Coupling the solar dynamo and the corona: solar wind, mass and momentum losses during the solar cycle Rui PINTO ([email protected])1,2, L. JOUVE4, R. GRAPPIN2,3, S. BRUN1 1Service d’Astrophysique - CEA / DSM / Irfu, Saclay, France 2LUTh, Observatoire de Paris-Meudon, France 3LPP, Ecole Polytechnique, UPMC, France 4DAMTP, Cambridge, UK

Abstract 3. Solar wind speed and magnetic polarity

We study the connections between the sun’s convection zone evolution and the dynamics of dw1_b00; (u_//)/cs; nt= 15.00 dw1_b06x01b; (u_//)/cs; nt= 35.00 the solar wind and corona. We input the magnetic fields generated by a 2.5D axisymmetric 0.6 0.6 1 1.50 0 1.50 kinematic dynamo code (STELEM) into a 2.5D axisymmetric coronal MHD code (DIP). -1 The computations were carried out for an 11 year cycle. We show that the solar wind’s 0.5 0.5 1.00 1.00

velocity and mass flux vary in latitude and in time in good agreement with the well known 1 time-latitude assymptotic wind speed diagram. Overall sun’s mass loss rate, momentum 0.4 0.500.4 0.50 flux and magnetic breaking torque are maximal near the solar minimum. 0 0.3 0.000.3 -1 0.00 1 1 0 1

-0.50 1 -0.50 1. Introduction 0.2 0.2 1 0 0 1 1 -1.00 0 0 -1.00

0.1 0.1 0 Coupling two MHD axi-symmetric numerical codes -1.50 -1.50 0 0.0 0 1 0 0.0 1 STELEM: Kinematic dynamo (CZ) DIP: Solar wind and corona 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 dw1_b07c; (u_//)/cs; nt= 33.00 dw1_b20; (u_//)/cs; nt= 17.00

0.6 1 0.6 0 -1 1.50 1.50 É Convective zone, tachocline É Isothermal corona (γ = 1, T = 1.3 MK) É Meridional circulation 0.5 0.5 Stationary potential 1.00 1.00 É Able to reproduce the É 0 “butterfly diagram” “external” field B + induced -1 field b 0.4 0.4 -1 É Potential B-field for r > R 0.50 0.50 É Solar wind develops within (Jouve & Brun, 2007) the open-field regions (coronal holes) 0.3 0 0.000.3 0.00

É Closed field regions -1 -0.50 -0.50 (streamers) = stagnant 0.2 0.2 zones transmit potential field↓ evolution B (t) 0 1 10 0 -1.00 -1.00 (11 year cycle) → 0.1 0 0.1

0 -1.50 -1.50 0.0 0 1 0.0 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (Grappin et al., 2000; Pinto et al., 2010) Figure 5: Evolution of the coronal magnetic field during the solar cycle. Only the first 3 R and the northern hemisphere are shown. Each snapshot corresponds to t = 0, 3, 3.5, 10 years (from left to right, top to bottom). White∼ lines are magnetic field lines. The colorscale Motivation: represents the wind flow velocity (Mach number) projected onto the signed magnetic field. This quantity serves as a tracer for the B-field’s polarity. Red/orange means positive polarity, while green/blue means negative polarity. The black contours trace Mach numbers 0, 1. É Study the influence of the dynamo generated fields on the global solar wind properties

É Evaluate the topology of the non-potential coronal magnetic field during the solar cycle É Predict solar wind speed from open flux tube geometry variations Ur [km/s]; r = 14 R_sun CoronalB magnetic [G]; r = field 14 R_sun É How does the polarity inversion happen in the corona? 10 É10The coronal (poloidal) magnetic field is globally dipolar 450.00 faraway from the surface, except during the polarity 1.16 reversal É How does the streamer/current sheet/coronal hole distribution evolve over a cycle? É Close to the surface, the magnetic topology is much more É Does the sun’s mass and moment loss rates vary during the cycle and how? 433.33 0.77 8 complex8

É The polarity reversal happens quickly in the corona, even if 416.67 the underlying B-field evolves slowly. 0.39 2. Mass, momentum flux and breaking torque 6 6 É For r R , B Br , as the solar wind “opens up” the field 400.00 0.00 lines. Furthermore,| | ≈ Br is nearly independent of the latitude faraway from the sun.

time [yr] 4 time [yr] 4 383.33 É The magnetic field decomposes mainly in low-order -0.39 multipolar components at the minimum, while if higher order 10 8 2 2 2 366.67 components2 appear at the maximum -0.77 5 7 6 É The multiple current sheets shown in fig. 5 may be 0 5 350.00 interpreted as a highly warped current sheet in the -1.16 -5 0 0 z [R_sun] 4 1 1 non-axisymetric corona -10 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

< R_alfven > [R_sun] co-latitude co-latitude -9·104 2·1011 4·1011 6·1011 0 2 4 6 8 10 R_alfven [R_sun] time [yr] Figure 6: Assymptotic wind speed during the solar cycle. Wind velocity 0 0 Time-latitude diagram. É The time-latitude diagram for the assymptotic wind speed shows good agreement with current observations and in situ 10 8·1017 measures at 1 AU 5 (cf. time-latitude plots in Wang & Sheeley, 2006) 6·1017 É Fast solar wind originates essentially from high latitude -1 -1 0 17 regions, while the slow wind flows mostly aside the streamers -5 4·10 z [R_sun] at lower latitudes. Exceptions may occur, though. -10 2·1017 É Wind speed and flux tube expansion factor are well correlated

0 5.0·1017 1.0·1018 1.5·1018 Angular momentum flux 0 2 4 6 8 10 at all latitudes -2 -2 Angular momentum flux time [yr]

-0.0 0.5 1.0 1.5 2.0 -0.0 0.5 1.0 1.5 2.0 Figure 2: Alfvén radius rA and angular momentum flux 2 〈 〉 Figure 1: Alfvén surfaces (black contours) at the solar minimum (left) Ω rA . 4. Conclusions and at about the polarity inversion (right). White lines are magnetic 〈 〉 field lines, and the colorscale represents the wind’s poloidal Mach number. The distance unit is 7.1 R . 1. We used two 2.5D MHD models – STELEM for the the solar minimum, as so is the resulting magnetic convection zone dynamo and DIP for the wind and breaking torque. Note that this is the moment when corona – to study the quasi-steady evolution of the the total photospheric surface that is magnetically 1.5 1.5 Magnetic breaking torque sun’s coronal environment in response to the cyclic connected open flux regions (where the solar wind -0.2 evolution of the dynamo source field. flows) is maximal. At the same time the coronal 1.0 1.0 magnetic field is at it’s least multipolar -0.4 2. Surface magnetic field, coronal topology and configuration, meaning that it decays the slowest assymptotic wind speed computed are all in good -0.6 with radial distance and the alfvén radius (lever arm 0.5 0.5 agreement with observations. length) is naturally the highest. -0.8 -tau = -d/dt M * Omega 3. Overall sun’s mass loss rate is dominated by 0.0 0.0 5. Variations in the radial and transverse large scale -1.0 outward flows originating at low latitudes. The 0 2 4 6 8 10 time [yr] predominance of these depends on the time-varying magnetic gradients may be pertinent to the study of ˙ 2 the triggering of eruptive phenomena. -0.5 -0.5 Figure 4: Magnetic breaking torque τ = MΩ rA magnetic field geometry and how it restrains the (normalised). − 〈 〉 solar wind from “opening-up” outflow channels. 6. Future work will include a more detailed treatment -1.0 -1.0 4. Global angular momentum flux is maximum near of the chromospheric dense layers. From solar minimum to solar maximum: -1.5 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 5. References 2 É global B increases Figure 3: Mass flux ρVr r sin θ in the meridional plane at the same | | instants as in fig. 1 (above). The factor r 2 sin θ is due to the spherical É rA decreases Grappin, R., Léorat, J., & Buttighoffer, A. 2000, Astronomy and Astrophysics, 362, 342 〈 〉 expansion of a surface element normal to the radial direction. Outflows É total surface open flux decreases Jouve, L. & Brun, A. S. 2007, Astronomy and Astrophysics, 474, 239 which originate at lower latitude dominate the global mass loss rate. Matt, S. & Pudritz, R. E. 2008, The Astrophysical Journal, 678, 1109 É coronal hole / streamer boundaries approach the poles Pinto, R., Grappin, R., & Leorat, J. 2010, in SW12, Vol. 1216 (Saint-Malo, (France): AIP), 80–83 Wang, Y. & Sheeley, N. R. 2006, Astrophysical Journal, 653, 708 É M˙ falls Weber, E. J. & Davis, L. 1967, Astrophysical Journal, 148, 217 É Angular momentum flux and breaking torque τ weaken

É τ < 0, all the time (see, e.g Matt & Pudritz, 2008; Weber & Davis, 1967)

Coupling the solar dynamo and the corona [email protected]