MULTISPECTRAL EMISSION OF THE SUN DURING THE FIRST WHOLE SUN MONTH: MAGNETOHYDRODYNAMIC SIMULATIONS

The MHD approximation is appropriate for long-scale, low-frequency phenomena in magnetized plasmas such as the solar corona. To reproduce the Sun's emission properties during Whole Sun Month, we have solved the following set of viscous and resistive MHD equations:

$$\begin{equation} \mathbf{ \nabla }\times \mathbf{B} = \frac{4 \pi }{c} \mathbf{ J}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \mathbf{ \nabla }\times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \mathbf{ E} + \frac{\mathbf{ v}\times \mathbf{ B}}{c} = \eta \mathbf{ J}, \end{equation} \tag{ 3 }$$

$$\begin{equation} \frac{\partial \rho }{\partial t} +\mathbf{ \nabla\ \cdot}\ (\rho \mathbf{ v}) =0, \end{equation} \tag{ 4 }$$

$$\begin{equation} \frac{1}{\gamma -1} \left(\frac{\partial T}{\partial t}+ \mathbf{v}\cdot \mathbf{\nabla } T \right) = -T\mathbf{\nabla } \cdot \mathbf{v} +\frac{m}{k \rho } S, \end{equation} \tag{ 5 }$$

$$\begin{equation} \rho \left(\frac{\partial \mathbf{ v}}{\partial t} + \mathbf{ v \ \cdot }\ \mathbf{ \nabla } \mathbf{ v} \right) = \frac{1 }{c} \mathbf{J}\times \mathbf{B}- \mathbf{ \nabla } (p+p_w) + \rho \mathbf{g} + \mathbf{\nabla }\ \cdot\ (\nu \rho \mathbf{ \nabla v}), \end{equation} \tag{ 6 }$$

$$\begin{equation} S= (- \mathbf{\nabla }\cdot \mathbf{q} - n_en_p{Q(T)}+{H_{\mathrm{ch}}}), \end{equation} \tag{ 7 }$$

where B is the magnetic field, J is the electric current density, E is the electric field, ρ, v, p, and T are the plasma mass density, velocity, pressure, and temperature, respectively, $\mathbf{g}=-g_0 R_\odot ^2 \mathbf{\hat{r}}\big/r^2$ is the gravitational acceleration, η is the resistivity, and ν is the kinematic viscosity. Equation (7) contains the radiation loss function Q(T) as in Athay (1986), n_e and n_p are the electron and proton number density (which are equal for a hydrogen plasma), γ = 5/3 is the polytropic index, H_ch is the coronal heating term (see Section 2.2), and q is the heat flux. A collisional (Spitzer's law) or collisionless (Hollweg 1978) formulation is used according to the radial distance,

$$\begin{equation} \mathbf{q}=\left\lbrace \begin{array}{@{}ll@{}} - \kappa _0 T^{5/2} \mathbf{\hat{b}} \mathbf{\hat{b}} \cdot \nabla T & \hbox{if}\quad R_\odot \le r \lesssim 10\;R_\odot \\ \ms \alpha n_ekT \mathbf{v} & \hbox{if}\quad r \gtrsim 10\;R_\odot \end{array} \right., \end{equation} \tag{ 8 }$$

where κ₀ = 9 × 10⁻⁷ erg K^−7/2 cm⁻¹ s⁻¹, $\mathbf{\hat{b}}$ is the unit vector along B, and α is a parameter, which was set to 1. The transition between the two forms occurs smoothly between 7.5 R_☉ and 12.5 R_☉ (Mikić et al. 1999). The wave pressure term p_w in Equation (6) represents the contribution due to Alfvén waves (Jacques 1977). It is evolved using the Wentzel-Kramers-Brillouin (WKB) approximation² for time-space-averaged Alfvén wave energy density ,

$$\begin{equation} \frac{\partial \epsilon }{\partial t}+ \mathbf{\nabla } \cdot \mathbf{F} =\mathbf{v} \cdot \mathbf{\nabla } p_w -D, \end{equation} \tag{ 9 }$$

where F = (3/2v + v_A) is the Alfvén wave energy flux, $v_A=B/\sqrt{4\pi \rho }$ is the Alfvén speed, and p_w = /2. The Alfvén wave velocity is $\mathbf{v}_A=\pm v_A \mathbf{\hat{b}}$; in a multidimensional implementation, it is necessary to transport two Alfvén wave fields (waves parallel and antiparallel to B), which are combined to give . The dissipation term D, which expresses the nonlinear dissipation of Alfvén waves in interplanetary space and is modeled phenomenologically (Hollweg 1978), was set to zero for the present investigation. The boundary conditions on the velocity are determined from the characteristic equations along B. The surface magnetic flux at r = R_☉ is specified from a synoptic magnetic map; for this case we use a smoothed NSO Kitt Peak map for Carrington Rotation (CR) 1913 (1996 August 22 to September 18). At r = R_☉ we also specify the Alfvén wave pressure. At the upper radial boundary, which is placed beyond all critical points, the characteristic equations are used as well.

For the present cases, we have used a nonuniform grid in r × θ × ϕ of 131 × 101 × 151 points. The smallest radial-grid interval at r = R_☉ was ∼300 km; the angular resolution was highest in the area containing the largest active region and the southward extension of the northern coronal hole (which was dubbed "the elephant's trunk") and was slightly larger than 1°. A uniform resistivity η was used, corresponding to a resistive diffusion time τ_R ∼ 4 × 10³ hr, which is much lower than the value in the solar corona. This is necessary to dissipate structures that cannot be resolved since they are smaller than the cell size. The Alfvén travel time at the base of the corona (τ_A = R_☉/V_A) for |B| = 2.205 G and n₀ = 10⁸cm⁻³, which are typical reference values, is 24 minutes (Alfvén speed V_A = 480 km s^-1), so the Lundquist number τ_R/τ_A is 1 × 10⁴. A uniform viscosity ν is also used, corresponding to a viscous diffusion time τ_ν such that τ_ν/τ_A = 500. Again, this value is chosen to dissipate unresolved scales without substantially affecting the global solution. In all the simulations we have used fixed chromospheric values of density and temperature at the base of the domain of n₀ = 2 ×  10¹² cm^-3 and T₀ = 20, 000 K, respectively. This is an overestimation of the pressure that does not affect the coronal solution (Mok et al. 2005). In fact, at this value of temperature, radiation loss (n_en_pQ(T) in Equation 7) becomes very small. The heat flux also tends to become very small, and a layer with uniform temperature develops near r = 1R_☉, roughly similar to the "temperature plateau" at the top of the chromosphere. The density at the lower boundary is set to a fixed chromospheric value that is large enough (for the specified heating) to produce a temperature plateau. If the specified chromospheric density (and pressure) are too large, the thickness of the temperature plateau will increase slightly (since the scale height in the plateau region is very small, equal to 1200 km at T = 20,000 K). If the chromospheric density is too low, the transition region may evaporate. It is not crucial to estimate the required chromospheric density accurately; it is best to overestimate it. An overestimation will simply produce a slightly thicker plateau region, without affecting the overall solution. Figure 1 demonstrates this property by showing one-dimensional loop solutions for different base densities. The coronal parts of the solution are essentially identical. The geometrical properties of this loop were extracted from the three-dimensional solution as described in Section 3. The loop launch point and position in the global corona can be seen in Figure 2.

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A principal difficulty in computing three-dimensional coronal solutions using Equations (3)–(7) is related to the extremely steep temperature and density gradients in the transition region, a consequence of the balance between the conduction of heat from the hot corona and radiative loss in the transition region. In order to efficiently model the coupling between the transition region and corona, we have developed a technique that artificially broadens the transition region, while maintaining accuracy in the corona (Z. Mikić et al. 2008, in preparation). We have found that if the temperature dependence of the thermal conductivity κ(T) and radiation loss function Q(T) is modified in such a way as to keep the product κ(T)Q(T) unchanged, then the coronal solution is not changed significantly. Looking at Equation (7), we see that the balance of radiative losses with thermal conduction in the transition region can be expressed in a dimensional analysis as

$$\begin{equation} n_e n_p{Q(T)} \sim \kappa (T)*T/L^2, \end{equation} \tag{ 10 }$$

where L is the length scale associated with the temperature gradient. We see that

$$\begin{equation} L \sim \sqrt{\frac{\kappa (T)*T}{n_e n_p{Q(T)}}}. \end{equation} \tag{ 11 }$$

To broaden the transition region we need to increase κ and decrease Q at low temperatures. We have modified κ(T) to be constant for temperatures below T_c = 500, 000 K. Accordingly, we reduce Q to keep κQ unchanged.

With this modification, the smallest transition region length scale is expected to be 400 km (for a loop p = 1 dyn cm^-2). Indeed, when we simulate a one-dimensional loop with this modification, we find that the transition region is broadened significantly (by a factor of ∼400), and yet the coronal solution remains virtually unchanged. Figure 3 shows a comparison of loops calculated with and without this modification. The geometry of the loop is the same as that of Figure 1. We find that the emission of the loop in EUV is not significantly modified for coronal temperatures (above ∼500, 000 K). More accurate calculations at lower temperatures can be achieved by lowering the temperature at which the modification occurs, with the consequence of a requirement for higher resolution.

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In the thermodynamic MHD model we have just described, it is necessary to specify explicitly a coronal heating function H_ch(r, θ, ϕ). The physical mechanism that heats the solar corona and powers its emission in the X-ray and EUV bands continues to be a hotly debated topic of research. Although there is general consensus that what heats the corona must ultimately involve conversion of magnetic energy into heat, it is not clear whether the main mechanism can be identified as the dissipation of high-frequency waves or rather in the slow energy build-up through photospheric motions followed by its rapid release, as proposed by Parker (1979). Mandrini et al. (2000), Priest et al. (2000), and Aschwanden (2001) have summarized the characteristics and physical implications of a number of models that are currently believed to be viable. Besides these physics-based models, phenomenological heating models have also been developed, starting from studies of correlations between radiation losses and magnetic flux in the Sun and other stars (Gurman et al. 1974; Schrijver et al. 1985; Fisher et al. 1998). Pevtsov et al. (2003) concluded that there is a universal linear correlation between magnetic flux and X-ray radiance.

Typically, coronal heating mechanisms have been constrained by calculating the X-ray and EUV emission of coronal loops and comparing it with observations. Plasma loops, first observed in Skylab, can be studied with simple, one-dimensional models. Initially, one-dimensional loops were investigated with static models (Rosner et al. 1978; Craig et al. 1978; Hood & Priest 1979; Vesecky et al. 1979; Serio et al. 1981), and then with numerical HD simulations (Wu et al. 1981; Cheng et al. 1983; Klimchuk et al. 1987; Mok et al. 1991; see also Bray et al. 1991 for a review). From the loop density and temperature distributions provided by the models, the emission (at least for optically thin lines) can be calculated (e.g., Karpen et al. 2001) using the CHIANTI database (Dere et al. 1997). As discussed in the introduction, amalgamations of loop atmosphere calculations for several thousand loops have been used to simulate emission from active regions (Lundquist et al. 2004; Warren & Winebarger 2006). The most comprehensive attempt to date to constrain coronal-heating models with the appearance of the entire corona is that of Schrijver et al. (2004).

For the present study, we show results for three heating models. The heating fluxes, in erg cm⁻² s⁻¹, calculated at the base of the corona for each model are shown in Figure 4.

• 1.  
  The first heating function we considered is a simple exponential decay law:

  $$\begin{equation} H_{\mathrm{exp}} = H_0\exp {-(r-R_\odot)/\lambda _0}, \end{equation} \tag{ 12 }$$

  where H₀ = 4.9128 × 10⁻⁷erg cm^-3 s⁻¹ and λ₀ = 0.7R_☉. This heating function yields fast wind in one-dimensional models (Withbroe 1988) when used in conjunction with an Alfvén wave pressure at the base of the corona of p_w = 8.36 × 10⁻²dyn cm^-2. The total power injected into the corona is 4.93 ×  10²⁷erg s^-1.
• 2.  
  The second model is based on that of Schrijver et al. (2004), who calculated one-dimensional loop solutions based on the following heat flux at the base of the corona:

  $$\begin{equation} F_{\mathrm{ch}}=4\ \times\ 10^{14}B/L \exp[-(B/500)^2 ]\, \mathrm{erg\;cm^{-2}\;s^{-1}}, \end{equation} \tag{ 13 }$$

  where B is the magnetic field strength at the base and L is the half-length of the loop. Despite its elegance and deceptive simplicity, the implementation of this model in a three-dimensional computation is not straightforward. First, we note that in principle the heat flux can become infinite as the length of the loops tends to zero near the neutral line. Hence, in calculating F_ch, a minimum loop-length criterion must be imposed. Second, as we will see in the discussion given below, the constant in front of the right-hand side applies only when using magnetic-flux distributions with the same resolution of the magnetogram of Schrijver et al. (2004) and generally must be rescaled at other resolutions. Third, this model was developed for one-dimensional calculations and requires field-line tracing to deposit the heat in the three-dimensional corona. Not only can this be a computationally intensive task, if it is done at every time step of a three-dimensional time-dependent calculation, but it also requires special care when calculating lengths of loops in regions where small initial errors may give large uncertainties in the final result (i.e., near quasi-separatrix layers, Titov et al. 2008).To deposit heat in the volume for the Schrijver et al. (2004) heating specification, we used the following method: first, a heat-flux map is constructed for each (θ, ϕ) grid point at the lower boundary from Equation (13). To deposit this heat in the calculation volume (i.e., to obtain H_ch), a field line is traced from each point on the heat-flux map and the grid cells intersected by this field line are identified. If the field line is closed (returns to the solar surface), then we deposit the following volumetric heating contribution into each intersected cell:

  $$\begin{equation} \Delta H_{\mathrm{ch}}^{(j)}=\frac{F_{\mathrm{ch}}^{(j)} \Delta A_j}{\sum _k \Delta V_k }, \end{equation} \tag{ 14 }$$

  where ΔA_j is the area associated with each point j on the heat-flux map and ΔV_k is the volume of the kth computational cells intersected by the jth field line. This corresponds to assuming uniform heating along each loop, as in Schrijver et al. (2004). Each cell volume may be intersected by zero, one, or several field lines and the total deposited heating at x_j is the sum of all contributions:

  $$\begin{equation} H_{\mathrm{ch}} (\mathbf{x}_j) = \sum _j \Delta H_{\mathrm{ch}}^{(j)}. \end{equation} \tag{ 15 }$$

  Since this formula applies only to closed-field regions and we are also modeling coronal holes, we have added to H_ch the heating calculated with model 1. For computational simplicity, we did not recompute the heating at every time step, but used the magnetic field configuration obtained with heating model 1 and held the heating constant in time. While the emission was greatly affected by changes in the heating model, changes to the topology and structure of the magnetic field were relatively small, justifying this approximation.In our first attempt to obtain the volumetric heating H_ch, we used the magnetic flux distribution of Figure 2(a) to calculate the heat flux F_ch according to Equation (13). With this method, the total power deposited in the corona was 3.77 × 10²⁷erg s^-1, which turned out to be insufficient to heat the corona and gave very dim emission and a poor comparison with observations. The reason for this poor result is that the smoothed magnetic map used for the calculation has a smaller amount of unsigned magnetic flux than the maps used by Schrijver et al. (2004). If we chose a higher resolution magnetic map than Schrijver et al., we would in general obtain too high a heat flux. A further complication is that the heat flux depends on the accuracy of the field line tracing. A higher accuracy tracing routine (such as the one we use) captures more short-scale loops and yields a higher heat flux than obtained by Schrijver et al. (2004). To develop a heating that matched as closely as possible the Schrijver et al. heating, we obtained a heat-flux map from Dr. Schrijver that was developed for this time period, as shown in Figure 4(b). This was based on a source-surface model using a magnetic map for CR1913 with the same resolution used by Schrijver et al. (2004) and had an integrated total power of 2.65 × 10²⁸erg s^-1. After computing H_ch with the method described above, the total power deposited in the volume was 2.41 × 10²⁸erg s^-1. This does not exactly match the power obtained with Equation (13) because in general the open- and closed-field regions from the MHD solution are different from those in the source-surface model. In addition, to provide heating in the coronal hole regions, we have added to H_ch the heating calculated with model 1, bringing the total power to 2.90 × 10²⁸erg s^-1. We have also added the same Alfvén wave pressure of model 1.
• 3.  
  While the solutions that we show in this paper are found by integrating the equations to steady state, in general we are interested in investigating time-dependent solutions, for example for studying coronal mass ejections (CMEs). The magnetic field structure will then evolve considerably during the computation, and if we are using heating model 2, we must frequently recalculate the computationally expensive Equations (14) and (15). Therefore, we have developed an empirical heating model that depends only on local variables and can be computed rapidly. This third heating specification comprises three terms, the previous exponential heat function of Equation (12) plus two new terms representing the quiet Sun and active region heating:

  $$\begin{equation} H_{\mathrm{ch}} = H_\mathrm{exp}+H_\mathrm{QS}+H_\mathrm{AR}, \end{equation} \tag{ 16 }$$

  $$\begin{equation} H_\mathrm{QS}=H^0_{\mathrm{QS}} f(r) \frac{ B_t^2 }{B\big(|B_r|+B_r^c \big)}, \end{equation} \tag{ 17 }$$

  $$\begin{equation} H_\mathrm{AR}= H_{\mathrm{AR}}^0 g(B) \left(\frac{B}{B_0}\right)^{1.2}, \end{equation} \tag{ 18 }$$

  where $B_t=\sqrt{\vphantom{A^A}\smash{\hbox{$B^2_\theta +B^2_\phi$ }}}$ is the tangential magnetic field. For this case, the best results were obtained for H⁰_QS = 1.18 × 10⁻⁵erg cm^-3 s⁻¹, B^c_r = 0.55 G, H⁰_AR = 1.87 × 10⁻⁵erg cm^-3 s⁻¹, and B₀ = 1 G. The first term (H_exp) is necessary to accelerate the solar wind. However, we found that this term does not deposit enough energy to realistically heat the closed-field regions, in particular along neutral lines. To deposit more heat in the quiet Sun, a second term (H_QS) is introduced. This term mimics the 1/L behavior of heating in the quiet Sun, since it is large along neutral lines, where B_r = 0, and the short loops that surmount them. A third term (H_AR) depends on the strength of the magnetic field and adds more heating in active regions, where the magnetic field is larger and emission stronger. The dimensionless functions f(r) and g(B) are included in Equations (17) and (18) to limit the effect of the terms H_QS and H_AR to their intended regions. For this case,

  $$\begin{eqnarray} f(r) &=& \frac{1}{2} \left(1+\tanh \frac{ 1.7-r/R_\odot }{0.1} \right) \exp \left(- \frac{r/R_\odot -1}{0.2} \right), \nonumber \\ g(B) &=& \frac{1}{2} \left(1+\tanh \frac{B-18.1}{3.97} \right), \nonumber \end{eqnarray}$$

  which serve to turn off the heating when r ≳ 1.7R_☉ and B ≲ 18 G, respectively. The same Alfvén wave pressure of model 1 has also been added to this model. The relative importance of Alfvén wave and thermal pressure can be appreciated in Figure 5, where they are plotted as a function of height above the surface, starting from the North Pole and from the main active region.Although this model is still dependent on the magnetogram resolution as is heating model 2, it does not involve field-line tracing, making it suitable for three-dimensional computations.

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From the density and temperature calculated with our MHD model it is possible to produce synthetic emission images (Mok et al. 2005). The count registered by an instrument in a given configuration i is given by

$$\begin{equation} \begin{array}{lr} \displaystyle D = \int n_e^2(w) f_i(T(w),n_e(w)) \, d w &\mathrm{[DN\;s^{-1}\;pixel^{-1}]} \end{array}, \end{equation} \tag{ 19 }$$

where D is integrated along the line of sight w. The function f_i(T, n_e) takes into account atomic physics, geometry, and the properties of both the instrument and the filters. Since the dependence of f_i on the electron density n_e is rather weak, we have neglected it in this investigation. The shapes of f_i for the EIT 171, 195, 284 Å filters and for the SXT AlMg configuration are shown in Figure 6. These are the configurations in which processed emission images may be obtained using the EIT processing software³ or from the Yohkoh Legacy Data Archive.⁴

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