Vorticity is a mathematical concept used in fluid dynamics. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion It can be related to the amount of "circulation" or "rotation" (or more strictly, the local angular rate of rotation) in a fluid. In Fluid dynamics, circulation is the Line integral around a closed curve of the Fluid Velocity.

The average vorticity in a small region of fluid flow is equal to the circulation Γ around the boundary of the small region, divided by the area A of the small region. In Fluid dynamics, circulation is the Line integral around a closed curve of the Fluid Velocity.

$\omega_{av} = \frac {\Gamma}{A}$

Notionally, the vorticity at a point in a fluid is the limit as the area of the small region of fluid approaches zero at the point:

$\omega = \frac {d \Gamma}{dA}$

Mathematically, the vorticity at a point is a vector and is defined as the curl of the velocity:

$\vec \omega = \vec \nabla \times \vec v .$

In the field of flow of a fluid, vorticity ω is zero almost everywhere. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular cURL is a Command line tool for transferring files with URL syntax. It is only in special places, such as the boundary layer or the core of a vortex, that the vorticity is not zero. In Physics and Fluid mechanics, a boundary layer is that layer of Fluid in the immediate vicinity of a bounding surface V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a

## Fluid dynamics

In fluid dynamics, vorticity is the curl of the fluid velocity. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion cURL is a Command line tool for transferring files with URL syntax. In Physics, velocity is defined as the rate of change of Position. It can also be considered as the circulation per unit area at a point in a fluid flow field. In Fluid dynamics, circulation is the Line integral around a closed curve of the Fluid Velocity. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code 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 of a fluid element. Do not confuse with Angular frequency The unit for angular velocity is rad/s An irrotational fluid is one whose vorticity=0. In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. 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. V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a 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 translating, 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 Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio 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 (i. In Fluid dynamics, a potential flow is a Velocity field which is described as the Gradient of a scalar function the velocity potential 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 to do this rigorously). The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i. The vorticity equation is an important prognostic equation in the Atmospheric sciences. e. low mach number) fluids, with conservative body forces:

${D\omega \over Dt} = \omega \cdot \nabla u + \nu \nabla^2 \omega$

Even for real flows (3-dimensional and finite Re), the idea of viewing things in terms of vorticity is still very powerful. Mach number (\mathrm{Ma} or M (generally ˈmɑːk sometimes /ˈmɑːx/ or /ˈmæk/ is the speed of an object moving through air or any Fluid 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 1858 Hermann von Helmholtz published his seminal paper entitled "Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen" in In general, the presence of viscosity causes a diffusion of vorticity away from these small regions (e. Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement 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), the vorticity will be diffused throughout the flow field and it is probably simpler to look at the velocity field (i. 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 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. For the term 'vortex-tube' used in Fluid dynamics please see Vorticity The vortex tube, also known as the Ranque-Hilsch vortex The 'strength' of a vortex-tube 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 (or equivalently, of Kelvin's circulation theorem) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. In Fluid mechanics, Helmholtz's theorems describe the three-dimensional motion of fluid in the vicinity of vortex filaments. In Fluid mechanics, Kelvin's Circulation Theorem states " In an Inviscid, Barotropic flow with conservative body forces the circulation

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 (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

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

$\frac{D\vec\omega}{Dt} = \frac{\partial \vec \omega}{\partial t} + \vec V \cdot (\vec \nabla \vec \omega) = \vec \omega \cdot (\vec \nabla \vec V) - \vec \omega (\vec \nabla \cdot \vec V) + \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p + \vec \nabla \times \left( \frac{\vec \nabla \cdot \underline{\underline{\tau}}}{\rho} \right) + \vec \nabla \times \vec B$

where, $\vec V$ is the velocity vector, ρ is the density, p is the pressure, $\underline{\underline{\tau}}$ is the viscous stress tensor and $\vec B$ 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). A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin

## Atmospheric sciences

In the atmospheric sciences, vorticity is the rotation of air around a vertical axis. 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 Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five 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. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. 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 of the atmosphere is counter-clockwise so is associated with positive vorticity, and anti-cyclonic rotation is clockwise so is associated with negative vorticity. In Meteorology, a cyclone refers to an area of closed circular fluid motion rotating in the same direction as the Earth. In Meteorology, an anticyclone (that is opposite to a Cyclone) is a Weather phenomenon in which there is a descending movement of the air and In the Southern Hemisphere cyclonic rotation is clockwise with negative vorticity; anti-cyclonic rotation is counter-clockwise with positive vorticity. In Meteorology, a cyclone refers to an area of closed circular fluid motion rotating in the same direction as the Earth. In Meteorology, an anticyclone (that is opposite to a Cyclone) is a Weather phenomenon in which there is a descending movement of the air and

Relative and absolute vorticity are defined as the z-components of the curls of relative (i. cURL is a Command line tool for transferring files with URL syntax. e. , in relation to Earth's surface) and absolute wind velocity, respectively. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 Wind is the flow of Air or other Gases that compose an Atmosphere (including but not limited to the Earth's) In Physics, velocity is defined as the rate of change of Position.

This gives

$\zeta=\frac{\partial v_r}{\partial x} - \frac{\partial u_r}{\partial y}$

for relative vorticity and

$\eta=\frac{\partial v_a}{\partial x} - \frac{\partial u_a}{\partial y}$

for absolute vorticity, where u and v are the zonal (x direction) and meridional (y direction) components of wind velocity. The terms zonal and meridional are used to describe directions on a Globe. 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 (PV is a quantity which is proportional to the dot product of Vorticity and Stratification that following a parcel of air or water 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 (or potential temperature), the result is a conserved quantity of adiabatic flow, termed potential vorticity (PV). In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would acquire if adiabatically brought to a standard reference In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Because diabatic processes, which can change PV and entropy, occur relatively slowly in the atmosphere, PV is useful as an approximate tracer of air masses over the timescale of a few days, particularly when viewed on levels of constant entropy. See also Adiabatic process, a concept in Thermodynamics In Quantum chemistry, the Potential energy surfaces are obtained

The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 500 hPa geopotential height) over a limited amount of time (a few days). A simplified form of the Vorticity equation for an inviscid Divergence -free flow the barotropic vorticity equation can simply be stated as Rossby (or planetary) waves are giant Meanders in high-altitude winds that are a major influence on Weather. 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 ridge is a geological feature that features a continuous elevational crest for some distance Geopotential height is a vertical coordinate referenced to Earth's Mean sea level — an adjustment to geometric height ( Elevation above mean sea level using the In the 1950s, the first successful programs for numerical weather forecasting utilized that equation. Numerical weather prediction uses current weather conditions as input into Mathematical models of the atmosphere to predict the weather.

In modern numerical weather forecasting models and GCMs, vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation. Prognostic equation - in a physical (and especially geophysical simulation context is a prognostic equation predicts variables for some time in the future on the basis of the values at

## Other fields

Vorticity is important in many other areas of fluid dynamics. For instance, the lift distribution over a finite wing may be approximated by assuming that each segment of the wing has a semi-infinite trailing vortex behind it. In the context of a Fluid flow relative to a body the lift force is the component of the Aerodynamic force that is Perpendicular to the flow 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 one of the branches of Fluid mechanics that uses Numerical methods and algorithms to solve and analyze problems that involve The strengths of the vortices are then summed to find the total approximate circulation about the wing. In Fluid dynamics, circulation is the Line integral around a closed curve of the Fluid Velocity. According to the Kutta–Joukowski theorem, lift is the product of circulation, airspeed, and air density. The Kutta–Joukowski theorem is a fundamental theorem of Aerodynamics.