Pores and sunspots are some of the magnetic features observed on the surface of the solar photosphere. In this presentation numerical models are used to study idealised pores and sunspots. Vertical flux tubes are placed in a compressable convecting photosphere in an axisymmetric cylindrical box with radius up to 4 times its depth [1]

Botha, GJJ

Hurlburt, NE

Rucklidge, AM

White Rose Research Online

This is a repository copy of Dynamics of convection around axisymmetric magnetic flux tubes.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/84734/Version: Accepted VersionConference or Workshop Item:Botha, GJJ, Rucklidge, AM and Hurlburt, NE (2004) Dynamics of convection around axisymmetric magnetic flux tubes. In: 31st EPS Conference on Plasma Physics, 28 Jun - 02 Jul 2004, London. eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Dynamics of convection around axisymmetric magnetic fluxtubesG.J.J. Botha1, A.M. Rucklidge1, N.E. Hurlburt21 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK2 Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto CA 94304, USAPores and sunspots are some of the magnetic features observed on the surface ofthe solar photosphere. Magnetic flux is pushed to boundaries between convection cells(granules), where they form flux elements that have a lifetime of a few minutes, fieldstrengths of 1500G and magnetic fluxes in the order of 0.0001TWb. The flux elementscollect at the corners of supergranules (where the surface flows converge) and the mag-netic field strength grows (possibly) to form pores. Pore sizes range from 2000 – 4000km,with field strengths of approximately 2000G and magnetic fluxes of up to 7TWb. Thisfield strength seems not to depend on pore size. Pores may grow into sunspots. Mostsunspots tear apart within a few hours, but some have lifetimes of several weeks andgrow to diameter sizes of up to 50,000km. Their field strengths can be up to 3000G in theumbra and 1500G in the penumbra, with magnetic flux varying between 2 – 200TWb.The sunspot penumbra has a surface flow radially outward from the sunspot (Evershedflow), which is slightly elevated above the rest of the photosphere. The supergranularcell around a sunspot and its penumbra is called the moat cell, and features outwardconvection. In this presentation a numerical model is used to study idealised pores andsunspots. Vertical flux tubes are placed in a compressible convecting photosphere in anaxisymmetric cylindrical box with radius up to 4 times its depth [1]. The numerical boxis initialised with a temperature and density stratificationT = T0(1 + θz) and ρ = ρ0(1 + θz)m, (1)with the 0 subscript defining the quantity at the top of the box (z = 0), θ the initialtemperature gradient, and m the polytropic index (T0 = 1, ρ0 = 1). Fully compressible,nonlinear axisymmetric magnetoconvection is solved, using equations [2]∂ρ∂t= −∇ · (uρ) , (2)31st EPS Conference on Plasma Phys. London, 28 June - 2 July 2004 ECA Vol.28G, P-2.062 (2004)∂u∂t= −u · ∇u −1ρ∇P + θ(m + 1)ẑ −σζ0K2Qρj × B +σKρ∇ · τ , (3)∂T∂t= −u · ∇T − (γ − 1)T∇ · u +γKρ∇2T +σK(γ − 1)ρ(12τ : τ + ζ20QK2j2), (4)∂A∂t= (u × B)φ − ζ0K(∇× B)φ , (5)with auxiliary equationsP = ρT , B = ∇× (φ̂A) , ∇ · B = 0 , j = φ̂(∇× B)φ , (6)and where τ is the rate of strain tensor, γ the ratio of specific heats, σ the Prandtlnumber, K the dimensionless thermal conductivity, ζ0 the magnetic diffusivity ratio atz = 0, and Q the Chandrasekhar number. All other symbols have there usual meaning.The numerical domain is a 2D representation (r, z) of an axisymmetric cylinder withradius Γ. On the axis (r = 0) the boundary conditions are∂ρ∂r= Ur =∂Uz∂r= A = Br =∂Bz∂r= j =∂T∂r= 0. (7)At r = Γ the boundary is a slippery, perfectly conducting, thermally insulating wall. Atthe bottom boundary (z = 1) the temperature is fixed and the magnetic field is vertical.A potential field is placed at the top boundary (z = 0) together with a temperaturegoverned by Stefan’s law. The equations are evolved using fourth-order Bulirsch-Stoertime integration and sixth-order compact finite differencing [3]. We have explored thefollowing parameter values:105 ≤ R ≤ 107 ; 0.1 ≤ σ ≤ 0.3 ; 10 ≤ Q ≤ 104 ; 0.02 ≤ ζ0 ≤ 0.2 ;(γ, m) = (53, 1.45) ; (53, 1.495) ; (54, 3.9975) ;and θ = 10. For the values used in Figure 1, the temperature increases by a factor of11 in the vertical direction, and the density by a factor of 55.We found anti-clockwise (inward flow at the top) and clockwise (outward flow) con-vection cells – which may respectively give an indication of processes associated withpores and the perumbral flow around sunspots. Figure 1 is an example of clockwiserotation. With such a convection cell, there is always a smaller anti-clockwise rotatingcell forming at the left hand side of the large convection cell, which is time dependent.The fluctuation of the smaller cell shows periodic behaviour, seen clearly in the plasma31st EPS 2004; G.J.J.Botha et al. : Dynamics of convection around axisymmetric magnetic flux tubes 2 of 4                                                                                                                              Figure 1: The six 2D plots show a time sequence in the clockwise direction, which isthe same direction as movement in the main convection cell. Each plot consists of threeboxes in the (r, z) plane. The top (black and white) box is the potential field. Themiddle box shows the magnetic field lines as contours, the temperature in colour andthe velocity as arrows. The bottom box shows the magnetic field strength as arrows,the density as contour lines and the current in the φ direction in colour. These resultswere obtained with parameter values R = 106, Q = 100, σ = 0.1, ζ0 = 0.2, m = 1.45,and K = 0.529.31st EPS 2004; G.J.J.Botha et al. : Dynamics of convection around axisymmetric magnetic flux tubes 3 of 4Figure 2: Kinetic and magnetic energy during evolution of clockwise convection cell.kinetic and magnetic energy (Figure 2). As the small anti-clockwise cell forms, it pullsmagnetic field lines and a plume of colder plasma to the bottom of the box. Once thefield lines have moved underneath the cell, they are convected in the r direction towardsr = Γ. With this action the anti-clockwise cell is destroyed, and the cycle repeats itself.The increase in kinetic energy in Figure 2 is due to the slow growth of the large clockwiseconvection cell. The peaks in magnetic energy correspond to the formation of the smallanti-clockwise cell. With its destruction magnetic energy decreases.The structure of the flows was investigated in terms of the possible influence ofthe numerical boundaries of the domain. At the top boundary, the potential magneticfield was replaced by a uniform (in z direction) magnetic field, while Stefan’s law wasreplaced by a fixed temperature T0. Neither of these changes influenced the convectionin any significant way. The radius (Γ) of the numerical box was a deciding factor.For Γ > 3 more than one large convection cell formed, with the largest next to thestrong magnetic field at r = 0 and rotating in an anti-clockwise direction. The physicalparameters influencing the convection the most are Q and R. A large R causes strongconvection currents to sweep the magnetic field lines with the flow, while a strong Qtends to inhibit convection. This inhibition takes the form of many smaller convectioncells forming between the field lines. Numerical constraints have prevented us fromincreasing θ and m to large values.[1] N.E. Hurlburt & A.M. Rucklidge, 2000, MNRAS 314, 793-806.[2] N.E. Hurlburt & J. Toomre, 1988, ApJ 327, 920-32.[3] S.K. Lele, 1992, Journal of Computational Physics, 103, 16-42.31st EPS 2004; G.J.J.Botha et al. : Dynamics of convection around axisymmetric magnetic flux tubes 4 of 4

Dynamics of convection around axisymmetric magnetic flux tubes

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Dynamics of convection around axisymmetric magnetic flux tubes

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