The equilibrium structure of sunspots depends critically on its magnetic topology and is dominated by magnetic forces. Tension force is one component of the Lorentz force which balances the gradient of magnetic pressure in force-free configurations. We employ the tension term of the Lorentz force to clarify the structure of sunspot features like penumbral filaments, umbral light bridges and outer penumbral fine structures. We compute vertical component of tension term of Lorentz force over two active regions namely NOAA AR 10933 and NOAA AR 10930 observed on 05 January 2007 and 12 December 2006 respectively. The former is a simple while latter is a complex active region with highly sheared polarity inversion line (PIL). The vector magnetograms used are obtained from Hinode(SOT/SP). We find an inhomogeneous distribution of tension with both positive and negative signs in various features of the sunspots. The existence of positive tension at locations of lower field strength and higher inclination is compatible with the uncombed model of the penumbral structure. Positive tension is also seen in umbral light bridges which could be indication of uncombed structure of the light bridge. Likewise, the upward directed tension associated with bipolar regions in the penumbra could be a direct confirmation of the sea serpent model of penumbral structures. Upward directed tension at the PIL of AR 10930 seems to be related to flux emergence. The magnitude of the tension force is greater than the force of gravity in some places, implying a nearly force-free configuration for these sunspot features. From our study, magnetic tension emerges as a useful diagnostic of the local equilibrium of the sunspot fine structures.
Deep Dive into Magnetic Tension of Sunspot Fine Structures.
The equilibrium structure of sunspots depends critically on its magnetic topology and is dominated by magnetic forces. Tension force is one component of the Lorentz force which balances the gradient of magnetic pressure in force-free configurations. We employ the tension term of the Lorentz force to clarify the structure of sunspot features like penumbral filaments, umbral light bridges and outer penumbral fine structures. We compute vertical component of tension term of Lorentz force over two active regions namely NOAA AR 10933 and NOAA AR 10930 observed on 05 January 2007 and 12 December 2006 respectively. The former is a simple while latter is a complex active region with highly sheared polarity inversion line (PIL). The vector magnetograms used are obtained from Hinode(SOT/SP). We find an inhomogeneous distribution of tension with both positive and negative signs in various features of the sunspots. The existence of positive tension at locations of lower field strength and higher i
The equilibrium structure of sunspots is obviously dominated by magnetic forces. Early researches on this problem dealt mainly with the problem of the global equilibrium of sunspots (Meyer et al. 1977). The sunspot was modelled as a magnetic flux rope where the lateral force balance was envisaged as a pressure balance between the photospheric plasma pressure and the magnetic + plasma pressure inside the flux rope (e.g., Chitre 1963). Since the magnetic field is divergence free, the field lines of the flux rope must bend back into the photosphere resulting in a "closed" field topology. The resulting curvature of the field lines produces a magnetic tension that should basically have a downward vertical component. The model of Meyer et al. (1977) also put constraints on the field line curvature, since the equilibrium becomes unstable when the radius of curvature is shorter than a certain value. With the availability of high resolution magnetograms of sunspots, it has become clear that the earlier models of sunspots might no longer be adequate to explain the dynamical equilibrium of various fine structures seen in the umbra as well as in the penumbra. One simple diagnostic of the vertical equilibrium is the vertical component of the magnetic tension, which can be determined from the lateral gradients of the vector magnetic field. Information about the vertical component of the magnetic tension was already found to be very useful, as e.g. seen in the correlation of low tension force with large magnetic shear in early vector magnetograms (Venkatakrishnan et al. 1993) measured by the MSFC vector magnetograph (Hagyard et al. 1982). The magnetic tension measurements at the polarity inversion lines underlying filaments/prominences are like-wise very important since the vanishing of magnetic tension at these highly sheared locations makes the prominence structure extremely vulnerable to dynamical instabilities, via thermal instabilities (Venkatakrishnan 1990b).
Modern observations of sunspots have revealed the existence of different fine structures. In the umbra we have the umbral dots and light bridges Sobotka (1989); Sobotka et al. (1997b,a). In the penumbra we have spines (stronger, more vertical field) wrapping around the intraspines (weaker, more horizontal field) (Lites et al. 1993;Borrero et al. 2008). A few models have also been proposed to explain these structures (e.g., uncombed model of Solanki & Montavon (1993), the gappy model of Spruit & Scharmer (2006)).
In this paper, we investigate the height variation of the sunspot fine structure using the calculations of the vertical component of the magnetic tension force. The expression for computing the vertical component of tension force is given in Section 2. In Section 3, we describe the data sets used. Section 4 describes the analysis and results. Finally in Section 5 we present our conclusions.
In any plasma with magnetic field B and plasma pressure p, the equation for magneto-hydrostatic equilibrium is given by (Parker 1979),
where ρ is the plasma density and g is the acceleration due to gravity. The first term in Equation 1 is the Lorentz force, second term is the force due to plasma pressure and the last term is the force on the plasma due to gravity. We can split up the Lorentz force (say F) in two terms as,
The first term in this equation is the tension force (say T). The second term represents the force due to magnetic pressure. The vertical component of the tension term can be expanded in terms of the horizontal derivatives of the magnetic field as:
where, the last component is drawn from the condition,
The utility of the tension force as a diagnostic of dynamical equilibrium has not found much attention in the literature so far except in a restricted sense (Venkatakrishnan 1990a,b;Venkatakrishnan et al. 1993). We have computed tension force using Equation 3 and expressed it in the units of dynes/cm 3 .
We have used the vector magnetograms of NOAA AR 10933 observed on 05 January 2007 and NOAA AR 10930 observed on 12 December 2006. These data sets are obtained from the Solar Optical Telescope/Spectro-polarimeter (SOT/SP: (Tsuneta et al. 2008;Suematsu et al. 2008;Ichimoto et al. 2008;Shimizu et al. 2008)) onboard Hinode (Kosugi et al. 2007).
The Hinode (SOT/SP) data have been calibrated by the standard “SP PREP” routine developed by B. Lites and available in the Solar-Soft package. The “SP PREP” determines the thermal shifts in the spectral and slit dimensions and also applies the drift corrections for calibrating the data from level0 to level1. The prepared polarization spectra have then been inverted to obtain vector magnetic field components using an Unno-Rachkowsky (Unno 1956;Rachkowsky 1967) inversion under the assumption of Milne-Eddington (ME) atmosphere (Landolfi & Landi Degl’Innocenti 1982;Skumanich & Lites 1987). We have used the “STOKESFIT” inversion code 1 which is available in the Solar-Soft package. The latest version of the inve
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