numerical simulations of wave propagation in the solar

Mem. S.A.It. Vol. 81, 744c© SAIt 2010 Memorie della

Numerical simulations of wave propagation inthe solar chromosphere

C. Nutto, O. Steiner, and M. Roth

Kiepenheuer-Institut fur Sonnenphysik, Schoneckstr. 6, 79104 Freiburg, Germanye-mail: [email protected]

Abstract. We present two-dimensional simulations of wave propagation in a realistic, non-stationary model of the solar atmosphere. This model shows a granular velocity field andmagnetic flux concentrations in the intergranular lanes similar to observed velocity andmagnetic structures on the Sun and takes radiative transfer into account.We present three cases of magneto-acoustic wave propagation through the model atmo-sphere, where we focus on the interaction of different magneto-acoustic wave modes at thelayer of similar sound and Alfven speeds, which we call the equipartition layer. At this layeracoustic and magnetic mode can exchange energy depending on the angle between the wavevector and the magnetic field vector.Our results show that above the equipartition layer and in all three cases the fast magneticmode is refracted back into the solar atmosphere. Thus, the magnetic wave shows an evanes-cent behavior in the chromosphere. The acoustic mode, which travels along the magneticfield in the low plasma-β regime, can be a direct consequence of an acoustic source withinor outside the low-β regime, or it can result from conversion of the magnetic mode, possiblyfrom several such conversions when the wave travels across a series of equipartition layers.

Key words. MHD – waves – Sun: atmosphere – Sun: chromosphere – Sun: helioseismology– Sun: photosphere

1. Introduction

Classical helioseismology relies on waves withfrequencies below the acoustic cut-off fre-quency. These type of waves are trapped insidethe acoustic cavity of the Sun. High frequencywaves above the acoustic cut-off frequency,however, are able to travel freely into the so-lar atmosphere. Along their path through theatmosphere these waves interact with the com-plex magnetic field that is present in the pho-tosphere and the chromosphere. Being able toobserve these running waves in the atmosphere

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with high cadence instruments like MOTH(Finsterle et al. 2004a) a new field in helio-seismology is emerging. However, the analysistools at hand are still very basic (Finsterle et al.2004b; Haberreiter et al. 2007) and the char-acterization of the interaction of the magneto-acoustic waves with the magnetic field is stilldifficult. For the interpretation of the observa-tions, it is helpful to look at numerical sim-ulations of wave propagation through a mag-netically structured solar model atmosphere.In the past such simulations have always beencarried out for idealized atmospheric modelsand magnetic configurations (Rosenthal et al.

Nutto et al.: Numerical wave propagation in the solar atmosphere. 745

2002; Bogdan et al. 2003). However, in realitythe magnetic field is expected to have a com-plex structure.

Here we report on two-dimensional simu-lations of magneto-acoustic wave propagationin a model of the solar atmosphere includingrealistic granular velocity fields and complexmagnetic fields. In Sect. 2 we will give a shortdescription of the setup of the atmosphericmodel and of the excitation of the waves. InSect. 3 we will present and discuss snapshotsof our wave propagation simulations. Resultsare summarized in the last section.

2. Method

For the simulations of magneto-acoustic wavepropagation, we carry out two-dimensionalnumerical experiments with the CO5BOLDcode1. The code solves the magnetohydrody-namic equations for a fully compressible gasincluding radiative transfer. Thus, the modelatmosphere shows a realistic granular veloc-ity field and magnetic flux concentrations inthe intergranular lanes caused by the advec-tive motion of the granules. More details onthe application of the the CO5BOLD code forwave propagation experiments can be found inSteiner et al. (2007).

2.1. Two-dimensional model of the solaratmosphere

The model atmosphere used for the two-dimensional numerical wave propagation ex-periments is basically the same as was usedby Steiner et al. (2007). The computational do-main of the box covers the upper layer of theconvection zone from about −1 300 km belowmean optical depth unity all the way to themiddle layers of the chromosphere at about1 600 km above mean optical depth unity.Along the vertical direction an adaptive gridwith 188 grid points is used corresponding tocell sizes of 46 km for the biggest cells in theconvection zone and 7 km for the smallest cellsin the chromosphere. Transmitting boundary

1 see www.astro.uu.se/∼bf/co5bold main.html for the main pages of CO5BOLD

Fig. 1. Magnetic setup of the model atmosphererepresented in gray scales of log |B|. The white curvecorresponds to the equipartition layer. On the righthand side a flux sheet with strengths up to 2 300 Ghas evolved. On the left hand side in the photosphereare weak horizontal magnetic fields. The crosses in-dicate the location of the wave excitation, where thenumbers correspond to the three cases mentioned inthe text.

conditions are applied for the lower and upperboundary. The lateral dimension of the box is5 000 km, with a grid resolution of 123 gridpoints corresponding to a horizontal spatialresolution of 40 km. Periodic boundary condi-tions are applied in the lateral directions.

Figure 1 gives an impression of the mag-netic setup. The plot shows the absolute mag-netic field strength log |B|. Since we are espe-cially interested in the interaction of the waveswith the magnetic field, we use a model wherea strong flux sheet has evolved during the sim-ulation time. In the core of the flux sheet,which is present on the right hand side of thebox, magnetic field strengths up to 2 300 Gare reached. On the left-hand side weaker hor-izontal fields have evolved. The white contourshows the equipartition layer where the Alfvenspeed equals the sound speed, and approxi-mately coincides with the β = 1 layer. At thislayer the acoustic and magnetic mode of themagneto-acoustic waves can exchange energy(Cally 2007). This layer is known as the modeconversion zone, where an acoustic wave canchange to a magnetic mode and vice versa. Thestrength of this interaction depends on the an-gle between the wave vector and the magneticfield (attack angle), the wave number k, and the

www.astro.uu.se/~bf/co5bold_main.html

www.astro.uu.se/~bf/co5bold_main.html

746 Nutto et al.: Numerical wave propagation in the solar atmosphere.

width of the conversion zone (Cally 2007). Toinvestigate this interaction we excite waves atthree different positions in the computationaldomain. The excitation positions are indicatedby crosses in Fig. 1. The form of the excita-tion will be discussed in Sect. 2.2. The locationof the excitation is at a depth for which it wasshown that the excitation of waves in the Sunis most likely to take place namely between thesurface and 500 km in depth (Stein & Nordlund2001). The three lateral positions are chosen atlocations where interesting cases of wave andmagnetic field interactions occur:

i) where the magneto-acoustic waves interactwith the magnetic canopy of the flux sheet,

ii) where the magneto-acoustic waves are ex-cited inside the flux sheet, and

iii) where the magneto-acoustic waves interactwith weak horizontal magnetic fields.

2.2. Wave excitation

In order to excite waves in our simulationdomain we follow the description given byParchevsky & Kosovichev (2009), where thespatial and temporal behavior of the wavesource is modeled by the function:

f (r, t)=

A

[1− r2

R2src

]2(1 − 2τ2)e−τ

2for r ≤ Rsrc,

0 for r > Rsrc,(1)

where

τ =ω0(t − t0)

2− π with t0 ≤ t ≤ t0 +

4πω0,

Rsrc is the radius of the source, A is the ampli-tude of the disturbance, and ω0 = 2π f0 deter-mines the central frequency of the wave excita-tion. This function can now be used as a sourcefunction in the momentum equations by takingthe gradient of Eq. 1, S = ∇ f . Then, the com-ponents of the source function S can be com-bined with the components of the pressure gra-dient. Thus, the source function can be seen asa perturbation to the pressure gradient, launch-ing an acoustic wave with a central frequencyof ω0.

For our simulations we use a central fre-quency of f0 = 20 mHz and a radius of the

Fig. 2. a) Temporal behavior of the wave source. b)Power of the wave source. The pressure disturbanceexcites a non-monochromatic wave with a centralfrequency of f0 = 20 mHz.

source of Rsrc = 50 km. The temporal behav-ior of the wave source is plotted in panel a) ofFig. 2. The panel b) shows the power spectrumof the excitation source. A non-monochromaticwave is excited with most of the power aroundf0 = 20 mHz.

3. Results

In this section we present snapshots taken fromour simulations of wave propagation in a com-plex magnetically structured atmosphere. Wewill focus on the interaction between the waveand the magnetic field that is taking place inthe mode conversion zone.

Panel a) of Fig. 3 shows snapshots of thepropagation of a wave that is excited just out-side of a flux sheet underneath the magneticcanopy. The propagation of both wave modes,the predominantly acoustic and predominantlymagnetic mode, can be tracked in the pertur-bation of the square root of the energy den-sity δε1/2 = ρ1/2δv. Because of the positionof the wave source beneath the β = 1 layer,


Fig. 3. a) Perturbation of the square root of the energy density, ρ1/2δv, caused by a wave that is excitedoutside in the vicinity of the flux sheet. Shown are three snapshots taken at 80 s, 126 s, and 137 s into thesimulation run. b) Perturbation δB of the magnetic field. The snapshots are taken at the same time as inpanel a). In all plots, the solid contour corresponds to the equipartition layer indicating the conversion zone.

in a region of high plasma β, the applied wavesource excites a wave that is predominantly anacoustic mode. Upon reaching the conversionzone, indicated by the solid contour in Fig. 3,energy is transferred from the acoustic to themagnetic mode (Cally 2007). Because of thelarge attack angle between the local wave vec-tor and the magnetic field vector, most of theenergy is converted to the fast magnetic mode.Transmission of the acoustic mode is not ap-parent. The magnetic mode is best seen in theperturbation δB of the magnetic field, plottedin panel b) of Fig. 3. The second plot in panelb), at 126 s into the simulation, shows the fastmagnetic mode emerging from the conversionzone. Because of the gradient of the Alfvenspeed, the wave is refracted back down into thephotosphere.

Next, we will focus on the wave propaga-tion when the wave source is located inside

the flux sheet close to the equipartition layer.Figure 4 shows again snapshots of the pertur-bation of the square root of the energy den-sity δε1/2, panel a), and the perturbation δB ofthe magnetic field, panel b). Since the wavesource is located above the equipartition layerin the low-β regime, the fast magnetic wavemode is excited directly and together with theslow acoustic mode, different from the previ-ous case. Because of the higher Alfven speedabove the equipartition layer, the fast magneticwave runs ahead of the slow acoustic mode.This fast magnetic mode can again be easilyidentified in the perturbation δB of the mag-netic field, plotted in panel b) of Fig. 4. As inthe previous case, the gradient of the Alfvenspeed causes the fast magnetic wave to re-fract to a degree that it travels back towardsthe lower atmosphere. The slow acoustic waveemerges from the location of the wave exci-


Fig. 4. a) Perturbation δε1/2 caused by a wave that is excited inside of the flux sheet close to the equipartitionlayer. Shown are three snapshots taken at 80 s, 105 s, and 180 s into the simulation run. b) Perturbation δBof the magnetic field. The snapshots are taken at the same time as in panel a). In all plots, the solid contourcorresponds to the equipartition layer indicating the wave-conversion zone.

tation at a later time (last plot in panel a) ofFig. 4), and is then guided along the magneticfield lines into the higher layer of the atmo-sphere. As the acoustic wave travels up theatmosphere the wave front is steepening andstarts to shock.

Finally, we excite a wave just below weakhorizontal magnetic fields. Snapshots of thewave propagation are plotted in Fig. 5, wherepanel a) shows again the perturbation of thesquare root of the energy density δε1/2. Panelb) shows the perturbation of the pressureδp/p0, with p0 being the local unperturbedpressure. As the excitation location is belowthe equipartition layer in the high-β regime,primarily the fast acoustic wave mode is ex-cited. At the first equipartition layer, the lo-cal wave and magnetic vector are almost per-pendicular to each other. Thus, most of the

acoustic wave energy is converted into the fastmagnetic mode. A transmitted acoustic waveis not visible at first. However, the existenceof multiple equipartition layers in the path ofthe magneto-acoustic waves, adds to the com-plexity of the interpretation. Multiple equipar-tition layers have never been addressed in for-mer investigations. Due to the large attack an-gle, one expects that no acoustic wave shouldbe transmitted through the mode conversionzone. However, by looking at the pressure dis-turbance δp/p0 in Fig. 5 it can be seen thatan acoustic wave can be transmitted throughthe multiple equipartition layers by a series ofmode conversion. In the middle plot of panel b)a strong pressure disturbance starts to reemergeabove the multiple conversion zones. This canbe seen as an indication of a transmitted acous-


Fig. 5. a) Perturbation δε1/2 caused by a wave that is excited beneath weak horizontal magnetic fields.Shown are three snapshots taken at 80 s, 137 s, and 145 s into the simulation run. b) Perturbation of thepressure, δp/p0. The snapshots are taken at the same time as in panel a). In all plots, the solid contourcorresponds to the equipartition layer indicating the wave-conversion zone.

tic wave in spite of the horizontal magneticfields.

4. Summary

We have shown numerical experimentsof wave propagation in a realistic two-dimensional model atmosphere. We focusedon the interaction between acoustic and mag-netic wave modes at the equipartition layerwhere the Alfven speed is similar the soundspeed. When the location of the wave sourceis below the equipartition layer in the high-βregime, the launched wave is predominantlyacoustic in nature. In the first and third ofthe presented cases, the large attack angle ofthe local wave vector with the magnetic fieldvector results in the conversion of the fastacoustic mode into the fast magnetic mode.When there are multiple equipartition layers,

an acoustic mode can be transmitted into thehigher layers of the atmosphere by a seriesof mode conversions. In all three cases thefast magnetic mode is usually refracted backtowards the lower atmosphere because of thegradient in Alfven speed.

If the location of the wave source is closeto the equipartition layer, both, the acousticand the magnetic mode, are excited. The slowacoustic wave trails the fast magnetic wave.While the latter shows an upper turning pointagain, the slow acoustic wave is guided alongthe magnetic field into the higher layers of theatmosphere where the wave starts to shock.

The simulations confirm that the propa-gation of magneto-acoustic waves in the so-lar atmosphere is strongly influenced by theequipartition layer. In comparison to investiga-tions where static background models are used,our simulations demonstrate that the equiparti-


tion layer can show a complex topography andit is highly dynamic itself.

Acknowledgements. The authors acknowl-edge support from the European Helio- andAsteroseismology Network (HELAS), which isfunded as Coordination Action by the EuropeanCommission’s Sixth Framework Programme. C.Nutto thanks NSO for the support of a travel grantto attend the workshop.

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numerical simulations of wave propagation in the solar

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