One-dimensional Shallow-water Model Research Articles

Impact dynamics of mud flows against rigid walls

• Mud-flows may cause serious damages to people and structures. • The evaluation of the impact of mud flows against obstacles is crucial for damage assessment. • The impact force depends on both the fluid rheology and the geometry. • In some conditions the peak force does not exceed the time-asymptotic hydrostatic value. • Useful indications on how to evaluate the peak force have been provided. Mud flows represent one of the major causes of natural hazards in mountain regions. Similarly to debris flows, they consist of a hyper-concentrated mixture of water and sediments flowing down a slope and may cause serious damages to people and structures. The present paper investigates the force produced by a dam-break wave of mud impacting against a rigid wall. A power-law shear-thinning model is used to describe the rheology of the hyper-concentrated mixture. A one-dimensional shallow water model is adopted and a second-order Finite Volume scheme is employed to numerically solve the governing equations. The results indicate that depending on the fluid rheological parameters and on the bottom slope, there exists a minimum value of the wall distance above which the peak force does not exceed the asymptotic value of the hydrostatic final condition. For two different values of the channel slope, the dimensionless value of this lower bound is individuated for several values of the power-law exponent and of a dimensionless Basal Drag coefficient. An estimation of the maximum peak force for wall distance smaller than the minimum value is also provided.

Meteotsunami Observed in Japan Following the Hunga Tonga Eruption in 2022 Investigated Using a One-Dimensional Shallow-Water Model

Meteotsunami Observed in Japan Following the Hunga Tonga Eruption in 2022 Investigated Using a One-Dimensional Shallow-Water Model

Temporal Discretization Algorithms for the Continuity Equation of the One-dimensional Shallow Water Model

Temporal Discretization Algorithms for the Continuity Equation of the One-dimensional Shallow Water Model

Large-time behavior of solutions to an outflow problem for a shallow water model

We investigate the existence and the time-asymptotic stability of boundary layer solutions for a one-dimensional shallow water model. Under the subsonic condition on the far-field equilibrium state, we show that there exists a one-parameter family of boundary layer solutions satisfying the outflow constant-flux boundary condition. We also show that there is a unique time-asymptotic boundary layer profile for a given initial data, and prove its stability using standard energy methods, provided that the initial perturbation is sufficiently small. Our stability result is obtained without the zero mass condition nor “artificial” condition on the profile strength. Rather the time-asymptotic profile is determined alone by the initial perturbation from the end state so that it satisfies the zero mass condition and its amplitude is appropriately small simultaneously.

Application of One-Dimensional Shallow Water Model to Flows in Open Channels with Bends

Application of One-Dimensional Shallow Water Model to Flows in Open Channels with Bends

An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part II: Observing System Simulation Experiments with Advanced Research WRF (ARW)

Abstract An ensemble-based four-dimensional variational data assimilation (En4DVAR) algorithm and its performance in a low-dimension space with a one-dimensional shallow-water model have been presented in Part I. This algorithm adopts the standard incremental approach and preconditioning in the variational algorithm but avoids the need for a tangent linear model and its adjoint so that it can be easily incorporated into variational assimilation systems. The current study explores techniques for En4DVAR application in real-dimension data assimilation. The EOF decomposed correlation function operator and analysis time tuning are formulated to reduce the impact of sampling errors in En4DVAR upon its analysis. With the Advanced Research Weather Research and Forecasting (ARW-WRF) model, Observing System Simulation Experiments (OSSEs) are designed and their performance in real-dimension data assimilation is examined. It is found that the designed En4DVAR localization techniques can effectively alleviate the impacts of sampling errors upon analysis. Most forecast errors and biases in ARW are reduced by En4DVAR compared to those in a control experiment. En3DVAR cycling experiments are used to compare the ensemble-based sequential algorithm with the ensemble-based retrospective algorithm. These experiments indicate that the ensemble-based retrospective assimilation, En4DVAR, produces an overall better analysis than the ensemble-based sequential algorithm, En3DVAR, cycling approach.

An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part I: Technical Formulation and Preliminary Test

Abstract Applying a flow-dependent background error covariance (𝗕 matrix) in variational data assimilation has been a topic of interest among researchers in recent years. In this paper, an ensemble-based four-dimensional variational (En4DVAR) algorithm, designed by the authors, is presented that uses a flow-dependent background error covariance matrix constructed by ensemble forecasts and performs 4DVAR optimization to produce a balanced analysis. A great advantage of this En4DVAR design over standard 4DVAR methods is that the tangent linear and adjoint models can be avoided in its formulation and implementation. In addition, it can be easily incorporated into variational data assimilation systems that are already in use at operational centers and among the research community. A one-dimensional shallow water model was used for preliminary tests of the En4DVAR scheme. Compared with standard 4DVAR, the En4DVAR converges well and can produce results that are as good as those with 4DVAR but with far less computation cost in its minimization. In addition, a comparison of the results from En4DVAR with those from other data assimilation schemes [e.g., 3DVAR and ensemble Kalman filter (EnKF)] is made. The results show that the En4DVAR yields an analysis that is comparable to the widely used variational or ensemble data assimilation schemes and can be a promising approach for real-time applications. In addition, experiments were carried out to test the sensitivities of EnKF and En4DVAR, whose background error covariance is estimated from the same ensemble forecasts. The experiments indicated that En4DVAR obtained reasonably sound analysis even with larger observation error, higher observation frequency, and more unbalanced background field.

Externalizing the lateral boundary conditions from the dynamic core in semi-implicit semi-Lagrangian models

Research is still undertaken to develop so-called transparent lateral boundary conditions (LBC) for limited-area numerical weather prediction models. In the widely used semi-implicit semi-Lagrangian schemes, this naturally leads to LBC formulations that are intrinsically intertwined with the numerics of the dynamic core. This may have profound consequences for the implementation and the maintenance of future model codes. For instance, scientific development on the dynamics may be hindered by constraints coming from today’s choices in the LBC formulation and vice versa. Building further on the work of Aidan McDonald, this paper proposes an approach where (1) the LBCs can be imposed by an extrinsic numerical scheme that is fundamentally different from the one used for the dynamic core in the interior domain and (2) substituting one such LBC scheme for another is possible in a manner that leaves the Helmholtz solver unmodified. It is argued that this concept may provide the necessary frame for formulating transparent boundary conditions in spectral limited-area models. Since this idea touches all aspects of the LBC problem, its feasibility can only be established by a rigorous systematic approach. As a first step, this paper provides promising experimental support in a one-dimensional shallow-water model.

Mass Residuals as a Criterion for Mesh Refinement in Continuous Galerkin Shallow Water Models

Mass balance error has been computed traditionally by using conventional fluxes derived from the conservation of mass equation, but recent literature supports a method based on fluxes that are consistent with the discretization of the governing equations. By comparing the mass residuals from these two methods to the truncation errors produced by the discretization of the governing equations, we show that the conventional fluxes produce mass residuals that are more descriptive of the overall behavior of the model, i.e., they are better correlated with truncation error. Then we demonstrate that these mass residuals can be used as a criterion for mesh refinement. In an example using a one-dimensional shallow water model, we demonstrate that, by moving nodes from regions with large mass residuals to regions with small mass residuals, a mesh can be developed that shows less truncation error than a mesh developed by using localized truncation error analysis. And, in an example using a two-dimensional shallow water model, we demonstrate that the computed solution can be improved in regions with large mass residuals through mesh refinement.

Externalizing the lateral boundary conditions from the dynamic core in semi-implicit semi-Lagrangian models

Research is still undertaken to develop so-called transparent lateral boundary conditions (LBC) for limited-area numerical weather prediction models. In the widely used semi-implicit semi-Lagrangian schemes, this naturally leads to LBC formulations that are intrinsically intertwined with the numerics of the dynamic core. This may have profound consequences for the implementation and the maintenance of future model codes. For instance, scientific development on the dynamics may be hindered by constraints coming from today’s choices in the LBC formulation and vice versa. Building further on the work of Aidan McDonald, this paper proposes an approach where (1) the LBCs can be imposed by an extrinsic numerical scheme that is fundamentally different from the one used for the dynamic core in the interior domain and (2) substituting one such LBC scheme for another is possible in a manner that leaves the Helmholtz solver unmodified. It is argued that this concept may provide the necessary frame for formulating transparent boundary conditions in spectral limited-area models. Since this idea touches all aspects of the LBC problem, its feasibility can only be established by a rigorous systematic approach. As a first step, this paper provides promising experimental support in a one-dimensional shallow-water model.

Analysis of the Friction Term in the One-Dimensional Shallow-Water Model

The numerical simulation of unsteady open channel flows is very commonly performed using the one-dimensional shallow-water model. Friction is one of the relevant forces included in the momentum equation. In this work, a generalization of the Gauckler-Manning friction model is proposed to improve the modeling approach in cases of dominant roughness, unsteady flow, and distorted cross-sectional shapes. The numerical stability conditions are revisited in cases of dominant friction terms and a new condition, complementary to the basic Courant-Friedrichs-Lewy condition, is proposed. Some test cases with measured data are used to validate the quality of the approaches.

CONSERVATION LAWS FOR ONE-DIMENSIONAL SHALLOW WATER MODELS FOR ONE AND TWO-LAYER FLOWS

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

On the Critical Plunger Speed and Three-Dimensional Effects in High-Pressure Die Casting Injection Chambers

During the initial slow stage of the injection process in high-pressure die casting machines with horizontal cold chamber, a plunger pushes the molten metal which partially fills the injection chamber, causing the formation of a gravity wave. The evolution of the wave surface profile, which depends on the plunger acceleration law, may trap air in the molten metal, causing porosity when the metal solidifies. In this work, a one-dimensional shallow-water model, which is solved numerically using the method of characteristics, and a three-dimensional numerical model, based on a finite element formulation and the volume of fluid (VOF) method for treating the free surface, are used to analyze the flow of molten metal in an injection chamber of circular cross section. The results for the evolution of the free surface obtained from both models for different plunger motion laws and initial filling fractions of the injection chamber were in good agreement for broad ranges of operating conditions. The existence of a critical plunger speed, above which the reflection of the wave of molten metal against the chamber ceiling might appreciably increase air entrapment effects, is investigated. The results for the wave profiles in chambers of circular cross section are compared with those obtained in an equivalent two-dimensional configuration of the injection chamber, for which the shallow-water model is solved analytically. It is shown how the results obtained by applying the one-dimensional model to a two-dimensional chamber configuration can be used to reproduce, with an acceptable degree of accuracy, the salient characteristics of the flow of molten metal in a real injection chamber of circular cross section.

A Partitioned Kalman Filter and Smoother

A new approach is advanced for approximating Kalman filtering and smoothing suitable for oceanic and atmospheric data assimilation. The method solves the larger estimation problem by partitioning it into a series of smaller calculations. Errors with small correlation distances are derived by regional approximations, and errors associated with independent processes are evaluated separately from one another. The overall uncertainty of the model state, as well as the Kalman filter and smoother, is approximated by the sum of the corresponding individual components. The resulting smaller dimensionality of each separate element renders application of Kalman filtering and smoothing to the larger problem much more practical than otherwise. In particular, the approximation makes high-resolution global eddy-resolving data assimilation computationally viable. The approach is described and its efficacy demonstrated using a simple one-dimensional shallow water model.

A Shallow-Water Model of the Diurnal Dryline

This study explores the diurnal variation in the movement and structure of the dryline using a one-dimensional shallow-water model. The model is adapted to test some common theories of dryline motion including the diurnal variation in surface friction, static stability, inversion erosion, and momentum mixing aloft. These mechanisms of diurnal variation are first studied individually and then in unison. A diurnal variation in the surface friction produces a model dryline that moves westward during the day (in disagreement with observations) and has a southerly wind maximum near midnight. A diurnal variation in the static stability produces a model dryline that steepens in slope and moves eastward during the day and then surges westward at night with a southerly wind maximum 6 to 9 h after the minimum stability. Inversion erosion during the day produces a nearly vertical model dry front that moves eastward during the day with surface southwesterlies. At sunset the model dryline surges westward with a southerly wind maximum before midnight. A diurnal variation of the momentum mixing aloft has no significant effect on the model dryline. Results show that the combined case with a diurnal variation of surface friction, inversion erosion, and static stability with terrain most accurately describes the observed dryline system. The westward surge depicted in the model is compared to the flow evolution of the corresponding dam-break problem for a rotating fluid.

Synchronization in chaotic Hamiltonian systems and a geophysical application

This paper addresses the question of the rate of synchronization of two identical systems as a function of the inserting time interval Delta t between inserted variables of the driving system in the role of the same variables of the driven system in a simplified Hamiltonian system and its application to a simplified geophysical model. We start by analyzing the synchronization in a simplified two-degree Hamiltonian system. The synchronization rate turns out to be a decreasing function of the inserting time interval Delta t up to a certain limit Delta t(o) where the process reverses and the synchronization rate becomes slower as the inserting frequency decreases. The key point of the analysis uses a second-order Taylor expansion of the system resolvent which indicates that synchronization rate is basically of order O(Delta t(2)) for small Delta t. The study is then extended to include a simplified geophysical system. A nonlinear one-dimensional shallow-water model on a periodic domain meant to represent a latitudinal circle around 45 degrees N is used. It is found that when the zonal wind is inserted, the maximum synchronization rate is obtained when the inserting time interval is approximately 4 h. When the meridional wind is inserted, it is obtained at slightly less than 4 h. It is shown, in particular, that the synchronization rate depends on the latitude (or the Coriolis parameter). A low-order simplified dynamical system derived from the one-dimensional shallow-water model is used to show that this optimum time interval Delta t(o) when the zonal wind and the geopotential, for example, are inserted varies approximately as square root of [2]/2 Omega sin phi to accuracy O(Delta t(3)). Analyses performed with a linear version of the shallow-water model reveal that this latter can be used to explain the observed convergence behavior in the nonlinear model. The only point is the choice of the stationary state for linearization purposes. It is then suggested that in more complicated geophysical systems, the closest stationary state to the climatology can be used to estimate the crossover point Delta t(o).

The Implementation of the Semi-Implicit Scheme in Cell-Integrated Semi-Lagrangian Models

ABSTRACT The cell-integrated semi-Lagrangian method, in which trajectories from the corner points of a grid cell define its extent at a previous time, may be applied to a set of equations in Lagrangian form derived from the complete set of primitive equations to construct a numerical model which conserves exactly the discrete forms of the global integral constraints of mass, momentum, entrophy and total energy. Due to the fulfilment of these integral constraints one should expect the numerical model to be absolutely stable. Experiments with a simple one-dimensional shallow water model show, however, that a time step dependent instability develops when a CFL criterion for gravity waves is exceeded. Analysis of experiments with one-dimensional shallow water models unveils the mechanism of the instability. Using again for simplicity one-dimensional models a main achievement is a successful implementation of the semi-implicit time stepping scheme in the cell-integrated models.

Optimal determination of nudging coefficients using the adjoint equations

The adjoint equations of a numerical model can be used for model-parameter estimation. In this study, a general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow-water model in the context of analysis-nudging four-dimensional data assimilation (FDDA): the weighting coefficients used by the Newtonian nudging technique are determined such that the model error during the assimilation period is optimally reduced subject to some constraints. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated using this simple grid-point model in an Observing Systems Simulation Experiments (OSSE) mode. The results show that in principle, it is feasible to determine a set of nudging weights which minimize the model error over the period covered by the observations. It is demonstrated, however, that the magnitude and distribution of these “optimal” nudging weights are sensitive to the prescribed estimate of the nudging weights and the corresponding coefficient matrix which define a penalty term in the cost function. The penalty term is a weak constraint on the size and distribution of the optimal nudging weights while the model is the strong constraint. The fit of the model to the data is greater when this constraint on the nudging weights is weaker, but then the nudging weights may be too large or even negative. Thus the “optimal” solution for this model parameter is not unique because specification of this penalty term in the cost function introduces a new uncertainty into the nudging FDDA framework. Nevertheless, this optimal-nudging approach does show promise, but the sensitivity of the technique to the penalty term requires further investigation under more realistic conditions. DOI: 10.1034/j.1600-0870.1993.t01-4-00003.x

The effect of model errors in variational assimilation

A linearized, one-dimensional shallow water model is used to investigate the effect of model errors in four-dimensional variational assimilation. A suitable initialization scheme for variational assimilation is proposed. Introducing deliberate phase speed errors in the model, the results from variational assimilation are compared to standard analysis/forecast cycle experiments. While the latter draws to the data and reflects the model errors only in the datavoid areas, variational assimilation with the model used as strong constraint is shown to distribute the model errors over the entire analysis domain. The implications for verification and diagnostics are discussed. Temporal weighting of the observations can reduce the errors towards the end of the assimilation period, but may deteriorate the subsequent forecasts. An extension to variational assimilation is proposed, which seeks not only to determine the initial state from the observations but also some of the tunable parameters of the model. The potentional usefulness of this approach for parameterization studies and for a separation of forecast errors into model- and analysis errors is discussed. Finally, variational assimilations with the model used as weak constraint are presented. While showing a good performance in the assimilation, forecasts can suffer severely if the extra term in the equations up to which the model is enforced are unable to compensate for the real model error. In the discussion, an overall appraisal of both assimilation methods is given. DOI: 10.1034/j.1600-0870.1992.t01-3-00002.x

Optimal nonlinear objective analysis

Some practical problems in objective analysis for numerical weather prediction are best solved by using nonlinear analysis equations. They include the existence of nonlinear prior constraints on the analysis, and the use of observations which are nonlinearly related to the analysis parameters, or which have non-Gaussian error distributions. Bayesian methods are used to derive equations for the optimal (maximum likelihood) nonlinear analysis. It is shown how to incorporate a strong constraint that the four-dimensional evolution of the analysis should be consistent with a NWP model, by reduction of the control variable to the space-dimensioned initial field for the model. The iterative solution of the nonlinear analysis equation then involves the integration of the NWP model, and its adjoint. The behaviour of the nonlinear equations is demonstrated with a simple one-dimensional shallow-water model. It is shown that time-tendency information, and indirect observations such as wind speed, or the movement of a tracer, can be used in the analysis. The resulting forecasts are better than those made from an analysis from a traditional analysis-forecast cycle. The nonlinear method is shown to be capable of ‘moving’ a discontinuity similar to a front, to fit observations defining its position, thus giving an analysis with more detail than would be expected from the spatial resolution of the observations. The incorporation of additional nonlinear constraints, such as that used in initialization, is demonstrated. The method can be used to effectively reject observations with gross errors, by specifying a non-Gaussian error distribution. However, this generates multiple minima which complicate the search for the best analysis, so the complex decision-taking algorithms associated with other methods of quality control are not avoided. The convergence properties of iterative methods of solution, and approximations to the ideal equations, are studied, in order to provide some indication as to whether the nonlinear effects might be allowed for in a practical analysis scheme.