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Topic: Primitive equations

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	I
		The equation for the vertical component of velocity (parallel to the gravitational force vector) is replaced by a diagnostic relationship on the assumption that, on the spatial and temporal scales of interest, motions are in hydrostatic balance.
		This equation indicates that the acceleration of horizontal atmospheric motion results from the sum of the pressure gradient force, the Coriolis force due to the rotating frame of reference, and the sum of all frictional forces.
		The equations (3), (4), (9), (10) and (11) are the continuous form of the equations used to predict vorticity, divergence, temperature, surface pressure and atmospheric water vapor in the GCM.
	grads.iges.org /agcm/agcm_hydro.html (2250 words)

	Primitive equations - Wikipedia, the free encyclopedia
		The primitive equations are a version of the Navier-Stokes equations which describe hydrodynamical flow on the sphere under the assumptions that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere.
		The precise form of the primitive equations depends on the vertical coordinate system chosen, such as pressure coordinates, log pressure coordinates, or sigma coordinates.
		The analytic solution to the primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.
	en.wikipedia.org /wiki/Primitive_equations (392 words)

	4.2 The primitive equations
		The continuous equations solved by MOM are given by
		is latitude, which increases northward and is zero at the equator.
		4.2.2 Hydrostatic pressure and the equation of state
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node4.html (69 words)

	[No title]
		Equations for the spherical harmonic coefficients are obtained by multiplying the 2 1.
		The terms being treated implicitly are the Laplacian of the geopotential in the divergence equation 1.22 and the vertical heat flux terms in the thermodynamic equation 1.15.
		In equation 1.56 zu and zb are the pressure altitude of the upper boundary of the analysis and the lower boundary of the model, respectively.
	www.ubka.uni-karlsruhe.de /vvv/fzk/6278/6278.text (8504 words)

	Primitive equations information - Search.com
		The primitive equations are a version of the Navier-Stokes equations which describe hydrodynamical flow on the sphere under the assumptions that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere.
		The precise form of the primitive equations depends on the vertical coordinate system chosen, such as pressure coordinates, log pressure coordinates, or sigma coordinates.
		The analytic solution to the primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.
	www.search.com /reference/Primitive_equations (1095 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.
		Theoretically, the equations are reformulated in a mathematically convenient way, revealing the existence of an underlying Monge-Ampere equation, a nonlinear diagnostic equation for one of the primitive variables.
		In a GFD context, the parent model is the primitive equations, the slow dynamics consists of vortical motion and the fast dynamics consists of inertia-gravity waves.
	www.ima.umn.edu /geoscience/abstracts/2-11abs.html (2100 words)

	5.1.1 Vertical modes in MOM and their relation to eigenmodes
		As discussed in Section 6.11 of Gill (1982), the linearized primitive equations for a stratified fluid can be partitioned into a countably infinite (i.e., discrete) number of orthogonal eigenmodes, each with a different vertical structure.
		Consequently, the depth averaged mode of a rigid lid ocean model corresponds directly to the barotropic mode of the linearized rigid lid primitive equations.
		In contrast, the ocean model's depth averaged mode cannot fully describe the free surface primitive equation's barotropic mode, which is weakly depth dependent.
	www.ocgy.ubc.ca /~yzq/books/MOM3/s2node35.html (679 words)

	Balanced Models and Hamiltonian Models
		Three-dimensional primitive equations are analyzed for flows with various ranges of Burger number including the asymptotic regime of strong stable stratification and weak rotation.
		Nonlinear wave-vortex interactions and shear-stratified pancake dynamics are investigated via large scale massively parallel simulations of both the full primitive equations and the asymptotic limit equations (in the case of stable strong stratification).
		The governing equations are the non-hydrostatic 3D `Primitive' equations under the Boussinesq approximation.
	home.hetnet.nl /~e.c.neven/bal01bal.html (2783 words)

	Pn-Pz
		Ertel's potential vorticity equation is the governing equation for an important class of motion where the equation becomes the sole prognostic equation determining the time evolution of the flow, with all other variables expressed in terms of the potential vorticity by diagnostic equations.
		A set of filtered equations obtained from the fundamental equations of motion of a fluid by applying the hydrostatic approximation and neglecting the viscosity.
		They comprise three prognostic and three diagnostic equations, the former of which are the x and y (or horizontal) components of the momentum equation and the thermodynamic equation of energy, and the latter the continuity equation, the hydrostatic equation and the equation of state.
	stommel.tamu.edu /~baum/paleo/ocean/node31.html (6255 words)

	\documentclass[10pt,twoside]{article} (Site not responding. Last check: 2007-10-21)
		The assumptions of the model agree well with the characteristics of the tropical area: the circulation is there characterized by steady zonal currents varying montly or seasonally and longwaves propagating westward along the equator known as "equatorial waves" and superimposed to the steady mean currents.
		Belmiloudi, Asymptotic Behaviour of the Perturbation of the primitive equations of the ocean with vertical viscosity.
		Belmiloudi, Mathematical analysis and optimal control problems for the perturbation of the primitive equations of the ocean with vertical viscosity, J. Appl.
	www.insa-rennes.fr /~abelmilo/abstract6.htm (570 words)

	OC2910: Physical Oceanography Basic Concepts (Site not responding. Last check: 2007-10-21)
		The hydrostatic approximation is a simplification of the equation governing the vertical component of velocity.
		The geostrophic approximation is a simplification of the equations governing the horizontal components of velocity.
		It is valid when the largest terms in the equations of motion are those involving the Coriolis force and the pressure gradient.
	www.oc.nps.navy.mil /nom/day1/partd.html (1082 words)

	Elizabeth Bennett Abstract (Site not responding. Last check: 2007-10-21)
		The primitive group notion was emphasized later by Galois, who defined primitive groups in connection with primitive equations and who was the first to show the close relation which exists between the theory of substitution grojups and the solution of algebraic equations.
		From his time mathematicians have recognized the fact that the determination of all the substitution groups of a given degree is a fundamental problem of algebra.
		Since the determination of intransitive groups is based on a knowledge of the transitive groups of lower degree and the determination of imprimitive groups requires a knowledge of primitive groups of lower degree, it is evident that the determination of the primitive groups of different degrees is important for the solution of this fundamental problem.
	www.agnesscott.edu /lriddle/WOMEN/abstracts/bennett_abstract.htm (291 words)

	Abstracts (Site not responding. Last check: 2007-10-21)
		Such equations arise, for example, in the study of thin films, for which planar waves correspond with fluid coating a pre-wetted surface.
		An interesting feature of these equations is that both compressive and undercompressive planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, M \ddot{\textrm{u}} nch, and Shearer indicates that undercompressive waves can be nonlinearly stable.
		We prove the global existence of strong solutions to the Navier-Stokes equations when the initial data and the external forces are in large sets (in the sense, it is large as the thickness is small).
	www.math.tamu.edu /~cbhu/abstract.html (613 words)

	APPENDIX A: Terms and Definitions
		Continuous derivatives in differential equations are replaced by finite difference approximations at a discrete set of points in space and time.
		Primitive Equations Models: Mathematical models which use the primitive equation system to approximate the dynamic, thermodynamic and static state of the atmosphere were called primitive equation models.
		Early primitive equation models used a system of six equations; three prognostic equations (the x and y components of the momentum equation, and the thermodynamic energy equation) and three diagnostic equations (the continuity equation, the hydrostatic approximation, and the equation of state) (Holton, 1979).
	www.nrlmry.navy.mil /~chu/chap5/ch5apa.htm (1795 words)

	Surface Pressure Poisson Equation Formulation of the Primitive Equations: Numerical Schemes
		Numerical methods for the primitive equations (PEs) of oceanic flow are presented in this paper.
		First, a two-dimensional Poisson equation with a suitable boundary condition is derived to solve the surface pressure.
		Consequently, we derive a new formulation of the PEs in which the surface pressure Poisson equation replaces the nonlocal incompressibility constraint, which is known to be inconvenient to implement.
	epubs.siam.org /sam-bin/dbq/article/39628 (205 words)

	The MOMA Code (Site not responding. Last check: 2007-10-21)
		The above equations are then integrated over each box to give an equation in which the advection and diffusive terms are replaced by the fluxes through the boundaries of the box and the other terms written in terms of averages over the box.
		In the primitive equation system of equations, the vertical velocity is required at two places, once to advect tracer quantities (like heat and salt) in the vertical, and once to transport momentum in the vertical.
		When solving the velocity equations, the flux of momentum through the side face of a grid box is calculated from the two velocity points on either side of the face.
	www.mth.uea.ac.uk /ocean/SEA/moma-report.html (5649 words)

	Geostrophic Wave Circulations
		Although the large-scale waves may be illustrated approximately by the wave solutions of linearized perturbation equations, the atmospheric circulations are governed by the nonlinear primitive equations.
		When the geostrophic perturbation solutions are applied for the primitive equations, we obtain the remainder equations containing nonlinear terms of perturbations, which are generally smaller in the order of magnitude than the linear terms in the perturbation equations.
		Thus, the time-averaged circulations cannot be illustrated completely by simplified linear equations, and the heat and momentum balances in the time-averaged circulations should be calculated using the primitive equations.
	wave.prohosting.com /nkpub/Introduction.html (1832 words)

	Peter Suber, "Recursive Function Theory"
		Primitive recursion is a method of defining a new function, h, through old functions, f and g.
		This circle is the essence of primitive recursion.
		The use of the terms "recursive" and "primitive recursive" for components for the overall theory, and again for the the overall theory, is confusing and regrettable but that's the way the terminology has evolved.
	www.earlham.edu /~peters/courses/logsys/recursiv.htm (3359 words)

	On the Regularity of Three-Dimensional Rotating Euler-Boussinesq Equations (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: : The 3D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification.
		The resolution of resonances and a non-standard small divisor problem are the basis for error estimates for such fast singular oscillating limits.
		Existence on infinite time intervals of regular solutions to the viscous 3D "primitive" equations is proven for initial data in H ff, ff 3=4.
	citeseer.ist.psu.edu /263274.html (524 words)

	This is Cheng Wang's home page.
		A 2-D Poisson equation with suitable boundary condition is derived to solve the surface pressure.
		Consequently, we derive a new formulation of the PEs in which the surface pressure Poisson equation replaces the nonlocal incompressibility constraint, which is known to be inconvenient to implement.
		In the fourth order scheme, the mean vorticity equation is approximated by compact difference scheme due to the difficulty of the mean vorticity boundary condition, while the fourth order long-stencil approximation is utilized to deal with the transport type equation for the sake of computational convenience.
	www.math.utk.edu /~wang/gfd.html (796 words)

	Five Basic Laws Used by Meteorologists: Primitive Equations
		Ideal Gas Law (Equation of State) -- expresses the relationship of the pressure a gas exerts to the volume it occupies and its temperature.
		Hydrostatic Law (Obtained from the Equation of Vertical Motion) -- the upwards directed pressure gradient acceleration acting on an air parcel (explained in class) is balanced by the acceleration of gravity.
		Conservation of Mass Applied to the Atmosphere (Equation of Continuity) – the fractional rate of increase experienced by an air parcel (or air column, following its motion), is equal to the convergence (negative divergence).
	www.geosciences.sfsu.edu /Geosciences/classes/m201/PrimtiveEquations/Primitive_Equations.htm (473 words)

	WWW interactive multipurpose server
		Equation d'une droite dans le plan, trouver une équation définissant une droite d'après les coordonnées de deux points.
		Equations d'une droite dans l'espace, trouver deux équations définissant une droite d'après les coordonnés de deux points.
		Equation d'une droite dans le plan, trouver une équation définissant une droite d'apres un point et un vecteur directeur.
	wims.unice.fr /wims/wims.cgi (5458 words)

	Center for Atmospheric Sciences : Education
		Quantitative description of electromagnetic energy, derivation of the equation of radiative transfer; applications to nadir and limb geometries; scattering, absorption and emission processes, Earth radiation balance considerations, Earth radiation budget satellite data studies.
		The basic governing equations for a rotating, compressible fluid on a sphere will be developed from first principles with discussion of the following topics: noninertial reference frames; apparent forces; conservation properties; and scale analysis.
		Shallow-water and quasi-geostrophic approximations to the primitive equations and their application will be demonstrated.
	cas.hamptonu.edu /education/3graduate.html (564 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-21)
		The primitive equations for ocean and atmospheric dynamics is a variant of the Benard convection system with rotation.
		Therefore, in studying the question of global regularity and well-posedness of the primitive equations for ocean and atmospheric dynamics one faces the same mathematical difficulties as in the case of three dimensional Navier-Stokes equations with rotation.
		However, due to the rotation and other geophysical constraints, such as the shallowness of the oceans and the atmosphere, geophysists took advantage of certain geophysical balances, such as geostrophic balance or hydrostatic balance, to derive reasonable, yet simplified, balanced models.
	www.math.unl.edu /pi/colloquia/abstract-20031030.txt (189 words)

	1.5.1 Equations of motion for the ocean
		But, because the ``compressible'' terms are linearized, the pressure equation 1.80 can be integrated implicitly with ease (the time-dependent term appears as a Helmholtz term in the non-hydrostatic pressure equation).
		Though necessary, the assumptions that go into these equations are messy since we essentially assume a different EOS for the reference density and the perturbation density.
		form of these equations that are used throughout the ocean modeling community and referred to as the primitive equations (HPE).
	mitgcm.org /pelican/online_documents/node27.html (487 words)

	Amazon.com: "hydrostatic primitive equations": Key Phrase page (Site not responding. Last check: 2007-10-21)
		Chapter 1 offers a brief introduction to the derivation of the oceanic equations of motion (the hydrostatic primitive equations) and various often- used approximate systems.
		Beginning with the traditional equations for conservation of mass, momentum, mechanical energy and heat,...
		The approximated models studied include the hydrostatic primitive equations, the shallow water equations, the barotropic vorticity equation, several approximately-geostrophic models and some acoustically-filtered models which permit buoyancy modes.
	www.amazon.com /phrase/hydrostatic-primitive-equations (312 words)

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