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Improving numerical accuracy for the viscous-plastic formulation of sea ice
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Abstract
Accurate modeling of sea ice dynamics is critical for predicting environmental variables and is important in applications such as navigating ice breaker ships. Research for both modeling and simulating sea ice dynamics is ongoing, with the most widely accepted model based on the viscous-plastic (VP) formulation introduced by Hibler in 1979. Due to its highly nonlinear features, this model is intrinsically challenging for computational solvers. In particular, sea ice simulations often significantly differ from satellite observations. This study therefore focuses on improving the numerical accuracy of the VP sea ice model. Since the poor convergence observed in existing numerical simulations stems from the nonlinear nature of the VP formulation, this investigation proposes using the celebrated weighted essentially non-oscillatory (WENO) scheme -- as opposed to the frequently employed centered difference (CD) scheme -- for the spatial derivatives in the VP sea ice model. We then proceed to numerically demonstrate that WENO yields higher-order convergence for smooth solutions, and that furthermore it is able to resolve the discontinuities in the sharp features of sea ice covers -- something that is not possible using CD methods. Finally, our proposed framework integrates a potential function method that utilizes the phase field method to naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations, resulting in a modified transport equations which includes additional forcing terms. Our method does not require post-processing, thereby avoiding the possible introduction of discontinuities and corresponding negative impact on the solution behavior. Numerical experiments are provided to demonstrate the efficacy of our new methodology.
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As Arctic conditions rapidly change, human activity in the Arctic will continue to increase and so will the need for high‐resolution observations of sea ice. While satellite imagery can provide high spatial resolution, it is temporally sparse and significant ice deformation can occur between observations. This makes it difficult to apply feature tracking or image correlation techniques that require persistent features to exist between images. With this in mind, we propose a technique based on optimal transport, which is commonly used to measure differences between probability distributions. When little ice enters or leaves the image scene, we show that regularized optimal transport can be used to quantitatively estimate ice deformation. We discuss the motivation for our approach and describe efficient computational implementations. Results are provided on a combination of synthetic and satellite imagery to demonstrate the ability of our approach to estimate dynamics properties at the original image resolution.
Accurate estimates of sharp features in the sea ice cover, such as leads and ridges, are critical for shipping activities, ice operations and weather forecasting. These sharp features can be difficult to preserve in data fusion and data assimilation due to the spatial correlations in the background error covariance matrices. In this article, a set of data fusion and data assimilation experiments are carried out comparing two objective functions, one with a conventional l2-norm and one that imposes an additional l1-norm on the derivative of the ice thickness state estimate. The latter is motivated by analysis of high resolution ice thickness observations derived from an airborne electromagnetic sensor demonstrating the sparsity of the ice thickness in the derivative domain. Data fusion and data assimilation experiments (using a 1 D toy sea-ice model) are carried out over a wide range of background and observation error correlation length scales. Results show the superiority of using an l1–l2 regularisation framework. For the data fusion experiments it was found when both background and observation error correlation length scales are zero, the ice thickness root mean squared error for the l1–l2 method was 0.16 m as compared to 0.20 m for the l2 method. The differences between the methods were greater when the background error correlation length scale was relatively short (approximately five times the analysis grid spacing), and were not significant for larger background error correlation length scales (e.g. 10 times the analysis grid spacing). For data assimilation experiments it was found that openings in the ice cover were captured better with the l1–l2 regularisation, with reduced errors in ice thickness, concentration and velocity. In addition, the ice thickness derivatives in the analyses were found to be more sparse when the l1–l2 method was used and are closer to the those from the true model run.
New numerical solvers are being considered in response to the rising computational cost of properly solving the sea ice momentum equation at high resolution. The Jacobian free version of Newton's method has allowed models to obtain the converged solution faster than other implicit solvers used previously. To further improve on this recent development, the analytical Jacobian of the 1D sea ice momentum equation is derived and used inside Newton's method. The results are promising in terms of computational efficiency. Although robustness remains an issue for some test cases, it is improved compared to the Jacobian free approach. In order to make use of the strong points of both the new and Jacobian free methods, a hybrid preconditioner using the Picard and Jacobian matrices to improve global and local convergence, respectively, is also introduced. This preconditioner combines the robustness and computational efficiency of the previously used preconditioning matrices when solving the sea ice momentum equation.
A new rheological model is developed that builds on an elasto-brittle (EB)
framework used for sea ice and rock mechanics, with the intent of
representing both the small elastic deformations associated with fracturing
processes and the larger deformations occurring along the faults/leads once
the material is highly damaged and fragmented. A viscous-like relaxation term
is added to the linear-elastic constitutive law together with an effective
viscosity that evolves according to the local level of damage of the
material, like its elastic modulus. The coupling between the level of damage
and both mechanical parameters is such that within an undamaged ice cover the
viscosity is infinitely large and deformations are strictly elastic, while
along highly damaged zones the elastic modulus vanishes and most of the
stress is dissipated through permanent deformations. A healing mechanism is
also introduced, counterbalancing the effects of damaging over large
timescales. In this new model, named Maxwell-EB after the Maxwell rheology,
the irreversible and reversible deformations are solved for simultaneously;
hence drift velocities are defined naturally. First idealized simulations
without advection show that the model reproduces the main characteristics of
sea ice mechanics and deformation: strain localization, anisotropy,
intermittency and associated scaling laws.
Current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations. Because the ice strength is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence, noise develops in the numerical solution and these models can even become numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit–EXplicit (IMEX) time integration method. This IMEX method was developed in the framework of an already implemented Jacobian-free Newton–Krylov solver. The basic idea of this IMEX approach is to move the explicit calculation of the sea ice thickness and concentration inside the Newton loop such that these tracers evolve during the implicit integration. To obtain second-order accuracy in time, we have also modified the explicit time integration to a second-order Runge–Kutta approach and by introducing a second-order backward difference method for the implicit integration of the momentum equation. These modifications to the code are minor and straightforward. By comparing results with a reference solution obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger. Results show that the second-order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error.
Scientific observations of sea ice began more than a century ago, but detailed sea-ice models appeared only in the latter half of the last century. The high albedo of sea ice is critical for the Earth's heat balance, and ice motion across the ocean's surface transports fresh water and salt. The basic components in a complete sea-ice model must include vertical thermodynamics and horizontal dynamics, including a constitutive relation for the ice, advection and deformational processes. This overview surveys topics in sea-ice modeling from the global climate modeling perspective, emphasizing work that significantly advanced the state of the art and highlighting promising new developments.
[1] In multicategory sea ice models the compressive strength of the ice pack is often assumed to be a function of the potential energy of pressure ridges. This assumption, combined with other standard features of ridging schemes, allows the ice strength to change dramatically on short timescales. In high-resolution (∼10 km) sea ice models with a typical time step (∼1 hour), abrupt strength changes can lead to large internal stress gradients that destabilize the flow. The unstable flow is characterized by large oscillations in ice concentration, thickness, strength, velocity, and strain rates. Straightforward, physically motivated changes in the ridging scheme can reduce the likelihood of abrupt strength changes and improve stability. In simple test problems with flow toward and around topography, stability is significantly enhanced by eliminating the threshold fraction G* in the ridging participation function. Use of an exponential participation function increases the maximum stable time step at 10-km resolution from less than 30 min to about 2 hours. Modifying the redistribution function to build thinner ridges modestly improves stability and also gives better agreement between modeled and observed thickness distributions. Allowing the ice strength to increase linearly with the mean ice thickness improves stability but probably underestimates the maximum stresses.
We investigate the convergence properties of the nonlinear solver used in viscous-plastic (VP) sea ice models. More specifically, we study the nonlinear solver that is based on an implicit solution of the linearized system of equations and an outer loop (OL) iteration (or pseudo time steps). When the time step is comparable to the forcing time scale, a small number of OL iterations leads to errors in the simulated velocity field that are of the same order of magnitude as the mean drift. The slow convergence is an issue at all spatial resolution but is more severe as the grid is refined. The metrics used by the sea ice modeling community to assess convergence are misleading. Indeed, when performing 10 OL iterations with a 6 h time step, the average kinetic energy of the pack is always within 2% of the fully converged value. However, the errors on the drift are of the same order of magnitude as the mean drift. Also, while 40 OL iterations provide a VP solution (with stress states inside or on the yield curve), large parts of the domain are characterized by errors of 0.5–1.0 cm s−1. The largest errors are localized in regions of large sea ice deformations where strong ice interactions are present. To resolve those deformations accurately, we find that more than 100 OL iterations are required. To obtain a continuously differentiable momentum equation, we replace the formulation of the viscous coefficients with capping with a tangent hyperbolic function. This reduces the number of OL iterations required to reach a certain residual norm by a factor of ∼2.
We present a new modeling framework for sea-ice mechanics based on elasto-brittle (EB) behavior. The EB framework considers sea ice as a continuous elastic plate encountering progressive damage, simulating the opening of cracks and leads. As a result of long-range elastic interactions, the stress relaxation following a damage event can induce an avalanche of damage. Damage propagates in narrow linear features, resulting in a very heterogeneous strain field. Idealized simulations of the Arctic sea-ice cover are analyzed in terms of ice strain rates and contrasted to observations and simulations performed with the classical viscous–plastic (VP) rheology. The statistical and scaling properties of ice strain rates are used as the evaluation metric. We show that EB simulations give a good representation of the shear faulting mechanism that accommodates most sea-ice deformation. The distributions of strain rates and the scaling laws of ice deformation are well captured by the EB framework, which is not the case for VP simulations. These results suggest that the properties of ice deformation emerge from elasto-brittle ice-mechanical behavior and motivate the implementation of the EB framework in a global sea-ice model.
1] Sea ice drift and deformation from coupled ice-ocean models are compared with high-resolution ice motion from the RADARSAT Geophysical Processor System (RGPS). In contrast to buoy drift, the density and extent of the RGPS coverage allows a more extensive assessment and understanding of model simulations at spatial scales from $10 km to near basin scales and from days to seasonal timescales. This work illustrates the strengths of the RGPS data set as a basis for examining model ice drift and its gradients. As it is not our intent to assess relative performance, we have selected four models with a range of attributes and grid resolution. Model fields are examined in terms of ice drift, export, deformation, deformation-related ice production, and spatial deformation patterns. Even though the models are capable of reproducing large-scale drift patterns, variability among model behavior is high. When compared to the RGPS kinematics, the characteristics shared by the models are (1) ice drift along coastal Alaska and Siberia is slower, (2) the skill in explaining the time series of regional divergence of the ice cover is poor, and (3) the deformation-related volume production is consistently lower. Attribution of some of these features to specific causes is beyond our current scope because of the complex interplay between model processes, parameters, and forcing. The present work suggests that high-resolution ice drift observations, like those from the RGPS, would be essential for future model developments, improvements, intercomparisons, and especially for evaluation of the small-scale behavior of models with finer grid spacing.
1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice momentum equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear momentum equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice momentum equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
We have implemented the Jacobian-free Newton–Krylov (JFNK) method to solve the sea ice momentum equation with a viscous-plastic (VP) formulation. The JFNK method has many advantages: the system matrix (the Jacobian) does not need to be formed and stored, the method is parallelizable and the convergence can be nearly quadratic in the vicinity of the solution. The convergence rate of our JFNK implementation is characterized by two phases: an initial phase with slow convergence and a fast phase for which the residual norm decreases significantly from one Newton iteration to the next. Because of this fast phase, the computational gain of the JFNK method over the standard solver used in existing VP models increases with the required drop in the residual norm (termination criterion). The JFNK method is between 3 and 6.6 times faster (depending on the spatial resolution and termination criterion) than the standard solver using a preconditioned generalized minimum residual method. Resolutions tested in this study are 80, 40, 20 and 10km. For a large required drop in the residual norm, both JFNK and standard solvers sometimes do not converge. The failure rate for both solvers increases as the grid is refined but stays relatively small (less than 2.3% of failures). With increasing spatial resolution, the velocity gradients (sea ice deformations) get more and more important. Nonlinear solvers such as the JFNK method tend to have difficulties when there are such sharp structures in the solution. This lack of robustness of both solvers is however a debatable problem as it mostly occurs for large required drops in the residual norm. Furthermore, when it occurs, it usually affects only a few grid cells, i.e., the residual is small for all the velocity components except in very localized regions. Globalization approaches for the JFNK solver, such as the line search method, have not yet proven to be successful. Further investigation is needed.
Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.
In the computation of discontinuous solutions of hyperbolic conservation laws, TVD (total-variation-diminishing), TVB (total-variation-bounded), and the recently developed ENO (essentially non-oscillatory) schemes have proven to be very useful. In this paper two improvements are discussed: a simple TVD Runge-Kutta type time discretization, and an ENO construction procedure based on fluxes rather than on cell averages. These improvements simplify the schemes considerably-especially for multi-dimensional problems with forcing terms. Preliminary numerical results are also given.
A generalized numerical model which allows for a variety of non-linear rheologies is developed for the seasonal simulation of sea-ice circulation and thickness. The model is used to investigate the effects (such as the role of shear stress and the existence of a flow rule) of different rheologies on the ice-drift pattern and build-up in the Arctic Basin. Differences in local drift seem to be closely related to the amount of allowable shear stress. Similarities are found between the elliptical and square cases and between the Mohr-Coulomb and cavitating fluid cases. Comparisons between observed and simulated buoy drift are made for several buoy tracks in the Arctic Basin. Correlation coefficients to the observed buoy drift range from 0.83 for the cavitating fluid to 0.86 for the square rheology. The average ratio of buoy-drift distance to average model-drift distance for several buoys is 1.15 (square), 1.18 (elliptical), 1.30 (Mohr-Coulomb) and 1.40 (cavitating fluid).
Sparse regularization plays a central role in many recent developments in imaging and other related fields. However, it is still of limited use in numerical solvers for partial differential equations (PDEs). In this paper we investigate the use of regularization to promote sparsity in the shock locations of hyperbolic PDEs. We develop an algorithm that uses a high order sparsifying transform which enables us to effectively resolve shocks while still maintaining stability. Our method does not require a shock tracking procedure nor any prior information about the number of shock locations. It is efficiently implemented using the alternating direction method of multipliers. We present our results on one and two dimensional examples using both finite difference and spectral methods as underlying PDE solvers.
The plastic wave speed is derived from the linearized 1-D version of the widely used viscous-plastic (VP) and elastic-viscous-plastic (EVP) sea-ice models. Courant–Friedrichs–Lewy (CFL) conditions are derived using the propagation speed of the wave. 1-D numerical experiments of the VP, EVP and EVP⁎ models successfully recreate a reference solution when the CFL conditions are satisfied, in agreement with the theory presented. The IMplicit–EXplicit (IMEX) method is shown to effectively alleviate the plastic wave CFL constraint on the timestep in the implicitly solved VP model in both 1-D and 2-D. In 2-D, the EVP and EVP⁎ models show first order error in the simulated velocity field when the plastic wave is not resolved. EVP simulations are performed with various advective timestep, number of subcycles, and elastic-wave damping timescales. It is found that increasing the number of subcycles beyond that needed to resolve the elastic wave does not improve the quality of the solution. It is found that reducing the elastic wave damping timescale reduces the spatial extent of first order errors cause by the unresolved plastic wave. Reducing the advective timestep so that the plastic wave is resolved also reduces the velocity error in terms of magnitude and spatial extent. However, the parameter set required for convergence to within the error bars of satellite (RGPS) deformation fields is impractical for use in climate model simulations. The behavior of the EVP⁎ method is analogous to that of the EVP method except that it is not possible to reduce the damping timescale with α=βα=β.
Most dynamic sea ice models for climate type simulations are based on the viscous–plastic (VP) rheology. The resulting stiff system of partial differential equations for ice velocity is either solved implicitly at great computational cost, or explicitly with added pseudo-elasticity (elastic–viscous–plastic, EVP). A recent modification of the EVP approach seeks to improve the convergence of the EVP method by re-interpreting it as a pseudotime VP solver. The question of convergence of this modified EVP method is revisited here and it is shown that convergence is reached provided the stability requirements are satisfied and the number of pseudotime iterations is sufficiently high. Only in this limit, the VP and the modified EVP solvers converge to the same solution. Related questions of the impact of mesh resolution and incomplete convergence are also addressed.
The standard model for sea ice dynamics treats the ice pack as a visco–plastic material that flows plastically under typical stress conditions but behaves as a linear viscous fluid where strain rates are small and the ice becomes nearly rigid. Because of large viscosities in these regions, implicit numerical methods are necessary for time steps larger than a few seconds. Current solution methods for these equations use iterative relaxation methods, which are time consuming, scale poorly with mesh resolution, and are not well adapted to parallel computation. To remedy this, the authors developed and tested two separate methods. First, by demonstrating that the viscous–plastic rheology can be represented by a symmetric, negative definite matrix operator, the much faster and better behaved preconditioned conjugate gradient method was implemented. Second, realizing that only the response of the ice on timescales associated with wind forcing need be accurately resolved, the model was modified so that it reduces to the viscous–plastic model at these timescales, whereas at shorter timescales the adjustment process takes place by a numerically more efficient elastic wave mechanism. This modification leads to a fully explicit numerical scheme that further improves the model's computational efficiency and is a great advantage for implementations on parallel machines. Furthermore, it is observed that the standard viscous–plastic model has poor dynamic response to forcing on a daily timescale, given the standard time step (1 day) used by the ice modeling community. In contrast, the explicit discretization of the elastic wave mechanism allows the elastic–viscous–plastic model to capture the ice response to variations in the imposed stress more accurately. Thus, the elastic–viscous–plastic model provides more accurate results for shorter timescales associated with physical forcing, reproduces viscous–plastic model behavior on longer timescales, and is computationally more efficient overall.
Numerical convergence properties of a recently developed Jacobian-free Newton–Krylov (JFNK) solver are compared to the ones of the widely used EVP model when solving the sea ice momentum equation with a Viscous-Plastic (VP) formulation. To do so, very accurate reference solutions are produced with an independent Picard solver with an advective time step of 10 s and a tight nonlinear convergence criterion on 10, 20, 40, and 80-km grids. Approximate solutions with the JFNK and EVP solvers are obtained for advective time steps of 10, 20 and 30 min. Because of an artificial elastic term, the EVP model permits an explicit time-stepping scheme with a relatively large subcycling time step. The elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the VP solution. Results show that residual elastic waves cause the EVP approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined. Compared to the reference solution, additional shear lines and zones of strong convergence/divergence are seen in the EVP approximate solution. The approximate solution obtained with the JFNK solver is very close to the reference solution for all spatial resolutions tested.
Behavior of the elastic-viscous-plastic (EVP) model for sea ice dynamics is explored, with particular attention to a necessary numerical linearization of the internal ice stress term in the momentum equation. Improvements to both the mathematical and numerical formulations of the model have moderated the impact of linearizing the stress term; simulations with the original EVP formulation and the improved version are used to explain the consequences of using different numerical approaches. In particular, we discuss the model behavior in two regimes, low ice concentration such as occurs in the marginal ice zone, and very high ice concentration, where the ice is nearly rigid. Most of these results are highly relevant to the viscous-plastic (VP) ice dynamics model on which the EVP model is based. We provide examples of certain pathologies that the VP model and its numerical formulations exhibit at steady state.
In this tutorial paper we will give a brief introduction of phase field method as a mathematical tool for describing interfaces and their motion. This method was originally developed as a model of solidification, which enabled us to simulate complicated structures like dendritic crystals easily. The principle of this method is simple and can be applicable to various situations, a few examples will be demonstrated.
A numerical model for the simulation of sea ice circulation and thickness over a seasonal cycle is presented. This model is used to investigate the effects of ice dynamics on Arctic ice thickness and air-sea heat flux characteristics by carrying out several numerical simulations over the entire Arctic Ocean region. The essential idea in the model is to couple the dynamics to the ice thickness characteristics by allowing the ice interaction to become stronger as the ice becomes thicker and/or contains a lower areas percentage of thin ice. The dynamics in turn causes high oceanic heat losses in regions of ice divergence and reduced heat losses in regions of convergence. TO model these effects consistently the ice is considered to interact in a plastic manner with the plastic strength chosen to depend on the ice thickness and concentration. The thickness and concentration, in turn, evolve according to continuity equations which include changes in ice mass and percent of open water due to advection, ...
Sea ice models contain transport equations for the area, volume, and energy of ice and snow in various thickness categories. These equations typically are solved with first-order-accurate upwind schemes, which are very diffusive; with second-order-accurate centered schemes, which are highly oscillatory; or with more so- phisticated second-order schemes that are computationally costly if many quantities must be transported (e.g., multidimensional positive-definite advection transport algorithm (MPDATA)). Here an incremental remapping scheme, originally designed for horizontal transport in ocean models, is adapted for sea ice transport. This scheme has several desirable features: it preserves the monotonicity of both conserved quantities and tracers; it is second-order accurate except where the accuracy is reduced locally to preserve monotonicity; and it efficiently solves the large number of equations in sea ice models with multiple thickness categories and tracers. Remapping outperforms the first-order upwind scheme and basic MPDATA scheme in several simple test problems. In realistic model runs, remapping is less diffusive than the upwind scheme and about twice as fast as MPDATA.
In polar oceans, seawater freezes to form a layer of sea ice of several metres thickness that can cover up to 8% of the Earth’s surface. The modelled sea ice cover state is described by thickness and orientational distribution of interlocking, anisotropic diamond-shaped ice floes delineated by slip lines, as supported by observation. The purpose of this study is to develop a set of equations describing the mean-field sea ice stresses that result from interactions between the ice floes and the evolution of the ice floe orientation, which are simple enough to be incorporated into a climate model. The sea ice stress caused by a deformation of the ice cover is determined by employing an existing kinematic model of ice floe motion, which enables us to calculate the forces acting on the ice floes due to crushing into and sliding past each other, and then by averaging over all possible floe orientations. We describe the orientational floe distribution with a structure tensor and propose an evolution equation for this tensor that accounts for rigid body rotation of the floes, their apparent re-orientation due to new slip line formation, and change of shape of the floes due to freezing and melting. The form of the evolution equation proposed is motivated by laboratory observations of sea ice failure under controlled conditions. Finally, we present simulations of the evolution of sea ice stress and floe orientation for several imposed flow types. Although evidence to test the simulations against is lacking, the simulations seem physically reasonable.
High order accurate weighted essentially nonoscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous solutions. A porous medium equation (PME) is used as an example to demonstrate the algorithm structure and performance. By directly approximating the second derivative term using a conservative flux difference, the sixth order and eighth order finite difference WENO schemes are constructed. Numerical examples are provided to demonstrate the accuracy and nonoscillatory performance of these schemes.
In this paper we introduce a new version of ENO (essentially non-oscillatory) shock-capturing schemes which we call weighted ENO. The main new idea is that, instead of choosing the ''smoothest'' stencil to pick one interpolating polynomial for the ENO reconstruction, we use a convex combination of all candidates to achieve the essentially non-oscillatory property, while additionally obtaining one order of improvement in accuracy. The resulting weighted ENO schemes are based on cell averages and a TVD Runge-Kutta time discretization. Preliminary encouraging numerical experiments are given. (C) 1994 Academic Press, Inc.
In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher and Chan. It was shown by Liu et al. that WENO schemes constructed from the r-th order (in L1 norm) ENO schemes are (r+1)-th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5-th order WENO scheme for the case r = 3, instead of the 4-th order with the original smoothness measurement by Liu et al. This 5-th order WENO scheme is as fast as the 4-th order WENO scheme of Liu et al., and both schemes are about twice as fast as the 4-th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest computing the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which do not contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy wave interaction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.
The 12-layer UCLA general circulation model encompassing troposphere and stratosphere (and superjacent 'sponge layer') is described. Prognostic variables are: surface pressure, horizontal velocity, temperature, water vapor and ozone in each layer, planetary boundary layer (PBL) depth, temperature, moisture and momentum discontinuities at PBL top, ground temperature and water storage, and mass of snow on ground. Selection of space finite-difference schemes for homogeneous incompressible flow, with/without a free surface, nonlinear two-dimensional nondivergent flow, enstrophy conserving schemes, momentum advection schemes, vertical and horizontal difference schemes, and time differencing schemes are discussed.
CICE: the Los Alamos sea ice model documentation and software user's manual version 5.1
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