A bound on the vertical transport of heat in the ‘ultimate’ state of slippery convection at large Prandtl numbers

2013 ◽  
Vol 729 ◽  
pp. 103-122 ◽  
Author(s):  
Xiaoming Wang ◽  
Jared P. Whitehead
Keyword(s):  
Quadratic Form ◽  
Boundary Conditions ◽  
Heat Transport ◽  
Upper Bound ◽  
Vertical Transport ◽  
Three Dimensions ◽  
Free Boundaries ◽  
Ultimate State ◽  
Vertical Heat ◽  
Prandtl Numbers

AbstractAn upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’$R{a}^{1/ 2} $scaling and, in the limit of infinite$Pr$, agrees with the bound of$R{a}^{5/ 12} $recently derived in that limit for stress-free boundaries.

What is the heat uptake potential of Antarctic Bottom Water?

2021 ◽  
Author(s):  
Jan Zika ◽  
Abhishek Savita ◽  
Ryan Holmes ◽  
Taimoor Sohail
Keyword(s):  
Heat Transport ◽  
Upper Bound ◽  
Bottom Water ◽  
Climate Models ◽  
Deep Ocean ◽  
Meridional Overturning Circulation ◽  
Antarctic Bottom Water ◽  
Heat Uptake ◽  
Vertical Heat ◽  
Highly Correlated

<p>Antarctic Bottom Water (AABW) is a cold dense water mass which sinks around Antarctica keeping the abyssal ocean relatively cool. Recent observations have suggested a component of recent deep ocean warming is linked to AABW. Here we explore how much changes in AABW could affect changes in vertical ocean heat transport in a warming climate. If the AABW circulation were to be completely extinguished, for example due to increases in upper ocean thermal stratification, AABW would cease to cool the deep ocean and hence lead to an effective warming of the abyss. Therefore, we propose that long term mean vertical heat transport of the AABW circulation is an effective upper bound on the change in heat transport that can be affected by changes in AABW. We call this upper bound the ‘heat uptake potential’. We analyse AABW circulations in an ensemble of numerical climate models. We find that the AABW circulation contributes between 0.05Wm<sup>-2</sup> and 0.15Wm<sup>-2</sup> to the global vertical heat balance in the model’s pre-industrial states. Indeed, under abrupt CO<sub>2</sub> forcing changes, AABW heat transport systematically reduces (in some cases completely), with the largest reductions occurring in models with the largest pre-industrial mean heat transports. The AABW circulation vertical heat transport is found to be highly correlated with the minimum of the Meridional Overturning Circulation at 50<sup>o</sup>S in the models, suggesting there may be observable constraints on the heat uptake potential of AABW.</p>

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

scholarly journals Eigenvalues of the negative Laplacian for arbitrary multiply connected domains

1996 ◽  
Vol 19 (3) ◽  
pp. 581-586
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Conditions ◽  
Spectral Function ◽  
Three Dimensions ◽  
Bounded Domains ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Asymptotic Formulae ◽  
Negative Laplacian

The purpose of this paper is to derive some interesting asymptotic formulae for spectra of arbitrary multiply connected bounded domains in two or three dimensions, linked with variation of positive distinct functions entering the boundary conditions, using the spectral function∑k=1∞{μk(σ1,…,σn)+P}−2asP→∞. Further results may be obtained.

scholarly journals Vertical heat transport in eddying ocean models

2008 ◽  
Vol 35 (23) ◽  
Author(s):  
C. L. Wolfe ◽  
P. Cessi ◽  
J. L. McClean ◽  
M. E. Maltrud
Keyword(s):  
Heat Transport ◽  
Vertical Heat ◽  
Ocean Models

scholarly journals Mixed insulating and conducting thermal boundary conditions in Rayleigh–Bénard convection

2017 ◽  
Vol 835 ◽  
pp. 491-511 ◽  
Author(s):  
Dennis Bakhuis ◽  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Heat Transport ◽  
Bottom Plate ◽  
Bulk Temperature ◽  
Stripe Pattern ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

A series of direct numerical simulations of Rayleigh–Bénard convection, the flow in a fluid layer heated from below and cooled from above, were conducted to investigate the effect of mixed insulating and conducting boundary conditions on convective flows. Rayleigh numbers between $Ra=10^{7}$ and $Ra=10^{9}$ were considered, for Prandtl numbers $\mathit{Pr}=1$ and $\mathit{Pr}=10$. The bottom plate was divided into patterns of conducting and insulating stripes. The size ratio between these stripes was fixed to unity and the total number of stripes was varied. Global quantities, such as the heat transport and average bulk temperature, and local quantities, such as the temperature just below the insulating boundary wall, were investigated. For the case with the top boundary divided into two halves, one conducting and one insulating, the heat transfer was found to be approximately two-thirds of that for the fully conducting case. Increasing the pattern frequency increased the heat transfer, which asymptotically approached the fully conducting case, even if only half of the surface is conducting. Fourier analysis of the temperature field revealed that the imprinted pattern of the plates is diffused in the thermal boundary layers, and cannot be detected in the bulk. With conducting–insulating patterns on both plates, the trends previously described were similar; however, the half-and-half division led to a heat transfer of about a half of that for the fully conducting case instead of two-thirds. The effect of the ratio of conducting and insulating areas was also analysed, and it was found that, even for systems with a top plate with only 25 % conducting surface, heat transport of 60 % of the fully conducting case can be seen. Changing the one-dimensional stripe pattern to a two-dimensional chequerboard tessellation does not result in a significantly different response of the system.

Studies in collapse analysis of rigid-plastic plates with a square yield diagram

1957 ◽  
Vol 241 (1226) ◽  
pp. 311-338 ◽  
Keyword(s):  
Reinforced Concrete ◽  
Boundary Conditions ◽  
Upper Bound ◽  
Plastic Plate ◽  
Rigid Plastic ◽  
Concentrated Load ◽  
Arbitrary Boundary ◽  
Loaded Plate ◽  
Concrete Plate ◽  
Collapse Analysis

The collapse loads and mechanisms of a rigid-plastic plate with a square yield diagram, such as continuously reinforced concrete plate, are considered. Particular attention is paid to the case of a single concentrated load applied to a plate of arbitrary plan and with arbitrary boundary conditions. Upper-bound solutions are also given for a uniformly loaded plate of regular polygonal plan.

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