The Sun s Internal Magnetic Field

The Sun s Internal Magnetic Field... and Rotation and Stratification Toby Wood & Michael McIntyre DAMTP, University of Cambridge Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 1 / 14

Lots of Waves Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 2 / 14

The Sun s structure and rotation convection zone radiation zone Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 3 / 14

The Sun s structure and rotation convection zone radiation zone tachocline Source: Schou et al. 1998 Radiation zone rotates uniformly with a period of 27 days Convection zone rotates differentially, poles 30% slower than equator Thin interface ( 0.05R ) the tachocline Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 3 / 14

The Sun s rotation Differential rotation in convection zone implies systematic long-range angular momentum transport Slow rotation at high latitudes must gyroscopically drive a poleward current, c.f. Orbital decay of satellites Accretion discs Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 4 / 14

The Sun s rotation Differential rotation in convection zone implies systematic long-range angular momentum transport Slow rotation at high latitudes must gyroscopically drive a poleward current, c.f. Orbital decay of satellites Accretion discs Poleward current leads to a meridional circulation Circulation burrows through convection zone, spreading differential rotation Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 4 / 14

What stops the burrowing? Uniform rotation of radiation zone could be explained by global-scale primordial magnetic field, B (Walén 1946, Mestel & Weiss 1987) Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 5 / 14

What stops the burrowing? Uniform rotation of radiation zone could be explained by global-scale primordial magnetic field, B (Walén 1946, Mestel & Weiss 1987) Non-uniform rotation produces torsional oscillation of the field with frequency field strength, B For a strong field ( 10 3 G) rapid oscillations are damped uniform rotation of each field line Ferraro s Law of Isorotation Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 5 / 14

An internal magnetic field Brun & Zahn 2006 Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 6 / 14

An internal magnetic field What prevents this field from diffusing into the convection zone? Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 7 / 14

An internal magnetic field What prevents this field from diffusing into the convection zone? Meridional circulations within the tachocline could hold the interior field in advection diffusion balance (Gough & McIntyre 1998) Same circulations that spread differential rotation through the convection zone Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 7 / 14

An internal magnetic field What prevents this field from diffusing into the convection zone? Meridional circulations within the tachocline could hold the interior field in advection diffusion balance (Gough & McIntyre 1998) Same circulations that spread differential rotation through the convection zone Magnetic confinement most critical in high latitudes Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 7 / 14

The high-latitude tachocline The meridional circulation confines the magnetic field to a thin magnetic confinement layer The retrograde Coriolis force on the circulation is balanced by a prograde Lorentz force from the magnetic field Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 8 / 14

The high-latitude tachocline The meridional circulation confines the magnetic field to a thin magnetic confinement layer The retrograde Coriolis force on the circulation is balanced by a prograde Lorentz force from the magnetic field Helium settling out of the convection zone leads to compositional stratification, which holds the tachopause flat Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 8 / 14

The model Model the magnetic confinement layer as a rapidly rotating, Boussinesq, MHD fluid Non-dimensionalise using magnetic advection diffusion length L η/u and timescale t L 2 /η... and assume B 2 2Ωη Ro Du Dt + ˆΩ u = p Ra T ĝ + ( B) B u = 0 DB = B u + 2 B Dt B = 0 η DT κ Dt ĝ u = 2 T where Ra is the modified Rayleigh number Ra N2 L 2 2Ωκ Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 9 / 14

Confinement layer solutions In the limit Ra find perfectly flat, self-similar solutions 4 4 2 2 0 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 Only dimensionless parameter is Self-solutions valid within radius Λ B2 2Ωη r/l (Ra ) 1/2 max(λ 1/2,Λ 1/2 ) at which tilting of stratification surfaces becomes significant For L 0.001R and Λ 1 we find r 6.4R Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 10 / 14

Summary The Gough & McIntyre tachocline model can be described, in the polar regions, by a simple set of self-similar solutions z / δ 4 3 2 1 0 r These solutions could be valid over the entire high-latitude tachocline Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 11 / 14

Numerical results 4 z 3.5 3 2.5 2 1.5 1 0.5 0 0.3 0.2 0.1 0 0.1 0.2 0.3 r Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 12 / 14

The helium sliplayer Du Dt + 2Ω u = 1 ( T p µ ρ 0 T 0 u = 0 B t + u B = B u + η 2 B B = 0 DT Dt T 0N 2 g 2 g u = κ 2 T Dµ Dt = χ 2 µ µ 0 ) g + ( B) B Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 13 / 14

The helium sliplayer Du Dt + 2Ω u = 1 ( T p µ ρ 0 T 0 u = 0 B t + u B = B u + η 2 B B = 0 DT Dt T 0N 2 g 2 g u = κ 2 T Dµ Dt = χ 2 µ µ 0 ) g + ( B) B Velocity discontinuities resolved over thin helium diffusion layer Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 13 / 14

The helium sliplayer Du Dt + 2Ω u = 1 ( T p µ ρ 0 T 0 u = 0 B t + u B = B u + η 2 B B = 0 DT Dt T 0N 2 g 2 g u = κ 2 T Dµ Dt = χ 2 µ µ 0 ) g + ( B) B Velocity discontinuities resolved over thin helium diffusion layer Viscosity unimportant! Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 13 / 14

The helium sliplayer 2Ω(Λ + Λ 1 ) u r z Dµ Dt = g µ 0 µ r = χ 2 µ Toby Wood & Michael McIntyre (DAMTP) The Sun s Internal Magnetic Field 14 / 14