Identifying and quantifying nonconservative energy production/destruction terms in hydrostatic Boussinesq primitive equation models

This paper seeks to illustrate the point that physical inconsistencies between thermodynamics and dynamics usually introduce nonconservative production/destruction terms in the local total energy balance equation in numerical ocean general circulation models (OGCMs). Such terms potentially give rise to undesirable forces and/or diabatic terms in the momentum and thermodynamic equations, respectively, which could explain some of the observed errors in simulated ocean currents and water masses. In this paper, a theoretical framework is developed to provide a practical method to determine such nonconservative terms, which is illustrated in the context of a relatively simple form of the hydrostatic Boussinesq primitive equation used in early versions of OGCMs, for which at least four main potential sources of energy nonconservation are identified; they arise from: (1) the “hanging” kinetic energy dissipation term; (2) assuming potential or conservative temperature to be a conservative quantity; (3) the interaction of the Boussinesq approximation with the parameterizations of turbulent mixing of temperature and salinity; (4) some adiabatic compressibility effects due to the Boussinesq approximation. In practice, OGCMs also possess spurious numerical energy sources and sinks, but they are not explicitly addressed here. Apart from (1), the identified nonconservative energy sources/sinks are not sign definite, allowing for possible widespread cancellation when integrated globally. Locally, however, these terms may be of the same order of magnitude as actual energy conversion terms thought to occur in the oceans. Although the actual impact of these nonconservative energy terms on the overall accuracy and physical realism of the oceans is difficult to ascertain, an important issue is whether they could impact on transient simulations, and on the transition toward different circulation regimes associated with a significant reorganization of the different energy reservoirs. Some possible solutions for improvement are examined. It is thus found that the term (2) can be substantially reduced by at least one order of magnitude by using conservative temperature instead of potential temperature. Using the anelastic approximation, however, which was initially thought as a possible way to greatly improve the accuracy of the energy budget, would only marginally reduce the term (4) with no impact on the terms (1), (2) and (3).

Thermodynamics/dynamics coupling in weakly compressible turbulent stratified fluids

In traditional and geophysical fluid dynamics, it is common to describe stratified turbulent fluid flows with low Mach number and small relative density variations by means of the incompressible Boussinesq approximation. Although such an approximation is often interpreted as decoupling the thermodynamics from the dynamics, this paper reviews recent results and derive new ones that show that the reality is actually more subtle and complex when diabatic effects and a nonlinear equation of state are retained. Such an analysis reveals indeed: (1) that the compressible work of expansion/contraction remains of comparable importance as the mechanical energy conversions in contrast to what is usually assumed; (2) in a Boussinesq fluid, compressible effects occur in the guise of changes in gravitational potential energy due to density changes. This makes it possible to construct a fully consistent description of the thermodynamics of incompressible fluids for an arbitrary nonlinear equation of state; (3) rigorous methods based on using the available potential energy and potential enthalpy budgets can be used to quantify the work of expansion/contraction B in steady and transient flows, which reveals that B is predominantly controlled by molecular diffusive effects, and act as a significant sink of kinetic energy.

Estimating Lorenz's reference state in an ocean with a nonlinear equation of state for seawater

The study of the mechanical energy budget of the oceans using Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically re-arranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill-defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which we illustrate using climatological data, we show that compressibility effects are in fact minor. The reference state can be regarded as a well defined one-dimensional function of depth, which forms a surface in temperature, salinity and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. We show that the reference state obtained by standard sorting methods is equivalent, though computationally more expensive, to the volume frequency distribution approach. The approach we present can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.

Numerical and experimental analysis of natural convection in a cavity with flush mounted heat sources on a side wall

This work considers a problem of interest in several technological applications such as the thermal control of electronic equipment. It is also important to study the heat transfer performance of these components under off-normal conditions, such as during failure of cooling fans. The effect of natural convection on the flow and heat transfer in a cavity with two flush mounted heat sources on the left vertical wall, simulating electronic components, is studied numerically and experimentally. The influence of the power distribution, spacing between the heat sources and cavity aspect ratio have been investigated. An analysis of the average Nusselt number of the two heat sources was performed to investigate the behavior of the heat transfer coefficients. The results obtained numerically and experimentally, after an error analysis, showed a good agreement.

Estudo numerico da convecção natural e forçada em cavidades tridimensionais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Anisotropic viscous models of large-deformation Mohr-Coulomb failure

We have developed a way to represent Mohr-Coulomb failure within a mantle-convection fluid dynamics code. We use a viscous model of deformation with an orthotropic viscoplasticity (a different viscosity is used for pure shear to that used for simple shear) to define a prefered plane for slip to occur given the local stress field. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We again assume the Boussinesq approximation, neglecting any effect of dilatancy on the stress field. An additional criterion is required to ensure that deformation occurs along the plane aligned with maximum shear strain-rate rather than the perpendicular plane, which is formally equivalent in any symmetric formulation. We also allow for strain-weakening of the material. The material can remember both the accumulated failure history and the direction of failure. We have included this capacity in a Lagrangian-integration-point finite element code and show a number of examples of extension and compression of a crustal block with a Mohr-Coulomb failure criterion. The formulation itself is general and applies to 2- and 3-dimensional problems.

An anisotropic viscous representation of Mohr-Coulomb failure for use is modelling coupled mantle-continent dynamics

In mantle convection models it has become common to make use of a modified (pressure sensitive, Boussinesq) von Mises yield criterion to limit the maximum stress the lithosphere can support. This approach allows the viscous, cool thermal boundary layer to deform in a relatively plate-like mode even in a fully Eulerian representation. In large-scale models with embedded continental crust where the mobile boundary layer represents the oceanic lithosphere, the von Mises yield criterion for the oceans ensures that the continents experience a realistic broad-scale stress regime. In detailed models of crustal deformation it is, however, more appropriate to choose a Mohr-Coulomb yield criterion based upon the idea that frictional slip occurs on whichever one of many randomly oriented planes happens to be favorably oriented with respect to the stress field. As coupled crust/mantle models become more sophisticated it is important to be able to use whichever failure model is appropriate to a given part of the system. We have therefore developed a way to represent Mohr-Coulomb failure within a code which is suited to mantle convection problems coupled to large-scale crustal deformation. Our approach uses an orthotropic viscous rheology (a different viscosity for pure shear to that for simple shear) to define a prefered plane for slip to occur given the local stress field. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We again assume the Boussinesq approximation - neglecting any effect of dilatancy on the stress field. An additional criterion is required to ensure that deformation occurs along the plane aligned with maximum shear strain-rate rather than the perpendicular plane which is formally equivalent in any symmetric formulation. It is also important to allow strain-weakening of the material. The material should remember both the accumulated failure history and the direction of failure. We have included this capacity in a Lagrangian-Integration-point finite element code and will show a number of examples of extension and compression of a crustal block with a Mohr-Coulomb failure criterion, and comparisons between mantle convection models using the von Mises versus the Mohr-Coulomb yield criteria. The formulation itself is general and applies to 2D and 3D problems, although it is somewhat more complicated to identify the slip plane in 3D.

The three dimensional model with the affection of stratified density (Caspian Sea)

The hydro dynamical actions in big Lakes directly influence dynamic, physical and chemical affairs. The circulation's models and temperature have something to do with the movements of fluids, and analysis for circulation in Caspian sea is because of the lack of observation through which the circulations and out comings are determined. Through the studies, three dimensional simulations (Large- Scale) are planned and performed, according to Smolakiewicz and Margolin works. This is a non- hydrostatic and Boussinesq approximation is used in its formulation is used in its formulation on the basis of Lipps (1990) theorem and curve lines, the fluid is constant adiabatic and stratified, and the wind power is considered zero. The profile of speed according to previous depth and before ridge can be drawn on the basis of density available between northern and southern ridges. The circulation field is drawn from 3 cm/s to 13 cm/s on the plate z= 5 cm , the vertical changes of speed on the plate is 0.02 m/s. Vertical profile , horizontal speed in previous on, and after the ridges on are drawn on different spaces. It changes from 0.5 cm/s to 1 cm/s before ridges.

Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state

In this paper, the concept of available potential energy (APE) density is extended to a multicomponent Boussinesq fluid with a nonlinear equation of state. As shown by previous studies, the APE density is naturally interpreted as the work against buoyancy forces that a parcel needs to perform to move from a notional reference position at which its buoyancy vanishes to its actual position; because buoyancy can be defined relative to an arbitrary reference state, so can APE density. The concept of APE density is therefore best viewed as defining a class of locally defined energy quantities, each tied to a different reference state, rather than as a single energy variable. An important result, for which a new proof is given, is that the volume integrated APE density always exceeds Lorenz’s globally defined APE, except when the reference state coincides with Lorenz’s adiabatically re-arranged reference state of minimum potential energy. A parcel reference position is systematically defined as a level of neutral buoyancy (LNB): depending on the nature of the fluid and on how the reference state is defined, a parcel may have one, none, or multiple LNB within the fluid. Multiple LNB are only possible for a multicomponent fluid whose density depends on pressure. When no LNB exists within the fluid, a parcel reference position is assigned at the minimum or maximum geopotential height. The class of APE densities thus defined admits local and global balance equations, which all exhibit a conversion with kinetic energy, a production term by boundary buoyancy fluxes, and a dissipation term by internal diffusive effects. Different reference states alter the partition between APE production and dissipation, but neither affect the net conversion between kinetic energy and APE, nor the difference between APE production and dissipation. We argue that the possibility of constructing APE-like budgets based on reference states other than Lorenz’s reference state is more important than has been previously assumed, and we illustrate the feasibility of doing so in the context of an idealised and realistic oceanic example, using as reference states one with constant density and another one defined as the horizontal mean density field; in the latter case, the resulting APE density is found to be a reasonable approximation of the APE density constructed from Lorenz’s reference state, while being computationally cheaper.

DISSIPATIVE BOUSSINESQ SYSTEM OF EQUATIONS IN THE BENARD-MARANGONI PHENOMENON

By using the long-wavelength approximation, a system of coupled evolution equations for the bulk velocity and the surface perturbations of a Benard-Marangoni system is obtained. It includes nonlinearity, dispersion, and dissipation, and it can be interpreted as a dissipative generalization of the usual Boussinesq system of equations. As a particular case, a strictly dissipative version of the Boussinesq system is obtained.

New approximation for free surface flow of groundwater: capillarity correction

An existing capillarity correction for free surface groundwater flow as modelled by the Boussinesq equation is re-investigated. Existing solutions, based on the shallow flow expansion, have considered only the zeroth-order approximation. Here, a second-order capillarity correction to tide-induced watertable fluctuations in a coastal aquifer adjacent to a sloping beach is derived. A new definition of the capillarity correction is proposed for small capillary fringes, and a simplified solution is derived. Comparisons of the two models show that the simplified model can be used in most cases. The significant effects of higher-order capillarity corrections on tidal fluctuations in a sloping beach are also demonstrated. (c) 2004 Elsevier Ltd. All rights reserved.

Two-dimensional approximation for tide-induced watertable fluctuations in a sloping sandy beach

The prediction of watertable fluctuations in a coastal aquifer is important for coastal management. However, most previous approaches have based on the one-dimensional Boussinesq equation, neglecting variations in the coastline and beach slope. In this paper, a closed-form analytical solution for a two-dimensional unconfined coastal aquifer bounded by a rhythmic coastline is derived. In the new model, the effect of beach slope is also included, a feature that has not been considered in previous two-dimensional approximations. Three small parameters, the shallow water parameter (epsilon), the amplitude parameter (a) and coastline parameter (beta) are used in the perturbation approximation. The numerical results demonstrate the significant influence of both the coastline shape and beach slopes on tide-driven coastal groundwater fluctuations. (c) 2004 Elsevier Ltd. All rights reserved.

Influence of memory in deterministic walks in random media: Analytical calculation within a mean-field approximation

Consider a random medium consisting of N points randomly distributed so that there is no correlation among the distances separating them. This is the random link model, which is the high dimensionality limit (mean-field approximation) for the Euclidean random point structure. In the random link model, at discrete time steps, a walker moves to the nearest point, which has not been visited in the last mu steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number n of points explored by the walker with memory mu=2, as well as the transient and period joint distribution. This result enables us to explain the abrupt change in the exploratory behavior between the cases mu=1 (memoryless walker, driven by extreme value statistics) and mu=2 (walker with memory, driven by combinatorial statistics). In the mu=1 case, the mean newly visited points in the thermodynamic limit (N >> 1) is just < n >=e=2.72... while in the mu=2 case, the mean number < n > of visited points grows proportionally to N(1/2). Also, this result allows us to establish an equivalence between the random link model with mu=2 and random map (uncorrelated back and forth distances) with mu=0 and the abrupt change between the probabilities for null transient time and subsequent ones.

Approximation to density functional theory for the calculation of band gaps of semiconductors

The local-density approximation (LDA) together with the half occupation (transitionstate) is notoriously successful in the calculation of atomic ionization potentials. When it comes to extended systems, such as a semiconductor infinite system, it has been very difficult to find a way to half ionize because the hole tends to be infinitely extended (a Bloch wave). The answer to this problem lies in the LDA formalism itself. One proves that the half occupation is equivalent to introducing the hole self-energy (electrostatic and exchange correlation) into the Schrodinger equation. The argument then becomes simple: The eigenvalue minus the self-energy has to be minimized because the atom has a minimal energy. Then one simply proves that the hole is localized, not infinitely extended, because it must have maximal self-energy. Then one also arrives at an equation similar to the self- interaction correction equation, but corrected for the removal of just 1/2 electron. Applied to the calculation of band gaps and effective masses, we use the self- energy calculated in atoms and attain a precision similar to that of GW, but with the great advantage that it requires no more computational effort than standard LDA.

Nonuniversal behavior for aperiodic interactions within a mean-field approximation

We study the spin-1/2 Ising model on a Bethe lattice in the mean-field limit, with the interaction constants following one of two deterministic aperiodic sequences, the Fibonacci or period-doubling one. New algorithms of sequence generation were implemented, which were fundamental in obtaining long sequences and, therefore, precise results. We calculate the exact critical temperature for both sequences, as well as the critical exponents beta, gamma, and delta. For the Fibonacci sequence, the exponents are classical, while for the period-doubling one they depend on the ratio between the two exchange constants. The usual relations between critical exponents are satisfied, within error bars, for the period-doubling sequence. Therefore, we show that mean-field-like procedures may lead to nonclassical critical exponents.