Simulations of Flux Emergence in Cool Stars: What s Convection, Rotation, and Stellar Structure got to do with it?

Simulations of Flux Emergence in Cool Stars: What s Convection, Rotation, and Stellar Structure got to do with it? Maria Weber,2, Matthew Browning 3, Nicholas Nelson 5, Yuhong Fan 6, Mark Miesch 7, Ben Brown 8,9, Suzannah Boardman, Joshua Clarke, Samuel Pugsley, Edward Townsend University of Chicago, 2 Adler Planetarium, 3 University of Exeter, former Mphys students at the University of Exeter, 5 California State University, Chico, 6 HAO/NCAR, 7 NOAA, 8 LASP, 9 Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder @SolarisMaria

a Persistent polar starspot 23 imaging of ζ And (Aitoff projection) Temperatu 27 9 Starspot Evolution on 3 3 b Roettenbacher+ 26 =. =.25 Transient starspots =.25 Milliarcseconds 8 =.375 t = 58. (days) 9 =.5 - =.75 =.625 =.875 Morris+ 27 Davenport+ 25.5. Figure 2 Surface images of ζ And from September 23 with fourteen nights of data using SURFING. a and b are presented as in Fig., except.5 that the 2-K contours of the Aitoff projection (a) range from 3,6 K to,6 K. The polar spot is observed to have evolved between the two sets of. Relative Flux Milliarcseconds observations. The lower-latitude spots present in the 23 data set, with the n equator, emphasizing the spot-latitude -.5 Maehara+ 22 advances in visible interferometry will allow for similar resolution on 9. Kővári, Zs. et al. Doppler imaging of stell giant binary ζ Andromedae. Astron. As -. more stars (down to θ. mas). For stars that cannot be resolved in. KBerdyugina, S. Starspots: a key to the s detail, combining interferometrically observed photocentre shifts due 8 (25). -.5. Monnier, J. D., Berger, J.-P., Millan-Gabe to rotation of starspots in and out of view with Doppler imaging would..2..6.8.. The Michigan Phase Infrared Combiner (MIRC resolve the degeneracies inherent in the Doppler images, allowing for Proc. SPIE 59, 37 378 (2). more accurate surface maps. By acquiring a number of these maps on 2. ten Brummelaar, T. A. et al. First results Figure. Top: orthographic projections of the model star, with an inclination ofatthe instrument. Astrophys. J. 628, 5 several stars or a few observation epochs of the same targets, could the 5 we day time window starting BJD - 25833.567 = 58. days (left) and by the black arrow. Bottom: phase-folded H. light for the dataprimary in the sam 3. Korhonen, etcurve al. Ellipsoidal investigate how the changing magnetic field affects our determinations overlaid (blue solid line), and the contributions from both the higher latitude Investigation using high-resolution spe starspots o set forbarnes+ 2 of stellar parameters (including mass and age)3,26. In addition, theclarity. Astron.25 Astrophys. 55, A (2).

Model Schematic: Equation of Motion: ρ dv dt = 2ρ(Ω v) (ρ e ρ)g + l s ( B 2 8π ) + B2 π k Spruit 98 C d ρ e (v v e ) (v v e ) (πφ/b) /2, () B r θ Φ toroidal Solar Ω, 5Ω Tachocline interface dynamo Browning+ 26 M dwarf, Solar 3Ω Distributed dynamo Browning 28 M Dwarf Rapid Ω # Brown+ 2 Sun.3M M Dwarf Convection ASH Solar 3

Convection modulates flux emergence Sun-like stars Dynamo-generated magnetic loops (a) (b) - + Direction of Rotation Tilt Angle adapted from Schrijver & Zwaan 2 22 Mx 5 kg TFT+convection gure 8. Buoyant magnetic loops evolving from small-scale wreath sections amplified by turbulent intermitten Thin flux tubes apex Nelson+ 23 Weber & Fan 25 Weber+ 2, 23a, 23b Weber & Fan 25 Convection & magnetic buoyancy work in concert to promote flux emergence.72r Polar View.92R.85R Weber, Nelson, Browning - in prep footpoint B (G) Downflows naturally induce loops ~5 2 apart Dynamogenerated loops ] Nelson+ 2 Convection introduces a statistical spread in tilt angles

Convection can also suppress flux emergence Fully convective M dwarf Average suppression depth " Weber+ 27 3 kg.5r 5.725R.75R 3 kg.75r 5 The global rise of TFTs is more strongly suppressed by convective flows when the flux tube is initiated: in the deeper interior at lower latitudes with a weaker magnetic field strength Weber & Browning 26, Weber+ 27 5

Rotation alters emergence properties No convection Ω 3 29 days No convection 3Ω 3 57 days 7 days Sun, Ω Tilt Angles 2 days 2 - kg 2 Mx 2 Mx 22 Mx based on Weber+ 23 Weber, Nelson, Browning - in prep.72r.8r.92r Sun, 5Ω Brown+ 28 Due to the Coriolis force, more rapid rotation: Lengthens the rise time Leads to poleward deflection Increases tilt angles Weber Thesis 2 6

tube is not given a cross-sectional extent. in Weber & Browning 26. The flux tube initial configuration is show i reaching the simulation upper boundary in black. The azimuthal axis has the apex of the tube at the upper boundary is on the right-hand side. U mesh spheres represent.95r and the initial flux tube,initia resp of Fig.radius. r3 kgthe flux tubes initiated at.5r, (Center) initiated at.75r, (Right)&same as center image, in Weber Browning 26. Th tube is not given a cross-sectional extent. reaching the simulation upper bo Stellar structure impacts emergence latitudes and more Tachocline or not? Emergence Latitude Probability Functions 6 kg 3 kg kg the apex of the tube at the upp mesh spheres represent.95r and initiated at.5r, (Center) initiat tube is not given a cross-sectiona 2 kg No tb based on Weber+ 23 3 kg.5r.75r Flux Tubes in Fully Convective Stars -( =.75R -( =.5R Azimuthal velocity Figure 3. (a) Meridional plot of the longitudinal velocity v φ for the fast differential rotation profile, averaged over 6 days with contour intervals every m s around zero relative to the rotating frame. Dashed lines are at radii of.5r and.75r. (b) Angular velocity Ω averaged over the same time interval as a function of radius along indicated latitudinal cuts for the fast ( Ω/Ω 22%) differential rotation profile, and the slow ( Ω/Ω 2%) differential rotation profile approximated using Equation 8. Case T TL ATL THE M f s C 7 Parameters TEQ, vφ =, Rad. Heat. TEQ, vφ = vφℓ, Rad. Heat. TEQ, vφ = vφℓ, Adiabatic TEQ, vφ = vφhe, Rad. Heat. MEQ, Rad. Heat. Fast Diff. Rot., Ω/Ω 22% Slow Diff. Rot., Ω/Ω 2% indicates convective field tac ho cli n 2.5! Fig. 2. kg flux tubes initiated at θ = 5 in temperature equilibrium with the star (Ω=2.6 6 rad s ). Upper and lower mesh spheres repre the base of the convection zone, respectively. (Left).M, (Center).5M,.! Evolution of the 3 kg flux tubes are similar in trajectory. Table Flux tube simulation parameters. Those in TEQ have a density deficit following Eq. 7, with an internal azimuthal speed () vφ = co-rotating with Ω, (2) vφ = vφℓ co-rotating with the local longitudinal velocity v φ corresponding to either the fast or slow differential rotation profile, or (3) vφ = vφhe, the azimuthal velocity required for the flux tube to be in horizontal force equilibrium following Eq. 2. Those in MEQ have a neutral buoyancy and a prograde vφ following Moreno-Insertis et al. (992). Flux tubes evolve either with radiative heating following Eq. 6 or adiabatically such that ds/dt =. The presence of an applied velocity field (see Sec. 2.3) is represented by C. Bu il based on Weber & Browning 26 refer to the two profiles as fast (f) and slow (s), corresponding to angular velocity contrasts Ω/Ω of 22% and 2%, respectively. ( For simplicity in referring to a set of simulations with particular initial conditions, we have a adopted a naming scheme given in Table. For example, the Case ATLf simulations discussed briefly in Section 3.2 refer to flux tubes that evolve adiabatically (A), are initially in thermal equilibrium (T), and have an internal azimuthal speed vφ corresponding to the local longitudinal velocity v φ of the fast differential rotation profile (Lf). The Case TLsC simulations discussed in Section correspond to flux tubes that evolve with the influence of radiative heating, where the tube is initially in thermal equilibrium (T) and co-rotating with the slow differential rotation profile (Ls). The application of the suffix C indicates the presence of time-varying convective flows (C), where the applied longitudinal velocity profile v φ always corresponds to either the slow or fast profile as indicated. t in e.6! Fig. 2. kg flux tubes initiated at θ = 5 in temperature equilibrium and co-rotating with the star (Ω=2.6 6 rad s ). Upper and lower mesh spheres represent.95r and the base of the convection zone, respectively. (Left).M, (Center).5M, (Right).6M. Evolution of the 3 kg flux tubes are similar in trajectory. Fully Convective M Dwarf, ' = 6 here allows for more direct comparison between TFT simulations, removing any effects that may arise because of stochastic variations in the radial vr and latitudinal vθ velocity fields. Figure 3b shows the angular velocity Ω (nhz) from the original ASH hydrodynamic case as a function of radius for latitudinal cuts at, 5, and 6. Shown on the same plot is the angular velocity Ω approximated using Equation 8 for a contrast of Ω/Ω 2%. This simplistic approach creates a differential rotation profile very similar to Case Cm in Browning (28) with the same angular velocity contrast of 2%, where the presence of equipartition-strength magnetic fields quenches the differential rotation. The presence of magnetic fields in Case Cm does affect the distribution of angular momentum. However, we note that the amplitudes of v shown in Figure 2 are commensurate uil 2 6 kg kg Sun, '( = 2 t in Fig. 2. kg flux tubes initia with the star (Ω=2.6 6 rad the base of the convection zone, r Evolution of the 3 kg flux tubes tac ho cli n e 3. Flux tubes initiated at θ = 5 in mechanical Unlike solar case, in M dwarf there is a tendency forfig.high latitude emergence (> 3 )force equilibrium. The azimuthal axis has been rotated so the apex of the tube at the upper boundary is on the right-handof flux tube Assumption side. Upper and lower mesh spheres represent.95r and the base of the convection zone, Exceptions when flux tubes initiated closer to the surface and of sufficiently weak region, and thereby respectively. (Top) 3 kg flux tubes, (Bottom) kg flux tubes. generating (Left).M, (Center).5M, (Right).6M. All kg tubes shown here have developed one buoyant loop. The ( 3 kg) or strong field strengths ( 2 kg) initial thermodynamic Fig. 3. Flux tubes initiated at θ = 5 in mechanical force equilibrium 3 kg flux tubes develop one buoyant loop in the.m star, and two loops in the.5 and axis has been rotated so the apex of the tube at the upper boundary is.6m. It is likely that convective motions will modulate the shape properties, of especially the 3 kg matter Increased density in M dwarfs leads to longer flux tube rise times by x side. Upper and lower mesh spheres represent.95r and the base of7the flux tubes, removing any preference for these low order m= and m=2 unstable modes (i.e. 3. FLUX TUBES IN A QUIESCENT CONVECTIVE

Summary Convection, rotation, and stellar structure are all important contributing factors to the overall trend of flux emergence. Ω 3 Ω Fully Convective M dwarf Solar Convection modulates flux emergence Fluid motions both suppress and promote the rise of magnetism Convection introduces a statistical spread in emergence properties Due to the Coriolis force, rapid rotation: Lengthens the rise times Leads to poleward emergence Increases tilt angle Tendency for polar flux emergence in M dwarfs, unlike in solar-like stars Increased density in M dwarfs leads to longer rise times Assumptions about flux tube generating region (i.e. tachocline or not) has consequences for flux emergence This work is a step toward linking magnetic flux emergence, convection, and dynamo action along the lower end of the main sequence. 8