STRUCTURESTRUCTURE && DYNAMICSDYNAMICS OFOF THETHE SOLARSOLAR ATMOSPHEREATMOSPHERE
SIRAJ HASAN, IIA
National Symposium on Astrophysics “Sun, Stars & Extragalactic Universe” August 8-10, 2002 Background The structure and dynamics of the Sun and other stars is closely related to the presence of magnetic fields which are responsible for various kinds of activity involving a complex interaction between the field and plasma. Examples: ► Sunspots that are dark cool features, responsible for a rich variety of magnetic activity. They exhibit a 11 year cycle ► Faculae that consist of bright magnetic regions which along with sunspots are the main constituents of active regions. Faculae are arranged in plages and in an irregular network called the enhanced network ► Prominences or filaments which are cool dense structures that can become active and erupt ► Solar flares representing a sudden (< 1 hr) release of large energy (1029 -1032 erg) ► Coronal mass ejection (CMEs) which represent the ejection of coronal matter into space
Questions An understanding of the solar atmosphere must necessarily focus on the physics of magnetic fields at different locations from the deep convection zone to the upper atmosphere. Some fundamental questions that arise are:
►How is the magnetic field generated, maintained and dispersed? ►What are its properties such as structure, strength and geometry? ►What are the dynamical processes associated with magnetic fields? ►What role do magnetic fields play in energy transport in the solar atmosphere? Magnetic Field Generation & Emergence ►The magnetic field of the Sun emerges through the surface in compact bipolar regions ►The frequency with which such regions appear is modulated on a 11 year time scale ►Bipolar regions emerge primarily with a east-west orientation ►These regions occur within a latitude belt ±30º ►The magnetic field is generated through a dynamo mechanism by the combined effect of differential rotation and flows (meridional circulation) in the convection zone ►The “seat” of the dynamo is likely to be at the base of the convection zone in the convective overshoot layer which co- incides with the tachocline The solar magnetic field in time through magnetograms which depict opposite polarities as darker and brigher regions Solar Dynamo The essential idea is that a magnetic field is maintained by currents induced in a plasma by its motion across field lines. V Induced electric field Ohm’s law Ampere’s law B V × B/c j = σ ( E+V × B/c ) curl B = 4π/c j
Maxwell’s equation c curl E = -∂B/∂t j × B/c Lorent’z force
2 ∂ B = ∇ × (V × B ) + η∇ B Induction equation ∂t
The dynamo involves the generation of a toroidal (azimuthal) component of the field from a poloidal (meridional) field due to differential rotation (ω effect). The next step is the regeneration of the poloidal field from the toroidal field (α effect). Flux Emergence Once the field has been amplified by the dynamo it needs to be “released” into the convection zone where it is transported upwards through magnetic buoyancy. Consider an isolated flux tube in pressure balance with its surroundings: B 2 p + = p i 8π e
Assuming Ti=Te (thermal equilibrium), then since pi < pe, ρi < ρe. The tube being less dense than its surroundings will float up due to buoyancy. The rise of a flux tube is influenced by buoyancy, aero- dynamic drag and the Coriolis force. If the field is not strong enough (<105 G), the inertial forces would make the tube rise in high latitudes, contrary to observations. Schematic diagram of the likely sub-photospheric magnetic structure of an emerging active region. Surface Phenomena ►The surface magnetic field is highly structured consisting of discrete elements varying in horizontal scales from 50000 km (sunspots) to 100 km in intense flux tubes with kilogauss field strengths ► Extended structures (plages) and the network consist of assemblies of intense flux tubes . In plages flux tubes fill up about 30% of the total area. Outside active regions (the quiet Sun), magnetic elements have a filling factor of abut 1% Calculations show that magnetic fields emerge through the surface with a field strength typically of around 400 G (equipartition value), which is much lower than the observed values. Consequently, some local process must be present which intensifies the field to kilogauss strengths. Convective collapse
The atmosphere inside and outside the tube is initially in hydrostatic and
thermal equilibrium (T i= Te). Consider a small downwards adiabatic displacement ∆z of a fluid element. Assuming that this element remains in pressure equilibrium with the ambient medium, it will experience negative buoyancy if the temperature gradient inside the tube is super- adiabatic The relationship between the critical photospheric radius aph atmosphere inside and photospheric field strength Bph (solid line) which demarcates the separation between convectively stable and unstable tubes. The top scale denotes the magnetic flux. Superposed are data from observations: squares are from Solanki et al. (1996); the light shaded region is from Lin (1995), Lin & Rimmele (1999); the dark shaded region is from Sanchez et al. (2000). All points above the solid curve denote stable configurations while those below are unstable. A flux tube that is unstable will trace out a vertical path. Magnetic carpet ►The solar surface is covered with mixed polarity magnetic fields from the largest to scales down to the resolution limit of current solar telescopes ►There are 4 basic processes: – Emergence (appearance of new flux) – Cancellation (merging of opposite polarities) – Coalescence (merging of same polarities) – Fragmentation (splitting into smaller flux concentrations)
►The continuous emergence of ephemeral regions together with the constant cancellation of merged and fragmented flux concentrations means that the magnetic field outside of active regions is highly dynamic ►Estimates reveal that the magnetic flux is replaced every 40 hours! ►This implies that there is a very large amount of energy available for heating the solar atmosphere through magnetic reconnection Potential field extrapolations for mixed polarity fields. The field locations are selected randomly on a mesh simulating supergranulation. Dynamical processes in the Chromosphere ►The quiet chromosphere consists of magnetic and non- magnetic regions. Although magnetic fields are found everywhere on the Sun they are dynamically unimportant in the interior of supergranulation cells.
►Strong magnetic elements occur at the cell boundaries and constitute the “magnetic network”. Vertical tubes in pressure equilibrium with the outside medium expand upward to conserve magnetic flux. From a low filling factor (< 1%) in the photosphere the tubes spread to 15% in the layers of formation of the emission features in the H and K lines of ionized calcium (at a height of 1 Mm) and to 100% in the so-called “magnetic canopy”.
►The remaining quiet Sun outside the network is called the internetwork, sometimes also referred to as cell interior.
MAGNETIC NETWORK ►Magnetic elements can be identified with bright points in G-band (4305 Å) images, which are co- spatial with Ca II and TRACE (UV) structures ►High resolution observations suggest that these network bright points (NBPs), located in intergranular lanes, are in a highly dynamical state, due to the buffeting effect of random convective motions ►A histogram of the velocities of NBPs shows a mean speed of 1.4 km s-1 but there were several instances of motions with speeds as high as 3 km s-1 .
Observations ►Ground-based observations of the Ca II H and K lines, which are formed in the low chromosphere, reveal that the chromosphere in the magnetic network of the quiet Sun oscillates with periods of around 7 min., in contrast to the internetwork which exhibits oscillations at higher frequencies. Recent observations suggest the presence of multiple peaks in the power spectrum with periods in the 4-15 min. range.
►Space observations have also confirmed the above picture and the absence of significant power in oscillations with frequencies above 3 mHz Velocity power spectra in the network & inter- network (after Lites et al. 1993)
Power spectrum of a NBP at different thresholds (after McAteer et al 2002) THEORETICAL ASPECTS ►The magnetic field in the network can be idealized in terms of thin isolated vertical flux tubes which fan out with height, due to the decrease in pressure with height. A ``thin'' flux tube can support sausage or longitudinal oscillations and kink or transverse oscillations.
►The earliest studies on MHD wave excitation used extensions of the Lighthill mechism. More recently, several studied have examined the generation of longitudinal and transverse waves in a flux tube through turbulent motions in the convection zone. ►An alternative scenario, based on observations is that the excitation of transverse (kink) waves occurs through the footpoint motions of magnetic elements. Rapid motions with velocities larger than 2 km s-1 can excite transverse oscillations which carry adequate energy for coronal heating
►Calculations show that transverse waves get converted to longitudinal waves in the chromosphere. The latter can easily dissipate through shock formations and contribute to heating the atmosphere
►Many numerical simulations of dynamical effects associated with the interaction of magnetic fields and convection have been carried out These simulations clearly show the bending of flux sheath through the buffeting action of granules. This interaction can excite MHD oscillations in the magnetic element which can propagate upwards and heat the chromosphere and corona The convective flow of gas in the photosphere sweeps magnetic lines of force into narrow tubes of magnetized material. In such flux tubes the field reaches a strength of 2000 G. The tubes are permanently pressed and shaken; they do not settle to a steady equilibrium. Along the tubes, MHD waves transmit energy into the upper part of the atmosphere (after Steiner et al. 1998) PERTURBATIONS IN A FLUX TUBE Sausage mode Kink mode
ω ω
2 2 2 2 2 2 2 2 = cλ k + ωλ = cκ k + ω κ Linear 1-D Model Consider a vertical thinα isothermal flux tube. The linear oscillations of this tube areα given by a Klein-Gordon equation of the form: α ∂ 2Q ∂ 2Q − c2 α −ω 2Qα = F ∂t 2 ∂z 2 α
-z/4H where Qα=ξα e , ξα is the Lagrangian displacement, H is the scale height (α=λ,κ for longitudinal & transverse waves). The waves have theω following dispersion relation:
2 2 2 2 = cα k + ω α
tube speed cutoff frequency Typical parameters
-1 Sound speed: Cs = 6.6 km s Plasma beta: β = 0.3 (B ≈ 1600 G at z = 0)
Tube speeds are: -1 Cκ = 5.7 km s (transverse waves) -1 Cλ = 5.9 km s (longitudinal waves)
Cutoff periods are:
Pκ = Pa 2γ (1+ 2ββ) ≈ 7 min. (transverse waves)
Pλ = Pa (60 + 50 ) /(63 + 48β ) ≈ 3 min. (longitudi nal waves) where P = 4πH/cs=180 s is the acoustic cutoff period. LINEAR RESULTS ► The generic behaviour is the same for transverse and longitudinal wave excitation: the buffeting action of a granule on a flux tube impulsively excites a pulse that travels away from the source region (with the kink or longitudinal tube speed). After the passage of the pulse, the atmosphere oscillates at the cutoff period of the mode, with an amplitude that slowly decays in time. ► The initial pulse carries most of the energy ---- subsequently the atmosphere oscillates as a whole in phase, without energy transport. The wave period observed in the magnetic network is interpreted as the cutoff period of transverse waves, which leads naturally to an oscillation at this period (typically in the 7 min. range). ► Velocities in transverse and longitudinal waves excited by granules are comparable in magnitude for typical values of β (β<<1) . However, the vertical energy flux in transverse waves is an order of magnitude larger than that in longitudinal waves for network field strengths. (N.B. The displacements and velocities in the flux tube scale linearly and the energy fluxes scale quadratically with the applied external velocity.) Difference between G band and continuum images obtained at the Swedish Solar Observatory on Oct. 5, 1995. Black corks are tracers for tracking NBPs (after van Ballegooijen et al. 1998).
Time variation of the vertical energy flux in a single flux tube at a height of 750km generated due to footpoint motions taken from obser-vations (after Hasan et al. 2000). NONLINEAR ANALYSIS Nonlinear effects become important in the chromosphere, where the velocity amplitudes become comparable with the tube speed, leading to an efficient coupling of the kink and longitudinal modes.
A horizontal motion is applied impulsively at the base, due to which a transverse wave is generated. Due to nonlinear coupling, a longitudinal wave is generated. After the passage of the impulse, the two modes get decoupled and generate motions at their respective cutoff frequencies (after Hasan et al. 2002) NONLINEAR RESULTS ► When the transverse velocities are significantly less than the kink wave speed, there is essentially no excitation of longitudinal waves. However, at heights where the two become comparable, longitudinal wave generation becomes efficient.
► A large transverse pulse generates a longitudinal wave through mode coupling (similar to for the coupling of torsional Alfvén waves with longitudinal waves).
► After the passage of the pulse, “wakes” are generated, that represent decoupled kink and longitudinal waves oscillating at their respective cutoff frequencies.
► Transverse waves lose energy to longitudinal waves through mode coupling.. The fractional energy in longitudinal motions increases rapidly at first with the forcing transverse velocity at the base, before eventually saturating to a value when there is almost equipartition of energy between the two modes. SUMMARY ►The structure and dynamics of the solar atmosphere is directly related to the presence of magnetic fields ►The field is generated through a dynamo action which most likely occurs in a shear layer below the convection zone. ►Magnetic fields emerge through the surface as large coherent flux elements with strengths of a few hundred Gauss. This field is further strengthened by convective collapse to kilogauss strengths ►Flux tubes in the magnetic network play an important role in heating the upper atmosphere. Their footpoints located in the subphotosphere, are constantly shaken by convective motions due to which MHD oscillations are excited. ►Detailed calculations have identified the nature of the observed low frequency waves observed in the network