• Bao, J.-W., and Y.-H. Kuo, 1995: On–off switches in the adjoint method: Step functions. Mon. Wea. Rev.,123, 1589–1594.

  • Bates, J. R., and A. McDonald, 1982: Multiply-upstream, semi-Lagrangian advective schemes: Analysis and application to a multi-level primitive equation model. Mon. Wea. Rev.,110, 1831–1842.

  • Côté, J., and A. Staniforth, 1988: A two-time-level semi-Lagrangian semi-implicit scheme for spectral models. Mon. Wea. Rev.,116, 2003–2012.

  • ——, M. Roch, A. Staniforth, and L. Fillion, 1993: A variable-resolution semi-Lagrangian finite-element global model of the shallow-water equations. Mon. Wea. Rev.,121, 231–243.

  • 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.

  • Daley, R., 1995: Estimating the wind field from chemical constituent observations: Experiments with a one-dimensional extended Kalman filter. Mon. Wea. Rev.,123, 181–198.

  • Evensen, G., 1992: Using the Extended Kalman Filter with a multilayer quasi-geostrophic ocean model. J. Geophys. Res.,97, 17 905–17 924.

  • Gauthier, P., P. Courtier, and P. Moll, 1993: Assimilation of simulated wind Lidar data with a Kalman filter. Mon. Wea. Rev.,121, 1803–1820.

  • Lacarra, J.-F., and O. Talagrand, 1988: Short-range evolution of small perturbations in a barotropic model. Tellus,40A, 81–95.

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

  • 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.

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

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

  • ——, ——, and A. Staniforth, 1996b: Linearizing iterative processes for 4D data assimilation schemes. Priprints, Proc. 11th Conf. on Numerical Weather Prediction, Norfolk, VA, Amer. Meteor. Soc., 243–245.

  • 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.

  • Rabier, F., and P. Courtier, 1992: Four-dimensional assimilation in the presence of baroclinic instability. Quart. J. Roy. Meteor. Soc.,118, 649–672.

  • Ritchie, H., 1986: Eliminating the interpolation associated with the semi-Lagrangian scheme. Mon. Wea. Rev.,114, 135–146.

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

  • Robert, A., T. L. Yee, and H. Ritchie, 1985: A semi-Lagrangian and semi-implicit numerical integration for multilevel atmospheric models. Mon. Wea. Rev.,113, 388–394.

  • Tanguay, M., P. Bartello, and P. Gauthier, 1995: Four-dimensional data assimilation with a wide range of scales. Tellus,47A, 974–997.

  • Thépaut, J.-N., P. Courtier, G. Belaud, and G. Lemaître, 1996: Dynamical structure functions in a four-dimensional variational assimilation: A case study. Quart. J. Roy. Meteor. Soc.,122, 535–561.

  • Vukicevic, T., 1991: Nonlinear and linear evolution of initial forecast errors. Mon. Wea. Rev.,119, 1602–1611.

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

  • Zou, X., I. M. Navon, and J. G. Sela, 1993: Variational data assimilation with moist threshold processes using the NMC spectral model. Tellus,45A, 370–387.

  • Zupanski, D., 1993: The effects of discontinuities in the Betts–Miller cumulus convection scheme on four-dimensional variational data assimilation. Tellus,45A, 511–524.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 63 25 2
PDF Downloads 29 11 0

Temporal Accumulation of First-Order Linearization Error for Semi-Lagrangian Passive Advection

Monique TanguayMeteorological Research Branch, Atmospheric Environment Service, Dorval, Quebec, Canada

Search for other papers by Monique Tanguay in
Current site
Google Scholar
PubMed
Close
,
Saroja PolavarapuMeteorological Research Branch, Atmospheric Environment Service, Downsview, Ontario, Canada

Search for other papers by Saroja Polavarapu in
Current site
Google Scholar
PubMed
Close
, and
Pierre GauthierMeteorological Research Branch, Atmospheric Environment Service, Dorval, Quebec, Canada

Search for other papers by Pierre Gauthier in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by , who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, this study is extended by considering the impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to the use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem. In particular, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number.

For a constant zonal wind leading to an integral value of the Courant number, the first-order linearization errors are seen to amplify linearly in time and to resemble the second-order derivative of the advected field for linear interpolation and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lagrange linearization errors are smaller and affect small scales. For this to happen requires a wind configuration leading to a persistent integral value of the Courant number. Regions where sharp gradients of the advected tracer field are present are the most sensitive to this error, which is nevertheless observed to be small. Finally, passive tracers experiments driven by winds obtained from a shallow-water model integration confirm that higher-order interpolation schemes (whether correct or not) give similar negligible linearization errors since the probability of having the upstream point being located exactly on a grid point is vanishingly small.

Corresponding author address: Dr. Monique Tanguay, Recherche en Prevision Numerique, 2121, Route Trans-canadienne No. 508, Dorval, PQ H9P 1J3 Canada.

Abstract

The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by , who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, this study is extended by considering the impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to the use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem. In particular, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number.

For a constant zonal wind leading to an integral value of the Courant number, the first-order linearization errors are seen to amplify linearly in time and to resemble the second-order derivative of the advected field for linear interpolation and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lagrange linearization errors are smaller and affect small scales. For this to happen requires a wind configuration leading to a persistent integral value of the Courant number. Regions where sharp gradients of the advected tracer field are present are the most sensitive to this error, which is nevertheless observed to be small. Finally, passive tracers experiments driven by winds obtained from a shallow-water model integration confirm that higher-order interpolation schemes (whether correct or not) give similar negligible linearization errors since the probability of having the upstream point being located exactly on a grid point is vanishingly small.

Corresponding author address: Dr. Monique Tanguay, Recherche en Prevision Numerique, 2121, Route Trans-canadienne No. 508, Dorval, PQ H9P 1J3 Canada.

Save