Helicity of a toroidal vortex with swirl
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Helicity of a Toroidal Vortex with Swirl E. Yu. Bannikovaa,b*, V. M. Kontorovicha,b, and S. A. Poslavskyb aInstitute
of Radio Astronomy, National Academy of Sciences of Ukraine, Chervonopraporna ul. 4, Kharkov, 61002 Ukraine b Karazin Kharkov National University, pl. Svobody 4, Kharkov, 61022 Ukraine *e-mail: [email protected] Received October 7, 2015
Abstract—Based on the solutions of the Bragg–Hawthorne equation, we discuss the helicity of a thin toroidal vortex in the presence of swirl, orbital motion along the torus directrix. The relation between the helicity and circulations along the small and large linked circumferences (the torus directrix and generatrix) is shown to depend on the azimuthal velocity distribution in the core of the swirling ring vortex. In the case of nonuniform swirl, this relation differs from the well-known Moffat relation, viz., twice the product of such circulations multiplied by the number of linkages. The results can find applications in investigating the vortices in planetary atmospheres and the motions in the vicinity of active galactic nuclei. DOI: 10.1134/S1063776116040026
1. INTRODUCTION In nature, toroidal vortices often have a swirl [1], orbital motion along the torus directrix. These apparently include the attached ring vortices of tropical cyclones, hurricanes, and tornados [2] as well as the solar toroidal vortices [3] responsible for the 11-year activity cycle and many others (see, e.g., [4]). A topological helicity integral arises in the presence of swirl [5]. This can increase the vortex stability, which is confirmed in laboratory experiments [6]. As is well known, the helicity for two linked vortex contours must be equal to the product of the circulations multiplied by twice the number of linkages [1, 5, 7, 8]. Using an example admitting a hydrodynamic solution, we will show that this relation for a toroidal vortex with swirl has a different form reflecting the spatial distribution of vorticity.
2 ⎛p ⎞ V × curl V = ∇ ⎜ + V ⎟ , ⎝ρ 2 ⎠
where in cylindrical coordinates (r, φ, z)
ω = curl V ∂(rV ϕ ) ⎞ ⎛ ∂V ∂V ∂V i z ⎟. = ⎜ − ϕ i r , ⎛⎜ r − z ⎞⎟ i ϕ, 1 ∂r ⎠ r ∂r ⎝ ∂z ⎝ ∂z ⎠
∂ψ Vr = − 1 , r ∂z
∂ψ Vz = 1 . r ∂r
(2.3)
In this case, the continuity equation divV = 0 is satisfied identically. The orbital velocity component can now be represented as
Vϕ =
f (ψ) , r
(2.4)
where f(ψ) is a known function. Substituting expressions (2.3) and (2.4) into (2.2), we obtain
∂ψ Δψ ∂ψ ⎞ ⎛ curl V = ⎜ − 1 f ' ir, − i ϕ, 1 f ' i z ⎟, ∂z ∂r ⎠ r r ⎝ r
(V ⋅ ∇)V = −∇ p/ ρ. Given that
(2.2)
Let us introduce the Stokes stream function ψ defined by the relations
2. THE BRAGG–HAWTHORNE EQUATION AND ITS SOLUTION FOR A TOROIDAL VORTEX Consider the axisymmetric steady flow of an ideal incompressible fluid in the absence of mass forces. The Euler equation in this case is
(2.1)
where
( )
(V ⋅ ∇)V = grad(V 2 /2) − V × curl V,
∂ψ ∂ ψ Δ ≡ r ∂ 1 + 2, ∂r r ∂r ∂z
we rewrite it as 769
2
f'≡
df . dψ
(2.5)
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