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Beljadid A.,University of Ottawa | Mohammadian A.,University of Ottawa | Charron M.,Recherche en Prevision Numerique Atmospherique | Girard C.,Recherche en Prevision Numerique Atmospherique
Monthly Weather Review | Year: 2014

In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangianmethods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear termin the second step makes the proposed class of schemes competitive in terms of efficiency compared to somewell-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi- Lagrangian semi-implicit schemes and semi-implicit predictor-corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies. © 2014 American Meteorological Society.

Charron M.,Recherche en prevision numerique atmospherique | Zadra A.,Recherche en prevision numerique atmospherique | Girard C.,Recherche en prevision numerique atmospherique
Quarterly Journal of the Royal Meteorological Society | Year: 2014

A four-dimensional tensor formalism suitable for the equations of motion of a classical fluid in the presence of a given external gravitational field is presented. The formalism allows for arbitrary time-dependent transformations of spatial coordinates. Some well-known conservation laws are derived in covariant form. The metric tensor and the associated Christoffel symbols are calculated for coordinate systems useful in meteorology. The vertical momentum equation employed in the Canadian operational weather forecasting model is obtained using the proposed tensor formalism. © 2013 Her Majesty the Queen in Right of Canada. Quarterly Journal of the Royal Meteorological Society published by John Wiley & Sons Ltd on behalf of the Royal Meteorological Society.

Zadra A.,Recherche en Prevision Numerique Atmospherique | Charron M.,Recherche en Prevision Numerique Atmospherique
Quarterly Journal of the Royal Meteorological Society | Year: 2015

Three manifestly invariant Lagrangians are presented from which the covariant equations of motion for inviscid classical fluids are derived using the least action principle. Invariance and covariance are here defined with respect to synchronous, but otherwise arbitrary, coordinate transformations, i.e. supposing that time intervals are absolute as required by Newtonian mechanics. In the first Lagrangian, the flow is formulated in terms of fluid particles, but conservation of mass and entropy is assumed apriori. In the second Lagrangian, the flow is described by fields, and conservation of mass and entropy is obtained with Lagrange multipliers. The third Lagrangian is also based on a field formulation, but has no Lagrange multipliers and produces all the desired equations of motion, including conservation of mass and entropy. The differences and similarities between these formulations are discussed. Hydrostatic equations are rederived from an asymptotic expansion of the action using the covariant field formulation. © 2015 Royal Meteorological Society.

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