MAGNETOHYDRODYNAMICS Vol. 38 (2002), No. 1-2, pp

Transcription

1 MAGNETOHYDRODYNAMICS Vol. 38 (22), No. 1-2, pp NON-STATIONARY SCREW FLOW IN A TOROIDAL CHANNEL: WAY TO A LABORATORY DYNAMO EXPERIMENT P. Frick, V. Noskov, S. Denisov, S. Khripchenko, D. Sokoloff, R. Stepanov, A. Sukhanovsky Institute of Continuous Media Mechanics, Korolyov 1, Perm, RUSSIA A possibility to implement a new type of dynamo experiment, namely, a nonstationary dynamo in a braked toroidal channel is considered. We suggest that a helical flow with the magnetic Reynolds number sufficient for screw dynamo action can develop in a toroidal channel with liquid sodium (about 1 kg) accelerated up to the frequency of 5 RPS and braked abruptly. The growth of the resulting magnetic field is expected to be sufficient to isolate dynamo effect. In this paper we present the results of hydrodynamic experiments with water prototypes. The spatial and temporal profiles of the helical flow excited in a braked channel by a diverter are investigated. Numerical simulation of the magnetic field evolution is performed for a helical flow with a profile extracted from the experimental data and scaled to the parameters of MHD apparatus. The data from the hydrodynamic experiment and the results of MHD numerical simulations are used to optimize the MHD apparatus under development. Introduction. At the very end of the XX century the long-standing experimental efforts to implement the self-excitation of magnetic fields by MHD dynamo was crowned with success. The MHD dynamo experiments performed in Riga [1] and Karlsruhe [2] are a real breakthrough in the development of experimental verification of the dynamo theory predictions. This verification seems to be of crucial importance for geophysics and astrophysics because dynamo is thought to be responsible for generation and maintenance of magnetic fields in cosmic bodies, e.g., the Sun and the Earth [3, 4]. Furthermore, the dynamo experiments allow one to construct geophysical and astrophysical dynamo models not only on the first principles basis, but also on the experimental results. The available dynamo experiments reproduce only a part of the aspects encountered in astrophysical dynamos. Therefore, many additional experiments are needed to gain a more penetrating insight into the MHD dynamo phenomenon [5]. The realization of MHD dynamo is an extremely complicated and expensive experiment, since this phenomenon is a threshold process requiring high magnetic Reynolds numbers. The lowest theoretical estimate [6] of the critical value, Rm =17.7, was obtained for the so-called Ponomarenko dynamo (see below), representing a screw motion of a rigidly rotating cylinder in an infinite conductive medium. However, even such a relatively low magnetic Reynolds number can be obtained by moving tons of liquid sodium and hundred kilowatts to drive the pumps. The first attempt to perform the MHD dynamo experiment by pumping liquid sodium through tubes of specific shape was made in Riga in 1987 [7]. In this experiment, a decrease in magnetic field decrement with increasing velocity was observed, but the generation threshold has not been reached yet. The second generation of dynamo experiments based on the same physical idea of pumping liquid sodium is currently undertaken by several scientific teams, and the first results have been recently obtained. In the Riga experiments the beginning of 143

2 the field amplification was observed in November 1999 [1]. A magnetic field generation with subsequent stabilization of the magnetic field was observed in the Karlsruhe experiment in December 1999 [2] (see also the corresponding papers in this issue). Thus it can be safely said that the possibility of dynamo action has been proved by laboratory experiments. The next objective of experimental dynamo studies is to extend the experimental scheme to mimic different types of dynamos in astrophysics. In particular, a quasistationary magnetic field as it occurs in the nonlinear stage of the Karlsruhe experiment can be compared with the Earth magnetic field. The magnetic field of the Sun existing in the form of propagating dynamo waves shows a somewhat different behaviour. Dynamo waves propagating as sharp magnetic fronts are discussed for galaxies. In addition, the nonlinear galactic dynamo gives an important example of nonstationary dynamo action because its time-scale is comparable with the life-time of the galaxy. The experimental realization of a propagating dynamo wave in a nonstationary dynamo seems to be the next milestone in the development of dynamo experiments. A new way of performing dynamo experiment without power consuming steady pumping has been recently suggested in [8]. The experiment is based on the idea that a nonstationary screw flow can exist for a limited time in a rapidly rotating torus with liquid sodium if the torus shell is abruptly braked. All estimates were made for a toroidal channel (radius of the cross-section r =.1m, median radius of the torus R =.5 m) filled with liquid sodium of about 1 kg. A relatively weak engine of the power of a dozen kilowatts proves to be sufficient to slowly accelerate this torus up to the frequency f = 5 RPS. Apart from a transient regime the liquid metal rotates more or less rigidly and accumulates enough energy so that, for a limited time after the abrupt braking of the wheel, a strong flow is sustained. The brake system must absorb about 1 6 J of the rotational energy of the solid part of the apparatus for.1 s, i.e., about 1 7 W, which is comparable with the power of a heavy aircraft brake. In principle, this experimental scheme allows us to obtain large magnetic Reynolds numbers with a relatively low power consumption. In the limiting case of the instantaneous braking, the magnetic Reynolds number Rm defined as Rm = σµµ U r =2πσµµ fr R, (1) can reach at the first moment the value of approximately 1, with upper estimation of characteristic velocity U =2πfR, the relative magnetic permeability µ = 1 and the sodium conductivity σ. The corresponding hydrodynamic Reynolds number Re = U r /ν is about 1 7,whereνis the kinematic viscosity. The transformation of rotation into a screw motion by a kind of diverter (see below) leads to some energy losses and a corresponding reduction of Rm and Re. The Ponomarenko dynamo, which forms the basis for many schemes of dynamo experiments, is a well studied topic in the dynamo theory (see, e.g., [6, 9, 1, 11, 12]). However, the proposed experimental scheme leads to some new aspects, which have not been studied previously. These are the mechanical constrains imposed on the rotating shell, which require its careful optimization. In addition, the nonstationary nature of the flow and the toroidal shape of the cavity should be considered in the context of the Ponomarenko dynamo. The aim of this paper is to check the feasibility of nonstationary dynamo experiment and to provide answers to the arising questions. At the present stage of research we are interested in finding estimates for the basic parameters of the MHD apparatus required for the dynamo experiment. Our estimations are based on the integral measurements of hydrodynamic flow characteristics for a reduced 144

3 water model, on simple hydrodynamic models allowing to extrapolate the obtained results to the real scale of MHD apparatus and on relatively simple dynamo simulations for cylindrical tubes. The outline of the paper is as follows. A brief summary of the screw dynamo properties is given in Sec. 1. In Sec. 2 the linear dynamo problem for realistic screw flows in a cylindrical tube with the finite wall is investigated and the optimal parameters of the toroidal channel for the dynamo apparatus are discussed. Sec. 3 describes the experimental water apparatus. In Sec. 4 the experimental results for a nonstationary (but non-screw) flow in the braked torus are presented. The experimental investigation of the screw flow in the toroidal channel is described and a simple model for the nonstationary screw flow is developed in Sec. 5. The results of numerical simulations performed for the MHD device are given in Sec. 6. The nonstationary dynamo problem is discussed in Sec. 7, and a general discussion is presented in Sec Screw dynamo. A simple example of a helical flow, which can be a dynamo if the magnetic Reynolds number is sufficiently high, was suggested by Ponomarenko as early as 1973; a marginally stable magnetic field in this flow was considered by Lortz even in The flow considered by Ponomarenko can be presented as a superposition of differential rotation ω andmotionwithavelocity v z along the axis z of differential rotation (below we use the cylindrical coordinate system r, φ, z). Being interested in an explicit analytical solution, Ponomarenko considered a particular form of ω and v z, which do not vanish for r r,and vanish for r>r. The dynamo action is related here to the velocity jump at r = r. Of course, it looks plausible that main features of this particular example remain valid provided the jump is smoothed out. On the one hand, however, an explicit dispersion relation for the dynamo growth rate obtained by Ponomarenko occurs quite complicated and, on the other hand, a rigidly rotating jet with a velocity jump on the boundary does not look as a most general swirling flow, so the Ponomarenko example was developed to include rotating jets with various velocity profiles. It is important to stress a fine distinction between the general case and the particular example suggested by Ponomarenko: the first one is referred to as a screw dynamo. A dynamo in a rotating jet of a conducting fluid is an example of the so-called slow dynamos [13], i.e., its self-excitation cycle involves a slow diffusive process. To be more precise, the radial magnetic field component H r, which is affected by differential rotation ω(r), gives rise to the azimuthal magnetic field component H φ, and magnetic diffusion produces H r from H φ, which closes the self-excitation cycle. The last link of this chain is possible because the diffusion of the vector magnetic field is different from the diffusive transfer of the scalar field. In particular, the diffusive transfer mixes the components H r and H φ. The magnetic field transfer in a screw flow V(r) =(,rω(r),v z (r)) is governed by the induction equation H r t H φ t + ω H r φ + v H r z z + ω H φ φ + v H φ z z = = 1 [( 1r ) Rm 2 1 Rm H r 2 ] H φ r 2 φ [( 1r 2 ) H φ + 2 r 2 H r φ, (2) ] + r ω r H r. (3) The component H z can be derived from the solenoidality condition. The whole length is measured in units of the radius r, the longitudinal velocity in units of U = v z (), the angular velocity in units of U /r and the time in units of r /U. 145

4 The properties of the growing solution H(r, φ, z, t) =h(r)exp(γt+ikz +imφ) can be estimated from Eqs. (2), (3) as follows [11]. We suppose that the radial magnetic field component is regenerated by the diffusion term 2Rm 1 r 2 H φ / φ Rm 1 mh r, which in this case should be comparable with H r / t = γh r. Since the azimuthal magnetic field component is regenerated by differential rotation, one may expect that rh r ω/ r H φ / t, because for an eigensolution H φ / t = γh φ, these two relations give for a mode with m = O(Rm ) the estimate Re γ Rm 1/2,whereRe stands for the real part (growth rate). Of course, one must show that other terms participating in Eqs. (2, 3) do not destroy this balance. As it appears from the following analysis, one has to use v z to balance other terms of our equations, so a screw dynamo is impossible without v z. Dynamo action is a self-excitation of a magnetic field with no contribution of external currents, so that the dynamo generated magnetic field should decay at spatial infinity and be regular on the symmetry axis (here we introduce the boundary conditions into the analysis). Correspondingly, the diffusive terms in Eqs. (2, 3) should be comparable with the generation terms, that gives l Rm 1/4,where l is the radial scale of the growing mode. In addition, the advective terms like ω H r / φ + v z H r / z should not exceed the generation terms, i.e., they should vanish in two leading orders (O(1) = Rm and Rm 1/4 ). From this follows a condition ω(r )m + v z (r )k =, relating the wavenumbers m and k, and a condition relating the oscillation rate Im γ and r (Im stands for the imaginary part). Note that here v z is explicitly used. As a result, an asymptotic representation of the growing modes of screw dynamo in the form of dynamo waves propagating along cylinders has been obtained [11]. The qualitative estimates of this paragraph can be supported by the results of numerical simulations (see [14]). The above analysis is valid for the screw motion with smooth ω and v z, while the original Ponomarenko dynamo exploits its discontinuous profiles. This results in slightly different scaling laws for the Ponomarenko dynamo in comparison with the general screw dynamo. Another set of scaling laws appears in the case of short wave solution with very large m and k [9]. However, the experience of numerical investigation of screw dynamos [12] shows that the scaling Re γ Rm 2q, l Rm q, H r H φ, Rm 2 (4) with q.25 is valid for reasonable velocity profiles with comparable rotation and propagation velocity (v z ωr ). By performing an asymptotic decomposition of the growth rate up to the second nonvanishing term, Re γ(rm) = Γ 1 Rm 1/2 Γ 2 Rm 1/4, from the equation Re γ(rm ) = we can obtain an asymptotic estimate of Rm. It is apparent that these screw dynamo results, being rather academic, need to be verified under special experimental conditions. Extensive numerical investigations of this type were undertaken by A. Gailitis and his collaborators (see, e.g., [6]) for the Ponomarenko dynamo. They demonstrated that Rm estimate as well as scalings (4) are stable in the case of screw motions in a finite tube. Of course, each specific experimental device requires investigation of particular excitation conditions and configuration of the excited magnetic field (see below). In addition to a quantitative modification of the excitation condition due to the boundary effects, a new physical effect has been demonstrated in [15]. A screw dynamo usually appears as a convective rather than absolute instability, i.e., starting from a spatially bounded initial condition there occurs a magnetized blob with a growing magnetic field propagating with the fluid motion. This blob is carried out from the dynamo region of a size L in a time L/U, so that the 146

5 magnetic field in the blob is expected to grow by a factor exp(re γl/u ) only. Under some condition a non-propagating magnetic field can be generated behind the blob (absolute instability), that, however, requires a fine tuning of the velocity profile. In this respect, a toroidal channel looks much more attractive than a cylindrical tube because the magnetized blob does not leave the channel. In the context of the above theory the screw flow is treated as a laminar one although it is apparent that a screw flow with high Reynolds numbers should have a turbulent component. It would appear natural that the screw dynamo theory is applicable to this case provided that the magnetic field is averaged over turbulent pulsations and the turbulent contribution into magnetic diffusivity is considered [16]. However, none of the available experimental studies has taken into account the role of turbulent diffusivity. In Sec. 8 we give an estimate of the turbulent contribution of magnetic diffusivity based on the results obtained with a water model. An attempt to generalize the asymptotic theory of screw motion to the nonlinear situation has been made in [1]. However, it is still unclear which kind of the balance results in saturation of the screw dynamo. The simplest estimate of the magnetic field strength, which can be obtained for screw dynamo, is the equipartition estimate H 2 /8π = ρu 2 /2. Taking for liquid sodium ρ =.93 g/cm 3 and v =2πR f with radius R =.5m and frequency f = 5 RPS, one obtains H G. A more sophisticated estimate could be derived on the basis that the back action of the magnetic field on the sodium motion should not involve a change of the main flow shape, but only a destruction of fine tuning in the asymptotic balance. The asymptotic expansion up to the third order provides an estimate of the steady magnetic field, H s Rm 1/2 H. Reasoning from the fact that the Lorenz force contains first spatial derivatives of the magnetic field, we obtain H s Rm 3/4 H. For Rm = 1 we get H s H /3. Hence, one can expect an amplification of the magnetic field up to the level of 1 3 G. This estimate, when compared with 2 G achieved in the Karlsruhe experiment [5], seems to be still too optimistic. Let us estimate the dynamo time-scale τ. From Eq. (4) we obtain τ = r /U Rm 1/2 so that for the parameters accepted above τ =.3 s. If the screw flow is sustained during.5 s, we expect to have about 15 dynamo time-scales to observe the magnetic field self-excitation. On the other hand, about 13 dynamo time-scales are required to amplify the Earth magnetic field (.5 G) taken as a seed magnetic field up to H. Since all the estimates mentioned deliver us no more than the order of magnitude of relevant quantities, the experiment clearly needs a detailed quantitative verification as presented below. 2. Screw dynamo in real flows in a conducting channel. We first consider a linear dynamo problem for the screw flow in a cylindrical conducting tube with the finite wall thickness d = r 1 r (the inner radius of the tube is r, the outer r 1 ), surrounded by the air. The ratio of the wall conductivity to that of the fluid is denoted by σ 1. The magnetic field in the outer domain can be represented according to H = P by a potential P satisfying P =. This is true only if the total current along the cylinder vanishes. Writing P = p(r)exp(γt + imφ + ikz) weobtain p + 1 ( ) m 2 r p r 2 + k2 p =. (5) The solution of (5), which remains finite as r,readsp(r)=ch m (1) (i k r), where H m (1) is the Hankel function of order m and C is the constant. The continuity 147

6 of H across the outer boundary of the conducting tube results in the boundary conditions h r (r 1 )mh m (1) (i k r 1) = h φ (r 1 ) k r 1 H m (1) (i k r 1), (6) ( k h r (r 1 )+r 1 h 2 r (r r 2 ) 1 1) = i m +m h φ (r 1 ). (7) The regularity of H at the axis of the cylinder leads to h r () = h φ () =, for m = ±1, h r () = h φ () =, for m ±1. (8) The field generation occurs if Re γ>. Numerical solutions of the eigenvalue problem were calculated using the QRalgorithm and up to 8 grid points. To check the code, we compare our Rm with the known results obtained for an infinitely thick, highly conducting or nearly insulating wall (we took for that d =.3, and σ 1 =.1 or σ 1 = 1). Using the velocity profiles given in [12], we reproduced the corresponding Rm with an accuracy of about 5%. The analytical solutions [6] were reproduced with an accuracy <, 1%. Two kinds of profiles for v z were used in our further simulations (see Fig. 1). First, a family of functions v z (r) = cosh(ξ) cosh(rξ) cosh(ξ) cosh() (9) was considered, which describes the whole spectrum of profiles from the Poiseille solution (ξ = 1) up to a rigid-body motion (ξ ; in practice, for ξ = 1, d =5andσ 1 =1Rm differs from that of the Ponomarenko dynamo by less than, 1%). Second, the logarithmic profile [17] ( ) k2 v z (r) =1 5.75k 1 ln (1) k 2 r was used, where k 1 and k 2 are defined by the relations k 1 (2.5ln(k 1 Re) + 5.5) = 1andv z (1) =. Expression (1) is considered to be a correct description of the mean velocity profile for a turbulent flow with the hydrodynamical Reynolds number being Re > 1 5. In the dynamo experiment under discussion the Reynolds numbers are expected to be as high as vz(r) r Fig. 1. Velocity profiles used in simulations. Thick curves show the profiles (9): ξ = 1 (dotted), ξ =18 (dashed), ξ = 1 (solid). Thin curves show profiles (1): Re = 1 6 (solid), Re = 1 7 (dashed).

7 Rm ξ Fig. 2. Rm versus the velocity parameter ξ. The neutral curves for k = 1, m = 1, are calculated for different thickness of the wall of the tube: d = (triangle), d =.1 (romb), d =.3(star),d= 1(box). For the angular velocity we use the relation ω(r) =χv z (r). (11) In the limit of solid body rotation, 2πχ 1 is the screw pitch. Our water experiments described below confirm the applicability of the logarithmic approximation for toroidal channels. On the other hand, the experimental data concerning the velocity profile do not allow us to define exactly the structure of the stream. Moreover, the logarithmic profile (1) does not describe the velocity in the vicinity of the wall, where it should be corrected to satisfy the boundary condition. Therefore, we prefer to consider the family of profiles described by (9). The wall thickness is another critical parameter to be specified. Fig. 2 shows Rm versus the flow parameter ξ for three different wall thicknesses. The thickness d is given in units of the radius r. For a thin conducting wall (d =.1), the increase of parameter ξ leads to a monotonous increase of Rm, whereas for a thick conducting wall (e.g., the case d = 1 in Fig. 2), the larger is ξ (quasi-solid motion), the lower is Rm. For d =.3, the neutral curve shows a minimum at ξ 5. The excitation threshold Rm depends on the conductivity of the wall. The results for profile (9) at ξ = 18 and various wall thicknesses are presented in Fig. 3 by solid lines. The Ponomarenko dynamo cannot exist in the limit of an ideally conducting medium [6] and the neutral curve Rm (σ 1 ) has a smooth minimum with a pronounced increase in Rm for σ 1 > 1. For smooth velocity profiles, the position of the minimum is shifted into the domain of moderate conductivity σ 1. For d =.3, the minimum arises at σ 1 =3.5. From the practical viewpoint, a low threshold at a relatively thin wall is of a particular interest. We suggest d =.15 as an optimal value. Then the minimum of the neutral curve Rm =27 corresponds to σ 1 = 5.5. The ratio of copper to sodium conductivities yields approximately a similar value. Similar results with a slightly higher optimal value 7 Fig. 3. Rm versus σ 1. Neutral curves for different wall thicknesses for velocity profiles (9) with ξ =18 (solid curves) and (1) for Re = (dashed curves): d =.3 (triangles), d =.15 (stars), d =.5 (boxes). 6 Rm σ 1 149

8 Fig. 4. Apparatus for water experiment: 1 plexiglass cylinder, 2 toroidal channel, 3 brakes, 4 braking disc, 5 braking control, 6 electromotor, 7 tachometer, 8 lighting, 9 computer, 1 diverter, 11 free rotating flat blade, 12 optical fiber, 13 optical probe, 14 video camera. for the critical magnetic Reynolds number (Rm = 3) were obtained for the velocity profile (1). 3. Hydrodynamical experiment: Apparatus. The density and viscosity of liquid sodium are very close to those of water. This fact simplifies the experimental study of hydrodynamical characteristics of the MHD screw flow to be examined. A general scheme of the apparatus for water experiments is shown in Fig. 4. The toroidal channel 2 for a water flow was cut through a plexiglass cylinder 1, composed of two halves and installed on a hub of a car wheel incorporating the braking system 3 5. The hub was rotated by an electromotor 6 (2 kw) with a frequency up to 5 RPS. The braking system allows us to vary the braking force, i.e., the amplitude and the time of deceleration. The angular velocity of the channel was measured by a tachometer 7. Two channels (called hereafter channel A and channel B) were used (Fig. 5). Their parameters as well as the inferred parameters of the MHD apparatus are given in Table 1. The screw flow is created by the 6-blades (channel A) or 8-blades (channel B) diverters shown in Fig. 6. The typical braking time was about.2.5 s and the typical time of subsequent decay about 1 2 s. The flow was visualized by polystyrene particles sus- B A Fig. 5. Two channels: A (small) and B (large), used in water experiments. Fig blade flow diverters. The left diverter is fitted with two flat freely rotating blades, which enable one to measure azimuthal velocities upstream and downstream the diverter. 15

9 Table 1. Model A Model B MHD apparatus (estimations) R,m r,m Mass of the channel, kg Mass of fluid, kg Inertia moment of the channel, kg m Inertia moment of fluid, kg m Frequency of rotation, RPS Maximal linear velocity, m/s Nominal Re Effective Re Effective Rm 4 Minimal braking time, s Energy of rotation, J Dissipated power, W pended in a weak NaCl solution. The size of particles was about 2 mm because we are interested in the visualization of a large-scale flow, and smaller particles would follow the small-scale flow perturbations. The large-scale structure of the screw flow is illustrated in Fig. 7a by a photograph taken with a relatively long exposure (.1 s) at a later stage of the evolution when light dispersion does not hinder visualization of the tracks. The photograph in Fig. 7b is taken at the same stage of flow evolution using kalliroscopic fluids for visualization. Then the small-scale turbulent structure can be observed well ehough. Observations were recorded by a video camera (25 frames per second). We studied two velocity components defined in local cylindrical coordinates related to the axis of the channel. The longitudinal component of velocity v z (along the axis of the channel) was evaluated by measuring the corresponding projection of Fig. 7. Screw flow in channel A at the later stage of evolution (1.5 s after full stop). On the left panel, the polystyrene particles display a large-scale screw structure of the flow. On the right panel, small kalliroscopic particles show a small-scale structure of the turbulent flow. Both snapshots are taken for the same moment of evolution. One diverter is used, which can be seen in the lower part of the channel as a light body. 151

10 track lengths of polystyrene particles. To evaluate the angular velocity v ϕ,twoflat blades (in front and behind the diverter) were used, which freely rotated around the axis of the diverter (Fig. 5). The revolutions of the blade in the channel were counted by the optical probe 13 placed opposite the cylinder axis. Two optical fibers 12 incorporated into the plexiglass body transferred the light impulses from the blades to the counter. 4. Flow in a braked torus. The first stage of the experiments concerns the flow driven in the braked torus without diverters. This flow being nonscrew is unable to give rise to dynamo action. However, before proceeding to the discussion on the role of diverters, we must be sure that the flow exists for a sufficiently long time after braking. The flow is supposed to be turbulent because the corresponding Reynolds number is very high (see the Table 1). The turbulent flows in cylindrical tubes at large Reynolds numbers are a well studied topic [17]. Assuming that a toroidal channel can be considered in the first order approximation as a cylindrical one (approximation of a thin torus), we apply the available results to cylindrical tubes to construct a simple model, which describes the velocity evolution in the toroidal channel. The mean velocity U of inertial motion in a smooth tube of radius r can be related to the tangential wall stress τ by du dt = R dω dt 2τ r ρ, (12) where the velocity U is measured in the frame of reference moving with the channel walls, Ω is the angular velocity of the channel, ρ is the mass density of the fluid, and τ = ρv 2. Here, v is the so-called dynamic velocity. For Re > 1 5, the dynamic velocity is related to the mean velocity by an empirical law [17] ( U = v 2.5ln r ) v (13) ν Using Eqs (12) and (13), we can calculate the time evolution of the mean flow velocity in the braked torus. Some examples of numerical simulations based on Eqs (12, 13) are given in Fig. 8. The panel (a) shows the results of numerical simulations done for two U,m/s c ccccc c c ccc c s s s s s c c s s s s s s s s c s s s c c c c s s s s s s s s c c c c c c s s s s t,s (a) s c c c c c c U,m/s (b) t,s Fig. 8. Flow evolution in a braked torus mean velocity U versus time. Experiments (points) and results of simulation (lines). (a) channel A, f = 5 RPS. Two regimes of braking are shown: white points braking time T b =.25 s, black points T b =.7s. (b) simulation for the flow in the channel with R =.5m, r =.1m and f =5RPS. 152

11 different braking times using the experimental data for the angular velocity Ω of the channel. Both curves fit well the values of the mean velocity U measured in the experiment. The higher are the Reynolds numbers, the better is the agreement between the experimental data and approximation Eqs (12, 13). Fig. 8b shows the results of a simulations made for the nonscrew flow in the channel with the parameters of the proposed MHD device (last column in Table 1). The family of curves describing the flow evolution were obtained for the same initial angular velocity and different braking times to illustrate the fact that the time of a supercritical flow can be markedly varied by changing the acceleration regime. In these simulations the channel velocity was assumed to be defined by a purely linear law, which results in a well pronounced break of the curves at the end of braking. Although an increase of the braking time causes a decrease in the maximal value of velocity, it extends the time interval in which the velocity is higher than the critical value (about 3 m/s in the given case). The velocity profile is an essential point in determining Rm (see Sec. 2). The model (12, 13) implies that the profile describing a stationary turbulent flow in a cylindrical tube is established in the essentially nonstationary flow under discussion. Measuring the length of the tracks for a given coordinate y on the projection of the channel, one averages the results corresponding to different coordinates r. To reconstruct the radial velocity distribution, we apply the technique commonly used in optics to axisymmetric object reconstruction from its projection [18]. In the simplest version, this technique involves a step-like distribution of the unknown function f(r) (for the ring number n, f(r) =f n ). Then, calculating the beam length in each zone, one obtains a linear algebraic system for recalculation of the values of f n. The case under consideration is more complicated than the optical problem because the particles are distributed inhomogeneously along the radius r. We firstly reconstruct the particle density, using the above algorithm, and then accomplish the reconstruction of the velocity profile with the same algorithm taking into account the particle distribution. Noting that due to the toroidal geometry of the channel the velocity distribution slightly differs from the axisymmetric one, we introduce a ϕ-dependence in the form f n (ϕ) =f n () + f n sin ϕ. Because of the nonstationary nature of the problem, the statistically valid results for each moment of evolution can be obtained only by averaging over a set of realizations. Moreover, an essential variation of velocity during the decay causes certain difficulties with the choice of universal exposure time: the particle tracks are too long and knotted at the earlier stage of evolution, and at the later stages they appear to be too short for taking measurements. Thus it is necessary to make separate recordings for each period of evolution, that multiplies the number of realizations for every operating regime. Fig. 9a shows the reconstructed velocity profile in the channel B at the instant t =.76 s after the start of the experiment shown in Fig. 8a. Hereyis the distance from the axis of the channel (negative values correspond to the direction towards the wheel axis). The cross section of the channel was divided in 1 zones. The observations of the particle distribution have shown that the particles practically do not penetrate the boundary layer, and all points describe the velocity inside the flow core, where the profile is practically flat. The general inclination reflects the toroidal geometry of the flow (the outer liquid particles move faster than the inner ones). In addition, this figure displays the theoretical profile (9) used in the simulations of Sec. 2 at ξ = 18 (solid line). The profile (1) for the actual Reynolds number value is represented by a dotted line. Both curves are determined by minimizing the r.m.s. deviation from the experimental points. Fig. 9(a) clearly demonstrates the advantage of the profile (9) (r.m.s.=.9 m/s in contrast 153

12 U, m/s 8 6 U, m/s (a).5 (b) y, mm y, mm Fig. 9. Velocity profile of a nonscrew (a) and a screw (b) flows. Circles experimental data, solid line profile (9) with ξ = 18, dotted line profile (1). to r.m.s.=1.7 m/s for (1)), which will be used as a basic profile in all simulations that follow. The reconstruction of the velocity profile of a helical flow (Fig. 9(b)) shows that the boundary layer becomes thinner. The local maximum of U at the center of the channel reflects the existence of a hole in the center of the diverter (Fig. 6). Now we can estimate the turbulent contribution to magnetic diffusivity in the screw flow in the experimental apparatus. First of all, in the case of high conductivity and low viscosity the turbulent contribution β to magnetic diffusivity is equal to the turbulent contribution ν T to kinematic diffusivity. The latter can be estimated from the properties of the turbulent boundary layer. According to [17], ν T =.16λU, whereuis the mean velocity of a turbulent flow in a tube and λ is the width of the turbulent boundary layer. The experiments with water models demonstrate that λ does not exceed 1/1 of r. Scaling this estimate to the sodium apparatus, we obtain λ 1 2 m, so for U = 1 m/s one gets β 1 2 m 2 /s, while the coefficient of molecular magnetic diffusion for liquid sodium is.1m 2 /s. Thus, the contribution of turbulence into the magnetic diffusion coefficient seems to be negligible. 5. Hydrodynamics of a screw flow. The screw flow is generated by diverters (Fig. 6) installed in the channel. Each diverter introduces the azimuthal velocity and influences the velocity profile, thus producing an additional total flow resistance. In the case of a slow laminar inertial flow, the diverter dams the channel cross-section and drives a screw flow with helicity approaching the order value of U 2 /r. At a high value of initial velocity and strong braking, the screw flow evolution is governed by a complicated law. Though during the whole time of braking the fluid goes several times around the channel, a large scale screw flow is observed only in a limited area behind the diverter (we refer this area as a screw zone ). The blade arranged upstream the diverter does not rotate until braking is completed. This result holds even for a very slow initial channel rotation (a few RPS). The destruction of a large scale screw flow in the rotated channel can be explained by conservation of the angular momentum (or by the Coriolis force, if one prefers to describe the evolution in the frame related to the channel). Note that the destruction of the large scale screw flow does not result in total dissipation of the helicity introduced in the flow by the diverter. It can be expected that helicity should be transferred from the scale of channel cross-section r to smaller scales. The evolution of the screw flow was studied in details in the channel B. We started with one diverter and a relatively soft regime of braking (T b =.7s). The 154

13 Fig. 1. Propagation of the screw zone. (left) t =.8 s prior to full stop, (middle) at the instant of full stop, (right) t =.8 s after full stop. T b =.7s,exposureis1/25 s. screw zone behind the diverter is a highly nonequilibrium structure, the extent of which can be characterized by an angle δ (see Fig. 1). During braking δ fluctuates about a mean value, which can be estimated in angular units as <δ>=9 12. After the stop of the channel the screw zone propagates with the mean velocity of the flow, and at a time t 1 T b +3πR /2U the screw motion spreads across the whole channel. A typical time evolution of the screw motion in the channel is shown in Fig. 11. The angular velocity of the blade installed behind (downstream) the diverter is denoted by black circles and the angular velocity of the blade installed in front (upstream) of the diverter by white circles. Three stages can be distinguished: screw flow propagation, main stage, and decay (in Fig. 11 separated by vertical lines). The main stage is defined as the interval, in which the energy of the rotational motion upstream the diverter exceeds half of its maximal value. The log-log plot (Fig. 11) demonstrates that at the third stage the decay follows a power law. A thin solid line indicates the slope of 1.7, that corresponds to energy evolution E(t) t The dependence of flow evolution on the braking intensity is illustrated in Fig. 12, which shows the evolution of the upstream blade for four different braking times T b. In contrast to the nonscrew flow (compare with Fig. 8), the reduction of the braking time does not cause an increase in the life-time of the screw flow with Rm > Rm. Slower braking corresponds to shorter life-time of the screw flow in the channel. It seems that the first stage of the flow evolution can be reduced by installing additional diverters in the channel. Fig. 13a shows the evolution of the screw motion in the channel B for the braking time being T b =.39 s and Fig. 11. Evolution of angular velocities of the downstream (full circles) and upstream (open circles) blades, T b =.36 s. ω, 1/s t, s 155

14 14 12 ω, 1/s Fig. 12. Angular velocity of the upstream blade for different braking times: T b =.2s (crosses,; T b =.5s (full circles), T b =.8s (open circles), T b =1.3s (triangles) t, s several numbers N of the diverters. The upstream blade (white circles) moves very slowly up to a moment t = t 1 (in the case of one diverter t 3 =.5s) when the screw zone approaches the entrance of the next diverter. It remains at rest throughout the braking time, when two and three diverters are used, and starts to move soon after the beginning of braking, if four diverters are installed in 2 a 2 b 1 N=1 1 N=1 2 2 ω, s 1 1 N=2 ω,s 1 1 N= N=3 1 N= N=4 1 N= t,s t,s Fig. 13. Evolution of angular velocities of the downstream (full circles) and upstream (open circles) blades in the channel B with different numbers N. (a) T b =.39 s, (b) T b =.19 s. Lines represent the corresponding angular velocities obtained from numerical simulations. 156

15 the channel. This fact contributes to the above estimate of δ obtained from the track observations. Fig. 13b shows similar measurements performed for a shorter braking time (T b =.19s). Note that the maximum angular velocity at the exit of the diverter decreases essentially, if more diverters are installed in the channel (ω = 29 s 1 for one diverter and ω = 17 s 1 for four diverters). The maximal angular velocity at the diverter entrance remains practically the same for any number of diverters. The simple model introduced in Sec. 4 for a nonscrew flow in the braked torus can be extended for a flow in a channel with diverters. Let W (z,t) <ω>r/2 2 be the angular momentum averaged over the given channel cross-section z = R θ corresponding to an angle θ (θ = is the diverter position), <ω>is the mean angular velocity of the fluid, which corresponds to the rotation velocity of a flat blade. W propagates due to the mean velocity U and decays due to friction at the wall. Friction losses for W are assumed to be proportional to those for the mean velocity U. Additional decay of rotation is caused by the action of the Coriolis force during the braking period and is proportional to W Ω. As a result, we get the following equation W t + U W ( ) 2 z = µ 2v W CWΩ, (14) Ur where µ and C are the dimensionless empirical coefficients. We find that µ =1. and C = 2. give reasonable accord with the experimental data. Two additional terms, which describe the action of the diverter on the flow, are introduced into Eq. (12): the first term defines the frontal resistance and the second one characterizes the transformation of U into W : du dt = R dω dt 2v2 η NU2 r L N ( W 2 r 2L () W 2 (L) ), (15) where L = 2πR is the channel circumference and η denotes a dimensionless empirical coefficient (we use η =.1). The action of the diverter on W is given by the boundary condition W () = W (L) +ζ[ur /2 W(L)] because for our diverter the inclination of velocity line in the swirling flow near the tube wall is about 45 and the empirical coefficient ζ =.75 describes the efficiency of the diverter. Fig. 13 illustrates the results of flow evolution simulation Eqs (14), (15). 6. Parameter estimates for the MHD apparatus. The proposed model can be used to estimate the characteristics of a nonstationary MHD flow in the MHD apparatus. A numerical simulation was carried out resting upon Eqs (14, 15) for a channel with two diverters (N =2),R =.5m, r =.1m and different braking times T b. Fig. 13 shows that after full stop of the torus a regime with a relatively stable value of velocity is established within a time designated by T eff. This regime corresponds approximately to the second stage shown in Fig. 11. This regime can be characterized by the effective magnetic Reynolds number Rm eff = σµ r 2U2 +4W(L) 2. (16) The values of Rm eff and T eff were calculated with the given T b for each simulation. The results are presented in log-log scale in Fig. 14, which shows that the dependence of both characteristics on the braking time obeys the power laws. In the case of two diverters the corresponding fits are Rm eff =7.5T 2/3 b, T eff =.6T 2/3 b s. (17) 157

16 Fig. 14. Characteristics of a nonstationary MHD flow in a braked torus with two diverters. Rm eff (open circles) and T eff (full circles) versus T b (from numerical simulations). Solid lines show the fits (18). The results of water experiment are illustrated with triangles, which indicate the artificial Rm (defined by the velocity of water and magnetic diffusivity of sodium) obtained for the model B with various braking times T b. The dotted line traces the slope 2/3. u e u u e u e u e u u e e u e 1 e.1 e Rm u e eæ T eæ, s T b, s These fits are shown in Fig. 14 by solid lines and allow us to conclude that the product Rm eff T eff for a given channel and initial energy (channel rotation rate) is constant. The same simulations for one diverter have led to Rm eff =8.6T 2/3 b, T eff =.9T 2/3 b s. (18) The system displays the same power-law dependence on the braking time but with different prefactors. Thus, one diverter slightly improves Rm eff and markedly increases the duration of the main stage. The product Rm eff T eff for one diverter is 72% larger than that for two diverters, that provides a strong argument for using only one diverter in the MHD apparatus. To check this conclusion, we have experimentally investigated the dependence of Rm eff on the braking time in a water model. The corresponding results are also shown in Fig. 14 and support the validity of the predicted power law (18). The choice of parameters for the MHD-apparatus is governed by the wish to maximize Rm eff for realistic values of a sodium volume and the inertia momentum of the solid wall of a channel. The inertia momentum is directly related to the energy accumulated by the wall, thar requires a corresponding braking system. Fig. 15 illustrates this optimization problem. Three families of lines show Rm eff (calculated for the case of f = 5 RPS and T b =.1s) on the plane (R,r ), the volume of liquid sodium (V R r 2 ) and the energy accumulated by a solid shell of the channel (E Rr 5 ). 2 Thus R =.4m and r =.12 m can be R, m r,m Fig. 15. Channel optimization: Rm eff (short-dashed lines), the volume of liquid sodium in liters (solid lines, the values are given above the panel) and the energy of rotation in 1 5 J(longdashed lines, values are given on the right of the panel) in dependence on large radius R and small radius r of the channel.

17 .4 Fig. 16. Dependence of γ = Re γ on k with Rm = 2 (box), Rm = 4 (romb), Rm = 6 (asterisk), Rm = 8 (cross), Rm = 12 (triangle). γ * k reasonably proposed as the optimal values for the channel geometry. Then we have Rm eff 4, V.115 m 3 and E 1 6 J. The parameter estimates for the MHD apparatus for the case under discussion are presented in the third column of Table Nonstationary dynamo problem. In the examined case some problems which usually are not discussed in the context of screw dynamo, deserve a more careful consideration because the dynamo process develops during a limited time under varying hydrodynamic and magnetic Reynolds numbers. In this section we present some numerical results concerning the dynamo action during the screw flow decay. It is known that the screw dynamo process is sensitive to χ and the growth rate Re γ depends on the value of the magnetic Reynolds number Rm. Fig. 16 shows a variation of Re γ with the wave number for various values of Rm. The simulations are done for the velocity profile (9) with ξ = 18, d =.15 and σ 1 =5.7. The lowest curve corresponds to a subcritical regime, i.e., Rm < Rm. The maximum of this curve (which corresponds to the slowest decay, but to the fastest growth in case of Rm > Rm ), is located at k =.8. For Rm = 4, the fastest growth of the magnetic field corresponds to k =.88. Further growth of Rm is characterized by a slow trend of the maximum to the value k = 1, and Re γ varies from.2 to.4. It means that the expected time of supercritical state at the entrance of the diverter (.2s) is only about 5 1 times longer than the typical growth time. In order to examine the time evolution of the magnetic field in the nonstationary screw flow, we consider Eqs (2, 3) with a time-dependent velocity V(r, t), taking H(r, φ, z, t) =h(r, t)exp(ikz + imφ) and applying initial conditions h r (r, ) = h r (r),h φ(r, ) = h φ (r). Three factors should be taken into account: the change of the mean velocity U, the variation of the flow profile and the variation of χ. Thus v z (r, t) = U(t)ṽ z (r, ξ(t)) (19) ω(r, t) = χ(t)v z (r, t), (2) where the function ṽ z (r, ξ(t)) is determined by the velocity profiles (9) with ξ depending on time. We used the experimental data concerning the time evolution of the mean longitudinal velocity and that of the mean angular velocity over the cross-section, rescaled to the MHD device using (18). It allows us to define χ(t). We assume also that ξ develops in time as ξ(t) =ξ e t +1 with ξ = 18. The corresponding time-dependence of the flow characteristics is presented in Fig. 17. Fig. 18 shows the results of simulations performed for R =.4m,r =.12 m, d =.15, σ 1 =5.5, f =5RPS,T b =.2 s. The simulations were done for two different scenarios of velocity evolution: the first one corresponds to the velocity at 159

18 U( t), ω(,t)r, m/s t, s Fig. 17. Time-dependence of U(t) (thin solid line) and ω(,t)r (thick solid line). The dashed line corresponds to the level of U for the generation threshold Rm = 27. the exit of the diverter (left), the second to the velocity evolution at the entrance of the diverter (right). It should be emphasized that our simulations do not take into account the variation of the velocity field along the channel axis z. Therefore, we assume that the angular velocity throughout the channel is similar to that measured at the entrance or at the exit of the diverter (the longitudinal velocity is really homogeneous along the channel). Fig. 18 shows the evolution of energy for the azimuthal magnetic field component near the channel wall (the azimuthal component is generated mainly in the shear flow near the wall and the maximum of the radial component is on the channel axis). The left panel shows the behavior of the magnetic field in the first case, that corresponds to the point in the channel providing the most favorable conditions for field generation because the screw flow arises here immediately after the braking begins and the diverter provides the best χ. The worse generation is expected at the entrance of the diverter - here, at the peak of kinetic energy, the flow is characterized by negligible χ. As a result, the magnetic field dissipates until t =.3 s. In Fig. 18 solid lines show the results obtained for pure sodium (µ 1). One can see that the flow at the exit of the diverter has some time to amplify the magnetic energy by about 1 4 times, but at the entrance of the diverter only a weak amplification of the magnetic field with further decay is observed (solid line in the right panel). The increasing of the MHD device size is an undesirable way because it leads to fast growth of the energy, which must dissipate during braking. The best way to achieve an essential growth of the magnetic Reynolds number is to increase h φ h φ t, s t, s Fig. 18. Magnetic field evolution in a nonstationary flow. Energy density of the azimuthal component near the channel wall is simulated for two cases: velocity corresponds to the data obtained at the exit of the diverter (left) and at the entrance of the diverter (right). Three values of effective magnetic permeability are considered: µ = 1 (solid line), µ =1.5 (dotted lines), µ = 2 (dashed lines). 16

19 the magnetic permeability. The permeability of liquid metal can be increased by admixing iron powder. We demonstrated in [19] that a turbulent screw flow perfectly provides a homogeneous suspension of iron particles of a size up to hundreds micrometers and that the effective permeability of this suspension at low concentration follows the formula µ 1+5.3c (c is the volume fraction of iron) exceeding this estimate as c>.12. Fig. 18 also presents the results of a simulations made for various values of magnetic permeability: µ = 1.5, which corresponds to the volume fraction of iron powder of about 1% (dotted lines) and µ = 2 (about 2% of iron powder), illustrated by dashed lines. The last case presents the limiting volume fraction, which really can be effectively admixed in the channel. The case with c =.1 seems to be realistic. Then the flow at the exit of the diverter amplifies the magnetic energy by a factor of 1 9 (of course this estimate is obtained in the framework of linear problem) and the flow at the diverter entrance amplifies the magnetic energy by a factor of about 1. The real behaviour seems to take place between these two limiting cases. 8. Conclusions. To sum up, the results presented in this paper confirm the feasibility of the idea of the dynamo experiment in a braked torus. However, our analysis has revealed some constraints on the preliminary order-of-magnitude estimates done in [8]. The main constraint is associated with the finite lifetime of the dynamo regime. The magnetic energy in the apparatus is expected to exceed its initial level by a factor of about 1. This means that the problem of dynamo action detecting in the experiment should not be restricted only to the measurements of magnetic field strength. It should also include the determination of magnetic field configuration and its comparison with the predictions of the dynamo theory. From this follows that the next step towards the experiment is the development of magnetic field registration facilities. On the other hand, the limited magnetic field growth allow us to restrict the theoretical investigations of dynamo activity to kinematic treatment. A prolongation of dynamo action requires an increase of the magnetic Reynolds number attainable in the apparatus. A possible way is to enlarge magnetic permeability of the fluid using a mixture of sodium and ferromagnetic particles. Our preliminary results in this direction allow us to believe that the magnetic permeability can be increased by a factor of 2. The limited time of dynamo action makes the experimental situation similar to the situation in galaxies, where the galactic lifetime also exceeds the dynamo timescale by a factor of about 1 only. The use of an iron admixture to enlarge the magnetic permeability makes this similarity more pronounced. The interstellar medium contains several phases differing in physical properties. This fact should be taken into account more carefully as it has been done so far in the conventional galactic dynamo theory. An unexpected result of the water experiment is the observation of a relatively long period during braking and just after its comletion when the general screw motion has not yet established. There are some indications that during this period the turbulent motion is formed by small-scale vortices with a dominant sign of helicity. This delay in development of general screw motion essentially restricts the duration of the screw dynamo action. However, this provides a gentle hint of a possibility to observe generation of the mean field based on small-scale helical turbulence rather than on general screw motion. We note that the mean-field generation by helical turbulence is much more attractive from the astrophysical viewpoint than a pure screw dynamo, yet, a simulation of this mechanism under laboratory conditions is traditionally considered as a hopeless task. 161