SOLAR ATMOSPHERIC DYNAMIC COUPLING DUE TO SHEAR MOTIONS DRIVEN BY THE LORENTZ FORCE

Transcription

1 The Astrophysical Journal, 666: , 2007 September 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. SOLAR ATMOSPHERIC DYNAMIC COUPLING DUE TO SHEAR MOTIONS DRIVEN BY THE LORENTZ FORCE Ward Manchester IV Center for Space Environment Modeling, University of Michigan, Ann Arbor, MI Received 2006 October 27; accepted 2007 May 18 ABSTRACT Recent theoretical and numerical works have shown that shearing motions observed during magnetic flux emergence are driven by the Lorentz force. The Lorentz force results from the nonuniform expansion of the magnetic field in a highly pressure-stratified atmosphere. This deformation of the flux system causes the axial component of the field to become weaker (along field lines) in the corona than in the lower atmosphere, which produces the shearing Lorentz force. The shear flows result in the magnetic field being drawn parallel to the polarity inversion lines of active regions as observed at the photosphere and in filaments. In order to arrive at an equilibrium state after flux emergence, the axial field must evolve to be constant along field lines. This nonlocal requirement for force balance results in a coupling by shear flows that act to equilibrate the axial magnetic field. This paper will elaborate on the dynamic coupling such shearing flows cause between the layers of the solar atmosphere. This coupling occurs by way of self-organized flows that transport magnetic flux and energy from below the photosphere into the corona. This work explains how coronal magnetic fields become energized to produce coronal mass ejections and two-ribbon flares. Subject headinggs: MHD Sun: coronal mass ejections (CMEs) 1. INTRODUCTION 2002; Yang et al. 2004; Liu et al. 2005). Higher in the atmosphere, the magnetic field is difficult to measure directly, but its direction Coronal mass ejections (CMEs) are energetic expulsions of plasma from the solar corona that are driven by the release of can be inferred from plasma structures formed within the field. Seen in chromospheric H absorption, filaments form only over magnetic energy typically in the range of ergs. The majority photospheric inversion lines (Zirin 1983) in magnetic fields of CMEs originate from the eruption of preexisting large- scale helmet streamers ( Hundhausen 1993). Less common fast CMEs typically come from smaller, more concentrated locations of magnetic flux referred to as active regions. In this case, the CMEs often occur shortly after the flux has emerged at the photosphere, but can also happen even as the active region is decaying. While CMEs occur in a wide range of circumstances, there are common features that suggest that the coronal magnetic fields that spawn CMEs and large flares are coupled to the solar interior. In the case of homologous CMEs, more than two dozen eruptions can occur from the same active region, often separated by only a few hours ( Takasaki et al. 2004). Such repetitive behavior suggests that the magnetic field can be continuously recharged over a relatively short period of time. A connection of CMEs to the solar interior dynamo is suggested by the role CMEs play in restructuring the global coronal magnetic field during the solar cycle by expelling magnetic flux and helicity (Hundhausen 1993; Rust 1994; Low 1996). In this paper we demonstrate that a strong coupling between the coronal and subphotosphere magnetic field naturally occurs as a result of shearing motions driven along polarity inversion lines by the Lorentz force as magnetic fields emerge through the photosphere. Thesignificanceofthecouplingprovidedbyshearingmotions stems from the fact that all CMEs originate above photospheric magnetic polarity inversion lines (neutral lines), which exhibit strong magnetic shear. Shear implies that the magnetic field has a strong component parallel to the photospheric line that separates magnetic flux of opposite sign, and in this configuration, the field possesses significant free energy. In contrast, a potential field runs perpendicular to the inversion line and has no free energy. There is enormous evidence for the existence of highly sheared magnetic fields associated with CMEs and large flares. At the photosphere, magnetic shear is measured directly with vector magnetograms (e.g., Hagyard et al. 1984; Zirin & Wang 1993; Falconer et al. nearly parallel to the inversion line ( Leroy 1989). Fibrils and H loops that overlay photospheric bipolar active regions are also indicative of magnetic shear ( Foukal 1971). Comparisons between vector magnetograms and H images show that the direction of the sheared photospheric magnetic field coincides with the orientation of such fibril structures (Zhang 1995). Higher in the corona, evidence of magnetic shear is found in loops visible in extreme-ultraviolet ( Liu et al. 2005) and X-ray sigmoids ( Moore et al. 2001). These structures are seen to run nearly parallel to the photospheric magnetic inversion line prior to CMEs that are followed by the reformation of closed bright loops that are much more potential in structure. Finally, observations in the extremeultraviolet by the Transition Region and Coronal Explorer show that 86% of two-ribbon flares show a strong-to-weak shear change of the ribbon footpoints that indicates the eruption of a sheared core of flux (Su et al. 2007). In addition to sheared magnetic fields, shear flows are observed to be aligned along magnetic inversion lines. Here magnetic elements of opposite polarity move parallel to the inversion line in opposite directions. Such shearing motions are most frequently found as magnetic flux emerges at the solar photosphere ( Zirin 1983; Strous et al. 1996; Yang et al. 2004; Deng et al. 2006). Particularly intense high-velocity shear flows at the photosphere have recently been observed in direct association with CMEs (Meunier & Kosovichev 2003; Yang et al. 2004). Higher in the atmosphere, shear flows are observed at greater speeds. Chromospheric Dopplergrams made in the H have revealed velocity shearatamagnitudeof5kms 1 (Malherbe et al. 1983), while those made in C iv lines of the transition region show shear flows with a magnitude of 20 km s 1 (Athay et al. 1982, 1985). More recent observations made with the SUMER instrument aboard the Solar and Heliospheric Observatory have shown evidence of velocity shear in active region loops above the solar limb with magnitudes in the range of km s 1 (Chae et al. 2000). 532

2 COUPLING BY SHEAR FLOWS 533 Sheared magnetic fields are at the epicenter of solar eruptive behavior. Large flares are preferentially found to occur along the most highly sheared portions of magnetic inversion lines ( Hagyard et al. 1984; Zhang 1995). More recent analysis by Schrijver et al. (2005) found that shear flows associated with flux emergence drove enhanced flaring. Similarly, active region CME productivity is also strongly correlated with magnetic shear as shown by Falconer (2001) and Falconer et al. (2002, 2006). It is not coincidental that large flares and CMEs are strongly associated with filaments, which are known to form only along sheared magnetic inversion lines ( Zirin 1983). The buildup of magnetic shear is essential for energetic eruptions, and for this reason, it is of fundamental importance to understanding solar activity. The observed photospheric shearing motions are self-consistently explained as a response to the Lorentz (tension) force that arises during flux emergence. The motion takes the form of large-amplitude shear Alfvén waves that occur when the magnetic field buoyantly rises in a gravitationally stratified atmosphere as was first shown by Manchester & Low (2000) and Manchester (2000, 2001) and found in later simulations by Fan (2001), Magara & Longcope (2003), and Archontis et al. (2004). This shearing mechanism offers explanations to many aspects of active regions. First, it explains why magnetic shear naturally coincides with the magnetic inversion line. Second, this mechanism explains the shear velocity observed at different heights in the solar atmosphere in terms of the velocity amplitude of the Alfvén waves ( Manchester 2001). Furthermore, this shearing process distributes the axial flux to self-consistently create the pattern of differential shear observed in bipolar active regions (Manchester 2001) as well as initiate eruptions such as flares and CMEs ( Manchester 2003; Manchester et al. 2004). What we will now show more clearly is that Lorentz force driven shearing motions provide a mechanism by which regions of the solar atmosphere and interior connected by field lines are nonlinearly coupled and exhibit an organized behavior that works to expel magnetic flux from the solar interior. Here, we describe time-dependent magnetohydrodynamic ( MHD) simulations of nonlinear buoyancy instabilities exhibited by a horizontal magnetic layer. In this paper we examine in detail the strong coupling of the emerging flux to the subsurface magnetic field. This coupling occurs because the shear flows transport magnetic flux from the layer into an expanding magnetic field and will persist until the shear component of the magnetic field is equilibrated along magnetic field lines. This coupling process allows multiple areas of emerging flux to interact so strongly that larger loops (arcades) can draw flux from their neighbors, causingthemtocollapse. This paper is organized in the following way. After the introduction, the system of governing equations are discussed in x 2, while initial and boundary conditions are specified in x 3. The stability of the initial states is then addressed in x 4. Detailed descriptions of the simulations are presented in x 5. Finally, we conclude in x 6 with a general discussion of our work in terms of its physical significance and its relation to observed solar active regions. 2. GOVERNING EQUATIONS OF MHD AND MATHEMATICAL MODELS We model the buoyancy instabilities discussed here with the assumption that the system is composed of magnetized plasma that behaves as an ideal gas. The plasma is taken to have infinite electrical conductivity so that the magnetic field is frozen into the plasma. The gravitational acceleration g is constant in the negative z-direction [we use Cartesian coordinates (x; y; z)]. With these assumptions, the evolution of the system is governed by the following ideal MHD þ :=(v) ¼ þ (v =:)v ¼ :pþ 1 (:<B) < B 4 þ :=(ev) ¼ ð3þ p ¼ ( ¼ :<(v < B); where is the mass density, p is the plasma pressure, e is the internal energy density (per unit volume), v is the velocity field, and B is the magnetic field. The simulations are invariant in the x-direction so that (@/@x) ¼ 0. However, nonzero velocity and magnetic field components in the x-direction are allowed and are critical to the dynamics of the emerging flux. In these simulations, the polytropic index is set to a value of 5/3. These equations are solved with the ZEUS-2D code, available from the Laboratory for Computational Astrophysics at the University of California, San Diego. In this application, ZEUS-2D is configured to explicitly solve the equations of ideal MHD with a uniform gravity on a fixed Eulerian grid. ZEUS-2D is compiled using the rotation option to evolve v x, which allows for motion out of the plane of variation. Tests and accuracy of ZEUS-2D are described in Stone & Norman (1992). 3. INITIAL CONDITIONS The equilibrium state we employ for an initial condition is a member of a family of two-dimensional equilibria described in Low & Manchester (2000, hereafter LM00). The configuration is magnetostatic in which the field is in pressure balance with the surrounding plasma and partially supports its weight. For the purpose of modeling flux emergence, we place the magnetic layer 2 3 pressure scale heights below an atmosphere that increases in temperature to roughly model the solar chromosphere and corona. The field is in a sheared configuration oriented (on average) at a 45 angle to the plane of variation in our two-and-a-halfdimensional (2.5D) numerical simulations. In this case, the perturbation to the layer can be described by e i(k = r!t), where k is oblique to the field (B) in the layer. This distinguishes the mode of instability from a Parker mode (Parker 1966) in which k is parallel to B or an interchange mode in which k is perpendicular to B. When k is oblique to B, the instability can be thought of as a mixed mode, which has been the subject of recent numerical simulations (e.g., Cattaneo et al. 1990; Matsumoto et al. 1993; Kusano et al. 1998; Manchester 2001). This work has shown that the mixed mode has unique properties not found in the Parker mode or interchange mode. Cattaneo et al. (1990) found the formation of a resonant surface that controlled how much of the magnetic layer was buoyantly disrupted. Kusano et al. (1998) determined that the mixed mode had a reduced linear growth rate compared to the Parker mode, but evolved more rapidly in the nonlinear regime. In Manchester (2001), arcades buoyantly rising from the magnetic layer experienced large-scale shear flows driven by the Lorentz force that occurred as opposite sides of the arcades moved in opposite directions parallel to the inversion line. In this paper we expand on the work of Manchester (2001) to more fully understand the transport and coupling properties of the Lorentz force driven shear flows. The two-dimensional initial states allow us to study the nonlinear interaction of multiple Fourier ð4þ

3 534 MANCHESTER Vol. 666 cells during the development of the instability. Here, the size of the computation domain is chosen to accommodate six complete Fourier cells. In this case, we give two of the cells small velocity perturbations to initiate the growth of the buoyancy instability in two discrete locations. The initial growth of the arcades is uncoupled, but eventually becomes strongly coupled by shearing motions that spread from the arcades throughout the magnetic layer. For comparison, we perform a simulation using a planar initial state that has the special property that it possesses ideal MHD invariants that are identical to those of the chosen undulating layer. That is, the planar state possesses the same magnetic flux and distribution of plasma on the field lines. In addition, the planar equilibrium state also possesses the same magnetic, gravitational, and thermal potential energy of the two-dimensional initial state (Manchester & Low 2000). These initial states give us an opportunity to study the evolution of a single system (as defined by the MHD invariants), set in two distinct equilibrium configurations, given the same perturbation. As shown in Manchester & Low (2000), the magnetically sheared, undulating state can be arrived at by ideal MHD displacements to a planar state. The initial planar field is defined as B y ¼ da dz ¼ da d d dz ; ð6þ where the flux coordinate takes the form ¼ expð z=hþ; ð7þ where H is the pressure scale height. The remaining field components for the planar state are B x ¼ B y and B z ¼ 0. The field points in a uniform horizontal direction that lies 45 from the y-axis. For the simulations to be presented, we use the following functional form of the flux profile, A() ¼ B 0 H ( 0 2) 2 2 : ð8þ This particular choice of A() produces physically desirable features in the initial state. First, it allows the magnetic field to go smoothly to zero on the line ¼ 0, so that on this line we can match the solutions to a field-free atmosphere above. In addition, the field strength falls off rapidly with depth, so as to become insignificant a few pressure scale heights below the uppermost field line. This construction spatially isolates the magnetic field in a layer placed away from the upper and lower boundaries of the computational domain, which minimizes the boundaries influence on the simulations. For the two-dimensional equilibrium state, the flux coordinate is defined as s ¼ exp z y þ cos : ð9þ H H Using the same flux function, A, with s, the two-dimensional state takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffi s 0 B x ¼ B 0 2 s 1 ; ð10þ 3 s s 0 B y ¼ B 0 3 s s 0 B z ¼ B 0 3 sin y s H exp z ; ð11þ H ; ð12þ P( s ) ¼ P 0 B s 3 3 s s þ s : ð13þ Fig. 1. Magnetic structure of the initial state. Black lines show the direction of the magnetic field projected onto the plane of variation ( y-z plane). The angle of the magnetic field away from the plane of variation is shown in color. The perturbed undulations are marked with white arrows. The planar state takes the simpler form 0 B y ¼ B x ¼ B 0 2 ; ð14þ while the plasma pressure defined by P() takes the form of equation (13), with s now replaced with ¼ exp ( z). The magnetic structure of the two-dimensional initial state is presented in Figure 1. Here, black lines show the direction of the magnetic field projected onto the plane of variation ( y-z plane), while the angle of the magnetic field away from the plane of variation is shown in color. The magnetic field is horizontally oriented 45 away from the y-axis and is more sheared in the crests of the undulations. The third and sixth undulations (from the left) are given velocity perturbations with the sixth undulation being perturbed at 20% greater velocity. The magnitude of the perturbations is a few percent of the sound speed. The parameters chosen to specify the initial states are as follows, 0 ¼ 1:8, B 0 ¼ 6:0 ; 10 5 G, T 0 ¼ 6:0 ; 10 3 K, 0 ¼ 3:3 ; 10 6 gcm 3, g ¼ 2:734 ; 10 4 cm s 2,andm ¼ 1:3m H,wherem H is the molecular weight of hydrogen. For this choice of parameters, the minimum plasma- of the magnetic layer is approximately We will typically present most quantities in units characteristic of the system with length, time, and velocity given in units of H, ¼ H /c s,andc s, which is the sound speed. For these models, the characteristic scales have the values H ¼ 1:40 ; 10 7 cm, ¼ 22:6 s,andc s ¼ 6:18 ; 10 5 cm s 1. The magnetic field is contained in an isothermal plasma that extends from the bottom boundary of the computational domain placed at z min ¼ 6:0H to a height of z ¼ 2:0H, where the density is 4 ; 10 7 gcm 3, roughly corresponding to the photosphere. A temperature-stratified field-free atmosphere is established to model roughly the solar corona-chromosphere system. Here, the temperature increases linearly from 6000 K and reaches a value of 4:5 ; 10 4 Katz ¼ 25H, which represents the temperature increase of the chromosphere. The temperature then increases at a steeper rate, reaching 2:0 ; 10 5 Katz ¼ 29H, representing the transition region. Finally, the temperature increases to approximately 3:0 ; 10 5 K at the upper boundary of the domain placed at z ¼ 107H. The grid configuration for the simulations is Cartesian, with nonuniform spacing in the vertical direction chosen to provide high resolution of the magnetic layer and photospheric pressure stratification. A much sparser grid distribution is applied to the upper atmosphere. We place 30 points in the interval 6:0H < z < 3:6H, 70 points in the interval 3:6H < z < 0:36H and 150 points in the interval 0:36H < z < 107H. In the last interval,

4 No. 1, 2007 COUPLING BY SHEAR FLOWS 535 the distance between successive points expands by a factor of Grid spacing in the horizontal direction is uniform with the number of grid points set at 200. We assume periodic boundaries for y ¼ y min and y ¼ y max and free (out flow) boundaries for z ¼ z min and z ¼ z max, which effectively allow waves to exit the system. 4. STABILITY OF THE PLANAR STATE The stability of plane-stratified magnetostatic atmospheres presents a much more tractable problem than do two-dimensional equilibria. Such planar states have received much attention over the years (see Newcomb 1961; Parker 1966; Gilman 1970). Newcomb treated the stability of equilibrium states with nonuniform horizontal magnetic fields and first arrived at the condition that is both necessary and sufficient for stability against all linear displacements. The condition can be written as d dz > 2 g p ; ð15þ where g is uniform, and both p and are modified by the presence of the magnetic field. For our purposes, we can write the above inequality for the planar atmosphere in terms of with the use of equations (7), (13), and (14). We have d d 2 (A 0 ) 2 1 < Z 4P 0 A 02 d A 02 : ð16þ From this expression, we see that stability is ensured provided that the magnetic pressure does not decrease more rapidly than a critical rate determined by g( 1)/. We evaluate the stability for ¼ 5/3 by substituting the flux profile expressed in equation (8) into equation (16), and we arrive at the criterion for stability, þ þ 4P 0 B 2 5 > 0: 0 ð17þ Figure 2 provides plots of the stability criterion (eq. [17]) and the magnetic field strength as functions of height (z) with dashed and solid lines, respectively. The plot of equation (17) reveals that the system is unstable in a narrow region ( 0:87H < z < 0:69H ) less than one pressure scale height in depth. Although determining the instability of an atmosphere from equation (15) is straightforward, finding the particular modes of instability and their associated growth rates requires numerical analysis that is outside the scope of this work. 5. RESULTS OF NUMERICAL SIMULATIONS In this section we present the results of the numerical simulations of buoyancy instabilities that are designed to demonstrate in detail the coupling properties of shear flows driven by the Lorentz force. Here, we examine the evolution of an undulating magnetic layer that has been locally perturbed in two locations. Figure 3 shows the time evolution of the magnetic and velocity fields following the perturbation of the initial state with sequences of frames at t ¼ 39:1,49.6,53.1, and The top row of Figure 3 shows the shear velocity v x. The middle row of Figure 3 shows the field geometry with field lines projected on the y-z plane and the shear angle away from the y-z plane shown in color. Finally, the bottom row of Figure 3 shows the shape of a single field line projected onto the y-z and x-y planes in black and red, respectively, with the magnitude of B x on the line plotted in green. Fig. 2. Magnetic field strength of the initial state plotted as a function of height with a solid line. The plot applies to the LM00 solution at a point halfway between the crests and valleys and to the equivalent planar state. The dashed line shows the stability criterion for the planar atmosphere plotted as a function of height for ¼ 5/3. The magnetic layer is unstable in the narrow region where the dashed line dips below zero. Examining Figure 3, we see that, as the instability progresses, the magnetic field goes from a configuration of six undulations at t ¼ 0 to a configuration of two rising arcades at t ¼ 39:1 where the field has been perturbed. The ascending arcade on the right grows faster, having been given a larger perturbation than the one on the left. The color representation of the field angle shows that as the magnetic arcades rise, they become increasingly sheared near the center, reaching a maximum shear angle near 90.Thetop row of Figure 3 starkly shows strong velocity shear with the horizontal velocity (v x ) reversing directions across the expanding arcades. In the early stages, the arcades evolve independently, but after they have risen several pressure scale heights, they become dynamically coupled by these shearing motions. By time t ¼ 49:6, the shear flow of the larger arcade extends through the layer and then begins to reverse the flow in the smaller arcade as seen in Figure 3b. By time t ¼ 53:1, the shear flow is almost completely reversed in the smaller arcade, which consequently begins to shrink. By time t ¼ 68:1, the smaller arcade has completely collapsed, and the field within it has unsheared. The images of the field line projected on the x-y plane graphically show how the line becomes sheared, drawn nearly parallel to the direction of invariance (inversion line) in the rising arcades. At the same time, the portion of the field line that is in the magnetic layer becomes nearly perpendicular to the direction of invariance. After time t ¼ 53:1, the field line in the larger arcade (projected on the x-y plane) takes on a distinctive sigmoid shape. The sheared geometry of the field lines is more clearly illustrated in the threedimensional representation given in Figure Dynamic Coupling by the Shear Flow It is instructive to examine the strength of the shear component of the magnetic field during the evolution of the instability. As the two magnetic crests rise, the expansion causes the magnitude of B x (axial component) to decrease in the arcades as seen

5 Fig. 3. Evolution of the buoyancy instability of an undulating magnetic layer. Depicted is an evolutionary sequence of images at times t ¼ 39:1, 49.6, 51.3, and 68.1 in units of ¼ cs /g. The top row shows the shear velocity (v x ) in color, and the middle row shows magnetic field lines projected on the y-z plane in black and the angle between the magnetic field and y-axis is shown in false color. The evolution of a single field line is shown in the bottom row. The shape of the line projected on the y-z plane is shown in black, while the field line seen from above projected on the x-y plane is shown in red. The magnitude of Bx along the field line is plotted in green. The images show the evolution of a pair of arcades ascending from the magnetic layer. The arcade on the right grows faster, having been given a larger perturbation than the loop on the left. The continued growth of arcades is driven largely by shearing motions and the accumulation of Bx flux. In the early stages, the arcades evolve independently, but by time t ¼ 49:6, shearing motions begin to couple the two arcades as the shear flow of the larger arcade extends through the layer and then begins to reverse the flow in the smaller arcade as seen in (b). By time t ¼ 53:1, the shear flow is almost completely reversed in the smaller arcade, which consequently begins to shrink. By time t ¼ 68:1, the smaller arcade has completely collapsed, and the field within it has unsheared as seen in (l ). Fig. 4. Three-dimensional view of the 2.5D system. Field lines of the emerged arcade (right-hand side of Fig. 3) are drawn for time t ¼ 68:1 above the photosphere where the shear flow (v x ) is illustrated in color. The field lines are shown as seen from above and from the side in the left and right panels, respectively. Note that the field lines low down in the arcade are nearly parallel to the inversion line, while the highest loops cross the inversion line at a 45 angle.

6 COUPLING BY SHEAR FLOWS 537 Fig. 5. Development of Lorentz force shear found in numerical simulations of flux emergence. This figure shows the central cross section of a magnetic flux rope emerging from the convection zone into the corona reported in Manchester et al. (2004). The magnetic field direction in the y-z plane is illustrated with white lines, and the direction of the electric current on the same plane is shown with black lines. The magnitude of B x is shown in color. In this case, a large vertical gradient in the axial (B x ) component of the magnetic field forms as a result of the field s nonuniform expansion in the gravitationally stratified plasma. This deformation of the field produces a horizontal component of the electric current that is oblique to B. The magnetic field crosses the current in opposite directions on opposite sides of the rope producing Lorentz forces in the x-direction that shear the rope. in the green line plotted in Figures 3i, 3k, and3l. Asfirstdescribed in Manchester & Low (2000) and Manchester (2000), the gradient in B x along the field line results in the Lorentz force that shears the expanding arcades. Examining the expression for the x-component of the Lorentz force for this 2.5D system (eq. [18]), we see reason for the shear. The gradient in B x is negative moving up the arcade in the direction of B on one side of the arcade, while on the opposite side the gradient of B x along B is positive. So we have F x ¼ 1 @y ¼ 1 4 :B x = B T : ð18þ Fig. 6. Transport of B x fluxinthesystembytheshearflowsillustratedwithflux distributions plotted as functions of time. The percentage of the B x flux on the lefthand side of the domain is plotted as a function of time with a solid line. The percentage of emerged B x (above z ¼ 0) on the left- and right-hand sides are plotted with dashed and dotted lines, respectively, while total emerged flux is plotted with a dot-dashed line. The plots show the transport of B x flux into the ascending arcades and the final accumulation of that flux to the dominant arcade by the action of shear flows. Force balance in the x-direction requires B x to be constant along field lines. The reason that B x appears as a maximum in the arcades in Figure 3j is that the shear flows have momentum, causing them to overshoot the equilibrium point and temporarily causing an inversion in the strength of B x. Similar behavior was reported in Manchester (2001). This same shearing process is found in more complex geometries such as three-dimensional emerging flux ropes (Fan 2001; Magara & Longcope 2003; Archontis et al. 2004) as shown in Figure 5. Here, the rope (from Manchester et al. 2004) is shown as it expands into the corona from the convection zone. The axial field strength of the rope is shown in color, while the direction of the magnetic field and current density (confined to the plane) are shown with white and black lines, respectively. The deformation of the magnetic field in the stratified atmosphere causes the electric current to be oblique to the magnetic field, producing the Lorentz force that drives the horizontal shear flow. In the buoyancy instabilities shown here, shearing motions transport axial flux from the magnetic layer into the expanding arcades. This transport of flux strongly couples the emerging flux to the subphotospheric magnetic field. To measure this flux transport, we integrate the shear (B x ) flux and plot the values in Figure 6 to show how the distribution changes as a function of time. Here, the percentage of B x flux that has risen into the left and right arcades above the height z ¼ 0 are plotted with dashed and dotted lines, respectively. The sum of these two quantities is plotted with a dot-dashed line, and the B x fluxontheleft-handsideis plotted with a solid line. From these plots it can be seen that a greater amount of flux accumulates in the larger arcade on the right. Given that both the left and right arcades contain the same number of field lines (drawn as contours of constant A), it clearly means that (along these field lines) there is a preferential accumulation of B x flux in the larger arcade by the action of shearing motions. This flux is drawn from the smaller arcade, moves through the layer, and accumulates in the larger arcade. This lateral transport is evident in the plot of the percentage of flux on the left side of the domain, which decreases from 50% to 37% by the end of the simulation. The flux transport is graphically seen in the shape of the field line projected on the x-y plane in Figure 3. As the system evolves to a new equilibrium, B x must equilibrate along field lines by shearing motions. In addition, the constraints of ideal MHD tell us that the magnetic flux of the system is preserved. These two facts together imply that the system evolves in such a way that the magnitude of B x is inversely proportional to the total area between adjacent field lines. This nonlocal evolution of B x by shear flows accumulates flux in the expanding arcade and dynamically couples the emerging flux to the subphotospheric field. What is truly remarkable in this example is that the magnitude of B x continues to decrease in the larger arcade even as the flux accumulates there. Here, B x becomes weaker because of the enormous shear-induced expansion of the arcade even as the flux accumulates there. Thus, there is a very pronounced positive feedback between the expansion of the field and shearing motions, which tends to accumulate all the flux to an expanding arcade where there is a global minimum in B x as is seen in Figure 3l.

7 538 MANCHESTER Vol. 666 Fig. 7. Time evolution of the planar initial state. Two images display the magnetic field, projected on the y-z plane, as black lines superposed on a color representation of the shear angle of the field at times t ¼ 77:4 and The conditions for the planar initial state are identical to the two-dimensional case including the invariants ideal MHD, namely, M(A) andb x (A). The system shows the same phenomena of arcade interaction by shear flows. This next simulation demonstrates the shear-flow coupling in buoyancy instabilities exhibited by a simpler planar initial state. The planar state is equivalent to the two-dimensional state used in the previous simulation in that it possesses the same ideal MHD invariants. Figure 7 shows the system at two times t ¼ 77:4 and Magnetic field lines, projected on the y-z plane, are drawn in black superposed on a false color image of the shear angle. By comparison to Figure 3, we find the only difference between the evolution of the two systems is that the instability of the planar state develops somewhat slower than the undulating state with an e-folding time of 7.0 compared to 4.2. We also note that at t ¼ 77:4, the maximum size of the arcades is smaller than that found in Figure 3. From these results, we infer that the preexisting undulations of the LM00 equilibrium state render the system more susceptible to buoyancy instabilities compared to the planar state. However, what is more remarkable is that the long-term evolution of the system beginning from a planar state exhibits the same pattern of coupling by shear flows. 6. DISCUSSION AND CONCLUSIONS The buildup of magnetic energy in active regions is essential to the onset of CMEs and flares. The magnetic stress must pass from the convection zone into the corona in the form of nonpotential fields and, in doing so, couples the two layers of the atmosphere. Observations show that the formation of such nonpotential fields occurs almost exclusively along magnetic inversion lines where the magnetic field is in a highly sheared state. This magnetic shear is essentialtocmesandflaresasshown by the very strong correlation between photospheric shear flows, flux emergence, and the onset of CMEs and large flares (Meunier & Kosovichev 2003; Yang et al. 2004; Schrijver et al. 2005). These shear flows are found to be strongest along the magnetic inversion line precisely where flares are found (Yang et al. 2004; Deng et al. 2006). There are observed features of magnetic shear in active regions that indicate a strong coupling between the corona and subphotospheric magnetic field. Notable is the persistence of the magnetic shear at the photospheric inversion line during and after flares (Wang et al. 1994; Liu et al. 2005), while higher in the corona, magnetic shear loss is apparent. In instances of two-ribbon flares (Zirin & Tanaka 1981; Zirin 1984), prior to the eruption H arches over the inversion line are highly sheared; afterward, the arches are nearly perpendicular to the inversion line. Su et al. (2006) found a similar pattern of magnetic shear loss in the apparent motion of footpoints in two-ribbon flares. Magnetic shear lost in the corona is replenished from below the photosphere, which is particularly clear in homologous CMEs where the replenishment is rapid, often during flux emergence (Meunier & Kosovichev 2003). These facts strongly suggest that in these circumstances magnetic shear (flux parallel to the neutral line) constantly makes its way through the photosphere to reside in the corona. On a larger scale, evidence for a connection between magnetic shear and the subphotospheric field is found in the axial fields in polar crown filaments, which are inconsistent in sign with the field produced by differential rotation at the photosphere acting on preexisting coronal magnetic fields. (Rust 1967; Leroy 1979; Leroy et al. 1984). Van Ballegooijen & Martens (1990) pointed out that the observed axial field can be produced by differential rotation below the photosphere and then transferred to the corona by a process involving magnetic reconnection and flux cancellation. This observed coupling between the subphotosphere and the corona is readily explained by shear flows driven by the Lorentz force for which the most compelling evidence comes from recent helioseismic observations of active regions. The results of which show strong shear flows with a magnitude of 1 2 km s Mm below the photosphere during flux emergence (Kosovichev & Duvall 2006). Furthermore, the helioseismic observations reveal that subphotospheric shear flows are associated with CMEs and flares as was found in the case of AR 9393 where strong shear flows occurred in the central part of the active region 3 days before a very strong flare (Kosovichev & Duvall 2006). The speed of these subphotospheric flows is consistent with the subsurface shear flow found in our simulation, as shown in Figure 3. We find that the maximum shear velocity is typically half the local Alfvén speed. At the photosphere, the maximum shear velocity is in the range of 3 4 km s 1 and reaches a value of 20 km s 1 in the low corona. From this model we can predict that the shear velocity will continue to decrease in the solar interior as the plasma- increases and the Alfvén speed decreases. The timescale for the shear flows

8 No. 1, 2007 COUPLING BY SHEAR FLOWS 539 to provide coupling is on the order of the length of the system divided by the Alfvén speed. At the photosphere, this amounts to a matter of minutes. At the base of the convection zone, where the density is 0.2 g cm 3, the temperature is 2:3 ; 10 6 K, and the magnetic field strength is G, the timescale would be of the order of a month. Shear flows driven by the Lorentz force in flux emerging in the convection zone may have implications for torsional oscillations, which occur as zonal bands extending to depths of 60 Mm. These flows are faster than the ambient rotation at lower latitudes and slower at higher latitudes by approximately 6 m s 1 (Howard & LaBonte 1980). The positioning of the torsional oscillations corresponds with flux emergence, which suggests a dynamic relationship with the magnetic field (Howe et al. 2000; Ulrich & Boyden 2005). We note here that the overall flow pattern of the torsional oscillations is consistent with a shear flow driven by the Lorentz force in the emerging flux. However, to rigorously demonstrate this physical cause, global simulations must be performed that include both thermodynamic and rotation effects. Shear flow driven by the Lorentz force readily explains many characteristics of active regions outlined above, synthesizes the observations, and gives them meaning in a larger context. The simulations presented here clearly illustrate how shear flows in this physical picture dynamically couple emerging flux and the subphotospheric field. This coupling occurs for two reasons: the transport of axial flux by shear flows and the requirement that the axial field strength be equilibrated along field lines to achieve equilibrium. In this case, shear flows transport axial flux from regions of high strength ( below the photosphere) into regions of low strength (the corona) until the axial field is equilibrated along field lines. This process results in the highly sheared magnetic fields near the polarity inversion line and causes a dynamic coupling in which the magnetized system exhibits a collective behavior that expels axial flux (toroidal flux on a global scale) from the solar interior. A strong feedback may occur when the axial field strength in an expanding arcade continues to weaken even as it accumulates axial flux. When the axial field cannot be equilibrated, it naturally leads to an eruption from a shearing catastrophe, which naturally explains the initiation of CMEs (Manchester 2003; Manchester et al. 2004). Numerous models of CMEs prescribe photospheric shearing motions to magnetic arcades to drive an eruption (e.g., Choe & Lee 1992; Mikic & Linker 1994; Wolfson 1995; Antiochos et al. 1999; Amari et al. 2003). These models have treated the corona apart from the solar interior, which seems a natural assumption given the enormous difference in plasma- in the two regions. However, this simplification misses the transition in pressure that the magnetic field must make, which self-consistently produces shearing motions. Our results also have significant implications for flux rope models of CMEs that rely on the accumulation of azimuthal flux (twist) to drive eruptions. Currently, these models produce an eruption by artificially advecting a flux rope through the bottom boundary of the corona until enough twist accumulates that there is a loss of equilibrium (Fan 2005; Gibson & Fan 2006). In contrast, realistic emergence of a rope through a stratified atmosphere (see Fig. 5) results in strong shear flows that lead to an accumulation of axial flux (rather than azimuthal flux) that drives eruptions such as flares and CMEs (Manchester et al. 2004). There are a number of simplifying assumptions about these simulations that should be mentioned. The simulations have been made for a magnetic layer in close proximity to the solar surface, covering no more than 5 ; 10 3 km in the horizontal direction, which is smaller than the size of a typical active region. Such restrictions in size are necessary to be able to properly resolve the photospheric pressure scale height and make the simulations numerically feasible, which is particularly true in three spatial dimensions. Certainly, the small scale treated in these simulation limits the ability of the emerging flux to produce large-scale eruptions (Manchester 2003; Manchester et al. 2004), but illustrates the process by which larger eruptions such as CMEs occur. In addition, this model uses a simple two-dimensional geometry and does not contain an active convection zone. However, the physical effect of dynamic coupling by shear flows demonstrated in this limited context certainly appears to play a role in the larger more complex magnetic structures of the Sun. Within active regions, there is strong evidence that, through the shearing process outlined here, the subphotospheric magnetic field plays a vital role in the buildup of magnetic shear and free energy necessary for flares and CMEs. In this picture, the subsurface field primarily acts as a reservoir of flux that is transferred to higher layers of the atmosphere by shearing flows driven by the Lorentz force. 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