A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.     

A Particle Tracking Algorithm for Arbitrary Unstructured Grids

A new semi-analytical algorithm for generating the local groundwater velocity field within irregular three-dimensional grid cells (such as a Voronoi cell) is presented. The method supports particle tracking in the MODFLOW-USG software package for a variety of supported unstructured grid formats. It is shown to replicate the results from the fully analytical Pollock's method when used on a rectilinear grid, and is demonstrated for difficult test cases where there is high spatial variability in the flow field within individual cells. The method is now available as an option in the mod-PATH3DU software package and may be used with most MODFLOW-USG packages without additional intervention.     

An Efficient and Robust Tsunami Model on Unstructured Grids. Part I: Inundation Benchmarks

A modern multi-purpose baroclinic circulation model (SELFE) has been recently extended to include the ability to simulate tsunami propagation and inundation. The core model is based on the 3-D nonlinear shallow-water wave (NSW) equations, which are solved on unstructured grids, using the finite-element method. A semi-implicit method is used to solve all equations to enhance numerical stability, thus bypassing the most stringent CFL restriction on the time step. Further aided algorithmically by an Eulerian-Lagrangian solution of the advection terms in the momentum equation and by a simple yet effective inundation algorithm, SELFE is very efficient and robust in both quasi-2-D (with two vertical layers) and 3-D modes. A quasi-2-D version of the model is being used to update and expand the characterization of tsunami hazards along the Oregon coast. As a part of a rigorous testing procedure that includes multiple types of coastal problems, we present in this paper a quantitative assessment of performance of the quasi-2-D SELFE for two challenging open benchmark problems proposed in the 3^rd International Workshop on Long-wave Runup Models. Satisfactory results are obtained for both problems.     

Injection Effects on Sediment Transport in Closed-Conduit Flows

Experiments were conducted to investigate injection effects on sediment transport in closed-conduit flows. The results show that the sediment transport rate essentially remains unchanged when the ratio of the injection velocity and that at boiling, Vi/Vcr < 10. However, significant sediment transport rate is observed when Vi/Vcr increases beyond this limit. In the literature, three semi-empirical models have been developed to relate seepage effects on the sediment transport rate. The experimentally measured data in the pre- and post-boiling condition (Liu and Chiew 2014, and the present study, respectively) are compared with these models. The results show that the models of Francalanci et al. (2008) and Nielsen et al. (2001) perform poorly in predicting injection effects on the sediment transport. Although Yang’s (2013) model could reasonably predict the influence of injection on the sediment transport rate in the post-boiling condition, it similarly fails when applied to the pre-boiling condition.     

A Godunov-type Finite Volume Scheme for Meso- and Micro-scale Flows in Three Dimensions

This short note reports the extension of the f-waves approximate Riemann solver (Ahmad and Lindeman, 2007; LeVeque, 2002; Bale et al., 2002) for three-dimensional meso- and micro-scale atmospheric flows. The Riemann solver employs flux-based wave decomposition for the calculation of Godunov fluxes and does not require the explicit definition of the Roe matrix to enforce conservation. The other important feature of the Riemann solver is its ability to incorporate source term due to gravity without introducing discretization errors. The resulting finite volume scheme is second-order accurate in space and time. The finite-difference schemes currently used in atmospheric flow models are neither conservative nor able to resolve regions of sharp gradients. The finite volume scheme described in this paper is fully conservative and has the ability to resolve regions of sharp gradients without introducing spurious oscillations in the solution. The scheme shows promise in accurately resolving flows on the meso- and micro-scales and should be considered for implementation in the dynamical cores of next generation meso- and micro-scale atmospheric flow models.     

Automatic Generation of Locally Refined Composite Grids for Efficient Solute Transport Modeling

Widely used numerical models of solute transport processes in subsurface aquifers are limited to nonlocally refined rectangular, or logically rectangular, structured grids. This presents an unsuitable option to efficient numerical simulations maintaining an acceptable level of accuracy. Optimal selection of locally refined cells for efficient solute transport models is challenging to the current generation of numerical models. We present a novel and relatively simple to implement algorithm addressing these shortcomings. This method operates in four steps involving travel times simulations, a grid coarsening stage followed by a selective local grid refinement based on a cell-wise indicator, and a final postprocessing step. The refinement index is the sum of weighted logarithmic distributions of scaled forward and backward travel times. We calculate representative flow and transport properties at the two scales of the composite grid with a flow-based upscaling technique. We present two test problems to demonstrate the performances of this new gridding algorithm. We obtain the most important speedups for composite grids generated with the highest indicator thresholds. When hydrodynamic dispersion effects increase, we obtain less important speedups. An important outcome of this work is that grid design depends on nature and strength of the underlying flow and solute transport processes. Therefore, we suggest developing solute transport workflows integrating this grid generation algorithm as an integral component to build comprehensive and efficient groundwater models.     

Unstructured Gridding for MODFLOW from Prior Groundwater Flow Models: A New Paradigm

This work introduces a new unstructured gridding approach relying on feedback from a previous groundwater flow model. All cells in a relatively coarse model using a rectilinear grid are recursively subdivided following a cell wise specific discharge-based indicator to generate quadtree, octree or Voronoï grids. This technique leverages the full potential of the latest MODFLOW engines. The suitability of this approach is demonstrated on challenging single and multilayered heterogeneous formations. The proposed method is straightforward to implement in existing software packages. It supports iterative updating of groundwater flow models from the legacy rectilinear to unstructured grids.     

Influence of Atmospheric Radon Transport on a Radon Flux from the Surface

Radon-222 is inert radioactive gas with a half-life period of 3.8 days, it is a decay product of radium-226. Being escaped from minerals and underground waters into pore space of rocks and soil, it is transported to the surface by diffusion and advection and gone to the atmosphere. When modeling processes of atmosphere radon transport one sets value of radon flux from the surface, depending on contents of radium-226 in rocks and conditions of radon transport in soil, especially on soil porosity and humidity. The impact of radon turbulent transport in the atmospheric boundary layer (ABL) on radon flux density from the surface is estimated in this paper. It is shown that both for stationary state and for typical diurnal variations of the radon volumetric activity (VA) in the ABL, the correction to the radon flux density caused by its turbulent transport in the ABL is negligible (less than 1%) and doesn’t exceed measurement errors. Thus, when calculating radon VA in the ABL it is really possible to set an average value of the radon flux density on the surface as a boundary condition.     

A Tracer Methodology for Identifying Ambient Flows in Boreholes

Identifying flows into, out of, and across boreholes is important for characterizing aquifers, determining the depth at which water enters boreholes, and determining the locations and rates of outflow. This study demonstrates how Single Borehole Dilution Tests (SBDTs) carried out under natural head conditions provide a simple and cheap method of identifying vertical flow within boreholes and determining the location of in‐flowing, out‐flowing, and cross‐flowing fractures. Computer simulations were used to investigate the patterns in tracer profiles that arise from different combinations of flows. Field tracer tests were carried out using emplacements of a saline tracer throughout the saturated length of boreholes and also point emplacements at specific horizons. Results demonstrated that SBDTs can be used to identify flowing fractures at the top and bottom of sections of vertical flow, where there is a change in vertical flow rate within a borehole, and also where there are consistent decreases in tracer concentration at a particular depth. The technique enables identification of fractures that might be undetected by temperature and electrical conductance logging, and is a simple field test that can be carried out without pumping the borehole.     

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     

Scalar and vector probabilistic seismic hazard analysis: application for Algiers City

This study deals with the application of probabilistic seismic hazard analysis (PSHA) for a rock site located in Algiers city. For this purpose, recent ground motion prediction equations developed in the world for similar sismotectonic context are used through logic tree in PSHA framework; the obtained results reflect clearly the high seismicity of the considered region. Moreover, deaggregation analysis is conducted to obtain the mean scenario in terms of magnitude and distance. In addition to the scalar-PSHA, a new method named vector-PSHA developed in recent years is performed in this study. Based on the multivariate probability theory, the software used in scalar approach is modified allowing the application of this approach for a real site in Algiers city with a vector of two and three parameters of intensity measure. The results are presented in terms of the joint annual rate of exceeding several thresholds such as PGA, PSA(T) of multiple vibration periods, peak ground velocity and Arias intensity and comparison between results of PSHA and V-PSHA is done.     

Atmospheric stability effect on subgrid-scale physics for large-eddy simulation

Field measurements in the atmospheric boundary layer were carried out to identify the effect of atmospheric stability on subgrid-scale physics for large-eddy simulation. The basic instrumentation setup consisted of 12 three-dimensional sonic anemometers arranged in two parallel horizontal arrays (seven sensors in the lower array and five sensors in the upper array). Data from this setup are used to compute the subgrid-scale (SGS) heat fluxes and SGS dissipation of the temperature variance under stable and unstable stability conditions. The relative contribution of the SGS vertical flux to the total turbulent flux increases when going from unstable to stable conditions. The relative importance of negative SGS dissipation (backscatter) events becomes larger under stable conditions. The model coefficients for two well-known SGS models (eddy-viscosity and non-linear) are computed. Model coefficients are found to depend strongly on stability. Under both stable and unstable conditions, large negative SGS dissipation is associated with the onset of ejection events while large positive SGS dissipation tends to occur during the onset of sweep events. These findings are also supported by conditionally sampled 2D velocity and temperature fields obtained using the 12 anemometers placed in a vertical array.