Random-Walk Simulation of Transport in Heterogeneous Porous Media: Local Mass-Conservation Problem and Implementation Methods

1996 ◽  
Vol 32 (3) ◽  
pp. 583-593 ◽  
Author(s):  
Eric M. LaBolle ◽  
Graham E. Fogg ◽  
Andrew F. B. Tompson
Keyword(s):  
Porous Media ◽  
Random Walk ◽  
Mass Conservation ◽  
Heterogeneous Porous Media ◽  
Local Mass ◽  
Conservation Problem ◽  
Local Mass Conservation ◽  
Random Walk Simulation

Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part II: Numerical Results

2007 ◽  
Vol 64 (6) ◽  
pp. 1794-1810 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
Michael E. McIntyre
Keyword(s):  
Potential Vorticity ◽  
Wave Speed ◽  
Invariant Manifolds ◽  
Mass Conservation ◽  
Flow Speed ◽  
Local Mass ◽  
Potential Vorticity Inversion ◽  
Divergence Equation ◽  
Local Mass Conservation ◽  
Derivatives Of

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-δδ, plain-γγ, and plain-δγ PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-δδ, hyper-γγ, and hyper-δγ PBMs, respectively. As proved in Part I, except for the leading-order plain-γγ each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton. Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ϵ = ||QPBM − QPE||2/||QPE||2, where QPBM and QPE denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||2 is the L2 norm. At 30 days the most accurate PBMs have ϵ ≈ 10−2 with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing ϵ over all orders and truncation types δδ, γγ, and δγ. Contrary to McIntyre and Norton’s conjecture, the minimal ϵ values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of ϵ suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5. As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error ϵ. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

scholarly journals The Local Mass Conservation in SPH

2003 ◽  
Vol 208 ◽  
pp. 411-412
Author(s):  
Yusuke Imaeda ◽  
Shu-ichiro Inutsuka
Keyword(s):  
Fluid Velocity ◽  
Mass Conservation ◽  
Local Mass ◽  
Term Evolution ◽  
Conservation Property ◽  
Local Mass Conservation ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
The Mean

Smoothed particle hydrodynamics (SPH) is one of the widely used methods to calculate the various astrophysical fluid dynamics. However, standard SPH cannot accurately describe the long-term evolution of shear flows: The large density error emerges within a dynamical timescale, and the amplitude of the error becomes larger than the value of density itself (Δρ ≳ ρ), when we take the mean separation of the particles as the smoothing length. The origin of error is due to the inaccurate description of the continuity equation in the standard SPH formalism. To ensure the local mass conservation property, we have reformulated SPH, in which we distinguish the particle velocity and the fluid velocity for the updation of the particle positions. We find that the present modification provides an accurate description of the density evolution in SPH.

Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part I: The Hyperbalance Equations

2007 ◽  
Vol 64 (6) ◽  
pp. 1782-1793 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
Michael E. McIntyre
Keyword(s):  
Velocity Field ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Mass Conservation ◽  
Velocity Fields ◽  
Local Mass ◽  
Local Mass Conservation ◽  
The Difference

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby–Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, v and vm, the first to advect PV and the second to advect mass. The difference v − vm is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called “hyperbalance equations,” whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field v to advect mass as well as to advect and evaluate the exact PV.

scholarly journals Inherently mass-conservative version of the semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model dynamical core

2013 ◽  
Vol 6 (3) ◽  
pp. 4809-4832
Author(s):  
V. V. Shashkin ◽  
M. A. Tolstykh
Keyword(s):  
Baroclinic Instability ◽  
Mass Conservation ◽  
Atmospheric Model ◽  
Seasonal Forecasts ◽  
Absolute Vorticity ◽  
Dynamical Core ◽  
Local Mass ◽  
Local Mass Conservation ◽  
Long Time ◽  
Global And Local

Abstract. The semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at Hydrometeorological centre of Russia. The distinct feature of SL-AV dynamical core is the semi-implicit semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. Semi-implicit semi-Lagrangian approach allows for long time steps while violates the global and local mass-conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no aposteriori mass-fixing algorithms are applied. However, the global mass-fixing algorithms degrade the local mass conservation. The inherently mass-conservative version of SL-AV model dynamical core presented in the article ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume semi-Lagrangian discretization for continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). The numerical experiments show that the presented new version of SL-AV dynamical core combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain induced Rossby wave test and baroclinic instability test for mass-conservative dynamical core are found to be in agreement with the results available in literature.

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