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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. 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Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. 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Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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A note on the energetics of a double-diffusive system
- GEORGE VERONIS
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- Journal of Fluid Mechanics / Volume 567 / 25 November 2006
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- 19 October 2006, pp. 111-116
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Adjacent oceanic water masses with the same density but different concentrations of heat and salt generate interleaving accompanied by double-diffusive processes. Laboratory experiments with salt and sugar concentrations are used to study the interleaving process. Most double-diffusive studies have treated vertical configurations in which one of the two components contains a destabilizing feature, salt above fresh water for salt fingers or warm underlying cold for the diffusive case. However, when the fluid lacks any gravitationally unstable feature, i.e. no gravitational potential energy is available in either component, the question arises as to what the source of energy is to drive the system. Such a case is discussed here and it is shown that the ultimate source of the energy is the chemical potential associated with the different property distributions. Diffusion creates a destabilizing property distribution and then enables the resulting potential energy to be released.
Experiments on double-diffusive sugar–salt fingers at high stability ratio
- John R. Taylor, George Veronis
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- Journal of Fluid Mechanics / Volume 321 / 25 August 1996
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- 26 April 2006, pp. 315-333
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In a series of laboratory experiments the growth of double-diffusive salt fingers from an initial configuration of two homogeneous reservoirs with salt in the lower and sugar in the upper layer was investigated. For most of the experiments the stability ratio was between 2.5 and 3, where the latter value is at the upper limit (the ratio of salt to sugar diffusivities) for which fingers can exist. In these experiments long slender fingers are generated at the interface. Essentially all theories or physical bases for models of salt fingers presuppose such a configuration of long fingers. Our measurements show that the length of fingers at high stability ratio increases with time like t1/2, with a coefficient that is consistent with the diffusive spread of the faster diffusing component (salt). When the initial stability ratio is closer to unity, fingers penetrate into the reservoirs very rapidly carrying with them large anomalies of salt and sugar which give rise to convective overturning of the reservoirs. The convection sweeps away the ends of the fingers, and when it is intense enough (as it is when the sugar anomaly is large) it can reduce the finger height to a value less than the width. After this initial phase the finger length grows linearly with time as has been found in previous studies. These results show that salt fingers can evolve in quite different ways depending on the initial stability ratio and must cast doubt on the use of simple similarity arguments to parameterize the heat and salt fluxes produced by fingers.
The role of the buoyancy layer in determining the structure of salt fingers
- George Veronis
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- Journal of Fluid Mechanics / Volume 180 / July 1987
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- 21 April 2006, pp. 327-342
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An initial state consisting of sugar solution lying above a denser salt solution in a Hele-Shaw cell is unstable to disturbances that evolve into long, slender fingers. An analysis of the structure of fully evolved (infinitely long) fingers that are independent of the vertical coordinate concludes that fingers with a width of the order of the buoyancy-layer thickness have maximum growth rate. Since effective gravity can be altered by inclining the Hele-Shaw cell toward the horizontal, fingers of different preferred widths can be established. An abrupt change of the angle of inclination changes the preferred width. A stability analysis of the resulting initial-value problem shows that perturbations with a vertical scale of the order of the buoyancy-layer thickness grow, and fluid from each finger penetrates laterally into the two adjacent fingers. The unstable modes resemble those observed experimentally by Taylor & Veronis (1986). It turns out that all vertically uniform fingers, even ones with the preferred width of the basic state, are unstable to a non-oscillatory peturbation that changes straight fingers to ones that have a vertically wavy structure. In all cases the vertical scale of the most unstable disturbance is of the order of the buoyancy-layer thickness. Also included is a discussion of the need for a model describing the transient evolution of fingers and particularly one that contains an analysis of the role of the transition region between the salt-finger zone and the reservoirs above and below.
The source-sink flow in a rotating system and its oceanic analogy
- Han-Hsiung Kuo, George Veronis
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- Journal of Fluid Mechanics / Volume 45 / Issue 3 / 15 February 1971
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- 29 March 2006, pp. 441-464
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Laboratory analogues of theoretical models of wind-driven ocean circulation are based on ideas presented by Stommel (1957). A particularly simple demonstration of the applicability of these ideas is contained in a paper by Stommel, Arons & Faller (1958). The present work develops the source-sink laboratory analogue of ocean circulation models to a point where chosen parametric values allow one to simulate the theoretical models of Stommel (1948) and Munk (1950) exactly. The investigation of the flow in a rotating cylinder generated by a source of fluid near the outer wall leads to a detailed description of the roles of the various boundary layers which occur. This knowledge is used to analyse the more complex source-sink flow in a pie-shaped basin. The laboratory analogue to the Stommel circulation model is analyzed in detail. It is shown that the change in the flow pattern brought about by a radial variation of the position of the eastern boundary in the pie-shaped basin is confined to the interior flow and the boundary layer is largely unaffected. When the bottom of the pie-shaped container slopes, the circulation pattern is changed significantly. For the particular case treated, the depth of the basin along the western boundary is unchanged and the maximum depth occurs at the southeast corner. The circulation generated by a source introduced at the apex of the pie has a gyre whose centre is shifted more toward the southwest corner than the corresponding centre of the gyre for a flat-bottomed basin. Two experiments are reported showing that the western boundary may separate because of the effect of bottom topography or because of the pressure of a cyclonic and an anti-cyclonic gyre generated by suitably placed sources and sinks.
The spin-up of a homogeneous fluid bounded below by a permeable medium
- John Kroll, George Veronis
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- Journal of Fluid Mechanics / Volume 40 / Issue 2 / 3 February 1970
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- 29 March 2006, pp. 225-239
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The spin-up of a homogeneous rotating fluid bounded at the top and/or bottom by a permeable medium has been proposed by Bretherton & Spiegel (1968) as a model for the spin-up in natural flows where turbulent processes transmit the direct effect of the boundaries deeper into the fluid than does the laminar Ekman layer. The theoretical analysis for the spin-up of a laterally unbounded fluid bounded by a permeable medium below is presented here. In addition, an experimental study of the process is presented. Theory and experiment agree reasonably well with a maximum difference of about 8% in the predicted and measured spin-up times. The effects of the side-wall boundary have been studied theoretically by Howard (1969). Experimental observations in the side-wall boundary layer confirm qualitatively the results of Howard's theory.
Spin-up of a stratified fluid: theory and experiment
- George Buzyna, George Veronis
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- Journal of Fluid Mechanics / Volume 50 / Issue 3 / 14 December 1971
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- 29 March 2006, pp. 579-608
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Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively.
A multi-scaling analysis of the spin-up problem
- Jean-Pierre St-Maurice, George Veronis
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- Journal of Fluid Mechanics / Volume 68 / Issue 3 / 15 April 1975
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- 29 March 2006, pp. 417-445
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The response of a contained rotating fluid to a small, abrupt change in the rotation rate is analysed by multi-scaling methods. The procedure makes use of the fact that three different physical processes (inertial oscillations, spin-up response, diffusion) give rise to three different time scales. Since the flow is known to have a boundary-layer character, the variables are derived into interior and boundary-layer parts. The pertinent parameter separating the magnitudes of the amplitudes and the different time scales is the square root of the Ekman number E½, so an expansion in powers of E½ is used. The solution for a homogeneous fluid is derived first and is shown to be consistent with the solution of Green-span & Howard (1963). The results are given in two forms: one is a direct deduction of the expansion method and is valid to O(E); the other is obtained by regrouping the terms to derive a form apparently valid for indefinitely long times. When the fluid is stratified, the physical structure of the system is substantially more complicated, and so is the analysis. Exact results can be obtained for the case where the buoyancy N and the rotational Ω frequencies are the same. For the general case F = N/Ω ≠ 1, results valid for t [Gt ] 1 can be obtained (where t is measured in units of Ω−1). In both cases the exact lowest-order solution for the interior can be derived since it is independent of short time t. For the stratified fluid the elementary spin-up solution of Holton (1965) is part of the solution at O(E½). The remaining part includes the long-time behaviour towards which the system tends as diffusive processes become dominant. The formulation of the long-time problem is complete a t O(E), but parts of it emerge from the analysis at lower order, and it is necessary to treat the lower-order system to obtain a consistent formulation at O(E). In particular, it is possible to show that the thermal boundary condition, which does not affect the elementary spin-up solution, should be satisfied only by the long-time part of the problem. The complete, lowest-order response of the system includes a diffusive part which is quantitatively significant even for times of the order of one spin-up time. It is suggested here that the diffusive contribution may be responsible for parts of the discrepancy between elementary spin-up theory and recent experiments.
Nonlinear source-sink flow in a rotating pie-shaped basin
- George Veronis And, C. C. Yang
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- Journal of Fluid Mechanics / Volume 51 / Issue 3 / 8 February 1972
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- 29 March 2006, pp. 513-527
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Source-sink flows in a rotating pie-shaped basin provide a laboratory analogue of wind-driven ocean circulation (Stommel, Arons & Faller 1958). Experiments and theory are presented here for flows which are mildly nonlinear. Theory and experiment show satisfactory agreement for the intense flow in the western boundary-layer region which contains the strongest nonlinear effects. The strengths of the sources and sinks were increased in the experiments in an attempt to induce an instability in the western boundary layer. However, the western boundary layer was always stable, even for relatively large Rossby numbers. Photographs from experiments with a basin of semicircular cross-section show the difference between eastern and western boundary layers in a striking manner.
Large-amplitude Bénard convection in a rotating fluid
- George Veronis
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- Journal of Fluid Mechanics / Volume 31 / Issue 1 / 8 January 1968
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- 28 March 2006, pp. 113-139
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Linear stability theory of Bénard convection in a rotating fluid (Chandrasekhar 1961) has shown that fluids with large ([ges ] 1) Prandtl number, σ, exhibit behaviour markedly different from that of fluids with σ [les ] 1. This difference in behaviour extends also into the finite-amplitude range (Veronis 1959, 1966I). In this paper we report a numerical study of two-dimensional Be´nard convection in a rotating fluid confined between free boundaries, with σ = 6·8 and σ = 0·2 for the range of Taylor number 0 [les ] [Fscr ]2 [Lt ] 105 and for Rayleigh numbers, R, extending an order of magnitude from the critical value of linear stability theory. The behaviour of water (σ = 6·8) is dominated by the rotational constraint even for relatively moderate values (∼ 103) of [Fscr ]2. A study of the resultant velocity and temperature fields shows how rotation controls the system, with the principal behaviour reflected by the thermal wind balance; i.e. the horizontal temperature gradient is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. A fluid with a small Prandtl number (σ = 0·2) becomes unstable to finite-amplitude disturbances at values of the Rayleigh number significantly below the critical value of linear stability theory. The subsequent steady vorticity and temperature fields exhibit a structure which is quite different from that of fluids with large σ. The rotational constraint is balanced primarily by non-linear processes in a limited range of Taylor number ([Fscr ]2 [les ] 103·6). For larger values of [Fscr ]2 the system first becomes unstable to infinitesimal oscillatory disturbances but a steady, finite-amplitude flow is established at supercritical values of R which are none the less smaller than the values that one would expect from linear theory. The ranges of Taylor number in which the above phenomena occur are different from those which were estimated on the basis of an earlier study (Veronis 1966 I) which made use of a minimal representation of the finite-amplitude velocity and temperature fields. No subcritical, finite-amplitude oscillatory motions were found in the present study. Comparison with some of the experimental features observed and reported by Rossby (1966) is also discussed and it is pointed out that some of the differences between theory and experiment may be traced to the restrictive conditions (two-dimensionality and free boundaries) of the present study.
Effect of a stabilizing gradient of solute on thermal convection
- George Veronis
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- Journal:
- Journal of Fluid Mechanics / Volume 34 / Issue 2 / 12 November 1968
- Published online by Cambridge University Press:
- 28 March 2006, pp. 315-336
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A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.
Large-amplitude Bénard convection
- George Veronis
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- Journal:
- Journal of Fluid Mechanics / Volume 26 / Issue 1 / September 1966
- Published online by Cambridge University Press:
- 28 March 2006, pp. 49-68
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Calculations are presented for two-dimensional Bénard convection between free bounding surfaces for ranges of Rayleigh and Prandtl numbers. The variables are expanded in a series consisting of the eigenfunctions of the stability problem and the system is truncated to take into account only a limited number of terms. The amplitudes of the eigenfunctions are evaluated by numerical integration of the resulting non-linear equations. In all cases considered, the system achieves a steady state with the motion consisting of a single large cell. Results for Nusselt number vs. Rayleigh number are given for a range of Prandtl number varying between 0·01 and 100 and show that heat flux increases slightly with decreasing Prandtl number. The calculations agree with those of Kuo where the ranges of Rayleigh number overlap. A simple heuristic argument based on the assumption that turbulent boundary layers exist is also given and the conclusions of the latter indicate that heat flux should decrease with decreasing Prandtl number. Thus the behaviour is qualitatively different from that of the calculations. The reason appears to be associated with the fact that the single large cell in the computed cases enables the fluid to accelerate through repeated cycles until it achieves a steady state with the amplitude of the motion much larger than could be acquired by a single turbulent blob free-falling in the gravitational field.