The Astrophysical Journal, 598:L125–L128, 2003 December 1 䉷 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
CORONAL LOOPS HEATED BY TURBULENCE-DRIVEN ALFVE´N WAVES Xing Li1 and Shadia Rifai Habbal1,2 Received 2003 June 11; accepted 2003 October 16; published 2003 November 11
ABSTRACT A two-fluid dynamic model of long-lived coronal loops is presented, whereby heating of the confined plasma is achieved by turbulence-driven Alfve´n waves. It is assumed that the nonthermal motions inferred from spectral line observations in the transition region are due to Alfve´n waves. It is also assumed that the turbulence is already fully developed when the waves are injected at the footpoint of the loop while the wave/turbulence energy is readily absorbed by the proton gas. The Coulomb coupling between protons and electrons subsequently heats the electron gas. The model produces a fairly uniform electron temperature in the coronal segment of the loop even though the heating is nonuniform. The model also reproduces electron densities of (1–4) # 10 9 cm⫺3, in the range inferred from observations, as well as a moderate flow speed around 10 km s⫺1 along the loop. The turbulence heating mechanism adopted in this Letter, however, cannot produce stable loops with temperatures T ≤ 1.3 # 10 6 K. Subject headings: Sun: corona — Sun: transition region velocities, their approach, although not self-consistent, yielded turbulent heating predominantly near the footpoints of the loops. By comparing model results with observations, numerical models provide an essential exploratory tool for understanding the physics of coronal heating and establishing the viability of the specific mechanisms explored. Although coronal loop models abound, self-consistent hydrodynamic numerical coronal loop models are rare (e.g., Antiochos 1994; see also recent review by Aschwanden 2001). The inclusion of the chromosphere and transition region in such models, while essential, remains challenging. In this Letter, the consequences of heating by Alfve´nic turbulence are explored as a mechanism for accounting for the observed properties of long-lived coronal loops (i.e., loops that can exist for at least a few hours). Nonthermal motions, inferred from spectral line observations, are assumed to be produced by Alfve´n waves propagating along coronal loops. Although the observational support for this assumption is not fully developed (Hara & Ichimoto 1999), the advantage of Alfve´nic turbulence is that these waves can carry a sufficiently large energy flux due to a large Alfve´n speed in the corona. Turbulence cascades wave energy from low to high frequencies, where waves are dissipated to heat the plasma by ion cyclotron resonance or proton viscosity (Braginski 1965). This process is fast enough to replenish the energy losses due to electromagnetic radiation and thermal conduction losses to the chromosphere. The model presented is a self-consistent turbulencedriven coronal loop model. It is assumed that the turbulence is already developed when the waves are injected at the footpoint below the transition region. The plasma mass, momentum, energy equations, and Alfve´n wave energy equations are solved simultaneously to explore the conditions necessary to establish stable loops, i.e., long-lasting loops, with steady flows.
1. INTRODUCTION
High spatial resolution space-based observations of the solar corona in the ultraviolet, extreme ultraviolet, and X-rays, as well as ground-based observations, made either with coronagraphs or during total solar eclipses, reveal the preponderance of arcsecond to subarcsecond wide looplike structures in the corona coexisting with filamentary structures extending outward. Both open and closed density magnetic structures reflect the intricate role of physical processes responsible for the high temperature of the corona. The plasma properties characteristic of looplike structures, such as distribution of temperature along the loop, density and density scale heights, and flow speeds along the loops, are primarily inferred from spectral observations, using either density or temperature line ratio techniques, and Doppler shifts. Despite advances in the accuracy of these inferences with the improved spatial and spectral resolution of recent observations, the mechanisms responsible for coronal heating remain elusive. Clues to the coronal heating processes are also often inferred from spectral line observations, in particular from the broadening of the lines beyond the thermal temperature of the confined plasma. Nonthermal motions in the transition region and corona have been observed at both disk center (Mariska, Feldman, & Doschek 1978; Chae, Schu¨hle, & Lemaire 1998) and above the limb (Hassler et al. 1990; Tu et al. 1998). Although the origin of the observed nonthermal motions is not known, it is often assumed that they are due to MHD turbulence. The application of turbulence as a heating mechanism for closed magnetic structures, or coronal loops, has been extensively explored (Dmitruk & Go´mez 1997, 1999; Heyvaerts & Priest 1992; Inverarity, Priest, & Heyvaerts 1995; Inverarity & Priest 1995). Aschwanden (2001), however, pointed out that these turbulence models are not compatible with observational findings of coronal heating localized near the footpoints of loops. More recently, Chae, Poland, & Aschwanden (2002) revisited the problem. Using a heating rate with a functional dependence on temperature, which was inferred from observed nonthermal
2. THE MODEL
We model a coronal loop as a thin semicircular cylindrical tube of an electron-proton plasma with constant cross section following a bundle of magnetic field lines anchored in the photosphere. The equations governing the one-dimensional time-dependent continuity, momentum, and energy equations
1 Institute of Mathematical and Physical Sciences, University of Wales, Aberystwyth SY23 3BZ, UK;
[email protected]. 2 Smithsonian Astrophysical Observatory, MS 15, Cambridge, MA 02138.
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k p p 3.2 # 10⫺9 Tp5/2 are the electron and proton heat flux conductivities. The npe is the Coulomb collision frequency given by Braginski (1965). The Coulomb logarithm ln L is taken to be 23 to compute npe. It is assumed that Alfve´n waves originating below the transition region enter the loop from one footpoint only, referred to as the left footpoint, chosen at the reference s p 0. The L rad is the optically thin radiation loss parameterized by Rosner, Tucker, & Vaiana (1978). The magnetic field B in the coronal loop is fixed at 80 G; Q is the turbulent heating rate. Chae et al. (1998) argued that Kolmogorov-type turbulence is preferable for coronal heating rather than Kraichnan-type turbulence. In such a case, the turbulent heating rate Q can be written as (Hollweg 1986) QpG
Fig. 1.—Coronal loop heated by Alfve´nic turbulence. (a) Flow speed (solid line) and wave amplitude (dashed line). (b) Proton (solid line) and electron (dotted line) temperature. (c) Electron density. (d) Turbulent heating rate as a function of distance along the loop. In this model, the wave velocity amplitude y p 10 km s⫺1 at s p 0, and the turbulence driving scale is l p 200 km.
for electrons and protons, and the energy equation for Alfve´n waves, can be written as ⭸r 1 ⭸(r va) ⫹ p 0, ⭸t a ⭸s
(1)
⭸v 1 ⭸( pe ⫹ pp ⫹ pw ) ⭸v ⫹v p⫺ ⫺ gk, ⭸t ⭸s r ⭸s
(2)
2 ⭸ ⭸T ⭸Te ⭸T 2T ⭸( va) ⫹v e⫹ e p ake e ⭸t ⭸s 3a ⭸s 3nka ⭸s ⭸s
(
)
⫹ 2npe (Tp ⫺ Te ) ⫺
2 L , 3nk rad
2 ⭸ ⭸T ⭸Tp ⭸T 2T ⭸( va) ⫹v p⫹ p p ak p p ⭸t ⭸s 3a ⭸s 3nka ⭸s ⭸s
(
(3)
)
2 ⫹ 2npe (Te ⫺ Tp ) ⫹ Q, 3nk ⭸pw 1 ⭸ v ⭸pw Q ⫹ [a(1.5 v ⫹ vA )pw] ⫺ ⫹ p 0, ⭸t a ⭸s 2 ⭸s 2
ry3 ry3 p , L l
(6)
where G is a constant close to unity and L is the driving scale of the turbulence. We will use l p L/G as a free parameter in this work. The constant G is absorbed into l, and l is simply called the turbulence driving scale. The model calculations start from the base of the lower transition region where the temperature is 2 # 10 4 K at the two footpoints. The wave amplitude is fixed at one footpoint (s p 0) but is free to change at the other footpoint. The two footpoints are treated as free boundaries for the plasma densities. The idea is that when waves enter a coronal loop from one footpoint, the energy flux can support a certain amount of material and drive a flow in the loop. The chromosphere below simply acts as a reservoir of material. The flow velocity at the footpoints is also allowed to change freely. The loop length is fixed at 72,000 km. Equations (1)–(6) form a closed set of equations. The timedependent equations of the two-fluid plasma system are integrated with variable step sizes in space and time using a fully implicit scheme described by Hu, Habbal, & Esser (1997). The step ds gradually increases from 35 m at the two footpoints to about 500 km at the apex of the loop, and 1340 grid points are used. With a trial guess for the initial coronal loop parameters, the solution, which is independent of the initial state, converges very rapidly to a steady state. 3. RESULTS
(4)
(5)
where r p ne me ⫹ n p m p ≈ n p m p is the plasma mass density, ne and n p are the electron and proton density, and me and m p are the electron and proton mass. The component of solar gravity parallel to the loop is gk. The coronal plasma is assumed to be quasi-neutral, with ne ≈ n p p n. The electron and proton temperatures are given by Te and Tp, pe p nkTe and pp p nkTp are the electron and proton pressure, and k is the Boltzmann constant. The plasma flow speed is v. The cross section of the coronal loop, a, is assumed to be a constant. The turbulent pressure is pw p ry2/2, y is the Alfve´n wave velocity amplitude, and vA p B/ (4pr)1/2 is the Alfve´n speed; ke p 7.8 # 10⫺7 Te5/2 and
A steady state solution is presented in Figure 1 with y p 10 km s⫺1 at s p 0. This choice is dictated by the nonthermal line broadening observations in the quiet Sun from Mariska et al. (1978), and more recently from the Solar Ultraviolet Measurement of Emitted Radiation (SUMER) instrument (Chae et al. 1998), which suggest that y is about 10 km s⫺1 in the quiet Sun when Te p Tp p 2 # 10 4 K. The model yields a flow speed of about 10 km s⫺1 in the middle of the loop where the minimum in the electron density is 2.2 # 10 9 cm⫺3. Although protons only are heated directly by turbulence, electrons (Fig. 1b, dotted line) and protons (Fig. 1b, solid line) end up with roughly the same temperature along the loop owing to the strong Coulomb coupling between them. Because Alfve´n waves enter the loop from the left footpoint s p 0 and lose power as they propagate along the loop, the wave amplitude decreases substantially when the waves arrive at the right footpoint (Fig. 1a, dashed line). The heating rate along the loop is highly asymmetrical, with the heating occurring primarily at the transition region (Fig. 1d). This is because the Alfve´n wave amplitude is higher in the left half of the loop. The max-
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Fig. 3.—Minimum electron density, maximum electron temperature, and maximum flow speed as a function of turbulence driving scale l. Here y p 10 km s⫺1 at s p 0.
Fig. 2.—Total gas and Alfve´n wave pressure as a function of distance along the loop.
imum temperature does not occur at the apex (middle) of the loop. One interesting feature of Figures 1b and 1d is that, although the heating in the right half of the loop is 1 order of magnitude smaller than that in the left half of the loop, the loop temperature does not show a comparable difference: the loop temperature is relatively flat throughout owing to the high electron heat conduction along magnetic field lines when the electron temperature is beyond 106 K. Figure 1 suggests that by primarily heating one footpoint alone, it is possible that a bright coronal loop can be formed. Recently, several observations have found that coronal loops are often nearly isothermal along their coronal segments (Aschwanden et al. 1999, 2000; Lenz et al. 1999; Brkovic´ et al. 2002). Aschwanden, Schrijver, & Alexander (2001) found that only a nonuniform heating function can explain the observations. However, most of the loops they investigated were not in quasi-static equilibrium. Since the turbulence driving scale in Figure 1 is 200 km, the frequency of the injected waves is f p vA/2pl ∼ 0.25 Hz. Hence, this turbulence-driven coronal loop model requires high-frequency waves. The model produces a moderate steady flow, with an approximately constant flow speed of 10 km s⫺1 in the coronal part of the loop. Since gravity is accelerating the plasma in the right half of the loop (the positive flow direction is defined from the left to the right footpoint), one may ask why the plasma in the loop is decelerated near the right footpoint. It turns out that the Alfve´n wave pressure gradient force plays a key role in accelerating the plasma above the left footpoint and decelerating the plasma above the right footpoint (see Fig. 2). Recent Transition Region and Coronal Explorer and SUMER observations found steady flows of 15–40 km s⫺1 along active region loops (Winebarger et al. 2002). It will be shown in Figures 3 and 4 that larger wave amplitudes or smaller driving scales lead to faster flows. The two footpoints have different densities and pressures. The pressure along the loop is not constant. The total gas pressure pe ⫹ pp and Alfve´n wave pressure are shown in Figure 2. Most previous hydrostatic coronal loop models assumed a constant pressure along a coronal loop. This assumption is not valid in
a hydrodynamic model. Interestingly, just above the left footpoint (i.e., above the transition region), there is a significant increase (0.2 dyn cm⫺2) in the gas pressure over a very small distance (the pressure decrease above the right footpoint is less pronounced). This means that the gas pressure gradient force is not accelerating (decelerating) the plasma over the same distance just above the left (right) footpoint. Hence, over the same distance the Alfve´n wave pressure gradient force is the dominant force accelerating (or decelerating) the plasma. This rapid wave pressure change is a well-known feature of small-amplitude Alfve´n waves propagating in an inhomogeneous medium. The minimum pressure occurs roughly in the middle of the loop, corresponding to the maximum flow speed in the region (see Fig. 1a). The rapid change in gas pressure in the transition region shown in Figure 2 may be evidence for waves playing a critical role in coronal heating. Since the driving scale l is unknown, it is informative to show results for different l-values. Figure 3 shows the minimum electron density, the maximum electron temperature, and the maximum flow speed as a function of l from the steady state loop solutions. For these solutions, the wave velocity amplitude at the base, y(s p 0), is fixed at 10 km s⫺1. For a smaller
Fig. 4.—Minimum electron density, maximum electron temperature, and maximum flow speed as a function of y(s p 0). Here l p 300 km.
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driving scale, waves can lift more mass from the chromosphere into the coronal portion of the loop. More localized heating may also lead to a more sufficient acceleration of plasmas (Mariska 1988). Figure 3 also shows that for a fixed y, there is a maximum temperature of Te p 2.5 # 10 6 K that electrons can reach in a steady state loop. When l 1 300 km, a smaller driving scale leads to an enhanced heating (eq. [6]) and a higher electron temperature in the loop. However, for l ! 200 km, this is not the case anymore. The heating becomes strongly localized at the left footpoint, and the heating scale height is small. As a result, the energy loss from heat conduction is significant and most of the loop is heated by electron heat conduction. At l ≈ 1350, the loop is almost in hydrostatic equilibrium with practically no flow. For larger values of l, a steady state solution cannot be found. If the amplitude of waves entering a loop is different, these waves carry a different energy flux into the loop, which may produce different physical parameters there. Indeed, there is some evidence that nonthermal motions in active regions are stronger than in the quiet Sun (Chae et al. 2000). Figure 4 shows the minimum electron density, the maximum electron temperature, and the maximum flow speed as a function of the wave amplitude y at s p 0 when l p 300 km is assumed. Figure 4 shows that these parameters are quite sensitive to the wave amplitude. As can be expected, a higher wave amplitude means more energy will be dissipated as heat in a loop, more material can be lifted into the loop, and plasmas in the loop become more energetic (faster flows and higher temperatures). Chae et al. (2000) found that dynamic loops have large bulk motions and large nonthermal line broadenings. Note that in Figure 4 all the maximum electron temperatures are beyond 1.3 # 10 6 K. If y(s p 0) is smaller than 6.5 km s⫺1, the loop
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will be too cool and a steady state solution cannot be found. Both Figures 3 and 4 suggest that cold loops with temperatures ≤1.3 # 10 6 K are not stable with the heating mechanism explored in this Letter. 4. CONCLUDING REMARKS
This Letter demonstrates that if the well-known nonthermal motions frequently observed in transition region spectral lines are due to Alfve´n waves, these waves, if dissipated by a turbulent cascade process, are capable of producing coronal electron densities, temperatures, and flow speeds that are within the ranges of values often observed in active region loops (Aschwanden et al. 1999, 2000; Chae et al. 2000; Winebarger et al. 2002). To produce observed densities greater than 109 cm⫺3, the driving scales for these waves must be hundreds of kilometers or less. This parameter study has concentrated on the wave velocity amplitude and loop length. However, the actual parameter space is quite large, and other factors, which may affect coronal loop modeling, are ignored in this study. For example, coronal loops may have varying cross section, and varying magnetic field strength, along their length. The loop inclination angle may also have some effect on the dynamics of the loop. Minor ions, especially helium, were not taken into account here. However, it is known from solar wind models that ion-cyclotron resonance may significantly affect parameters of minor ions and coronal major species. These aspects will be the subject of future work. This work was supported in part by a PPARC rolling grant to the University of Wales at Aberystwyth and by NASA grant NAG5-10873 to the Smithsonian Astrophysical Observatory.
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