Magnetic power spectrum in a dynamo model of Jupiter Yue-Kin Tsang - PowerPoint PPT Presentation

Magnetic power spectrum in a dynamo model of Jupiter Yue-Kin Tsang School of Mathematics, University of Leeds Chris Jones University of Leeds

Let’s start on Earth . . . CRUST various types of rocks MANTLE magnesium-iron silicate OUTER CORE liquid iron + nickle INNER CORE solid iron + nickle CMB (not to scale) core-mantle boundary (CMB): sharp boundary between the non-conducting mantle and the conducting outer core location of CMB r dyn : the depth at which dynamo action starts seismic waves observation gives r dyn ≈ 3486 km another way to estimate r dyn : magnetic power spectrum

Gauss coefficients g lm and h lm Outside the dynamo region, r dyn < r < a : j = 0 j = 0 dynamo region ∇ × B = µ 0 j = 0 = ⇒ B = −∇ Ψ r dyn ⇒ ∇ 2 Ψ = 0 ∇ · B = 0 = a a = radius of Earth Consider only internal sources, ∞ l � l + 1 ˆ � a � � Ψ( r , θ, φ ) = a P lm ( cos θ )( g lm cos m φ + h lm sin m φ ) r l = 1 m = 0 ˆ P lm : Schmidt’s semi-normalised associated Legendre polynomials g lm and h lm are determined from magnetic field measurements on the surface ( r ≈ a )

The Lowes spectrum Average magnetic energy over a spherical surface of radius r 1 1 � | B ( r , θ, φ ) | 2 sin θ d θ d φ E B ( r ) = 2 µ 0 4 π Inside the source-free region r dyn < r < a , ∞ l � a � 2 l + 4 � � � g 2 lm + h 2 � 2 µ 0 E B ( r ) = ( l + 1 ) lm r l = 1 m = 0 Lowes spectrum (magnetic energy as a function of l ): l � a � 2 l + 4 � � g 2 lm + h 2 � R l ( r ) = ( l + 1 ) lm r m = 0 � a � 2 l + 4 = R l ( a ) (downward continuation) r

Estimate CMB using the Lowes spectrum � a � 2 l + 4 R l ( a ) R l ( c ) = R l ( a ) c (Robert Parker, UCSD) a = Earth’s radius c = 3486 km ≈ 0 . 55 a white noise source hypothesis : turbulence in the core leads to even distribution of magnetic energy across different scales l R l ( c ) independent of l = ⇒ c ≈ r dyn

Interior structure of Jupiter 8 10 7 10 6 10 5 10 4 10 3 10 a b i n itio (HSE) 2 10 σ [S/m] Li u et a l. 2008 1 Röpke & Redmer 1989 10 0 Stevenson 1977 10 Lee & More 1984 -1 10 Gómez- P érez et a l. 2010 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 0 0.5 R J R J (NASA JPL) (French et al. 2012) low temperature and pressure near surface ⇒ gaseous molecular H/He extremely high temperature and pressure inside ⇒ liquid metallic H core? conductivity σ ( r ) varies smoothly with radius r

Lowes spectrum for Jupiter: observations 8 10 7 10 6 10 5 10 4 10 3 10 a b i n itio (HSE) 2 10 σ [S/m] Li u et a l. 2008 1 Röpke & Redmer 1989 10 0 Stevenson 1977 10 Lee & More 1984 -1 10 Gómez- P érez et a l. 2010 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 0 0.5 R J R J (Ridley & Holme 2016) (French et al. 2012) g lm and h lm for l max ∼ 4 − 7 computed from magnetic measurements of various missions (e.g. Pioneer 10 & 11, Voyager 1 & 2, JUNO: l max = 12) downward continuation to a flat Lowes spectrum ⇒ 0 . 73 R J � r dyn � 0 . 90 R J For a continuous conductivity profile σ ( r ) : dynamo action in transition region? dynamo radius r dyn well-defined? estimation of r dyn from Lowes spectrum reliable?

A numerical model of Jupiter spherical shell of radius ratio r in / r out = 0 . 0963 (small core) ρ, ¯ anelastic: linearise about a hydrostatic adiabatic basic state (¯ T , ¯ p , . . . ) rotating fluid with electrical conductivity σ ( r ) forced by buoyancy convection driven by secular cooling of the planet dimensionless numbers: Ra , Pm , Ek , Pr ∇ · (¯ ρ u ) = 0 � ∂ u � EkRaPm S d ¯ Ek � z × u = −∇ Π ′ + 1 � T r + Ek F ν ∂ t + ( u · ∇ ) u + 2 ˆ ρ ( ∇ × B ) × B − d r ˆ ¯ ρ ¯ Pm Pr ∂ B ∂ t = ∇ × ( u × B ) − ∇ × ( η ∇ × B ) � ∂ S � � � + Pm Pr Q ν + 1 + Pm ρ ¯ ¯ T ∂ t + u · ∇ S Pr ∇ · F Q = Ek Q J Pr H S RaPm Boundary conditions: no-slip at r in and stress-free at r out , S ( r in ) = 1 and S ( r out ) = 0, electrically insulating outside r in < r < r out (Jones 2014)

A numerical model of Jupiter spherical shell of radius ratio r in / r out = 0 . 0963 (small core) ρ, ¯ anelastic: linearise about a hydrostatic adiabatic basic state (¯ T , ¯ p , . . . ) rotating fluid with electrical conductivity σ ( r ) forced by buoyancy convection driven by secular cooling of the planet dimensionless numbers: Ra , Pm , Ek , Pr a Jupiter basic state: C.A. Jones / Icarus 241 (2014) 148–159 1 ρ ( r ) ¯ η ( r ) = 5000 14 µ 0 σ ( r ) 4500 12 4000 10 3500 Density (kg/m 3 ) log 10 η (m 2 /s) 3000 8 2500 6 2000 cut−off 4 + Data points from French et al. 2012 1500 2 Hyperbolic fit 1000 + Data points from French et al. 2012 0 500 Rational polynomial fit cut−off 0 −2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 x 10 7 x 10 7 (a) radius (metres) (b) radius (metres)

Ra = 2 × 10 7 , Ek = 1 . 5 × 10 − 5 , Pm = 3 , Pr = 0 . 1 radial magnetic field r = r out r = 0 . 75 r out -1.350 1.350 -9.500 9.500 8000 zonal velocity 6000 4000 2000 0 -2000 -4000 r = r out -6000 -12500.0 12500.0 -8000

Dynamo radius from Lowes spectrum R l ( r ) in nT 2 r dyn = 0.87 r J 0.2 10 12 0.15 10 11 0.1 10 10 0.05 slope 10 9 0 0.96 r J 10 8 0.93 r J -0.05 0.90 r J 10 7 0.87 r J -0.1 0.83 r J R l ( r ) 10 6 -0.15 0 10 20 30 40 0.7 0.8 0.9 1 l B r at the surface r = r out = ⇒ g lm , h lm = ⇒ R l ( r out ) downward continuation into ‘source-free’ j = 0 region: � r out � 2 l + 4 R l ( r ) = R l ( r out ) r r dyn = 0 . 87 r J , how reliable is this estimate?

Magnetic power spectrum, F l ( r ) R l ( r ) in nT 2 r dyn = 0.87 r J 0.2 10 12 0.15 10 11 0.1 10 10 0.05 slope 10 9 0 0.96 r J 10 8 0.93 r J -0.05 0.90 r J 10 7 0.87 r J -0.1 0.83 r J R l ( r ) 10 6 -0.15 0 10 20 30 40 0.7 0.8 0.9 1 l 2 µ 0 E B ( r ) = 1 � | B ( r , θ, φ ) | 2 sin θ d θ d φ 4 π ∞ � = F l ( r ) l = 1 j = 0 exactly = ⇒ R l ( r ) = F l ( r )

R l ( r ) versus F l ( r ) F l ( r ) ( ! ) and R l ( r ) (- -) in nT 2 r dyn = 0.87 r J 0.2 10 12 0 -0.05 0.15 10 11 -0.1 0.1 -0.15 10 10 0 0.5 1 0.05 slope 10 9 0 0.96 r J 10 8 0.93 r J -0.05 0.90 r J 10 7 0.87 r J R l ( r ) -0.1 0.83 r J F l ( r ) 10 6 -0.15 0 10 20 30 40 0.7 0.8 0.9 1 l j � = 0 ( R l deviates from F l ) starting at about 0 . 9 r J Lowes spectrum R l prediction deeper than actual r dyn numerical model produces r dyn consistently with observations transition layer (moderate σ ) not contributes to dynamo action F l ( r ) ∼ flat in a large range 0 . 5 r J < r < 0 . 9 r J : white-noise source