Kirchhoff s Elliptical Vortex

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Loren Brianne Bell
5 years ago
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1 1 Figure 1. An elliptical vortex oriented at an angle φ with respect to the positive x axis. Kirchhoff s Elliptical Vortex In the atmospheric and oceanic context, two-dimensional (height-independent) vortices like the Rankine vortex are commonly found at large scales (> a few tens of kilometres in the oceans, and a few hundreds of kilometres in the atmosphere). As a result, twodimensional vortices have a special significance. The Rankine vortex is the simplest such vortex. But its circular shape is easily deformed, for example via interactions with other distant vortices. As we will see below, the leading-order shape deformation is elliptical. Moreover, this shape tends to be preserved in time, i.e. the vortex remains approximately elliptical though its aspect ratio and orientation may vary. Here, we examine the velocity field induced by an elliptical vortex. Consider then a vortex of semi-major axis length a and semi-minor axis length b centred at the origin (cf. figure 1). To determine the flow induced by such a vortex, it is sufficient to take its semi-major axis to be coincident with the x axis, i.e. φ = 0 in figure 1. The vortex boundary ( x, ȳ) is described by the relation x 2 a 2 + ȳ2 b 2 = 1. (0.1) The vorticity is purely axial and non-zero (ω = ζ 0 k) only inside the vortex boundary. Hence, the corresponding flow field is purely two-dimensional u = (u, v, 0) and derivable from the streamfunction ψ (via u = ψ/ y, v = ψ/ x). This vortex was first studied by Gustav Robert Kirchhoff in 1876 (see www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Kirchhoff.html). Formally, we may calculate ψ directly from the Green function integral ψ = ζ 0 2π x 2 /a 2 +y 2 /b 2 <1 log x x dx dy (0.2) where again x and x are two-dimensional vectors here, or one may reduce this to a contour integral. However, in this case it is perhaps simplest to work in the natural

2 2 elliptical coordinate system, particularly for the flow exterior to the vortex where 2 ψ = 0. For the interior flow, where 2 ψ = ζ 0, we may try ψ = Ax 2 + By 2 + C (0.3) with A + B = ζ 0 /2 and C as yet arbitrary. Because only the sum A + B is constrained at this point, this form for ψ implicitly contains an as yet arbitrary solution of 2 ψ = 0, namely 1 2 (A B)(x2 y 2 ), which is needed for matching the velocity field at the vortex boundary to the exterior field. Moreover, the quadratic dependence on x and y is consistent with the elliptical symmetry. (No arguments like this are needed if one uses the Green function integral, but the calculations are tedious.) For the exterior flow, we use elliptical coordinates (ϱ, ϑ), defined by the relations where x = (ϱ + εϱ 1 ) cos ϑ y = (ϱ εϱ 1 ) sin ϑ, (0.4) ε = a b (0.5) is the eccentricity of the ellipse. For any given ϱ, the relations in Eq. 0.4 parametrise an ellipse. At the boundary, ϱ = ϱ, we must have x = a cos ϑ and ȳ = a sin ϑ according to Eq This implies that ϱ = 1 2 (a+b), i.e. the average of the semi-major and semi-minor axis lengths. In this notation, we recover cylindrical polar coordinates directly when ε = 0. Note that the isolines of ϱ and ϑ are orthogonal. Moreover, the smallest value of ϱ = ε corresponds to the line connecting the two focal points either at ( c, 0) & (c, 0) when a > b, or at (0, c) & (0, c) when a < b, where c = a 2 b 2. These coordinates are introduced to solve 2 ψ = 0 in the domain exterior to the ellipse. Using Eq. 0.4, one may express the Laplacian in elliptical coordinates as 2 ψ = 1 [ h 2 ϱ ( ϱ ψ ) ] + 2 ψ ϱ ϱ ϑ 2 (0.6) where h 2 = (ϱ εϱ 1 ) 2 cos 2 ϑ + (ϱ + εϱ 1 ) 2 sin 2 ϑ is a geometrical factor equal to (dx/dϑ) 2 +(dy/dϑ) 2 along coordinate lines of constant ϱ. This factor does not matter for the solution of 2 ψ = 0. In fact, there are many possible solutions of this equation, and those relevant to the exterior flow are proportional to ϱ m e imϑ, for any positive integer m. There is also a solution for m = 0, namely log ϱ. The solutions are, in fact, the same as for the flow exterior to a circle (replacing ϱ by r and ϑ by θ), a result which follows directly from conformal theory (using the transformation re iθ ϱe iϑ ). Not all of these solutions are needed for the present problem. To determine which ones are, we express the interior flow (cf. Eq. 0.3) in elliptical coordinates at the vortex boundary, ϱ = ϱ = 1 2 (). There, ψ = Aa 2 cos 2 ϑ + Bb 2 sin 2 ϑ = 1 2 (Aa2 Bb 2 ) cos 2ϑ (Aa2 + Bb 2 ) + C. (0.7) Hence, only the m = 0 and m = 2 exterior solutions of Laplace s equations are needed, so for ψ there we may take ψ = Dϱ 2 cos 2ϑ + E log ϱ (0.8) where E and F are two additional constants to be determined presently. At the vortex

3 boundary, ψ and the velocity u must be continuous for all values of ϑ (this follows from the Green function integral: though ω is discontinuous, the integration of Gω gives both continuous ψ and u). First of all, continuity of ψ implies both 1 2 (Aa2 Bb 2 ) = D ϱ 2 and 1 2 (Aa2 + Bb 2 ) + C = E log ϱ. The normal component of the velocity (in the direction of increasing ϱ) is equal to h 1 ψ/ ϑ, and its continuity is implied by the continuity of ψ. The tangential component (in the direction of increasing ϑ) is equal to h 1 ψ/ ϱ, so it is sufficient to demand continuity of ϱ ψ/ ϱ at the vortex boundary. Therefore, using Eq. 0.3 along with Eq. 0.4, one may show that continuity of the cos 2ϑ term implies ab(a B) = 2D ϱ 2 while continuity of the ϑ-independent term implies ab(a+b) = E. These four relations, along with A + B = 1 2 ζ 0, determine all five coefficients as follows: A = 1 2 ζ b 0 B = 1 2 ζ a 0 C = 1 [ 2 ζ 0ab log 1 2 () 1 ] (0.9) 2 D = 1 16 ζ 0ab(a 2 b 2 ) E = 1 2 ζ 0ab. (0.10) First of all, note that the circulation of the ellipse is equal to ζ 0 πab, so the term 1 2 ζ 0ab is equal to the vortex strength κ (= Γ/2π). So, far from the vortex, ψ κ log ϱ κ log r; in other words the flow becomes circular and depends only on the vortex strength. This is always true for an isolated vortex in an unbounded domain. Second, note that the vortex core is not rigidly rotating, since A B. The velocity field there is given by u = ψ y = ζ 0ay 3 v = ψ x = ζ 0bx. (0.11) At the vortex boundary, however, ū(ϑ) = ū m cos ϑ and v(ϑ) = ū m sin ϑ, where ū m = ζ 0 ab/(). The boundary is evidently moving, since ū is not tangent to the boundary. The normal velocity component is equal to ū = 1 ψ h ϑ = 1 2 ζ 0abεh 1 sin 2ϑ (0.12) where ε is the eccentricity (cf. Eq. 0.5) and h 2 = b 2 cos 2 ϑ + a 2 sin 2 ϑ. But now consider the normal velocity generated by rotating the boundary rigidly at some rate Ω e. This flow may be represented by ψ = 1 2 Ω e(x 2 + y 2 ), and so its normal component at the vortex boundary is equal to ū = 1 ψ h ϑ = 1 2 Ω e(a 2 b 2 )h 1 sin 2ϑ (0.13) Comparing with Eq. 0.12, one can see that this has the same dependence on ϑ; hence we can conclude that the self-induced velocity of the vortex causes it to rotate steadily at the rate Ω e given by Ω e = ζ 0abε a 2 b 2 = ζ 0ab () 2. (0.14) Now let s examine this flow in a reference frame rotating with the ellipse. Then, the interior streamfunction takes the form ψ = (A Ω e /2)x 2 + (B Ω e /2)y 2 + C = 1 ( ) 2 ( ) ab x 2 2 ζ 0ε a 2 + y2 b 2 + C (0.15) from which it is immediately clear that ψ is constant on the vortex boundary (cf. Eq. 0.1).

4 4 Moreover, ψ is constant on concentric, equal eccentricity ellipses, i.e. the curves x 2 /a 2 + y 2 /b 2 = α 2, where the constant α 1. Fluid particles move along these curves at the speed u (ϑ; α) = Ω e h = Ω e α(b 2 cos 2 ϑ + a 2 sin 2 ϑ) 1/2. (0.16) But since h = dx/dϑ = ds/dϑ (s denoting the arc length) along a given ellipse, the time taken for a fluid particle to circuit the ellipse is T = ds/u = 2π/Ω e, independent of α. That is, all particles have equal periods of rotation; moreover, this period is equal to the period of rotation of the vortex itself. While particles all rotate at the same rate within the vortex, they accelerate and decelerate during this motion. As they accelerate, two particles on adjacent ellipses α and α + dα starting say from ϑ = 0 move closer together, while as they deccelerate, they move farther apart. Note that they always keep the same value of ϑ. This motion is consistent with the fact that the fluid is incompressible ( u = 0), but it also indicates that fluid elements experience strain, or deformation, as they move around the vortex. The motion of each particle (X(t), Y (t)) inside the vortex is governed by the following equations: dx dt = u = a b Ω ey dy dt = v = + b a Ω ex. (0.17) These of course trace out an ellipse in time. Both X and Y exhibit simple harmonic motion with period 2π/Ω e. Writing X = αa cos ϑ(t) and Y = αb sin ϑ(t), it is easy to see that dϑ/dt = Ω e, i.e. a constant rate of change of the elliptical angle (which is one reason why elliptical coordinates are so useful). But note that we can also write dx = (Ω + γ)y dt dy = +(Ω γ)x (0.18) dt for some appropriately chosen values of Ω and γ. That is, the motion consists of a rigid rotation (Ω) plus a uniform strain (γ). Using Eq. 0.14, one may show Ω = 1 2 ζ 0 Ω e γ = 1 2 εζ 0. (0.19) Hence, the strain rate γ is proportional to the vortex eccentricity; in particular, γ vanishes for a circular vortex. Just as the circular vortex is unique in having rigid rotation, the elliptical vortex is unique in having uniform strain. This turns out to be a very important property, since the leading-order effect of distant vortices (not considered in this example) on a given vortex is well approximated by a rotating strain flow, like that in Eq. 0.18, except that the orientation of the strain (its extensional axis) may be arbitrary. For the elliptical vortex, the extensional axis of strain is oriented at the angle θ s = 45 for a > b, and at θ s = 45 for a < b. One may therefore anticipate that an external strain acting an elliptical vortex preserves its elliptical shape. However, in general, it leads to variations in the vortex eccentricity and its rate of rotation.

5 Figure 2. A general two-dimensional strain flow consists of a superposition of rigid rotation (dashed circles) and uniform strain (solid curves). Here, the extensional axis is oriented at the angle θ s = 30. The bold and thin arrows respectively indicate the flow direction associated with positive strain γ and rotation Ω individually. 5

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