Friday, 22 November 2013

Energy Loss of Solar f- and p-Modes due to MHD tube wave excitation.

I'm currently running numerical MHD simulations of the dynamics in the solar corona generated by Solar Global oscillations.

Here are some key references I'm studying

  • Helioseismology by J. Christensen-Dalsgaard
  • Lecture Notes on Stellar Oscillations by J

  • Today there was an interesting seminar entitled "Energy Loss of Solar f- and p-Modes due to MHD tube wave excitation" as it seems fairly close to some of the areas I'm investigating this seemed to be a must attend event! The talk was a discussion of the generation and propagation of sausage tube waves within the solar convection zone and chromosphere by the buffeting of p modes. Evidence of p-modes and solar global oscillations is nowadays easily observed from satellite imagery. The images below from SDO for the 15th November 2013 are the Magnetogram, HMI Intensity gram and Dopplergram.


    The Magnetogram shows the magnetic strengths and opposite polarities in the vicinity of sunspots. The right hand image below is a dopplergram. It is interesting to note that the dopplergram shows a suppression of the surface velocity at the at sunspot locations and suggesting that the intensity of p modes are reduced by magnetic fields. The observed power spectrum for p-mode oscillations is shown in the diagram below the intense region at 3.3mHz corresponds to the ubiquitous 5 minute oscillations. It is important to note the ridges present in this diagram. 

    The period versus horizontal wavelength diagram obtained by the Michelson Doppler Imager (MDI) on the SOHO spacecraft. Since only waves with specific combinations (related to the Sun's interior structure) of period and horizontal wavelength resonate within the Sun, they produce the fine-tuned `ridges' of greater power. (Image source: SOHO). The power spectrum obtained from  the MDI Medium-l data for the modes averaged over the azimuthal order m. The power concentrates in ridges corresponding to solar acoustic (p) modes. The lowest weak ridge corresponds to the fundamental (f) mode.

    Simulations may be used to excite a rich spectrum of p-mode oscillations (left), very similar to the MDI diagram (right). The dark line is the theoretical f-mode.

    The power spectrum above represents p-mode frequency oscillations of the Sun. The x and y axes are wave numbers which are calculated from the degree l of the spherical surface harmonic modes. The z axis is the frequency of oscillation.  In the diagram above the rings cut out of the z-axis plane are 3D representations of the p-mode ridges. Six rings equal six modes. Eight rings equal eight p-modes, and so on. The frequency value of the top of the z plane is the Nyquist frequency, the high-frequency limit given by the time resolution of the signal, which here is 8334 microHz. The densest part of the ridge pattern is roughly 3000 microHz. The frequency resolution is 5.71 microHz. This data was acquired using the Michelson Doppler Imager instrument on board the SOHO spacecraft.

    For solar oscillations, the power is not evenly distributed in the k-omega plane, but instead corresponds to ridges. Each of these ridges corresponds to a fixed number of wave nodes in the radial direction. The ridges are seen here as "rays" in the cut planes facing you.

    Introduction to helio-seismology

    Tube waves propagate along the many magnetic fibrils which are embedded in the convection zone and expand into the chromosphere due to the fall in density with height of the surrounding plasma.   The magnetic fibrils form a waveguide for these waves to freely propagate up and down the tube, those waves propagating upward pass through the photosphere into the chromosphere and enter the upper atmosphere, where they can be measured as loop oscillations and other forms of  propagating coronal waves. We treat the magnetic fibrils as vertically aligned, thin flux tubes embedded in a two region polytropic-isothermal atmosphere to investigate the coupling of p-mode driven sausage waves; which are excited in the convection zone and propagate into the overlying chromosphere. 

     Observational evidence for tube modes

    Observational evidence of tube modes presented by Srivistava et al (ApJ 2013)

    The excited tube waves carry energy away from the p-mode cavity resulting in a deficit of p-mode energy which we quantify by computing the associated damping rate and absorption coefficient of the driving p modes.  We calculate the damping rates/absorption coefficients and compare them with observations and previous theoretical studies of this nature.

    It is understood that magnetic field lines suppress the p-mode oscillations this occurs as a result of sausage mode oscillations being driven by the p-modes. The mode conversion from acoustic to magneto acoustic tube mode results in a substantial absorption of energy

    Friday, 8 November 2013

    Solar atmospheric MHD flux tube equilibria

    Fred Gent's presentation reviewed an area which is vital for correct modelling solar phenomena driven by p-mode oscillations. Numerical modelling of wave phenomena in the solar atmosphere require information about the structure of magnetic fields therein. Such information is gleaned from magnetograms or from the analysis of Zeeman splittings or Stokes profiles. Improvements are expected as coronal seismology increases in maturity and can be used to infer information about the magnetic field. Even when armed with this range of sophisticated spectropolarimetric methods it is challenging to construct a numerical model of the magnetic fields pervading the solar atmosphere and which can be used for simulations of solar wave phenomena. For an example see Shelyag et al 2010.

    The seminar described an analytical method for constructing the magnetic field using the magnetohydrostatic balance equation. Currently we use a numerical approach based on the self-similarity assumption ensuring that the divergence of the constructed magnetic field is zero everywhere. This construction method was considered by Schlüter, A., & Temesváry, S. 1958 and Schüssler & Rempel 2005. The construction is shown below.

    The objective is to construct a 3D magnetic flux tube in magnetohydrostatic equilibrium within a realistic stratified solar atmosphere. The ambition is to extend this to model multiple flux tubes in quasi-equilibrium. The magnetic configuration is comprised of vertical flux tubes expanding with height in response to the fall in plasma pressure, and the adjustment of the plasma pressure and density distributions arise from the analytic solution of the pressure balance equation.

    The plot above shows interpolated 1D fits to vertical hydrostatic atmospheric profiles (Vernazza et al. 1981; McWhirter et al. 1975, former up to 2.3 Mm, latter above 2.4 Mm): thermal pressure p (Pa) (dotted, light blue to blue), plasma density ρ ( μg m− 3) (dashed, purple to yellow) and temperature T ( K) (dash–dotted, red to green). Using these profiles it is possible to construct density and pressure profiles for a solar atmosphere in quasi-hydrostatic equlibrium. The profiles shown below are initially computed for a field free solar atmosphere the reference values for density and pressure, these are obtained from the measured data.

    The next stage in the process is to consider the contribution from the magnetic pressure
    Using a modified form of the fields for the self similarity method, it is possible to explicitly integrate the expressions for the magnetostatic balance. A single open magnetic flux tube spanning the solar photosphere (solar radius ≃ R) and the lower corona (R + 10 Mm) was modelled in magnetohydrostatic equilibrium within a realistic stratified atmosphere subject to solar gravity.  The results of the analysis for an axially symmetric 3D structure, with magnetic field strength, plasma density, pressure and temperature all consistent with observational and theoretical estimates is shown in the diagrams below;

    The left-hand diagram above shows a 3D rendition of the magnetic flux tube including the magnetic field lines (reducing field strength, turquoise–blue). The rear and bottom surfaces display the thermal pressure (reducing, brown–yellow) and the isosurfaces depict plasma-β (purple–green ≃277,1,0.08,0.025and0.016). A vertical 2D slice of the magnetohydrostatic background magnetic pressure is illustrated in the middle image. Some representative field lines are overplotted in blue. The box (black, dotted) encloses the region magnified for display in the image on the right.

    Solar Flux tubes  are observed to remain relatively stable for up to a day or more, and one of the objectives here is to apply the model as the background condition for numerical studies of energy transport mechanisms from the surface to the corona. 

    The diagram above shows a vertical 2D slice log profile of the magnetohydrostatic background (a) thermal pressure p (b) density ρ and (c) temperature T. Magnetic field lines (solid, blue) are overplotted in (a) and (b). The diagram below shows a vertical 2D slice of the log magnetohydrostatic background plasma-β: the ratio of thermal to magnetic pressure. 

    The self-similar construction ensures the magnetic field is divergence free. The equation of pressure balance for this particular set of flux tubes can be integrated analytically to find the pressure and density corrections required to preserve the magnetohydrostatic equilibrium. The model includes a number of free parameters, which makes the solution applicable to a variety of other physical problems and it may therefore be of more general interest. The presentation generated a lot of discussion one of the questions was around  the use of a consistency rule which implied a current free region. What was particularly exciting was the presentation of magnetic field configurations featuring multiple flux tubes 

    Ref: F.Gent et Al Monthly Notices of the Royal Astronomical Society, Volume 435, Issue 1, p.689-697
    (full journal article here)

    (arxIv version )