large-scale solar corona, extending from the coronal base to 15 solar radii. ... inner corona to investigate the energy output when the magnetic field is given as a ...
The Astrophysical Journal, 589:1040–1053, 2003 June 1 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
TWO-DIMENSIONAL MAGNETOHYDRODYNAMIC MODELS OF THE SOLAR CORONA: MASS LOSS FROM THE STREAMER BELT Eirik Endeve1 and Egil Leer1 Institute of Theoretical Astrophysics, P.O. Box 1029 Blindern, Oslo N-0315, Norway
and Thomas E. Holzer2 High Altitude Observatory, NCAR, P.O. Box 3000, Boulder, CO 80307 Received 2002 December 18; accepted 2003 February 14
ABSTRACT The equations of magnetohydrodynamics (MHD) are used to study an axially symmetric model of the large-scale solar corona, extending from the coronal base to 15 solar radii. We use a uniform heating of the inner corona to investigate the energy output when the magnetic field is given as a dipole at the coronal base. The heat input produces a large-scale magnetic field structure similar to that found by Pneuman and Kopp, with coronal holes in the polar regions and a helmet streamer around the equator. We pay special attention to the energy balance in the system, and find that the role of heat conduction is important in determining the thermal structure in magnetically closed regions. Insufficient energy loss to the transition region leads to a high temperature inside the closed region. In the coronal holes a solar wind is accelerated into interplanetary space, and the temperature is lower. As the difference in pressure scale height along open and closed flux tubes is large, the helmet streamer does not relax to a steady state; it opens periodically to eject mass into interplanetary space. These mass ejections may contribute significantly to the mass and energy flux in the solar wind. Subject headings: methods: numerical — MHD — solar wind — Sun: atmosphere — Sun: corona — Sun: magnetic fields The MHD equations have been widely used to study the interaction between ionized coronal gas and the solar magnetic field. Pneuman & Kopp (1971) solved the steady-state MHD equations to construct the first two-dimensional model of the solar corona with a dipole-like magnetic field. They considered an isothermal electron-proton gas and found that the gas escapes along open field lines at high latitudes. At low latitudes the plasma is trapped near the Sun by the closed magnetic field. In their steady-state solution the open and closed regions are separated by a current sheet to balance the pressure difference. Endler (1971) studied a similar problem by using a time-relaxation method to solve the time-dependent MHD equations, and obtained results similar to those of Pneuman & Kopp. Robertson (1983) studied a two-dimensional MHD model in which heat conduction from the inner corona supplies the energy needed to drive the solar wind into interplanetary space, and Steinolfson (1988) considered an adiabatic gas with heating to model coronal mass ejections. A polytropic equation of state has also been used in many MHD models of the solar corona (e.g., Steinolfson et al. 1982; Washimi et al. 1987; Wang et al. 1993; Linker & Mikic 1995; Wu et al. 1995; Keppens & Goedbloed 1999; Usmanov et al. 2000). As the energy flux is proportional to the mass flux in the polytropic model, there is no energy-loss mechanism in the static, magnetically closed regions. Therefore, heating of plasma in closed regions leads to a steady increase in pressure and opening of the magnetic field. In order to model these static regions of the solar corona, one must allow heat conduction to balance the energy input. Suess et al. (1996) introduced heat conduction in a model with coronal heating, and found that the heat conduction is not efficient enough to suppress the opening of field from the top of the helmet streamer. They pointed out that the opening of the closed field may lead to an unsteady streamer structure, and that mass from
1. INTRODUCTION
The dynamics and structure of the solar corona are governed by gas–magnetic field interactions. During periods of low solar activity, the large-scale coronal magnetic field is dominated by a significant dipole component. In this field configuration, magnetic field lines at high latitudes extend from the solar surface into interplanetary space to form coronal holes. The solar wind is accelerated along these open field lines. At lower latitudes around the equator, the plasma and magnetic field form a helmet streamer configuration. The helmet streamers are seen as bright, domeshaped structures that extend 2–3 RS (solar radii) into the corona. It is generally accepted that they are regions of closed magnetic field confining the coronal plasma. It has been suggested that helmet streamers are a source of the slow solar wind (Gosling et al. 1981). In this case, the streamer must open up to interplanetary space in order to release material. ‘‘ Opening ’’ of helmet streamers has been observed by Sheeley et al. (1997). Their observations indicate that the top of the streamer is apparently torn open to release mass into the solar wind. In the extreme case the helmet streamer can be completely disrupted, and a portion of it is ejected into interplanetary space as a coronal mass ejection (CME) (Illing & Hundhausen 1986). To understand the nature and physical structure of these relatively large scale closed-field regions, one must consider the force and energy balance in such a system. The use of numerical experiments, and a detailed analysis of the results, may be a fruitful approach in order to gain further understanding of this physical system. 1
Also at the High Altitude Observatory. The National Center for Atmospheric Research (NCAR) is funded by the National Science Foundation. 2
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then reconnected in the heliospheric current sheet. During this periodic process a significant amount of mass is released into the solar wind. 2. DESCRIPTION OF THE MODEL
To model an axially symmetric solar corona, we apply a one-fluid MHD description of an electron-proton plasma. We assume charge neutrality, ne ¼ np ¼ n, and equal electron and proton velocities, ue ¼ up ¼ u. Here n is the particle density and u is the flow velocity. The subscripts e and p refer to electrons and protons, respectively. The time-dependent equations are then given by: Continuity equation: @ þ @t
D
x ðuÞ
¼0
ð1Þ
Momentum equation: D
@ðuÞ þ @t
x ðuu
D
sÞ ¼ P
GMS er þ J µ B ð2Þ r2
Energy equation: x ðPuÞ
¼ ð 1Þ P xu þ
D
@P þ @t
D
D
xq
Qm Qvisc Qjoule
ð3Þ
Ampe`re’s law: J¼
1 l0
D
µB
ð4Þ
Ohm’s law: E ¼ ðu µ BÞ þ J
ð5Þ
Induction equation: @B ¼ µE; @t D
ð6Þ
where ð¼ mp nÞ; P; q; J; B, and E are the mass density, gas pressure, heat flux density, current density, magnetic field, and electric field, respectively; G, MS , mp, and l0 are the gravitational constant, the solar mass, the proton mass, and the magnetic permeability, r is the radial distance from the center of the Sun, s ¼ 12 ð u þ uT Þ is the viscous stress tensor, and and are the kinematic viscosity and the magnetic resistivity, respectively. Heating terms due to viscous dissipation and joule dissipation are consistently represented in the energy equation by the terms Qvisc and Qjoule , respectively. The heating terms due to dissipative effects are relatively small, and do not contribute significantly to the energy balance in the system. Dissipative effects are primarily included for the stability of the numerical solution to the equations. We are able to solve the equations without explicitly introducing resistivity, but it is included in order to control the reconnection rate of field lines with opposite polarity. The gas pressure is given by the ideal gas law, P ¼ Pe þ Pp ¼ nkðTe þ Tp Þ ¼ 2nkT, where we have defined the average temperature as T ¼ 12 ðTe þ Tp Þ. The heat flux density is defined as q ¼ qe þ qp . In the energy equation we have defined an explicit coronal heating term given by Qm ¼ Qme þ Qmp . In equation (3) the ratio of specific heats, , is set to 5=3. D
D
the streamer belt may leak into the solar wind. In a subsequent study of a two-fluid MHD model, Suess et al. (1999) found results very similar to the results obtained with their one-fluid model. Lionello et al. (2001) constructed a steadystate solution to the MHD equations in a model of the solar corona, where they adjusted the transition region pressure to the amount of heat flux from the corona to include the energy balance in the chromosphere-corona transition region (e.g., Rosner et al. 1978; Withbroe 1988; Hansteen & Leer 1995). In the present study we consider a one-fluid model of an axially symmetric electron-proton corona in a dipole magnetic field. The coronal plasma is heated and we allow for heat conduction along the field to balance the heat input in the closed regions (radiative losses are generally small in the coronal structures we will consider in this study, and are not included in the model). In the coronal holes the solar wind is the important energy sink. Our model is similar to the model considered by Suess et al. (1996, 1999) and Lionello et al. (2001). The essential difference between their studies and our study is that we run our models on long timescales, and we discover results not reported by these authors. The main focus of the study is on the stability of the equatorial streamer belt: in a steady state, the high pressure inside the closed region is balanced by magnetic and gravitational forces, and all the heat input in a magnetically closed region must be lost as heat conductive flux at the inner boundary. This also requires that the temperature maximum is at the top of the closed flux tubes. If the temperature maximum does not coincide with the top of the closed region, outward heat conduction will build up the temperature and pressure there until the outward heat flux is shut off. To study the time evolution of the streamer–coronal hole system, we run our numerical models on relatively long timescales. We do not adjust the coronal base pressure to the inward heat flux; we keep the base pressure uniform in latitude and fixed in time during the simulations. This somewhat limits the application of the model results to the real solar corona. However, we believe that we can gain insight into the force and energy balance of the system, even when we keep the base pressure fixed. As the coronal ions seem to have a higher temperature than the electrons (Kohl et al. 1998), the protons may carry a significant fraction of the heat flux. At equal electron and proton temperatures, the proton conduction is almost a factor of 25 smaller than the electron conduction. The classical expression for the heat flux density is proportional to T 7=2 =LT , where LT is the temperature length scale. Therefore, if the proton temperature is a factor of 2–3 larger than the electron temperature, the heat fluxes are comparable. In our model the effect of preferential proton heating (Tp > Te ) is taken into account by using a heat conductivity in the model that is reduced compared to the conductivity in an electron gas. This leads to a higher temperature inside the closed region. By varying the conductivity, we investigate the effects of both proton and electron heating in the onefluid model, and we find strikingly different results; in models in which the heat flux is carried by the electrons, the corona relaxes to a steady state. For a sufficiently reduced heat flux in the corona the pressure inside the streamer is high, and the system does not relax to a steady state. Periodically, the magnetic field bounding the helmet streamer is dragged into interplanetary space and
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The thermal coupling between electrons and protons is strong (Te ¼ Tp ) in the very inner corona, but the temperatures may be quite different in the extended corona and in the solar wind. In this study we use a one-fluid description, with T ¼ 12 ðTe þ Tp Þ, and must find a reasonable expression for the heat flux. If the coronal electrons are heated, a classical expression for the electron heat flux may be applicable to the solar wind (Lie-Svendsen et al. 1997). If most of the heat is deposited into the protons, the proton heat flux may be more important. As the protons become collisionless somewhere in the corona, the classical expression for the proton heat flux density cannot be used in the outer corona and in the solar wind. We assume that the classical expression for the heat flux may be used for both electrons and protons in the inner corona, and we use a classical Spitzer conductivity out to r ¼ rcoll . Beyond rcoll we use a heat flux density that is proportional to the enthalpy flux (Hollweg 1974), i.e., T 5=2 ð T x bÞb for r rcoll ; q¼ ð7Þ 3nkTu for r > rcoll : D
Here is the heat flux coefficient, and b ¼ B=jBj is the unit vector parallel to the magnetic field. In our models the transition between the two expressions for the heat flux density is smooth over 2 RS, starting at r ¼ rcoll . Typically we set rcoll 5 RS . The parameter is set equal to 1. This corresponds to a polytropic gas with a polytrope index of 4=3. A heat flux proportional to the enthalpy flux, with ¼ 1, results in a somewhat reduced heat flux compared to the classical expression, but in the supersonic solar wind heat conduction is not the dominating energy flux, and our simplified expression for the heat flux in the outer corona does not affect the solar wind mass flux and asymptotic flow speed. This expression is also much more convenient to deal with in a numerical code, and this is the primary reason to introduce it. In the classical expression for the heat conduction the heat flux coefficient, , is larger for the electrons than for the protons by about a factor of 24.4. For electrons and protons the heat conductive coefficients are given by (Braginskii 1965) e ¼ 7:8 1012 J m1 s1 K7=2
ð8Þ
p ¼ 3:2 1013 J m1 s1 K7=2 ;
ð9Þ
and
respectively. In this study we present models in which the value of has been varied between p and e . We want to investigate the effects of heating in open and closed magnetic regions of the solar corona. The specific heating mechanisms present in the solar corona are not known, and to make the problem as simple as possible we use a simplified heating term given by D
Qm ¼
x f m;
ð10Þ
where f m ¼ fm ðrÞer is a radial energy flux density emanating from the lower regions of the solar atmosphere, 2 RS r RS fm ðrÞ ¼ fm0 exp : ð11Þ r Hm Here an energy flux density fm0 enters the corona at the coronal base, r ¼ RS . This energy flux is dissipated in the
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solar corona over a scale height Hm . This corresponds to a heating rate given by Qm ðrÞ ¼
fm ðrÞ : Hm
ð12Þ
Note that we use a heating rate that is uniform in latitude and longitude. It only depends on the heliocentric distance, r. 3. METHOD OF SOLUTION
To solve our model problem we use a MHD code developed by Nordlund & Galsgaard (1995).3 The code is a timeexplicit approach to the solution of the system of nonlinear partial differential equations. The equations are discretized on a staggered grid, on which scalar quantities are located in the center of a grid cell and vector quantities are located on the cell faces or cell edges. Sixth-order approximations are used to calculate spatial derivatives, and a fifth-order interpolation method is used, when needed, to interpolate values between the different locations on a grid cell. For the time integration we use a third-order predictor-corrector method to find the solution at time tnþ1 ¼ tn þ Dt, when the solution for time tn is known. The code conserves mass, momentum, energy, and magnetic flux divergence. Kinematic viscosity and magnetic resistivity are explicitly put into the system of equations to stabilize the numerical scheme. The diffusion is localized in space to effectively smooth out high-frequency oscillations and sharp gradients. Where the smoothing is not needed the diffusion is minimal. To model magnetic reconnection, the resistivity in the vicinity of a thin current sheet is set to ¼ c l0 DxvA ;
ð13Þ
where c is a dimensionless number, Dx is the size of a grid cell, and vA is the Alfve´n speed. The resistive timescale is defined as R ¼ l0 L2 =, where L is a length scale defined by the spatial change in the magnetic field, which may be very small in a thin current sheet. We then find a typical reconnection speed in a current sheet, L Dx ¼ c ð14Þ vA : vR R L In our models the reconnection speed scales with the Alfve´n speed just outside the current sheet. On our grid very thin current sheets are resolved by only a few grid points, so the ratio (Dx=L) is close to 1. We typically set c ¼ 0:1. This will model fast reconnection in a current sheet where the reconnection speed is about 10% of the local Alfve´n speed (cf. Petschek 1964). The equations are discretized on a 129 65 (r, h) grid. The grid is uniformly spaced in the h direction, with D ¼ 2=77. In the radial direction we use a logarithmic grid with the smallest grid spacing in the inner corona, where gradients are steep. In the inner corona we use Dr ¼ 0:01 RS ; Dr increases with heliocentric distance, and for the outermost grid cell Dr is 0:41 RS . The computational domain extends from 1 to 15 RS in r, and from 0 to 180 in 3 MHD code developed by Nordlund & Galsgaard (1995) is available at http://www.astro.ku.dk/kg.
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h. On this computational domain there are only two physical boundaries; the inner boundary, r ¼ 1 RS , and the outer boundary, r ¼ 15 RS . At the poles, ¼ 0 and 180 , we use symmetry conditions to calculate values outside the computational domain. At the inner boundary the flow velocity is calculated from a condition of mass conservation, and we allow the heat flux to flow through the inner boundary. The density and temperature are set a priori, and we do not adjust the coronal base pressure to the amount of heat conducted into the transition region from the corona. (For a realistic treatment of the transition region–corona–solar wind system, one must pay attention to the interaction with the lower regions of the solar atmosphere, and the electron density at the inner boundary must be adjusted to the amount of heat conducted back into the transition region from the corona [Hansteen & Leer 1995]). The magnetic field at the inner boundary is fixed to a dipole during the simulation. At the outer boundary the flow is supersonic, and there we use simple extrapolations to calculate the boundary conditions. The strategy for obtaining a solution to the full system of equations is done in multiple steps. Every new step is initiated by the solution obtained in the previous step. The problem is an initial boundary value problem, and our experience indicates that the computational time needed to obtain a solution to the full set of equations depends on our initial guess. Using an arbitrary coronal configuration as an initial guess to the full set of equations has shown not to be the best strategy; this strategy may even fail to yield any solution at all. Typically, we first solve the equations for a spherically symmetric isothermal solar wind, without a magnetic field, to obtain a steady-state Parker wind solution (Parker 1958). In this step we omit the energy equation, by setting ¼ 1. In the next step we include a dipole magnetic field with a polar field strength B0 at the coronal base. In this step we use the isothermal spherically symmetric solar wind solution obtained in the first step and a dipole magnetic field as our initial condition to integrate the full set of MHD equations for an isothermal atmosphere ( ¼ 1) to a steady state. In the final step we include the full energy equation by increasing the value of from 1 to 5=3 to solve the complete set of equations. (Note that is not the polytrope index. When > 1 all the terms in the energy equation are introduced.) We can then integrate the equations forward in time to study the time evolution of the system. The isothermal solution is still far from the solution we are seeking, and it is our experience that to do the last step in one or two substeps, integrating the equations in time using some intermediate value for , before setting ¼ 5=3, is beneficial; the time step allowed in the explicit code is limited by the stiffness introduced in the energy equation from the thermal conduction term. For < 5=3 a larger time step may be used. All the results presented in this study are for time integrations in which ¼ 5=3. 4. RESULTS
First we present a base model for this study in which a constant energy flux Fm0 ¼ 4 R2S fm0 , with fm0 ¼ 100 W m2 , enters the solar corona at r ¼ 1 RS . The energy flux, for simplicity chosen to be uniform in h, is dissipated in the inner corona over a scale length Hm ¼ 1 RS . The electron density and temperature at the coronal base, also uniform in h, are set equal to n0 ¼ 1 1014 m3 and T0 ¼ 7 105 K, respec-
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tively. (An average energy flux density of fm0 ¼ 100 W m2 at the coronal base is sufficient to drive a solar wind with a proton flux density and a flow speed comparable to that observed at 1 AU [McComas et al. 1995], and supply heat flux to the transition region in order to maintain a base pressure of P0 0:002 Pa [Rosner et al. 1978].) The magnetic field strength at the inner boundary is B0 ¼ 5 104 T at the pole. This corresponds to a plasma ¼ P=ðB2 =2l0 Þ of about 2 102 . At the equator the dipole field strength at the inner boundary is a factor of 2 smaller than at the pole, and the plasma is 8 102 . We set the heat conduction coefficient to ¼ 5p in r < 5 RS . In r > 7 RS the heat flux is proportional to the enthalpy flux, with ¼ 1, and in the interval 5 RS < r < 7 RS we have a smooth transition between the two expressions for the heat flux density. A heat flux coefficient lower than e allows us to use a larger time step in the explicit code, but the main reason for using a reduced heat conductive coefficient is that the coronal protons seem to be heated more than the electrons. As the thermal coupling between electrons and protons is relatively weak, the heat transport in the plasma is less efficient than in an electron gas. (Later in this study we present models with a heat conductive coefficient ¼ e and ¼ p .) Given the energy input in the solar corona, our boundary conditions at the coronal base, and an initial condition for the plasma and magnetic field parameters in the solar atmosphere, we have integrated the equations in time. Our initial state of the solar atmosphere is far from an equilibrium state, and consequently, disturbances will initially propagate through the computational domain at characteristic speeds of the plasma. In coronal models in which the temperature and the electron density are of the order T 1 106 K and n 1 1014 m3 , and a magnetic field strength B 5 104 T, the Alfve´n speed, vA ¼ 1=2 ðB2 =l0 Þ , is of the order of 1000 km s1 . The sound speed, cs ¼ ðP=Þ1=2 , is of the order of 100 km s1 . Therefore, we expect sound waves to propagate through the computational domain (107 km) on a timescale of the order of 30 hr before the solar corona relaxes to a near equilibrium state. In Figures 1a–1c we plot a snapshot of the solution to equations (1)–(6) at t ¼ 30 hr after initiation of the numerical calculation. On this timescale the coronal plasma interacts with the magnetic field to form coronal holes in the polar regions and a magnetically closed region around the equator. This magnetically confined region where the plasma is in hydrostatic equilibrium, referred to as a helmet streamer, extends out to about 3 RS along the equator. In the open regions around the pole the solar wind expands along the magnetic field into interplanetary space. (See Fig. 4 for a plot of the large-scale magnetic field configuration.) The open and closed regions are separated by a thin current sheet bounding the helmet streamer. In the equatorial plane beyond the top of the helmet streamer, where the magnetic field changes polarity, there is a thin current sheet separating the northern and southern hemisphere. This heliospheric current sheet is connected to the top of the helmet streamer in a cusp-type neutral point (e.g., Pneuman & Kopp 1971). The coronal hole, the source of the solar wind, extends down to a colatitude of about ¼ 43 at the inner boundary, r ¼ 1 RS . Thus, the two coronal holes cover about 27% of the solar surface. At the outer boundary, r ¼ 15 RS , the gas from the two coronal hole regions fills the whole heliosphere. Figure 1a plots the radial flow velocity,
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Vol. 589
Fig. 1.—Snapshot of the solution to eqs. (1)–(6) at time t ¼ 30 hr after initiation of numerical calculation. Here an energy flux Fm0 ¼ 4 R2S fm0 is dissipated uniformly into the solar corona over a length scale Hm ¼ 1 RS , and fm0 is set equal to 100 W m2 . We plot (a) the radial flow velocity, (b) the electron density, and (c) the temperature vs. heliocentric distance for the pole (solid line) and the equator (dashed line).
Figure 1b the electron density, and Figure 1c the temperature versus heliocentric distance for the pole (solid line) and the equator (dashed line). Note that along the polar axis the field is radial and values are plotted along a polar field line. Along the equator, inside the helmet streamer, values are plotted across field lines. Beyond the neutral point values are plotted along the heliospheric current sheet. In the polar region the solar atmosphere has relaxed to a steady state after 30 hr. Along the polar axis the solar wind is rapidly accelerated into a supersonic flow. The critical point is located at r ¼ 2:7 RS , where the flow speed is 210 km s1 . At the outer boundary, r ¼ 15 RS , the flow velocity reaches 460 km s1 . The Alfve´n point, where the flow becomes super-Alfve´nic, lies outside the computational domain for colatitudes < 60 . The Alfve´n speed at r ¼ 15 RS is 510 km s1 at the pole. Along the equator the atmosphere has not reached a steady state after 30 hr, and disturbances are still propagating through the computational domain. Inside r ¼ 2:5 RS the plasma is in hydrostatic equilibrium, confined by the magnetic field. Beyond the top of the streamer the flow is gradually accelerated into a supersonic wind. At r ¼ 3 RS the radial flow speed is 14 km s1 . The Alfve´n point is located at r ¼ 5 RS . At the critical point, located at r ¼ 9:5 RS , the flow velocity is about 200 km s1 , and at the outer boundary it is 304 km s1 . As a result of the heating of the inner corona there is a sharp increase in temperature along both the pole and the equator from 7 105 K at the coronal base. Along the open field lines in the polar region the solar wind is the dominating energy loss; only a few percent of the input energy is lost as heat conduction through the inner boundary. The temperature along the polar axis increases to a maximum of 1:75 106 K, at r ¼ 2 RS . In the outer corona, r > 7 RS , the energy equation is given as a polytropic equation of state with a polytrope index of 4=3, and the temperature falls off
roughly as r2=3 . At the outer boundary the temperature along the pole is down to 8:2 105 K. Inside the helmet streamer around the equator the plasma is in hydrostatic equilibrium, and the heat input is only balanced by heat conduction through the inner boundary. This leads to a higher temperature inside the closed region than along open flux tubes. Along the equator the temperature maximum, located at r ¼ 2:2 RS , is 3:4 106 K (i.e., about a factor of 2 higher than the temperature maximum along the polar axis). In the heliospheric current sheet the temperature decreases with heliocentric distance to about 7 105 K at the outer boundary. Because of the relatively large difference in temperature between open and closed regions in the inner corona, the density and pressure scale heights are very different in these two regions. In the extended corona a high gas pressure along the equatorial current sheet must balance the magnetic pressure at higher latitudes (B2 0 along the equatorial current sheet). Inside r ¼ 1:2 RS the temperature profiles are close and the exponentially decreasing electron density profiles are aligned, but beyond r 2 RS the electron density along the equator is about a factor of 10 higher than the electron density along the pole. In the outer corona the electron density profiles fall off nearly as r2 . The proton flux density mapped to the orbit of earth, ðnuÞE , is about 2:2 1012 m2 s1 in the polar region. At the equator the proton flux density at 1 AU is about a factor of 7 larger, ðnuÞE ¼ 1:5 1013 m2 s1 . After 30 hr the streamer–coronal hole system seems to approach a steady state; the temporal changes in the corona are relatively small compared to what they are during the initial relaxation phase. When the coronal system reaches an equilibrium state, there is force and energy balance everywhere in the system; the magnetic field configuration is consistent with the thermodynamic structure of the coronal plasma. The high gas pressure on closed field lines in the
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inner corona is balanced by gravitational and magnetic forces, and in the outer corona, beyond the neutral point, a high gas pressure along the equatorial current sheet balances the magnetic pressure at higher latitudes. The temperature maximum must be located at the top of the closed flux tubes. The energy dissipated inside the closed region is balanced by thermal conduction, and along the open field the input energy is balanced by both the solar wind and downward conduction. If the gas pressure on closed field lines is some what too large, the top of the streamer will expand and open up some closed magnetic field, and if the amount of open magnetic flux is too large, some field lines will reconnect in the equatorial current sheet in order to settle to the equilibrium. In the state reached after 30 hr the temperature maximum is inside the top of the closed region. This results in a heat-conductive energy transport along the magnetic field to the inner boundary and to the top of the streamer. At the top of the streamer there is no explicit energy sink. Therefore, the heating will cause the thermal pressure to build up with time. The pile-up of thermal energy at the top of the magnetically closed region builds up the pressure until the magnetic forces are unable to withstand the pressure gradient force. Magnetic field lines confining the plasma at top of the streamer are then transported into interplanetary space along the heliospheric current sheet. As the closed magnetic field lines, anchored to the solar surface on each hemisphere, are transported along the equator the expansion leads to a decrease in plasma pressure along the field. This causes field lines with opposite polarity to move together and reconnect in the equatorial current sheet, seeking to re-form the streamer consistent with a reduced pressure. When the equations are integrated on a long timescale (140 hr), we find that an out-of-equilibrium state persists, in which massive plasmoids are repeatedly ejected into the solar wind. In Figure 2 we plot the solar mass-loss rate versus time for three heliocentric distances r ¼ 1:05, 3, and _ Þ, is given by 15 RS. Here the solar mass-loss rate, ðM I Z _ Þ¼ ðM u x dS ¼ 2 ur r2 sin d ; ð15Þ S
0
where S is a spherical shell with radius r. From Figure 2 (solid line) we see how the solar mass-loss rate at the outer boundary, r ¼ 15 RS , evolves with time.
Fig. 2.—Solar mass-loss rate vs. time for three heliocentric distances (r ¼ 1:05, 3, and 15 RS).
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For t < 30 hr the mass-loss rate varies between 1:25 109 _ Þ¼ and 1:9 109 kg s1 . For t ¼ 30 hr we find ðM 1:5 109 kg s1 . For t > 30 hr, field lines at the top of the streamer expand and reconnect, and the solar mass-loss rate varies periodically with time. In the interval 30 hr < t < 70 hr the size of the peaks in the solar mass-loss rate increases. _ Þ has a local maximum of about For t ¼ 36 hr ðM 1:9 109 kg s1 . For t ¼ 68 hr the maximum in the massloss rate profile has increased to 3 109 kg s1 . For t > 70 _ Þ has sharp peak maxima. Consecutive peaks have hr ðM more or less the same shape. The period is about 14 hr, and the mass loss varies between 1:1 109 and about 4 109 kg s1 . There is a slight increase in the peak values for t > 70 hr. We have run the model for t > 140 hr and find that the peak values do not increase. Closer to the Sun the variations in mass flux are smaller. For r ¼ 3 RS (Fig. 2, dashed line) small periodic variations begin to appear for t > 35 hr. For t > 70 hr the mass-loss rate varies periodically between 1:4 109 and 1:7 109 kg s1 . Near the solar surface, at r ¼ 1:05 RS , the variations in the solar mass-loss rate are less than 10%. Note also that at r ¼ 3 RS the time variation in the solar mass-loss rate is smooth, whereas at the outer boundary, r ¼ 15 RS , the variations appear as sharp peaks. These periodic variations are due to mass eruptions from the top of the streamer into the equatorial plane. In our axially symmetric models a torus of mass is ejected into interplanetary space every 14 hr or so. Figure 2 shows that the eruptions result in a significant contribution to the total mass flux in the solar wind. About 15% of the total solar mass loss is concentrated in the sharp peaks. The periodic mass ejections from the top of the streamer are localized to the equatorial region. In Figure 3 we plot the flow velocity (Fig. 3a) and the proton flux density (Fig. 3b) at heliocentric distance r ¼ 15 RS versus time for three colatitudes, ¼ 0 , 70 , and 90 . In the polar region, ¼ 0 , the solar atmosphere reaches a steady state after about 30 hr. For t > 30 hr the flow velocity and proton flux density are constant in time, about 460 km s1 and 4:5 1014 m2 s1 , respectively. [Scaled to 1 AU, the proton flux density in the polar region is ðnuÞE ¼ 2:2 1012 m2 s1 .] At lower latitudes,
Fig. 3.—(a) Radial flow velocity at r ¼ 15 RS vs. time. (b) Proton flux density, nur , at r ¼ 15 RS vs. time. In both panels we plot values for ¼ 0 (solid line), ¼ 70 (dotted line), and ¼ 90 (dashed line).
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closer to the equatorial current sheet, periodic variations appear in both the flow velocity and the proton flux density. For ¼ 70 (Fig. 3, dotted lines) the periodic variations are present for t > 50 hr. The period is about 14 hr. The flow velocity at ¼ 70 varies between 380 and 450 km s1 at the outer boundary. The proton flux density varies between 4:7 1014 and 6:8 1014 m2 s1 . At the equator (Fig. 3, dashed lines) the periodic variations are visible for t > 30 hr. The flow velocity at the outer boundary, at ¼ 90 , varies between 265 and 425 km s1 , but the variations in the proton flux density are large. During one period of about 14 hr the proton flux density at the outer boundary varies from about 6:0 1014 to about 2:8 1016 m2 s1 , i.e., almost 2 orders of magnitude. The energy dissipated into the magnetically open regions of the solar corona is either conducted back into the transition region, or used to drive the supersonic solar wind. In the closed region around the equator the plasma is in hydrostatic equilibrium. The heating in this region leads to a higher temperature than along the polar axis. The energy dissipated into the helmet streamer is conducted back into the transition region, or it is conducted to the top of the streamer where it is lost in the mass eruptions into the solar wind. In a steady state we can write the integrated energy equation for our model problem as I 1 2 5 GMS u þ P u x dS þ Fq þ Fm ¼ const ¼ FE ; 2 r S 2 ð16Þ where S is the spherical shell with radius r. Here the heatconductive energy flux, Fq , and the input energy flux, Fm , are given as I Fq ¼ q x dS ð17Þ S
and I Fm ¼
f m x dS :
ð18Þ
S
The solar wind energy flux is I 1 2 5 GMS u þ P u x dS : Fsw ¼ 2 r S 2
ð19Þ
At the inner boundary, r ¼ 1 RS , the energy flux is FE0 ¼ Fsw0 þ Fq0 þ Fm0 ;
ð20Þ
where Fsw0 ¼
GM _ Þ: ðM RS
ð21Þ
At the outer boundary the energy flux is FE1 ¼ Fsw1 þ Fq1 Fsw1 ; where the solar wind energy flux is I 1 2 x Fsw1 ¼ 2 u u dS :
ð22Þ
ð23Þ
S1
In a steady state, where FE0 ¼ FE1 , the energy added to the solar corona is either conducted back into the transition
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region or it is lost as solar wind energy flux, Fm0 ¼ Fq0 þ ðFsw1 Fsw0 Þ :
ð24Þ
It should be pointed out that our system does not relax to a steady state, but when averaged over times that are large compared to the period between two consecutive eruptions, equation (24) is a good approximation. In our base model, where fm0 ¼ 100 W m2 , the total energy flux flowing into the solar corona at the inner boundary is Fm0 ¼ 4 R2S fm0 6:1 1020 W. When this energy flux is dissipated uniformly over a scale height Hm ¼ 1 RS , we find that 78% is used to drive the supersonic solar wind and the mass ejections from the top of the streamer. About 22% of the input energy flux is conducted back into the transition region. About 95% of the heat flux through the inner boundary emanates from the closed region, and this heat flux varies by less than 5% during the streamer eruptions. 4.1. Evolution of One Eruptive Event We have now described the time evolution of the coronal system on a timescale of about 140 hr. This timescale is relatively large compared to the initial relaxation phase of the system. Some 30 hr after the initiation of the numerical calculation, the heating of the corona leads to energy release from the top of the streamer in the form of a mass eruption. The magnetic field lines, confining the plasma at the top of the streamer, are forced to expand because of the high pressure produced by the heating of the helmet streamer. About every 14 hr a plasmoid is ejected into interplanetary space, and the field lines reconnect to re-form the original streamer in the corona. The eruptions result in a significant contribution to the total solar mass-loss rate. We now turn our attention to one of these eruptive events. Figures 4 and 5 are schematic plots of one such event, starting at an arbitrarily chosen time t0 ¼ 85 hr after the initiation of the numerical calculation. In Figure 4 we plot the magnetic field configuration and the relative electron density for four consecutive times: t0 , t0 þ 4:0 hr, t0 þ 7:5 hr, and t0 þ 10:0 hr. Here we have defined the relative electron density as the ratio n=ns , where ns is the electron density in a static, spherically symmetric, isothermal corona at a temperature T ¼ 1 106 K and an electron density n0 ¼ 1 1014 m3 at the inner boundary. In Figure 4 the brighter areas correspond to a higher relative electron density. The large-scale magnetic field configuration shown in Figure 4 is, for all consecutive time steps, qualitatively unchanged during the eruption; at higher latitudes the magnetic field is open to interplanetary space, and around the equator in the inner corona the magnetic field lines are closed. Only in the equatorial region, beyond a heliocentric distance r 3 RS , is the gas–magnetic field interaction dynamic. Inside r ¼ 2:5 RS the coronal plasma is in hydrostatic equilibrium. For r > 2:5 RS some of the closed magnetic field is pulled out into interplanetary space, releasing mass into the solar wind. This region of activity extends from a colatitude 70 in the northern hemisphere to 110 in the southern hemisphere. In Figure 4 (upper left panel) we plot the initial phase, t ¼ t0 , of the streamer eruption. At this stage a plasmoid containing dense coronal plasma is confined by magnetic field lines connected to the solar surface. The relative electron density is largest at the center of the plasmoid, which is located at about r ¼ 5 RS in the equatorial plane. The
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Fig. 4.—Time evolution of one eruptive event starting at time t0 ¼ 85 hr after initiation of numerical calculation. In each panel we plot the magnetic field configuration and the relative electron density, n=ns . Brighter areas correspond to a higher relative electron density.
pressure gradient force exceeds the gravitational force and the J µ B force, and the plasmoid expands in the radial direction. As a result of the expansion, the gas pressure along the magnetic field starts to decrease. For t ¼ t0 þ 4:0 hr (Fig. 4, upper right panel) the plasmoid has moved farther out in the solar corona. The center of the plasmoid is now located at about r ¼ 7 RS . The plasma pressure behind the eruption is low, and as the plasmoid moves in the radial direction the plasma and magnetic field with opposite polarity move into the current sheet behind the eruption. In the equatorial current sheet the field lines connected to the solar surface in the northern hemisphere and field lines connected to the solar surface in the southern hemisphere reconnect behind the center of the plasmoid. The reconnection process breaks up the equatorial current sheet, and the plasmoid is disconnected from the solar surface and forms a magnetic ‘‘ island ’’ in the equatorial region. Note the magnetic field enclosing the center of the plasmoid. In the next time step, t ¼ t0 þ 7:5 hr, magnetic field lines with opposite polarity continue to reconnect in the equatorial current sheet, behind the outward-moving plasmoid. The reconnection process reproduces the closed field, with a relatively low gas pressure, on the resistive timescale, R . At the X-shaped reconnection point, r ¼ 7 RS , R is of the order of 104 s, and the speed of a field line into the equatorial current sheet is about 10% of the local Alfve´n speed. The center of the plasmoid is located at r ¼ 9:5 RS . On the solar side of the reconnection point the magnetic field connected to the solar surface
evolves toward re-formation of the helmet streamer configuration. Outside the reconnection point the magnetic field disconnected from the solar surface moves with the plasmoid in the radial direction into the solar wind. In the last panel in Figure 4, t ¼ t0 þ 10:0 hr, the erupted material moves freely with the solar wind toward the outer boundary of the computational domain. The center of the plasmoid is located at r ¼ 13:5 RS . A new streamer eruption is about to form in the corona. The center of the new plasmoid is located at r ¼ 4 RS . The dynamic behavior, in which plasmoids are periodically ejected into the solar wind, is the result of the streamer–coronal hole system seeking to reach an equilibrium state. The thermal structure along magnetic field lines is very different across the streamer–coronal hole boundary, and during the expansion/reconnection process the plasma along the streamer boundary adjusts between the two states. The intrinsic inertia of the system leads to an oscillation around the equilibrium. As the outermost field lines bounding the helmet streamer expand into the equatorial current sheet, and the pressure along this field decreases, the weight on the lower lying closed field is also reduced. This leads to expansion of the lower lying field lines as well, and to an expansion of too many closed field lines compared to the equilibrium. As a flow is set up along these expanding field lines and mass is released into the solar wind, the pressure along the field adjusts to a low pressure consistent with an open field. As the pressure exerted on the field is reduced,
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Fig. 5.—Time evolution of the same eruptive event as in Fig. 4. In the left column we plot radial flow velocity vs. heliocentric distance. In the right column we plot the electron density vs. heliocentric distance. In all panels we plot three colatitudes ¼ 0 (solid line), ¼ 70 (dotted line), and ¼ 90 (dashed line).
the field lines start to reconnect in the equatorial current sheet. Immediately after a field line is closed by reconnection, the pressure begins to adjust to closed-field conditions on timescales determined by the advective term and the heating term in the energy equation. In the expanding part of the helmet streamer the advective timescale, a ¼ jP=ðu x PÞj, and the heating timescale, m ¼ j3P=2Qm j, are of comparable size, and much larger than the conductive timescale, c ¼ j3P=ð2 x qÞj. (At r ¼ 3 RS we find that a and m are of the order of 104 s, whereas c is of the order of 102 s.) The reconnection process continues in the equatorial current sheet until the amount of closed flux is consistent with the increasing pressure inside the closed region. When the reconnection process stops, the amount of closed field exceeds the equilibrium value because of the slowness of the pressure increase along a newly formed closed flux tube. The top of the streamer is then pulled outward again to repeat the process. In Figure 5 we plot the same eruptive event as in Figure 4. In the left column we plot the radial flow velocity versus heliocentric distance, and in the right column we plot the electron density versus heliocentric distance. The results are plotted for colatitudes ¼ 0 (solid line), 70 (dotted line), and 90 (dashed line). During the eruption from the helmet streamer, the flow velocity and the electron density along the pole are unaffected by the dynamic behavior along the equator. Thus, the flow velocity profile and the electron density profile shown in Figure 5 are the same as the profiles for the flow velocity and the electron density in Figure 1 (solid
lines in Figs. 1a and 1b, respectively). At a lower latitude, ¼ 70 , the atmosphere is only slightly affected by the eruption in the equatorial plane. The variations are barely visible in the flow velocity profile and the electron density profile in Figure 5. The solar corona is in hydrostatic equilibrium out to r ¼ 1:5 RS . Beyond this point the radial flow speed increases with heliocentric distance. The critical point is located at r ¼ 3:5 RS , where the flow speed is about 220 km s1 . At the outer boundary the flow speed varies between 380 and 450 km s1 during the eruption. The electron density profile for ¼ 70 follows the electron density profile along ¼ 90 out to r ¼ 1:5 RS . For r > 3 RS the electron density profile follows the electron density profile along the polar axis, and falls off roughly as r2 in the outer corona. Along the equator, ¼ 90 , the eruption is clearly visible. For all consecutive time steps, shown in Figure 5, the atmosphere is in hydrostatic equilibrium inside r ¼ 2:5 RS . For r ¼ 3 RS the radial flow velocity is around 10 km s1 . The electron density in this static region falls off from 1 1014 m3 at the inner boundary to about 8 1011 m3 at r ¼ 3 RS . Beyond r ¼ 3 RS the flow velocity profile and the electron density profile are very time dependent. For t ¼ t0 the radial flow speed along the equator increases monotonically with heliocentric distance for r > 3 RS , and reaches 330 km s1 at the outer boundary. The electron density profile decreases monotonically with heliocentric distance. In the next time plot, t ¼ t0 þ 4:0 hr, the magnetic reconnection has set in. In the region 3 RS < r < 4 RS the magnetic tension force pulls the plasma toward the Sun, and for
D
D
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r ¼ 3:5 RS the radial flow speed is about 9 km s1 . For r > 4:5 RS the flow is outward. At r ¼ 6 RS the flow speed has a local maximum of about 180 km s1 . A maximum in the electron density profile has formed at r 7 RS . This is the center of the erupting plasmoid. The electron density at r ¼ 7 RS is 1:2 1011 m3 . For t ¼ t0 þ 7:5 hr, the plasmoid is disconnected from the solar surface. In the inner corona, around r 5 RS , the plasma is flowing toward the Sun with a flow velocity of about 10 km s1 . For r > 5:5 RS the radial flow speed is positive. The flow velocity profile has a maximum of 310 km s1 at r ¼ 8:5 RS . The electron density profile for t ¼ t0 þ 7:5 hr decreases with heliocentric distance out to r ¼ 8 RS . For r ¼ 9:5 RS the electron density has a local maximum of 1:2 1011 m3 . For r > 9:5 RS the electron density decreases with heliocentric distance. For t ¼ t0 þ 10 hr the plasmoid is approaching the outer boundary of the computational domain. The radial flow velocity along the equator is positive throughout the computational domain. At r 11:5 RS , the flow velocity has a maximum of 380 km s1 . The local maximum in the electron density, n ¼ 1 1011 m3 , has moved out to r 13:5 RS . During the eruption from the top of the streamer the plasmoid is accelerated into the solar wind. As it moves outward in the radial direction the flow velocity behind the plasmoid is larger than at the center. The mass of the plasmoid leaving the outer boundary of the computational domain is approximately 1013 kg. This is the mass of the entire torus in the axially symmetric model. It should also be mentioned that during one eruption the size of the polar coronal hole at the
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inner boundary increases from a minimum colatitude of 39 to a maximum colatitude of 45 , i.e., during one eruption a magnetic flux tube with an angular size of D 6 at the inner boundary expands into interplanetary space. It is mass from this flux tube that is released into the solar wind as plasmoids. 4.2. Varying the Heat Conductive Coefficient, In this one-fluid description of the solar corona we have not made the assumption of equal electron and proton temperatures; we have simply defined the temperature to be the average of the two. If the protons are heated, rather than the electrons, most of the heat-conductive energy flux may be carried by the protons. In the model described above we have, quite arbitrarily, used a heat flux coefficient ¼ 5p . We now vary the heat conductive coefficient from ¼ p to ¼ e to see how this affects the behavior of the streamer– coronal hole system. In all these models the input energy to the solar corona is the same as for the base model: an energy flux Fm0 ¼ 4 R2S fm0 , with fm0 ¼ 100 W m2 , is dissipated uniformly into the corona over a scale height Hm ¼ 1 RS . The electron density and the temperature at the inner boundary are set to 1 1014 m3 and 7 105 K, respectively. At the inner boundary the dipole magnetic field strength is B0 ¼ 5 104 T at the pole. In Figure 6 we plot solutions to the MHD equations 30 hr after initiation of the numerical calculation. The heat conductive coefficient has been given the values ¼ p , ¼ 5p , and ¼ e . We plot, from left to right, the radial
Fig. 6.—Snapshot of solution to eqs. (1)–(6) at time t ¼ 30 hr for the models in which the heat flux coefficient has been varied from ¼ p to ¼ e . We plot, from left to right, flow velocity, electron density and temperature along the pole (solid lines) and the equator (dashed lines). In the top row we plot results for ¼ p , in the middle row we plot results for ¼ 5p , and in the bottom row we plot results for a heat flux coefficient set to ¼ e .
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flow velocity, the electron density, and the temperature versus heliocentric distance. Solid lines show values along the polar axis, ¼ 0 . Dashed lines show values along the equator, ¼ 90 . In the top row, Figures 6a–6c, we plot results for ¼ p . The middle row, Figures 6d–6f, shows results for ¼ 5p (this is, the base model presented in Fig. 1; included for easy reference). Finally, in the bottom row, Figures 6g– 6h, we plot results for ¼ e . After 30 hr the solutions for both models, ¼ p and ¼ e , are qualitatively similar to the base model; in the inner corona, around the equator, the atmosphere is in hydrostatic equilibrium. At higher latitudes the solar wind is accelerated along the open field lines. Inside the helmet streamer the temperature is high, and the input energy is balanced by heat conduction to the coronal base. Along open field lines most of the input energy is lost in the solar wind, and the temperature is lower. The pressure scale height inside the helmet streamer is larger than in the coronal holes. Along the equatorial current sheet the gas pressure balances the magnetic pressure at higher latitudes. For the base model, in which ¼ 5p , we found that about 22% of the input energy flux to the solar corona, Fm0 , is carried as heat flux through the inner boundary from the corona to the transition region. The remaining energy flux, 78%, is used to drive the supersonic solar wind and the mass ejections from the top of the streamer. In the models in which we vary we find that the conductive energy loss to the inner boundary varies significantly. In the model with ¼ e , around 33% is lost as inward heat flux, and in the model with ¼ p only about 7% is lost as heat flux into the transition region. When the heat conductive coefficient is reduced the coronal temperature increases. Inside the helmet streamer, where only heat conduction balances the input energy, the temperature increase is larger than in the coronal holes. Along open field lines an increase in temperature also leads to an increase in the solar wind energy flux. For the model in which ¼ p the temperature profile along the polar axis increases from 7 105 K at the inner boundary to a maximum of 2:3 106 K at r ¼ 2:2 RS . (This is an increase in the maximum temperature of about 31% compared to the base model.) Along the equator the temperature maximum is located at about r ¼ 2:0 RS , where the temperature is 4:85 106 K (i.e., an increase of about 42% compared to the base model). The temperature profiles decrease monotonically with heliocentric distance in the outer corona. The electron density profiles decrease exponentially with heliocentric distance in the inner corona, and in the outer corona they fall off nearly as r2 . Only in the very inner corona, where the temperature profiles are close, are the two electron density profiles aligned. Beyond r 2 RS the electron density along the equator is about a factor 10 higher than the electron density along the pole. The flow speed along the polar axis increases rapidly with heliocentric distance. The critical point is located at r ¼ 2:3 RS . At the critical point the flow velocity is about 250 km s1 , and at the outer boundary the flow velocity is 575 km s1 . For this model the flow is sub-Alfve´nic inside the computational domain for colatitudes < 50 . Along the equator the atmosphere is in hydrostatic equilibrium out to about r ¼ 2:2 RS . The Alfve´n point is located at r ¼ 4:5 RS . The critical point is located at r ¼ 8 RS , where the radial flow speed is 250 km s1 . At the outer boundary the solar wind speed is 360 km s1 . The proton flux density, scaled to 1 AU, is about 2 1012 m2 s1 in
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the polar region, and in the equatorial plane the proton flux density mapped to the orbit of earth is ðnuÞE ¼ 1 1013 m2 s1 . When the coronal temperature increases, the critical point moves closer to the coronal base, and a larger fraction of the solar wind energy flux leaves the outer boundary as kinetic energy rather than potential energy (Leer & Holzer 1980). In the model in which we increase the heat conductive coefficient by almost a factor of 5, from ¼ 5p to ¼ e , the temperature in the solar corona decreases. Along the polar axis the temperature increases from the coronal base to a maximum of 1:55 106 K at r ¼ 2:5 RS (a decrease of about 12% compared to the base model). Beyond the temperature maximum the temperature is fairly constant out to about 7 RS . For r > 7 RS the temperature falls off as r2=3 . Along the equator, ¼ 90 , the coronal temperature is larger. The temperature increases from the coronal base to a maximum of about 2:5 106 K, at r ¼ 3 RS . (Compared to the base model the temperature maximum inside the streamer is reduced by 26%). In the outer corona the temperature profile along the equator follows the temperature profile along the pole. The electron density profiles decrease rapidly with increasing heliocentric distance in the inner corona. They fall off roughly as r2 in the outer corona. A smaller difference in the temperatures along open and closed flux tubes leads to a smaller difference in pressure scale heights. Beyond r 2 RS the electron density along the equator stays about a factor of 5 higher than the electron density along the polar axis. Along the polar axis the solar wind is accelerated into a supersonic flow from the coronal base. At the critical point, located at r ¼ 3 RS , the radial flow velocity is about 200 km s1 , and at the outer boundary the solar wind speed is 440 km s1 . In this model we find that the flow is sub-Alfve´nic inside r ¼ 15 RS down to a colatitude of about 30 . The atmosphere is in hydrostatic equilibrium out to about r ¼ 3 RS along the equator. At r ¼ 4:5 the flow becomes super-Alfve´nic. The critical point is located at r ¼ 8:5 RS , where the flow velocity is 200 km s1 . At the outer boundary the solar wind speed is 290 km s1 . In the polar region the proton flux density scaled to the orbit of earth is ðnuÞE ¼ 3:6 1011 m2 s1 . The proton flux density in the equatorial region, scaled to 1 AU, is about 5:6 1012 m2 s1 . In this model a smaller fraction of the input energy flux is lost in the solar wind, and compared to the base model both the solar wind speed at r ¼ 15 RS and the proton flux at 1 AU are reduced. The lower coronal temperature in the model with ¼ e is consistent with a larger closed field region around the equator, and consequently smaller coronal holes. In the model with ¼ e the coronal hole at the inner boundary extends down to a colatitude 40 in the northern hemisphere. For the model with ¼ p the coronal hole extends down to 45 . In Figure 2 we presented results for the model with ¼ 5p , run on a timescale of more than 100 hr. We have also run the models with heat conductive coefficient ¼ p and ¼ e on long timescales. In Figure 7 we plot the solar _ Þ, at heliocentric distance r ¼ 15 RS mass-loss rate, ðM versus time for the three models presented in Figure 6. Figure 6a is the mass-loss rate for the model with ¼ p , Figure 6b is for the base model with ¼ 5p (same profile as solid line in Fig. 2, but included for easy reference), and in Figure 6c we plot the solar mass-loss rate for the model with ¼ e .
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Fig. 7.—Solar mass-loss rate vs. time for the models in which the heat flux coefficient, , has been varied: (a) ¼ p , (b) ¼ 5p , and (c) ¼ e .
In the model in which is reduced to ¼ p , the time evolution of the streamer–coronal hole system is similar to the base model. For t < 30 hr the corona relaxes to a steady state in the polar region. During this relaxation phase the solar mass-loss rate at r ¼ 15 RS varies between 1:15 109 _ Þ¼ and 1:5 109 kg s1 . For t ¼ 30 hr we find ðM 1:4 109 kg s1 . For t > 30 hr sharp peaks in the solar mass-loss rate (also found in the base model) start to appear. Then the solar mass-loss rate varies periodically between 9:5 108 and 3:5 109 kg s1 . The period is about 11 hr, and the mass in each eruption is approximately 8 1012 kg. (For the base model we found a period between the peaks in the solar mass loss of about 14 hr and a mass of 1013 kg in each eruption.) In the case in which we set the heat conductive coefficient to ¼ e , Figure 7c, the massive eruptions from the top of the streamer are not present. In the initial relaxation phase the solar mass-loss rate increases from 8:5 108 kg s1 for t ¼ 0 hr to about 1:05 109 kg s1 for t ¼ 30 hr, but for t > 30 hr the solar mass-loss rate is approximately constant. There are some small ( 60 hr the variations have died out, the system has reached an equilibrium state, and the solar mass-loss rate is constant in time within 1%. In this model the difference in thermal structure along open and closed flux tubes is smaller than for the models in which is reduced. This means that in the case of expansion (reconnection) of the magnetic field bounding the helmet streamer (present in the models with a smaller value for ) the plasma along the streamer–coronal hole boundary is able to adjust to open (closed) field conditions in a shorter time, and the system reaches an equilibrium state in which the thermodynamic structure of the coronal plasma is consistent with the magnetic field structure. 4.3. Varying the Energy Input, fm0 , the Dipole Field, B0 , and the Resistivity, In the previous section we found that when electrons carry the heat flux the streamer eruptions are not present; after an initial relaxation phase of about 30 hr the system relaxes to a steady state in which there is force and energy balance everywhere in the corona. We now briefly describe
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results from models in which we have varied the input energy flux, the dipole field strength at the coronal base, and the magnetic resistivity. We take the base model with fm0 ¼ 100 W m2 , ¼ 5p , B0 ¼ 5 104 T, and c ¼ 0:1 (cf. eq. [13]) as our reference model and vary one parameter at a time. The boundary conditions at the coronal base are unchanged: the electron density and the temperature at the inner boundary are set to n0 ¼ 1 1014 m3 and T0 ¼ 7 105 K, respectively. First we present some models in which we have varied the energy input to the solar corona. We have studied one model in which we reduce the input energy flux to fm0 ¼ 50 W m2 , and one model in which we set fm0 ¼ 150 W m2 . In both models the input energy flux is dissipated over a scale height Hm ¼ 1 RS . The results from these model calculations are qualitatively the same as for the base model with fm0 ¼ 100 W m2 . However, a larger input energy flux results in a higher coronal temperature and a somewhat smaller closed magnetic field region around the equator. After an initial relaxation phase of about 30 hr, the polar regions have reached a steady state. In these models we find that the fraction of the input energy flux lost as heat conduction through the inner boundary, and the fraction used to drive the solar wind and the eruptions from the streamer are about the same as for the base model. After the initial relaxation phase the nonequilibrium phase sets in, in which plasmoids are periodically ejected into the solar wind from the top of the streamer. The mass and energy contained in each eruption are about the same as for the base model, but the period between the eruptions changes. For the base model we found a mass of 1013 kg in each eruption and a period of 14 hr. For the model with fm0 ¼ 50 W m2 the period between two consecutive eruptions increases to 21 hr, and the mass in each eruption is reduced to about 7 1012 kg. For the model with fm0 ¼ 150 W m2 the period is about 9 hr and the mass in each eruption is about 1013 kg. The increase in period is consistent with the increase in heating rate; as the expanding field is closed by reconnection, a stronger heating of the coronal plasma reduces the time needed to adjust to a high pressure consistent with a closed magnetic field. The base model has also been run with a weaker and a stronger magnetic field strength at the coronal base. We have calculated one model with B0 ¼ 3:5 104 T at the pole, and one model in which the dipole field strength at the pole is B0 ¼ 7 104 T. For the model with a weaker field the plasma at the coronal base is increased by a factor of 2, and for the model with a stronger field the plasma is reduced by a factor of 2. The results from these calculations are qualitatively similar to the base model, but a stronger field results in a somewhat larger closed field region around the equator (and smaller polar coronal holes). The larger closed field region around the equator leads to a slight increase (only a few percent) in the amount of energy lost as heat conduction through the coronal base. When the nonequilibrium state sets in after about 30 hr the period between eruptions is unchanged for both models (14 hr). In the model with a weaker magnetic field the mass contained in each eruption is about the same as for the base model (1013 kg), but for the model with a stronger magnetic field the mass contained in each plasmoid is somewhat reduced (5 1012 kg). Finally, we have varied the resistivity in the model. A smaller value for the magnetic resistivity leads to a slower
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reconnection rate in the equatorial current sheet. To investigate the effect of a slower reconnection rate in the equatorial current sheet, we have run the base model with a lower resistivity, and we find similar results. When we reduce the resistivity the period between the eruptions increases, and the mass contained in each eruption is nearly unchanged. When c is reduced by a factor 10, to 0.01, we find that 22% of the input energy flux is carried as heat flux through the inner boundary (same as for the base model). The rest is used to drive the solar wind and eruptions from the streamer. The period between the eruptions from the helmet streamer increases to about 20 hr, and the mass contained in each eruption increases by 20%, to about 1:2 1013 kg. When we slow down the reconnection rate in the equatorial current sheet, the plasma along the expanding part of the helmet streamer reaches a lower pressure before the field lines reconnect and close in the equatorial current sheet. When the magnetic field is closed by reconnection, the adjustment to a high gas pressure consistent with closed field conditions (determined by the advective timescale, a , and the heating timescale, m ) takes a longer time, and this leads to an increase in the period between the eruptions. 5. DISCUSSION AND CONCLUSION
This study shows that a constant, uniform heating of the inner solar corona with a dipole-like magnetic field and uniform boundary conditions at the coronal base produces an atmosphere with coronal holes in the polar regions and a magnetically closed region around the equator. This quasiequilibrium state is produced after integrating the MHD equations some 30 hr from an initial condition. From the coronal holes a steady solar wind is accelerated into interplanetary space, and in the closed region the atmosphere is in hydrostatic equilibrium. The gas pressure along the open field lines is lower than along the closed magnetic field. When the equations are integrated on a long timescale (>100 hr) the heating of the closed region leads to massive eruptions from the top of the helmet streamer for models with < e . The outermost part of the helmet streamer periodically expands into interplanetary space, releasing plasmoids with a mass of about 1013 kg into the solar wind. Behind the eruption the magnetic field reconnects, seeking to re-form the original helmet streamer configuration, but the coronal heating eventually leads to a new eruption. The period between the eruptions depends on the heating rate of the solar corona and the reconnection rate in the equatorial current sheet. The eruptions contribute significantly to the solar wind mass and energy flux. The presence of these massive eruptions is sensitive to the value chosen for the heat flux coefficient . In the model in which the heat flux is carried by the electrons, ¼ e , the solar corona relaxes to a steady equilibrium state in which there is force and energy balance everywhere in the system; the gravitational force, inertial forces, pressure gradient force, and magnetic forces balance each other exactly to form this equilibrium. The input energy is balanced by downward conduction in the closed region, and in the coronal holes the input energy is mainly balanced by the solar wind. In models in which protons are heated, and a significant fraction of the heat conductive energy is carried by the protons, the eruptions are present. The pressure scale height in the inner corona is very sensitive to the temperature structure. In the model in which the
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electrons carry the heat flux the temperature maximum inside the closed region is almost a factor of 2 lower than the temperature maximum inside the helmet streamer for the model in which the protons carry the heat flux (see Fig. 6). In coronal holes the temperature variations due to variations in are smaller, since the solar wind is the dominating energy-loss mechanism. The fact that the temperature inside the closed region is so sensitive to the heat flux coefficient results in a larger difference in the thermal structure along open and closed flux tubes for models in which is small. This difference seems to determine whether our system reaches an equilibrium state or not. In the models in which protons carry some of the heat flux, the plasma along the outermost flux tube confining the helmet streamer is heated to such a high temperature that the J µ B force is no longer able to withstand the pressure gradient force, and the top of the streamer expands into interplanetary space. When the top of the streamer expands, and mass and energy is released into the solar wind, the thermal structure along the expanding part of the streamer adjusts to conditions consistent with an open magnetic field. As the expanding flux tube is ‘‘ emptied,’’ and the ejected mass moves into the solar wind, the pressure on the opened field is reduced. When the pressure on the dipole field is reduced, magnetic reconnection in the equatorial current sheet causes the magnetic field to close. Because of the large difference in thermal structure along open and closed flux tubes, the time needed for the coronal plasma to adjust to a high pressure consistent with a closed field in equilibrium with the gas is too long to prevent too many field lines from closing up. When the helmet streamer expands into the solar wind the amount of opened magnetic flux exceeds the equilibrium value, and when the magnetic field lines have reconnected in the equatorial current sheet the amount of closed magnetic flux exceeds the equilibrium. Because of the inertia of the streamer–coronal hole system it oscillates around an equilibrium, and the amplitude is large enough to cause massive eruptions. The presence of the eruptions is very interesting in itself. They are present whenever the inward conductive energy transport is an insufficient energy sink. In the case of a reduced conductivity, the eruptions from the streamer are a more effective energy release mechanism than heat conduction to the coronal base. More interesting is the driving mechanism behind the eruptions. Although the parameters used in our numerical experiments (Qm , , and c ) may differ from values present on the Sun, this does not rule out the possibility that this physical process can occur in the solar corona. Our parameter study illustrates that the stability of the streamer–coronal hole system is marginal when there is a large difference in thermal structure across the streamer– coronal hole boundary. This difference is produced by the different energy-transport mechanisms dominating in these two regions of the inner corona. We have intentionally set up the system in a conservative manner; uniform boundary conditions and a uniform heating should not tend to drive this time-dependent behavior. If the helmet streamer is slightly out of equilibrium, and there is a large difference in pressure along open and closed field regions, it may not be able to restore an equilibrium state and undergoes a process in which mass and energy are released into interplanetary space. By solving as simple a problem as possible we are able to understand the basic physics behind the driving mechanism, which may be helpful in constructing more realistic models of the large-scale solar corona.
No. 2, 2003
MHD MODELS OF THE SOLAR CORONA
The heating function we have used in this study is chosen to only depend on heliocentric distance; the heating does not distinguish between open and closed regions, even though the physical conditions in these two regions are very different. A heating function that is independent of time, electron density, and the magnetic field configuration is probably not a realistic representation of the heating present in the real solar corona. To study this process in further detail and apply the results from this study to the Sun, one should include heating mechanisms that are some what more realistic. More realistic models of the solar corona should also include solving the two-fluid problem, in which the energy equations for the electrons and the protons are solved separately, and which includes collisional interactions between the two particle species. However, we believe that we will find similar results in a two-fluid model. The thermal coupling between electrons and protons becomes weak rather close to the coronal base, and in the case of proton heating the electron temperature will be relatively low. In this case, the protons will play a more important role (than electrons) for the energy balance in the corona. One should also include more realistic boundary conditions at the coronal base; the electron density should be adjusted to the amount of heat conducted back into the transition region from the coronal base. A more realistic treatment of the inner boundary would lead to a higher base pressure in the closed region than we have used in our models, and the base pressure in the coronal holes may be reduced. Consequently, the pressure difference between open and closed flux tubes may be even larger, and the inclusion of a more
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realistic inner boundary should not weaken the effect reported in this study. A final comment should be made about the resolution we have used in our numerical calculations. A poor spatial resolution tends to reduce the accuracy of the results, and may introduce significant numerical diffusion. On the other hand, a high resolution would make these models too expensive to run on long timescales. A dominating numerical diffusion would tend to smooth out the difference in thermal structure across the streamer–coronal hole boundary, and this damps out the oscillations and the mass ejections. The sixth-order approximations to the spatial derivatives in our code reduces some of the numerical diffusion. However, we have done one test in which we have improved the spatial resolution; we have calculated the base model with an increased h-resolution (128 128). From this calculation we get the exact same results as when the model is calculated on the 129 65 grid. When we reduce the resolution significantly (64 32), we find that the large-scale structure of the solution is similar to the models with a higher resolution, but the streamer eruptions are not present. There are some small (
