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Clive Temperton

Abstract

It is shown that nonlinear normal mode initialization (NMI) can be implemented without knowing the normal modes of a model. The implicit form of nonlinear NMI is particularly useful for models whose normal modes cannot readily be found; for example, if the underlying linear equations are nonseparable.

An implicit nonlinear NMI scheme is formulated for the shallow-water equations on a polar stereographic projection. The linear equations which define the implicit normal modes include most of the beta terms as well as variable Coriolis parameter and map scale factor. Even in this nonseparable case, the equivalence between implicit and conventional nonlinear NMI is shown to be exact.

The scheme is implemented in a regional model on a quasi-hemispheric domain, which uses a finite-element discretization on a nonuniform grid. The well-posed lateral boundary conditions of this model lead to consistent boundary conditions for the initialization. Results are presented not only for the implicit form of Machenhauer's nonlinear NMI technique, but also for the implicit form of Tribbia's corresponding second-order scheme which results in an even better initial balance.

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Clive Temperton

Abstract

We compare scalare and vector transform methods for global spectral models of the shallow-water equations. For the scalar transform methods, we demonstrate some economies in the number of Legendre transforms required. It is shown that the vector transform method is algebraically equivalent to the more usual scalar transform methods, and the choice of transform grid is discussed.

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Clive Temperton

Abstract

The application of variational normal mode initialization to the ECMWF multilevel grid-point model is described. The technique involves minimizing a variational integral of the changes made by the initialization to the analyzed mass and wind fields, suitably weighted with a view to exerting some control over the relative magnitudes of the adjustments to these fields. We show how to construct an appropriate three-dimensional integral, and describe an efficient procedure for solving the minimization problem in the simple case where the weights depend only on latitude.

Experimental results are presented to demonstrate that, in comparison with unconstrained initialization, the changes made to the analyzed mass field can be considerably reduced without undue damage to the wind field, and without compromising the benefits of initialization in providing a forecast free of high-frequency oscillations. In the case studied here, the use of constrained normal mode initialization has no significant impact on the results of a subsequent 5-day forecast.

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Clive Temperton

Abstract

Orthogonal vertical normal modes are found for a multilevel sigma-coordinate model, linearized about a state of rest with a nonisothermal mean temperature profile. Orthogonality permits the partitioning of energy into the vertical modes, and simplifies the application of variational techniques to normal mode initialization in the multilevel case. Improved procedures are described for the vertical transforms required during normal mode initialization. We derive relationships between the time derivatives of the energy computed in vertical normal mode space and in gold point space, and analogous relationships between the changes made by initialization to the vertical normal mode coefficients and the corresponding changes made in grid point space.

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Clive Temperton

Abstract

Implicit nonlinear normal mode initialization schemes enable nonlinear NMI to be performed in models whose normal modes cannot readily be computed. Such schemes are algebraically equivalent to conventional (“explicit”) NMI based on the normal modes of a set of linearized equations which is slightly different from the usual choice. In this paper we apply implicit NMI to a barotropic spectral model whose normal modes are easily found, thus permitting a direct comparison to be made between conventional and implicit NMI schemes. Both first-order (Machenhauer) and second-order (Tribbia) variants of implicit nonlinear NMI are formulated for a spectral model and compared with their explicit counterparts.

Experimental results show that the differences between explicit and implicit nonlinear NMI am insignificant except at the very largest horizontal scales. Besides validating the concept of implicit nonlinear NMI, this study suggests a practical approach to initializing very high resolution spectral models for which the normal modes require an inconveniently large amount of storage.

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Luc Fillion and Clive Temperton

Abstract

It is shown that implicit normal mode initialization can be combined with a variational technique, in order to control the relative magnitudes of the changes to the analyzed mass and wind fields. Since the initialization procedure is expressed entirely in physical space, the use of locally varying weights in the variational integral becomes more straightforward than in previous efforts to combine variational methods with normal mode initialization.

We present details of the application to a finite-element model of the shallow water equations on a stereographic projection. It is demonstrated that the use of variational initialization can change the slowly evolving component of the subsequent forecast, as well as eliminate the unrealistic fast component.

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Clive Temperton and Michel Roch

Abstract

In a previous study based on the shallow-water equations, it was shown that nonlinear normal mode initialization (NMI) can be implemented without knowing the normal modes of a model; this implicit form of nonlinear NMI is particularly useful in models for which computing the horizontal normal modes is impracticable. The present paper extends the technique to the multilevel Canadian Operational Finite-Element Regional Model. This paper shows that the method yields well-balanced initial conditions and consistent vertical velocity fields. Forecasts from these initial conditions using a semi-Lagrangian time-integration scheme with relatively large time steps are free from unrealistic high-frequency oscillations.

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Andrew Staniforth and Clive Temperton

Abstract

We present the firm application of the semi-implicit semi-Lagrangian integration technique to a finite-element barotropic model using the shallow-water equations on a variable-resolution. Two schemes based on this approach, differing only in their treatment of the rotational part of the wind field, are formulated and analyzed. A set of comparative experiments was performed using carefully balanced initial conditions (to eliminate spurious high-frequency oscillations); an Eulerian control integration was run at high resolution on a uniform grid. Both schemes are stable with timesteps at least six times longer than the limiting timestep of the corresponding Eulerian scheme using the same variable-resolution mesh. However, one scheme is consistently more accurate than the other. These results were explained by a theoretical analysis of the stability and accuracy of the schemes. We conclude that the semi-implicit semi-Lagrangian scheme is a promising technique for finite-element models as well as for finite-difference models.

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David L. Williamson and Clive Temperton

Abstract

In Part II of this paper, we describe the nonlinear normal mode initialization applied to the ECMWF multilevel global grid-point model and show that the procedure is highly successful in eliminating spurious high-frequency oscillations from forecasts made by the model. We determine the number of vertical modes that can be included in the procedure and demonstrate insensitivity to minor changes in the definitions of the modes. Attempts to include physical parameterizations within the initialization procedure are described as are the problems which arise with such attempts. It is shown that adiabatic nonlinear initialization is adequate to eliminate high-frequency gravity mode oscillations from a forecast by a model which includes non-adiabatic processes.

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Clive Temperton and David L. Williamson

Abstract

In Part I of this paper we review initialization methods for numerical weather prediction models, leading up to the development of schemes based on the normal modes of the forecast model. We present the derivation of the normal modes of ECMWF's multilevel global grid-point model, and compare the horizontal normal modes with those obtained using alternative finite-difference schemes. The impact of stability-enhancing Fourier filtering procedures on the normal modes is also discussed. Finally in Part I we apply linear normal mode initialization to a nine-level version of the model with 3.75° horizontal resolution. The application of nonlinear normal mode initialization to this model is presented in Part II.

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