Download Photospheric processes and magnetic flux tubes

Solar Physics Winter School at Kodaikanal Solar Observatory, December 10–22, 2006 Photospheric processes and magnetic flux tubes Oskar Steiner Kiepenheuer-Institut für Sonnenphysik, Freiburg i.Br. [email protected] http://www.uni-freiburg.de/ ˜steiner toc ref Solar Physics Winter School at Kodaikanal Solar Observatory, December 10–22, 2006 The magnetic fine structure of the quiet solar photosphere Oskar Steiner Kiepenheuer-Institut für Sonnenphysik, Freiburg i.Br. [email protected] http://www.uni-freiburg.de/ ˜steiner toc ref Houses I and II of the Kiepenheuer-Institut in Freiburg toc ref § 1 The concept of magnetic flux tubes A magnetic flux tube or magnetic flux bundle is defined by the surface generated by the set of field lines that intersect a simple closed curve. Flux tubes are the building blocks of a magnetic configuration, but they must not be thought of as independent isolated structures. toc ref The concept of magnetic flux tubes (cont.) The magnetic flux, crossing a section S of the flux tube is given by F = Z S2 B · dS S F2 S1 F1 −F1 Z V zZ }| { Z z }| { Z 0 z }| { ∇ · B dx3 = − B ds + B ds + B · n̂ dσ = 0 S1 As a consequence of ∇ · B toc F2 ref = 0: S2 F1 = F2 . tube surface The concept of magnetic flux tubes : Examples of magnetic flux tubes: Coronal loop Coronal flux tubes in emission at Fe IX, 17.1 nm, corresponding to gas in the temperature range 600 000 to 1 200 000 K. From the TRACE homepage. Active regions observed on August Active region seen on May 19, 1998 19, 1998 at 171 Å at 171 Å toc ref The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot Sunspots as examples of photospheric magnetic flux tubes Scharmer & Langhans, Swedish Solar Telescope, SST, La Palma toc ref The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot Courtesy, N. Bello Gonzales German Vacuum Tower Telescope (VTT), Tenerife toc ref →Evershed flow The concept of magnetic flux tubes : toc ref The concept of magnetic flux tubes : Examples of magnetic flux tubes: Pore Pores have no penumbra and are much smaller than sunspots. Their size is that of one or several granules. 11:08:30 20 arcsec 15 10 5 0 White-light image from the 0 5 10 15 arcsec 20 Swedish vacuum tower Pore (left) and corresponding Dopplergram telescope. M. Sobotka et (right). A downdraft exists at the periphery of al. (1999) the pore (dark blue). Kashia Mikurda (KIS) toc ref The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot & magnetic element 70 60 seconds of arc 50 40 30 20 10 0 0 20 40 seconds of arc 60 80 Sunspot and network magnetic elements. Pit Sütterlin, Dutch Open Telescope, DOT toc ref § 2 The discovery of small-scale magnetic flux concentrations The advent of the magnetograph around 1950 (H.W. Babcock, K.O. Kiepenheuer) enabeled researchers to investigate ever smaller magnetic structures than pores, today generally called magnetic elements. → Zeeman effect → Stokes Polarimetry – In 1968 Beckers and Schröter (SP, 4, 142) discover small patches of magnetic field with a strength of 600 – 1400 Gauß and a diameter of around 1.3”, which they named magnetic knots. With a contrast of 0.88 in the continuum, they are barly visible in white light. They are located in active regions. → magnetic knot – Magnetic elements even smaller than knots exist, which cannot be resolved with a magnetograph. Their field strength must be determined by indirect methods, or by inversion techniques. toc ref → filling factor → line ratio § 2a Observations of magnetic flux concentrations at 0.1” 2 hours movie captured with the Swedish 1-m Solar Telescope (SST) on La Palma by Luc Rouppe van der Voort & Michiel van Noort, University of Oslo. “G-band continuum” with a 1 nm bandwidth filter centered at 436.4 nm. Trailing part of an active region. Rouppe van der Voort et al. 2005, A&A 435, 327 toc ref Observations of magnetic flux concentrations at 0.1” (cont.) “Ribbon-like” structure. Berger, Rouppe van der Voort, Löfdahl et al. 2005, A&A 428, 613 with the new 1 m Swedish Solar Telescope on La Palma toc ref Observations of magnetic flux concentrations at 0.1” (cont.) Contrast profile of a Observed filtergram with contrast profile of a “ribbon” medium-sized KGB-model. structure. Knölker & Schüssler 1988, A&A, 202, 275 ρe pe → more ρi pi+p mag flux tube boundary τ c =1 ‘solar surface’ → isothermes toc ref 200 km 1000 km Observations of magnetic flux concentrations at 0.1” (cont.) “Loops” and “flowers”. Berger, Rouppe van der Voort, Löfdahl et al. A&A 428, 613 with the Swedish Solar Telescope on La Palma. → more → 0.01” Magnetograms ? toc ref Observations of magnetic flux concentrations at 0.1” (cont.) G-band movie of a time span of 2 h 35” taken with from the HINODE (Solar-B) satellite toc ref § 3 What confines a magnetic flux tube? Consider an isolated magnetic flux tube, embedded in s^ flux tube boundary strength at the flux-tube boundary can be described with Bi internal θ(x) with θ(x) = 0 for x < 0 and θ(x) = 1 for x > 0. In the coordinates of the local frame given by ŝ and n̂, where ŝ is tangential to the magnetic field of the flux-tube surface, B is given by the step function Be ξ a field-free medium. The discontinuity in magnetic field n^ external B = (0, 0, Bi − [Bi − Be ]θ(ξ)). toc ref § 3 What confines a magnetic flux tube? Consider an isolated magnetic flux tube, embedded in s^ Bi flux tube boundary a field-free medium. The discontinuity in magnetic field strength at the flux-tube boundary can be described with θ(x) with θ(x) = 0 for x < 0 and θ(x) = 1 for x > 0. In the coordinates of the local ξ n^ internal frame given by ŝ and n̂, where ŝ is tangential to the magexternal netic field of the flux-tube surface, B is given by c j and B = (0, 0, Bi − [Bi − Be ]θ(ξ)). Applying Ampère’s law: ∇ × B = 4π ′ using θ (ξ) = δ(ξ) (Dirac’s δ -distribution) we get: c (0, [B]δ(ξ), 0), where [B] = Bi − Be . j= 4π Be the step function Integration over an ε-range and letting ε → 0, leads to the sheet current which flows perpendicular to the ŝ-n̂ plane. toc ref c j = [B] 4π ∗ What confines a magnetic flux tube? (cont.) Lorentz force z }| { 1 dv = −∇p + (j × B) +ρg⊙ From the equation of motion ρ dt c c and Ampère’s law: ∇ × B = j , we obtain in the static case 4π 1 ∇p = 4π ((∇ × B) × B) + ρg and further using the vector idendity 1 (∇ × B) × B = (B · ∇)B − ∇(B · B) 2 1 B2 ) + ρg + (B · ∇)B ∇p = −∇( 8π 4π toc ref What confines a magnetic flux tube? (cont.) Lorentz force z }| { 1 dv = −∇p + (j × B) +ρg⊙ From the equation of motion ρ dt c c and Ampère’s law: ∇ × B = j , we obtain in the static case 4π 1 ∇p = 4π ((∇ × B) × B) + ρg and further using the vector idendity 1 (∇ × B) × B = (B · ∇)B − ∇(B · B) 2 1 B2 ) + ρg + (B · ∇)B ∇p = −∇( 8π 4π We decompose the last term into a component parallel (ŝ) and perpens^ boundary ξ n^ toc ref = B ŝ follows: ∂ ∂ B · ∇ = B · ŝ · ∇ = B and further (B · ∇)B = (B )B = ∂s ∂s 2 ∂B ∂ B ∂ 2 ∂ŝ 2 n̂ ŝ + B = ( )ŝ + B , B (Bŝ) = B ∂s ∂s ∂s ∂s 2 Rc where Rc is the curvature radius of the field line. Thus, we obtain: dicular (main normal n̂) to a surface field line: From B What confines a magnetic flux tube? (cont.) B2 ∂ B2 B 2 n̂ ∇p = −∇( ) + ρg + ( ) · ŝ + 8π ∂s 8π 4π Rc Multiplication of this equation with n̂ yields the force balance perpendicular to the tube ∂p ∂ B2 B2 1 surface: =− ( ) + ρg · n̂ + ∂n ∂n 8π 4π Rc and integration over a small intervall [−ǫ, ǫ] across the tube surface gives: s^ flux tube boundary Bi pi Be pe ^ n ξ ε ε toc Zpe 2 Be /8π dp pi | {z } pe −pi Z = B̄ 2 B2 ) +ρg · n̂2ǫ + 2ǫ d( 8π 4πRc Bi2 /8π | {z 1 (B 2 −B 2 ) − 8π e i from which we obtain: ref } Be2 Bi2 pe + = pi + 8π 8π § 4 A microscopic picture of the sheet current Stronger magnetic field, smaller radius of curvature + gradient dB0 dy Electrons and ions in a magnetic field with a transverse gradient, Magnetic field out of page − showing gradinet B drift motion. After Krall & Trivelpiece, 1973 Weaker magnetic field, larger radius of curvature toc ref § 4 A microscopic picture of the sheet current Stronger magnetic field, smaller radius of curvature + gradient dB0 dy Electrons and ions in a magnetic field with a transverse gradient, Magnetic field out of page showing gradinet − B drift motion. After Krall & Trivelpiece, 1973 Weaker magnetic field, larger radius of curvature Magnetic field out of plain Higher density of plasma Net current Lower density of plasma toc ref Drift current in a plasma with a density gradient. 1111 0000 0000 1111 0000 1111 Due to the inbal- ance of gyrating particles, a current results without a net transport of charges. After Krall & Trivelpiece, 1986 A microscopic picture of the sheet current (cont.) Higher density of plasma − − − Field−free plasma − − − Lower density of plasma net current Magnetic field out of page At the boundary of an isolated flux tube we have both effects: Drift current because of a sharp gradient in magnetic field. This drift current is not cancelled by gyrating particles within the flux tube because of the reduced particle number-density there. toc ref § 5 The equations for a hydrostatic flux tube In this paragraph, we derive the equations for computing the magnetic structure of a vertical, axisymmetric flux tube without twist. toc ref The equations for a hydrostatic flux tube (cont.) Consider a vertical axisymmetric flux tube without twist, embedded in an external medium of pressure pe . The magnetohydrostatic system of equations to be solved is: 0 = −∇p + ρg + j × B , (1) ∇ × B = 4πj , (2) ∇·B=0. (3) We decompose (1) in components parallel and perpendicular to the magnetic field. Thus, multiplying (1) by B gives B · (∇p − ρg) = 0 , (1-a) and taking the cross product of (1) with B gives j= 1 B × (∇p − ρg) , 2 B where we used that B × (j × B) toc ref = B 2 j − (B · j) · B = B 2 j (1-b) The equations for a hydrostatic flux tube (cont.) Using cylindrical„ coordinates and the solenoidality of the magnetic field « 1 ∂Az ∂Aφ ∂Ar ∂Az 1 ∂ ∂Ar B=∇ × A= − , − , ( (rAφ ) − ) , r ∂φ ∂z ∂z ∂r r ∂r ∂φ « „ 1 ∂ ∂Aφ , 0, (rAφ ) for an axisymmetric flux tube which reduces to B = − ∂z r ∂r « „ 1 ∂Ψ 1 ∂Ψ and further without twist. With Ψ := rAφ ⇒ B = − , 0, r ∂z « r ∂r „ 1 ∂2Ψ 1 ∂Ψ 1 ∂2Ψ ∇×B= 0, − + 2 , 0 so that Ampères law for the − 2 2 r ∂z r ∂r r ∂r φ-component (the r- and z -components being zero), becomes (Grad-Shafranov): ∂2Ψ 1 ∂Ψ ∂2Ψ − = −4πrjφ + ∂r2 r ∂r ∂z 2 The magnetic field components can be recovered from the scalar potential Ψ by: 1 ∂Ψ 1 ∂Ψ Br = − , Bz = . r ∂z r ∂r Note that curves of Ψ = const. describe field lines of the system. toc ref The equations for a hydrostatic flux tube (cont.) If gravity acts along the negative z -axis and s measures the δz s^ θ distance along magnetic field lines, inclined at an angle θ to δs n^ the vertical: g B · (∇p − ρg) = 0 ⇒ dp ds + ρg cos θ = 0 Ψ p = nkB T = (ρ/m̄)kB T ⇒ ρ = m̄p/(kB T ), thus, dp m̄g dp m̄g 1 m̄g + p=0 ⇒ =− dz With := and dz kB T p kB T H kB T integration from a reference height 0 to z we obtain  p = p0 (Ψ) exp − ZΨ,z Ψ,0 toc ref dz ′ H(T (Ψ, z ′ ))   The equations for a hydrostatic flux tube (cont.) In component notation B Using it in Eq. (1-b): = (Br , 0 , Bz ) and ∇p = (∂p/∂r , 0 , ∂p/∂z). 1 jφ = 2 B From » – ∂p ∂p p Bz − Br ( + ) . ∂r ∂z H 2 p = p0 (Ψ) exp 4− ZΨ,z dz H(T (Ψ, z ′ )) Ψ,0 ˛ ∂p p ∂p ˛˛ ∂Ψ we obtain =− + ∂z H ∂Ψ ˛z ∂z above equation for jφ reduces to and ˛ ∂p ˛˛ jφ = r ∂Ψ ˛z toc ref ′ 3 5 , ˛ ∂p ∂p ˛˛ ∂Ψ = so that the ˛ ∂r ∂Ψ z ∂r The equations for a hydrostatic flux tube (cont.) At the surface of the tube the magnetic field will, in general, be discontinuous, thus resulting in the existence of a sheet current there. Using 1 1 2 2 pe − pi = (Bi − Be ) = (Bi − Be )(Bi + Be ) 8π 8π and from Ampère’s law c (Bi − Be ) j = 4π ∗ we finally get jφ∗ toc ref 2(pe − pi ) = Bi + Be The equations for a hydrostatic flux tube: Solution procedure 1. Specify initial magnetic configuration 2. Calculate pressure distribution consisten with this field configuration 2 p = p0 (Ψ) exp 4− ZΨ,z dz ′ H(T (Ψ, z ′ )) Ψ,0 3 5 3. Evaluate the volume and sheet current respectively ˛ ∂p ˛˛ jφ = r ∂Ψ ˛z and jφ∗ = 2(pe − pi ) Bi + Be 4. Integrate Ampère’s equation ∂2Ψ 1 ∂Ψ ∂2Ψ − = −4πrjφ + 2 2 ∂r r ∂r ∂z and go back to step 2 and iterate 2–4. toc ref The equations for a hydrostatic flux tube (cont.) Boundary conditions for solving Ampère’s equation: – 0 W Ψ=0 L Ψ = r B0 z (r) dr merging with neighbouring flux tube Ψ =0 z W is given by the magnetic filling factor f : p W = R0 / f – Free boundary problem z r 0 0 R0 r Ψ = r B0 z (r) dr 0 toc ref W The equations for a hydrostatic flux tube (cont.) Pizzo (1990) used a “body-fitted” nonorthogonal coordinate system to map the physical domain into a unit square computational domain. A multigrid elliptic solver is used at each iteration stage for solving Ampère’s equation. Fiedler & Cally (1991) use a similar mesh in which contours of constant Ψ (field lines) constitute one coordinate, the normalized arc length along field lines the second one. Computational meshes for a model sunspot. Left: Constant arc length collocation. Right: Mesh with auxiliary “internal gridding surface”. From Pizzo, 1990 toc ref The equations for a hydrostatic flux tube (cont.) Jahn (1989) and Jahn & Schmidt (1994) use a similar method for the construction of sunspot models. From Jahn & Schmidt (1994) toc ref The equations for a hydrostatic flux tube (cont.) In case of horizontal temperature equilibrium, Ti (z) = Te (z) = T (z), we have (neglegting any horizontal variation in ionization degree) Hi (z) = He (z) = H(z). Then, 2 pi (z) = p0i exp 4− Zz dz ′ H(T (z ′ )) 0 from which follows that pi (z) 3 2 5 pe (z) = p0e exp 4− Zz dz ′ H(T (z ′ )) 0 < pe (z) ∀z assuming that pi does not depend on radius (thin tube approximation) because p0e − p0i = B02 /8π > 0. Since Ti (z) = Te (z) it follows for the densities that ρi (z) < ρe (z) ∀z . Since above conditions can be expected to hold very well in the photosphere we can say that photospheric flux tubes are rarefied , one also says partially evacuated . toc ref 3 5, § 6 The magnetic structure of a hydrostatic flux tube Consider the most simple case of a constant axial field strength and gas pressure at the base level, z0 , of the flux tube, where the radius is R0 . Then 8 < Bz0 r2 /2 for r ≤ R0 Ψ(r, z0 ) = : Bz0 R02 /2 for r ≥ R0 from Bz = 1 ∂Ψ r ∂r and from pe − pi = , pi0 2 2 (R0 ) + Br0 Bz0 = pe0 − 8π 1 (Bi2 − Be2 ). Note, that the final 8π Br0 (R0 ) is not known from the very beginning so that the boundary condition for the pressure needs to be adjusted in the course of the iteration. If we further assume that the temperature at a given hight level is constant, then the gas ˛ ∂p ˛˛ =0 pressure is constant too, so that the volume current is jφ = r ˛ ∂Ψ z and we have a potential field inside the flux tube. However, the sheet current at the tube surface, toc jφ∗ 2(pe − pi ) = , remains. Bi + Be ref The magnetic structure of a hydrostatic flux tube (cont.) Flux tube with a field strength of 1500 Gauß and a radius of 100 km at the base of the photosphere (τc = 1). Superimposed are plots of the radial variation of Br (dashed curve) and of Bz (solid curve) at different heights, both normalized to the value of Bz at the axis, indicated in Gauß. The flux tube merges with neighbouring fluxtubes at a height of ≈ 500 km. The filling factor is f = (R0 /W )2 = 0.1. From Steiner, Pneuman & Stenflo (1986) toc ref The magnetic structure of a hydrostatic flux tube (cont.) Left: A flux tube with non-uniform base pressure and magnetic field strength. p0 (r) increases parabolically from its value at the axis to three times that value at the surface. Right: The electric current shows a volume component in the interior and a sheet current at the surface of the flux tube. From Steiner, Pneuman & Stenflo (1986) toc ref The magnetic structure of a hydrostatic flux tube (cont.) Introducing the scalar function G = rBφ one can treat the case of an axisymmetric flux tube with twist. Ampère’s law then becomes „ If G 2 2 ∂G ∂ Ψ 1 ∂Ψ ∂G ∂ Ψ − , − , + ∂z ∂r2 r ∂r ∂z 2 ∂r « = −4πrj . = G(Ψ) (torque-free condition) we need only solve the φ-component, for which: ˛ 1 ∂p ˛˛ ∂G + jφ = r G . ˛ ∂Ψ z 4πr ∂Ψ In the absence of an external magnetic field, the magnitude of the sheet current 2(pe − pi ) , directed perpendicular to the field Bi lines at the surface and, hence, no longer purely azimuthal. The φ-component of the remains the same as before: |j∗ | = sheet current is: jφ∗ = 2(pe − pi ) Bi r 1−( The solution procedure remains the same as before. toc ref Bφ 2 ) . Bi The magnetic structure of a hydrostatic flux tube (cont.) Uniformly twisted flux tube with B0 τ0 r for r ≤ R0 and B0φ = R0 τ0 = 0.3. Right: Distribution of Bz and Bφ . Left: 8 field lines 45◦ apart on the flux-tube surface. From Steiner, Pneuman & Stenflo (1986) toc ref The magnetic structure of a hydrostatic flux tube (cont.) Curiously, there is a maximum twist that can be applied to such a flux tube. For the region far enough above the merging height the radial component of the magnetic field vanishes and the gas pressure becomes negligible so that Ampères law reduces to 1 ∂Ψ ∂G ∂2Ψ − + G =0. 2 ∂r r ∂r ∂Ψ For the case of a uniform twist at the flux-tube base, when G(Ψ) = 2τ0 Ψ, the R0 solution of this equation is 2τ0 Ψ = C1 rJ1 ( r) , R0 which has a maximum at rmax = with λ λR0 , 2τ0 ≈ 2.4 being the first zero oft the zeroth order Bessel function J0 . rmax may not be surpassed by the maximal tube radius, W , as long as we have have no return flux. toc ref The magnetic structure of a hydrostatic flux tube (cont.) Hence, for a given allowed flux tube expansion defined by the filling factor f = (R0 /W )2 , we must have λp τ0 ≤ f. 2 Physically, this limit can be understood in terms of the force balance between a point on the axis and a point where the maximum twist occurs. At the axis there is no twist and Bz2 /(8π) alone must be in force balance with the magnetic pressure excerted by Bφ max . toc ref § 7 The magnetic canopy The expansion rate of hydrostatic flux tubes critically depends on the combination of external to internal atmosphere. The figure to the left shows the expansion if we arbitrarily increase the internal temperature for z (above τc ≥0 = 1) by a factor, indicated for each curve. Likewise, the figure to the right shows expansion rates for various plasma β and two different temperature factors. Note the horizontal spreading of the flux-tube surface at low altitudes in the most extreme cases. From Steiner & Pizzo (1989) toc ref The magnetic canopy (cont.) A critical hight at which the flux tube must expand in horizontal direction can be derived from the expressions for gas pressure close to the tube surface along a magnetic field line, pi , and in the surrounding atmosphere, 0 1 pe : pi,e (z) = p0i,e exp @− B 02 ln 8π 0 ln p p 0e i e z crit Zz 0 dz ′ A . ′ Hi,e (z ) If Te < Ti ⇒ He < Hi and the pressure difference, pe −pi , and with it the surface field strength, decreases rapidly with height. Correspondingly, the flux tube expands until a critical height, zcrit , at which the field assumes a horizontal direction. toc ref The magnetic canopy (cont.) A critical hight at which the flux tube must expand in horizontal direction can be derived from the expressions for gas pressure close to the tube surface along a magnetic field line, pi , and in the surrounding atmosphere, 0 1 pe : pi,e (z) = p0i,e exp @− zZcrit Zz 0 dz ′ A . With pe (zcrit ) = pi (zcrit ) ′ Hi,e (z ) (β0 +1)/β0 1 1 p0e ( − )dz = ln He Hi p0i 0 B 02 ln 8π 0 ln p p 0e i e z crit If H=const. ⇒ zcrit z}|{ Hi He p0e = ln Hi − He p0i Te < Ti ⇒ He < Hi and the pressure difference, pe −pi , and with it the surface field strength, decreases rapidly with height. Correspondingly, the flux tube expands until a critical height, zcrit , at which the field assumes a horizontal direction. toc ref The magnetic canopy (cont.) A strongly expanding flux tube with a low critical height ensues when embedded in a cool atmosphere with a temperature that decreases beyond the traditional temperature minimum (COOLC or RE in the figure on the right). Such atmospheres were suggested on the basis of infrared observations in lines of CO. The figure to the left shows the gas pressure of the RE-atmosphere (dot-dashed) in combination with the ′ gas pressure of the flux-tube atmosphere (C of the above figure), for various base field-strengths. Critical heights of 780, 860, and 1000 km correspond to base field-strengths of 1300, 1500, and 1700 Gauß. toc ref The magnetic canopy (cont.) ′ Flux tube with a C -atmosphere embedded in the cool RE-atmosphere. The base field-strength is 1500 Gauß. The fieldlines spread into a horizontally extending canopy field at a height of 900 km and merge with the field of neighboring flux tubes. toc ref The magnetic canopy (cont.) Chromospheric magnetograms of a unipolar network region show near the limb a fringe pattern in polarity. They show a polarity inversion across a line that coincides with the limbward edge of a unipolar magnetic network field. The figure shows an example magnetogram taken in the line of Mg I b2 (5173 Å). In the vicinity of “C” one can see a fringe pattern. In the photosphere, this region sows a unipolar enhanced network. From R.G. Giovanelli: 1980, SP 68, 49 The fringe pattern can be understood in terms of the magnetic canopy. The canopy “roots” in the network and overlies the internetwork region. toc ref The magnetic canopy (cont.) ff θ zc ff 0 τ 50 limb b =1 a From Steiner (2000) Lines of sight passing through the network field receive a magnetic field component toward the observer and so do lines of sight passing through the canopy field to the left (disk center) side of the network (dashed line of sight). The canopy field to the right (limbward) side of the network, however, gives rise to a line of sight component of opposite direction (solid line of sight), which causes the fringe of opposite polarity. toc ref The magnetic canopy (cont.) The corresponding magnetogram of the magnetic network at the photospheric level would be of exclusively positive polarity. Towards the limb, the network fields apparently move closer together with increasing line-of-sight aspect angle θ , giving rise to an ever narrower fringe pattern of alternating polarity in the magnetogram. From observations, such as shown above, Giovanelly & Jones derived canopy heights of 600–1000 km in quiet-Sun regions and as low as 200 km in active regions. toc ref § 8 The thin flux-tube approximation The approximation of thin flux tubes is valid as long as all length scales along the tube are large in comparison to the tube diameter. For an axisymmetric vertical tube this means that R/H << 1, kR << 1 , where H is the scale height of any quantity, e.g., pressure in the tube, k is the vertical wave number of any perturbation propagating along the tube, and R the tube radius. In this case we can expand all physical quantities in radial direction and fully maintain their height dependency. toc ref The thin flux-tube approximation (cont.) Consider a magnetohydrostatic, axisymmetric, vertical magnetic flux tube and assume, that we can neclect any variation of Bz in radial direction (zeroth order approximation). If the temperature, and hence the pressure scale height Hp , is the same inside and outside the tube, then 2 B0z /8π z }| { z Bz2 pe − pi = (pe0 − pi0 ) exp(− )= Hp 8π from which we can compute Bz (z). Using the conservation of magnetic flux Φ = 2π R(z) R Bz (z)rdr = 2πR2 (z)Bz (z) = const = 2πR02 (z)Bz 0 we can compute the flux-tube radius as a function of height and finally obtain z z ) and Bz (r, z) = B0z exp(− ). R(z) = R0 exp( 4Hp 2Hp It takes about four pressure scale heights for the tube radius to expand by a factor of e. toc ref The thin flux-tube approximation (cont.) In order to satisfy solenoidality we can compute the first order term of the radial field expansion. With Br (r) = Br1 r we obtain 1 ∂ Bz (z) ∂Bz 1 ∇·B= (rBr ) + = 0 = 2Br − , r ∂r ∂z 2Hp hence, Br (r, z) = Bz (z) r. 4Hp However, note, that the boundary condition Bz2 z is zeroth-order accurate only. )= pe − pi = (pe0 − pi0 ) exp(− Hp 8π toc ref The thin flux-tube approximation (cont.) Generally, we can expand the full system of the MHD-equation: ∇·B=0, ∂B = ∇ × (v × B) , ∂t ∂ρ + ∇ · (ρv) = 0 , ∂t ∂v 1 ρ + (v · ∇)v = − ∇p + ρg + (∇ × B) × B , ∂t 4π γp ∂ρ ∂p + v · ∇p = + v · ∇ρ . ∂t ρ ∂t toc ref The thin flux-tube approximation (cont.) All pertaining quantities are expanded as a power series in r : vr (r, z, t) = rvr1 (z, t) + r3 vr3 (z, t) + · · · , vφ (r, z, t) = rvφ1 (z, t) + r3 vφ3 (z, t) + · · · , vz (r, z, t) = vz0 (z, t) + r2 vz2 (z, t) + · · · , Br (r, z, t) = rBr1 (z, t) + r3 Br3 (z, t) + · · · , Bφ (r, z, t) = rBφ1 (z, t) + r3 Bφ3 (z, t) + · · · , Bz (r, z, t) = Bz0 (z, t) + r2 Bz2 (z, t) + · · · , pz (r, z, t) = p0 (z, t) + r2 p2 (z, t) + · · · , ρz (r, z, t) = ρ0 (z, t) + r2 ρ2 (z, t) + · · · . toc ref The thin flux-tube approximation (cont.) The fact that the radial and the azimuthal components of v and B are odd series in r , while the axial components vz and Bz as well as the scalars p and ρ are even series is a general property of scalars and vector fields of an axisymmetric system. This was shown by Ferriz Mas & Schüssler, 1989. Introducing this expansion into the system of MHD-equations and sorting the different orders (k = 0, 1, . . . , 2n), one obtains a system of 9n + 5 equations (where primes denote derivatives after z ): toc ref The thin flux-tube approximation (cont.) Even orders: k = 0, 2, . . . , 2n ∂Bzk − =(k + 2) ∂t ∂ρk − =(k + 2) ∂t X i+j=k = −p′k ∂vzj + ρi ∂t 1 − gρk + 4π X (4 × (n + 1) equations) X i+j=k+1 X (vri Bzj − Bri vzj ) , ρi vri + i+j=k+1 ′ ρi vzj vzl i+j+l=k+1 (ρi vzj )′ , i+j=k + X jρi vzj vrl i+j+l=k+1 i+j+l=k X X iBzi Brj 1 X ′ ′ ), + Bri Brj − (Bφi Bφj 4π i+j=k X ∂ρi ∂pj −γ pj ) + (j − iγ)ρi pj vrl (ρi ∂t ∂t i+j+l=k+1 i+j=k X + (ρi p′j − γρ′i pj )vzl = 0 . X i+j+l=k toc ref The thin flux-tube approximation (cont.) Odd orders: k = 1, 3, . . . , 2n − 1 (4 × n equations) X ∂Brk − = [(vzi Brj )′ − (Bzi vrj )′ ] , ∂t i+j=k X X ∂Bφk ′ ′ − = [(vzi Bφj ) − (Bzi vφj ) ] + (k + 1) (vri Bφj − Bri vφj ) , ∂t i+j=k X i+j=k ∂vrj ρi + ∂t i+j=k+1 X ′ ′ vzl ρi vrj X + i+j+l=k+1 i+j+l=k ρi (lvrj vrl − vφj vφl ) 1 X [jBzi Bzj + (1 + j)Bφi Bφj ] , = −(k + 1)pk+1 + 4π i+j=k X i+j=k ∂vφj ρi + ∂t X ′ ρi vzj vφl i+j+l=k 1 X 1 ′ = Bzi Bφj + 4π 4π i+j=k toc ref + X (l + 1)ρi vrj vφl i+j+l=k+1 X i+j=k+1 (j + 1)Bri Bφj . The thin flux-tube approximation (cont.) These equations are complemented with the solenoidality condition: Even orders: k = 0, 2, . . . , 2n ′ + (k + 2)Br(k+1) = 0 . Bzk However, these equations are not independent. Instead use the potential A for the poloidal part od B: ∂A 1 ∂(rA) Bp = (Br , 0, Bz ) = (− , 0, . ∂z r ∂r Then, the r - and z -components of the induction equation reduce to the same expression: − vr ∂(rA) ∂A = vz A′ . ∂t r ∂r The expansion of A is of the form A(r, z, t) = rA1 (z, t) + r3 a3 (z, t) + . . . toc ref The thin flux-tube approximation (cont.) The solenoidality condition as well as the r -components of the induction equation drop out and are replaced by the above equation for A, where Br(2m+1) = −A′2m+1 , Bz(2m) = (2m + 2)A2m+1 m = 0, 1, 2, . . . , n . The MHD equations are then reduced to a system of 7n + 4 first order partial differential equations for 7n + 5 unknown functions (one more because of Bz2n that is not given by the n odd order equations for A(r, z, t)). To close the system one has to specify the kind of problem one wishes to consider with the corresponding boundary conditions. These are, e.g., for a magnetic flux tube embedded in a field-free plasma: ˛ B ˛ p+ 8π ˛ 2˛ toc ref = pe (R) . r=R The thin flux-tube approximation (cont.) Applying the recipe outlined above to leading order one obtains the following 5 equations for the 5 unknown functions ρ0 , p0 , A1 , vz0 , and vr1 : ′ ) = −p′0 − ρ0 g , ρ0 (v̇z0 + vz0 vz0 (1) ρ̇0 + (ρ0 vz0 )′ + 2ρ0 vr1 = 0 , (2) Ȧ1 = −vz0 A′1 − 2vr1 A1 , p0 γ ′ ṗ0 + vz0 p0 = (ρ˙0 + vz0 ρ′0 ) , ρ0 2 2 p0 + A1 = pe . 4π (3) (4) (5) Note that even in the lowest order, the system will keep essential non-linearities of the full MHD-equations. toc ref The thin flux-tube approximation (cont.) In the hydrostatic case, Eqs. (1)–(5) reduce to −p′0 − ρ0 g = 0 , 1 ′ p0 + Bz0 = pe , 8π γp0 d p0 = = c2 , dρ0 ρ0 which is, together with Φ = 2πR2 (z)Bz0 = const what we have derived at the beginning of this paragraph. toc ref The thin flux-tube approximation (cont.) This figure demonstrates for a vertical hydrostatic flux tube embedded in a standard solar atmosphere the deviation of the zeroth-order flux-tube approximation from the full (numerical) solution as a function of height and spatial scale . Each curve shows the height at which the field strength at the tube wall deviates from the axial value by the specified fractional amount D = |Bz (r = 0) − Bz (r = R)|/Bz (r = 0) as a function of the tube radius Rτe =1 . toc ref § 9 Magnetic flux tube in radiative equilibrium There are two basic modes of energy transport in the solar photosphere and convection zone: radiative and convective. When all of the energy is transported by radiation, we have radiative equilibrium, conversly, pure convective transport is called convective equilibrium. In a stationary transport process, the frequency distribution of the radiation, or the partioning of energy between the radiative and the convective mode of transfer, may be altered; but the energy flux as a whole is rigorously conserved. Formally, this is expressed by ∇ · Ftot = 0; Ftot = Frad + Fconv . In radiative equilibrium: ∇ · Frad = 0 , Fconv = 0 , In convective equilibrium: ∇ · Fconv = 0 , Frad = 0 . toc ref Magnetic flux tube in radiative equilibrium (cont.) In the solar photosphere radiative energy transfer by large prevails so that ∇ · Frad = 0 is a good approximation. With I(r, n̂, ν) being the radiative intensity with frequency ν propagating in direction n̂ at location r in a multidimensional coordinate frame, the total radiative flux is given by Frad = Z Z∞ I(r, n̂, ν)n̂ dν dω . 4π 0 I(r, n̂, ν) has dimension ergs cm−2 s−1 hz−1 sr−1 , correspondingly has Frad dimension ergs cm −2 −1 s . The radiation field follows as a solution of the radiative transfer equation (n̂ · ∇)I(r, n̂, ν) = η(r, ν) − κ(r, ν)I(r, n̂, ν) . The emissivity η(r, ν) is given by η(r, ν) function. toc ref = κ(r, ν)S(r, ν) with S being the source Magnetic flux tube in radiative equilibrium (cont.) From these equations we obtain: Z Z∞ (n̂ · ∇)I(r, n̂, ν) dν dω = ∇ · Frad 4π 0 Z Z∞ ! (κ(r, ν)S(r, ν) − κ(r, ν)I(r, n̂, ν) = 0 . = 4π 0 1 With J(r, ν) = 4π Z I(r, n̂, ν) dω being the mean intensity, we finally obtain the 4π constraint equation for radiative equilibrium: Z∞ 0 toc ref κ(r, ν)S(r, ν) dν = Z∞ 0 κ(r, ν)J(r, ν) dν . Magnetic flux tube in radiative equilibrium (cont.) In general we do not know the temperature distribution T (r) that satisfies radiative equilibrium. If we start with a guess T (0) (r) for which we have calculated the correct source function S(r, ν), we will find that the constraint equation for radiative equilibrium is not satisfied. It is therefore necessary to iteratively adjust T (r) until the requirement of radiative balance is satisfied. toc ref Magnetic flux tube in radiative equilibrium (cont.) Assume a given temperature distribution T (r) of the magnetohydrostatic configuration for which we also know the pressure p(r) and density ρ(r). This allows us to derive LTE-values of opacities κ(r, ν). We then are able to compute the radiation field at any point r and any frequency ν by evaluation of the formal solution of the radiative transfer equation Jν (r) = Λν (r, r′ )Bν (r′ ) + Gν (r) , where Λν is the integral operator which adds the intensities at r caused by emision at ′ all the points r in the considered computational domain, and where Gν is the transmitted mean intensity due to the incident radiation field into this domain. We also use strict local thermodynamic equilibrium (LTE) (Sν = Bν ), although a scattering component could be included in the solution method that follows. toc ref Magnetic flux tube in radiative equilibrium (cont.) Defining the integral operator K so that Kφ := R∞ κν φdν , where φ is a scalar 0 function, we can now write the constraint equation for radiative equilibrium as: KBν = KΛν Bν + KGν However, with a given initial temperature distribution, T (0) (r), this equation will in general not be satisfied but we can compute a correction ∆T (r) demanding that KBν (T (0) + ∆T ) = KΛν Bν (T (0) ) + KGν Expanding Bν (T + ∆T ) ≈ Bν (T ) + (∂Bν /∂T )|T ∆T , we can compute the temperature correction: ∆T = toc ref K(Λν − I)Bν (T ) + KGν K(∂Bν /∂T )|T Magnetic flux tube in radiative equilibrium (cont.) For most practical purposes this iterative scheme is very slow. The next better scheme to use is a Jacobi-like iteration, in radiative transfer known as accelerated Λ-iteration : ∆T = K(Λν − I)Bν (T ) + KGν , ∗ K(1 − λν )(∂Bν /∂T )|T ∗ where λν is the diagonal element of the matrix representing Λν (r). Note that ∆T = ∆T (r) and that the above equation must be evaluated for each r. The formal solution of the transfer equation must be computed in each iteration. An efficient way to do this consists in using the so called method of short characteristics and exploiting the symmetry properties . toc ref Magnetic flux tube in radiative equilibrium (cont.) Two-dimensional slab atmosphere. The rectangle in the upper left corner represents a magnetic flux sheet with a rarified atmosphere. The incident radiation on the bottom and on the right side of the computational domain are computed from the undisturbed, plane-parallel atmosphere with κ = κe . The left side is an axis of symmetry of the configuration. The top boundary has no incident radiation. κ e = const I in = I undisturbed toc ref I in = I undisturbed κ i = 0.2 κ e I in = I out axis of symmetry I in = 0 Magnetic flux tube in radiative equilibrium (cont.) Two-dimensional slab atmosphere. The rectangle in the upper left corner represents a magnetic flux sheet with a rarified atmosphere. The incident radiation on the bottom and on the right side of the computational domain are computed from the undisturbed, plane-parallel atmosphere with κ = κe . The left side is an axis of symmetry of the configuration. The top boundary has no incident radiation. Right: Resulting isotherms. κ e = const I in = I undisturbed toc ref I in = I undisturbed κ i = 0.2 κ e I in = I out axis of symmetry I in = 0 Magnetic flux tube in radiative equilibrium (cont.) Magnetic flux tube with Bz (r = 0, z = 0) = 1600 Gauß and a radius of z [km] R(z = 0) = 100 km in radiative equilibrium. Contour lines of 4800 constant temperaure and 6000 7200 8400 are r [km] toc ref indicated. Magnetic flux tube in radiative equilibrium (cont.) Magnetic flux tube with Bz (r = 0, z = 0) = 1600 Gauß and a radius of z [km] R(z = 0) = 100 km log τ in radiative equilibrium. −2.0 Contour lines of 4800 −1.0 constant temperaure and constant optical −2.0 6000 0.0 7200 −1.0 8400 depth 1.0 are r [km] toc ref indicated. Magnetic flux tube in radiative equilibrium (cont.) Magnetic flux tube with Bz (r = 0, z = 0) = 1600 Gauß and a radius of z [km] R(z = 0) = 100 km log τ in radiative equilibrium. −2.0 Contour lines of 4800 −1.0 constant temperaure and constant optical −2.0 6000 0.0 depth and the domain of 1.0 prescribed, fixed 7200 −1.0 8400 temperature are r [km] toc ref indicated. Magnetic flux tube in radiative equilibrium (cont.) Why is there a temperature elevation in the photospheric layers of a magnetic flux tube? Consider two points P1 and P2 . P1 receives a higher intensity of radiation from the flux tube’s hot walls as compared to P2 . Both points receive no radiation from the top. Therefore, the mean radiation is higher in P1 than in P2 , J1 radiative equilibrium J =S= R > J2 . In Bν dν ∝ T 4 and hence, T1 > T2 . P1 P2 P3 P4 τ c =1 toc ref Magnetic flux tube in radiative equilibrium (cont.) Temperature profile along the flux tube axis, left as a function of log τc , right as a function of height. Dashed and dot-dashed curve correspond to the temperature along the axis of flux tubes with a field strength of 1500 and 1600 Gauß at z = 0, respectively. The dotted curve is from the semi-empirical model of Keller et al. 1989. The solid curve represents the surrounding atmosphere. toc ref Magnetic flux tube in radiative equilibrium (cont.) Semi-empirical atmospheres for plage flux-tubes (dottet) and network flux-tubes (dashed) by Solanki & Brigljević (1992). The solid line corresponds to model C (average quiet Sun) of Fontenla et al. (1999). toc ref § 10 The physics of faculae The magnetic elements of the network are hardly visible at disk center, except in potospheric spectral lines or in the wings of chromospheric lines, or in special wavelength ranges like the G-band. They become increasingly brighter towards the limb for heliographic angles µ = cos(θ) larger than about 0.6, where they appear as extended bright areas, called faculae. Facular points of an area of 20 × 27 2 arcsec at disk center. Photograp by Mehltretter (1974 !) with the R. B. Dunn Telescope on Sacramento Peak at 393.4 ± 0.8 nm toc ref The physics of faculae (cont.) Speckle reconstructed image of facular region taken with the 1 m Swedish Solar Telescope in the continuum at 487.5 nm. Field of view approximately 80′′ × 80′′ . From Hirzberger & Wiehr (2005), A&A 438, 1059 toc ref The physics of faculae (cont.) There exists a long list of center to limb measurements of the continuum contrast of faculae. The measurements are contradictory because they are spatial resolution dependent and because of selection effects. There exists an equally long list of models most notable the “hot wall” model of Spruit (1976). From Spruit (1976), Sol. Phys. 50, 269 toc ref The physics of faculae (cont.) Center-to-limb variation (CLV) of continuum contrast (500 nm) Continuum contrast (575 nm) of 880 net- of a flux tube in radiative equi- work bright points as a function of helio- librium as shown in § 9.8 along graphic angle. Squares represent the con- with observed (dashed and trast of granulation. From Auffret and Muller dot-dashed) values. (1991), Observatoire du Pic du Midi toc ref The physics of faculae (cont.) Faculae at θ = 61◦ in the continuum at 587.5 nm (left) and in the G-band (right). Solar ′′ limb is right. Tickmarks indicate 1 distances. The facular brightening occurs on the disk-center side of granules limbward of a dark “facular lane”. From Hirzberger & Wiehr (2005), A&A 438, 1059 toc ref The physics of faculae (cont.) Mean spatial scan through faculae at θ = 61◦ in the 587.5 nm continuum (top) and in the G-band (bottom). Note the flat limbward decrease and the centerward dark “facular lane”. From Hirzberger & Wiehr (2005), A&A 438, 1059 toc ref The physics of faculae (cont.) - From a location at the solar surface and lateral to the flux sheet one “sees” a more transparent sky in the direction to the flux sheet compared to a direction under equal zenith angle but away from it. τ c =1 - Correspondingly, from a wide area surrounding the magnetic flux sheet or flux tube, radiation escapes more easily in the direction of the flux sheet/tube. - A single flux sheet/tube impacts the radiative escape in a cross-sectional area (“radiative cross section” ) that is much wider than the magnetic field concentration proper. toc ref The physics of faculae (cont.) From Steiner (2005) A&A 430, 691 • Double humped contrast profile at disk center • Sharp disk-center side increase due to “‘hot wall” • Gentle limbward decline at µ = cos θ = 0.5 due to lines of sight • profiles with 30◦ ≤ θ ≤ 60◦ show within a narrow region a “dark lane” centerward (in front of) the facular that traverse flux sheet in photospheric brightening • Lines of sight of dark lane layers (left of LOS) • Contrast enhancement travel through internal atmosphere of low wider at µ temperature gradient • The dark lane is the = 0.5 than at disk center • Smooth distribution of polarization signal toc ref manifestation of the “cool bottom” of faculae The physics of faculae (cont.) c) Equivalent width of total polarization of FeI 630.25 nm b) Continuum-contrast profiles for θ = ±45◦ a) Two surfaces of constant optical τc = 1 and zenith angle θ = ±45◦ . Lines of sight mark depth region of dark facular lane Results: Part of the dark lane phenomenon is caused by the cool deep layers of the flux sheet interior but also the downflow at the flux-sheet interface contributes to it. Close to the flux sheet the downflow is seen, which is dark like an intergranular lane. toc ref The physics of faculae (cont.) Appearance of faculae in 3-D MHD-simulations: Center to limb variation of the G-band intensity emanating from a simulation box of 6 × 6 Mm at cos θ = µ = 1.0, 0.8, 0.6, and 0.4. Right: Observation at µ = 0.63. From Carlsson, Stein, and Nordlund (2004) ApJL, 610, L137 toc ref The physics of faculae (cont.) Appearance of faculae in 3-D MHD-simulations: Comparison of observed faculae (top) with faculae from the simulation box of 6 × 6 Mm at µ = 0.5 with hBi = 400 G (middle) and 200 G (bottom). Right: Contrast profiles of facula of the middle panel (top) and the bottom panel (bottom). Keller, Schüssler, Vögler, and Zakharov (2004) A&A 607, L59 toc ref The physics of faculae (cont.) Rapid temporal variability of faculae. De Pontieu et al. (2006), ApJ 646, 1405 “Dark bands” in observations and in simulations Dark bands form naturally in the course of the evolution of a granule. They correspond to the dark lane that forms when a granule is about to split, similar to the central dark region of an exploding granule. toc ref § 10a Faculae and irradiance variability The enhanced radiation output from magnetic elements is believed to determine the long-term solar-irradiance variability – the variability of the “solar constant” with the solar cycle. The center-to-limb variation of the facular contrast influences the precise behaviour of this variability and the relation between solar irradiance and solar luminosity. A question in the physics of solar irradiance variability is: What is the connection between the variation in solar brightness and the evolution of the magnetic field at the solar surface? See also the Nature review of Foukal, Fröhlich, Spruit, and Wigley (2006) for a recent view on this topic. toc ref Faculae and irradiance variability (cont.) Composite from solar irradiance measurements. From the Website of the World Radiation Center http://www.pmodwrc.ch/ toc ref Faculae and irradiance variability (cont.) In an attempt to model solar irradiance variability from magnetograms, Fligge, Solanki and Unruh (2000) use model atmospheres of faculae, sunspots, and the quiet Sun to compute corresponding spectra as a function of disk position. They assign each pixel (i, j) on the solar disk a facular filling-factor, αfi,j (Φ, t), and a spot filling-factor , αsi,j (Φ, t), depending on the magnetic flux Φ and time t to obtain tot (λ) = (1 − αsi,j (Φ, t) − αfi,j (Φ, t)) · I q (µ(i, j), λ) Ii,j + αsi,j (Φ, t) · I s (µ(i, j), λ) + αfi,j (Φ, t) · I f (µ(i, j), λ) Pixel (i, j) is considered to lie within an active region if its magnetic flux density surpasses a center-to-limb dependent threshold value Φth (µ(i, j)). If the continuum intensity of this pixel is more than 10σ lower than that of the quiet Sun at equal µ it belongs to a sunspot, otherwise to a facular region. toc ref Faculae and irradiance variability (cont.) Top: Magnetogram (left) and corresponding white-light image (right) from MDI on SOHO. Bottom: Extracted maps for faculae (left) and sunspots (right). From Fligge et al. 2000 toc ref Faculae and irradiance variability (cont.) Measured (dashed) and modelled (solid) solar total and spectral irradiance variations from the time between 15 August (day 228) and 11 September (day 255) 1996. The model is able to reproduce the double humped structure originating from the CLV of facular contrast. From Fligge et al. (2000) toc ref Faculae and irradiance variability (cont.) Reconstruction (solid) of the total solar irradiance composed by Fröhlich & Lean (1998). The long-term behaviour, i.e., the difference between maximum and minimum (a) as well as shorter-term variations (b) are well reproduced. From Fligge et al. (1998) toc ref § 11 The interchange instability of magnetic flux tubes Consider a section perpendicular to the axis of a straight flux-tube. It can be shown, that a perturbation of the flux-tube boundary in such a way that the volumes V1 and V2 are equal, does not change the total energy of the configuration. unperturbed boundary V1 perturbed boundary V2 unperturbed boundary The instability that evolves from this perturbation is called interchange instability, as the magnetic field and gas of volume 1 is interchanged with the magnetic field and gas of volume 2. It is also called flute instability because of the shape of the perturbed surface (like the vertical parallel grooves on a classical architectural column, called flute). toc ref The interchange instability of magnetic flux tubes (cont.) Consider a small flux-tube section with a small perturbation ξ . The grey shaded area, Vi , be the magnetic flux tube, Ve is field-free. Vi ξ n n is the sur- S Ve face normal pointing out of the field-free plasma. In order that there is a net restoring force on the displaced surface we must, for the indicated displacement ξ , have p0itot + δpitot > p0e + δpe since for the equilibrium configuration p0itot ⇒ δpitot > δpe = p0e . From this follows the condition for stability: B2 ). |ξ · n|n · ∇pe < |ξ · n|n · ∇(pi + 8π (1) This criterion also follows from a more general energy principle due to Bernstein et al. (1958). It is both, necessary and sufficient for stability. toc ref The interchange instability of magnetic flux tubes (cont.) Within the flux tube we have B2 1 ∇pi + ∇( ) = ρi g + (B · ∇)B 8π 4π as was derived from the momentum equation in § 3. In the external atmosphere ∇pe = ρe g . Using these equations in the previously derived stability criterion (1) we obtain: n· » – 1 (B · ∇)B − (ρe − ρi )g > 0 . 4π From the first two equations we have B2 1 (B · ∇)B − (ρe − ρi )g = ∇(pi + − pe ) 4π 8π | {z } =0 on S which means that the bracketed vector in (2) is parallel to n on the surface S . toc ref (2) The interchange instability of magnetic flux tubes (cont.) Therefore, gravity can be eliminated taking the horizontal component of this vector. If h is a horizontal vector pointing out of the fluxtub into the field-free plasma we get h · [(B · ∇)B] < 0 . (3) Along any field line in S the magnitude of the component of B in any fixed outward horizontal direction must decrease upwards. For an untwisted axisymmetric flux tube this criterion can be expressed in cylinder coordinates as: ˛ dBr ˛˛ <0. ˛ dz S (4) B2 From n · (B · ∇)B = − (see derivation in § 3), where Rc is the curvature radius Rc of the surface in axial direction, results another useful form of (2) for an axisymmetric untwisted flux tube (χ is the inclination of S w.r.t the vertical): B2 − + (ρe − ρi )g sin χ > 0 . 4πRc toc ref (5) The interchange instability of magnetic flux tubes (cont.) We could compute dBr /dz from the results obtained with the thin flux-tube approximation in § 8. However, there we assumed a constant pressure scale height for both, the internal and external atmosphere. When using a realistic model atmosphere, Meyer et al. (1977) came to te conclusion that only flux tubes with Φ > 1019 Maxwell are stable against the flute instability. Sunspots and pores have flux in excess of 10 mx. With typical values of R 19 = 100 km and B = 1000 Gauß resulting in Φ ≈ 3 · 1017 G, magnetic elements are liable to the flute instability! Indeed, from G-band-bright-point movies (e.g., Rouppe van der Voort et al.) one gets the impression that magnetic elements are subject to fluting. On the other hand their typical life time of 6–8 minutes is still in excess of the crossing time for Alfvén or sound waves of about 100 s. Also there are bright points that undergo continual fragmentation and merging in a relatively stable location, persisting over several hours. toc ref The interchange instability of magnetic flux tubes (cont.) One remedy would consist in introducing twist, which would effectively supress the intergange instability. Another remedy was proposed by Schüssler (1984) who investigated the effect of various flows within and around a flux tube. Of all these, the most efficient in supressing the z B=0 Vi n flute instability is a whirl flow surrounding the flux χ tube. Such a flow may arise from the “bathtub effect” in the intergranular downdrafts. The cor- g responding stability criterion now reads r B=0 Ve toc S ve = vB eΦ A ˛ vφ2 dBr ˛˛ 1 − ρe Bz <0. 4π dz ˛S R Br that grows with height destabilizes, while vφ stabilizes. ref (6) The interchange instability of magnetic flux tubes (cont.) A typical stability diagram shows the maximal whirl flow velocity, vφ , needed to stabilize a magnetic flux tube as a function flux Φ. Three stability curves from Bünte et al. (1993). The two, requiering fastes flow velocities are derived from the model atmosphere of Meyer et al. (1977) and a from a modern model atmosphere, both using the thin flux-tube approximation. A model flux-tube from a numerical solution requires a vφ max of only 2 km/s. Magnetic tension forces indirectly act to supress the interchange instability. toc ref § 12 The formation of flux tubes by flux expulsion Consider a kinematic flow consisting of stationary rolls in a twodimensional box of size L × L: πx πz πx πz u = U [− sin( ) cos( ) , 0 , cos( ) sin( )] L L L L toc ref The formation of flux tubes by flux expulsion (cont.) Imposing an initial vertical magnetic field and that the field must be vertical at all boundaries leads to the sequence opposite (Galloway & Weiss, 1981). The clockwise flow first advects the field but later, when boundary layers have evolved, gradients are large enough so that the diffusion term dominates in the induction equation. The magnetic flux is thus expelled from the cell interior and concentrates in flux sheets near the boundary. toc ref The formation of flux tubes by flux expulsion (cont.) We can estimate the width, d, of the flux sheet. From the resistive induction equation ∂B = ∇ × (v × B) + η∇2 B . ∂t The time scale of field diffusion is τd = d2 /η , while the time scale for field advection is τad = L/v . In the steady state field decay must be balanced by field advection. Therefore τd = τad from where √ Rm = vL/η . d = L/ Rm We may also estimate the field strength in the flux sheet assuming that the whole flux across the box is advected into flux sheet: B0 L = Bd For a flux tube we would get B toc ref ⇒ = Rm B0 . √ B = Rm B0 . The formation of flux tubes by flux expulsion (cont.) As the field becomes stronger, it counteracts the motion and the kinematic regime changes to the dynamic regime. In this regime the magnitude of the Lorentz force becomes of the order of of the inertial force in the momentum equation: 1 (∇ × B) × B ≈ ρv · ∇v . 4π From this the order of magnitude of the maximal attainable field strength is: v2 1 B2 =ρ 4π d d For solar surface values of v ⇒ Bmax = Beqp p = 4πρ v . = 2 km/s and ρ = 3 · 10−7 g cm−3 we obtain Beqp = 388 Gauß. The equipartition field strength is considerabely smaller than the observed kGauß fields. However, Galloway, Proctor, & Weiss, (1977) derived in compressible, resistive, viscous numerical simulations Bmax /Beqp toc ref p = ν/η = Pr mag . § 13 Flows in magnetic elements: Observations In the seventies and the early eighties downdrafts in magnetic flux tubes with velocities up to 2.2 km s−1 were reported. But Solanki & Stenflo (1986) noted that a spurious zero-crossing shift is measured as a consequence of insufficient spectral resolution. V Spectral smearing of an asymmetric Stokes V profile with the blue lobe being larger than the red one (which is ab Ab usually the case for plage and network Ar vzc toc ref λ ar magnetic elements at disk center) results in a redshifted zero-crossing wavelength. § 13 Flows in magnetic elements: Observations In the seventies and the early eighties downdrafts in magnetic flux tubes with velocities up to 2.2 km s−1 were reported. But Solanki & Stenflo (1986) noted that a spurious zero-crossing shift is measured as a consequence of insufficient spectral resolution. V Spectral smearing of an asymmetric Stokes V profile with the blue lobe being larger than the red one (which is ab Ab usually the case for plage and network Ar vzc toc ref λ ar magnetic elements at disk center) results in a redshifted zero-crossing wavelength. Flows in magnetic elements: Observations (cont.) Subsequently, high spectral resolution measurements, carried out with the Fourier transform spectrometer at Kitt Peak resulted in no significant zero-crossing velocities. “Epur si muove” – And yet it does move! Grossmann-Doerth et al. (1996) and Sigwarth et al. (1999) measured a mean downflow −1 velocity of 0.7 – 0.8 km s . These mean velocities are averages over up to 93,000 magnetic elements with Stokes V amplitudes > 0.15%. Average over the weakest −1 network elements (amplitudes below 1%) would result in a velocity of 1 km s −1 however, that weak magnetic elements show a scatter of up to ±5 km s . Note, . The difference with respect to older measurements resulting in no significant zero-crossing shift is due to the high sensitivity of the employed polarimeters and to the better spatial resolution compared to the FTS data. Magnetic elements with a Stokes V signal (≥ 2%) show negligible zero-crossing shifts, also in these latest measurements. toc ref § 14 Flows in magnetic elements: Theory Consider a slender, vertical flux tube with Br ≪ Bz and vr ≪ vz so that the zeroth-order flux-tube approximation is applicable. We then need to consider motion in vertical direction, only. From mass conservation we have, A ρv (z+ δz) ∂ ∂ (Aρ) + (Aρv) = 0 ∂t ∂z and from magnetic flux conservation A(z+ δz) A(z) A ρv (z) toc ref BA = Φ = const , which can be summarized to ∂ ρ ∂ ρv ( )+ ( )=0. ∂t B ∂z B (Walén’s equation in 1-D) (1) Flows in magnetic elements: Theory (cont.) Furthermore, we have the momentum equation ρ dv dt =− ∂p − ρg , ∂z (2) and, quasi in lieu of the Lorentz force in the momentum equation, B2 p+ = pe , 8π (3) and the equation for isenthropic flow: ∂p ∂p +v = c2 ∂t ∂z which can be expressed as dp dρ = „ where c is the adiabatic speed of sound. toc ref „ dp ∂ρ ∂ρ +v ∂t ∂z « dρ s = c2 , « , (4) Flows in magnetic elements: Theory (cont.) Equations (1) – (4) are perturbed around the static equilibrium, p = p0 + p̃eiωt and similarly for ρ, v, and B , while v = ṽ exp(iωt). Substitution into (1) – (4) and linearization leads, for example for Eq. (3), to p + δp + 1 1 (B + δB)2 = p + δp + (B 2 + 2BδB + δB 2 ) = pe + δpe . 8π 8π It is assumed that the perturbation has no effect on the external atmosphere, so that δpe = 0. Thus, 1 B0 B̃ = 0 . 4π 2 A further simplification is that β = p/(B /(8π)) = const. for the equilibrium state. p̃ + toc ref Flows in magnetic elements: Theory (cont.) For linearizing the momentum equation we write ∂δv ∂(p + δp) ∂δv + δv )=− − (ρ + δρ)g . (ρ + δρ)( ∂t ∂z ∂z Using that for the unperturbed state −p ˙ + δρg obtain ρδv ′ − ρg = 0 and neglecting 2nd-order terms we + δp′ = 0 . Proceeding in the same way with Walén’s equation and the equation for isentropic flow leads to the following system: ˙ − ρδB ˙ + ρ′ δvB + ρδv ′ B − δvρB ′ = 0 , δρB ˙ + δρg + δp′ = 0 , ρδv 1 δp + BδB = 0 , 4π ˙ + ρ′ δv) = 0 . ˙ + p′ δv − γp (δρ δp ρ toc ref Flows in magnetic elements: Theory (cont.) Using the time separation Ansatz δq = q̃eiωt for δq = δρ, δB, δv, δp, we eliminate time derivatives and obtain the following system, equivalent to Roberts & Webb (1978): iω ρ̃B − iωρB̃ + ρ′ ṽB + ρṽ ′ B − ρB ′ ṽ = 0 , iωρṽ + ρ̃g + p̃′ = 0 , 1 p̃ + B B̃ = 0 , 4π iω p̃ + ṽp′ − c2 (iω ρ̃ + ρ′ ṽ) = 0 . We get from (2) from (3) and from (4) toc ref 1 ′ ρ̃ = − (p̃ + iωρṽ) , g 4π p̃ B̃ = − , B iω p̃ + ṽp′ ′ − ρ ṽ . iω ρ̃ = 2 c (1) (2) (3) (4) (5) (6) (7) Flows in magnetic elements: Theory (cont.) In Eq. (1) we substitute B̃ from (6) and iω ρ̃ from (7) to obtain ff  2 1 B ′ 2 ′ ′ p̃ = p )ṽ − ρB ṽ , (ρB B − 2 B2 c iω( c2 + 4πρ) (8) while (5) into (4) gives iω(g p̃ + c2 p̃′ ) . ṽ = 2 2 2 ′ ′ c ω ρ + gc ρ − gp (9) Furthermore we substitute in (1) ρ̃ and B̃ with help of (5) and (6) in order to obtain an equation in terms of ṽ and p̃ (and derivatives) only: iω ′ − (p̃ + iωρṽ)B 2 + iω4πρp̃ + ρ′ B 2 ṽ + ρB 2 ṽ ′ − ρBB ′ ṽ = 0 g into which we can either substitute (8) or (9) in order to obtain an ordinary homogeneous second order differential equation for ṽ or p̃, respectively. Doing the former we arrive after some algebra at the equation of toc ref (10) Flows in magnetic elements: Theory (cont.) Spruit & Zweibel (1978) who have allowed for variation with p and T of the mean molecular weight, µ, and the specific heats (cp , cv ), which they consider essential: ′′ ṽ + „ −1 + 2Hp γ ′ c2T γc2 « ′ ṽ +  2 » 1+β ω γ + δ(∇ − ∇a ) + c2T 2Hp2 p′0 =− p0 « „ ∂ ln T ∂ ln T − ∇ − ∇a = ∂ ln p ∂ ln p S Hp−1 1 1 1 1 + γβ/2 = + = c2T c20 c2A0 c20 „ ′ Hp c2T γc2 –ff ṽ = 0 ∗) « γp ∂p 1 = ρ ∂ρ S 1 + βγ/2 « „ ∂ ln µ δ =1 − ∂ ln T p c2 = β =8πp0 /B02 Derivatives are taken with respect to the vertical coordinate, z , counted positive in radial direction away from the Sun. toc ref Flows in magnetic elements: Theory (cont.) Equation (∗) must be complemented with boundary conditions. We are interested in the region of strong superadiabaticity, i.e., where δ is largest. Spruit & Zweibel use ṽ(z0 ) = ṽ(z1 ) = 0 , where z0 and z1 are sufficiently far away from this region, i.e., for the lower boundary depth z1 = 5000 km, while the upper boundary is set to the temperature minimum region at 500 km height. Equation (∗) together with the boundary condition then defines an eigenvalue problem : O ṽ = ω 2 ṽ . For a given atmospheric model and a given value of β , there exist solutions only for certain values of ω , ωi i = 1, 2, 3, . . .. The solution for ω strongly depends on the parameter β , which is a measure for the magnetic field strength in the flux tube. toc ref Flows in magnetic elements: Theory (cont.) For large β , there always exist negative eigenvalues ω 2 ⇒ For weak fields the static state is unstable. Only if β is smaller than a critical value βc , are all eigenvalues positive, which is necessary for stability. The result is βc = 1.83 corresponding to a critical field strength of about 1000 Gauß. From Spruit & Zweibel (1979). (η toc ref 2 = −ω 2 ) Flows in magnetic elements: Theory (cont.) Sketch of the field intensification throught convective collapse (Spruit, 1979). The initially unstable layer at z = z0 undergoes convective collapse and moves a distance ξ to an end state in which β < βc at z . Assuming the transition to evolve quasi static, we can compute ξ . dz 0 z0 B0 ρ0 p0 ξ dz z B ρp toc ref Flows in magnetic elements: Theory (cont.) For an initial state of β0 < 2, the flux tube is stable and the final (βs ) is equal to the initial state (slope +1). For an initial state with β0 > 2 (dashed line), the final state is reached where the vertical crosses the solid line below the dashed one with the corresponding final field strength βs . From Spruit (1979) toc ref Flows in magnetic elements: Theory (cont.) Try this descriptive explanation: In the limit of very strong magnetic field (β ≪ 1) the descending cool gas bubble can not further contract since it is “frozen into” the stiff magnetic flux tube. In this limit the scale height of the radius is 4 pressure scale height of the external atmosphere (see §8). Correspondingly the volume of the descending bubble within the flux tube contracts more slowly than for a descending bubble in the surrounding atmosphere. Vi 1 = Vi 2 Tblob = Tsurr After some distance the density of the bubble within the flux tube reaches the density of the Tblob < Tsurr ρblob < ρsurr Vf 1 > Tblob < Tsurr ρblob > ρsurr Vf 2 surrounding material and becomes even lighter at which point buouancy takes over and stops further descending. toc ref § 15 The convective collapse: Numerical simulation The process of field intensification in magnetic flux tubes through the superadiabatic effect (Parker, 1978) was subject of numerous numerical simulations. Linear stability analyses, others than that by Spruit & Zweibel (1979) and Spruit (1979), were also carried out by Webb & Roberts (1978), Unno & Ando (1979), Webb & Roberts (1980), Venkatakrishnan (1986), and Hasan (1986). Nonlinear numerical simulations in one spatial dimension were carried out by Venkatakrishnan (1983) and Hasan(1984). Hasan (1985) and Venkatakrishnan (1986) have taken non-adiabatic effects into account. Hasan (1986) showed, that the end state of the convective collapse is not a static flux tube, rather a state of overstable oscillation. toc ref The convective collapse: Numerical simulation (cont.) Variation of B (solid curve), v (dashed curve), and T (dotted curve) as a function of time at a height of 50 km in the photosphere. Initially, β = 7. The amplitude of the oscillation increases with time from 300 G to 450 G, from 550 K to 650 K, and from 1.2 −1 km s toc −1 to 1.9 km s ref for B , T , and v , respectively. From Hasan, 1985 The convective collapse: Numerical simulation (cont.) Radiative exchange permits the oscillation to extract energy from the surrounding medium and as a result the amplitude grows in time. However, if radiative exchange occurs too rapidly it can lead to a dispersal of the magnetic flux. Thus, for the convective collapse to be effective, the time scale for convective instability, tcon , should be less than the time scale for radiative exchange, trad , in the layers driving the instability. toc ref The convective collapse: Numerical simulation (cont.) A very careful linear stability analysis of the radiative effects on the onset of convective collapse and on the overstable oscillatory solution was carried out by Rajaguru & Hasan (2000). Among other results they found a critical flux limit of approx. 1 × 1018 Mx that demarcates small-scale flux-tubes into two groups: – Flux concentrations above this limit collapse unhindered by radiative effects to a field strength > 1160 G, and – flux concentrations below this limit are subject to the inhibiting action of radiation. They exhibit a flux-flux density relation. Strong tubes (high flux density) of large enough sizes (> 300 km) are radiatively damped, hence are not subject to the overstability. toc ref The convective collapse: Numerical simulation (cont.) Flux/field-strength relation for tubes with a Flux/field-strength relation from infrared 1.3 × 1018 Mx separating the un- observations. From Solanki et al. (1996) flux < stable (to the left) from the stable region (to the right). From Rajaguru & Hasan (2000) ApJ 544, 522 toc ref A&A 310, L33 The convective collapse: Numerical simulation (cont.) Growth rates (solid curves) of unstable Growth rates of convective and over- and frequencies (dotted curves) of the stable modes (fundamental mode) as a overstable mode for closed bottom (cb) function of surface (photosphere) radius and open bottom (ob) boundaries. From a0 of flux tube. From Rajaguru & Hasan Rajaguru & Hasan (2000) (2000) toc ref The convective collapse: Numerical simulation (cont.) Numerical simulation of the convective intensification of surface magnetic fields in 2D. Boundary condition for thermal variables and velocity: ∂T =0; v=0 ∂y ”wave absorbing layer”: addition of a diffusion term ε · d2 q/dy 2 to each equation 2400 x 1400 km periodic periodic y x pg = pg (t) ; inflow: s = sin (t) ∂(ρv) ∂s = 0 ; outflow: =0 ∂y ∂y toc ref The convective collapse: Numerical simulation (cont.) Boundary condition for the magnetic field: ∂By Bx = 0 ; =0 ∂y periodic periodic ∂By Bx = 0 ; =0 ∂y toc ref The convective collapse: Numerical simulation (cont.) 0s toc 50 s ref Binit = 400 G 100 s The convective collapse: Numerical simulation (cont.) 0s toc 50 s ref Binit = 400 G 100 s The convective collapse: Numerical simulation (cont.) 0s toc 50 s ref Binit = 400 G 100 s The convective collapse: Numerical simulation (cont.) 150 s Binit = 400 G 250 s Density (top), magnetic lines of force (middle), and velocity (bottom). In panel e (top) the density is replaced by a plot of isothermals. The horizontally running curve corresponds to optical depth one. From Grossmann-Doerth, Schüssler, & Steiner (1998) A&A 337, 928 toc ref The convective collapse: Numerical simulation (cont.) 150 s Binit = 400 G 250 s Density (top), magnetic lines of force (middle), and velocity (bottom). In panel e (top) the density is replaced by a plot of isothermals. The horizontally running curve corresponds to optical depth one. From Grossmann-Doerth, Schüssler, & Steiner (1998) A&A 337, 928 toc ref The convective collapse: Numerical simulation (cont.) t = 50 s t = 150 s t = 100 s t = 250 s Left: Plasma beta as a function of height. Right: Vertical velocity as a function of height along the central field line of the flux sheet toc ref The convective collapse: Numerical simulation (cont.) Temporal evolution of the magnetic field strength at two continuum optical depths (left) and of the continuum contrast (right) between 1600 km ≤ x ≤ 2400 km. Times in s in the left-hand margin. Note that the peak in the magnetic field is considerabely wider than the corresponding intensity signal toc ref The convective collapse: Numerical simulation (cont.) Binit = 100 G Binit = 100 G, else same initial condition as in 400 G run. a – d correspond to 60 s, 115 s, 175 s, and 230 s, respectively. After instant b the flux sheet extends through the open bottom of the computational domain toc ref The convective collapse: Numerical simulation (cont.) 525 612 551 Sequence of Stokes V (left) 1521 and Stokes I (right) profiles of 1443 Fe I 1.5648 µm. Times are indicated in the left margin of each panel. The numbers to 1661 the left give the field strength 1356 as derived from the peak separation of Stokes V toc ref The convective collapse: Numerical simulation (cont.) Stokes V zero-crossing shift, represented as Doppler velocity in km s−1 for three different spectral lines. Positive shift indicates downflow toc ref The convective collapse: Numerical simulation (cont.) Stability diagrams from 1D convective collapse simulations: St = stable, Us = unsable, Sh = shock formation. From Takeuchi (1999) Tubes of strong enough initial field strength are stable. Sufficiently weak flux tubes of lare enough radius produce a rebound shock. toc ref § 16 The convective collapse: Observation In a search for the “convective collapse”, Bellot Rubio et al. (2001) obseved with the Teneriffe Infrared Polarimeter (TIP) at the VTT a quiet Sun region, where they expected to find most easily isolated magnetic elements. The data consist of spectra of two Fe I lines at 1.565 µm over a spatial distance of 34” taken with a cadence of 5.6 s over a time period of 1h. The effective spatial and temporal resolution is 28 s and 0.8”, respectively. They found one event which possibly is a convective collapse. The image shows the temporal evolution of the Stokes V profile of this event. Time goes from bottom to top, dispersion in the horizontal direction toc ref The convective collapse: Observation (cont.) Bellot Rubio et al. distinguish three phases: 1.) ≈ 9 min, moderate redshift, slightly increasing magnetic signal, 2.) ≈ 4 min, strong redshift and increase of magnetic signal, 3.) toc ref ≈ 2 min, large blueshift and disappearence of Stokes-V signal. The convective collapse: Observation (cont.) Stokes-V profiles of the third phase, showing the appearance of a new, blue-shifted component, while the initial, red-shifted profile weakens and disappeares. From Bellot Rubio et al. (2001) toc ref The convective collapse: Observation (cont.) Temporal evolution of the internal velocity, the external velocity, the temperature, and the field strength at two different heights, 0 and 100 km, in the atmosphere. Results from “Stokes inversion” analysis. From Bellot Rubio et al. (2001) toc ref The convective collapse: Observation (cont.) Top: Internal velocity as a function of height at = 13.1 min. Middle: Location of the discontinuity as function of time. Bottom: Velocity difference across the discontinuity as function of time. From Bellot Rubio et al. (2001) toc ref The convective collapse: Observation (cont.) The observations show all the characteristics of the convective collapse seen in MHD simulations. However, quantitatively, there remain differences. In the observation the field intensifies from about 400 Gauß to only maximal 600 Gauß. Convective collapses in numerical simulations typically lead to kG-flux tubes. There are other questions, e.g.: Why is the velocity in the external medium so much similar to the flow within the magnetic element? How often are events like this only one observed? toc ref § 17 Interaction of magnetic flux tubes with convective motion G-band filtergram of an area of 32 × 22 arcs of the solar surface. The red line drawn across an elongated bright structure corresponds to the width of the two-dimensional simulation domain of 2400 km. The red box shows the size of the three-dimensional simulation domain of 4800 × 4800 km. Image by K. Mikurda, F. Wöger, and O. von der Lühe with the German Vacuum Tower Telescope (VTT) at Tenerife toc ref Interaction of magnetic flux tubes with convective motion (cont.) τ=1 surface computational domain 1.4 Mm τ=1 2.8 Mm 1.4 Mm convection zone 4.8 Mm 4.8 Mm convection zone base Typical size of a three-dimensional computational box (left) on scale with the convection zone boundaries (right) toc ref Interaction of magnetic flux tubes with convective motion (cont.) The ideal MHD-equations can be written in conservative form as: ∂U +∇·F =S, ∂t where the vector of conserved variables U , the source term S due to gravity and radiation, and the flux tensor F are U = (ρ, ρv, B, E) , 0 S = (0, ρg, 0, ρg · v + qrad ) , ρv B “ ” B ·B B ρvv + p + B8π I − BB 4π B F =B B vB − Bv B @ “ ” ·B 1 E + p + B8π v − 4π (v · B) B The tensor product of two vectors a and b is the tensor ab toc ref 1 C C C C C. C C A = C with elements cmn = am bn . Interaction of magnetic flux tubes with convective motion (cont.) The total energy E is given by E = ρǫ + ρ v·v B ·B + , 2 8π where ǫ is the thermal energy per unit mass. The additional solenoidality constraint, ∇ · B = 0, must also be fulfilled. The MHD equations must be closed by an equation of state which gives the gas pressure as a function of the density and the thermal energy per unit mass p = p(ρ, ǫ) , usually available to the program in tabulated form. The radiative source term is given by qrad = 4πρ 1 Jν (r) = 4π toc ref I Iν (r, n)dΩ , Z κν (Jν − Bν )dν , I(r, n) = I0 e−τ0 + Z 0 τ0 “σ π ” T 4 (τ ) e−τ dτ Interaction of magnetic flux tubes with convective motion (cont.) Likewise as done in § 9 we could formally write Jν (r) = Λν (r, r′ )Bν (r′ ) + Gν (r) , where Λν is the integral operator which adds the intensities at r caused by emision at ′ all the points r in the considered computational domain, and where Gν is the transmitted mean intensity due to the incident radiation field into this domain. In the formula I(r, n) = I0 e−τ0 + Z 0 τ0 “σ π ” T 4 (τ ) e−τ dτ τ0 is the optical depth from the boundary to location r along direction n and dτ = κ ds, where κ is the total opacity and s the spatial distance along a line through location r in direction n. Here, the index ν has been dropped for the quantities I0 , τ0 , τ , and κ. In practice the frequency integration is either replaced by using frequency mean quantities (Rosseland mean opacity) or it is approximated by a method of multiple frequency bands. For a detailed description of the latter method see Ludwig (1992). toc ref Interaction of magnetic flux tubes with convective motion (cont.) In practice it is not the ideal MHD-equations that are solved but rather some kind of a viscous and resistive form of the equations with flux tensor 0 ρv B « „ B BB B · B B I− −σ ρvv + p + B 8π 4π F=B B Bv − vB − η[∇B + (∇B)T ] B B „ « @ B·B 1 E+p+ v− (v · B) B + η(j × B) − σv + qturb 8π 4π where σ 1 C C C C C, C C C A = νρ[(∇v) + (∇v)T − (2/3)(∇ · v)I] is the viscous stress tensor, η = (ν/Prm ) = 1/(4πσ) the magnetic diffusivity with σ being the electric conductivity, and η(j number. q turb is a turbulent diffusive heat flux, which would typically be proportional to the entropy gradient: toc × B) = (η/4π)(∇ × B) × B . Prm is the magnetic Prandtl ref q turb = −(1/Pr)νρT ∇s, where Pr is the Prandtl number. Interaction of magnetic flux tubes with convective motion (cont.) Typically, ν is not taken to be the molecular viscosity coefficient but rather some turbulent value that takes care of the dissipative processes that cannot be resolved by the computational grid. Such subgrid-scale viscosities should only act where velocity gradients are strong causing srong turbulence. Therefore, they typically depend on velocity gradients like in the Smagorinsky-type of turbulent viscosity where ( "„ «2 „ «2 „ «2 # ∂vx ∂vy ∂vz t + + + ν =c 2 ∂x ∂y ∂z „ ∂vy ∂vx + ∂y ∂x «2 + „ ∂vx ∂vz + ∂z ∂x «2 + „ ∂vy ∂vz + ∂z ∂y «2 )1/2 , where c is a free parameter. This parameter is normally chosen as small as possible just in order to keep the numerical integration stable and smooth, but otherwise having no effect on large scales. toc ref Interaction of magnetic flux tubes with convective motion (cont.) Some numerical (high resolution) schemes feature an inherent dissipation that acts like the explicite dissipative terms shown in the flux tensor above. This artificial viscosity is made as small as possible but just large enough in order to keep the numerical scheme stable. One then only has to program the ideal equations. Of course, in this case it is difficult to quote the actual Reynolds and Prandtl numbers because they change form grid cell to grid cell depending on the flow. Therefore, for some applications it might be preferable to explicitly include the dissipative terms in the equations using constant dissipation coefficients, which then allows for well defined dimensionless numbers. However one integrates the ideal equations on a discrete computational grid one is locked with a discretization error that normally assumes a form similar to the dissipative terms in the non-ideal equations. See LeVeque, Mihalas, Dorfi, & Müller (1998) for more on computational methods for astrophysical fluid flow. toc ref Interaction of magnetic flux tubes with convective motion (cont.) Typical boundary conditions for thermal variables and velocity: ∂vx,y ∂z = 0 ; vz = 0 ; ∂ǫ ∂z = 0 or ∂ 2ǫ ∂z 2 =0 periodic z y periodic x ∂vx,y ∂z Z = 0 ; ρvz dσ = 0 ; outflow: ∂s ∂z =0 inflow: s = const Example of a simulation of solar granulation. The emergent intensity over an area of 12 × 12 Mm is shown. The computation was carried out with the CO5 BOLD-code in a domain of 400 × 400 × 165 grid cells. The granular contrast is 16.65% at 620 nm. Courtesy, M. Steffen, AIP toc ref Interaction of magnetic flux tubes with convective motion (cont.) Typical boundary conditions for the magnetic field: Bx,y = 0 ; ∂Bz ∂z =0 periodic z y periodic =0 periodic z periodic y x x Bx,y = 0 ; ∂Bx,y,z ∂z ∂Bz ∂z =0 outflow: ∂Bx,y,z ∂z =0 inflow: By = Bz = 0, Bz = const. Example of a simulation of magneto-convection in the solar photosphere by Vögler, Shelyag, Schüssler, Cattaneo, Emonet, and Linde, A&A 429, 355. 1 h movies of the continuum intensity, the vertical magnetic field, the temperature, and the vertical velocity toc ref → PDFs Interaction of magnetic flux tubes with convective motion (cont.) Edge-on view of a thin magnetic sheet. Note the spreading of field lines in the photosphere. At a depth of ≈ 300 km below the surface the sheet is disrupted. From Vögler et al. (2005) A&A 429, 335 toc ref Interaction of magnetic flux tubes with convective motion (cont.) Good agreement with the thin tube approximation. hpgas ii (solid), hpgas ie (dashed), hpgas ii + hpmag ii (dotted). Top: h|B|ii (solid), h|B|ie (dotted), p 2µ(hpgas ie − hpgas ii ) (dashed) Bottom: → more toc ref Interaction of magnetic flux tubes with convective motion (cont.) Two-dimensional simulation of the interaction of a magnetic flux sheet with convective motion (Steiner, Grossmann-Doerth, Knölker, and Schüssler, 1998). Initial condition: - Magnetic flux tube embedded in a standard model atmosphere - T = T (y), density reduced within flux sheet so that β = pmag /pgas = 1 - Smooth transition between flux sheet and surrounding plasma - Small velocity perturbation Methods of numerical integration: - Numerical integration with flux corrected transport (FCT) scheme of Zalesak (1979) - Integration of the induction equation with the FCT scheme of DeVore (1991) - Constraint transport (CT) for maintaining ∇ · B =0 - Adaptive mesh refinement of Berger & Oliger (1984), Berger & Colella (1989) toc ref Interaction of magnetic flux tubes with convective motion: Snapshot toc ref Interaction of magnetic flux tubes with convective motion: Animations Evolution of temperature and magnetic field: QuickTime movie. From t ≃ 2 : 00 to 5 : 00 min, a growing “granule” pushes the flux sheet to the left. Shock fronts in the non-magnetic atmosphere are well visible between 4:30 and 7:30 min. From 8:00 to 9:00 min a strong shock propagates vertically within the flux sheet. The bending of the flux sheet to the right reaches a maximum at about 10:00 min, followed by a fast motion to the left. Note the bow shock associated with this swaying. Strong shocks in the flux sheets are again visible between 15:00 and 17:00 min. Tracer particles follow plasma motion: QuickTime movie. Note the persistent downflows on both sides of the flux sheet. Within the flux sheet, oscillatory motion along the field lines and the propagation of shocks can be seen; shocks are reflected at the top due to the closed boundary condition. For the same reason, the downflows are forced to bend sideways near the bottom of the box. toc ref Interaction of magnetic flux tubes with convective motion: Animations Stokes I , V , and Q of the spectral line Fe I 5250.2: QuickTime movie. Horizontally-averaged Stokes profiles for vertically incident line of sight, normalized to the average continuum intensity, are given in three panels placed above the temperature/fieldline panel. At t ≃ 8 : 00 min and 15:50 min Stokes V takes on a complex shape due to the passage of a shock wave in the flux sheet (“shock signature”). At 10:20 min the Q signal becomes large caused by the sizeable horizontal field component of the strongly inclined flux sheet. The full movie, published in the accompanying video to volume 495 of ApJ, is available here as QuickTime movie. toc ref Interaction of magnetic flux tubes with convective motion: Continuum contrast Normalized continuum intensity for vertical incidence as a function of horizontal position. The eight curves are separated in time by about 100 s each. Each curve shows a local maximum at the position of the magnetic flux sheet, which is embedded in darker downflow regions. A strong inclination of the flux sheet exposes the “hot wall” of the flux sheet at a favourable angle toward the observer, leading to the strong and extended brightening visible in curve e for 1400 km < toc ref x < 1900 km Interaction of magnetic flux tubes with convective motion: Shock front Upward propagating shock wave in a magnetic flux sheet. Height profiles of the vertical velocity (vy , lower curve) and temperature (T , upper curve) are given at horizontal location x = 1180 km at time t = 16.5 min. The shock front located at a height of 360 km has heated the post-shock material by about 1600 K. It rapidly cools off due to expansion and radiation toc ref Interaction of magnetic flux tubes with convective motion: Shock transit Time sequence of Stokes I (left) and Stokes V (right) of Fe I 5250.2, reflecting the transit of a shock front. Time increases from bottom to top and consecutive profiles are separated by 10 seconds each. The superposition of a redshifted pre-shock and a blueshifted post-shock profile leads to the complex V -profiles i–o during the shock transit. The instanteneous height positions (in km) of the shock front are given to the right of the corresponding Stokes V profiles. Stokes I originates mainly in the field free region outside the flux sheet and it gets only weakly affected in the far blue wing. toc ref Interaction of magnetic flux tubes with convective motion: Downflow jet Left: Vertical velocity component in two heights (solid: y = 0 km, dashed: y = −200 km) as function of the horizontal coordinate (x) in the central part of the flux sheet at t = 14.5 min. Downflow jets adjacent to the flux sheet are visible. Right: Horizontal velocity at height y on the flux sheet toc ref = 0. The downflows are maintained by horizontal flows impinging Interaction of magnetic flux tubes with convective motion: Red wing The downflow jets drag some fluid within the magnetic flux sheet with them downwards, leading to a downflow of magnetized material. This results in a redshifted component in the synthetic Stokes profile. Observed Stokes V profiles show an extended “tail” in the red wing, too. The figure shows a comparison between observed Stokes V profiles of Fe I 5250.2 and 5250.6 and synthetic profiles of these lines from the simulation run. The “red tails” of the synthetic profiles are in qualitative agreement with the corresponding features in the observed profiles. toc ref § 18 Stokes analysis of the simulation result Stokes V profile asymmetries provide a useful diagnostic for the gas motion within magnetic elements and in their close surroundings. V ab Ab − Ar δA := Ab + Ar Ab Ar vzc λ δa := ar ab − ar ab + ar A relation between the sign of the area asymmetry and the gradients of the absolute magnetic field strength and the velocity along a line of sight is given by: sign(δA) = −sign( d|B| dv(τ ) · ) dτ dτ (Solanki & Pahlke, 1988; Sanchez Almeida et al., 1989) toc ref (1) Stokes analysis of the simulation result (cont.) The simulation shows often an accelerating downflow or upflow within the flux tube. According to the Eq. (1) and, as illustrated in the following figure, this situation corresponds to a negative area asymmetry, which is at variance with the observed positive mean value of δA = 6% of Sigwarth et al. (1999). v d|B| dτ > 0 dv(τ ) dτ > 0      v ⇒ δA < 0 d|B| dτ > 0 dv(τ ) dτ > 0      ⇒ δA < 0 Sign of the Stokes V area asymmetry formed along the flux-tube axis with an accelerating downflow (left) or upflow (right) toc ref Stokes analysis of the simulation result (cont.) The situation changes completely for lines of sight that pass through the expanding boundary of the flux tube as sketched in the figure. Then, the area asymmetry is positive. LOS v=0 v d|B| dτ < 0 dv(τ ) dτ > 0      ⇒ δA > 0 Sign of the Stokes V area asymmetry formed along lines of sight that pass through the expanding flux-tube boundary How realistic are these sketches? We have no means to answer this question observationally, because it addresses the internal structure of magnetic elements that is far beyond the spatial resolution capabilities of present time solar telescopes, but we may attempt an answer by examination of snapshots from a simulation. toc ref Stokes analysis of the simulation result (cont.) + – + + + – + + + – + + – + – – + – – + – + + – + + – + + – + + - + Stokes V profiles that emanate from vertical lines of sight distributed over the horizontal interval between 600 and 1380 km of the simulation snapshot. Positive area asymmetries result in the two regions labeled with “+”, while the lines of sight in the middle region labeled with “−” contribute negative area asymmetries, exactly as expected from the simple sketches above. From Steiner (1999) toc ref Stokes analysis of the simulation result (cont.) We conclude that the measured, spatially unresolved Stokes V area asymmetry of network elements results from a delicate balance of negative and positive contributions due to the internal structure of the elements. The simulations also suggest that the amplitude asymmetry, δa, is not subject to a similar balance and therefore shows larger and almost always positive values. A statistical analysis similar to the ones made by Sigwarth et al. (1999) has been carried out with synthetic profiles of the simulation series. Considering the individual snapshots to be independent single magnetic elements allows for the determination of mean values of δA, δa, and vzc . The result is compiled in the table, where the first two rows are derived from synthetic profiles of simulation models, the rest from observations. toc ref Stokes analysis of the simulation result (cont.) Ref. Instrument Region FeI δa δA vzc 1 Simulation temporal distrib. 6302.5 14.9 0.4 0.74 1 Simulation temporal distrib. 5250.2 14.1 -2.6 0.67 2 ASP Quiet Sun average 6302.5 15.0 6.0 0.73 2 ASP Active region 6302.5 9.4 0.7 0.48 3 Zimpol Quiet Sun average 5250.2 14.8 3.9 1.0 4 ASP Plage average 6302.5 10.0 3.0 0.2 5 FTS Network 5250.2 22.4 7.4 <0.3 5 FTS Plage 5250.2 16.2 6.6 <0.3 1 Steiner et al.: 1998, ApJ 495, 468; 2 Sigwarth et al.: 1999, A&A, 349, 941; 3 Grossmann-Doerth et al.: 1996, A&A 315, 610; ApJ 474, 810; toc ref 4 Martı́nez Pillet et al.: 1997, 5 Solanki S.K.: 1993, Space Science Review 63, 1 § 19 MHD-simulation from the convection zone to the chromosphere At the Kiepenheuer-Institut, Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag have carried out a three-dimensional MHD-simulation that encompasses the integral layers from the top of the convection zone to the mid-chromosphere. We use CO5 BOLD, a finite volume code for solving the hydrodynamic equations in two or three spatial dimensions. It is based on Riemann solvers and higher order reconstruction schemes. For MHD we use a constrained transport scheme for the magnetic field and a 2nd-order accurate HLL Riemann solver . toc ref MHD-simulation from the convection zone to the chromosphere (cont.) z = 60 km z = 1300 km 4000 4000 3000 3000 y [km] y [km] z = -1210 km 2000 2000 1000 1000 1000 0 2000 3000 x [km] 1 log |B| (G) 4000 2 1000 0 2000 3000 x [km] 1 log |B| (G) 4000 2 1000 0.2 0.4 2000 3000 x [km] 0.6 0.8 log |B| (G) 4000 1.0 1000 2000 3000 x [km] 4000 1.2 Horizontal sections through 3-D computational domain. Color coding displays log |B| with individual scaling for each panel. Left: Bottom layer at a depth of 1210 km. Middle: Layer 60 km above optical depth τc = 1. Right: Top, chromospheric layer in a height of 1300 km. White arrows indicate the horizontal velocity on a common scaling. Longest arrows in the panels from left to right correspond to 4.5, 8.8, and 25.2 km/s, respectively. Rightmost: Emergent intensity . toc ref MHD-simulation from the convection zone to the chromosphere (cont.) 2.5 500 2.0 0 1.5 -500 1.0 -1000 0.5 z [km] 1000 log |B| (G) 3.0 0.0 1000 2000 3000 x [km] 4000 Snapshot of a vertical section showing log |B| (color coded) and velocity vectors projected on the vertical plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed and solid black contours β = 1 and 100, respectively. movie with β = 1 surface Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2005, in Chromospheric and Coronal Magnetic Fields, Innes, Lagg, Solanki, & Danesy (eds.), ESA Publication SP-596 toc ref MHD-simulation from the convection zone to the chromosphere (cont.) 35 1300 30 30 800 20 600 10 400 200 400 600 z [km] 1200 1000 25 1100 20 1000 900 15 800 10 700 5 800 1000 1200 1400 x [km] 3400 3600 3800 4000 x [km] 4200 4400 Two instances of shock induced magnetic field compression. Absolute magnetic flux density (colors) with velocity field (arrows), Mach β = 1-contour (white solid). toc ref = 1-contour (dashed) and |B| [G] 40 |B| [G] z [km] 1200 MHD-simulation from the convection zone to the chromosphere (cont.) 2 1 3.2 vs vx 0.8 log ρ p Bz cs cA β toc −10.4 8.6 12.6 6.0 4.0 1.31 ref −3.8 −10.7 2.5 5.2 3.9 2.4 2.14 2.4 7.0 v2 v1 ρ2 v B ≅ 2 ≅ 1 ρ1 v2 B1 2 2 2 v 1 > cs 1 + cA 1 MHD-simulation from the convection zone to the chromosphere (cont.) 2.5 500 2.0 0 1.5 -500 1.0 z [km] 1000 log |B| (G) 3.0 0.5 -1000 0.0 1000 2000 3000 x [km] 4000 Snapshot of a vertical section showing log |B| (color coded) and B projected on the vertical plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed and solid black contours β = 1 and 100, respectively. Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2005, in Chromospheric and Coronal Magnetic Fields, ESA Publication SP-596 toc ref MHD-simulation from the convection zone to the chromosphere (cont.) The formation of the small-scale canopy field proceeds by the action of the expanding flow above granule centers. This flow transports “shells” of horizontal magnetic field to the upper photosphere and lower chromosphere, where layers of different field directions may come close together, leading to a complicated meshwork of current sheets in a height range from approximately 400 to 900 km. toc ref MHD-simulation from the convection zone to the chromosphere (cont.) Logarithmic density, current log |j|, in a vertical cross section (top panel) and in four horizontal cross sections in a depth of 1180 km below, and at heights of 90 km, 610 km, and 1310 km above the average height of optical depth unity from left to right, respectively. The arrows in the top panel indicate the magnetic field strength direction. toc ref and MHD-simulation from the convection zone to the chromosphere (cont.) Estimate of the ohmic dissipation of the current sheets: The typical current density in the height range from 400 to 4 1000 km is 10 statamp cm i.e., 0.03 A m −2 −2 , . From Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2006 The electrical conductivity in the photosphere and the lower chromosphere is about 10...100 A/Vm (Stix, The Sun). Using this value, the ohmic dissipation is 1 2 1 Pj = j ≈ 0.032 ≈ 10−5 . . . 10−4 W m−3 . σ 10 . . . 100 When integrating over a height range of 500 km this heat deposition leads to an energy flux of 10 toc −5 . . . 10−4 · 5 × 105 = 5 . . . 50 W m−2 . ref § 20 Future directions in numerical simulations Future numerical simulations on solar convection and the magnetic coupling to the outer atmosphere will concentrate on: § 20.1. More physics: Dynamic hydrogen ionization § 20.2. More Space: Towards the simulation of supergranulation cells and the magnetic coupling from the convection zone to the corona § 20.3. Boundary conditions: Advecting magnetic field across boundaries § 20.4. Numerical experiments toc ref § 20.1 More physics: Dynamic hydrogen ionization Under the condition of the solar chromosphere the assumption of LTE (local thermodynamic equilibrium) is not valid. Even the assumption of statistical equilibrium in the rate equations is not valid. Kneer (1980) showed that the relaxation timescale for the ionization of hydrogen varies from 100 s to 1000 s in the middle to upper chromosphere. In order to compute the time dependent hydrogen ionization in a three-dimensional environnment, simplifications are needed. We employ the method of fixed radiative rates. We solve the time-dependent rate equations n n j6=i j6=i l l X X ∂ni Pij + ∇ · (ni v) = nj Pji − ni ∂t Pij = Cij + Rij . toc ref Future directions (cont.): More physics: Dynamic hydrogen ionization Effect of dynamic H-ionization in the upper part of a 2-D simulation. Left column: LTE ionization degree and electron density. Right column: Corresponding time-dependent NLTE quantities. Bottom left: Gas temperature, which is the same for the LTE and the time-dependent case. Leenaarts & Wedemeyer-Böhm 2006 toc ref →3-D Future directions (cont.): More physics: Dynamic hydrogen ionization The non-equilibrium electron number density is needed, . e.g., for computing synthetic maps at (sub-)millimetre wavelengths, as the opacity at these wavelengths is due to thermal free-free transitions of hydrogen, including H− . Chromospheric dynamics as we hope to see it with the Large Milimeter Array (ALMA). Courtesy, Sven Wedemeyer-Böhm, KIS Brightness temperature maps (upper row) and contribution function (lower row) for the LTE case (left) and with non-equilibrium electron densities (right). toc ref § 20.2 Towards the simulation of supergranulation cells Efforts are underway to increase the simulation box so as to accommodate a supergranulation cell. Recently, Stein et al. started a simulation of 48 × 48 × 20 Mm using 5003 grid cells. With this simulation they hope to find out more about the origin of supergranulation and to carry out helioseismological experiments. Courtesy, R.F. Stein toc ref Future directions (cont.) Recently, Hansteen et al. have carried out simulations over a hight range from the top layers of the convection zone to the corona. They seek to understand the formation of jets such as dynamic fibrils, mottles, and spicules in the solar chromosphere, which is one of the most important, but also most poorly understood, phenomena of the Sun’s magnetized outer atmosphere. Courtesy, M. Carlsson toc ref § 20.3 Boundary conditions: Advecting magnetic field across boundaries Recently, we have relaxed the condition of vanishing horizontal magnetic field components at the top and bottom boundary. Now, horizontal magnetic field of a specified flux density may be advected across the bottom boundary into the box toc ref § 20.4 Numerical experiments Time sequence of a two-dimensional simulation of magnetoconvection starting with an initial homogeneous vertical magnetic field of 10 G. This sequence was repeated with a plane parallel, oscillatory velocity perturbation at the bottom with an amplitude of 50 m/s and a frequency of 100 mHz. The logarithm of the absolute velocity difference of the two runs shows the propagation and perturbation of plane waves through the presence of a magnetic flux concentration. toc ref § 21 Waves carried by magnetic flux tubes The movie sequence of § 17 suggests that the interaction of a magnetic flux tube with convective motion excites transverse tube waves. Consider a straight vertical tube and let us first compute the restoring force for a small z transversal displacement, ξ(z, x), such that the flux-tube axis remains in the x, z -plane. ^ l tangent vector to the axis, n̂ the principal normal. ξ (z,t) n^ l̂ is the From § 3.3 we know that the magnetic tension force in the direction of the principal normal is 2π/ k (B 2 /4π)n̂/Rc , where Rc is the curvature radius for which (1/Rc ) = ξ ′′ , where ξ ′ = ∂ξ/∂z . The buoyancy force component in direction of n̂ is x toc ref (ρi − ρe )gξ ′ , where ξ ′ = tan α ≈ α. Waves carried by magnetic flux tubes (cont.) From this, we find for the restoring force 2 B Fn = (ρi − ρe )gξ ′ + ξ ′′ , 4π Fdyn = (ρ + ρe )ξ̈ . Here, we incorporate the effect of the external medium on the dynamics by adding ρe to which must be oposed by the inertial force of the displacement: the internal intensity, which alone would do it if there were no external medium, which, however, adds to the inertia. Hence, the equation of the transverse tube wave is: ρi ρi − ρe ′ ¨ gξ + c2A ξ ′′ , ξ= ρi + ρe ρi + ρe where cA = B/(4πρ)1/2 is the Alfvén speed. If the atmosphere is isothermal we have a constant pressure scale height, Hp , of same size inside and outside the flux tube. Then, ρe /ρi = pe /pi = 1 + 1/β and using βc2A = 2gHp leads to a 2nd-order homogeneous ODE with constant coefficients, which has the solution: toc ref Waves carried by magnetic flux tubes (cont.) ξ ∼ exp(iωt + ikz + z/4Hp ) , with the dispersion relation gHp ω = β + 1/2 2 „ 1 k + 16Hp2 2 « , from which we see that no transverse tube wave can propagate if ω is smaller than the cutoff frequency, ωc , ωc2 g 1 = . 8Hp 2β + 1 Using these equations, Choudhuri et al. (1993a) investigate the propagation of a kink wave initiated by the movement of its footpoint according to vx (z = 0, t) = v0 e toc ref −bt2 . Waves carried by magnetic flux tubes (cont.) Displacement, ξ(z, t), of the magnetic flux tube as a function of height (s = z/(4Hp )) at various instants (τ = ωc t, where ωc is the p cutoff frequency) for (1/ωc ) b/π =: λ = 0.5. Choudhuri et al. (1993a) Choudhuri et al. compute the total energy that is injected into the system by footpoint motion. They find for a velocity of v0 = 3 kms−1 with a total displacement of about 750 km and typical solar values for density, pressure scale-height, etc., values of several times 10 26 erg. They stress the importance of occasional rapid footpoint motions over slow motions. Choudhuri et al. (1993b) come to similar conclusions when incorporating a temperature jump simulating transition to coronal temperatures. toc ref Waves carried by magnetic flux tubes (cont.) We now reformulate the equation of the transverse tube wave using the reduced displacement Qk (z, t) defined by ξ(z, t) = ez/(4Hp ) · Qk (z, t) . Substituting ξ in the equation of the transverse tube wave leads to a partial differential equation of the type of the Klein-Gordon-equation in one spatial dimension, the Klein-Gordon-equation for transversal tube waves: ∂ 2 Qk 1 ∂ 2 Qk 1 Qk = 0 , − 2 − ∂z 2 ckink ∂t2 16Hp2 c2kink = toc ref β 2gHp = c2A 2β + 1 2β + 1 and k2 = 1 . 2 16Hp (1) Waves carried by magnetic flux tubes (cont.) Turning now to longitudinal waves we may take advantage of the equation for the longitudinal velocity perturbation in § 14.7. Assuming an isothermal atmosphere with Hp = const. we also have γ = const.. Furthermore, we continue to use β = const. Then, Eq. (∗) reduces to  2 ff 1 ′ 1 1 ω ṽ ′′ − + ( − 1)(1 + β) ṽ = 0 , ṽ + 2 2 2Hp cT 2Hp γ which, as for the transversal tube wave is a linear, homogeneous, 2nd-order ODE, so that we can proceed as before. The general solution again is v ∼ exp(iωt + ikz + z/4Hp ) , with the dispersion relation  ω 2 = c2T k2 + toc ref 1 2Hp2 „ «ff 1 1 + (1 + β)(1 − ) , 8 γ Waves carried by magnetic flux tubes (cont.) from which we see that no longitudinal tube wave can propagate if ω is smaller than the cutoff frequency, ωc , ωc2 = With v̈ c2T 2Hp2  ff 1 1 + (1 − )(1 + β) . 8 γ = −ω 2 v we can rewrite the equation for longitudinal waves in a isothermal, vertical flux tube: 1 ′ 1 1 v̈ v − ( − 1)(1 + β)v = 0 . v − 2 + 2 2Hp cT 2Hp γ ′′ Again we define a reduced displacement, Qλ (z, t): ξk (z, t) = ez/(4Hp ) · Qλ (z, t) , i.e., v(z, t) = ez/(4Hp ) · Q̇λ (z, t) , toc ref Waves carried by magnetic flux tubes (cont.) to obtain the Klein-Gordon-equation for the longitudinal tube wave (Rae & Roberts, 1982): ∂ 2 Q̇λ 1 ∂ 2 Q̇λ 2 − − k λ Q̇λ = 0 , 2 2 2 ∂z cT ∂t (2) 1 1 + γβ/2 1 1 = = , + 2 2 2 2 cT c cA c0 » – 1 1 1 1 2 kλ = 2 + (1 − )(1 + β) . Hp 16 2 γ Hasan & Kalkofen (1999) solve Eqs. (1) and (2) with an additional forcing term on the right hand side resulting from an external motion of granules. toc ref Waves carried by magnetic flux tubes (cont.) Response of a flux tube (β “interaction time” of τ = 0.3) on an initial footpoint motion with v⊥ = 1 km s−1 over an = 50 s. Left: Variation of the reduced velocity, Q̇, of the kink and longitudinal wave at two different heights in the solar atmosphere. After passage of the primary impulse, the atmosphere oscillates as a whole with the cutoff period associated with the wave. Left: Wave-energy flux in the vertical direction, where f0 denotes the filling factor of magnetic fluxtubes at z = 0. With t → ∞ velocity and pressure are out of phase by 90◦ so that the wave carries no enery in this limit. Transverse waves are more efficiently excited than longitudinal waves. From Hasan & Kalkofen (1999) toc ref Waves carried by magnetic flux tubes (cont.) When taking the footpoint motion from actual observations of G-band bright points, the corresponding vertical energy flux in transverse waves takes place in brief and intermittant bursts, quite in contradiction with the observed persitency of emission from the network region (left). Therefore, Hasan, Kalkofen, & van Ballegooijen (2000) argue that unobserved random footpoint motion occuring on a time-scale of a few seconds and a displacement-scale of a few km must be present. Adding such high-frequency motion results in a much larger and less intermittent energy flux (right). They conclude, that for transverse waves to provide sustained chromospheric heating, the main contribution must come from high-frequency motions with periods of 5–50 s. toc ref Waves carried by magnetic flux tubes (cont.) These calculations, however, are only valid in the linear regime. In the non-linerar regime mode coupling can take place. As a transversal tube wave propagates upwards in the vertical direction, the velocity amplitude increases because of the exponentially decreasing density and energy conservation and as it becomes comperable to the tube speed, coupling to the longitudinal magneto-acoustic wave can take place and transfer energy to them. Since the longitudinal mode can form shocks, the coupling ensures an ample source of energy for heating the atmosphere. Numerical simulation of mode coupling have been carried out by Ulmschneider, Zähringer, & Musielak (1991). Mode coupling does probably also contribute to the generation of the longitudinal flow and associated shock waves within the flux sheet in the simulation shown in § 17. toc ref An external magnetic field removes the degen- +3 +2 +1 0 −1 −2 −3 J=3 ' & The Zeeman effect eracy of the energy levels of electrons in an atom, which split into sublevels of distinct mag- ∆ M = −1 ∆ M=0 ∆ M = +1 netic quantum numbers. Transitions between these levels are subject to selection rules with +2 +1 0 −1 −2 J=2 σ the consequence, that the emitted light has distinct polarization properties. π σ λ For example, light travelling in the direction of the magnetic field is left or right circularly polarized, depending on whether it stems from a ∆M or a λb λ0 λr = −1 ∆M = +1 transition. Depending on the selected transition it also has different wavelength Zeeman σ -components in emission → ∆λB → backto § 2 $ % e g ∗ λ2 B ∆λB = 4πcme → Stokes ' & The Zeeman splitting → backto § 2 $ % ' & Stokes V profile Absorption spectral line (Stokes I ) together with the corresponding Stokes V (schematic). Due to the Zeeman-effect, left or right cir- 1 cular polarization prevails in the flanks of a Stokes I spectral line. Stokes V is the difference of right and left circularly polarized light. 0 Stokes V 0 λ0 λ → backto § 2 $ % ' & Magnetic knot Example of a magnetic knot, recorded with the Teneriffe Infrared Polarimeter, TIP, at the German Vacuum Tower Telescope by R. Schlichenmaier, KIS. 0.17 T → backto § 2 $ % ' & Magnetic filling factor – solar microscopy 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 resolution element A res Twodimensional spectroscopy has a spatial resolution of ≈ 0.5 arcs at best. Therefore, Ares ≫ Aft . We can define a filling factor: flux tube A ft Aft α= . Ares With polarimetric measurements the state of the plasma in Aft can be explored without the need of resolving the flux tube, since only the light from Aft is polarized – This is the idea of solar microscopy (Stenflo). Note, however, that there might be additional unobservable magnetic flux in Ares with zero net polarization signal because of opposite polarity. → backto § 2 H H $ % ' & Magnetic filling factor – solar microscopy (cont.) I : I -profile from Ares (observable), Iα : I -profile from Aft , assuming that Landé g = 0 (non-observable), Iσ1,2 : the two σ -components of the I -profile from Aft (non-observable). 1 Iα Iσ I σ2 1 0 λ0 λ0 − ∆ λH λ0 λ0 + ∆ λH 1 V = [Iα (λ + ∆λH ) − Iα (λ − ∆λH )] 2 stems from Aft (observable) → backto § 2 H H $ % ' & Magnetic filling factor – solar microscopy (cont.) Expansion of the σ -components at λ = λ0 : ∂Iα 1 ∂ 3 Iα 1 ∂ 2 Iα 2 3 Iα (λ + ∆λH ) = Iα (λ) + + ∆λ ∆λ ∆λH + H H + ... 2 3 ∂λ 2 ∂λ 6 ∂λ ∂Iα 1 ∂ 3 Iα 1 ∂ 2 Iα 2 3 Iα (λ − ∆λH ) = Iα (λ) − − ∆λ ∆λ ∆λH + H H + ... 2 3 ∂λ 2 ∂λ 6 ∂λ = 4.67 · 10−13 · gλ2 B , [B] = Gauß , [λ] = Å , and k = 4.67 · 10−13 · λ2 we write ∆λH = k · g · B ⇒ – » 3 1 ∂ Iα ∂Iα 2 (kgB) + ... + V = kgB 3 ∂λ 6 ∂λ With ∆λH In a first approximation Iα ≈ αI , which would be correct if the atmosphere in Aft was identical with the atmosphere in Ares . If the I -profile from Aft and Ares differ only in the depression, not in the line shape, we may define a line-weakening factor w : → backto § 2 H H $ % ' & Magnetic filling factor – solar microscopy (cont.) Iα 1− Iα | {z c} rel. line depression in mag. region =w „ « I 1− . Ic | {z } rel. line depression in non-mag. region Ic and Iαc are the continuum from Ares and Aft , respectively. Then „ « Iα I α = I αc − w I αc − I c . Ic We set Iαc ∂I ∂Iα = wα , so that = αIc and obtain ∂λ ∂λ – » 3 1∂ I ∂I 2 (kgB) + ... , + V = kgBwα 3 ∂λ 6 ∂λ which can be solved for B if α and w are known. The latter two drop out in the line-ratio method. → backto § 2 $ % From the expansion of the σ -profiles we obtain (see Magnetic filling factor): V = kgBwα » 3 – ' & The line ratio method ∂I 1∂ I 3 (kgB) + ... , + 3 ∂λ 6 ∂λ from which we can in principle compute B if α and w were known. Taking a spectral-line ratio, we obtain » 3 – ∂I1 1 ∂ I1 2 (kg B) + ... + 1 3 6 ∂λ V1 w1 g1 ∂λ – » = 3 V2 w2 g2 ∂I2 1 ∂ I2 2 (kg B) + ... + 2 3 ∂λ 6 ∂λ and get so rid of the filling factor α. Taking two lines with small g -factors we can neglect higher-order terms and get V1 ∂I2 w1 g1 ∂I1 )/( ). ≈ ( V2 w2 g2 ∂λ ∂λ → backto § 2 H H $ % ' & The line ratio method (cont.) From (w1 /w2 )(λ) we can derive the g χe Fe 5247.06 2 0.09 eV Fe 5250.65 1.5 2.2 eV temperature structure of the flux-tube atmosphere. This has been done with lines of the “thermal ratio” (see table). Choosing a line pair with similar line parameters except as of the Landé g-factor, we have w1 ≈ w2 , thus, » 3 – 1 ∂ I1 ∂I1 2 (kg B) + ... + 1 3 6 ∂λ V1 g1 ∂λ » – , ≈ 3 V2 g2 ∂I2 1 ∂ I2 2 (kg B) + ... + 2 3 ∂λ 6 ∂λ from which expression we may derive B . A suitable choice for a “magnetic ratio” is given in the following table: g χe Fe 5247.06 2 0.09 eV Fe 5250.22 3 0.12 eV → backto § 2 H H $ % ' & The line ratio method (cont.) If the magnetic field were weak one would measure V5247 g5247 2 ≈ = . V5250 g5250 3 Typically measured values are V5247 /V5250 ≈ 1. This means that the higher-order terms are important (“Zeeman saturation”), and the magnetic field is strong. Scatter plot of apparent flux densities observed in the two lines Fe I 5247.06 and Fe I 5250.22. If the field were intrinsically weak, (≤ 500 G), the points would fall around the ◦ dashed 45 line. → backto § 2 $ % ' & Individual magnetic bright points adjacent to a “ribbon” structure. Berger, Rouppe van der Voort, Löfdahl et al. A&A 428, 613 with the new Swedish Telescope on La Palma → backto § 2a $ % ' & Formation of the bright and the dark ring of an ideal flux tube τc =1 T1 T3 T2 I/Iquiet 1 → backto § 2a $ % ' & More “flowers”. Berger, Rouppe van der Voort, Löfdahl et al. A&A 428, 613 with the new Swedish Telescope on La Palma → backto § 2a $ % ' & Magnetograms at 0.1” spatial resolution.... LOS ... maybe a disappointment. High expectations are at stake fueled by movies like the ones of Cattaneo & Emonet or Bz , at constant geometrical height, not → vector polarimetry → backto § 2a Vögler et al. Be aware that these show a magnetogram signal. $ % ' & Field strength |B|, filling factor β , and velocity v from Stokes inversion. From Beck, Schmidt, Bellot Rubio, Schlichenmaier & Sütterlin (2005) [A&A in press]. → backto § 2a H H $ % – + + – + + – + + – + – – + – – + – + + – + + – + + – + ' + - + & + + + Stokes V profiles that emanate from vertical lines of sight distributed over the horizontal interval between 600 and 1380 km of the simulation snapshot. Positive area asymmetries result in the two regions labeled with “+”, while the lines of sight in the middle region labeled with “−” contribute negative area asymmetries, exactly as expected from the simple sketches above. From Steiner (1999) → backto § 2a $ % ' & The critical height – the canopy height B 02 ln 8π 0 ln p z crit p 0e i e → backto § 7.1 $ % ' & Method of short characteristics α γ P a d b O g M e h f c β δ Short characteristic MO. The intensity incident on point O is a mini formal solution along the short characteristic MO. The physical variables at M and P are obtained by quadratic interpolation using points a . . . h (8 point stencil). Points α . . . δ are used additionally for the evaluation of the intensity incident at the cell boundary at M (12 point stencil). After Kunasz & Auer (1988) → backto § 9.6 H H $ % ' & Method of short characteristics Formal solution of the radiative transfer equation with short characteristics. For rays propagating from the lower left to the upper right one starts in the lower left corner cell and integrates along short characteristics in the bottom row cells from left to right, using the (blue) boundary intensities. Then, next row from left to right and so on → backto § 9.6 $ % ' & Observation by Sigwarth et al. 1999, A&A 349, 941 → backto § 13 $ % ' & Overstability If at the onset of instability oscillatory motion prevails, then one says, following Eddington, that one has the case of overstability. Eddington: “In the usual kinds of instability, a slight displacement provokes restoring forces tending away from equilibrium; in overstability it provokes restoring forces so strong as to overshoot the corresponding position on the other side of equilibrium.” → backto § 15 $ % ' & Net circulation of granular flow correlates with vortical flow at z = 550 km. vhorizontal and grey-scale map of Bz at z = 550 km. Top: Bottom: Same as top at z = 100 km. → backto § 17 H H $ % ' & Radiative channeling T [103 K] 4.0 6.0 8.0 10.0 12.0 0.2 [Mm] 0.0 -0.2 -0.4 0.0 0.2 0.4 0.6 0.8 1.0 [Mm] → backto § 17 $ % ' & Statistical properties of the magnetic field in a layer of 100 km thickness around optical depth unity. → backto § 17 $ % ' & What is the CO5 BOLD code? 5 CO BOLD stands for COnservative COde for the COmputation of COmpressible COnvection in a BOx of L Dimensions with L=2,3. 5 CO BOLD is designed for simulating hydrodynamics and radiative transfer in the outer and inner layers of stars. Additionally, it can treat magnetohydrodynamics, non-equilibrium chemical reaction networks, dynamic hydrogen ionization, and dust formation in stellar atmospheres. → backto § 19 H H $ % (Courtesy Sven Wedemeyer-Böhm) → backto § 19 ' & Application examples of CO5 BOLD H H $ % ' & 5 What is the CO BOLD code? (cont.) Simulation of solar granulation with Simulation of a red supergiant with CO5 BOLD. 400 × 400 × 165 grid cells, CO5 BOLD. 2353 grid cells, 11.2 × 11.2 Mm, Mean contrast at λ ≈ 620 nm is 16.65%. mstar = 12m⊙ , Teff = 3436 K, Rstar = 875R⊙ Courtesy M. Steffen, AIP Courtesy Bernd Freytag → backto § 19 H H $ % 5 CO BOLD works with ' & 5 What is the CO BOLD code? (cont.) - Cartesian (non-equidistant) grids, - realistic equation of state, - non-local, multidimensional radiation transport, - realistic opacities, - various boundary conditions 5 CO BOLD is programmed with - FORTRAN 90, - OpenMP directives, 5 The manual for CO BOLD can be found under http://www.astro.uu.se/˜bf/co5bold main.html Just type CO5BOLD in Google. → backto § 19 $ % ' & A shock capturing numerical scheme Two-dimensional radiation-hydrodynamic simulation of surface convection including the chromospheric layer. The dimensions of the computational domain are: Width, 5600 km; Height above the surface of τ = 1, 1700 km; Depth below this surface level: 1400 km. S. Wedemeyer et al. 2004, A&A 414, 1121 → backto § 19 H H $ % For a basic example of a shock capturing numerical scheme, consider a ' & A shock capturing numerical scheme (cont.) piecewise constant reconstruction with discontinuities at cell interfaces (S.K. Godunov, 1959). q 1 0 0 1 q 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 x 11 00 00 11 00 11 1 0 0 1 q xi 1 0 0 1 0 1 1 0 0 1 11 00 00 11 00 11 x i+1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 x → backto § 19 H H $ % The shock-tube problem ' & A shock capturing numerical scheme (cont.) t0 ρ ρ l ρ r x p p l p r x v vl = vr = 0 x → backto § 19 $ % The shock-tube problem t1 t0 ρ ρ ρ l ρl ρ* l ρ*r ρ r p p l ρ r x x p ' & A shock capturing numerical scheme (cont.) p l p* p p r r x x v* v v vl = vr vl = 0 x vr x → backto § 19 $ % ' & A shock capturing numerical scheme (cont.) The shock-tube problem t1 t0 ρ ρ ρ l ρl ρ* l ρ*r ρ r p p l r x x p ρ p t l q* p* p p r r x x v* v v vl = vr l vl = 0 x ql q *r qr x vr x → backto § 19 $ % ' & A shock capturing numerical scheme (cont.) 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