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1. Introduction THE ASTROPHYSICAL JOURNAL, 549:1212È1220, 2001 March 10 ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A. THE ZIGZAG PATH OF BUOYANT MAGNETIC TUBES AND THE GENERATION OF VORTICITY ALONG THEIR PERIPHERY T. EMONET1 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307-3000; emonet=hao.ucar.edu F. MORENO-INSERTIS2 Instituto deAstrof• sica de Canarias, E38200 La Laguna, Spain; fmi=ll.iac.es AND M. P. RAST High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307-3000; mprast=hao.ucar.edu Received 1999 September 10; accepted 2000 October 20 ABSTRACT We study the generation of vorticity in the magnetic boundary layer of buoyant magnetic tubes and its consequences for the trajectory of magnetic structures rising in the solar convection zone. When the Reynolds number is well above 1, the wake trailing the tube sheds vortex rolls, producing a von Ka rma n vortex street, similar to the case of Ñows around rigid cylinders. The shedding of a vortex roll causes an imbalance of vorticity in the tube. The ensuing vortex force excites a transverse oscillation of the Ñux tube as a whole so that it follows a zigzag upward path instead of rising along a straight vertical line. In this paper, the physics of vorticity generation in the boundary layer is discussed and scaling laws for the relevant terms are presented. We then solve the two-dimensional magnetohydrodynamic equations numerically, measure the vorticity production, and show the formation of a vortex street and the conse- quent sinusoidal path of the magnetic Ñux tube. For high values of the plasma beta, the trajectory of the tubes is found to be independent of b but varying with the Reynolds number. The Strouhal number, which measures the frequency of vortex shedding, shows in our rising tubes only a weak dependence with the Reynolds numbers, a result also obtained in the rigid-tube laboratory experiments. In fact, the actual values measured in the latter are also close to those of our numerical calculations. As the Rey- nolds numbers are increased, the amplitude of the lift force grows and the trajectory becomes increas- ingly complicated. It is shown how a simple analytical equation (which includes buoyancy, drag, and vortex forces) can satisfactorily reproduce the computed trajectories. The di†erent regimes of rise can be best understood in terms of a dimensionless parameter, s, which measures the importance of the vortex force as compared with the buoyancy and drag forces. For s2 > 1, the rise is drag dominated and the trajectory is mainly vertical with a small lateral oscillation superposed. When s becomes larger than 1, there is a transition toward a drag-free regime and epicycles are added to the trajectory. Subject headings: hydrodynamics È MHD È Sun: interior È Sun: magnetic Ðelds 1. INTRODUCTION total vorticity of the tube, the appearance of alternating aerodynamic lift forces, and a consequent sideways deÑec- The motion of magnetic Ñux tubes in astrophysical media tion of the upward motion (Fan, Zweibel, & Lantz 1998a). is a complex phenomenon in which the full two- or three- The rise of the tube is thus accompanied by a lateral oscil- dimensional magnetohydrodynamic structure of the tube lation, yielding a zigzag path. plays an important role. Much attention has been devoted Vorticity is generated in the magnetic Ñux-tube periphery to the dynamics of the tubes as one-dimensional deformable essentially through the curl of the Lorentz force, with the ropes with little consideration of the physics of the bound- buoyancy force being of secondary importance as a vor- ary layer at their periphery or of the surrounding external ticity source there (Emonet & Moreno-Insertis 1998). As Ñows (for a summary, see Moreno-Insertis 1997a, 1997b). with the kinetic energy of the Ñow, viscosity tends to dissi- Yet, the generation of vorticity in the boundary layer has pate the vorticity. However, for buoyant magnetic tubes in the potential to modify the trajectory as well as the struc- a stratiÐed medium, the ratio of viscous dissipation to mag- ture of the Ñux tube. As with high Reynolds number Ñows netic generation of vorticity, both in the boundary layer, around rigid cylinders in the laboratory, the vorticity gener- scales as(R/Hp)(Rem/Re) (see ° 2), where Re and Rem are the ated in the periphery of the tube accumulates in a trailing viscous and magnetic Reynolds numbers, respectively, R is wake, yielding a vonKa rma n vortex street. Shedding of the tube radius, andH is the local pressure scale height. oppositely signed vorticity leads to an imbalance of the p WheneverR/Hp > 1Re andm [ Re (both conditions apply for magnetic tubes in stellar interiors), the magnetic term is predominant and governs the vorticity budget and the for- 1 Current address: Department of Astronomy and Astrophysics, mation of the trailing wake. University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637; emonet=Ñash.uchicago.edu. This paper presents the results of an analytical and 2 Also at Departamento deAstrof• sica, Universidad de La Laguna, numerical study of the generation of vorticity in the bound- E-38200 La Laguna, Spain. ary layer surrounding a magnetic Ñux tube and discusses its 1212 BUOYANT MAGNETIC TUBES AND VORTICITY 1213 implications for the buoyant rise of the tube across a strati- tity of interest is the longitudinal vorticityuy ey. Its time Ðed medium. We Ðrst address the vorticity sources and evolution is governed by the following equation (Emonet & obtain the basic parameters of the problem and the corre- Moreno-Insertis 1998): sponding scaling laws (° 2). The magnetic, buoyancy, and viscous terms in the vorticity equation are found to be T1 T2 JMMKMML JMMKMML inversely proportional to the plasma beta. In ° 3, we present ~1 D u e *o (B Æ $ )(J e ) compelling empirical support for this b scaling from 2.5- o A y yB \ $ A B Âg] t t y y dimensional numerical simulations of both rectilinear and Dt o t o co zigzag rises. In ° 4, we examine the Reynolds number depen- k ] +2(u e ) dences. Despite the complexity of the process, an under- o y y standing of the results follows from a simpliÐed equation of GMHMI the motion for the tube (including buoyancy, aerodynamic T3 drag, and vortex forces). Various regimes of rise can be ] A1B C[ ] JÂB] 2 D delineated depending on a parameter s, which measures the $t  $*p k+ u . o c t importance of the lift force as compared with the aero- GMMMMMMMMMMHMMMMMMMMMMI dynamic drag and is proportional to the square of the vor- T4 (1) ticity generated in the boundary layer: for s2 > 1 the dynamics are drag-dominated, whereas for s2 ? 1 they are The subscript t stands for the component of a vector in the drag-free (° 5). The parameter s grows with increasing Rey- vertical plane, J is the current density, c is the speed of light, nolds numbers. A discussion of these results is presented in and k is the dynamic viscosity. The e†ect of compressibility ° 6. in the viscous term is safely neglected in this equation There are several recent publications concerning the two- because the rise velocity is assumed to be much smaller than and three-dimensional structure of buoyant magnetic tubes either theAlfve n(V )( or the soundc ) speed. An asymptotic A s ~1 (e.g., Moreno-Insertis & Emonet 1996; Emonet & Moreno- analysis based on the small parameters b and R/Hp Insertis 1998; Fan et al. 1998a, 1998b; Hughes & Falle (appropriate for buoyant magnetic tubes in the lower solar 5 ~2 1998). Their basic astrophysical motivation is to understand convection zone where typicallyb Z 10 and R/Hp [ 10 ) the transport of magnetic Ñux from the interior to the shows that the fourth term on the right-hand side of equa- ~1 surface of cool stars. Fan et al. (1998a) already noted the tion (1),T4, is O(b ) smaller than the other three, T1, T2, detachment of vortex rolls and the ensuing imbalance of andT3 (Emonet & Moreno-Insertis 1998). These Ðrst three vorticity as causing the tube to rise following a zigzag path. terms correspond to vorticity generation by buoyancy In their calculation, however, the vortex loss did not result (baroclinic production,T1)( and magnetic stressesT2), and from a simple instability of the wake but, rather, was to the viscous dissipation(T3), respectively. The viscous directly linked to the presence of a neighboring Ñux tube. term is the only one present in laboratory experiments with Also related to this paper is a Letter by Hughes & Falle Ñows around rigid cylinders. (1998), in which they present the results of a 2.5-dimensional If the magnetic Ðeld lines are assumed to be sufficiently numerical simulation of a rising Ñux tube at Reynolds twisted for the tube to maintain coherence during its rise, number D103 and b \ 10. The rise of their tube shows a but still witho Bt oo smaller thanBy o , then within the inte- trajectory deviating from the vertical soon after the begin- rior of the tube, the Ðrst two vorticity generation terms of ning of the rise. In addition, the tube subsequently stops equation (1) compete with each other in an oscillatory moving upward and, for a short while, even descends before fashion with time (Moreno-Insertis & Emonet 1996). In the resuming its rise. In this paper, we argue that these motions tube boundary layer (where the magnetic Ðeld distribution are just part of an incipient zigzag oscillation in an falls to zero), however, they reinforce each other, generating intermediate-drag regime, in which the zigzag and the iner- oppositely signed horizontal vorticity on either side of the tial oscillations of a buoyantly accelerated object subjected tube.
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