Available potential energy density for Boussinesq fluid flow

2013 ◽  
Vol 714 ◽  
pp. 476-488 ◽  
Author(s):  
Kraig B. Winters ◽  
Roy Barkan
Keyword(s):  
Energy Density ◽  
Potential Energy ◽  
Evolution Equations ◽  
Positive Definite Function ◽  
Boussinesq Approximation ◽  
Exact Expression ◽  
Positive Definite ◽  
Definite Function ◽  
Available Potential Energy ◽  
Boussinesq Fluid

AbstractAn exact expression ${\mathscr{E}}_{a} $ for available potential energy density in Boussinesq fluid flows (Roullet & Klein, J. Fluid Mech., vol. 624, 2009, pp. 45–55; Holliday & McIntyre, J. Fluid Mech., vol. 107, 1981, pp. 221–225) is shown explicitly to integrate to the available potential energy ${E}_{a} $ of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128). ${\mathscr{E}}_{a} $ is a positive definite function of position and time consisting of two terms. The first, which is simply the indefinitely signed integrand in the Winters et al. definition of ${E}_{a} $, quantifies the expenditure or release of potential energy in the relocation of individual fluid parcels to their equilibrium height. When integrated over all parcels, this term yields the total available potential energy ${E}_{a} $. The second term describes the energetic consequences of the compensatory displacements necessary under the Boussinesq approximation to conserve vertical volume flux with each parcel relocation. On a pointwise basis, this term adds to the first in such a way that a positive definite contribution to ${E}_{a} $ is guaranteed. Globally, however, the second term vanishes when integrated over all fluid parcels and therefore contributes nothing to ${E}_{a} $. In effect, it filters the components of the first term that cancel upon integration, isolating the positive definite residuals. ${\mathscr{E}}_{a} $ can be used to construct spatial maps of local contributions to ${E}_{a} $ for direct numerical simulations of density stratified flows. Because ${\mathscr{E}}_{a} $ integrates to ${E}_{a} $, these maps are explicitly connected to known, exact, temporal evolution equations for kinetic, available and background potential energies.

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

scholarly journals The Point Value Maximization Problem for Positive Definite Functions Supported in a Given Subset of a Locally Compact Group

2018 ◽  
Vol 61 (1) ◽  
pp. 179-200
Author(s):  
Sándor Krenedits ◽  
Szilárd Gy. Révész
Keyword(s):  
Extremal Problem ◽  
Positive Definite Function ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Maximization Problem ◽  
Definite Function ◽  
Locally Compact ◽  
Value Maximization ◽  
Compact Abelian Groups ◽  
Complex Exponential

AbstractThe century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

scholarly journals Positive definite functions and cut-off for discrete groups

2020 ◽  
pp. 1-17
Author(s):  
Amaury Freslon
Keyword(s):  
Random Walks ◽  
Quantum Groups ◽  
Discrete Group ◽  
Coxeter Groups ◽  
Free Groups ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Definite Function ◽  
Discrete Groups ◽  
Compact Quantum Groups

Abstract We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

Dependence of the Weyl coefficient on singular interface conditions

2009 ◽  
Vol 52 (2) ◽  
pp. 445-487 ◽  
Author(s):  
Matthias Langer ◽  
Harald Woracek
Keyword(s):  
Hamiltonian Function ◽  
Positive Definite Function ◽  
Canonical System ◽  
Positive Definite ◽  
Definite Function ◽  
Interface Conditions ◽  
Canonical Systems ◽  
Bessel Equation ◽  
Continuation Problem ◽  
Sturm Liouville

AbstractWe investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parametrizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular end point, and a Sturm–Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function.

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

2013 ◽  
Vol 735 ◽  
pp. 499-518 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Equation Of State ◽  
Kinetic Energy ◽  
Nonlinear Equation ◽  
Potential Energy ◽  
Reference State ◽  
Constant Density ◽  
Production Term ◽  
Reference Position ◽  
Available Potential Energy ◽  
Boussinesq Fluid

AbstractIn 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 to 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 affects 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 idealized 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.

scholarly journals On Godement's characterisation of amenability

1998 ◽  
Vol 57 (1) ◽  
pp. 153-158 ◽  
Author(s):  
Alain Valette
Keyword(s):  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Groups ◽  
Definite Function ◽  
Compact Groups ◽  
Locally Compact ◽  
Unimodular Group ◽  
Compactly Supported ◽  
Assembly Map

Motivated by a question related to the construction of the Baum-Connes analytical assembly map for locally compact groups, we refine a criterion of Godement for amenability: for a unimodular group G, our criterion says that G is amenable if and only if every compactly supported, positive-definite function has non-negative integral over G.

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