Misplaced Pages

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56-402: Let u ( x , t ) {\displaystyle \mathbf {u} (x,t)} be the velocity field and ∇ × u {\displaystyle \nabla \times \mathbf {u} } the corresponding vorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is inviscid ; (ii) either

112-416: A finite wing 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 . The strengths of the vortices are then summed to find

168-405: A continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity ω {\displaystyle {\boldsymbol {\omega }}} would be twice the mean angular velocity vector of those particles relative to their center of mass , oriented according to the right-hand rule . By its own definition,

224-508: A direction perpendicular to it. In a two-dimensional flow where the velocity is independent of the z {\displaystyle z} -coordinate and has no z {\displaystyle z} -component, the vorticity vector is always parallel to the z {\displaystyle z} -axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector z ^ {\displaystyle {\hat {z}}} : The vorticity

280-415: A phenomenon known as vortex stretching . This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents. 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

336-546: A term due to the Earth's rotation, the Coriolis parameter . The potential vorticity is absolute vorticity divided by the vertical spacing between levels of constant (potential) temperature (or entropy ). 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

392-488: Is shear (that is, if the flow speed varies across streamlines ). For example, in the laminar flow within a pipe with constant cross section , 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

448-415: Is a pseudovector (or axial vector) field that describes the local spinning 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 and provides a convenient framework for understanding a variety of complex flow phenomena, such as

504-457: Is a regular value , this is precisely the degree of the Gauss map (i.e. the signed number of times that the image of Γ covers the sphere). Isotopy invariance of the linking number is automatically obtained as the degree is invariant under homotopic maps. Any other regular value would give the same number, so the linking number doesn't depend on any particular link diagram. This formulation of

560-412: 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. Viscous effects introduce frictional losses and time dependence. In a three-dimensional flow, vorticity (as measured by the volume integral of the square of its magnitude) can be intensified when a vortex line is extended —

616-424: Is also related to the flow's circulation (line integral of the velocity) along a closed path by the (classical) Stokes' theorem . Namely, for any infinitesimal surface element C with normal direction n {\displaystyle \mathbf {n} } and area d A {\displaystyle dA} , the circulation d Γ {\displaystyle d\Gamma } along

SECTION 10

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672-487: Is an axial vector, it can be associated with a second-order antisymmetric tensor Ω {\displaystyle {\boldsymbol {\Omega }}} (the so-called vorticity or rotation tensor), which is said to be the dual of ω {\displaystyle {\boldsymbol {\omega }}} . The relation between the two quantities, in index notation, are given by where ε i j k {\displaystyle \varepsilon _{ijk}}

728-459: Is described by the vorticity equation , which can be derived from the Navier–Stokes equations . In many real flows where the viscosity can be neglected (more precisely, in flows with high Reynolds number ), 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

784-540: 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 , and negative vorticity is anticyclonic rotation ; 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

840-665: Is not the only component of severe thunderstorms , and these values are to be taken with caution. That is why the Energy Helicity Index ( EHI ) has been created. It is the result of SRH multiplied by the CAPE ( Convective Available Potential Energy ) and then divided by a threshold CAPE: This incorporates not only the helicity but the energy of the air parcel and thus tries to eliminate weak potential for thunderstorms even in strong SRH regions. The critical values of EHI: Vorticity In continuum mechanics , vorticity

896-418: Is over γ 2 {\displaystyle \gamma _{2}} . Also, a neighborhood of ( s , t ) is mapped under the Gauss map to a neighborhood of v preserving or reversing orientation depending on the sign of the crossing. Thus in order to compute the linking number of the diagram corresponding to v it suffices to count the signed number of times the Gauss map covers v . Since v

952-571: Is simplified to only use the horizontal component of wind and vorticity , and to only integrate in the vertical direction, replacing the volume integral with a one-dimensional definite integral or line integral : where According to this formula, if the horizontal wind does not change direction with altitude , H will be zero as V h {\displaystyle V_{h}} and ∇ × V h {\displaystyle \nabla \times V_{h}} are perpendicular , making their scalar product nil. H

1008-712: Is simply Gauss's linking integral. This is the simplest example of a topological quantum field theory , where the path integral computes topological invariants. This also served as a hint that the nonabelian variant of Chern–Simons theory computes other knot invariants, and it was shown explicitly by Edward Witten that the nonabelian theory gives the invariant known as the Jones polynomial. The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally, there exists higher dimensional topological quantum field theories. There exists more complicated multi-loop/string-braiding statistics of 4-dimensional gauge theories captured by

1064-416: Is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet regularization or renormalization is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself. Since the theory is Gaussian and abelian,

1120-402: Is the ideal irrotational vortex , 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

1176-526: Is the three-dimensional Levi-Civita tensor . The vorticity tensor is simply the antisymmetric part of the tensor ∇ v {\displaystyle \nabla \mathbf {v} } , i.e., In a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, in the central core of a Rankine vortex . The vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines , if there

SECTION 20

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1232-467: Is the vorticity vector in Cartesian coordinates . 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

1288-418: Is then positive if the wind veers (turns clockwise ) with altitude and negative if it backs (turns counterclockwise ). This helicity used in meteorology has energy units per units of mass [m/s] and thus is interpreted as a measure of energy transfer by the wind shear with altitude, including directional. This notion is used to predict the possibility of tornadic development in a thundercloud . In this case,

1344-412: 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. The evolution of the vorticity field in time

1400-411: Is true in the case of two-dimensional potential flow (i.e. two-dimensional zero viscosity flow), in which case the flowfield can be modeled as a complex-valued field on the complex plane . 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 of vorticity away from the vortex cores into

1456-492: The Gauss map Γ {\displaystyle \Gamma } from the torus to the sphere by Pick a point in the unit sphere, v , so that orthogonal projection of the link to the plane perpendicular to v gives a link diagram. Observe that a point ( s , t ) that goes to v under the Gauss map corresponds to a crossing in the link diagram where γ 1 {\displaystyle \gamma _{1}}

1512-665: The Wilson loop observable in U ( 1 ) {\displaystyle U(1)} Chern–Simons gauge theory . Explicitly, the abelian Chern–Simons action for a gauge potential one-form A {\displaystyle A} on a three- manifold M {\displaystyle M} is given by We are interested in doing the Feynman path integral for Chern–Simons in M = R 3 {\displaystyle M=\mathbb {R} ^{3}} : Here, ϵ {\displaystyle \epsilon }

1568-419: The perimeter of C {\displaystyle C} is the dot product ω ⋅ ( n d A ) {\displaystyle {\boldsymbol {\omega }}\cdot (\mathbf {n} \,dA)} where ω {\displaystyle {\boldsymbol {\omega }}} is the vorticity at the center of C {\displaystyle C} . Since vorticity

1624-463: The 1950s, the first successful programs for numerical weather forecasting utilized that equation. In modern numerical weather forecasting models and general circulation models (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation . Related to the concept of vorticity is the helicity H ( t ) {\displaystyle H(t)} , defined as where

1680-489: The appropriate J {\displaystyle J} , we can get back the Wilson loops. Since we are in 3 dimensions, we can rewrite the equations of motion in a more familiar notation: Taking the curl of both sides and choosing Lorenz gauge ∂ μ A μ = 0 {\displaystyle \partial ^{\mu }A_{\mu }=0} , the equations become From electrostatics,

1736-428: 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 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). In

Hydrodynamical helicity - Misplaced Pages Continue

1792-417: The curves are required to always be immersions or not), which is an example of an h -principle (homotopy-principle), meaning that geometry reduces to topology. This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail: There is an algorithm to compute

1848-406: The effective action is quadratic in J {\displaystyle J} , it is clear that there will be terms describing the self-interaction of the particles, and these are uninteresting since they would be there even in the presence of just one loop. Therefore, we normalize the path integral by a factor precisely cancelling these terms. Going through the algebra, we obtain where which

1904-478: The flow is incompressible ( ∇ ⋅ u = 0 {\displaystyle \nabla \cdot \mathbf {u} =0} ), or it is compressible with a barotropic relation p = p ( ρ ) {\displaystyle p=p(\rho )} between pressure p and density ρ ; and (iii) any body forces acting on the fluid are conservative . Under these conditions, any closed surface S whose normal vectors are orthogonal to

1960-634: The following alternative formula The formula n 1 − n 4 {\displaystyle n_{1}-n_{4}} involves only the undercrossings of the blue curve by the red, while n 2 − n 3 {\displaystyle n_{2}-n_{3}} involves only the overcrossings. Given two non-intersecting differentiable curves γ 1 , γ 2 : S 1 → R 3 {\displaystyle \gamma _{1},\gamma _{2}\colon S^{1}\rightarrow \mathbb {R} ^{3}} , define

2016-402: The following standard positions. This determines the linking number: Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy , which further requires that each curve be an immersion , not just any map. However, this added condition does not change the definition of linking number (it does not matter if

2072-417: The form of the linking integral . It is an important object of study in knot theory , algebraic topology , and differential geometry , and has numerous applications in mathematics and science , including quantum mechanics , electromagnetism , and the study of DNA supercoiling . Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of

2128-485: The formation and motion of vortex rings . Mathematically, the vorticity ω {\displaystyle {\boldsymbol {\omega }}} is the curl of the flow velocity v {\displaystyle \mathbf {v} } : where ∇ {\displaystyle \nabla } is the nabla operator . Conceptually, ω {\displaystyle {\boldsymbol {\omega }}} could be determined by marking parts of

2184-478: The general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation. 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 where ω = ( ω x , ω y , ω z ) {\displaystyle {\boldsymbol {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}

2240-484: The handedness (or chirality ) of the flow. Helicity is one of the four known integral invariants of the Euler equations; the other three are energy , momentum and angular momentum . For two linked unknotted vortex tubes having circulations κ 1 {\displaystyle \kappa _{1}} and κ 2 {\displaystyle \kappa _{2}} , and no internal twist,

2296-447: The helicity is given by H = κ 2 ( W r + T w ) {\displaystyle H=\kappa ^{2}(Wr+Tw)} , where W r {\displaystyle Wr} and T w {\displaystyle Tw} are the writhe and twist of the tube; the sum W r + T w {\displaystyle Wr+Tw} is known to be invariant under continuous deformation of

Hydrodynamical helicity - Misplaced Pages Continue

2352-533: The helicity is given by H = ± 2 n κ 1 κ 2 {\displaystyle H=\pm 2n\kappa _{1}\kappa _{2}} , where n is the Gauss linking number of the two tubes, and the plus or minus is chosen according as the linkage is right- or left-handed. For a single knotted vortex tube with circulation κ {\displaystyle \kappa } , then, as shown by Moffatt & Ricca (1992),

2408-404: The integral is over a given volume V {\displaystyle V} . In atmospheric science, helicity of the air motion is important in forecasting supercells and the potential for tornadic activity. Linking number In mathematics , the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space . Intuitively,

2464-469: The linking number of γ 1 and γ 2 enables an explicit formula as a double line integral , the Gauss linking integral : This integral computes the total signed area of the image of the Gauss map (the integrand being the Jacobian of Γ) and then divides by the area of the sphere (which is 4 π ). In quantum field theory , Gauss's integral definition arises when computing the expectation value of

2520-634: The linking number of two curves from a link diagram . Label each crossing as positive or negative , according to the following rule: The total number of positive crossings minus the total number of negative crossings is equal to twice the linking number. That is: where n 1 , n 2 , n 3 , n 4 represent the number of crossings of each of the four types. The two sums n 1 + n 3 {\displaystyle n_{1}+n_{3}\,\!} and n 2 + n 4 {\displaystyle n_{2}+n_{4}\,\!} are always equal, which leads to

2576-409: The linking number represents the number of times that each curve winds around the other. In Euclidean space , the linking number is always an integer , but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds , where linking numbers can also be fractions or just not exist at all). The linking number was introduced by Gauss in

2632-532: The path integral can be done simply by solving the theory classically and substituting for A {\displaystyle A} . The classical equations of motion are Here, we have coupled the Chern–Simons field to a source with a term − J μ A μ {\displaystyle -J_{\mu }A^{\mu }} in the Lagrangian. Obviously, by substituting

2688-499: The solution is The path integral for arbitrary J {\displaystyle J} is now easily done by substituting this into the Chern–Simons action to get an effective action for the J {\displaystyle J} field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. J = J 1 + J 2 {\displaystyle J=J_{1}+J_{2}} , with Since

2744-462: The total approximate circulation about the wing. According to the Kutta–Joukowski theorem , lift per unit of span is the product of circulation, airspeed, and air density. 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

2800-400: The tube. The invariance of helicity provides an essential cornerstone of the subject topological fluid dynamics and magnetohydrodynamics , which is concerned with global properties of flows and their topological characteristics. In meteorology , helicity corresponds to the transfer of vorticity from the environment to an air parcel in convective motion. Here the definition of helicity

2856-551: The vertical integration will be limited below cloud tops (generally 3 km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the storm in subtracting its motion: where ⁠ C → {\displaystyle {\vec {C}}} ⁠ is the cloud motion relative to the ground. Critical values of SRH ( S torm R elative H elicity) for tornadic development, as researched in North America , are: Helicity in itself

SECTION 50

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2912-446: The vorticity (that is, n ⋅ ( ∇ × u ) = 0 {\displaystyle \mathbf {n} \cdot (\nabla \times \mathbf {u} )=0} ) is, like vorticity, transported with the flow. Let V be the volume inside such a surface. Then the helicity in V , denoted H , is defined by the volume integral For a localised vorticity distribution in an unbounded fluid, V can be taken to be

2968-459: 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 Research ). In aerodynamics , the lift distribution over

3024-443: The vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ω {\displaystyle {\boldsymbol {\omega }}} , defined as the curl of the velocity field v {\displaystyle \mathbf {v} } describing the continuum motion. In Cartesian coordinates : In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in

3080-412: The vorticity vector is a solenoidal field since ∇ ⋅ ω = 0. {\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.} In a two-dimensional flow , ω {\displaystyle {\boldsymbol {\omega }}} is always perpendicular to the plane of the flow, and can therefore be considered a scalar field . Mathematically,

3136-400: The whole space, and H is then the total helicity of the flow. H is invariant precisely because the vortex lines are frozen in the flow and their linkage and/or knottedness is therefore conserved, as recognized by Lord Kelvin (1868). Helicity is a pseudo-scalar quantity: it changes sign under change from a right-handed to a left-handed frame of reference; it can be considered as a measure of

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