Vorticity

# Vorticity

Overview
Vorticity is a concept used in fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."

More formally, vorticity can be related to the amount of "circulation" or "rotation" (or more strictly, the local angular rate of rotation) in a fluid.
The average vorticity in a small region of fluid flow is equal to the circulation  around the boundary of the small region, divided by the area A of the small region.

Notionally, the vorticity at a point in a fluid is the limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

as the area of the small region of fluid approaches zero at the point:

Mathematically, vorticity is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

and is defined as the curl of the velocity field:

In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, vorticity is the curl of the fluid velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

.
Discussion

Recent Discussions
Encyclopedia
Vorticity is a concept used in fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."

More formally, vorticity can be related to the amount of "circulation" or "rotation" (or more strictly, the local angular rate of rotation) in a fluid.
The average vorticity in a small region of fluid flow is equal to the circulation  around the boundary of the small region, divided by the area A of the small region.

Notionally, the vorticity at a point in a fluid is the limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

as the area of the small region of fluid approaches zero at the point:

Mathematically, vorticity is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

and is defined as the curl of the velocity field:

## Fluid dynamics

In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, vorticity is the curl of the fluid velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. It can also be considered as the circulation per unit area at a point in a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

flow field. It is a vector quantity, whose direction is along the axis of the fluid's rotation. For a two-dimensional flow, the vorticity vector is perpendicular to the plane.

For a fluid having locally a "rigid rotation" around an axis (i.e., moving like a rotating cylinder), vorticity is twice the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

of a fluid element. An irrotational fluid has no vorticity. Somewhat counter-intuitively, an irrotational fluid can have a non-zero angular velocity (e.g. a fluid rotating around an axis with its tangential velocity inversely proportional to the distance to the axis has a zero vorticity); see also forced and free vortex.

One way to visualize vorticity is this: consider a fluid flowing. Imagine that some tiny part of the fluid is instantaneously rendered solid, and the rest of the flow removed. If that tiny new solid particle would be rotating, rather than just moving with the flow, then there is vorticity in the flow.

In general, vorticity is a specially powerful concept in the case that the viscosity is low (i.e. high Reynolds number). In such cases, even when the velocity field is relatively complicated, the vorticity field can be well approximated as zero nearly everywhere except in a small region in space. This is clearly true in the case of 2-D potential flow
Potential flow
In fluid dynamics, potential 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 approximation for several applications...

(i.e. 2-D zero viscosity flow), in which case the flowfield can be identified with the complex plane, and questions about those sorts of flows can be posed as questions in complex analysis which can often be solved (or approximated very well) analytically.

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity (may have to apply the 2nd Fundamental Theorem of Calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

to do this rigorously). In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i.e. low Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

) fluids, with conservative body forces:
with the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

.

Even for real flows (3-dimensional and finite Re), the idea of viewing things in terms of vorticity is still very powerful. It provides the most useful way to understand how the potential flow solutions can be perturbed for "real flows." In particular, one restricts attention to the vortex dynamics, which presumes that the vorticity field can be modeled well in terms of discrete vortices (which encompasses a large number of interesting and relevant flows). In general, the presence of viscosity causes a diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

of vorticity away from these small regions (e.g. discrete vortices) into the general flow field. This can be seen by the diffusion term in the vorticity transport equation. Thus, in cases of very viscous flows (e.g. Couette Flow
Couette flow
In fluid dynamics, Couette flow refers to the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other. The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates...

), the vorticity will be diffused throughout the flow field and it is probably simpler to look at the velocity field (i.e. vectors of fluid motion) rather than look at the vorticity field (i.e. vectors of curl of fluid motion) which is less intuitive.

Related concepts are the vortex-line, which is a line which is everywhere tangent to the local vorticity; and a vortex tube which is the surface in the fluid formed by all vortex-lines passing through a given (reducible) closed curve in the fluid. 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 at everywhere along the tube (because vorticity has zero divergence). It is a consequence of Helmholtz's theorems
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 filaments...

(or equivalently, of Kelvin's circulation theorem
Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem states In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time. The theorem was developed by William Thomson, 1st Baron Kelvin...

) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence.

Note however that in a three dimensional flow, vorticity (as measured by the volume integral of its square) can be intensified when a vortex-line is extended—known as vortex stretching
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....

(see say Batchelor, section 5.2). Mechanisms such as these operate in such well known examples as the formation of a bath-tub vortex in out-flowing water, and the build-up of a tornado by rising air-currents.

## Vorticity equation

Main article: Vorticity equation
Vorticity equation
The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components...

The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around.
In fluid mechanics this equation can be expressed in vector form as follows,

where, is the velocity vector, is the density, is the pressure, is the viscous stress tensor and is the body force term. The equation is valid for compressible fluid in the absence of any concentrated torques and line forces. No assumption is made regarding the relationship between the stress and the rate of strain tensors (c.f. Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

).

## Atmospheric sciences

In the atmospheric sciences
Atmospheric sciences
Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...

, vorticity is the rotation of air around a vertical axis. Vorticity is a vector quantity and the direction of the vector is given by the right-hand rule with the fingers of the right hand indicating the direction and curvature of the wind. When the vorticity vector points upward into the atmosphere, vorticity is positive; when it points downward into the earth it is negative. Vorticity in the atmosphere is therefore positive for counter-clockwise rotation (looking down onto the Earth's surface), and negative for clockwise rotation.

In the Northern Hemisphere cyclonic rotation
Cyclone
In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

of the atmosphere is counter-clockwise so is associated with positive vorticity, and anti-cyclonic rotation
Anticyclone
An anticyclone is a weather phenomenon defined by the United States' National Weather Service's glossary as "[a] large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere"...

is clockwise so is associated with negative vorticity. In the Southern Hemisphere cyclonic rotation
Cyclone
In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

is clockwise with negative vorticity; anti-cyclonic rotation
Anticyclone
An anticyclone is a weather phenomenon defined by the United States' National Weather Service's glossary as "[a] large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere"...

is counter-clockwise with positive vorticity.

A closely related phenomenon is helicity, which is vorticity in motion along a third axis in a corkscrew fashion. Helicity is important in forecasting supercell
Supercell
A supercell is a thunderstorm that is characterized by the presence of a mesocyclone: a deep, continuously-rotating updraft. For this reason, these storms are sometimes referred to as rotating thunderstorms...

s and the potential for tornadic
A tornado is a violent, dangerous, rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. They are often referred to as a twister or a cyclone, although the word cyclone is used in meteorology in a wider...

activity.

Relative and absolute vorticity are defined as the z-components of the curls of relative (i.e., in relation to Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's surface) and absolute wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

, respectively.

This gives

for relative vorticity and

for absolute vorticity, where u and v are the zonal
Zonal
Zonal can refer to:* Zonal and meridional, directions on a globe* Zonal and poloidal, directions in a toroidal magnetically confined plasma* Zonal polynomial, a symmetric multivariate polynomial...

(x direction) and meridional (y direction) components of wind velocity. The absolute vorticity at a point can also be expressed as the sum of the relative vorticity at that point and the Coriolis parameter at that latitude (i.e., it is the sum of the Earth's vorticity and the vorticity of the air relative to the Earth).

A useful related quantity is potential vorticity
Potential vorticity
Potential vorticity is a quantity which is proportional to the dot product of vorticity and stratification that, following a parcel of air or water, can only be changed by diabatic or frictional processes...

. The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the z direction. But if the absolute vorticity is divided by the vertical spacing between levels of constant entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

(or potential temperature), the result is a conserved quantity
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

of adiabatic flow, termed potential vorticity (PV). Because diabatic
Diabatic
A diabatic process is one in which heat transfer takes place, which is the opposite of an adiabatic process. In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born-Oppenheimer approximation...

processes, which can change PV and entropy, occur relatively slowly in the atmosphere, PV is useful as an approximate tracer
Tracer
Tracer may refer to:* Histochemical tracer, a substance used for tracing purposes in histochemistry, the study of the composition of cells and tissues...

of air masses over the timescale of a few days, particularly when viewed on levels of constant entropy.

The barotropic vorticity equation
Barotropic vorticity equation
This 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, no vertical wind shear of the geostrophic wind. It also implies that thickness contours are parallel to...

is the simplest way for forecasting the movement of Rossby wave
Rossby wave
Atmospheric Rossby waves are giant meanders in high-altitude winds that are a major influence on weather.They are not to be confused with oceanic Rossby waves, which move along the thermocline: that is, the boundary between the warm upper layer of the ocean and the cold deeper part of the...

s (that is, the troughs
Trough (meteorology)
A trough is an elongated region of relatively low atmospheric pressure, often associated with fronts.Unlike fronts, there is not a universal symbol for a trough on a weather chart. The weather charts in some countries or regions mark troughs by a line. In the United States, a trough may be marked...

and ridge
Ridge
A ridge is a geological feature consisting of a chain of mountains or hills that form a continuous elevated crest for some distance. Ridges are usually termed hills or mountains as well, depending on size. There are several main types of ridges:...

s of 500 hPa
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

geopotential height
Geopotential height
Geopotential height is a vertical coordinate referenced to Earth's mean sea level — an adjustment to geometric height using the variation of gravity with latitude and elevation. Thus it can be considered a "gravity-adjusted height"...

) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting utilized that equation.

In modern numerical weather forecasting models and general circulation models (GCM's), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation
Prognostic equation
Prognostic equation - in the context of physical 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....

.

## Importance

Modern fluid mechanics fully embraces the role of vorticity in fluid motion. Vortex dynamics has retained a characteristic "flavor" deriving from its particle-based (Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

) interpretation and from its frequently intuitive, "mechanistic" description of flow phenomena. For example, the entire process of blowing out a candle by a puff of air is readily explained by vortex dynamics but is much more complicated to explain using the usual primitive variables of fluid flow theory such as pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

and velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. In particular, the speed of the vortex ring
Vortex ring
A vortex ring, also called a toroidal vortex, is a region of rotating fluid moving through the same or different fluid where the flow pattern takes on a toroidal shape. The movement of the fluid is about the poloidal or circular axis of the doughnut, in a twisting vortex motion...

that propagates from the origin of the puff to the candle is only readily understood when the vortex motion is fully elucidated.

Vorticity is important in many other areas of fluid dynamics. For instance, the lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

distribution over a finite wing may be approximated by assuming that each 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
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

. The strengths of the vortices are then summed to find the total approximate circulation about the wing. According to the Kutta–Joukowski theorem
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

, lift is the product of circulation, airspeed, and air density.

• Barotropic vorticity equation
Barotropic vorticity equation
This 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, no vertical wind shear of the geostrophic wind. It also implies that thickness contours are parallel to...

In fluid dynamics, d'Alembert's 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 force is zero on a body moving with constant velocity relative to the fluid...

• Vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

• Vortex tube
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. It has no moving parts....

• Vortex stretching
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....

• Vortical
Vortical
In fluid dynamics, vortical means pertaining to a vortex or to vortices. The movement of a fluid can be said to be vortical if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis....

• Vorticity equation
Vorticity equation
The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components...

• Horseshoe vortex
Horseshoe vortex
The horseshoe vortex model is a simplified representation of the vortex system of a wing. In this model the wing vorticity is modelled by a bound vortex of constant circulation, travelling with the wing, and two trailing vortices, therefore having a shape vaguely reminiscent of a horseshoe...

• Kutta–Joukowski theorem
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

• Wingtip vortices
Wingtip vortices
Wingtip vortices are tubes of circulating air that are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure...

### Fluid dynamics

• Biot-Savart law
• Circulation
• Navier-Stokes equations
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

• Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. ISBN 0-521-81984-9
• Chorin, Alexandre J.
Alexandre Chorin
Alexandre J. Chorin is a professor of mathematics at the University of California, Berkeley who works in applied mathematics. He is known for his contributions to the field of Computational fluid dynamics....

, "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
• Majda, Andrew J.
Andrew Majda
Andrew Joseph Majda is an American mathematician and the Morse Professor of Arts and Sciences at the Courant Institute of Mathematical Sciences of New York University...

, Andrea L. Bertozzi, "Vorticity and Incompressible Flow". Cambridge University Press; 2002. ISBN 0-521-63948-4
• Tritton, D. J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0-19-854493-6
• Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0-12-059820-5