Kelvin waves with helical Beltrami flow structure

In this work we show that when an inviscid axisymmetric Rankine flow experiences a soft expansion, rotating Kelvin waves can be excited. Downstream of the region where the expansion occurs (the transition region) the resulting flow can be expressed as the addition of a Rankine and a Beltrami flow. The Beltrami constant is determined from the Rankine upstream flow, and the helix pitch of the n=1 mode results from the boundary conditions downstream. Finally, a discussion of the process leading to oscillatory flow and a conjecture about the topological background that sustains the Beltrami flow structure are offered.


I. INTRODUCTION
The problem of the structural modification of a vortex in an axisymmetric swirling flow passing from an upstream cylindrical tube to a downstream one, through a transition expansion region, was historically related to two kinds of flows in the downstream region: The formation of a vortex breakdown or to an oscillatory flow. 1,2 The formation of helical structures 3,4 was of special practical and theoretical interest.
In previous works 5, 6 we have studied some characteristics of this particular flow using the vortex filament solution. In this case, we focus on the study of the oscillatory flow and its connection with the features of the upstream flow. In doing so, our departure point is the well established Batchelor's analysis of the swirling flow. 1 This author took a Rankine flow upstream of an inviscid fluid and found the form of the steady Euler equations for the whole fluid field and the corresponding solutions for the downstream region. We show that the flow in transition and downstream regions is a Beltrami flow in a rotating and translating frame. As this resulted in a main characteristic of the overall flow, the consideration of this type of feature for the fluid is justified, more than assumed, as a simplification of the problem. 4,7,8 We also discuss deeper derivations of this result. Dritschel 7 studied helical symmetric rotating waves in a cylindrical tube with the assumption of a Beltrami flow. He established that Lord Kelvin's mathematical expressions of 1880, obtained in terms of Bessel functions 9 for the linear stability of a rigidly rotating flow in a pipe, are exact solutions of the Euler equations. Other works assuming the Beltrami flow hypothesis are not conclusive concerning the physical background which it is supposed to be. Here, we show that the Rankine flow determines the formation of downstream Beltrami flows. Still, we discuss the role that the topology plays in two important aspects, i.e., the frozen of the vorticity lines in an inviscid fluid, and the concept of helicity introduced in hydrodynamics by Moffatt. 10,11 At the same time, we argue about the possible origin of the structural change of the vortex, in the bifurcation of the basic flow.
We organize the sections as follows: In Sec. II we write the basic axisymmetric flow equations and show that in the transition and downstream regions they represent a Rankine plus a Beltrami flow field. In Sec. III, we introduce linear normal modes and show that helical rotating waves develop as a Beltrami flow, for a particular relationship between the frequency of the modes and the angular speed of the Rankine flow. In Sec. IV, we establish the boundary conditions downstream and we obtain the pitch of the helix for the n =1 mode. In Sec. V a discussion in relation to the breaking of symmetry of the axisymmetric flow is presented and a conjecture regarding the relationship between the topology of the flow and the Beltrami condition is made.

II. BASIC AXISYMMETRIC FLOW
We consider an inviscid fluid with a Rankine upstream flow scenario schematically shown in Fig. 1. In cylindrical coordinates the fluid is described as where ⍀, U 0 , and a 0 are constants, a 0 is the radius of the rotational core, and b 0 is the radius of the inlet stream. The upstream flow around the core is potential, and is = ٌ ϫ v = 0. The Kelvin circulation theorem assures that the flow in the transition and downstream regions is also potential. Thus, for the whole structure we can separate the flow in a rotational zone ͑corresponding to the core͒, and an irrota-tional one corresponding to the potential flow. 12 On account of the mass conservation and the axisymmetry, we may introduce the stream function in the form and the Euler equations are written as where H is the total head function. Thus, for the axisymmetric flow the constancy of the circulation C = rv is guaranteed. When the flow is steady, both the circulation and the total head are functions of the stream function, and the following relations are verified for the whole flow field: 1 The upstream flow described by Eq. ͑1͒ determines linear relations of C͑͒ and H͑͒ for the whole rotational zone with , Thus, from Eqs. ͑5͒ and ͑6͒, the following relations are obtained:

͑7͒
As can be seen from Eq. ͑7͒, while the upstream rotational zone is described with a Rankine flow of vorticity R =2⍀z, the transition and downstream rotational zones can be described as the joint contribution of a Rankine flow v R plus a Beltrami one v B , and is In fact, taking from Eqs. ͑7͒ and ͑9͒, we derive Eq. ͑8͒. Therefore, the main effect of the expansion is to produce a Beltrami flow superimposed to the Rankine one with the Beltrami constant ␥ determined by the associated Rankine flow upstream. Moreover, the potential flow of the irrotational zone can be seen as a Beltrami flow with the eigenvalue ␥ irrot = 0. In the downstream region the conservation laws for the axisymmetric solution give 1,6 a and b are the parameters of the downstream region ͑see Fig. 1͒. a is determined from the equation which was obtained using the condition that the axial speeds are the same at r = a. Equation ͑13͒ in a dimensionless form is written as where = b / b 0 is the expansion parameter, = a / a 0 is the nondimensional expanded vortex radius, = a 0 / b 0 , and = U 0 / ⍀b 0 are the Rossby numbers, with U 0 as the axial speed.
In Fig. 2 we plot the expanded vortex radius as a function of the Rossby number for different expansion parameters. As it was already pointed out in a previous work, 6 each line has two branches, an upper unstable one and a lower stable branch. The point where the two branches-of different stability-join corresponds to the critical Rossby number c . Thus, a saddle-node bifurcation is obtained. When Ͻ c the axisymmetric solutions Eqs. ͑10͒ and ͑11͒, are no longer possible and a breakdown can appear. Therefore, only when Ͼ c the lower branch solution can support perturbed oscillations. The figure shows that the value of c grows with which is also shown in Fig. 3. This means that when is increased, the loss of the axisymmetric solution occurs at a lower angular speed ⍀. A necessary condition for these axisymmetric solutions to hold, which is satisfied by the flows considered here, is that reversal axial flows are not allowed all over the transition expansion region. 1 Figure 4 shows the dependence of the expansion radius as a function of the Rossby number for different values of , the upstream vortex radius. Note that the parameter c increases with , and = 0.272 corresponds to the observational value as stated in Ref. 3.

III. HELICAL FLOW
We take infinitesimal perturbations of helical symmetry in the downstream flow, i.e., with velocities and pressure perturbations, as ͕␦v r ,␦v ,␦v z ,␦p͖ = ͕f͑r͒,g͑r͒,h͑r͒,͑r͖͒e in−it , ͑15͒ where = − kz is the helical coordinate with the helix pitch k, and is the frequency of the perturbation. Imaginary positive values of indicate growing perturbations in time and the instability of the basic flow. Thus, the linearization around the axisymmetric solution of the Euler equations take a downstream form of the type where ͑v , v r , v z ͒ are the velocity components of the basic flow. The linear stability of the basic flow is determined ͑taking into account boundary conditions͒ by the solutions of Eqs. ͑16͒-͑19͒ in the rotational and irrotational zones, i.e., solving the corresponding eigenvalue equation that relates to n.
Our aim is to show that there are some real values of for which the solutions of Eqs. ͑16͒-͑19͒ represent a helical Beltrami flow with perturbations of the form Eq. ͑15͒ corresponding to Kelvin waves that propagate in the axisymmetric core surrounded by an irrotational flow. We are also interested in the driving mechanism of these waves.
In order to solve Eqs. ͑16͒-͑19͒ in the rotational zone we choose = n⍀. ͑20͒ Note that if this condition is satisfied it can be easily verified that in the rotational downstream zone the solutions of Eqs. ͑16͒-͑19͒ associated with perturbations of the basic flow given by Eq. ͑10͒ match the Beltrami condition Eq. ͑7͒ and can be written as On the other hand, and for the frequency of Eq. ͑20͒, the solutions of Eqs. ͑16͒-͑19͒, associated with basic flow perturbations in the irrotational downstream zone ͓see Eq. ͑11͔͒, which satisfy the boundary condition ␦v irrot ͑r = b͒ = 0 can be written as In the next section we will come back to the necessary matching conditions ͑for the n = 1 case͒ at the frontier of the rotational and the irrotational zones. Defining the angular coordinate = − ⍀t, Eq. ͑15͒ represents a helical rotating wave. In the rotating frame with angular speed ⍀, the flow is steady. Therefore, the downstream flow for the rotational zone is v rot = ͑0,⍀r,U 0 ͒ + ͭ ͓0,␥AJ 1 ͑␥r͒,␥AJ 0 ͑␥r͔͒ The first term of Eqs. ͑28͒ and ͑29͒ represents a Rankine flow. The second one represents a steady Beltrami flow in a frame turning with angular speed ⍀, with constants ␥ and ␥ irrot = 0, respectively. In fact, as it was shown by Chandrasekhar and Kendall 13 or by Barberio-Corsetti, 14

IV. BOUNDARY CONDITIONS AND HELIX PITCH FOR THE n = 1 MODE
To find the helix pitch k as a function of the Rossby number for the basic flow cases of Sec. II we apply the boundary conditions at the frontier between the rotational and the irrotational zones to the n = 1 mode of the flows given by Eqs. ͑28͒ and ͑29͒. To this aim we follow the procedure performed in a previous work. 6 Writing the perturbed core radius as r = a + ⌬e in−it , after linearizing, the continuity of the pressure across this boundary implies On the other hand, the fact that the surface r = a + ⌬e in−it moves with the fluid gives the kinematics condition

͑31͒
which must be satisfied on both sides of the surface, with V 0 as the basic flow. Then, taking into account Eqs. ͑10͒, ͑11͒, ͑21͒-͑23͒, and ͑25͒-͑27͒ and the resonant condition Eq. ͑20͒ replaced in Eqs. ͑30͒ and ͑31͒ we obtain the equation for k as an implicit function of the Rossby number , which in terms of the dimensionless variables defined in Eq. ͑14͒ results in where f , g , h are the nondimensional velocity components f , g , h given by Eqs. ͑21͒-͑23͒ and ͑25͒-͑27͒ in the rotational and the irrotational zones. The helix pass is defined as the z length variation ͑for a given value of coordinate ͒ when changes an amount 2. In fact, ⌬ = ⌬ − k⌬z if ⌬ = 0 and ⌬ =2 it results in the H p = ⌬z =2 / k helix pass. Thus, numerically solving Eq. ͑32͒, we obtain Figs. 5 and 6 for the pitch k and the helix pass H p , respectively, as a function of the Rossby number for the mode n = 1. From the figures we see that the helix pitch increases and the helix pass decreases with increasing expansion and decreasing Rossby number , respectively. For a definite Rossby number we obtain, in accordance with Eq. ͑24͒, an upper limit of the pitch due to the relation k Յ 2 / .
As seen, a saddle-node bifurcation of an axisymmetric flow occurs when = c , and the basic flow disappears when Ͻ c . Thus, when subject to infinitesimal helical perturbations of the form Eq. ͑15͒, helical rotating flows can arise if Ͼ c together with the marginal stability of the lower branch of the basic axisymmetric flow. On the other hand, and for finite amplitude perturbations, the passage from axisymmetric flow to helical flow is also a bifurcation. Due to the rotating character this is a Hopf bifurcation whose Rossby critical value cc is cc Ͼ c .
As an example, to give account of this critical parameter, we can first consider the mode n = 1. We see that most of the helix pass values result in one or two orders of magnitude greater than the unit length value. Also, our results correspond ͑the same order of magnitude͒ to the observed experimental values. 3 So, taking these results into account, we chose the value H p = 10 as a limit parameter of the calculus. Secondly, in Eq. ͑15͒ we took the frequency = n⍀ + is with s → 0. Figure 7 shows the dependence of s with the parameter , note that cc Ϸ 0.565 due to the fact that if Ͻ 0.565, then s Ͼ 0 ͑unstable case͒, and that s Ͻ 0 when Ͼ 0.565 ͑stable͒. Thus, s ϰ cte͑ − cc ͒ near cc .

V. SUMMARY AND DISCUSSION
In what follows, we summarize the main results of our work and we present a discussion in relation to the formation of RKW and the Beltrami flow structure. ͑i͒ A Rankine upstream flow subject to a smooth expansion ͑see Fig. 1͒ turns to an axisymmetric flow that can be expressed as the addition of a Rankine and a Beltrami flow in the rotational zone at the transition and downstream regions. The value of the Beltrami eigenvalue ␥ = 2⍀ / U 0 is determined by the Rankine parameters ⍀ and U 0 ; in nondimensional variables the eigenvalue is 2 / , with the Rossby number.
͑ii͒ As shown in Fig. 2, we obtain a representation of the expanded vortex radius downstream in terms of a two branched function of the Rossby number: An upper unstable branch and a lower stable one joining at the critical Rossby number c , where a saddle-node bifurcation takes place. When Ͻ c the axisymmetric solutions are lost and a breakdown can appear. Therefore, to support perturbations of the lower branch solution the flow has to fulfill the relation Ͼ c . The value of c grows with the expansion parameter .
͑iii͒ The perturbations become Kelvin waves with a helical Beltrami flow structure if the resonant condition = n⍀ is satisfied by the frequency . The basic axisymmetric flow is marginally stable for infinitesimal helical perturbations.
͑iv͒ A main result obtained is that, as in the case of the axisymmetric flow, the eigenvalue ␥ = 2⍀ / U 0 is determined by the Rankine parameters ⍀ and U 0 .
͑v͒ We determined the helix pitch k and the helix pass H p as a function of the Rossby number represented in Figs. 5 and 6, respectively, for the mode n = 1 via the application of matching conditions between downstream rotational and irrotational zones and infinitesimal helical perturbations with the resonant condition Eq. ͑20͒. We defined the parameter cc as the critical Rossby number; as can be seen from Fig. 7, if = cc a Hopf bifurcation takes place. For this value, the flow is marginally stable; it becomes unstable if Ͻ cc and stable if Ͼ cc .
͑vi͒ From Fig. 5 we also note that the helix pitch grows with the expansion parameter and decays with the Rossby number . ͑vii͒ Helical rotating waves take place in the range ͑ c , cc ͒. In the example of Sec. IV the interval is ͑0.535, 0.565͒.
We now have elements to discuss about the structural  features that led to ͑a͒ the formation of helical Kelvin waves and ͑b͒ the fact that these waves are of a Beltrami flow type. ͑a͒ The scenario: When Ͼ cc , the Rankine flow passing trough the expansion region preserves its axisymmetric character. The expansion produces a Beltrami axisymmetric flow of eigenvalue ␥ = 2⍀ / U 0 determined by the upstream Rankine flow. If is decreased the same configuration is maintained until the critical value cc is reached. From a frame system turning with ⍀ and translating with U 0 , this value corresponds to the occurrence of a bifurcation-from a Beltrami axisymmetric flow to a Beltrami helical flowproduced by helical perturbations. The occurrence of a Hopf bifurcation for a marginally stable axisymmetric solution implies that, on the one hand, infinitesimal helical perturbations are sustained by the axisymmetric solutions when = cc , i.e., the saddle-node bifurcation controls the dynamic in the subspace of axisymmetric solutions. For finite amplitudes on the other hand the Hopf bifurcation controls the angular stability in the orthogonal subspace. A main feature of this flow is that the corresponding eigenvalue is always the same, ␥. Thus the whole new flow can be represented again by the addition of a Rankine and a Beltrami flow. As was seen from the linear approximation the superposition of modes is possible because modes of higher wave-number order have the same eigenvalue, ␥. Therefore, bifurcations other than the one associated with mode n = 1 could have a different critical parameter cc , but they will require an identical scenario. Kelvin waves with helical Beltrami flow structure are obtained while decreases until a critical value c is reached, giving rise to the possibility of a vortex breakdown. 1 ͑b͒ About the reasons why a Beltrami flow is a final state of the axisymmetric and helical symmetry cases.
The Woljter Theorem: 15 Woljter has introduced the concept of helicity in the context of the ideal magnetohydrodynamics, and proved that a force-free equilibrium ͑the analogous of the Beltrami one͒ can be obtained from a variational principle. In fact, he showed that this equilibrium configuration results from minimizing the magnetic energy restricted to the helicity conservation. The force-free magnetic equilibrium is expressed as ٌ ϫ B = ␣B, with B the magnetic field, ␣ constant and the helicity defined as the integral ͐A . B dV over the plasma volume, where A is the potential vector ٌ ϫ A = B.
Moffatt 10 was the first to introduce the concept of helicity in hydrodynamics, defined as ͐v . dV. Moreover, it was proved 11 that for the ideal hydrodynamics, the helicity that expresses topological properties of the flow, is conserved.
We present here a conjecture that will be the subject of a future research. Focusing on the topological analogy between the behavior of the magnetic field, in ideal magnetohydrodynamics, and the vortex field, in ideal hydrodynamics, we note that: a direct consequence of the Kelvin circulation theorem is that as the magnetic lines, the vortex lines are frozen to the fluid. First, in analogy to the magnetic energy M = ͐B 2 dV of Woljter's theorem we can define the enstrophy as the integral over a fluid volume of the square of the vorticity ⌽ = ͐ 2 dV. We assume a null surface integral at the boundary. The Appendix shows that the Beltrami flow is an extreme of the enstrophy subject to the helicity conservation.
The enstrophy variation is not affected if a frame with a uniform rotation and a uniform translation is adopted. In the magnetic case the physical conditions that lead to the forcefree equilibrium are the frozen of the magnetic field lines to the fluid and the helicity conservation. Then the variational principle leads to the equilibrium via the minimization of the magnetic energy. Both physical conditions are present in our case. So, this topological analogy suggests the conjecture that in a frame with uniform rotation and translation where this flow is steady, both conditions lead to the Beltrami equilibrium. The extreme of the enstrophy is the topological analog of the magnetic energy. The conjecture resembles the Kelvin's variational principle 16 for a two-dimensional fluid in the sense that considers the steady states in the same type of frame too.
Although the condition B = ٌ ϫ v B = ␥v B is determined by making zero this first variation of the enstrophy, the stability of the equilibrium is not guaranteed. It requires ␦ 2 ⌽Ն0 to assure a minimum value of ⌽. Nevertheless, we anticipate arguments in favor of stability. The variational principle gives all possible solutions of the Euler equations. When the helicity is conserved, the solution is the Beltrami flow for a given eigenvalue. In our case, the eigenvalue ␥ is determined previously by the Rankine flow. Then both, the helicity conservation, and the given ␥ eigenvalue determine the Beltrami flow downstream. However, some final features are determined ͑e.g., the helix pitch͒ by equilibrium parameters and boundary conditions. While Ͼ cc the final equilibrium is the Beltrami axisymmetric flow. Yet, when is in the range ͑ c , cc ͒ the final state is the rotating helical Beltrami flow.
On the other hand, the axisymmetric Beltrami flow corresponds to the stable axisymmetric steady solution. So, as Batchelor pointed out, axisymmetric waves of finite amplitude can arise. Nevertheless, the helical Beltrami perturbations lead to marginal stability, whenever the eigenvalue of the perturbations is the same ␥ eigenvalue as the basic Beltrami one. This marginal stability holds for infinitesimal amplitude perturbations but when reaches the critical value cc the finite amplitudes produce a Hopf bifurcation. The final state is a rotating wave with Beltrami flow structure of eigenvalue ␥. We conjecture that when a transition between steady states of different symmetries occur, stability against infinitesimal perturbations is guaranteed if the transition relates a Beltrami state to another of the same eigenvalue. In our case, for finite amplitude the dynamic is dominated by the equilibrium characteristics: the saddle-node bifurcation by Ͼ cc and the Hopf bifurcation by Ͻ cc . However, for both, the axisymmetric and the helical symmetric cases, when Ͼ c the final state is a flow with Beltrami field structure. Some authors 4 have discussed the reasons to assume a Beltrami flow. Our conjecture proposes that when a Rankine flow is assumed as the ideal flow in a tube, the most natural final state that follows the perturbations is a Beltrami flow.

ACKNOWLEDGMENTS
We are very grateful to Professor Fernando Minotti for his valuable comments. This paper is dedicated to the memory of our professor and guide Constantino Ferro Fontán, and also to the memory of our colleague and friend Aníbal Sicardi Schifino.

APPENDIX: VARIATIONAL DERIVATION OF THE EQUILIBRIUM BELTRAMI FLOW
Let ⌽ be the enstrophy defined as in the domain D with the conditions that n . = 0 and n . v =0 on ‫ץ‬D. And let ͐v . dV be the helicity with the condition ͵ v . dV = constant.

͑A2͒
Taking the first variation of ⌽ subject to Eq. ͑A2͒ and introducing the constant Lagrangian multiplier ␥, we obtain Integrating by parts and discarding the surface integrals we obtain ٌ ϫ = ␥. ͑A4͒ Equation ͑A4͒ corresponds to the vorticity field, = ␥v + ٌ, ٌ 2 = 0 in D, n . ٌ = 0 on ‫ץ‬ D.