Scalar advection schemes for ocean modelling on unstructured triangular grids

Several schemes for scalar advection on unstructured triangular grids are assessed for possible use in ocean modelling applications. Finite element, finite volume and finite volume–element approaches are evaluated. A series of tests, including a numerical order of convergence analysis, idealized rotating cone and cylinder experiments, and transport of a tracer through the Stommel Gyre representation of ocean basin-scale circulation, are carried out. Volume element Eulerian–Lagrangian and third-order Runge-Kutta discontinuous Galerkin schemes are recommended for use in tracer studies. Taylor–Galerkin and second-order Runge–Kutta discontinuous Galerkin are found to be robust and accurate second-order schemes. When positivity is required, a fluctuation redistribution scheme was found to be an easily implemented, accurate, and computationally efficient approach. Responsible editor: Phil Dyke     

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other authors.     

Sensitivity of Typhoon Vamei (2001) Simulation to Planetary Boundary Layer Parameterization Using PSU/NCAR MM5

This paper investigates the sensitivity of the numerical simulations of a near equatorial Typhoon Vamei (2001) to various planetary boundary layer (PBL) parameterization schemes in the Pennsylvania State University (PSU)/National Centre for Atmospheric Research (NCAR) non-hydrostatic mesoscale model (MM5). The numerical simulations are conducted on two domains at 45 and 15 km grids nested in a one-way fashion. Four different PBL parameterization schemes including the Blackadar (BLK) scheme, the Burk–Thompson (BURKT) scheme, the NCEP Eta model scheme and the NCEP medium range forecast (MRF) model scheme are investigated. Results indicate that the intensity and propagation track of the simulated near equatorial typhoon system is not very sensitive to the different PBL treatments. The simulated minimum central pressures and the maximum surface wind speeds differ by only 5–6 hPa and 6–8 ms⁻¹, respectively. Larger variations between the simulations occur during the weakening phase of the typhoon system. While all schemes simulated the typhoon with reasonable accuracy, the ETA scheme produces the strongest storm intensity with the largest heat exchanges over the marine environment and the highest warm moisture air content in the PBL around the core of the storm.     

High-order balanced CWENO scheme for movable bed shallow water equations

In this work the numerical integration of 1D shallow water equations (SWE) over movable bed is performed using a well-balanced central weighted essentially non-oscillatory (CWENO) scheme, fourth-order accurate in space and in time. Time accuracy is obtained following a Runge–Kutta (RK) procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained using WENO reconstructions of conservative variables and of flux and bed derivatives. An original treatment for bed slope source term, which maintains the established order of accuracy and satisfies the property of exactly preserving the quiescent flow (C-property), is introduced in the scheme. This treatment consists of two procedures. The former involves the evaluation of the point-values of the flux derivative, considered as a whole with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the expected regularity of the free surface elevation. The high accuracy of the scheme allows to obtain good results using coarse grids, with consequent gain in terms of computational effort. The well-balancing of the scheme allows to reproduce small perturbations of the free surface and of the bottom otherwise of the same order of magnitude of the numerical errors induced by the non-balancing. The accuracy, the well-balancing and the good resolution of the model in reproducing free surface flow over movable bed are tested over analytical solutions and over numerical results available in literature.     

一种时空域优化的高精度交错网格差分算子与正演模拟

如何有效压制数值频散是有限差分正演模拟研究中的关键问题之一.近年来,许多学者对二阶声波方程的差分算子开展了大量的优化工作,在压制频散方面取得不错的效果.一阶压强-速度方程广泛用于研究地震波在地下变密度模型中传播规律,目前针对一阶方程的优化工作大多只是在空间差分算子上展开.本文在前人研究的基础上,推导出一阶声波方程中压强场与偏振速度场之间的解析关系,据此在传统交错网格基础上给出一种高精度的显式时间递推格式,该递推格式将时间差分与空间差分算子结合在一起,并采用共轭梯度法得到精确时间递推匹配系数,实现时空差分算子的同时优化.在编程实现算法的基础上,通过频散分析与三个典型模型测试表明:本文方法能够较为有效地压制时间频散与空间频散,提高数值计算精度;同时对复杂模型也有很好适用性.     

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

An important part in the numerical simulation of tsunami and storm surge events is the accurate modeling of flooding and the appearance of dry areas when the water recedes. This paper proposes a new algorithm to model inundation events with piecewise linear Runge–Kutta discontinuous Galerkin approximations applied to the shallow water equations. This study is restricted to the one-dimensional case and shows a detailed analysis and the corresponding numerical treatment of the inundation problem.The main feature is a velocity based “limiting” of the momentum distribution in each cell, which prevents instabilities in case of wetting or drying situations. Additional limiting of the fluid depth ensures its positivity while preserving local mass conservation. A special flux modification in cells located at the wet/dry interface leads to a well-balanced method, which maintains the steady state at rest. The discontinuous Galerkin scheme is formulated in a nodal form using a Lagrange basis. The proposed wetting and drying treatment is verified with several numerical simulations. These test cases demonstrate the well-balancing property of the method and its stability in case of rapid transition of the wet/dry interface. We also verify the conservation of mass and investigate the convergence characteristics of the scheme.     

High-order schemes for 2D unsteady biogeochemical ocean models

Accurate numerical modeling of biogeochemical ocean dynamics is essential for numerous applications, including coastal ecosystem science, environmental management and energy, and climate dynamics. Evaluating computational requirements for such often highly nonlinear and multiscale dynamics is critical. To do so, we complete comprehensive numerical analyses, comparing low- to high-order discretization schemes, both in time and space, employing standard and hybrid discontinuous Galerkin finite element methods, on both straight and new curved elements. Our analyses and syntheses focus on nutrient–phytoplankton–zooplankton dynamics under advection and diffusion within an ocean strait or sill, in an idealized 2D geometry. For the dynamics, we investigate three biological regimes, one with single stable points at all depths and two with stable limit cycles. We also examine interactions that are dominated by the biology, by the advection, or that are balanced. For these regimes and interactions, we study the sensitivity to multiple numerical parameters including quadrature-free and quadrature-based discretizations of the source terms, order of the spatial discretizations of advection and diffusion operators, order of the temporal discretization in explicit schemes, and resolution of the spatial mesh, with and without curved elements. A first finding is that both quadrature-based and quadrature-free discretizations give accurate results in well-resolved regions, but the quadrature-based scheme has smaller errors in under-resolved regions. We show that low-order temporal discretizations allow rapidly growing numerical errors in biological fields. We find that if a spatial discretization (mesh resolution and polynomial degree) does not resolve the solution, oscillations due to discontinuities in tracer fields can be locally significant for both low- and high-order discretizations. When the solution is sufficiently resolved, higher-order schemes on coarser grids perform better (higher accuracy, less dissipative) for the same cost than lower-order scheme on finer grids. This result applies to both passive and reactive tracers and is confirmed by quantitative analyses of truncation errors and smoothness of solution fields. To reduce oscillations in un-resolved regions, we develop a numerical filter that is active only when and where the solution is not smooth locally. Finally, we consider idealized simulations of biological patchiness. Results reveal that higher-order numerical schemes can maintain patches for long-term integrations while lower-order schemes are much too dissipative and cannot, even at very high resolutions. Implications for the use of simulations to better understand biological blooms, patchiness, and other nonlinear reactive dynamics in coastal regions with complex bathymetric features are considerable.     

Fully coupled methods for multiphase morphodynamics

We present numerical methods for a system of equations consisting of the two dimensional Saint–Venant shallow water equations (SWEs) fully coupled to a completely generalized Exner formulation of hydrodynamically driven sediment discharge. This formulation is implemented by way of a discontinuous Galerkin (DG) finite element method, using a Roe Flux for the advective components and the unified form for the dissipative components. We implement a number of Runge–Kutta time integrators, including a family of strong stability preserving (SSP) schemes, and Runge–Kutta Chebyshev (RKC) methods. A brief discussion is provided regarding implementational details for generalizable computer algebra tokenization using arbitrary algebraic fluxes. We then run numerical experiments to show standard convergence rates, and discuss important mathematical and numerical nuances that arise due to prominent features in the coupled system, such as the emergence of nondifferentiable and sharp zero crossing functions, radii of convergence in manufactured solutions, and nonconservative product (NCP) formalisms. Finally we present a challenging application model concerning hydrothermal venting across metalliferous muds in the presence of chemical reactions occurring in low pH environments.     

The Elastic Strain Energy of Damaged Solids with Applications to Non-Linear Deformation of Crystalline Rocks

Laboratory and field data indicate that rocks subjected to sufficiently high loads clearly deviate from linear behavior. Non-linear stress–strain relations can be approximated by including third and higher-order terms of the strain tensor in the elastic energy expression (e.g., the Murnaghan model). Such classical non-linear models are successful for calculating deformation of soft materials, for example graphite, but cannot explain with the same elastic moduli small and large non-linear deformation of stiff rocks, such as granite. The values of the third (higher-order) Murnaghan moduli estimated from acoustic experiments are one to two orders of magnitude above the values estimated from stress–strain relations in quasi-static rock-mechanics experiments. The Murnaghan model also fails to reproduce an abrupt change in the elastic moduli upon stress reversal from compression to tension, observed in laboratory experiments with rocks, concrete, and composite brittle material samples, and it predicts macroscopic failure at stress levels lower than observations associated with granite. An alternative energy function based on second-order dependency on the strain tensor, as in the Hookean framework, but with an additional non-analytical term, can account for the abrupt change in the effective elastic moduli upon stress reversal, and extended pre-yielding deformation regime with one set of elastic moduli. We show that the non-analytical second-order model is a generalization of other non-classical non-linear models, for example “bi-linear”, “clapping non-linearity”, and “unilateral damage” models. These models were designed to explain the abrupt changes of elastic moduli and non-linearity of stiff rocks under small strains. The present model produces dilation under shear loading and other non-linear deformation features of the stiff rocks mentioned above, and extends the results to account for gradual closure of an arbitrary distribution of initial cracks. The results provide a quantitative framework that can be used to model simultaneously, with a small number of coefficients, multiple observed aspects of non-linear deformation of stiff rocks. These include, in addition to the features mentioned above, stress-induced anisotropy and non-linear effects in resonance experiments with damaged materials.     

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Scalar Transport

This is the first paper in a two-part series on the implementation of Godunov-type schemes on unstructured grids for atmospheric flow simulations. Construction of a high-resolution flow solver for the scalar transport equation is described in detail. Higher-order accuracy in space is achieved via a MUSCL-type gradient reconstruction after van Leer and the monotonicity of solution is enforced by slope limiters. Accuracy in time is maintained by implementing a multi-stage explicit Runge-Kutta time-marching algorithm. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. Five different benchmark test cases are simulated for the validation of the numerical scheme.     

The Homogeneous Finite-Difference Formulation of the P-SV-Wave Equation of Motion

Two different approaches to finite-difference modeling of the elastodynamic equations have been used: the heterogeneous and the homogeneous. In the heterogeneous approach, boundary conditions at interfaces are treated implicitly; in the homogeneous, they are explicitly discretized. We present a homogeneous finite-difference scheme for the 2-D P-SV-wave case. This scheme represents a generalization of earlier such schemes, being able to model media with arbitrary non-uniformities, provided only that all interfaces are aligned with the numerical grid. We perform a detailed comparison of the generalized homogeneous scheme with the analogous heterogeneous scheme, and show the two schemes to be identical for media with a spatially constynt Poisson's ratio. For media where Poisson's ratio is spatially varying, the schemes differ by terms first-order in the spatial step size. However, a comparison of the numerical results produced by the two schemes shows that the resulting differences are negligible for a wide range of values of the Poisson's ratio contrast.     

求解弹性波方程的辛RKN格式

将弹性波方程变换至Hamilton体系,构造适用于弹性波模拟的高效显式二阶辛Runge-Kutta-Nystrm(RKN)格式,运用根数理论得到此格式的阶条件方程组.通过给定系数的限定条件,得到方程的对称解.为了使时间离散误差达到极小,提出数值频率与真实频率比较,通过Taylor展开,得到关于辛系数的限定方程,求解方程组得到最小频散辛RKN格式.对比分析时间演进方程的稳定性,得到使库朗数达到极大值的限定方程,求解方程组得到最稳定辛RKN格式.发现此两种格式为同一格式.新得到的辛RKN格式不依赖于空间离散方法,为了对比的需要,选取有限差分法进行空间离散.在频散、稳定性分析中,与常见辛格式对比,从理论上分析了本文提出的格式在数值频散压制、稳定性提升等方面的优势,数值实验进一步证实了理论分析的正确性.     

A nearly analytic symplectic partitioned Runge–Kutta method based on a locally one-dimensional technique for solving two-dimensional acoustic wave equations

In this paper, we develop a new nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique for numerically solving two-dimensional acoustic wave equations. We first split two-dimensional acoustic wave equation into the local one-dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high-order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge–Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge–Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique, such as their computational efficiency, stability, numerical dispersion and long-term calculation capability.     

Why the <Emphasis Type="Italic">Euler</Emphasis> scheme in particle tracking is not enough: the shallow-sea pycnocline test case

During the last decades, the Euler scheme was the common “workhorse” in particle tracking, although it is the lowest-order approximation of the underlying stochastic differential equation. To convince the modelling community of the need for better methods, we have constructed a new test case that will show the shortcomings of the Euler scheme. We use an idealised shallow-water diffusivity profile that mimics the presence of a sharp pycnocline and thus a quasi-impermeable barrier to vertical diffusion. In this context, we study the transport of passive particles with or without negative buoyancy. A semi-analytic solutions is used to assess the performance of various numerical particle-tracking schemes (first- and second-order accuracy), to treat the variations in the diffusivity profile properly. We show that the commonly used Euler scheme exhibits a poor performance and that widely used particle-tracking codes shall be updated to either the Milstein scheme or second-order schemes. It is further seen that the order of convergence is not the only relevant factor, the absolute value of the error also is.     

Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed

This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.     

Evaluation of Short-Range Forecasts from a Mesoscale Model Over the Indian Region During Monsoon 2006

This study examines the short-range forecast accuracy of the Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model (MM5) as applied to the July 2006 episode of the Indian summer monsoon (ISM) and the model's sensitivity to the choice of different cumulus parameterization schemes (CPSs), namely Betts-Miller, Grell (GR) and Kain-Fritsch (KF). The results showed that MM5 day 1 (0–24 h prediction) and day 2 (24–48 h prediction) forecasts using all three CPSs overpredicted monsoon rainfall over the Indian landmass, with the larger overprediction seen in the day 2 forecasts. Among the CPSs, the rainfall distribution over the Indian landmass was better simulated in forecasts using the KF scheme. The KF scheme showed better skill in predicting the area of rainfall for most of the rainfall thresholds. The root mean square error (RMSE) in day 1 and day 2 rainfall forecasts using different CPSs showed that rainfall simulated using the KF scheme agreed better with the observed rainfall. As compared to other CPSs, simulation using the GR scheme showed larger RMSE in wind speed prediction at 850 and 200 hPa over the Indian landmass. MM5 24-h temperature forecasts at 850 hPa with all the CPSs showed a warm bias of the order of 1 K over the Indian landmass and the bias doubled in 48-h model forecasts. The mean error in temperature prediction at 850 hPa over the Indian region using the KF scheme was comparatively smaller for all the forecast intervals. The model with all the CPSs overpredicted humidity at 850 hPa. The improved prediction by MM5 with the KF scheme is well complemented by the smaller error shown by the KF scheme in vertical distribution of heat and mean moist static energy in the lower troposphere. In this study, the KF scheme which explicitly resolve the downdrafts in the cloud column tended to produce more realistic precipitation forecasts as compared to other schemes which did not explicitly incorporate downdraft effects. This is an important result especially given that the area covered by monsoon-precipitating systems is largely from stratiform-type clouds which are associated with strong downdrafts in the lower levels. This result is useful for improving the treatment of cumulus convection in numerical models over the ISM region.     

A 3D unstructured non-hydrostatic ocean model for internal waves

A 3D non-hydrostatic model is developed to compute internal waves. A novel grid arrangement is incorporated in the model. This not only ensures the homogenous Dirichlet boundary condition for the non-hydrostatic pressure can be precisely and easily imposed but also renders the model relatively simple in its discretized form. The Perot scheme is employed to discretize horizontal advection terms in the horizontal momentum equations, which is based on staggered grids and has the conservative property. Based on previous water wave models, the main works of the present paper are to (1) utilize a semi-implicit, fractional step algorithm to solve the Navier-Stokes equations (NSE); (2) develop a second-order flux-limiter method satisfying the max–min property; (3) incorporate a density equation, which is solved by a high-resolution finite volume method ensuring mass conservation and max–min property based on a vertical boundary-fitted coordinate system; and (4) validate the developed model by using four tests including two internal seiche waves, lock-exchange flow, and internal solitary wave breaking. Comparisons of numerical results with analytical solutions or experimental data or other model results show reasonably good agreement, demonstrating the model’s capability to resolve internal waves relating to complex non-hydrostatic phenomena.     

Climate change during the last 150 million years: reconstruction from a carbon cycle model

Variations of the atmospheric CO₂ level and the global mean surface temperature during the last 150 Ma are reconstructed by using a carbon cycle model with high-resolution input data. In this model, the organic carbon budget and the CO₂ degassing from the mantle, both of which would characterize the carbon cycle during the Cretaceous, are considered, and the silicate weathering process is formulated consistently with an abrupt increase in the marine strontium isotope record for the last 40 Ma. The second-order variations of the atmospheric CO₂ level and the global mean surface temperature in addition to the first-order cooling trend are obtained by using high-resolution data of carbon isotopic composition of marine limestone, seafloor spreading rate, and production rate of oceanic plateau basalt. The results obtained from this model are in good agreement with the previous estimates of palaeo-CO₂ level and palaeoclimate inferred from geological, biogeochemical, and palaeontological models and records. The system analyses of the carbon cycle model to understand the cause of the climate change show that the dominant controlling factors for the first-order cooling trend of climate change during the last 150 Ma are tectonic forcing such as decrease in volcanic activity and the formation and uplift of the Himalayas and the Tibetan Plateau, and, to a lesser extent, biological forcing such as the increase in the soil biological activity. The mid-Cretaceous was very warm because of the high CO₂ level (4–5 PAL) maintained by the enhanced CO₂ degassing rate due to the increased mantle plume activities and seafloor spreading rates at that time, although the enhanced organic carbon burial would have a tendency to decrease the CO₂ level effectively at that period. The variation of organic carbon burial rate may have been responsible for the second-order climate change during the last 150 Ma.     

Numerical modeling of 3D fully nonlinear potential periodic waves

A simple and exact numerical scheme for long-term simulations of 3D potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following nonorthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid and the fourth-order Runge–Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear 2D surface waves, generation of extreme waves, and direct calculations of nonlinear interactions.     

横向各向同性介质紧致交错网格有限差分波场模拟(英文)

针对有限差分数值模拟的频散问题,本文将交错网格技术和紧致差分格式相结合,推导了横向各向同性介质一阶速度一应力波动方程的紧致交错网格差分格式;对比分析了紧致交错网格差分格式、交错网格差分格式以及紧致差分格式的截断误差主项,并利用Fourier误差分析方法分析了上述三种差分格式的近似精度;在此基础上,分别采用上述三种差分格式进行了波场数值模拟。结果表明,当差分方程阶数相同时,紧致交错网格差分格式截断误差最小,数值频散最弱,差分精度最高,证实了该方法的有效性。