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Harold Ritchie

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Harold Ritchie

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Previous tests with grid-point numerical weather prediction models have shown that semi-Lagrangian schemes permit the use of time steps that are much larger than those permitted by the Courant-Friedrichs-Lewy (CFL) stability criterion for the corresponding Eulerian models, without reducing the accuracy of the forecasts. Thus model efficiency is improved because fewer time steps are needed to complete the forecast.

In a first step to see if similar results can be achieved in spectral models, Ritchie, in a previous study, applied interpolating and noninterpolating semi-Lagrangian treatments of advection to the problem of simple advection by a steady wind field on a Gaussian grid. This present paper combines these treatments of advection with the semi-implicit scheme in a spectral model of the shallow water equations expressed in vector momentum form. Model formulations are presented and intercomparison experiments are performed. It is shown that both interpolating and noninterpolating semi-Lagrangian schemes can be applied accurately and stably to a spectral model of the shallow water equations with time steps that are much larger than the CFL limit for the corresponding Eulerian model.

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Harold Ritchie

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Scale-dependent phase errors are characteristic of Eulerian advection schemes and can lead to significant erroneous dispersion, especially for features whose length scales are near the resolution limits of a numerical model. However, the corresponding errors are much smaller for a semi-Lagrangian scheme using a high-order interpolation technique. This property is exploited by introducing a semi-Lagrangian formulation in the moisture equation of the DRPN (Division de Recherche en Prévision Numérique) regional baroclinic finite element model. The impact of this change is assessed by comparing the resulting precipitation fields with those from the version of the model in which all variables are treated using the Eulerian formulation. It is found that the semi-Lagrangian scheme is better, with the improvement being partly attributable to the need for some horizontal diffusion to control noise associated with the Eulerian advection, while no such diffusion is necessary for the semi-Lagrangian scheme.

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Harold Ritchie

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There are several reasons why it is desirable to eliminate the interpolation associated with the conventional semi-Larangian scheme. Interpolation leads to smoothing and is also the most costly operation associated with the technique. Furthermore, its elimination produces a scheme that is more readily adaptable to a spectral model.

In the conventional semi-Lagrangian method, in order to predict a field value at grid point (Xi, Yj) it is necessary to calculate the trajectory over one time step for the fluid element that arrives at (Xi, Yj). One then moves along this trajectory in order to extract the field value at an upstream location that generally lies between the grid points, and hence requires the use of interpolation formulae.

This trajectory can be represented as a vector. In the new scheme, the trajectory vector is considered to be the sum of two other vectors—a first vector joining (Xi, Yj) to the grid point (Xu, Yu) nearest the upstream location, and a second vector joining (Xu, Yu) to the upstream location. The advection along the first vector is done via a Lagrangian technique that displaces the field from one grid point to another and, therefore, does not require interpolation. The advection along the second vector is accounted for by an Eulerian approach with the advecting winds modified in such a way that the Courant number is always less than one, thus retaining the attractive stability properties of the interpolating semi-Lagrangian method.

Here the noninterpolating scheme is applied to a model of the shallow water equations and its performance is assessed by comparing the results with those produced by one model which uses the interpolating semi-Lagrangian technique, and another model which uses a fourth-order Eulerian approach. Five-day integrations indicate that the scheme is stable, accurate, and appears to have efficiency advantages.

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Harold Ritchie

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The nondivergent barotropic vorticity equation is integrated numerically in order to investigate a potential resonance mechanism for Rossby waves on a shear flow in the presence of a nonlinear critical layer. lie numerical model, which uses a mixture of spectral and finite element techniques, simulates the propagation of a weakly forced Rossby wave on a semi-infinite beta plane. It is found that a large amplitude response can be obtained by “tuning” the geometry of the flow and that there is an associated increase in the thickness of the critical layer, showing that this mechanism gives a low latitude response to a midlatitude forcing. Nonlinear critical layers may thus have an important impact on large-scale atmospheric motions. The logarithmic phase shift is also investigated. It appears to develop an imaginary part as the nonlinearities come into force.

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Harold Ritchie

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The treatment of advection is related to the stability, accuracy and efficiency of models used in numerical weather prediction. In order to remain stable, conventional Eulerian advection schemes must respect a Courant-Friedrichs-Lewy (CFL) criterion, which limits the size of the time step that can be used in conjunction with a given spatial resolution.

In recent years, tests with gridpoint models have shown that semi-Lagrangian schemes permit the use of large time steps (roughly three to six times those permitted by the CFL criterion for the corresponding Eulerian models), without reducing the accuracy of the forecasts. This leads to improved model efficiency, since fewer steps are needed to complete the forecast.

Can similar results be achieved in spectral models? This paper examines the semi-Lagrangian treatment of advection on the Gaussian grid used in spectral models. Interpolating and noninterpolating versions of the semi-Lagrangian scheme are applied to the problem of solid body rotation on the globe, and their performance is compared with that of an Eulerian spectral treatment. It is shown that the semi-Lagrangian models produce stable, accurate integrations using time steps that far exceed the CFL limit for the Eulerian spectral model.

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Harold Ritchie and Christiane Beaudoin

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This study examines the extensions that have been made to a basic semi-Lagrangian semi-implicit multilevel spectral primitive equation model in preparing it for use as an operational data assimilation and medium-range forecast model. The authors present an optimized formulation using accurate approximations to alternate trigonometric calculations for finding the upstream positions and performing the transformations for treating a vector form of the equation of motion in spherical geometry. The impact of the order of accuracy of the interpolators used in the semi-Lagrangian algorithms is also examined. It is shown that the recommended approximations have no significant meteorological consequences, but in a typical model step they reduce the time spent in the semi-Lagrangian calculations from about 50% to about 30%.

Through a series of sensitivity tests, the authors establish the viability of the semi-Lagrangian semi-implicit method for spectral models with a more comprehensive physical parameterization package, and on a wider range of meteorological situations than in the previous study. First of all it is confirmed that even in these global runs with full physical parameterizations the sensitivity to the conversion from an Eulerian to a semi-Lagrangian formulation is acceptably small. In time truncation error tests with a T79 horizontal resolution it is found that, on average and with full physics, 30 min is about the upper limit for the time step based on acceptable time truncation errors. Since the Courant-Friedrichs-Lewy limit for the corresponding Eulerian model is about 10 min and the overhead of the semi-Lagrangian calculations is less than 30% per time step, there is a significant gain in efficiency by using the optimized semi-Lagrangian formulation. Results are also presented showing the improvement in the accuracy of the, 5-day forecasts obtained by raising the model top and by increasing the horizontal resolution.

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Harold Ritchie and Monique Tanguay

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It has been demonstrated previously by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-implicit semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. Rivest et al. recommended using a second-order instead of a first-order semi-implicit off-centering to eliminate the spurious resonances, the former being more accurate. The present study shows by a linear one-dimensional analysis that a first-order semi-implicit off-centering can be used more effectively to eliminate the spurious resonances when combined with a spatially averaged Eulerian instead of a semi-Lagrangian treatment of mountains. The analysis reveals that the resonance is much less severe with the spatially averaged Eulerian treatment of mountains and, hence, can be suppressed with a weaker first-order off-centering. This combination could represent a valid alternative to second-order off-centering that needs extra time levels. The study also reveals that a serious truncation error is present in the neighborhood of the twin resonances when a semi-Lagrangian treatment of mountains is used. With the spatially averaged Eulerian treatment of the mountains the numerical solution filters the corresponding waves. These various points are illustrated with both barotropic and baroclinic semi-implicit semi-Lagrangian spectral models. An important feature of the baroclinic model formulation is the inclusion of topography in the basic-state solution that is used for the semi-implicit treatment of the gravity-wave-producing terms. In tests run from real data it appears that, in current three-time-level models, simply changing from the semi-Lagrangian to the spatially averaged Eulerian treatment of mountains is sufficient to significantly reduce the topographic resonance problem, permitting the use of larger time steps that produce acceptable time truncation errors without provoking the fictitious numerical amplification of short-scale waves.

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Harold Ritchie and Anne-Marie Leduc

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Slow-start procedures were developed decades ago in order to provide a smoother evolution in numerical simulations performed with three-time-level integration schemes. The advantages of a slow start versus a conventional forward start should increase as the time step increases. With the advent of centered semi-Lagrangian semi-implicit schemes, we now have accurate numerical weather prediction models that can use much larger time steps than before, and the impact of the start-up procedure becomes more important. This paper examines the advantages of a slow-start procedure in the context of such models.

The impacts on Rossby and gravity waves are analyzed, and some of the advantages are illustrated through tests with an idealized solution. This is a Haurwitz-type solution that has been modified to satisfy the divergence equation rather than just the balance equation as in the Phillips solution. Experiments are performed with shallow-water models initialized with real data in order to assess under what conditions the quantitative impact of the start-up is significant. It is found that inserting a single start-up at the beginning of an integration does not appear to have a significant impact on medium-range forecasts, although a slow start is important in order to retain the benefit of a careful initialization procedure when the models are subsequently run with large time steps permitted by semi-Lagrangian semi-implicit algorithms. When a start-up is made every 6 h, as in an analysis cycle, the forward start introduces a significant error when the time step is large, but this error decreases rapidly as the time step is reduced. For all the time stops examined, the slow start is very effective in controlling the error due to the computational modes introduced by the start-up. Tests are also performed with an operational baroclinic model that uses a time step of 30 min, and it is found that, with this time step, even repeated start-ups with a conventional forward start do not generate significant computational modes.

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André Robert, Tai Loy Yee, and Harold Ritchie

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A complete multilevel atmospheric model of the primitive meteorological equations is integrated at high spatial resolution with a large time step of 90 min. Numerical stability is achieved by associating a semi-Lagrangian technique with the commonly used semi-implicit algorithm.

A detailed description of the method is given and some results are presented. From these runs, it seems possible to infer that the time truncation errors remain relatively small. Because of the 1arger time step, the semi-Lagrangian technique contributes to a significant enhancement of the efficiency of the semi-implicit integration scheme.

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