Three-dimensional numerical simulation of MHD waves observed … · 2017-10-20 ·...

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Three-dimensional numerical simulation of MHD waves observed by the Extreme Ultraviolet Imaging Telescope S. T. Wu, 1 Huinan Zheng, 1 S. Wang, 1,2 B. J. Thompson, 3 S. P. Plunkett, 4,5 X. P. Zhao, 6 and M. Dryer 7,8 Abstract. We investigate the global large amplitude waves propagating across the solar disk as observed by the SOHO/Extreme Ultraviolet Imaging Telescope (EIT). These waves appear to be similar to those observed in H in the chromosphere and which are known as “Moreton waves,” associated with large solar flares [Moreton, 1960, 1964]. Uchida [1968] interpreted these Moreton waves as the propagation of a hydromagnetic disturbance in the corona with its wavefront intersecting the chromosphere to produce the Moreton wave as observed in movie sequences of H images. To search for an understanding of the physical characteristics of these newly observed EIT waves, we constructed a three-dimensional, time-dependent, numerical magnetohydrodynamic (MHD) model. Measured global magnetic fields, obtained from the Wilcox Solar Observatory (WSO) at Stanford University, are used as the initial magnetic field to investigate hydromagnetic wave propagation in a three-dimensional spherical geometry. Using magnetohydrodynamic wave theory together with simulation, we are able to identify these observed EIT waves as fast mode MHD waves dominated by the acoustic mode, called magnetosonic waves. The results to be presented include the following: (1) comparison of observed and simulated morphology projected on the disk and the distance- time curves on the solar disk; (2) three-dimensional evolution of the disturbed magnetic field lines at various viewing angles; (3) evolution of the plasma density profile at a specific location as a function of latitude; and (4) computed Friedrich’s diagrams to identify the MHD wave characteristics. 1. Introduction Observations of wave phenomena propagating across the solar disk were made by a number of investigators in the early 1960s [Athay and Moreton, 1961; Moreton, 1960, 1964], and referred to as “Moreton waves.” This phenomenon was seen in movie sequences of the H emission, which showed successive semicircular “fronts” expanding from a flare with speeds rang- ing from a few hundred to well over 10 3 km s 1 . Higher speeds have been observed. An unusually extreme example of More- ton wave’s acceleration from 2500 to 4000 km s 1 for the X12/3B flare of June 4, 1991, has been reported by Sakurai et al. [1995]. Transient coronal EUV emission has also been observed prior to and during the initial stage of a solar flare [Neupert, 1989]. Neupert’s data showed the presence of a wave- like disturbance traveling with a speed of 760 km s 1 , ex- panding from a site along the neutral line. He suggested that it may have been the location of a chromospheric mass ejection, but he did not present the global morphology of this observed wave. The SOHO/Extreme Ultraviolet Imaging Telescope (EIT) images the solar disk and inner corona [Delaboudinie ´re et al., 1995]. On May 12, 1997, an Earth-directed coronal mass ejec- tion (CME) was observed by the SOHO/EIT and reported by Thompson et al. [1998]. The CME, originating north of the central solar meridian, was later observed by the Large Angle Spectrometric Coronagraph (LASCO) as a “halo” CME (i.e., a bright expanding ring centered about the occulting disk as reported by Plunkett et al. [1998]). Thompson et al. [1998] reported the details of the EIT observations of this CME event; we will not repeat them here. The focus of our study is to understand the physics of the observed, almost circularly shaped, wave propagating across the solar disk with a mea- sured average speed of 250 km s 1 . This particular speed is much slower than the speed of a Moreton wave in the chro- mosphere-corona transition region. These observed waves (i.e., Moreton waves and presumably, Neupert’s transient coronal emission waves) have been attrib- uted to MHD fast mode waves generated in the impulsive phase of the flare [Meyer, 1968; Uchida, 1968, 1974; Uchida et al., 1973]. This seems to be a reasonable interpretation; how- ever, the recently observed EIT waves have distinct character- istics which differ from previously observed Moreton waves [Moreton, 1960] and transient coronal emissions [Neupert, 1989]. These differences are (1) the speed of the EIT wave is much less than the Moreton wave with propagation speeds on 1 Center for Space Plasma and Aeronomic Research and Depart- ment of Mechanical Engineering, University of Alabama in Huntsville, Huntsville, Alabama, USA. 2 Permanently at Department of Earth and Space Science, University of Science and Technology of China, Hefei, Anhui, People’s Republic of China. 3 NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. 4 Universities Space Research Association, Washington, D. C., USA. 5 Also at Naval Research Laboratory, Washington, D. C., USA. 6 Hansen Experimental Physics Laboratory, Stanford University, Stanford, California, USA. 7 Space Environment Center, NOAA, Boulder, Colorado, USA. 8 Also at Exploration Physics International, Inc., Milford, New Hampshire, USA. Copyright 2001 by the American Geophysical Union. Paper number 2000JA000447. 0148-0227/01/2000JA000447$09.00 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. A11, PAGES 25,089 –25,102, NOVEMBER 1, 2001 25,089

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Three-dimensional numerical simulation of MHD wavesobserved by the Extreme Ultraviolet Imaging Telescope

S. T. Wu,1 Huinan Zheng,1 S. Wang,1,2 B. J. Thompson,3 S. P. Plunkett,4,5

X. P. Zhao,6 and M. Dryer7,8

Abstract. We investigate the global large amplitude waves propagating across the solardisk as observed by the SOHO/Extreme Ultraviolet Imaging Telescope (EIT). Thesewaves appear to be similar to those observed in H� in the chromosphere and which areknown as “Moreton waves,” associated with large solar flares [Moreton, 1960, 1964].Uchida [1968] interpreted these Moreton waves as the propagation of a hydromagneticdisturbance in the corona with its wavefront intersecting the chromosphere to produce theMoreton wave as observed in movie sequences of H� images. To search for anunderstanding of the physical characteristics of these newly observed EIT waves, weconstructed a three-dimensional, time-dependent, numerical magnetohydrodynamic(MHD) model. Measured global magnetic fields, obtained from the Wilcox SolarObservatory (WSO) at Stanford University, are used as the initial magnetic field toinvestigate hydromagnetic wave propagation in a three-dimensional spherical geometry.Using magnetohydrodynamic wave theory together with simulation, we are able to identifythese observed EIT waves as fast mode MHD waves dominated by the acoustic mode,called magnetosonic waves. The results to be presented include the following: (1)comparison of observed and simulated morphology projected on the disk and the distance-time curves on the solar disk; (2) three-dimensional evolution of the disturbed magneticfield lines at various viewing angles; (3) evolution of the plasma density profile at aspecific location as a function of latitude; and (4) computed Friedrich’s diagrams toidentify the MHD wave characteristics.

1. Introduction

Observations of wave phenomena propagating across thesolar disk were made by a number of investigators in the early1960s [Athay and Moreton, 1961; Moreton, 1960, 1964], andreferred to as “Moreton waves.” This phenomenon was seen inmovie sequences of the H� emission, which showed successivesemicircular “fronts” expanding from a flare with speeds rang-ing from a few hundred to well over 103 km s�1. Higher speedshave been observed. An unusually extreme example of More-ton wave’s acceleration from 2500 to 4000 km s�1 for theX12/3B flare of June 4, 1991, has been reported by Sakurai etal. [1995]. Transient coronal EUV emission has also beenobserved prior to and during the initial stage of a solar flare[Neupert, 1989]. Neupert’s data showed the presence of a wave-

like disturbance traveling with a speed of �760 km s�1, ex-panding from a site along the neutral line. He suggested that itmay have been the location of a chromospheric mass ejection,but he did not present the global morphology of this observedwave.

The SOHO/Extreme Ultraviolet Imaging Telescope (EIT)images the solar disk and inner corona [Delaboudiniere et al.,1995]. On May 12, 1997, an Earth-directed coronal mass ejec-tion (CME) was observed by the SOHO/EIT and reported byThompson et al. [1998]. The CME, originating north of thecentral solar meridian, was later observed by the Large AngleSpectrometric Coronagraph (LASCO) as a “halo” CME (i.e.,a bright expanding ring centered about the occulting disk asreported by Plunkett et al. [1998]). Thompson et al. [1998]reported the details of the EIT observations of this CMEevent; we will not repeat them here. The focus of our study isto understand the physics of the observed, almost circularlyshaped, wave propagating across the solar disk with a mea-sured average speed of �250 km s�1. This particular speed ismuch slower than the speed of a Moreton wave in the chro-mosphere-corona transition region.

These observed waves (i.e., Moreton waves and presumably,Neupert’s transient coronal emission waves) have been attrib-uted to MHD fast mode waves generated in the impulsivephase of the flare [Meyer, 1968; Uchida, 1968, 1974; Uchida etal., 1973]. This seems to be a reasonable interpretation; how-ever, the recently observed EIT waves have distinct character-istics which differ from previously observed Moreton waves[Moreton, 1960] and transient coronal emissions [Neupert,1989]. These differences are (1) the speed of the EIT wave ismuch less than the Moreton wave with propagation speeds on

1Center for Space Plasma and Aeronomic Research and Depart-ment of Mechanical Engineering, University of Alabama in Huntsville,Huntsville, Alabama, USA.

2Permanently at Department of Earth and Space Science, Universityof Science and Technology of China, Hefei, Anhui, People’s Republicof China.

3NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.4Universities Space Research Association, Washington, D. C., USA.5Also at Naval Research Laboratory, Washington, D. C., USA.6Hansen Experimental Physics Laboratory, Stanford University,

Stanford, California, USA.7Space Environment Center, NOAA, Boulder, Colorado, USA.8Also at Exploration Physics International, Inc., Milford, New

Hampshire, USA.

Copyright 2001 by the American Geophysical Union.

Paper number 2000JA000447.0148-0227/01/2000JA000447$09.00

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. A11, PAGES 25,089–25,102, NOVEMBER 1, 2001

25,089

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the order of the local Alfvenic speed; and (2) the morphologyof the EIT wave propagation can be almost circular across thesolar disk, while Moreton waves were typically confined to anarc rarely exceeding 120�.

In this paper, we present a three-dimensional, time-dependent magnetohydrodynamic model using observed mag-netic fields and assumed solar plasma properties (no observa-tions are available) to investigate the propagating, finiteamplitude, global MHD waves on the solar disk. We pursuethe approach in order to understand the observed EIT wavesand their relationship with Moreton waves. The MHD modeland initial boundary conditions will be described in section 2.Numerical results and analysis will be included in section 3.The concluding remarks are given in section 4.

2. Model DescriptionA three-dimensional, time-dependent ideal (nondissipative)

numerical MHD model is used. For simplicity, we approximatethe energy equation with an adiabatic energy process with thepolytropic index (�) being 1.67. The other equations includemass conservation, momentum conservation, and magnetic in-duction to take into account the nonlinear interaction betweenplasma flow and magnetic fields. The mathematical model ispresented in a matrix form in spherical coordinates for conve-nience of development of the numerical code, namely,

�u��t � Ar

�u��r � A�

1r

�u���

� A�

1r sin �

�u���

� F� � 0, (1)

Figure 1. The photospheric magnetogram synoptic map (in Gauss units) of Carrington rotation 1922 forApril and May 1997 (data provided by Stanford Wilcox Solar Observatory). The shaded dots represent thelocations at which the Friedrich’s wave diagrams are constructed as shown in Figure 7.

Figure 2. The initial plasma beta based on the measured magnetic field strength given in Figure 1 and theassumed constant values of density (no � 1.2 � 108 cm�3) and temperature (To � 1.6 � 106 K) asdiscussed in the text.

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with

u� � ��v r

v�

v�

Br

B�

B�

p

� , (2)

Ar � �v r � 0 0 0 0 0 00 v r 0 0 0 B�/� B�/� / 2�0 0 v r 0 0 �Br/� 0 00 0 0 v r 0 0 �Br/� 00 0 0 0 v r 0 0 00 B� �Br 0 0 v r 0 00 B� 0 �Br 0 0 v r 00 �p 0 0 0 0 0 v r

� ,

(3)

A� � �v� 0 � 0 0 0 0 00 v� 0 0 �B�/� 0 0 00 0 v� 0 Br/� 0 B�/� / 2�0 0 0 v� 0 0 �B�/� 00 �B� Br 0 v� 0 0 00 0 0 0 0 v� 0 00 0 B� �B� 0 0 v� 00 0 �p 0 0 0 0 v�

� ,

(4)

A� � �v� 0 0 � 0 0 0 00 v� 0 0 �B�/� 0 0 00 0 v� 0 0 �B�/� 0 00 0 0 v� Br/� �B�/� 0 / 2�0 �B� 0 Br v� 0 0 00 0 �B� B� 0 v� 0 00 0 0 0 0 0 v� 00 0 0 �p 0 0 0 v�

� ,

(5)

Figure 3. The spatial distribution of the pressure pulse. This pulse linearly increases to a maximum of afactor of 2 times the background pressure at 398 s. The pulse remains constant until 955 s, then decreases tothe initial pressure at 1910 s.

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F� �1r �

��2v r � v� cos � ���v�

2 � v�2� � �B�

2 � B�2�/� � �/r

v rv� v�2 cot � �BrB� B�

2 cot � �/�v��v r � v� cot � � B��Br � B� cot � �/�Br�2v r � v� cos � �v rB� � v��Br � B� cot � �v rB� � v��Br � B� cot � ��p�2v r � v� cos � �

� , (6)

where t , r , � , u� (vr, v�, v�), B� (Br, B�, B�), p are the time,radius, density, velocity vector, magnetic field vector and thermal

pressure, normalized by �A, Rs, �o, VA, Bo, po. Additionalsymbols are � (latitude), � (longitude), � 8�po/B0

2, VA �Bo/�4��o, � � GMs/RsVA

2 , �A � Rs/VA, where Rs is thesolar radius, Ms the solar mass, G the gravitational constant.

The numerical scheme used to develop the MHD model isthe fractional step method [Yanenko, 1971]. This numericalscheme consists of a preliminary step and correction steps. Theimplicit scheme [Roache, 1999] is used for maximum stability atthe preliminary step with respect to each coordinate (r , � , �).The explicit scheme [Roache, 1999] is used for the correctionstep for accuracy. The overall accuracy of this numerical

Figure 4. Comparison between the SOHO/EIT 195 Å running-difference images of the large scale wavetransient and numerically simulated density-enhancement (� � �o)/�o images of a large-scale wave transienton the solar disk. The simulated density enhancements compare very well with the EIT intensity enhancementat each of the four times shown here.

Figure 5. The measured and numerically simulated distance-time curves of the wavefronts shown in Figure4 in the north, south, southwest, and southeast directions. The numerically simulated wave fronts are identifiedby the 10% level of the density enhancement.

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scheme is tested by using an analytical solution given by Low[1984], which shows that the difference between the numericaland analytical solution is within 5% over the time period of themodel calculation.

The initial state of this model is a magnetohydrostatic solaratmosphere; thus, we can choose the current-free (i.e., poten-tial) magnetic field topology as the initial state. In this studythe measured photospheric global-scale, line-of-sight magneticfield map (Carrington rotation 1922) of May 12, 1997, shown inFigure 1 from the Stanford Wilcox Solar Observatory is usedtogether with the source-surface model [Hoeksema and Scher-rer, 1985; Hoeksema, 1989, 1991]. Because of the low spatialresolution of the magnetograph at WSO, the photosphericfield is concentrated into subarc second bundles of strong fieldthat could not be measured. That restriction gives the mea-sured photospheric weak field on the order of a gauss as shownin Figure 1. Physically, this representation of global field dis-tribution is a reasonable approximation, because at small dis-tances above the photosphere, the field is no longer concen-trated into small bundles of radial field. The magnetic fluxrapidly spreads into a more uniform distribution. This inter-pretation is supported by the change in field strength across thedisk of various features measured using spectral lines whichform at different heights [Hoeksema, 1984]. Because of the lackof number density (n) and temperature (T) measurementsjust above the solar surface at the coronal base, we simplychoose typical values of corona given by Feldman et al. [1999]:no � 1.2 � 108 cm�3 and To � 1.6 � 106 K. These values

are uniformly distributed on the solar disk at this level. Itshould be understood that this choice is not realistic becausethe density varies from pole to equator on the solar surface;however, the present interest is to examine the propagation ofMHD waves in a region away from the poles. This choiceshould be appropriate. We obtained, Figure 2, the plasma beta(i.e., the ratio of plasma pressure to the magnetic pressure16�nkT/B2) using the measured magnetic fields from Figure1 together with the assumed plasma density and temperature.The vertical row of numbered dots at Carrington longitude�135� will be discussed later.

The lower and upper boundary conditions in the radial di-rection are adopted from the time-dependent characteristicboundary conditions. This requirement means that the valuesfor all the physical quantities are updated for each time-stepaccording to the compatibility equations derived from the gov-erning equations [Hu and Wu, 1984; Wu and Wang, 1987]. Thenonreflecting conditions are used according to the direction ofcharacteristics in the radial direction. Periodic boundaries areused for the latitudinal and longitudinal directions. Specificmathematical expressions used for the radial boundary condi-tions are included in Appendix A.

3. Numerical Results and AnalysesFrom the observations, EIT images show an almost circular

expanding wave front, emanating from a central point near, orat, NOAA active region 8038 at N23W07 from 0450 to 0700

Figure 6. Relative density (� � �o)/�o profiles on the solar disk at a temporal cadence as a function oflatitude at various times at Carrington longitude 131.5�, where the disturbance occurred at 23� N of theequator. The two dash-dot lines represent south (left) and north (right) coronal hole boundaries.

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UT, May 12, 1997. To simulate this wave front, we introduce athermal pressure pulse of a factor 2 higher than the back-ground pressure. The temporal and spatial shapes of the pulseare shown in Figure 3. The total energy input corresponding tothis pulse is �5 � 1028 ergs. Physically, this pressure pulse maybe interpreted as representing active region heating which re-sults from complex processes that are not considered in thispaper. The EIT images (top) and model-simulated imagesshowing computed density enhancements (bottom) are shownin Figure 4. By comparing the simulation results with observa-tions, it is clearly indicated that the simulated and observedwave fronts are matched well as shown in Figure 4. The sim-

ulated wave is the representation of density enhancements dueto the given disturbance at four representative times. Further,we have plotted the measured and simulated distance and timecurves in various directions (i.e., south, north, southeast, andsouthwest) in Figure 5. These kinematical characteristics alsoexhibit remarkable agreement.

By noting the results shown in Figures 4 and 5, there is afeature which needs explanation, that is: both figures indicatethat the density enhancement does not move any further alongthe northern direction after it reaches a location �600 Mmfrom the initiation site (see Figure 5) where it remains. Thiscan be understood by examining the magnetic field measure-

Figure 7. The computed Friedrich’s wave diagrams at various positions marked in Figure 1 where the solid;the dotted; dashed and dot; and dashed lines represent the slow, Alfvenic, sonic and fast mode waves,respectively. The arrow represents the direction of the magnetic field at a specific location obtained from theWSO measurement. The value is plotted as a function of measured counter clockwise referring to themagnetic field line. The horizontal and vertical axes scale the magnitude of the wave speed.

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Figure 8. The computer evolution of the density enhancement, (� � �o)/�o viewed at central meridian (CM),45�, and 90� away from central meridian (where the explosion occurred) at 1910 and 2865 s, respectively.

Figure 9. MHD fast wave Mach number at 955, 1432, 1910, and 2865 s after introduction of the perturbationwith their maximum values being 0.58, 0.61, 0.53, and 0.48, respectively.

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ments shown in Figure 1, where it can be seen that this positioncorresponds to the southern boundary of the north polar coro-nal hole. The physical explanation of this “anomaly” is asfollows. When these large-amplitude nonlinear MHD waveshit the southern boundary of the north polar coronal hole, anyassociated mass motion across the magnetic field lines becauseof the frozen-in condition is impeded. This means that the fieldand plasma move together. By looking at the plasma beta ()distribution shown in Figure 2 it is clear that the is larger inthe neighborhood of the disturbance and, close to the coronalhole boundary, the becomes much smaller. In such a situa-tion, the large beta () plasma and field medium is easilymoved around by the plasma flow; whereas, in the small betaplasma and field medium, the motion is much more rigid andhence constrained. Therefore, when the disturbances reach thesouthern boundary of the north polar coronal hole the massmotion basically stops. That is why the plasma and field pile upagainst the coronal hole boundary. It can be seen from Figure4 that the northern MHD wave front brightens up as a result ofthe increased plasma density. However, fast mode MHD wavesleak out into the coronal hole as shown in Figure 6, whichshows the relative density enhancement (� � �o)/�o as afunction of latitudinal angle on the solar surface, oriented atthe center of the meridian of the observed location of theactive region at various times. The two dashed lines representthe northern and southern polar coronal hole boundaries. Themeasurements of Figure 1 show that the distances between thecenter of the active region (NOAA 8038; 23�N of disk center)

and the northern and southern boundaries of the coronal holeare 650 and 1200 Mm, respectively.

We now understand that as the wave front propagates in thenorthern direction, it asymptotically approaches a distance of�700 Mm from the initiation site as shown in Figure 5. On theother hand, the wave continues to propagate along the south,southeast and southwest directions because these waves havenot reached the boundary of the southern polar coronal hole.Figure 6 shows the evolution of relative density (� � �o)/�o

after introduction of the initial pressure pulse as shown inFigure 3. This pressure pulse reaches its peak at 955 s andgradually declines to zero at 1910 s. By looking at these results,we notice that the plasma has piled-up at the southern bound-ary of the northern polar coronal hole as indicated by theresults shown in Figures 4 and 5. It is also noted that there isno density enhancement at the northern edge of the southernpolar coronal hole because the wave, as yet, has not reachedthe southern polar coronal hole boundary. It is worth noting inFigure 6 that the compression and rarefaction of those wavesare still enhanced after the initial pulse was terminated. Be-cause of the frozen-in condition, the plasma moving with themagnetic fields hits the coronal hole boundary (very smallplasma beta) and bounces back (i.e., reflected) to cause thesecompressions and rarefactions. We have checked the totalmass which is conserved.

At this point we have identified the observed EIT waves asMHD waves and explained the northward and southwardpropagation of the density enhancements as they approach the

Figure 10. Radial flow speed (solid) and radial MHD fast mode speed (dotted) along the radial directionat specific latitude (�) and Carrington longitude (�) locations and t � 1433 s.

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coronal hole boundaries. However, we have not, as yet, men-tioned the modes of the MHD waves. It is well known fromMHD wave theory [Jeffrey and Taniuti, 1964] that when adisturbance appears in a magnetized plasma, three modes ofMHD waves, (i.e., Alfven, fast and slow modes) will always begenerated. In this study, we focus our attention on the twocompressive modes which are characterized by the fast andslow modes and their characteristic speeds as follows:

Cf,s2 �

12

a2 � b2 ��a2 � b2� 4a2b2 cos � , (7)

where a and b are the sonic and Alfven speeds, respectively,and is the angle between the magnetic field and propagatingwave (k� ).

In order to understand further these MHD wave character-istics in this particular case, we constructed Friedrich’s dia-grams [Bazer and Fleischman, 1954; Friedrichs, 1954; Friedrichand Kranzer, 1958] at the locations marked by the numbereddots on Figure 1, as shown in Figure 7. The Friedrich’s dia-grams are constructed by using the specific physical parameters(i.e., field strength and direction, density and temperature) ateach of the numbered positions (1–9) when the wave is gen-erated by a given perturbation at these specific positions. TheFriedrich’s diagram, representing the phase polars of hydro-

magnetic waves, explains the MHD waves’ generation andpropagation. Figure 7 represents the magnitude of the 4 char-acteristic speeds that propagate in a specific direction referringto the initial magnetic field direction as shown in each panel.The direction of the initial magnetic field is determined by theWSO measurements together with the potential field model.By looking at these results, we recognize that the fast-modeMHD wave, represented by the dashed line, propagates almostin a circular shape which implies isotropic wave propagation. Itis apparent that this fast-mode wave resembles the observedEIT waves. This is obvious from Figure 2 because the back-ground is a high plasma-beta medium. When the MHD wavepropagates in a high plasma beta medium, it degenerates to apressure wave as if it were to propagate in a nonmagnetizedmedium. Specifically, let us examine the Friedrich’s diagramsat location 4 where the disturbance occurs as observed. Thecharacteristic speed of the fast-mode MHD wave at this posi-tion is almost equal to the acoustic wave speed. This impliesthat this particular observed MHD fast-mode wave is domi-nated by the acoustic mode that is the magnetosonic wave. Thisresult is consistent with the plasma-beta (i.e., much largerthan unity) that was deduced from the measured magnetic fieldstrength shown in Figure 2.

Figure 11. Normalized tangential magnetic field (Bt), normalized density �, pressure p , and entropy s , inthe radial direction at specific locations of latitude (�) and Carrington longitude (�) referring to the region ofthe inner box shown in Figure 10 and t � 1433 sec. Plus signs indicate the location where the MHD fastshock-jump conditions are satisfied. Solid lines in the first (top) two rows represent the Bt. Solid/dotted/dashed lines in the second two rows are density (�), pressure ( p), and entropy (s), respectively. When thereis no plus sign, this implies that the shock jump condition is not satisfied. As shown in Figure 10, it correspondsto where the flow speed does not cross over the MHD fast wave speed.

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Let us examine this wave further in three dimensions. Figure8 shows the density enhancement (i.e., wave front) at two timeswith various viewing angles. When it is viewed from centralmeridian on the solar disk (leftmost panel), the wave frontresembles the wave front seen by EIT. When it is viewed at 45�and 90� from the central meridian, the three-dimensional fea-ture of these MHD waves are clearly indicated. It also could beseen that the edge of the wave front intersects the layer of solaratmosphere on the disk to mark the positions as the wave frontseen by EIT. By looking at Figure 8, it is apparent that thismagnetosonic fast wave has a three-dimensional shell topol-ogy. This phenomenon was also suggested by Uchida [1968]and modeled by using a linear ray-tracing analysis [Uchida etal., 1973] to interpret the generation and propagation of theMoreton wave. That is, the edge or “skirt” of a three-dimensional shell (wave) intersecting a certain layer of thesolar atmosphere (i.e., photosphere, chromosphere, and co-rona) generates the waves at that layer of the atmosphere.

Up-to-now, we have identified this observed EIT wave (May12, 1997) on the solar disk to be a magnetosonic wave. It isinteresting to investigate further whether this wave could be aMHD fast shock. To examine this aspect, we need to examinethe ratio of the plasma mass flow speed to the local MHDfast-mode characteristic speed. To obtain this ratio, we need toknow the plasma mass flow speed and local MHD fast-modecharacteristic speed on the solar disk. According to nonlinearwave theory [Jeffrey and Taniuti, 1964] the wave propagation

speed is the plasma mass flow speed plus the local character-istic wave speed. The local characteristic speed is the speed ofthe MHD fast-mode wave at latitude (�) and Carrington lon-gitude (�)-directions on the solar disk which can be computedfrom (7) by using the local field and plasma parameters on thesolar surface. Then, we take the ratio of plasma mass flowspeed to the local MHD fast-mode characteristic speed; if thisvalue is larger than unity, a MHD fast shock will appear. Thisratio could be defined as the MHD fast wave Mach number,and its results (i.e. contours of the MHD fast wave Machnumber) are given in Figure 9. Specifically, the results pre-sented here are the maximum possible MHD fast wave Machnumber for different times which are computed by usingplasma mass flow speeds (v� and v�) and the minimum fastspeed on the solar surface. By examining Figure 9, the highestvalue of the contours are 0.58, 0.61, 0.53, and 0.48, at thesetimes, respectively; thus these observed EIT waves are notMHD fast shocks on the solar disk.

However, it is interesting to know whether there is a possi-bility that this MHD wave could develop to a MHD fast shockalong the radial direction. We have plotted the radial MHDfast speed (Cf ,r) and radial flow speed along the radial direc-tion at specific locations (i.e., at a specific latitude (�) andlongitude (�) on the solar disk) in Figure 10 at t � 1433 s. Itshould be noted that there is a region (i.e., 13.5� � � � 32�;122� � � � 140�) as marked by the box (dashed-dotted lines)shown in Figure 10, where the radial flow speed would exceed

Figure 12. The computed evolution of the response of the magnetic field topology responds to the three-dimensional disturbance as viewed from central meridian at various times. We only select those foot pointseffected by the disturbance to draw the field lines.

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the local Cf ,r in some locations, and thus a MHD fast shockcould develop. It is interesting to note that the region wherethe shock may develop is directly above the initiation site atlatitude � � 23�N and the longitude, � � 135�. To examinefurther the existence of MHD fast-mode shocks in this partic-ular region, we test the magnetic field and plasma properties(i.e., density, pressure, and entropy) to see whether they satisfythe shock-jump conditions given in chapter 6, p. 219, andAppendix D of Jeffrey and Taniuti [1964]. These results areshown in Figure 11 for t � 1433 sec. The cross marks indicatewhere the shock-jump conditions are satisfied. By looking atthese results, we recognize that the MHD fast-shock jumpconditions are satisfied in specific locations in this region. Thismay be attributed to the local plasma conditions because thelocal values of plasma beta have deterministic effects on theshock formation [Whang, 1987]. In two locations (i.e., � �13.5�, � � 131.5� and � � 22.5�, � � 122.5�), we find theexistence of forward MHD fast-shock pairs. When these MHDfast shocks propagate outward beyond our computational do-main, it is not clear whether they would survive the low plasma-beta environment [Whang, 1987]. In short, we realize this ob-served EIT wave has not developed to a MHD fast shock onthe solar disk (i.e., at the layer of lower corona), but a weak fastMHD shock does appear when it propagates outward at �1.6solar radii in a limited region directly above this disturbanceregion. This is obvious, when a point disturbance propagatesoutward spherically, it will be attenuated away from the dis-turbance center line.

Finally, we show the disturbed magnetic field lines viewed

from central meridian at the various times depicted in Figure12. The small arrow indicates the location of the disturbance.It shows that the initial potential field used for this simulationhas been significantly affected and is now highly nonpotential.However, it is difficult to realize the relationship between thedensity enhancements and these disturbed magnetic field lines.However, the relationships could be recognized by examiningthe disturbed field lines away from central meridian. Figure 13shows the disturbed field lines at various viewing angles (left toright: central meridian, 45� and 90� from central meridian),respectively, for two different times (i.e., 1910 and 2856 s afterintroduction of the disturbance). It is easily recognized thatthese magnetic field lines correspond to the relative densityplots shown in Figure 8 where magnetic field lines formed theloops having the character of density enhancements. It is in-teresting to note that the field lines, viewed at the limb, showrising coronal loops similar to those shown in Figure 8, andwhich have been observed in some EIT limb events (B. J.Thompson, personal communication, 2000). Another observa-tional feature that could be explained by this simulation is thesuggestion of Thompson et al. [1999] that the EIT waves couldbe generated by the expanding field lines in the solar atmo-sphere. This feature can be seen simultaneously from the dis-turbed magnetic field (Figures 12 and 13) and density enhance-ment (Figure 8). That is, after introduction of the pressuredisturbance, the pressure force acts on the fields, then, as a resultof the large plasma beta, the magnetic field is not stiff and is easilymoved around to generate detection of the EIT waves.

Figure 13. The computed magnetic field topology corresponding to Figure 8.

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4. Concluding RemarksUsing a three-dimensional, time-dependent ideal MHD

model, the observed EIT waves (May 12, 1997) were, for thefirst time, identified as MHD-magnetosonic waves, i.e., MHDfast mode waves dominated by the acoustic mode, as shown bythe Friedrich’s diagram (Figure 7). The preevent state of thisthree-dimensional MHD model is in hydrostatic equilibriumwith measured global line-of-sight magnetic fields togetherwith a source-surface model given by the Stanford WilcoxSolar Observatory [Hoeksema and Scherrer, 1985]. RecentlyWang [2000] presented a study for these observed EIT wavesby using a linear analysis of the ray tracing method. He sug-gested that the fast-mode MHD wave hypothesis could ac-count for some properties of EIT bright fronts. He has pre-sented an excellent argument to examine the low propagationspeed of EIT waves. In our study the low propagation speed ofobserved EIT was achieved by using WSO global field mea-surements which only accommodate weak fields that lead to apropagation speed of �260 km s�1. We have run several sim-ulations by simply increasing the measured field by a factor of2 and 4, respectively to test the plasma beta effect on the wavepropagation speed. These tests show that the propagationspeed is much faster than the observed EIT wave speed. Thesimulation results could not match the observation. That isanother reason we believe this observed EIT wave is generatedin a large plasma beta environment and identified as a mag-netosonic wave.

The present MHD numerical simulations reproduced someobserved features of the May 12, 1997, event observed by EIT.From the observations, the measured propagation speed ofthis EIT wave is far less than the speed of Moreton waves asreported by Moreton [1961] and Neupert [1989]. We would liketo offer the following explanation for this fact.

On the basis of MHD theory [Jeffrey and Taniuti, 1964], thepropagation speeds of MHD waves depend on the values ofplasma beta. The Moreton wave is measured in a flaring activeregion environment and is directly related to the magnetic fieldstrength of the active region, where the plasma beta is small.Hence the measured speed of a Moreton wave is of the orderof the Alfven speed (�103 km/s). On the other hand, the EITwave propagates in a global-scale environment in which themeasured average global-scale magnetic field only accounts forweak fields. As we have shown in Figure 2, the plasma beta ismuch larger than unity. This leads to an acoustic speed of afactor 2 larger than the Alfven speed in this case. Therefore wesuggest that the earlier observed Moreton wave and Neupert’scoronal EUV transient are MHD fast waves propagating in alocal medium directly related to the active region fieldstrength, i.e., in a small plasma-beta medium. On the otherhand, the newly observed EIT wave [Thompson et al., 1998] isa MHD fast wave propagating on a global scale in a largeplasma-beta medium which is dominated by the acoustic modeand named “magnetosonic wave.” From our analysis this ob-served EIT wave has not developed into a MHD fast shock onthe disk, but there is a small region along the radial directionat �1.6 solar radii (Figure 11) where a forward MHD fastshock exists. We have also revealed the existence of forwardfast MHD shock pairs in the radial direction. It should benoted that we have not discovered slow-mode MHD waves inthe simulation. This may be explained by using the Friedrich’swave diagrams shown in Figure 7 where the slow-mode MHDwaves only appear in a very limited (or restricted) situation.

As a final remark, this three-dimensional magnetohydrody-namic simulation describes an outward bulk motion of plasmasand magnetic fields and an outwardly propagating density thatmimics the observed EIT wave. It may be considered as theearly stage of a coronal mass ejection.

Appendix A: Boundary ConditionsThe boundary conditions used for this simulation are the

time-dependent characteristic boundary conditions which arebased on previous work [Hu and Wu, 1984; Nakagawa et al.,1987; Wu and Wang, 1987]. However, the numerical proce-dures to implement these conditions are different than theprevious work; thus we briefly summarize below.

Using the governing equations (1)–(6) in the text, we seekthe characteristics along the projected normal in the r–t plane.Let us denote the left eigenvectors of Ar by Li, thus

LiAr � �Li, i � 1, 2, . . . , 8, (A1)

where the eigenvalues � i are

� i � vn, vn cs, vn bn, vn cf (A2)

with

vn � v r, Bn � Br, B2 � B�, B3 � B�, bn � Br2/� , cfs

2

�12

a2 � b2 ��a2 � b2�2 4a2bn2� , (A3)

a2 ��p2�

, b2 �Br

2 � B�2 � B�

2

�. (A4)

The projected, normal characteristics are described by theequations

drdt � vn, vn cs vn bn, vn cf. (A5)

The corresponding left eigenvectors are

� � vn, L1 � 0 0 0 0 1 0 0 0� , (A6)

L2 � � �p�

0 0 0 0 0 0 �1� , (A7)

� � vn cs,L4� 0 cs�bn2 cs

2�BnB2cs

BnB3cs

�0

B2cs

2

B3cs2

��

2��bn

2 cs2�� , (A8)

� � vn � cs, L4� 0 cs�bn2 cs

2�BnB2cs

BnB3cs

�0

�B2cs

2

��

B3cs2

2��bn

2 cs2�� , (A9)

� � vn bn L5 � 0 0 B3 sign �Bn�

� B2 sign �Bn� 0 B3

���

B2

��0� , (A10)

� � vn � bn L6 � 0 0 B3 sign �Bn�

� B2 sign �Bn� 0 �B3

��

B2

��0� , (A11)

� � vn cf, L7� 0 cf�bn2 cf

2�BnB2cf

BnB3cf

�0

B2cf

2

B3cf2

��

2��bn

2 cf2�� , (A12)

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� � vn � cf, L8� 0 cf�bn2 cf

2�BnB2cf

BnB3cf

�0

�B2cf

2

��

B3cf2

2��bn

2 cf2�� . (A13)

Let

Q � �Ar

�u�r A�

1r

�u��

A�

1r sin �

�u��

F . (A14)

The corresponding characteristic equations along these pro-jected normal characteristics are listed with their respectiveprojected normal characteristics.

drdt � vn, L1

�u�t � L1Q , (A15)

L2

�u�t � L2Q , (A16)

drdt � vn cs, L3

�u�t � L3Q , (A17)

drdt � vn � cs, L4

�u�t � L4Q , (A18)

drdt � vn bn, L5

�u�t � L5Q , (A19)

drdt � vn � bn, L6

�u�t � L6Q , (A20)

drdt � vn cf, L7

�u�t � L7Q , (A21)

drdt � vn � cf, L8

�u�t � L8Q , (A22)

Nonreflecting boundary conditions are stated as

L1

�u�t � 0, (A23)

L2

�u�t � 0, (A24)

L3

�u�t � 0, (A25)

L4

�u�t � 0, (A26)

L5

�u�t � 0, (A27)

L6

�u�t � 0, (A28)

L7

�u�t � 0, (A29)

L8

�u�t � 0. (A30)

The general approach in our simulation is that the character-istic equations along the outgoing characteristics are chosen

(compatibility equations). The nonreflecting boundary condi-tions are chosen along the incoming characteristics. The non-reflecting boundary conditions and compatibility equations arecombined into a complete system to determine the values of allthe dependent variables on the boundary.

For the bottom boundary, we have the following situations.1. When vn � cf, all projected characteristics are outgo-

ing, equations (A15)–(A22) are used.2. When �cf � vn � �bn, there are seven outgoing and

one incoming projected characteristics, equations (A15)–(A21) and (A30) are used.

3. When �bn � vn � �cs, there are six outgoing and twoincoming projected characteristics, equations (A15)–(A19)and (A21), (A28), and (A30) are used.

4. When �cs � vn � 0, there are five outgoing and threeincoming projected characteristics, equations (A15)–(A17)and (A19), (A21), (A26), (A28), and (A30) are used.

5. When 0 � vn � cs, there are three outgoing and fiveincoming projected characteristics, equations (A17)–(A19)and (A21), (A23), (A24), (A26), (A28), and (A30) are used.

6. When cs � vn � bn, there are two outgoing and sixincoming projected characteristics, equations (A19), (A21),(A23)–(A26), (A28), and (A30) are used.

7. When bn � vn � cf, there are one outgoing and sevenincoming projected characteristics, equations (A21), (A23)–(A28), and (A30) are used.

8. When vn � cf, all projected characteristics are incom-ing, equations (A23)–(A30) are used.

9. Top boundary treatment is similar to that in bottomboundary.

Acknowledgments. The authors would like to thank J. Gurman andE. Tandberg-Hanssen for reading the manuscript and giving valuablecomments. We also thank the referees for constructive comments andsuggestions. Work performed by S. T. Wu, H. Zheng, and S. Wang wassupported by Air Force Office of Scientific Research grant F49620-00-0-0204 and National Science Foundation grant ATM-0070385. H.Zheng and S. Wang were also partly supported by the National NaturalFoundation of China grant 49834030. S. P. Plunkett was partiallysupported by NASA grant NAG5-8116.

Janet G. Luhmann thanks the referees for their assistance in eval-uating this paper.

ReferencesAthay, R. G., and G. E. Moreton, Impulsive phenomena of the solar

atmosphere, I, Some optical events associated with flares showingexplosive phase, Astrophys. J., 133, 935–950, 1961.

Bazer, J., and O. Fleischman, Propagation of weak hydromagneticdiscontinuities, Phys. Fluids, 2, 366–373, 1959.

Delaboudiniere, J. P., et al., EIT: Extreme-ultraviolet imaging tele-scope for the SOHO mission, Sol. Phys., 164, 291–312, 1995.

Feldman, U., G. A. Doschek, U. Schuhle, and K. Wilhelm, Propertiesof quiet-Sun coronal plasmas at distances of 1.03 � RJ � 1.50along the solar equatorial plane, Astrophys. J., 518, 500–507, 1999.

Friedrichs, K. O., Non-linear wave motion in magnetohydrodynamics,Rep. 2105, Los Alamos Natl. Lab., Los Alamos, N. M., 1954.

Fredrichs, K. O., and H. Kranzer, Nonlinear wave motion in magne-tohydrodynamics, Rep. MH-8, Inst. of Math. Sci., New York Univ.,New York, 1958.

Hoeksema, J. T., Extending the Sun’s magnetic field through the three-dimensional heliosphere, Adv. Space Res., 9(4), 141–145, 1989.

Hoeksema, J. T., Large-scale solar and heliospheric magnetic fields,Adv. Space Res., 11(1), 15–24, 1991.

Hoeksema, J. T., and P. H. Scherrer, An atlas of photospheric mag-netic field observations and computed coronal magnetic fields:1976–1985, Sol. Phys., 105, 205–211, 1985.

Hu, Y. Q., and S. T. Wu, A full-implicit-continuous-Eulerian (FICE)

25,101WU ET AL.: NUMERICAL SIMULATION OF MHD WAVES

Page 14: Three-dimensional numerical simulation of MHD waves observed … · 2017-10-20 · Three-dimensional numerical simulation of MHD waves observed by the Extreme Ultraviolet Imaging

scheme for multi-dimensional, transient magnetohydrodynamic(MHD) flows, J. Comput. Phys., 55, 33–64, 1984.

Jeffrey, A., and T. Taniuti, Non-linear wave propagation with appli-cation to physics and magnetohydrodynamics, chap. 6, p. 219, andAppendix D, Academic, San Diego, Calif., 1964.

Low, B. C., Self-similar magnetohydrodynamics, Part four, The physicsof coronal transients, Astrophys. J., 281, 392–404, 1984.

Meyer, F., Flare-produced coronal waves, in Proc. IAU Sym 35: Struc-ture and Development of Solar Active Regions, edited by K. O.Kiepenheuer, p. 485, D. Reidel, Norwell, Mass., 1968.

Moreton, G. E., H� observations of flare-initiated disturbances withvelocities �1000 km/sec, Astron. J., 65, 494–501, 1960.

Moreton, G. E., Fast-moving disturbances on the Sun, Sky Telescope,21, 145–146, 1961.

Moreton, G. E., H� shock wave and winking filaments with the flare of20 September 1963, Astron. J., 69, 145–151, 1964.

Nakagawa, Y., Y. Q. Hu, and S. T. Wu, The method of projectedcharacteristics for the evolution of magnetic arches, Astron. Astro-phys., 179, 354–370, 1987.

Neupert, W. M., Transient coronal extreme ultraviolet emission beforeand during the impulsive phase of a solar flare, Astrophys. J., 504,344–353, 1989.

Plunkett, S. P., B. J. Thompson, R. A. Howard, D. J. Michels, O. C. St.Cyr, S. J. Tappin, R. Schwenn, and P. L. Lamy, LASCO observationsof an Earth-directed coronal mass ejection on May 12, 1997, Geo-phys. Res. Lett., 25, 2477–2480, 1998.

Roache, R. J., Fundamentals of Computational Fluid Dynamics, Her-mosa, Albuquerque, N. M., 1999.

Sakurai, T., M. Irie, M. Miyashita, K. Yamaguchi, and Y. Shiomi, AnX12/3B white-light flare of 1991 June 4 on NOAA Region 6659, inProceedings of the Second SOLTIP Symposium, Nakaminato, Japan,13–17 June 1994, STEP GBRSC Special Issue, vol. 5, pp. 33–36,Ibaraki Univ. Press, Mito, Japan, June 1995.

Thompson, B. J., S. P. Plunkett, H. B. Gurman, J. S. Newmark, O. C.St. Cyr, and D. J. Michels, SOHO/EIT observations of an Earth-directed coronal mass ejection on May 12, 1997, Geophys. Res. Lett.,25, 2465–2468, 1998.

Thompson, B. J., J. B. Gurman, W. M. Neupert, J. S. Newmark, J.-P.Delabourdiniere, O. C. St. Cyr, S. Stezelberger, L. P. Dere, R. A.Howard, and D. J. Michels, SOHO/EIT observations of the 1997April 7 coronal transient: Possible evidence of coronal Moretonwaves, Astrophys. J. Lett., 517, L151–L155, 1999.

Uchida, Y., Fast-mode wavefronts and Moreton’s wave phenomena,Sol. Phys., 4, 30–44, 1968.

Uchida, Y., M. D. Altschuler, and G. Newkirk, Flare-produced coronalMHD-fast-mode wave front and Moreton’s wave phenomenon, Sol.Phys., 28, 495–516, 1973.

Wang, Y. M., EIT waves and fast-mode propagation in the solarcorona, Astrophys. J., 543, L89–L93, 2000.

Whang, Y. C., Slow shocks and their transition to fast shocks in theinner solar wind, J. Geophys. Res., 92, 4349–4356, 1987.

Wu, S. T., and J. F. Wang, Numerical tests of a modified full implicitcontinuous Eulerian (FICE) scheme with projected normal charac-teristic boundary conditions for MHD flows, Comput. Meth. Appl.Mech. Eng., 64, 267–282, 1987.

Yanenko, N. N., The Method of Fractional Steps, The Solution of Prob-lems of Mathematical Physics in Several Variables, p. 149, Springer-Verlag, New York, 1971.

M. Dryer, NOAA/SEC, 325 Broadway, Boulder, CO 80305, USA.S. P. Plunkett, Universities Space Research Association, 300-D

Street SW, Washington, DC 20024, USA.B. J. Thompson, Code 682.3, NASA Goddard Space Flight Center,

Greenbelt, MD 20771, USA.S. Wang, S. T. Wu, and H. Zheng, Center for Space Plasma and

Aeronomic Research, University of Alabama in Huntsville, Huntsville,AL 35899, USA.

X. P. Zhao, Hansen Experimental Physics Laboratory, StanfordUniversity, Stanford, CA 94305, USA.

(Received December 4, 2000; revised April 23, 2001;accepted April 23, 2001.)

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