As a background model we use the tripartite model of Jahn & Schmidt (A&A 290, 295, 1994), which is based on the concept of interchange convection. The sunspot model is in magnetohydrostatic equilibrium. It consists of three different stratifications: umbra, penumbra and quiet sun.Two current sheets, namely the peripatopause and the magnetopause, generate the overall magnetic field. While the peripatopause thermally isolates the umbra, the magnetopause permits energy transmission from the hotter quiet sun into the penumbra. It is assumed that convective interchange of magnetic flux tubes distribute the leaked-in heat horizontally, thus leading to the surplus brightness of the penumbra.

In order to test the hypothesis of interchange convection, we investigate the behaviour, i.e. the onset of instability and the subsequent temporal evolution, of a thin magnetic flux tube being embedded in a sunspot model of Jahn & Schmidt.

Initial configuration:

We embed a thin magnetic flux in the sunspot model.

Initially, the tube lies along the magnetopause. The tube is in magnetohydrostatic equilibrium. The initial equilibrium becomes instable due to radiative heat exchange between the tube and its surrounding: The tube being at penumbral temperatures will get heated by the hotter plasma of the adjacent quiet sun.

This image displays the tube with its actual (simulated) size.  

First frame of animation:

For the animation of the tube's evolution, the tube is magnified and shown with constant radius. However, the tube's radius really increases upwards:


Tube magnified proportional to radius

The color coding represents the flow velocity along the tube: positive velocity points upwards along the tube. 

Onset of Instability: The adjacent hotter quiet sun heats up the tube. Thereby, the tube expands, gets less dense and becomes buoyant. Buoyancy forces overcome restoring magnetic forces and the tube starts to rise. We want to note, that it is essential that the magnetopause of the background model exceeds a critical inclination angle. Only then, the buoyancy forces can overcome restoring magnetic forces.

During the subsequent evolution (see video sequence below) the tube rises through the subphotospheric penumbra. There, the stratification is superadiabatic and convectively unstable.

Animation that shows the tube's evolution
Evolution beween upper and lower snapshot is shown (2.5 hours)
MPEG MPEG MPEG
MPEG movie AVI movie Quick Time movie
(1.4 MB) (1.5 MB) (3.3 MB)

Snapshot at the end of evolution (2.5 hours):

Tube in real size Tube magnified proportional to radius

The tube has risen through parts of the penumbra. Above the photosphere the tube ceases to rise, since there the stratification is convectively stable and, more important, the tube's surplus heat is lost very efficiently by radiation (in the photosphere, the radiative time scale has its minimum). During the rise the footpoint of the tube (i.e. the intersection of the tube with the photosphere at tau = 1) migrates inwards (towards the umbra), just as observed penumbral grains do.

As the tube rises, the magnetic field strength of the tube decreases more rapidly than the background magnetic field strength. Due to the condition of total (gas + magnetic) pressure equilibrium between the tube and its surroundings, a surplus gas pressure builds up within the tube. The resulting pressure gradient accelerate a flow along the tube that points upwards beneath the photophere and outwards towards the quiet sun above the photophere.

At this stage the flow exerts a centrifugal force at the turning point in the photosphere, which prevents the tube from sinking. In fact, the tube runs into a quasi-stationary equilibrium, in which the centrifugal force is balanced by the magnetic tension that tries to drag down the tube. In result, the footpoint stops migrating inward. Here, buoyancy forces are negligible, since the suphotospheric part of the tube is close to vertical.

 

 
Blow up of photospheric layers:

 

In order to compare our model with observations, let us focus on the photospheric part of the evolution.

The tube is displayed in real size

The depth extends from -1000 km up to +300 km. The (tau = 1)-level is marked with a horizontal black line and is placed at -150 km in the penumbra (-> Wilson depression). Radially the box extends from the outer umbra through the penumbra, out to the adjacent quiet sun.

 

Initial configuration in zoomed box:

Temperature (upper panel) and magnetic field strength (lower panel) of the tube and of the background model.

The black solid circles track the motion of mass elements.

The tube is magnified proportional to radius.

 

Animation that shows the tube's evolution in zoomed box
Evolution beween upper and lower snapshot is shown (2.5 hours)
MPEG MPEG MPEG
MPEG movie AVI movie Quick Time movie
(1.5 MB) (2.5 MB) (5.1 MB)

Snapshot at end of evolution in zoomed box for different physical variables:

Tube lies almost horizontal just above the photophere. As is seen in the upper panel, the upflow heats up the tube and makes it hotter than the background. Above the photosphere ( (tau = 1)-level), the upflow bends horizontal. As the plasma flows outwards, it cools due to radiative losses. After approx. 2000 km it finds itsself in temperature equilibrium with the surrounding background.

The footpoint of the tube (, i.e. the intersection of the tube with the photosphere at tau = 1) is the hottest photospheric part of the tube.

In the second panel it is seen that the magnetic field strength is substantially reduced near the footpoint. Since magnetic flux is conserved along the tube, that means (and it is also visible) that the tubes radius is increased near the footpoint.

The third panel shows the optical thickness of the tube. Here, the color coding of the background is artificially set to a optical thickness of 100 below the photosphere and to 0.01 above the photosphere. It can be seen that the tube is optically thick (~100) where the tube is hotter than the background and becomes transparent (<0.1) where the tube is in temperature equilibrium with the background.

The fourth panel shows the gas pressure inside the tube as well as in the background. From the tube's footpoint outwards, the plasma looses internal energy by radiative losses. Therby, the gas pressure decreases leading to a gas pressure gradient in the horizontal part of the tube. Since the vertical component of gravity vanishes in the horizontal part, the plasma is accelerated by that gas pressure gradient. As the fifth panel shows, the plasma is accelerated from approx. 4 km/s near the footpoint up to 14 km/s near the outer edge of the penumbra.

Consequenes and interpretation of our results:

Bright filaments and penumbral grains:

In white light images such a flux tube would appear as a radially elongated bright filament with a length of approx. 2000 km. In the umbral direction the filament would appear brighter and thicker. The outer parts of the filament would become thinner and less bright. Thus, our model tube has a comet like shape and perfectly matches the characteristics of observed penumbral grains. The tube's footpoint migrates inwards just as penumbral grains do. The proper velocity of the footpoint decreases inwards from 1 km/s down to 100 m/s. The slower the footpoint, the longer the bright tail. This suggests that penumbral grains and bright filaments are of the same physical nature: Both have a comet like shape, only that the tail of bright filaments is longer than the tail of the penumbral grains. And according to our model, penumbral grains should migrate faster than bright filaments, or in other words, the longer the filament, the slower it should migrate.

 

Surplus brightness of the penumbra as compared to the umbra:

Our model suggests that the surplus brightness of the penumbra is caused by upflows along magnetic flux tubes. These upflows result naturally during the dynamical evolution of a flux tube that rises through the subphotospheric part of the penumbra. This consequence is somewhat in conflict with the background model of Jahn & Schmidt: there, the process of interchange convection HORIZONTALLY distributes the heat that leakes-in through the magnetopause. According to the simulation presented here, a flux tube that lies along the magnetopause becomes instable and rises due to radiative heat exchange in the photosphere long before the tube can be heated in the optically thick subphotopheric layers. Hence, the surplus brightness of the penumbra is not due to horizontal heat transport from the magnetopause inwards, but due to upflows along magnetic flux tubes.

Image design: Daniel Müller

 

For the formation and presence of a sunspot penumbra we propose the following scenario: As magnetic flux emerges through the photosphere, it piles up forming a pore initially. As additional flux gathers around the pore, the magnetopause becomes more and more inclined. Once, the inclination of the magnetopause exceeds a critical value, the buoyancy forces caused by radiative heat exchange in the photophere overcome the restoring magnetic forces, and the tubes starts to rise. As the tubes rise, an upflow along the tubes develops. When these uflows reach the photophere they bend horizontally and cause penumbral grains and bright filaments.

Plasma flow in bright filaments:

The plasma flow in the bright part of the flux tube may not be detectable for the following reason: Flow velocities are deferred from Doppler shifts of photospheric absorption lines. Even the wings of absorption lines originate almost exlusively above the continuum. Since the bright part of the tube is optically thick, the plamsa inside the tube is not visible and the continuum originates from the outermost layer of the flux tube. Thus, line profiles are not shifted by such a bright flow channel.

The Evershed flow:

In the outer part of the penumbra, the tube constitues a thin transparent flow channel that is slightly elevated above the photosphere having a length of approx. 3000 km. Depending on the view angle, such a low flow channel Doppler-shifts the contribution functions of photospheric absorption lines that stem from inside the channel. Hence, photospheric line profiles become shifted and assymmetric. The degree and relative strength of line shift and line asymmetry depend on the formation heights of the contribution functions. Since the flow channel is thin, major contribution of photospheric lines stem from layers, in which no flow channels are present ( click for scetch ). Thus, taking into account, that such flow channels may not be spatially resolved (in our model the tubes typically have a diameter around 50 km), nor resolved in depth (the formation height of a photospheric absorption line spans at least approx. 200 km in depth), our model can reproduce the observed features of the EVERHED EFFECT.

 

 


Literature:

R. Schlichenmaier, K.Jahn, H.U. Schmidt: in ASP Conf. Series 118, 140 (1997) ( download pdf )

R. Schlichenmaier: PhD thesis (Die Dynamik magnetischer Flussröhren im Sonnenfleck: Ein Modell für den Evershed-Effekt und die penumbrale Feinstruktur), Ludwig-Maximilians-Universität München (1997) ( download pdf )

R. Schlichenmaier, K.Jahn, H.U. Schmidt: ApJ 493, L121-L125 (1998)

R. Schlichenmaier, K.Jahn, H.U. Schmidt: A&A 337, 897-910 (1998)

R. Schlichenmaier: in ASP Conf. Series 183, 91 (1999) ( download pdf )

R. Schlichenmaier, J. Bruls, M. Schüssler: A&A 349, 961 (1999)

R. Schlichenmaier: AN 323, 303 (2002) ( download pdf )

R. Schlichenmaier: in ASP Conf. Series 286, 211 (2003) ( download pdf )

R. Schlichenmaier, S.K. Solanki: A&A 411, 257 (2003)

MPEG Click here to get the entire video (34 MegaBytes!!) in QuickTime Format.