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In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, vorticity is a
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
that describes the local
spinning Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of
fluids In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of
vortex ring A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant flow in a vortex ring is said to be toroidal, ...
s. Mathematically, the vorticity \vec is the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
\vec: :\vec \equiv \nabla \times \vec\,, where \nabla is the
nabla operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
. Conceptually, \vec could be determined by marking parts of a continuum in a small
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity \vec would be twice the mean
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector of those particles relative to their
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, oriented according to the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
. In a two-dimensional flow, \vec is always perpendicular to the plane of the flow, and can therefore be considered a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
.


Examples

In a mass of continuum that is rotating like a rigid body, the vorticity is twice the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector of that rotation. This is the case, for example, in the central core of a
Rankine vortex The Rankine vortex is a simple mathematical model of a vortex in a Viscosity, viscous fluid. It is named after its discoverer, William John Macquorn Rankine. The vortices observed in nature are usually modelled with an Potential flow#Examples of ...
. The vorticity may be nonzero even when all particles are flowing along straight and parallel
pathline Streamlines, streaklines and pathlines are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field in three-dimensional space in the framework of ...
s, if there is
shear Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...
(that is, if the flow speed varies across streamlines). For example, in the
laminar flow In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
within a pipe with constant
cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest. Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal
irrotational vortex In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
, where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of continuum that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocity ''about their center of mass'' is zero. : Another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the continuum becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, then there is vorticity in the flow. In the figure below, the left subfigure demonstrates no vorticity, and the right subfigure demonstrates existence of vorticity. :


Mathematical definition

Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by \vec, defined as the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of the velocity field \vec describing the continuum motion. In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
: :\begin \vec = \nabla \times \vec &= \begin\dfrac & \, \dfrac & \, \dfrac\end \times \beginv_x & v_y & v_z\end \\ px &= \begin \dfrac - \dfrac & \quad \dfrac - \dfrac & \quad \dfrac - \dfrac \end \,. \end In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it. In a two-dimensional flow where the velocity is independent of the z-coordinate and has no z-component, the vorticity vector is always parallel to the z-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector \hat: :\begin \vec = \nabla \times \vec &= \begin \dfrac & \, \dfrac & \, \dfrac \end \times \beginv_x & v_y & v_z\end \\ px &= \left(\frac - \frac\right)\hat\,. \end The vorticity is also related to the flow's circulation (line integral of the velocity) along a closed path by the (classical)
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
. Namely, for any
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
surface element with normal direction \vec and area dA, the circulation d\Gamma along the
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...
of C is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
\vec \cdot (\vec \, dA) where \vec is the vorticity at the center of C.Clancy, L.J., ''Aerodynamics'', Section 7.11


Evolution

The evolution of the vorticity field in time is described by the vorticity equation, which can be derived from the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. In many real flows where the viscosity can be neglected (more precisely, in flows with high
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
), the vorticity field can be modeled by a collection of discrete vortices, the vorticity being negligible everywhere except in small regions of space surrounding the axes of the vortices. This is true in the case of two-dimensional
potential flow In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid appr ...
(i.e. two-dimensional zero viscosity flow), in which case the flowfield can be modeled as a
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
field on the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of vorticity away from the vortex cores into the general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation.


Vortex lines and vortex tubes

A vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation :\frac = \frac = \frac\,, where \vec = (\omega_x, \omega_y, \omega_z) is the vorticity vector in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. A vortex tube is the surface in the continuum formed by all vortex lines passing through a given (reducible) closed curve in the continuum. The 'strength' of a vortex tube (also called vortex flux) is the integral of the vorticity across a cross-section of the tube, and is the same everywhere along the tube (because vorticity has zero divergence). It is a consequence of
Helmholtz's theorems In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are ...
(or equivalently, of
Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states:In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the ...
) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence. In a three-dimensional flow, vorticity (as measured by the
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
of the square of its magnitude) can be intensified when a vortex line is extended — a phenomenon known as
vortex stretching In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. ...
. This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents.


Vorticity meters


Rotating-vane vorticity meter

A rotating-vane vorticity meter was invented by Russian hydraulic engineer A. Ya. Milovich (1874–1958). In 1913 he proposed a cork with four blades attached as a device qualitatively showing the magnitude of the vertical projection of the vorticity and demonstrated a motion-picture photography of the float's motion on the water surface in a model of a river bend. Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity" and "Fundamental Principles of Flow" by Iowa Institute of Hydraulic ResearchFilms by Hunter Rouse — IIHR — Hydroscience & Engineering
).


Specific sciences


Aeronautics

In
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
, the
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobile ...
distribution over a
finite wing A finite wing is an aerodynamic wing with tips that result in trailing vortices.http://www.aerospaceweb.org/question/aerodynamics/q0167.shtml This is in contrast to an infinite wing. According to John D. Anderson, Jr., finite wings experience ...
may be approximated by assuming that each spanwise segment of the wing has a semi-infinite trailing vortex behind it. It is then possible to solve for the strength of the vortices using the criterion that there be no flow induced through the surface of the wing. This procedure is called the vortex panel method of
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
. The strengths of the vortices are then summed to find the total approximate circulation about the wing. According to the
Kutta–Joukowski theorem The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed large enough so ...
, lift is the product of circulation, airspeed, and air density.


Atmospheric sciences

The relative vorticity is the vorticity relative to the Earth induced by the air velocity field. This air velocity field is often modeled as a two-dimensional flow parallel to the ground, so that the relative vorticity vector is generally scalar rotation quantity perpendicular to the ground. Vorticity is positive when – looking down onto the earth's surface – the wind turns counterclockwise. In the northern hemisphere, positive vorticity is called
cyclonic rotation Cyclonic rotation, or cyclonic circulation, is atmospheric motion in the same direction as a planet's rotation, as opposed to anticyclonic rotation. For Earth, the Coriolis effect causes cyclonic rotation to be in a counterclockwise Two-dimensi ...
, and negative vorticity is
anticyclonic rotation Anticyclonic rotation, or anticyclonic circulation, is atmospheric motion in the direction opposite to a planet's rotation. For Earth, this motion is in a clockwise direction in the Northern Hemisphere and counterclockwise in the Southern Hemispher ...
; the nomenclature is reversed in the Southern Hemisphere. The absolute vorticity is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, the
Coriolis parameter The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi. :f = 2 \Omega \sin \varphi.\, The rotation rate ...
. The
potential vorticity In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It i ...
is absolute vorticity divided by the vertical spacing between levels of constant (potential) temperature (or
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
). The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the vertical direction, but the potential vorticity is conserved in an adiabatic flow. As adiabatic flow predominates in the atmosphere, the potential vorticity is useful as an approximate tracer of air masses in the atmosphere over the timescale of a few days, particularly when viewed on levels of constant entropy. The
barotropic vorticity equation The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. I ...
is the simplest way for forecasting the movement of
Rossby wave Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby. They are observed in the atmospheres and ...
s (that is, the troughs and
ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...
s of 500 
hPa HPA may refer to: Organizations * Harry Potter Alliance, a charity * Halifax Port Authority, Canada * Hamburg Port Authority, Germany * Hawaii Preparatory Academy, a school in Hawaii, US * Health Protection Agency, UK * Heerespersonalamt, the Ger ...
geopotential height Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height (altitude above mean sea level) that accounts for the variation of gravity with latitude and altitude. Thu ...
) over a limited amount of time (a few days). In the 1950s, the first successful programs for
numerical weather forecasting Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in ...
utilized that equation. In modern numerical weather forecasting models and
general circulation model A general circulation model (GCM) is a type of climate model. It employs a mathematical model of the general circulation of a planetary atmosphere or ocean. It uses the Navier–Stokes equations on a rotating sphere with thermodynamic terms f ...
s (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a
prognostic equation Prognostic equation - in the context of physical (and especially geophysical) simulation, a prognostic equation predicts the value of variables for some time in the future on the basis of the values at the current or previous times. For instance, ...
. Related to the concept of vorticity is the helicity H(t), defined as :H(t) = \int_V \vec \cdot \vec \, dV where the integral is over a given volume V. In atmospheric science, helicity of the air motion is important in forecasting
supercell A supercell is a thunderstorm characterized by the presence of a mesocyclone: a deep, persistently rotating updraft. Due to this, these storms are sometimes referred to as rotating thunderstorms. Of the four classifications of thunderstorms (s ...
s and the potential for tornadic activity.


See also

*
Barotropic vorticity equation The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. I ...
*
D'Alembert's paradox In fluid dynamics, d'Alembert's paradox (or the hydrodynamic paradox) is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag for ...
*
Enstrophy In fluid dynamics, the enstrophy can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particu ...
* Velocity potential *
Vortex In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
*
Vortex tube The vortex tube, also known as the Ranque-Hilsch vortex tube, is a mechanical device that separates a compressed gas into hot and cold streams. The gas emerging from the hot end can reach temperatures of , and the gas emerging from the cold end ...
*
Vortex stretching In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. ...
*
Horseshoe vortex The horseshoe vortex model is a simplified representation of the vortex system present in the flow of air around a wing. This vortex system is modelled by the ''bound vortex'' (bound to the wing) and two '' trailing vortices'', therefore having ...
*
Wingtip vortices Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift.Clancy, L.J., ''Aerodynamics'', section 5.14 One wingtip vortex trails from the tip of each wing. Wingtip vortices are sometimes named ''trailing'' ...


Fluid dynamics

*
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
* Circulation * Vorticity equations *
Kutta–Joukowski theorem The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed large enough so ...


Atmospheric sciences

*
Prognostic equation Prognostic equation - in the context of physical (and especially geophysical) simulation, a prognostic equation predicts the value of variables for some time in the future on the basis of the values at the current or previous times. For instance, ...
*
Carl-Gustaf Rossby Carl-Gustaf Arvid Rossby ( 28 December 1898 – 19 August 1957) was a Swedish-born American meteorologist who first explained the large-scale motions of the atmosphere in terms of fluid mechanics. He identified and characterized both the jet ...
* Hans Ertel


References


Bibliography

* * * * * * Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London * "
Weather Glossary
'"' The Weather Channel Interactive, Inc.. 2004. * "

'". Integrated Publishing.


Further reading

* Ohkitani, K., "''Elementary Account Of Vorticity And Related Equations''". Cambridge University Press. January 30, 2005. * Chorin, Alexandre J., "''Vorticity and Turbulence''". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. * Majda, Andrew J., Andrea L. Bertozzi, "''Vorticity and Incompressible Flow''". Cambridge University Press; 2002. * Tritton, D. J., "''Physical Fluid Dynamics''". Van Nostrand Reinhold, New York. 1977. * Arfken, G., "''Mathematical Methods for Physicists''", 3rd ed. Academic Press, Orlando, Florida. 1985.


External links

* Weisstein, Eric W., "
Vorticity
'". Scienceworld.wolfram.com. * Doswell III, Charles A., "

'". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma. * Cramer, M. S., "''Navier–Stokes Equations -

Introduction''". Foundations of Fluid Mechanics. * Parker, Douglas, "''ENVI 2210 – Atmosphere and Ocean Dynamics

'". School of the Environment, University of Leeds. September 2001. * Graham, James R., "''Astronomy 202: Astrophysical Gas Dynamics''". Astronomy Department,
UC Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public university, public land-grant university, land-grant research university in Berkeley, California. Established in 1868 as the University of Californi ...
. ** "
The vorticity equation: incompressible and barotropic fluids
'". ** "

'". ** "

'". * "
Spherepack 3.1
'". (includes a collection of FORTRAN vorticity program) * "

Real-Time Model Predictions''". (Potential vorticity analysis) {{Meteorological variables Continuum mechanics Fluid dynamics fr:Tourbillon (physique)