Momenta of a vortex tangle by structural complexity analysis

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Author's personal copy2224 R.L. Ricca / Physica D 237 (2008) 2223–2227Fig. 1. The area A of the projected graph resulting from the projection p ν ofa vortex line L on the plane Π is proportional to the component of the linearmomentum of L in the ν-direction.arc-length s. Vorticity ω is defined on L i , and is simply given byω = ¯ωX ′ , where, in general, X = X(s, t) denotes the positionvector, ¯ω a constant and t ≡ X ′ the unit tangent to L i (the primedenoting the derivative with respect to s, and t is time). Thelinear momentum P = P(T ) corresponds to the hydrodynamicimpulse, which generates the motion of T from rest, and fromits standard definition [11] takes the formP = 1 ∫X × ωd 3 X = 1 N∑Γ i X × X2 T2i=1∫L ′ ds, (1)iwhere P is here intended per unit density, and Γ i representsthe circulation of L i . Similarly, for the angular momentumM = M(T ), that corresponds to the moment of the impulsiveforces acting on T ; we haveM = 1 ∫X × (X × ω)d 3 X3 T= 1 N∑Γ i X × (X × X3∫L ′ )ds, (2)ii=1where, again, M is intended per unit density. We remark thatunder both Euler’s equations and LIA, we havedPdt= 0,dMdt= 0. (3)2. Interpretation of momenta in terms of projected areaArms and Hama [8], who first proved the conservation ofthe integral on the right-hand side of Eq. (1) for a single vortexline, showed that this quantity admits interpretation in terms ofprojected area (see Fig. 1). Indeed, by direct inspection of theintegrand above, it is evident that under LIA the plane projectedarea of the vortex line is proportional to the component ofthe linear momentum of the vortex along the direction ofprojection.Let p = p ν denote the orthogonal projection onto the planeΠ along the direction ν, and L ν = p ν (L) be the graph diagramof a smooth space curve L under p ν . Evidently L ν depends onν. For the moment let L ν be a smooth planar curve with noself-intersections, but in general L ν will be a nodal curve withself-intersections, the latter resulting from the projection of theapparent crossings of L, when L is viewed along the line ofsight ν.By identifying the vortex line with its geometric support L,the projected graph diagram L ν will be oriented, the orientationbeing naturally induced by the vorticity vector. Let L xy , L yz ,L zx be the three graph diagrams of the projected vortex lineonto the mutually orthogonal planes x = 0, y = 0, z = 0,and let A xy = A(L xy ), A yz = A(L yz ), A zx = A(L zx ) be thecorresponding areas of the plane regions bounded by L xy , L yz ,L zx , respectively.By applying the results of Arms and Hama [8], from (1) wehaveP xy = Γ A xy , P yz = Γ A yz , P zx = Γ A zx , (4)where Γ is vortex circulation. Moreover:Definition 2.1. The resultant area A max is the maximal areaobtained by max ν p(A) = p max (A) (along the resultant axisν max ) among all possible projected areas A.The direction of the resultant axis ν max is clearly that of thelinear momentum. Hence, also from [8], we haveTheorem 2.2 (Maximal Area Interpretation). The resultantlinear momentum of a vortex line L, of circulation Γ , underLIA is given by P = Γ A max ν max , where A max is the resultantarea. The projected area of L on any plane perpendicular tothat of the resultant area is zero.Similar results hold true for the angular momentum. Withreference to the right-hand side of Eq. (2), the second integralcan be interpreted in terms of areal moment, according to thefollowing definition:Definition 2.3. The areal moment around any axis is theproduct of the area A multiplied by the distance d betweenthat axis and the axis a G , normal to A through the centroidG of A.For a vortex line L, the centroid G of the projected area A isthe center weighted with respect to the vorticity distribution ofL. As for the linear momentum, the components of the angularmomentum are determined by the areal moments:M xy = Γ d z A xy , M yz = Γ d x A yz , M zx = Γ d y A zx , (5)where evidently d x , d y , d z are the distances between therotational axis and the centroid axes through A yz , A zx , A xy ,respectively. Similar considerations apply to define the resultantareal moment of L:Definition 2.4. The resultant areal moment of L is the arealmoment around the resultant axis a G of the projected areas ofL onto two mutually orthogonal planes, parallel to a G .⋃These observations are easily extended to a tangle T =i L i of N vortex lines L i , provided we carefully define the
Author's personal copyR.L. Ricca / Physica D 237 (2008) 2223–2227 2225Fig. 2. (a) The number in the dashed region is the value of the index I P (C) according to the right-hand rule convention and the algebraic intersection numbercalculated by Eq. (6). (b) The oriented nodal curve, resulting, for example, from the standard projection of a figure-8 knot, has 5 bounded regions. Note that one ofthe interior regions has index 0, due to the opposite orientation of the strands crossed by ρ. (c) Contribution from each A(R j ) must be weighted according to thecirculations on the boundary ∂R j .area of the resulting oriented graph diagram. The difficulty hereis precisely in the correct calculation of such area.3. Signed area of oriented graph diagramsThe oriented graph diagram of a tangle of vortex linesis an oriented nodal curve (i.e. the “underlying universe”of the tangle) in R 2 , and this can easily attain considerablecomplexity, particularly as regards the localization of selfintersections.A necessary first step is to reduce nodal curves ofany complexity to good nodal curves, that have (at most) doublepoints. Nodal points are classified according to their degree ofmultiplicity µ(P) given by the number of arcs incident at thepoint of intersection P. If P is a double point, then µ(P) = 2.If P is a point of multiplicity µ(P) = n (n > 2), we can alwaysreduce its multiplicity by “shaking” the graph diagram (actuallyits pre-image) near P to get m = 1 2 (n2 − n) double points, byvirtual perturbations of the incident arcs from their location.Thus, if h (n) is the total number of points of multiplicity n, byapplying this shaking technique we can always replace theseh (n) points with h(n) = mh (n) (m ≥ 3) double points. We saythat a graph diagram is a good projection, when it has at mostdouble points. Hence, by the shaking technique, we can alwaysreduce highly complex graph diagrams to good nodal curves.Let C denote one such good nodal curve on Π , and let A(C)be the corresponding total area. In order to calculate this area,first we need to define the index I P (C) of C at the point P (forthis see, for example, [12]). Let P ∉ C, t the tangent to C and ρthe radiant vector with foot at P, that intersects C transversally.At each intersection point X ∈ ρ ∩ C assign the algebraicsign ɛ(X) = ±1, according to the standard convention givenby the right-hand rule, that is ɛ(X) = +1 when the frame{ρ, t} is positive (see Fig. 2(a)). If X is a double point, thenthe intersection is computed with one of the neighbouring pairsof the incident, equi-oriented arcs.Definition 3.1. The index I P (C) of C at P is the algebraicintersection number given byI P (C) =∑ɛ(X). (6)X∈ρ∩CHence, I P (C) ∈ Z.Let us now consider the Z sub-domains {R j } j=1,...,Zdetermined by C ∩ Π and bounded by C, and let A(R j ) > 0denote their standard area. Since every point P ∈ R j has thesame I P (C), we shall call I j the index associated with anypoint P ∈ R j and assign this value to each sub-domain R jof C ∩Π (see Fig. 2(b)). The signed area of an oriented graph, aconcept that can be traced back to Gauss [13], is thus given bythe following rule.Rule 3.2 (Signed Area). The signed area A(C) of an oriented,planar nodal curve C, is given byA(C) =Z∑I j A(R j ), (7)j=1where A(R j ) > 0 is the standard area of R j .4. Linear and angular momenta of a vortex tangle bystructural complexity analysisBy the signed area rule we can calculate the projectedarea of any nodal curve, be it the graph of a single vortexline, or that of a complex tangle of vortices. If the vorticeshave different circulations, a weighting factor defined in termsof contributions from each arc of ∂R j must be assigned toA(R j ) (see Fig. 2(c)). The simplest correction comes from thealgebraic weighting γ j . Let L j = L(∂R j ) = ∑ k=1,...,M L k, jdenote the total length of the boundary curve ∂R j made of Moriented arcs, the k-th arc having length L k, j and circulationΓ k . We haveDefinition 4.1. The circulation weighting factor γ j of R j isgiven byγ j =M∑Γ k L k, jk=1L j. (8)If all the vortices have same circulation Γ , then evidentlyγ j = Γ . Appropriate weighting of circulation is necessary todetermine the correct location of the centroid of the projectedarea. To summarize, we have the following result.
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Momenta of a vortex tangle by structural complexity analysis

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