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I. M. Navon

Abstract

A Sasaki variational approach is for the first time applied to enforce a posteriori conservation of potential enstrophy and total mass in long-term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a limited-area domain. The performance and accuracy of the variational approach is compared with that of a modified Bayliss-Isaacson a posteriori technique, also designed to enforce conservation of potential enstrophy and total mass, and with that of a periodic application of a two-dimensional high-order Shapiro filter.

While both the variational and the Bayliss-Isaacson a posteriori techniques yielded very satisfactory results after 20 days of numerical integration with regard to conservation of the integral constraints of the shallow-water equations and the accuracy of the solution, the high-order filtering approach performed in a less satisfactory way. This is attributed to the effects of the boundary conditions in the limited-area shallow-water equations models.

The Bayliss-Isaacson technique was found to be more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the updating procedure adopted for the Lagrange multiplier. The filtering technique is the most economical in terms of CPU time, but it is inadequate for limited-area domains with non-periodic boundary conditions and coarse meshes. In conclusion further research in this direction is suggested as these techniques provide viable alternatives to the rather complex conserving schemes proposed by other investigators.

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I. M. Navon
and
David M. Legler

Abstract

During the last few years new meteorological variational analysis methods have evolved, requiring large-scale minimization of a nonlinear objective function described in terms of discrete variables. The conjugate-gradient method was found to represent a good compromise in convergence rates and computer memory requirements between simpler and more complex methods of nonlinear optimization. In this study different available conjugate-gradient algorithms are presented with the aim of assessing their use in large-scale typical minimization problems in meteorology. Computational efficiency and accuracy are our principal criteria.

Four different conjugate-gradient methods, representative of up-to-date available scientific software, were compared by applying them to two different meteorological problems of interest using criteria of computational economy and accuracy. Conclusions are presented as to the adequacy of the different conjugate algorithms for large-scale minimization problems in different meteorological applications.

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I. M. Navon
and
R. de Villiers

Abstract

The Turkel–Zwas (T–Z) explicit large time-step scheme addresses the issue of fast and slow time scales in shallow-water equations by treating terms associated with fast waves on a coarser grid but to a higher accuracy than those associated with the slow-propagating Rossby waves. The T–Z scheme has been applied for solving the shallow-water equations on a fine-mesh hemispheric domain, using realistic initial conditions and an increased time step. To prevent nonlinear instability due to nonconservation of integral invariants of the shallow-water equations in long-term integrations, we enforced a posteriori their conservation. Two methods, designed to enforce a posteriori the conservation of three discretized integral invariants of the shallow-water equations, i.e., the total mass, total energy and potential enstrophy, were tested. The first method was based on an augmented Lagrangian method (Navon and de Villiers), while the second was a constraint restoration method (CRM) due to Miele et al., satisfying the requirement that the constraints be restored with the least-squares change in the field variables. The second method proved to be simpler, more efficient and far more economical with regard to CPU time, as well as easier to implement for first-time users. The CRM method has been proven to be equivalent to the Bayliss–Isaacson conservative method. The T–Z scheme with constraint restoration was run on a hemispheric domain for twenty days with no sign of impending numerical instability and with excellent conservation of the three integral invariants. Time steps approximately three times larger than allowed by the explicit CFL condition were used. The impact of the larger time step on accuracy is also discussed.

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I. M. Navon
and
R. de Villiers

Abstract

An augmented Lagrangian multiplier-penalty method is applied for the first time to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited-area domain. The method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems.

The computational efficiency and accuracy of the method is tested using two finite-difference solvers of the nonlinear shallow water equations on a β-plane. The method is also compared with a pure quadratic penalty approach. The updating of the Lagrangian multipliers and the penalty parameters is done using procedures suggested by Bertsekas. The method yielded satisfactory results in the conservation of the integral constraints while the additional CPU time required did not exceed 15% of the total CPU time spent on the numerical solution of the shallow water equations. The methods proved to be simple in their implementation and they have a broad scope of applicability to other problems involving nonlinear constraints; for instance, the variational nonlinear normal mode initialization.

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I. M. Navon
and
H. A. Riphagen

Abstract

An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form.

The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Padé differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained.

Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the overall fourth-order convergence rate of the more accurate interior approximation.

A fourth-order dissipative term was added to the equations to overcome the increased aliasing due to the fourth-order method. Alternatively, Wallington and Shapiro low-pass filters were applied.

The numerical integration of the shallow-water equations is performed in a channel corresponding to a middle-latitude band. A linearized version of this method is shown to be unconditionally stable.

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Mohan K. Ramamurthy
and
I. M. Navon

Abstract

A conjugate-gradient variational blending technique, based on the method of direct minimization, has been developed and applied to the problem of initialization in a limited-area model in the summer monsoon region. The aim is to blend gridded winds from a high-resolution limited-area analysis with a lower-resolution global analysis for use in a limited-area model that uses the, global analyst for boundary conditions. The ability of the variational matching approach in successfully blending meteorological analyses of varying resolutions is demonstrated. Reasonable agreement is found between the blended analyses and the imposed weak constraints, together with an adequate rate of convergence in the unconstrained minimization procedure. The technique is tested on the 1979 onset vortex vortex case using data from the FGGE Summer MONEX campaign. The results indicate that the forecasts made from the variationally matched analyses show positive impact and perform better than those from the unblended analyses.

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D. N. Daescu
and
I. M. Navon

Abstract

Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the order-reduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin–Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

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Yanqiu Zhu
and
I. M. Navon

Abstract

The full-physics adjoint of the Florida State University Global Spectral Model at resolution T42L12 is applied to carry out parameter estimation using an initialized analysis dataset. The three parameters, that is, the biharmonic horizontal diffusion coefficient, the ratio of the transfer coefficient of moisture to the transfer coefficient of sensible heat, and the Asselin filter coefficient, as well as the initial condition, are optimally recovered from the dataset using adjoint parameter estimation.

The fields at the end of the assimilation window starting from the retrieved optimal initial conditions and the optimally identified parameter values successfully capture the main features of the analysis fields. A number of experiments are conducted to assess the effect of carrying out 4D Var assimilation on both the initial conditions and parameters, versus the effect of optimally estimating only the parameters. A positive impact on the ensuing forecasts due to each optimally identified parameter value is observed, while the maximum benefit is obtained from the combined effect of both parameter estimation and initial condition optimization. The results also show that during the ensuing forecasts, the model tends to “lose” the impact of the optimal initial condition first, while the positive impact of the optimally identified parameter values persists beyond 72 h. Moreover, the authors notice that their regional impacts are quite different.

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F. H. M. Semazzi
and
I. M. Navon

Abstract

Application of the bounded-derivative and normal-mode methods to a simple linear barotropic model at a typical middle latitude shows that the two methods lead to identical constraints up to a certain degree of approximation. Beyond this accuracy the two methods may differ from each other.

When applied to a global nonlinear barotropic model using real data, again the two methods lead to similar balanced initial states. The parity oscillations in the unbalanced height field, which have amplitudes of up to 60 m with a dominant periodicity of about 5 to 6 h, are practically eliminated by both initialization methods. The rotational wind component is smooth even for the unbalanced initial state. The small-scale spatial features of the irrotational wind component are drastically reduced by initialization. Both the nonlinear normal-mode and the bounded-derivative initialization methods yield similar divergence field centered around the areas of highest orography.

The comparison shows that there is no significant loss of information in the man and momentum fields, despite the fact that the bounded-derivative method employs only the original, rotational wind component to construct a balanced initial state compared to the normal-mode method, which, in addition, makes use of the unbalanced divergent wind and height fields.

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X. Zou
,
I. M. Navon
, and
J. Sela

Abstract

Variational four-dimensional data assimilation, combined with a penalty method constraining time derivatives of the surface pressure, the divergence, and the gravity-wave components is implemented on an adiabatic version of the National Meteorological Center's 18-level primitive equation spectral model with surface drag and horizontal diffusion. Experiments combining the Machenhauer nonlinear normal-mode initialization procedure and its adjoint with the variational data assimilation are also presented. The modified variational data-assimilation schemes are tested to assess how well they control gravity-wave oscillations.

The gradient of a penalized cost function can be obtained by a single integration of the adjoint model. A detailed derivation of the gradient calculation of different penalized cost functions is presented, which is not restricted to a specific model.

Numerical results indicate that the inclusion of penalty terms into the cost function will change the model solution as desired. The advantages of the use of simple penalty terms over penalty terms including the model normal modes results in a simplification of the procedure, allowing a more direct control over the model variables and the possibility of using weak constraints to eliminate the high-frequency gravity-wave oscillations. This approach does not require direct information about the model normal modes. One of the encouraging results obtained is that the introduction of the penalty terms does not slow the convergence rate of the minimization process.

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