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Topic: Boussinesq approximation

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	Boussinesq approximation - Wikipedia, the free encyclopedia
		In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow.
		The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids.
		Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, katabatic winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating).
	en.wikipedia.org /wiki/Boussinesq_approximation (483 words)

	Oceanography Textbooks - Y to Z
		The quasi-static approximation 107, 7.1 The exact quasi-static equations 109, 7.2 Asymptotic analysis of the primitive equations 115, 7.3 The boundary layer phenomenon and the primitive equations 117, 7.4 Simplified primitive equations 119, 7.5 The hydrostatic balance adjustment problem (in an adiabatic atmosphere) 123, 7.6 Complementary remarks 1 130, 7.7 Complementary remarks 2 136, 8.
		The Boussinesq approximation 142, 8.1 The Boussinesq equations 144, 8.2 Some considerations concerning the singular nature of the Boussinesq approximation 147, 8.3 Three new forms of the Boussinesq equations 149, 8.4 Concerning a linear theory of the Boussinesq waves 155, 8.5 The problem of adjustment to the Boussinesq state 164, 8.6 Complementary remarks 168, 9.
		The deep convection approximation 202, 10.1 The "anelastic" equations of Ogura and Phillips 203, 10.2 The deep convection equations according to Zeytounian 205, 10.3 The relation between the Boussinesq and the deep convection approximations 210, 10.4 Complementary remarks 213, 11.
	stommel.tamu.edu /~baum/ocean-books-yz.html (1225 words)

	4.1 Assumptions (Site not responding. Last check: 2007-10-19)
		As discussed by many authors, including the original paper by Boussinesq (1903), as well as Spiegel and Veronis (1960), Chandrasekhar (1961), Gill (1982), and Müller (1995), the Boussinesq approximation is justified for large-scale ocean modeling on the basis of the relatively small variations in density within the ocean.
		Consistent with the above approximations, Bryan also made the thin shell approximation because the depth of the ocean is much less than the earth's radius, which itself is assumed to be a constant (i.e., a sphere rather than an oblate spheroid).
		The thin shell approximation amounts to replacing the radial coordinate of a fluid parcel by the mean radius of the earth, unless this cordinate is differentiated.
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node3.html (669 words)

	Introduction to Physical Oceanography : Chapter 7 - Some Mathematics: The Equations of Motion - Conservation of Mass: ...
		Density is nearly constant in the ocean, and Joseph Boussinesq (1842-1929) noted that we can safely assume density is constant except when it is multiplied by g in calculations of pressure in the ocean.
		The approximations are true for oceanic flows, and they ensure that oceanic flows are incompressible.
		The Boussinesq assumption is equivalent to assuming sea water is incompressible.
	oceanworld.tamu.edu /resources/ocng_textbook/chapter07/chapter07_07.htm (463 words)

	Non Boussinesq Natural Convection Club (Site not responding. Last check: 2007-10-19)
		It’s no doubt that the Boussinesq approximation is the most popular and convenient mathematical model for natural convection.
		Mihaljan, J.M. A rigorous exposition of the Boussinesq approximation applicable to a thin layer of fluid, Astrophys.
		To all appearances this paper of Mihaljan was the first publication on the verification of the Boussinesq approximation.
	www.math.rsu.ru /convection/review.html (346 words)

	Boussinesq Equations -- from Eric Weisstein's World of Physics
		With the addition of an equation for temperature, this will complete the Boussinesq equations.
		Let H be the conductive heat flux and J be the heat generated per unit volume.
		Depending on the value of the Péclet number, one of these terms maybe negligible with respect to the other.
	scienceworld.wolfram.com /physics/BoussinesqEquations.html (142 words)

	Re: Boussinesq approximation
		Boussinesq approximation is for the problems that the variations of temperature as well as the variations of density are small.
		Boussinesq approximation - Abdul Aziz Jaafar, Tue, 24 Nov 1998, 12:53 p.m.
		Re: Boussinesq approximation - Alex Ivancic, Mon, 7 Dec 1998, 5:26 a.m.
	www.cfd-online.com /Forum/main_archive_1998.cgi?read=1475 (115 words)

	Talk Abstracts: Reduced Descriptions of Coupled GFD Systems (Slow Manifolds in the Ocean and Atmosphere), February ...
		The Boussinesq equations are used to describe the dynamical behaviour of a rotating, stratified fluid, a prime example being the oceans.
		The Balance Equations are an asymptotically consistent (but nonunique) set of approximations for rotating, stably stratified flows, built around the quasi-static momentum balances of hydrostacy in the vertical and gradient-wind balance in the horizontal divergence.
		Numerical simulations are used to study the population of slow manifolds in rotating, stably stratified flow in the Boussinesq approximation, with rotation and stratification both in the vertical direction.
	www.ima.umn.edu /geoscience/abstracts/2-11abs.html (2100 words)

	Fluid Dynamics and the Navier-Stokes Equations
		This is called steady flow, and is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe.
		Boussinesq approximation - In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow.
		Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, downhill winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating).
	www.universe-review.ca /R13-10-NSeqs.htm (1617 words)

	Dynamics of Solar Convection
		We assume this section is a good approximation to the actual shape of the convection zone in that area.
		A further approximation is imposed that there is only two-dimensional fluid motion, that is, no fluid moves parallel to the axis of the cylinder.
		As with the spatial approximation, variations in the fields that occur in a shorter time period than the time step are not observed.
	www.tc.cornell.edu /~slantz/SPUR/SPUR95/Jason/report.html (3874 words)

	Non-Boussinesq HIM (Site not responding. Last check: 2007-10-19)
		It is common to use the Boussinesq approximation to reduce computational expense in ocean general circulation models.
		Models using the Boussinesq approximation are volume, not mass conserving and thereby steric effects are not captured.
		Choice of Boussinesq or non-Boussinesq is run time configurable.
	www.gfdl.noaa.gov /~wga/node1.html (152 words)

	Untitled Document (Site not responding. Last check: 2007-10-19)
		The mathematical model is a primitive equation with the Boussinesq approximation, since the horizontal scale is much larger than the vertical one.
		For the 3D basin we consider that under the condition of high aspect ratio we can assume the hydrostatic approximation in addition, which provides a fundamental equation connecting the pressure and density.
		We develop a velocity-pressure algorithm with a pressure Neumann condition in primitive variables, using a Boussinesq approximation and fixed Prandtl number.
	www.sci.sdsu.edu /compsci/Cordoba/ObidioRubioMercedes.htm (139 words)

	Boussinesq Flames
		The chosen gravity is too high to provide the parameter-independent solution, but still the two cases illustrate the key difference between 2-D and 3-D bubbles - the thickness of the tail.
		Vladimirova N, "Model Flames in the Boussinesq Limit: Rising Bubbles", preprint, PDF.
		Vladimirova N. and Rosner R., "Model Flames in the Boussinesq Limit: The Case of Pulsating Fronts", Phys.
	flash.uchicago.edu /~nata/flame/boussinesq (687 words)

	EXERCISE 2 SC200, Spring 04.
		It is known that sufficiently close to the onset of convection and in a sufficiently wide cell, the fluid motion is weakly three-dimensional in that only slow spatial and temporal modulations of locally two-dimensional roll pattern can occour.
		The five three-dimensional Boussinesq equations can then be replaced by a single two-dimensional equation that describes the evolution of a complex amplitude or envelope function whose phase and magnitude slowly modulate the sinusoidal oscillations of the rolls.
		A lowest-order nontrivial amplitude equation is obtained by multiple-scale analysis of Boussinesq equations near onset.
	www.ii.uib.no /~petter/SC200/oppgave2.html (1061 words)

	Appendix: Discussion of approximations
		In the proposed Boussinesq approximation, only the former term is retained (albeit the latter could be in principle included).
		The simplification provided by the Boussinesq approximation is considerable, essentially because it filters out the acoustic modes from the equations.
		Similarly, a surface whose position and shape is given (rather than calculated) avoids the problem of having to deal with surface gravity waves.
	www.unc.edu /~ascotti/mbiwe/paper/node19.html (638 words)

	Fall 2004: 12.800 Week 5 student questions (Site not responding. Last check: 2007-10-19)
		I have only a vague idea of what Boussinesq and anelastic means physically, and I'd be more comfortable if there was something I could read that had a good discussion of where they come from and when they're appropriate and how they break down.
		B1: I think it is pretty clear how the equations get simplified by using Boussinesq approximation, but what I didn't see that obvious from the classes is what is the situation without it.
		B1: One things I have always wonder about approximations, in general not only with Boussinesq, is when is the right moment in the derivations to use them.
	wind.mit.edu /~hansen/classes/F04-12.800.quest3.html (1495 words)

	Adres1 (Site not responding. Last check: 2007-10-19)
		Crucial tools in the analysis are the assumption of spatial periodicity of the flow and the identification of conserved quantities.
		Simultaneous transport of heat and solute in a porous medium, with emphasis on the Oberbeck-Boussinesq approximation, is studied.
		This approximation, in which density variations are neglected everywhere except where gravitation is concerned, is often used to simplify the system of equations.
	www.win.tue.nl /appliedanalysis/seminar/010523/abstracts.html (418 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		However, conventional GCMs make the Boussinesq approximation as a consequence of which mass is not conserved.
		Here we present a careful comparison of the Boussinesq and non-Boussinesq solutions ensuring that only non-Boussinesq effects can be responsible for the observed differences.
		We find that model errors (differences) due to the Boussinesq approximation are apparently smaller than the errors due to the hydrostatic approximation.
	www.ecco-group.org /posters/WOCE/woce_losch.txt (256 words)

	Lake Jesup Model Formulation (Site not responding. Last check: 2007-10-19)
		The EFDC model has been used extensively in the central area of the Indian River Lagoon (IRL) to assess the impacts of freshwater runoff on salinity regime (Zarillo and Surak, 1994) and to determine the influence of physical processes on fish recruitment through Sebastian Inlet (Shenker and Zarillo, 1996).
		Similar to other 3-D hydrodynamic models, the EFDC formulation assumes that water is incompressible, and invokes the hydrostatic and Boussinesq approximations.
		The Boussinesq approximation assumes that density is a constant except in terms involving a horizontal pressure gradient.
	www.fit.edu /acadres/dmes/zarillo/Lake_Jesup_Model_Formulation.html (448 words)

	Re: Boussinesq approximation
		y A. Giorgini (1978), The validity of the Boussinesq approximation for liquida and gases, Int.
		you can safely use the Boussinesq approximation for the next temp.
		Re: Boussinesq approximation - ZHONG ZENG, Thu, 26 Nov 1998, 11:08 p.m.
	www.cfd-online.com /Forum/main_archive_1998.cgi?read=1572 (133 words)

	Richard Greatbatch - 19999-2000 Biennial Report
		I am interested in the dynamics, modelling and predictability of the oceanic and atmospheric circulation, the influence of climate change associated with global warming, and fundamental aspects of the governing equations, including the Boussinesq approximation and the parameterization of transport processes associated with mesoscale eddies.
		GRACE is expected to measure bottom pressure variability in the ocean at unprecedented accuracy, leading to a demand for ocean circulation models that relax the Boussinesq approximation.
		Some 9 years ago now, McDougall and Garrett (1982) raised concern that the Boussinesq tracer equation carried by models could be in error by up to 100% compared to the diapycnal mixing term associated with water mass conversion.
	www.phys.ocean.dal.ca /people/po/Greatbatch_biennial.html (1371 words)

	Bn-Bz
		He also proposed a second type of steady, non-tidal current that flows against the wind during periods of strong winds, with the driving force being a hypothesized upward tilt of the sea surface downwind caused by waves piling up water there.
		Actually a set of filtering approximations (attributed to Boussinesq) that results in an equation set applied to almost all oceanic motions except sound waves.
		The name ``Boussinesq approximation'' is not always used identically with the above series of approximation steps and may or may not include the assumption of incompressibility.
	www-ocean.tamu.edu /~baum/paleo/paleogloss/node8.html (2454 words)

	AGU 2001 Spring Meeting (Site not responding. Last check: 2007-10-19)
		Although that is true, those terms are neglected in the Boussinesq approximation.
		Using the vorticity equation instead of the momentum equation may introduce additional terms.
		The additional terms introduced by using the vorticity equation instead of the momentum equation are neglected in the Boussinesq approximation.
	cires.colorado.edu /~mjones/gravity_wave/agu01.html (372 words)

	[59.01] Simulations of 2D Turbulence in the Anelastic Approximation (Site not responding. Last check: 2007-10-19)
		High resolution simulations of two-dimensional convection using the Anelastic approximation are presented.
		While many studies of this sort have been conducted for the Boussinesq approximation, these are the first to use the Anelastic Approximation in this turbulent regime.
		We find that the overall behaviour of the fluid is similar to the Boussinesq calculations, with very complex "superplumes" and branching plumes.
	www.aas.org /publications/baas/v34n2/aas200/753.htm (208 words)

	SC200 - Report 2 - Thierry Matthey, Inga Berre
		Such systems are governed by the Boussinesq equations.
		It is known that sufficiently close to the onset of convection and in a sufficiently wide cell, the fluid motion is weakly three-dimensional in that only slow spatial and temporal modulations of locally two-dimensional roll pattern can occur.
		The difference in F(tau) for the two methods after steady state is reached occures because of approximation error (the Finite Diffence method is less accurate than the Spectral Method).
	www.ii.uib.no /~matthey/sc200/02 (2478 words)

	Turbulent Flow and Heat Transfer in Circular Couette Flows in Concentric Annulus
		A Reynolds stress turbulence model is employed to determine three normal components of the Reynolds stress and its off-diagonal one.
		The turbulent heat flux is expressed by Boussinesq approximation in which the eddy diffusivity.for heat is given as functions of the temperature variance
		The governing boundarylayer equations are discretized by means of a control volume finite-difference technique and numerically solved using the marching procedure.
	www.hindawi.com /GetArticle.aspx?doi=10.1155/S1023621X98000049 (230 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #6534714 - Numerical study of the validity of the Boussinesq approximation for a fluid-saturated porous medium
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		Numerical study of the validity of the Boussinesq approximation for a fluid-saturated porous medium
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=6534714 (130 words)

	Non Boussinesq Natural Convection Club (Site not responding. Last check: 2007-10-19)
		Here you may find some information on the research done on various hydrodynamical phenomena on the basis of non-Boussinesq models as well as on testing the validity of the Boussinesq approximation in different problems.
		If you find that the information given is not complete, correct, or out-of-date please let us know
		IST - Fifth Framework Programme of the European Community (1998-2002)
	www.math.rsu.ru /convection (118 words)

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