scholarly journals Numerical and Physical Diffusion: Can Wave Prediction Models Resolve Directional Spread?

2005 ◽  
Vol 22 (7) ◽  
pp. 886-895 ◽  
Author(s):  
F. Ardhuin ◽  
T. H. C. Herbers
Keyword(s):  
Continental Shelf ◽  
Prediction Models ◽  
Spectral Band ◽  
Wave Spectrum ◽  
Single Frequency ◽  
Propagation Time ◽  
Time Step ◽  
Numerical Diffusion ◽  
Wave Prediction ◽  
Lagrangian Scheme

Abstract A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.

Development of a Fast Three-Dimensional Dynamic Radiative Transfer Solver for Numerical Weather Prediction Models


2021 ◽  
Author(s):  
Richard Maier ◽  
Bernhard Mayer ◽  
Claudia Emde ◽  
Aiko Voigt
Keyword(s):  
Radiative Transfer ◽  
Numerical Weather Prediction ◽  
Prediction Models ◽  
Weather Prediction ◽  
Three Dimensional ◽  
Transfer Model ◽  
Heating Rates ◽  
Time Step ◽  
Numerical Weather ◽  
Numerical Weather Prediction Models

<div> <div> <div> <div> <p>The increasing resolution of numerical weather prediction models makes 3D radiative effects more and more important. These effects are usually neglected by the simple 1D independent column approximations used in most of the current models. On top of that, these 1D radiative transfer solvers are also called far less often than the model’s dynamical core.</p> <p>To address these issues, we present a new „dynamic“ approach of solving 3D radiative transfer. Building upon the existing TenStream solver (Jakub and Mayer, 2015), radiation in this 3D model is not solved completely in each radiation time step, but is rather only transported to adjacent grid boxes. For every grid box, outgoing fluxes are then calculated from the incoming fluxes from the neighboring grid cells of the previous time step. This allows to reduce the computational cost of 3D radiative transfer models to that of current 1D solvers.</p> <p>Here, we show first results obtained with this new solver with a special emphasis on heating rates. Furthermore, we demonstrate issues related to the dynamical treatment of radiation as well as possible solutions to these problems.</p> <p>In the future, the speed of this newly developed 3D dynamic TenStream solver will be further increased by reducing the number of spectral bands used in the radiative transfer calculations with the aim to get a 3D solver that can be called even more frequently than the 1D two-stream solvers used nowadays.</p> <p>Reference:<br><span>Jakub, F. and Mayer, B. (2015), A three-dimensional parallel radiative transfer model for atmospheric heating rates for use in cloud resolving models—The TenStream solver, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 163, 2015, Pages 63-71, ISSN 0022-4073, . </span></p> </div> </div> </div> </div>

Quantitative Evaluation of Numerical Diffusion (Truncation Error)

1971 ◽  
Vol 11 (03) ◽  
pp. 315-320 ◽  
Author(s):  
R.B. Lantz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Truncation Error ◽  
Convective Diffusion ◽  
Block Size ◽  
Time Step ◽  
Partial Differential ◽  
Numerical Diffusion ◽  
Convection Diffusion

Abstract Numerical diffusion (truncation error) can limit the usefulness of numerical finite-difference approximations to solve partial differential equations. Many reservoir simulation users are aware of these limitations but are not as familiar with actually quantifying the magnitude of the truncation error. This paper illustrates that, over a wide range of block size and time step, the truncation error expressions for convective-diffusion partial differential equations are quantitative. Since miscible, thermal, and immiscible processes can be of the convective-diffusion equation form, the truncation error expressions presented can provide guidelines for choosing block size-time step combinations that minimize the effect of numerical diffusion. Introduction Truncation error limits the use of numerical finite-difference approximations to solve partial differential equations. In the solution of convection-diffusion equations, such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing two-phase fluid flow can also be rearranged into a convection-diffusion form. And, in fact, miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs, and probably all numerical analysts, recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this paper is to give the user more than just a qualitative feel for the importance of truncation error. In this paper, insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. Consequently, the reservoir simulator user can observe that these expressions are quantitative and can use them as guidelines for choosing block sizes and time steps that keep the numerical diffusivity small. DEVELOPMENT OF EXPRESSIONS FOR TRUNCATION ERROR APPLICATION TO CONVECTION-DIFFUSION EQUATION To illustrate the method of quantifying numerical diffusivity, consider a convective-diffusion equation of the form: ..............(1) Symbols are defined in the Nomenclature. The first term on the right-hand side represents the diffusion, and the second term represents convection. Such an equation describes the flow of either a two-component miscible mixture or heat in one dimension with constant diffusivity. EXPLICIT DIFFERENCE FORMS An explicit expression for the truncation error (the space derivatives are approximated at a known time level) can be developed by examining the Taylor's series expansion representing first- and second-order derivatives. For the time derivative: .....(2) SPEJ P. 315

Uncertainties of Wind Sea and Swell Prediction From the Torsethaugen Spectrum

2009 ◽  
Author(s):  
Elzbieta M. Bitner-Gregersen ◽  
Alessandro Toffoli
Keyword(s):  
Continental Shelf ◽  
Wave Spectrum ◽  
Single Mode ◽  
Offshore Structures ◽  
Reasonable Accuracy ◽  
Double Peak ◽  
Design Work ◽  
Practical Applications ◽  
Wind Sea ◽  
Norwegian Continental Shelf

In practical applications, it is usually assumed that the wave spectrum is of a single mode form, and well modelled by a JONSWAP or Pierson-Moskowitz spectrum. This assumption is of a reasonable accuracy for severe sea states. However, moderated and low sea states are often of a combined nature, consisting of both wind-sea and swell and should be characterized by a double peak spectrum. Bimodal seas can have a significant impact on the design and operability of fixed and floating offshore structures as well as LNG terminals. Although several separation procedures for the wave components exist the bimodal Torsethaugen spectrum is probably the only one well established in design work. This spectrum was developed primarily for one location at the Norwegian Continental Shelf (Statfjord Field) but in qualitative terms is expected to be of much broader validity. The present study discusses applicability of the Torsethaugen spectrum for locations outside the Norwegian Continental Shelf and uncertainties related to use of the spectrum.

Systematic Experimental Validation of High-Order Spectral Method for Deterministic Wave Prediction

2019 ◽  
Author(s):  
Marco Klein ◽  
Matthias Dudek ◽  
Günther F. Clauss ◽  
Norbert Hoffmann ◽  
Jasper Behrendt ◽  
...  
Keyword(s):  
Spectral Method ◽  
Experimental Validation ◽  
Linear Method ◽  
Wave Spectrum ◽  
High Order ◽  
Linear Wave ◽  
Short Term ◽  
Wave Prediction ◽  
Non Linear ◽  
High Order Spectral Method

Abstract The applicability of the High-Order Spectral Method (HOSM) as a very fast non-linear method for deterministic short-term wave prediction is discussed within this paper. The focus lies on the systematic experimental validation of the HOSM in order to identify and evaluate possible areas of application as well as limitations of use. For this purpose, irregular sea states with varying parameters such as wave steepness and underlying wave spectrum are addressed by numerical simulations and model tests in the controlled environment of a seakeeping basin. In addition, the influence of the propagation distance is discussed. For the evaluation of the accuracy of the HOSM prediction, the surface similarity parameter (SSP) is utilized, allowing a quantitative validation of the results. The results obtained are compared to linear wave prediction to discuss the pros and cons of a non-linear deterministic short-term wave prediction. In conclusion, this paper shows that the non-linear deterministic wave prediction based on HOSM leads to a substantial improvement of the prediction quality for moderate and steep irregular wave trains in terms of individual waves and prediction distance.

Performance of Formulations for Whitecapping in Wave Prediction Models

2002 ◽  
Author(s):  
Gerbrant Ph. van Vledder ◽  
David P. Hurdle
Keyword(s):  
Prediction Models ◽  
Wave Spectra ◽  
Third Generation ◽  
Wave Prediction

This paper describes work currently being carried out to examine possible methods to improve the computation of the dissipation by whitecapping in third generation wave prediction models. Such alternatives are needed to avoid unphysical dissipation behavior in the case of double-peaked wave spectra. First, an overview is given of the problems associated with the formulation for whitecapping that is now widely used in wave prediction models. Second, a summary is given of existing suggestions to improve the whitecapping formulation. Third, a number of examples are given with the new formulations to illustrate the potential improvements.

scholarly journals A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov–Poisson equation

2019 ◽  
Vol 33 (5) ◽  
pp. 924-947 ◽  
Author(s):  
Katharina Kormann ◽  
Klaus Reuter ◽  
Markus Rampp
Keyword(s):  
Massively Parallel ◽  
Benchmark Problems ◽  
Time Step ◽  
Decomposition Approach ◽  
Poisson Problem ◽  
Dimensional Phase Space ◽  
Communication Scheme ◽  
Parallel Case ◽  
Lagrangian Scheme ◽  
Efficient Parallelization

This article presents an optimized and scalable semi-Lagrangian solver for the Vlasov–Poisson system in six-dimensional phase space. Grid-based solvers of the Vlasov equation are known to give accurate results. At the same time, these solvers are challenged by the curse of dimensionality resulting in very high memory requirements, and moreover, requiring highly efficient parallelization schemes. In this article, we consider the 6-D Vlasov–Poisson problem discretized by a split-step semi-Lagrangian scheme, using successive 1-D interpolations on 1-D stripes of the 6-D domain. Two parallelization paradigms are compared, a remapping scheme and a domain decomposition approach applied to the full 6-D problem. From numerical experiments, the latter approach is found to be superior in the massively parallel case in various respects. We address the challenge of artificial time step restrictions due to the decomposition of the domain by introducing a blocked one-sided communication scheme for the purely electrostatic case and a rotating mesh for the case with a constant magnetic field. In addition, we propose a pipelining scheme that enables to hide the costs for the halo communication between neighbor processes efficiently behind useful computation. Parallel scalability on up to 65,536 processes is demonstrated for benchmark problems on a supercomputer.

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