

Search for other papers by S. Polavarapu in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

A semi-Lagrangian treatment of the 1D passive tracer equation using prescribed piecewise-continuous interpolating functions is considered. Whether the process of localizing the upstream position affects the adjoint counterpart of the associated tangent linear model when the Courant number varies spatially is evaluated. An analysis is done and the accuracy of the numerical adjoint is found to be reduced when the upstream positions are irregularly distributed around the grid point at which the adjoint is evaluated. This irregular distribution follows an acceleration or a deceleration pattern and is a consequence of a discontinuity in the integer value of the Courant number. The cubic Lagrange interpolation reduces this lack of accuracy when compared to the linear interpolation. Numerical 1D adjoint experiments confirm the analysis. The adjoint of the 2D passive tracer equation on the sphere is studied in the context of a solid body rotation. Numerous irregular distributions are induced by a sequence of discontinuities in the integer value of the longitudinal Courant number near the poles. Numerical 2D adjoint experiments reveal that the cubic spline interpolation could also be affected, but to a lesser extent than the cubic Lagrange interpolation. Those errors are visible even if the Lipschitz criterion giving bounds to the intensity in the variation of the Courant number is respected.

Corresponding author address: Recherche en Prévision Numérique, 2121 Route Trans-Canadienne, Dorval, Québec, H9P 1J3, Canada.

Email: monique.tanguay@ec.gc.ca

Abstract

A semi-Lagrangian treatment of the 1D passive tracer equation using prescribed piecewise-continuous interpolating functions is considered. Whether the process of localizing the upstream position affects the adjoint counterpart of the associated tangent linear model when the Courant number varies spatially is evaluated. An analysis is done and the accuracy of the numerical adjoint is found to be reduced when the upstream positions are irregularly distributed around the grid point at which the adjoint is evaluated. This irregular distribution follows an acceleration or a deceleration pattern and is a consequence of a discontinuity in the integer value of the Courant number. The cubic Lagrange interpolation reduces this lack of accuracy when compared to the linear interpolation. Numerical 1D adjoint experiments confirm the analysis. The adjoint of the 2D passive tracer equation on the sphere is studied in the context of a solid body rotation. Numerous irregular distributions are induced by a sequence of discontinuities in the integer value of the longitudinal Courant number near the poles. Numerical 2D adjoint experiments reveal that the cubic spline interpolation could also be affected, but to a lesser extent than the cubic Lagrange interpolation. Those errors are visible even if the Lipschitz criterion giving bounds to the intensity in the variation of the Courant number is respected.

Corresponding author address: Recherche en Prévision Numérique, 2121 Route Trans-Canadienne, Dorval, Québec, H9P 1J3, Canada.

Email: monique.tanguay@ec.gc.ca

Save
  • Bertsekas, D. P., 1982: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 395 pp.

  • Coté, J., S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998: The operational CMC–MRB Global Environmental Multiscale (GEM) Model. Part I: Design considerations and formulation. Mon. Wea. Rev.,126, 1373–1395.

  • Courtier, P., J. Derber, R. Errico, J.-F. Louis, and T. Vukicevic, 1993:Important literature on the use of adjoint, variational methods and the Kalman filter in meteorology. Tellus,45A, 342–357.

  • Haberman, R., 1987: Elementary Applied Partial Differential Equations. Prentice-Hall, 547 pp.

  • Lacarra, J.-F., and O. Talagrand, 1988: Short-range evolution of small perturbations in a barotropic model. Tellus,40A, 2–95.

  • LeDimet, F. X., and O. Talagrand, 1986: Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus,38A, 97–110.

  • Lewis, J. M., and J. C. Derber, 1985: The use of adjoint equations to solve a variational adjustment problem with advective constraints. Tellus,37A, 309–322.

  • Li, Y., I. M. Navon, P. Courtier, and P. Gauthier, 1993: Variational data assimilation with a semi-Lagrangian semi-implicit global shallow-water equation model and its adjoint. Mon. Wea. Rev.,121, 1759–1769.

  • McDonald, A., 1984: Accuracy of multiply-upstream, semi-Lagrangian advective schemes. Mon. Wea. Rev.,112, 1267–1275.

  • ——, 1991: Semi-Lagrangian methods. ECMWF Seminar 1991, Reading, Berkshire, United Kingdom, ECMWF, 257–272.

  • Ménard, R., 1994: Kalman filtering of Burgers’ equation and its application to atmospheric data assimilation. Ph.D. thesis, McGill University, 211 pp. [Available from Dept. of Atmospheric and Ocean Science, McGill University, 805 Sherbrooke Street West, Montreal, PQ H3A 2K6, Canada.].

  • Orszag, S. A., 1972: Comparison of pseudospectral and spectral approximations. Stud. Appl. Math.,51, 253–259.

  • Polavarapu, S., M. Tanguay, R. Ménard, and A. Staniforth, 1996: The tangent linear model for semi–Lagrangian schemes: Linearizing the process of interpolation. Tellus,48A, 74–95.

  • Pudykiewicz, J., and A. Staniforth, 1984: Some properties and comparative performance of the semi–Lagrangian method of Robert in the solution of the advection–diffusion equation. Atmos.–Ocean,22, 283–308.

  • ——, R. Benoit, and A. Staniforth, 1985: Preliminary results from a partial LRTAP model based on an existing meteorological forecast model. Atmos.–Ocean,23, 267–303.

  • Rabier, F., E. Klinker, P. Courtier, and A. Hollingsworth, 1996: Sensitivity of forecast errors to initial conditions. Quart. J. Roy. Meteor. Soc.,122, 121–150.

  • Ritchie, H., 1987: Semi-Lagrangian advection on a Gaussian grid. Mon. Wea. Rev.,115, 608–619.

  • Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological equations. Atmos.–Ocean,19, 35–46.

  • Smolarkiewicz, P. K., and J. Pudykiewicz, 1992: A class of semi-Lagrangian approximations for fluids. J. Atmos. Sci.,49, 2082–2096.

  • Talagrand, O., 1991: The use of adjoint equations in numerical modeling of the atmospheric circulation. Proc. First SIAM Workshop on Automatic Differentiation, Breckenridge, CO, Society for Industrial and Applied Mathematics, 169–180.

  • Tanguay, M., S. Polavarapu, and P. Gauthier, 1997: Temporal accumulation of first-order linearization error for semi-Lagrangian passive advection. Mon. Wea. Rev.,125, 1296–1311.

  • Thépaut, J.-N., D. Vasiljevic, and P. Courtier, 1993: Variational assimilation of conventional meteorological observations with a multilevel primitive-equation model. Quart. J. Roy. Meteor. Soc.,119, 153–186.

  • Vukićević, T., and R. M. Errico, 1993: Linearization and adjoint of parametrized moist diabatic processes. Tellus,45A, 493–510.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 136 33 2
PDF Downloads 57 19 2