Formation of Inhomogeneous Magnetic Structures in MHD Turbulence and Turbulent Convection

Formation of Inhomogeneous Magnetic Structures in MHD Turbulence and Turbulent Convection Igor ROGACHEVSKII and Nathan KLEEORIN Ben-Gurion University of the Negev Beer-Sheva, Israel Axel BRANDENBURG and Koen KEMEL NORDITA, Stockholm, Sweden Petri KÄPYLÄ and Maarit KORPI University of Helsinki, Finland

Magnetic Structures: Sunspots

Solar Magnetic Structures

What is the mechanism of formation of solar magnetic structures? How solar turbulent convection can form largescale magnetic structures? Solar dynamo mechanism can generate only weak (<< 1000 G) nearly uniform large-scale magnetic field. How is it possible to create strongly inhomogeneous magnetic structures from originally uniform magnetic field?

Outline Physics of the effect of turbulence on large-scale magnetic pressure (on large-scale Lorentz force) Direct numerical simulations of the effect of turbulence on large-scale Lorentz force Theory of the effect of turbulence on Lorentz Force Large-scale instability: numerical simulations Estimates for the solar convective zone

Different Effects of Turbulence Turbulent viscosity Turbulent diffusion Turbulent magnetic diffusion Alpha effect Turbulent diamagnetic or paramagnetic velocity Lambda effect generation of differential rotation Turbulent Thermal diffusion

Mean-Field Approach Instantaneous magnetic field: @H = r (v H r H) @t H = B + b; v = U + v; Mean magnetic field: @B B = hhi; U = hvi = r (U B + hu bi r B) @t Fluctuations of magnetic field: @b r (u b hu bi r b) = (B r)u (u r)b B(r u) @t Source of tangling magnetic fluctuations: I b = (B r)u (u r)b B(r u)

Turbulent Transport of Passive Scalar Instantaneous particle number density: @n @t + r (n v) D m n = 0 Mean particle number density: @N @t + r hn vi D m N = 0 Fluctuations of particle number density: n = N + n 0 ; N = hni @n 0 @t +r (n0 v hn 0 vi) D m n 0 = N (r v) (v r)n Source of fluctuations: I n 0 = N (r v) (v r)n

Methods and Approximations Quasi-Linear Approach or Second-Order Correlation Approximation (SOCA) or First-Order Smoothing Approximation (FOSA) Rm << 1, Re << 1 Steenbeck, Krause, Rädler (1966); Roberts, Soward (1975); Moffatt (1978) Path-Integral Approach (delta-correlated in time random velocity field or short yet finite correlation time) Kraichnan-Kazantsev model of random velocity field Zeldovich, Molchanov, Ruzmaikin, Sokoloff (1988) Rogachevskii, Kleeorin (1997) St = `=u 1 Tau-approaches (spectral tau-approximation, minimal tauapproximation) third-order or high-order closure Re >> 1 and Rm >> 1 Pouquet, Frisch, Leorat (1976); Vainshtein, Kitchatinov (1983); Kleeorin, Rogachevskii, Ruzmaikin (1990); Blackman, Field (2002) Renormalization Procedure (renormalization of viscosity, diffusion, electromotive force and other turbulent transport coefficients) - there is no separation of scales Moffatt (1981; 1983); Kleeorin, Rogachevskii (1994)

Mean-Field Theory Path-Integral Approach St = `=u 1 Solution: Wiener trajectory: @ t B = (B r)v (v r)b + B; B i (t; x) = hg ij (t; t 0 ;») B j (t 0 ; x)i w ;»(t; xjt 0 ) = x Green function: Z t t0 0 dg ij =d¾ = r j v i ; G ij (t; t 0 ;») = ± ij + v(t ¾;») d¾+(2 ) 1=2 w(t t 0 ); Z t t0 Kraichnan-Kazantsev model of random velocity field: µ t1 t hv i (t 1 ; x)v j (t 2 ; y)i / ± 2 0 B i (t = t 0 ; x) = B i (t 0 ; x); r j v i (t ¾;») d¾ + :::; c hw i (t)i w = 0; hw i (t + )w j (t)i w = ± ij ;

Tau Approach Equations for the correlation functions for: The velocity fluctuations The magnetic fluctuations The cross-helicity tensor ij ( k) ui u j ( M II ) ( II ) M ij ( k) bi bj ( II ) M ij ( k) bi u j The spectral t-approximation (the third-order closure procedure) u b DM ˆ ( III ) ( k) DM ˆ ( III ) 0 ( k) M ( II ) ( k) M t ( k) c ( II ) 0 ( k) III DM ˆ ( ) ij ( k) ui ( u ) u j u j ( u ) ui u

Renormalization Procedure The first step is the averaging over the scale that is inside the inertial range of turbulence. The next stage of the renormalization procedure comprises a step-by-step increase of the scale of the averaging up to the maximum scale of turbulent motions. This procedure allows the derivation of equations for the turbulent transport coefficients: eddy viscosity, turbulent diffusion, turbulent heat conductivity, electromotive force coefficients, etc. To apply this procedure an equation invariant under the renormalization of the turbulent transport coefficients must be determined.

Models of Background Turbulence Inhomogeneous compressible turbulence: hu i u j i (0) = E(k) 8¼ k 2 (1 + ¾ T =2) Anisotropic helical turbulence: hu i u j i (0) k = E(k) 8¼k 2 Density Stratified turbulence: hu i (!; k) u j (!; k)i (0) = Free parameters: ± ij k ij + ¾ T k ij + (1 + ¾ T ) i 2k 2 (k ir j k j r i ) hu 2 i ½ h(1 ²) ³ ±ij k ij + 2² ³ ±ij e i e j k? ij ¾ T = h[r u]2 i hjr uj 2 i ; hu 2 i E(k) 8¼ 2 k 2 c (! 2 + 2 c ) ²; E(k); k ij = k i k j k 2 ; i hu 2 i i ¾ k 2 " ijn k n ¹ v (k); ¹ v = hu (r u)i; k? ij = k? i k? j i = r i½ ½ r u = u i i ± ij k ij + i k 2 ( i k j j k i ) k 2? ;

where Momentum Equation @ @t ½ U = r j ij ij = ½ U i U j + ± ij (p + 1 2 B2 ) B i B j ¾ º ij (U) Averaged equation: where U = ¹U + u ; @ @t ¹½ ¹U = r j ¹ ij ¹ ij = ¹½ ¹U i ¹U j +± ij (¹p+ 1 2 ¹ B 2 ) ¹B i ¹B j ¹¾ º ij ( ¹U)+ 1 2 hb2 i ± ij hb i b j i+¹½hu i u j i Turbulent Maxwell and Reynolds stresses: B = ¹B + b ¾ T ij = 1 2 hb2 i ± ij hb i b j i + ¹½hu i u j i; Turbulent viscosity: hu i u j i = 2 º t ¹S ij (¹U)

Magnetic Fluctuations

Mean Momentum Equation @ @t ¹½ ¹U = r j ¹ ij where U = ¹U + u ; ¹ ij = ¹½ ¹U i ¹U j +± ij (¹p+ 1 2 ¹ B 2 ) ¹B i ¹B j ¹¾ º ij ( ¹U)+ 1 2 hb2 i ± ij hb i b j i+¹½hu i u j i Turbulent Maxwell and Reynolds stresses: B = ¹B + b Turbulent viscosity: ¾ T ij = 1 2 hb2 i ± ij hb i b j i + ¹½hu i u j i; hu i u j i = 2 º t ¹S ij (¹U) hb i b j i = f 1 ( ¹B) ± ij + f 2 ( ¹B) ¹B i ¹B j ; hu i u j i = f 3 ( ¹B) ± ij + f 4 ( ¹B) ¹B i ¹B j ; hb 2 i ± ij = (3f 1 ( ¹B) + f 2 ( ¹B)) ± ij ;

F e i where Effective Lorentz Force = 1 2 r i(1 q p )¹B 2 +(¹B r)(1 q s )¹B i = r j ¾ ¹ B ij ¾ ¹ B ij = 1 2 ¹ B 2 ± ij ¹B i ¹B j + ¾ e ij ( ¹B); Turbulence contribution to Lorentz Force: ¾ e ij ( ¹B) = ¾ T ij ( ¹B) ¾ T ij ( ¹B = 0) 1 2 q p ¹ B 2 ± ij q s ¹B i ¹B j ; Turbulent Maxwell and Reynolds stresses: ¾ T ij = 1 2 hb2 i ± ij hb i b j i + ½hu i u j i;

Equation of State for Isotropic Turbulence

Total Turbulent Energy

Conservation of Total Turbulent Energy

Change of Turbulent Pressure

Equation of State for Anisotropic Turbulence

Effective Magnetic Pressure

Magnetic Fluctuations

Theory of Effective Lorentz Force

F e = 1 2 r(1 q p)¹b 2 + (¹B r)(1 q s )¹B

The Effective Lorentz Force

Effective Lorentz Force - DNS F e i where = 1 2 r i(1 q p )¹B 2 +(¹B r)(1 q s )¹B i = r j ¾ ¹ B ij ¾ ¹ B ij = 1 2 ¹ B 2 ± ij ¹B i ¹B j + ¾ e ij ( ¹B); Turbulence contribution to Lorentz Force: ¾ e ij ( ¹B) = ¾ T ij ( ¹B) ¾ T ij ( ¹B = 0) 1 2 q p ¹ B 2 ± ij q s ¹B i ¹B j ; Turbulent Maxwell and Reynolds stresses: ¾ T ij = 1 2 hb2 i ± ij hb i b j i + ½hu i u j i;

Effective Lorentz Force - DNS F e i = 1 2 r i(1 q p )¹B 2 + (¹B r)(1 q s )¹B i Total Maxwell and Reynolds stresses: ¾ total ij = 1 2 ¹ B 2 ± ij ¹B i ¹B j + 1 2 hb2 i ± ij hb i b j i+½hu i u j i; Method: ¹B = ¹B 0 e x ¾ T ij = 1 2 hb2 i ± ij hb i b j i + ½hu i u j i; ¾ T xx( ¹B) = 1 2 hb2 i hb 2 xi + ½hu 2 xi; q p ( ¹B) = 2 ¹B 2[¾T yy( ¹B) ¾ T yy( ¹B = 0)]; ¾ T yy( ¹B) = 1 2 hb2 i hb 2 yi + ½hu 2 yi; q s ( ¹B) = 1 2 q p( ¹B) 1 ¹B 2[¾T xx( ¹B) ¾ T xx( ¹B = 0)]

DNS: 3D Forced Non-Stratified Turbulence Re = u rms º k f = 180 ; Rm = u rms k f = 45 k f = 5 k 1 All simulations are performed with the PENCIL CODE, which uses sixth-order explicit finite differences in space and a third-order accurate time stepping method. BOUNDARY CONDITIONS are periodic in 3D. A white noise non-helical homogeneous and isotropic random forcing. An isothermal equation of state P = ½ c 2 with constant sound speed: s Volume averaging yield: hb i b j i; hu i u j i;

Effective Lorentz Force - DNS F e = 1 2 r(1 q p)¹b 2 + (¹B r)(1 q s )¹B

DNS: 3D Stratified Forced Turbulence Rm = u rms Pm = º = 1 4 ; 1 ; 1; 2; 4; 8; 2 k f = 35; 70; 140; k f = (5 10) k 1 ½ bot ½ top = 535; BOUNDARY CONDITIONS at the top and bottom: U z = 0; r z U x = r z U y = 0 B z = 0; r z B x = r z B y = 0 BOUNDARY CONDITIONS: 1).The horizontal boundaries are periodic. 2). For the velocity we apply impenetrable, stress-free conditions. 3). For the magnetic field we use perfect conductor boundary conditions.

Effective Lorentz Force - DNS F e = 1 2 r(1 q p)¹b 2 + (¹B r)(1 q s )¹B We perform horizontal averages which show a strong dependence on height: hb i b j i; hu i u j i; We also perform time averaging in order to improve the statistics.

DNS: 3D Turbulent Convection Re = u rms d 2¼ º = 40 100 ; Rm = u rms d 2¼ = 10 50 ; L x = L y = 5L z = 5d; ½ bot ½ top = 300; BOUNDARY CONDITIONS: 1).The horizontal boundaries are periodic. 2). We keep the temperature fixed at the top and bottom boundaries. 3) For the velocity we apply impenetrable, stress-free conditions. 4). For the magnetic field we use vertical field conditions. BOUNDARY CONDITIONS: U z = 0; r z U x = r z U y = 0 B z 6= 0; B x = B y = 0

Effective Lorentz Force DNS: Turbulent Convection F e = 1 2 r(1 q p)¹b 2 + (¹B r)(1 q s )¹B We perform horizontal averages which show a strong dependence on height. We also perform time averaging in order to improve the statistics.

Large-Scale MHD-Instability

Large-Scale MHD-Instability - NS ¹½ ¹F = 1 2 r(1 q p)¹b 2 + (¹B r)(1 q s )¹B

Numerical Simulations: 3D-Magnetic Structures

Magnetic Structures -3D

Magnetic Structures Simulations Sunspots

Large-Scale MHD-Instability - NS

2D-STRUCTURE

Magnetic Structures -2D

Mechanism of Large-Scale MHD-Instability Let us consider an isolated flux tube of magnetic field lines. If the flux tube is lighter than the surrounding fluid, it moves upwards. The reason for continued upward floating of the magnetic flux tube is as follows. The decrease of the magnetic field inside the ascending tube is due to its expansion. This is accompanied by an increase of the magnetic pressure inside the tube. The latter is caused by the negative effective mean magnetic pressure. The decrease of the magnetic field inside the tube results in a decrease of the fluid density and causes a buoyancy of the magnetic flux tube, i.e., the excitation of the large-scale instability. The instability causes the formation of magnetic structures. The energy for this instability is supplied by the small-scale turbulence. In contrast, the free energy in Parker's magnetic buoyancy instability, is drawn from the gravitational field.

2D-STRUCTURE in Turbulent Convection

Growth Rate of Large-Scale MHD-Instability in Turbulent convection Horizontal Magnetic Field Vertical Magnetic Field

Large-Scale MHD-Instability

Parameters (top of Solar Convective Zone)

Parameters (bottom of CZ)

References I. Rogachevskii and N. Kleeorin, Phys. Rev. E 76, 056307 (2007) A. Brandenburg, N. Kleeorin and I. Rogachevskii, Astron. Nachr. 331, 5-13 (2010) A. Brandenburg, K. Kemel, N. Kleeorin and I. Rogachevskii, Astrophys. J., submitted (2011) P. Käpulä, A. Brandenburg, N. Kleeorin, M. Korpi and I. Rogachevskii, Month. Notes R. Astron. Soc., to be submitted (2011)

Conclusions Generation of magnetic fluctuations in a turbulence with mean magnetic field results in a strong modification of the large-scale Lorentz force. This study is the first DNS demonstration of the effect of turbulence on the mean Lorentz force (in 3D forced stratified turbulence and in 3D turbulent convection). This phenomenon causes formation of the large-scale magnetic structures even in originally uniform mean magnetic field. 3D simulations show that these magnetic structures have bipolar structures which are broadly reminiscent of sunspots and AR. MHD effects with strong small-scale turbulence should be reconsidered by taken into account effect of turbulence on the mean Lorentz force.

THE END

Equation of State for Isotropic Turbulence

Magnetic Fluctuations and Turbulent Pressure

Theory of Effect of Turbulence on Lorentz Force

Magnetic Fluctuations

Magnetic Fluctuations and Anisotropic Effects

Vertical Distribution of Mean Magnetic Field in a Stratified Turbulence

Mean-Field Modeling of Magnetic Flux Tubes in a Stratified Turbulence

Large-Scale MHD-Instability

Large-Scale MHD-Instability: Horizontal Field

Large-Scale MHD-Instability : Horizontal Field

Large-Scale MHD-Instability: Vertical Field

References N. Kleeorin, I. Rogachevskii and A. Ruzmaikin, Sov. Astron. Lett. 15, 274-277 (1989) Sov. Phys. JETP 70, 878-883 (1990) N. Kleeorin and I. Rogachevskii, Phys. Rev. E 50, 2716-2730 (1994) N. Kleeorin, M. Mond and I. Rogachevskii, Phys. Fluids B 5, 4128-4134 (1993) Astron. Astrophys. 307, 293-309 (1996) I. Rogachevskii and N. Kleeorin, Phys. Rev. E 76, 056307 (2007) A. Brandenburg, N. Kleeorin and I. Rogachevskii, Astron. Nachr. 331, 5-13 (2010) A. Brandenburg, K. Kemel, N. Kleeorin and I. Rogachevskii, Astrophys. J., submitted (2011) P. Käpulä, A. Brandenburg, N. Kleeorin, M. Korpi and I. Rogachevskii, Month. Notes R. Astron. Soc., to be submitted (2011)