Gravitational steady states of solar coronal loops. Abstract

Transcription

1 Gravitational steady states of solar coronal loops Linda E. Sugiyama Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge MA M. Asgari-Targhi Harvard-Smithsonian Center for Astrophysics, Cambridge MA Abstract Coronal loops on the surface of the sun appear to consist of curved, plasma-confining magnetic flux tubes or ropes, anchored at both ends in the photosphere. Toroidal loops carrying current are inherently unstable to expansion in major radius due to toroidal-curvature-induced imbalances in the magnetic and plasma pressures. An ideal MHD analysis of a simple isolated loop with density and pressure higher than the surrounding corona, based on the theory of magnetically confined toroidal plasmas, shows that the radial force balance depends on the loop internal structure and varies over parameter space. It provides a unified picture of simple loop steady states in terms of the plasma beta β o, inverse aspect ratio ϵ = a/r o, and the MHD gravitational parameter Ĝ ga/va 2, all at the top of the loop, where g is the acceleration due to gravity, a the average minor radius, and v A the shear Alfvén velocity. In the high and low beta tokamak orderings, β o = 2n o T/(B 2 o/2µ o ) ϵ 1 and ϵ 2, that fit many loops, the solar gravity can sustain nonaxisymmetric steady states at Ĝ ϵβ o that represent the maximum stable height. At smaller Ĝ ϵ2 β o, the loop is axisymmetric to leading order and stabilized primarily by the two fixed loop ends. Very low beta, nearly force-free, steady states with β o ϵ 3 may also exist, with or without gravity, depending on higher order effects. The thin coronal loops commonly observed in solar active regions have ϵ 0.02 and fit the high beta steady states. Ĝ increases with loop height. Fatter loops in active regions that form along magnetic neutral lines and may lead to solar flares and Coronal Mass Ejections, have ϵ and may fit the low beta ordering. Larger loops tend to have Ĝ > ϵβ o and be unstable to radial expansion because the exponential hydrostatic reduction of the density at the loop-top reduces the gravitational force ρĝ ˆR below the level that balances expansion, in agreement with the observation that most sufficiently large loops grow. 1

2 I. INTRODUCTION Solar coronal loops are prominent dynamic features of the sun s surface [1 3]. They appear to be curved magnetic flux tubes or ropes, anchored at both ends in the photosphere and lower regions [2]. Many loops are quasi-stable, with characteristic heights R = R o and plasma parameters such as magnetic field B, density n, and temperature T. Unstable loops can grow to great heights and directly affect the earth as sources of particles for the solar wind or as seeds for intermittent large events such as Coronal Mass Ejections (CMEs) [3 5]. The difficulty of directly measuring plasma and magnetic field properties means that most solar loop models use simplified approximations [2 4] that only partially match observations [7]. The magnetohydrodynamic (MHD) momentum equation is ρdv/dt = (1/µ o )J B p ρgĥ in mks units, with gravitational force ρgĥ, where ρ = nm i is the plasma mass density, g the acceleration due to gravity, and ĥ the solar-vertical unit vector. It is often split into two pieces, a force-free equation for the magnetic field, as in the nonlinear force-free field (NLFFF) model[8 11], where J B = 0 in a background field B = Φ, and a hydrostatic density balance driven by gravity, ρgĥ = p 2 kt n, with solution ρ = ρ o exp( h/h), where h is the height above the solar surface and H = 2kT/M i g. Due to curvature, toroidal plasmas carrying current are inherently unstable to expansion in major radius, because of the imbalance between the magnetic field and plasma pressures over the inside and outside radial halves of the torus, R < R o and R > R o (cf. Freidberg [12]). The forces appear at higher orders in an expansion in the inverse aspect ratio ϵ = a/r o of the torus. They have been considered for solar loops, but generally for larger loops and imbalances that might lead to CMEs [13 15]. The internal steady state balance of gravity, pressure, and variation over the minor radius and along the loop, has been neglected, in part because few measurements exist at this level of resolution. Toroidal magnetically confined plasmas for magnetic fusion have been studied in the laboratory for decades. The theoretical understanding and practical implementation of feedback control of the plasma equilibrium can maintain plasmas for long periods of time. A conducting shell and/or an externally generated vertical field B Z is used to balance the radial force. (Here Z is the coordinate along the toroidal axis of symmetry in an (R, Z, ϕ) cylindrical coordinate system, shown in Fig. 1.) The field produces a volumetric force on the plasma, J ϕ B Z in the - ˆR direction. Recent laboratory experiments [16] on curved magnetic 2

3 FIG. 1. Coronal loop model in cylindrical (R, Z, ϕ) coordinates, with solar-vertical height h = h( ˆR sin ϕ + ˆϕ cos ϕ). Dashed horizontal line marks the transition region between the corona and the chromosphere. I Z is a possible axial current. flux ropes have been the first to apply a vertical field, in the form of a transverse magnetic field arching across the main flux rope. On the sun, however, there is little evidence for transverse fields at the same height as many coronal loops. Gravity, however, is inescapable. The paper considers the ideal MHD steady states of a toroidal magnetic flux rope in a small inverse aspect ratio expansion. Gravity introduces an important source of nonaxisymmetry, not previously considered. The tension between plasma pressure, gravity, and the loop internal magnetic structure, anchored by the two loop ends fixed in the photosphere, defines a limited set of steady states. Section II describes the MHD steady state model and the inverse aspect ratio expansion. Section III discusses the analytical solutions. Section IV compares the model to observations of thin coronal loops in active regions, taken from Alfvén wave heating studies [17 20]. Section V discusses thicker loops and implications of the solutions. The final section is a summary. II. CORONAL LOOP MODEL A simple coronal loop is assumed to have an ideal MHD structure similar to a laboratory toroidal plasma, with nested magnetic surfaces that confine a plasma with density and pressure above the surrounding background. The model looks for steady state solutions for a single, isolated loop in the shape of a partial torus with its two ends fixed in the 3

4 photosphere, which joins smoothly to the surrounding plasma and fields. The magnetic field is assumed to have a nearly axisymmetric magnetic axis, with moderate magnetic winding number. The loop cross section is arbitrary. The model is shown in Fig. 1. To avoid strong gradients, only the upper part of the loop in the solar corona is analyzed, above the dashed horizontal line in Fig. 1. The location of the axis of symmetry R = 0 may vary, but for simplicity the discussion assumes that the loop height is R o and significantly larger than the distance between the base of the corona and the photosphere. The MHD equations for a magnetic torus can be non-dimensionalized using a small set of parameters. These are a reference magnetic field B o = B ϕo, the toroidal field at major radius R = R o and Z o = 0, a length L o taken to be the average minor radius a of the torus, the plasma density n o at (R o, Z o ), and a reference velocity v o, the poloidal shear Alfvén velocity v Aθ (a/r o )v A in terms of the standard Alfvén velocity v A = B o / µ o n o M i Z i, where M i and Z i are the average ion mass and charge and µ o is the permittivity of free space. The reference time t o = L o /v o is the Alfvén time τ A = a/v Aθ = R o /v A. The plasma pressure is normalized to the plasma beta at R o, p = ˆpβ o /2, where β o = p o /(Bo/2µ 2 o ) for p o = n e T e + n i T i = 2n o T o, where the Boltzmann constant k is absorbed into T. The current satisfies Ampere s law J = (1/µ 0 ) B. The ideal MHD force balance in nondimensional form becomes ϵ 2 ρ v/ t = ϵ 2 ρ(v )v + J B p ρĝĥ. (1) The hats on the scaled quantities have been dropped, except for the gravitational parameter Ĝ, Ĝ g n o M i (µ o a/b 2 o) = g a/v 2 A = ϵ 2 ga/v 2 Aθ, (2) where the acceleration g = m/s 2 is assumed to be constant over the corona. The magnetic field B can be written exactly [21, 22] as B = ψ ϕ + (1/R) F + R o I ϕ (3) F 2 = (R o /R)( Ĩ/ ϕ), (4) where ψ is an approximate poloidal magnetic flux and I = RB ϕ /R o a toroidal field variable. The nonaxisymmetric function F makes B = 0, leading to Eq. (4). (Axisymmetry is 4

5 defined as / ϕ = 0.) The perpendicular direction is defined relative to toroidal angle ϕ, direction ϕ = (1/R) ˆϕ, so = ˆR( / R) + Ẑ( / Z) and 2 = ( 2 / R 2 ) + ( 2 / Z 2 ). The equilibrium plasma flow v, assumed to be primarily toroidal, is small and is neglected for a first approximation. For single closed coronal loops, typical velocities v ϕ = 5 10 km/s correspond to ϵv ϕ /v Aθ = δ at the top of the loop (e.g., Table I). The momentum term ϵ 2 ρvϕ 2/R δ2 ϵ ϵ 3 ϵ 4 is small. The model assumes that the loop has a predominantly axisymmetric toroidal field of the form B ϕ = (R o B o /R)I with I = 1 + O(ϵ), where the notation O(ϵ) means order ϵ or smaller. Such a field might be driven by an effective axial Ẑ current through the central hole of the torus (Fig. 1; this current does not appear in the model.) The poloidal field ordering B θ /B ϕ ϵ gives a magnetic winding number near unity. The scaled MHD variables can be expanded in ϵ as I = 1 + ϵ(ĩ0 + ϵĩ1 +...) (5) ψ = ψ 0 + ϵψ (6) F = ϵf 1 + ϵ 2 F (7) p = ϵp 1 + ϵ 2 p 2 + ϵ 3 p (8) ρ = ρ 0 + ϵρ (9) Ĝ = Ĝ0 + ϵĝ1 + ϵ 2 Ĝ 2 + ϵ 3 Ĝ (10) The scaled current density J = B gives RJ ϕ = [ 2ψ (1/R)( ψ/ R)] (1/R)( F/ Z) and RJ = Ĩ ˆϕ + (1/R) ( ψ/ ϕ) (1/R) ( F/ ϕ) ˆϕ. Thus J ϕ ϵψ 0 ϵ and J ϵj ϕ. The model defines the order of the J B terms[12]. Term J ϕ B ϵ 2, but one term in J B ϕ (the first term in Eq. (11)) can reach ϵ 1, otherwise it is ϵ 2. At ϵ 1 it can only be balanced by p β o ϵ 1, leading to the high beta tokamak ordering. The low beta tokamak ordering has β o ϵ 2. Subsequently, high and low beta refer to these orderings. A third, nearly force-free, ordering at very low beta, β o O(ϵ 3 ), is also possible. The complete momentum equation becomes [ (R o I/R 2 ) Ĩ + (1/R) ( ψ/ ϕ) ˆϕ ] + (1/R) ( F/ ϕ) [ ] + (RJ ϕ /R 2 ) ψ + ( F/ R)Ẑ ( F/ Z) ˆR) (11) 5

6 + ˆϕ [ (1/R 2 ) Ĩ ψ ˆϕ (1/R) ( F/ ϕ) ψ ˆϕ + Ĩ F (1/R) ( F/ ϕ) F ] p ρĝ(sin ϕ ˆR + cos ϕ ˆϕ) ϵ 2 ρ(v )v = 0. III. STEADY STATE III.1. Hydrostatic density and temperature A flux rope requires the existence of a magnetic flux function ψ eff that satisfies B ψ eff = 0, so that a magnetic field line stays on a surface of constant ψ eff. Toroidal laboratory plasmas satisfy J B = p, so B p = 0. The loop gravity breaks this condition. In potential form, ρgĥ = ρ (gh) where h = R sin ϕ, and Choosing a pressure of the form B ( p + ρ gh) = 0. (12) p = P (ψ eff ) exp[ M i gh/2t (ψ eff )], (13) for some function P (ψ eff ) and temperature T = T (ψ eff ), leads to B ψ eff = 0, so ψ eff is a flux function. The density becomes n = N o (ψ eff ) exp( M i gh/2t (ψ eff )), where N o (ψ eff ) P (ψ eff )/2T (ψ eff ). From Eq. (3), B ψ eff = ψ eff ψ ˆϕ + (R o I/R) ψ eff / ϕ + F ψ eff ψ eff ψ. = 0, where the F term is O(ϵ) smaller than the first two. In near-axisymmetry, The factor exp( M i gh/2t ) introduces an important source of nonaxisymmetry. Writing M i gh/2t = (h/r o )/(H LR T (ψ eff )/T o ) defines a hydrostatic density scale height H LR ϵβ o /2Ĝ relative to the loop height R o, where T o is the peak interior value of T (ψ eff ). A gravity Ĝ ϵ2 β o leads to a large scale height H LR 1/2ϵ 1 and a nearly constant exponential around the loop. Larger gravity Ĝ ϵβ o gives H LR 1/2, significantly reducing the loop-top density compared to the loop feet (e at h/r o = 1 = T/T o ; H LR = 9.49 gives 0.9 and 4.48 gives 0.8). The variation is larger in the outer part of the plasma, where T < T o. Toroidal variation can also appear through the nonaxisymmetry of ψ eff ψ 0 + ψ 0s sin ϕ + ψ 0c cos ϕ in the profile functions P, T, and N o. Gravity imposes an extra steady state condition on the temperature. Besides T = T (ψ eff ), possibilities include T = T (Φ g ) for Φ g = gh, leading to p = P (ψ eff ) exp[ Φ g 0 dφ g /T (Φ g )], 6

7 but the solutions have similar properties. Observations of well defined, lasting loop footpoints and high temperature spectral emission curves suggest that a loop flux function exists. Physically, existence of a flux function makes a temperature form T (ψ eff ) likely, because the loop parallel thermal conductivity is typically large. Observationally, the Extreme UltraViolet (EUV) temperature of many coronal loops is nearly constant over the corona [7, 28], as is the temperature predicted by Alfvén wave coronal loop heating models [17 20]. III.2. Elliptic structure The internal magnetic structure is determined by perpendicular force balance. Equation (11) yields an elliptic second order differential equation for the flux ψ, ψ ψ = (R o I/R) ( ψ/ ϕ) ˆϕ (14) R o I Ĩ R 2 p R 2 ρĝ sin ϕ ˆR, to two orders in ϵ. Here 2 (1/R)( / R). The magnetic F terms do not contribute. In axisymmetry, Eq. (14) reduces to the Grad-Shafranov (GS) pseudo-differential equation [23] for ψ(r, Z) in terms of arbitrary flux functions p(ψ) and I(ψ) = 1 + ϵĩ(ψ), ψ = (R o I/2)(dĨ(ψ)/dψ) R2 (dp(ψ)/dψ). (15) At high beta, β o ϵ 1, p balances the Ĩ term at order ϵ 1, while ψ is order ϵ 2. At low beta, β o ϵ 2, all three terms are order ϵ 2. Sophisticated GS solvers are routinely used for fusion plasmas [24, 25], including helical equilibria [26], and analytical solutions continue to be developed. The axisymmetric GS equation is also used to model interplanetary CMEs (ICMEs) [27]. Many 2D solutions with nested surfaces exist for reasonable choices of the right hand side and loop boundary at fixed ϕ. This also holds for Eq. (14). In nonaxisymmetry, a second relation can be obtained from J = (J B + J B ) = 0. The lowest order relation, for high or low beta, [B + (R o Ĝ/B 2 )ρ cos ϕ] 2 ψ = (16) (2/B) p Ẑ (R oĝ/b2 ) sin ϕ ρ Ẑ, expresses the variation of 2 ψ RJ ϕ along the loop. Since B ϵ 1, at high beta all terms are the same order if Ĝ ϵβ o; larger Ĝ is not possible. At smaller Ĝ = O(ϵ2 )β o, the 7

8 equation formally becomes the high beta stellarator relation, [12] (B )( 2 ψ 0) = 2 p 1 Ẑ. It is trivially satisfied in axisymmetry. III.3. High beta, β o ϵ III.3.1. Large Ĝ ϵβ o ϵ 2 Each beta and Ĝ ordering requires a separate expansion in ϵ. Assuming gravity to be the only source of nonaxisymmetry, the ϵ-expansion is related to Fourier expansion in toroidal angle. From B ψ eff = 0 in Sec. III.1, the leading order nonaxisymmetric components of the fluxes ψ 0s,0c and ψ eff 0s,0c are either all nonzero or all zero. In the high and low beta tokamak orderings, the entire loop becomes nonaxisymmetric at leading order for Ĝ ϵβ o. For small enough ϵ, axisymmetry exists at Ĝ ϵ2 β o, but for ϵ it requires smaller Ĝ = O(ϵ 3 )β o, depending on H LR. At high beta, p β o ϵ 1, the force balance (11) at lowest order ϵ 1 yields components ˆϕ cos : 0 = (ρ 0 + ρ 0c2 /2)Ĝ1 (17) ˆR (1) : 0 = ( ˆR o I 0 / ˆR 2 )( Ĩ0/ R) ( p 1 / R) (18) (1/2)ρ 0s Ĝ 1. ˆR sin : 0 = ( ˆR o I 0 / ˆR 2 )( Ĩ0s/ R) ( p 1s / R) (19) (ρ 0 ρ 0c2 /2)Ĝ1. Here subscripts s and c identify the Fourier coefficients of sin ϕ and cos ϕ, respectively, and s2, c2 the sin 2ϕ and cos 2ϕ coefficients. Powers of ϵ have been divided out, creating the factors ˆR R/R o 1 in the denominators. The factor ˆR o I 0 1 identifies the J B ϕ terms. Equation (17) shows Ĝ1 = 0. Then the ˆR Eq. (18), combined with the corresponding Ẑ equation, where ( / Z) replaces ( / R) and Ĝ1 does not appear, gives the high beta tokamak relation for the diamagnetic reduction of B ϕ inside the plasma by the pressure, p 1 = R o I 0 / ˆR 2 Ĩ 0 or Ĩ0 p 1 p 1,bdy, where p 1,bdy is the value on the loop boundary. The hydrostatic scale height H LR = ϵβ o /2Ĝ2 1/2 induces general leading order nonaxisymmetry. The symmetry of the hydrostatic exponential (13) around the top of the loop, ϕ = π/2, leads to nonzero harmonics (1), sin ϕ, cos 2ϕ, sin 3ϕ, etc. At least the first three 8

9 may have similar magnitude for H LR 1/2. The mass density at the top of the loop is ρ o ρ 0 ρ 0s ρ 0c2, with ρ 0s < 0 and ρ 0c2 > 0, compared to the loop feet ρ o ρ 0 + ρ 0c2. From (19), the diamagnetic relation holds for the sin ϕ components Ĩ0s, p 1s, and similarly for cos and cos 2ϕ. Appendix A gives the ϵ 2 equations. They show that nonzero Ĝ2 requires ψ 0s and ψ 0c nonzero, and that this requires all variables to have sin and cos harmonics at leading order. In the ˆϕ cos Eq. (A2), repeated here, 0 = (ρ 0 + ρ 0c2 /2)Ĝ2 p 1s / ˆR + (1/ ˆR 2 ) (20) [ (Ĩ0 + Ĩ0c2/2) ψ 0c ˆϕ + Ĩ 0c ψ 0 ˆϕ], the ρ and p terms do not cancel in general for the forms of Sec. III.1, so that at least one of ψ 0c or Ĩ0c must be nonzero. The ˆϕ cos 2ϕ Eq. (A8) can be used to write the axisymmetric ˆϕ (A1) in two ways, 0 = (1/ ˆR 2 )[(1/2) (Ĩ0 Ĩ0c2/2) ψ 0 ˆϕ + Ĩ 0s ψ 0s ˆϕ] (21) 0 = (1/ ˆR 2 )[(1/2) (Ĩ0 + Ĩ0c2/2) ψ 0 ˆϕ + Ĩ 0c ψ 0c ˆϕ] ρ 0c Ĝ 2. (22) In general, (Ĩ0 ± Ĩ0c2/2) ψ 0 ˆϕ is nonzero for hydrostatic pressures p 1 and p 1c2, through the ϵ 1 diamagnetic relations. Assuming ψ 0c 0, Eq. (21) requires Ĩ0c nonzero. Since p 1s exists, Ĩ0s exists by (19) and Eq. (22) requires ψ 0s 0. Therefore ψ 0s and ψ 0c are nonzero, as is Ĩ0c. Then the ϵ 1 diamagnetic relations show p 1c is nonzero and therefore so is ρ 0c = p 1c /2T 1 (ψ eff ). Thus both sin and cos harmonics exist at leading order in the variables ρ, p, Ĩ, and ψ. Physically, the sin ϕ harmonic of ψ corresponds to expansion at the top of the loop in response to the reduction of the density by gravity. The cos ϕ harmonic describes the asymmetry between the two footpoints, which also appears in ρ 0c and p 1c. Observationally, asymmetry may exist, but has not been studied systematically. For ˆR, the cos 2ϕ Eq. (A10) can be used to eliminate ρ 0c Ĝ 2 from the axisymmetric Eq. (A4), yielding ˆR (1) : 0 = (1/ ˆR 2 )[ 2 ψ 0 ( ψ 0 / R) (23) 9

10 ( ˆR o I 0 / ˆR 2 )( / R)(Ĩ1 + Ĩ1c2) ( / R)(p 2 + p 2c2 ) (1/ ˆR 2 ) ψ 2 0c ( ψ 0c / R) (1/2 ˆR 2 )( / R)[Ĩ2 0c + (Ĩ0 + Ĩ0c2)Ĩ0c2/2)]. The sum of Eqs. (18), (23), and the corresponding Ẑ equations give the high beta elliptic equation (14). The axisymmetric relation 0 = Ĩ 0 ψ 0 ˆϕ or Ĩ0 = Ĩ0(ψ 0 ), Eq. (25), is replaced by Eq. (21). Gravity appears in the elliptic equation for 2ψ 0s, the leading order sin harmonic of ψ or J ϕ 1, ˆRsin : 0 = (ρ 0 ρ 0c2 /2)Ĝ2 p 2s / R (24) ( ˆR o I 0 / ˆR 2 )[( Ĩ1s/ R) + (1/ ˆR)( ψ 0c / Z)] (1/ ˆR 2 )[ ψ 2 0s ( ψ 0 / R) + ψ 2 0 ( ψ 0s / R)] (1/ ˆR 2 )( / R)[(Ĩ0 Ĩ0c2/2)Ĩ0s], and in (A2) for the hydrostatic sin pressure p 1s. In (24), the effective gravitational density (ρ 0 ρ 0c2 /2) is intermediate between the loop-top and footpoint values. The remaining equations are given by (A2) (A6) and (A7) (A10) in Appendix A. III.3.2. Smaller Ĝ ϵ2 β o ϵ 3 Smaller Ĝ ϵ2 β o yields a nearly axisymmetric solution at hydrostatic scale heights H LR = ϵβ o /2Ĝ 1/(2ϵ) 1 where the exponential is nearly axisymmetric. The flux function ψ eff ψ, since ψ eff ψ ˆϕ = O(ϵ). For Ĝ2 = 0 and loop footpoints axisymmetric to lowest order (cos harmonics ψ 0c = ρ 0c = p 1c = 0 and Ĩ0c = 0 by the ϵ 1 diamagnetic relation analogous to (19)), the ϵ 2 equations in Appendix A show all leading order nonaxisymmetric terms ψ 0s,0c, Ĩ0s,0c,0c2, p 1s,1c,1c2, ρ 0s,0c,0c2, and F 1s,1c to be zero. The ϵ 2 Eqs. (A2) (p 1s = 0) and (A7) (p 1c2 = 0) show that footpoint axisymmetry is linked to hydrostatic axisymmetry, H LR 1. The diamagnetic (19) and cos 2ϕ relations then give Ĩ 0s = Ĩ 0c2 = 0. From (A1), ˆϕ(1) : 0 = (1/ ˆR 2 ) Ĩ 0 ψ 0 ˆϕ, (25) so Ĩ0 = Ĩ0(ψ 0 ). From (A3), Ĩ 0 ψ 0s ˆϕ = 0, so in general ψ 0s = 0 (rather than Ĩ 0 = Ĩ0(ψ 0s ) with ψ 0 = ψ 0s ). Then from (25) and (18), p 1 = p 1 (ψ 0 ). 10

11 At ϵ 2, Eq. (A4) reduces to ˆR (1) : 0 = p 2 / R ( ˆR o I 0 / ˆR 2 )( Ĩ1/ R) (26) (1/ ˆR 2 ) ψ 2 0 ( ψ 0 / R). Equations (18) and (26), combined with their Ẑ components and Eq. (27) below, become the high beta axisymmetric GS equation (15) to two orders in ϵ. The ϵ 3 equations are given in Appendix B. Applying leading order axisymmetry, the main equations become ˆϕ (1) : 0 = (1/ ˆR 2 )[ Ĩ 1 ψ 0 ˆϕ (27) + Ĩ 0 ψ 1 ˆϕ] ˆϕ sin : 0 = p 2c / ˆR + (1/ ˆR 2 ) Ĩ 1s ψ 0 ˆϕ (28) ˆϕ cos : 0 = ρ 0 Ĝ 3 p 2s / ˆR (29) + (1/ ˆR 2 )[ Ĩ 1c ψ 0 ˆϕ + Ĩ 0 ψ 1c ˆϕ] ˆR sin : 0 = ρ 0 Ĝ 3 p 3s / R (30) ( ˆR o I 0 / ˆR 2 )[( Ĩ2s/ R) + (1/ ˆR) ( ψ 1c / Z)] (1/ ˆR 2 )[( / R)(Ĩ0Ĩ1s) + ψ 2 1s ( ψ 0 / R) + ψ 2 0 ( ψ 1s / R)] ˆR (1) : 0 = p 3 / R ( ˆR o I 0 / ˆR 2 )( Ĩ2/ R) (31) (1/ ˆR 2 )( / R)(Ĩ0Ĩ1) (1/ ˆR 2 )[( ψ 0 / R) ψ 2 1 +( ψ 1 / R) ψ 2 0 (1/ ˆR)( ψ 0 / R) 2 ]. Eqs. (A5) and (A6) give the diamagnetic relations ˆR sin : 0 = p 2s / R ( ˆR o I 0 / ˆR 2 )( Ĩ1s/ R) (32) ˆR cos : 0 = p 2c / R ( ˆR o I 0 / ˆR 2 )( Ĩ1c/ R). (33) The gravitational term ρ 0 Ĝ 3 again appears in the elliptic equation for the leading order sin components of ψ and p, now 2ψ 1s in (30) and p 2s in (29). The higher order variables are fully nonaxisymmetric and allow sufficient freedom to solve the system at succeeding orders. The structure of a high beta Ĝ ϵ2 β o coronal loop resembles a high beta tokamak[12, 29] with small nonaxisymmetric corrections. Radial stability is provided by the two fixed footpoints combined with axisymmetry. The two footpoint cross sections are completely 11

12 specified by external processes and set the axisymmetric and cos ϕ harmonics (the latter to zero). The hydrostatic density variation must be sufficiently small, H LR (T/T o ) 1, to make the entire loop axisymmetric to leading order. Its shape cannot deform, preventing radial expansion. In contrast, the cross section of a tokamak with no vertical field is only partially constrained. The gravitational radial force is at least one order higher in ϵ than the main J B ϕ and p forces and the force density two orders higher, smaller than a tokamak vertical field. III.4. Low beta, β o ϵ 2 At low beta, β o ϵ 2, the maximal gravity Ĝ ϵβ o ϵ 3 leads to a hydrostatic density scale height H LR 1/2, as at high beta, that produces similar nonaxisymmetry in the leading order pressure p 2, toroidal field Ĩ1, and ρ 0. The order ϵ 1 equations vanish. At ϵ 2, the ˆϕ equations vanish. Gravity Ĝ appears in the ϵ 3 ˆϕ equations and gives them strong similarities to the ϵ 2 high beta, maximal Ĝ equations (21) and (A2) (A3). All variables again require sin and cos nonaxisymmetry at their lowest orders. From Appendix B, with all harmonics p 1X = Ĩ0X 0, the ˆϕ equations are ˆϕ (1) : 0 = (1/ ˆR 2 )[ (Ĩ1 Ĩ1c2/2) ψ 0 ˆϕ (34) + Ĩ 1s ψ 0s ˆϕ] ˆϕ cos : 0 = (ρ 0 + ρ 0c2 /2)Ĝ3 p 2s / ˆR (35) + (1/ ˆR 2 )[ (Ĩ1 + Ĩ1c2) ψ 0c ˆϕ + Ĩ 1c ψ 0 ˆϕ] ˆϕ sin : 0 = p 2c / ˆR + (1/ ˆR 2 )[ Ĩ 1s ψ 0 ˆϕ (36) + (Ĩ1 Ĩ1c2/2) ψ 0s ˆϕ]. Here Eq. (34) has been reduced to the same form as Eq. (21) by using (B8). At ϵ 2, the ˆR equations express the low beta version of the elliptic relation (14), ˆR(1) : 0 = ( / R)(p 2 p 2c2 ) (37) ( ˆR o I 0 / ˆR 2 )( / R)(Ĩ1 Ĩ1c2) (1/ ˆR 2 )[ ψ 2 0 ( ψ 0 / R) + ψ 2 0s ( ψ 0s / R)] ˆRsin : 0 = p 2s / R ( ˆR o I 0 / ˆR 2 )( Ĩ1s/ R) (38) 12

13 ( ˆR o I 0 / ˆR 2 )[( Ĩ1s/ R) + (1/ ˆR)( ψ 0c / Z)] (1/ ˆR 2 )[ ψ 2 0s ( ψ 0 / R) + ψ 2 0 ( ψ 0s / R)] ˆRcos : 0 = p 2c / R (39) ( ˆR o I 0 / ˆR 2 )[( Ĩ1c/ R) (1/ ˆR)( ψ 0s / Z)] (1/ ˆR 2 )[ ψ 2 0c ( ψ 0 / R) + ψ 2 0 ( ψ 0c / R)] ˆRsin2 : 0 = (1/2 ˆR 2 )[ ψ 2 0s ( ψ 0c / R) (40) + ψ 2 0c ( ψ 0s / R)] ˆRcos2 : 0 = p 2c2 / R ( ˆR o I 0 / ˆR 2 ) Ĩ1c2/ R (41) (1/2 ˆR 2 )[ ψ 2 0c ( ψ 0c / R) ψ 2 0s ( ψ 0s / R)]. In Eq. (37), the 2ψ 0c term was eliminated using the cos 2ϕ Eq. (41). Gravity Ĝ3 appears at order ϵ 3 in Eq. (35) for the leading order sin pressure p 2s and field Ĩ 1s, which enter the elliptic equation (38) for the leading order sin ψ 0s. The ˆR equations at order ϵ 3 are given by Appendix B, setting Ĩ0 = Ĩ0s = Ĩ0c = 0. Low beta also allows more axisymmetric solutions at smaller gravity Ĝ O(ϵ2 )β o, stabilized by the fixed footpoints, analogous to the high beta cases of Sec. III.3.2. The variables are axisymmetric at leading order. Now Eq. (37) reduces to the low beta GS equation (15), while (34) gives Ĩ1 = Ĩ1(ψ 0 ) and (13) p 2 = p 2 (ψ 0 ). The gravitational radial force is at least one order higher than the J B and p forces and the force density ρĝ is two orders higher. III.5. Very low beta, β o = O(ϵ 3 ) Steady states are also possible in nearly force-free loops at very low beta. For J B = 0, the radially destabilizing hoop force due to J ϕ B is balanced by J B ϕ, both at order ϵ 2 in the momentum equation. Finite p and gravity appear at β o = O(ϵ 3 ) and Ĝ = O(ϵ4 ). The p is radially destabilizing for centrally peaked p and gravity is stabilizing. The third order components (J B ϕ ) 3 and (J ϕ B ) 3, however, can have either sign and radial stability must be determined by the full solution. For the maximal β o ϵ 3 and Ĝ ϵ4, H LR 1/2 and the loop is nonaxisymmetric. The absolute value of β o is small. For ϵ 0.2, an upper limit for the asymptotic expansion, β o ϵ

14 Loop R o a/r o n o T o B o v A τ A β o Ĝ Ĝ/ϵ 3 Ĝ/ϵ 2 H LR (10 7 m) (10 15 m 3 ) (10 6 K) (G) (10 6 m/s) (s) (10 2 ) (10 4 ) H H H1,H2: short hot r7 loops[20]. Loops 1-5,7-9 are warm loops[19] (3,5,9 are r1 cases, 8 is r4 ; 1,2,4 were not reported there; loop 6 did not converge in the AW heating model). TABLE I. Coronal loops from active region 11564, Sept. 7, 2012 IV. COMPARISON TO OBSERVATIONS THIN LOOPS Quasi-steady state coronal loops are commonly observed in solar active regions. They have thin cross sections, with ϵ 0.02 and typically satisfy the high beta ordering, β ϵ 1. Table I shows representative data for loops in active region NOAA on Sept. 7, 2012, pictured in Fig. 2(a). The numbers are taken from studies[19, 20] of loop heating by Alfvén Wave (AW) fluctuations[30]. The magnetic field line length, loop cross-sectional diameter, and field strength at the top of the loop were obtained from an NLFFF model based on flux rope insertion[9 11]. The warm loops, with T = 1 3 MK, are the best converged cases with the same numbers from [19] (see caption, Table I). Two hot loops, H1 and H2, are from an AW heating study [20] of short hot, T 5 MK, loops. Their values have greater uncertainty, due to the different observational analysis and small sample size. The gravitational model provides a good match to the data. The analytical solutions represent asymptotic limiting cases. The Ĝ solutions at ϵ2 β o and ϵβ o span the observed 14

15 4 (a) (b) H1 H (c) FIG. 2. Coronal loops in three active regions. (a) NOAA on Sept. 7, 2012, Table I. Horizontal width m. Short hot loops H1, H2 in the center. (b) NOAA on May 5, 2010, Table II. Horizontal width m. (c) NOAA on March 7, 2012, Table III; loops determined from potential field magnetic model. Horizontal width m. Gray-scale background shows spectral emission line observations from the Solar Dynamics Observatory (SDO) by AIA with logarithmic scaling, (a),(c) in the 171 A band and (b) in 193 A. Light green (negative) and red (positive) areas show photospheric magnetic flux regions of opposite polarity, directed into and out of the sun, from HMI line of sight magnetograms. Numbered lines show coronal loops that follow bright emission curves. Not all loops converge in the NLFF/potential field or AW heating models (see Tables). 15

16 range of heights. Overall, Ĝ orders the loops by height. Table I is ordered by height R o. Columns 2 6 give input data at the loop top, R o L cor /π from the NLFFF loop length L cor, density n o and temperature T o from converged AW heating values input n e and T final or T max. The field B o is the minimum NLFFF B min over the loop, which occurs at or near the loop-top. The middle columns show calculated MHD parameters v A, τ A, β o, and Ĝ. The last three columns give the ratios of Ĝ with ϵ3 and ϵ 2 and the hydrostatic scale height H LR. The high beta, small Ĝ ϵ 2 β o ϵ 3 ordering gives a good fit to the short hot loops (H1,H2) and the shortest warm loops (4,1,5). The longest, lower density loops (7,2) have larger Ĝ ϵβ o ϵ 2. Intermediate length loops (9,3,8) have intermediate values and properties. The horizontal line approximately separates the two Ĝ scalings. Although individual fits vary, the three ranges of increasing loop height correlate well with increasing Ĝ. Similar results are found for other active regions. Table II shows warm loops from active region NOAA observed on May 5, 2010, from AW study [17], and Table III from NOAA on March 7, 2012, from [18], pictured in Figs. 2(b) and (c), respectively. The last used the potential rather than NLFF field and tested a wider variety of magnetic loops. The fit is summarized in Fig. 3. Overall, Ĝ increases with height. Some systematic differences exist between data sets, but may reflect real differences between active regions. The result also justifies the use of a loop height R o that neglects the distance between the bottom of the corona and the photosphere, h/r o < 0.1 for the considered loops. Typical thin loop lifetimes[7] of some s, generally smaller for longer loops, last significantly longer than the gravity time scale τ G for radial displacement by a diameter 2a, ρg ρ v R / t ρ2a/τ 2 G. At ϵ = 0.02, shorter loops with Ĝ/β o ϵ 2 have τ g 2ϵ 1/2 τ A 100 s and for Ĝ/β o ϵ, τ g 2τ A 140s. The model uses the parameters at the top of the loop. The most sensitive quantity is probably the magnetic field, since β o and Ĝ are proportional to B 2 o. The fit uses magnetic and spectroscopic data from higher resolution observational instruments deployed starting in the late 1990 s, and relies on the force-free NLFF or potential field magnetic models implemented in the CMS code[10, 11] for the loop field. Earlier data for generic active region loops [31] (1992), shows similar trends (Table 2 there), but had uniformly larger values of B. 16

17 Loop R o a/r o n o T o B o v A τ A β o Ĝ Ĝ/ϵ 3 Ĝ/ϵ 2 H LR (10 7 m) (10 15 m 3 ) (10 6 K) (G) (10 6 m/s) (s) (10 2 ) (10 4 ) Loops from Alfvén Wave heating study[17]. Longer loops 2 and 9 did not converge in the AW model. TABLE II. Coronal loops from active region 11067, May 5, 2010 FIG. 3. Ĝ orders the thin loops of Tables I III by height R o. (a) Ĝ/ϵ2 for all loops. (b) Detail for small R o < m, corresponding to box in (a). Smallest R o loops have small Ĝ 3. Open circles/square have low B o (high β o ) and may correspond to larger R o (see Tables). Color online. 17

18 Loop R o a/r o n o T o B o v A τ A β o Ĝ Ĝ/ϵ 3 Ĝ/ϵ 2 H LR (10 7 m) (10 15 m 3 ) (10 6 K) (G) (10 6 m/s) (s) (10 2 ) (10 4 ) * Loops are r2 cases from Table I in AW study[18], using the potential field model. (Loop 4* is ordered by B o, not R o.) Loops 10,11 in Fig. 2(c) did not converge in the potential field model; 9,14,16,17,20,21,22 did not converge in the AW model. Loop 22 did not match an emission curve. They are omitted from the table. TABLE III. Coronal loops from active region 11428, March 7, 2012 V. DISCUSSION V.1. Low beta, fat loops Steady states also exist in the low beta tokamak ordering. Low beta is most easily achieved by fatter loops with larger ϵ, at similar absolute values of beta. At times, thicker loops appear in active regions, along the magnetic neutral line that separates two regions of opposite photospheric magnetic polarity. Some may grow to generate solar flares and Coronal Mass Ejections[3, 32 35]. The early loops[10, 32 34] have typical ϵ and satisfy the low beta ordering β ϵ 2 for betas of a few per cent. Some loops reach stable 18

19 post-flare or post-cme heights of some m, where they gradually decay. Others may shrink, while yet others continue to grow. Some neutral line loops undergo an apparent MHD kink instability during their initial growth, which twists the entire loop by some 180 degrees at mid-height [35]. After a flare, some stabilize at heights comparable to untwisted loops (e.g., m in [35]). Although not simple loops, the overall gravitational force balance is similar. More observational data is needed to evaluate thicker loop steady states. Observations of low, early stage neutral line loops suggests greater complexity and nonaxisymmetry than thin loops. As a first estimate, if the active region thick loops are assumed to have parameters similar to thin loops at the same height, or perhaps somewhat higher B o at comparable height [35], the Tables in Sec. IV suggest steady state heights for ϵ = 0.1 that range up to some 10 8 m. Observed after-flare steady state heights are broadly consistent with this limit. Other types of thicker loops exist, such as long-lived solar filaments[36] and prominences outside active regions, that may also lead to flares and/or CMEs. They have a complicated structure and are not discussed here. V.2. Radial expansion and nonaxisymmetry Simple loop steady states are parametrized in terms of (ϵ, β o, Ĝ), or (R o, a, n o, T o, B o ). In a torus[12], the radial expansion forces switch magnitudes in different regimes of β o and ϵ. Loop forces also depend on Ĝ and all three parameters determine the degree of loop nonaxisymmetry. The fixed footpoints, which specify the loop boundaries, magnetic fields, toroidal current, and internal profiles P (ψ eff ), T (ψ eff ), N o (ψ eff ) can constrain the radial stability, unlike the full torus. In loops with leading order axisymmetry at smaller Ĝ/β o < ϵ 2, radial stability is determined not by the local radial forces on each toroidal segment, but by the axisymmetry constraint and fixed footpoints. At larger values of ϵ, Ĝ/β o ϵ k with a k 3 that makes H LR 1/2ϵ k 2 1, may be required for sufficient axisymmetry. At the maximal Ĝ/β o ϵ, the loop is significantly nonaxisymmetric and the local radial force balance, including gravity, determines stability. Loop stabilization by fixed axisymmetric footpoints, without gravity, was recognized early[13], but differs from the present model. Stability was attributed to magnetic field line tension, assigned mainly to the J B ϕ term, which balanced the volume-integrated radial 19

20 forces. In an ϵ-ordering, the important factor is axisymmetry to leading order. Axisymmetry was assumed in early work[13, 14], making it equivalent to the very low beta case of Sec. III.5. When the loop forces and boundary conditions are axisymmetric to leading order, global axisymmetry dominates and the local radial forces on a toroidal segment need not balance. Sufficiently strong gravity breaks the axisymmetry and the local radial force balance, including gravity, again dominates. The local radial expansion force on the loop can be calculated from the ˆR component of the momentum equation, integrated over the plasma volume. It is now a function of ϕ. Terms p, J B ϕ, and J ϕ B give rise to the tire pressure, 1/R or B ϕ, and hoop or toroidal current expansion forces, respectively, due to the geometrical differences between the inside and outside of the torus[12]. For brevity, the paper considers only loops with density and pressure higher than the surrounding corona. Then in the low and high beta orderings, the sum of the p and J B ϕ radial forces over a small toroidal sector is destabilizing[12] (see below). The hoop force is destabilizing. Steady state requires an extra stabilizing process, such as gravity, an externally supplied vertical field, or fixed footpoints and axisymmetry. If instead the pressure has a minimum inside the loop, the minor-radius gradient r p > 0 is still destabilizing, but its contribution to Ĩ changes sign and J B ϕ becomes stabilizing, changing the force balance. The radial force density due to p is p R = (p R)+p 2 R = (p R)+p/R. The volume integral over a toroidal sector R ϕ is F P = d 3 x p R = ds n R p + d 3 x p/r. The surface integral vanishes, since ds n ˆR = 0 on the sides at constant ϕ and p = p bdy is small on the loop boundary, so F P d 3 x p/r ϕ p da, where d 3 x = R dϕ drdz = R dϕda. The 1/R force arises from J B ϕ ˆR = B ϕ J ˆϕ R (R o I/R 2 ) Ĩ R = (R o I Ĩ) (1/R) = (R oi Ĩ (1/R)) R oi Ĩ 2 (1/R), where I 1. The divergence vanishes on integration since Ĩbdy = 0 and F 1/R = d 3 x (R o I Ĩ/R3 ). It depends on the sign of Ĩ, which is zero on the plasma boundary and has an extremum in the interior. It has a positive, stabilizing contribution from J ϕ B and a negative one from a centrally peaked pressure with r p > 0 (Eqs. (14) or (15)). At high beta, F 1/R d 3 x (p p bdy )/R > 0 and for small p bdy < p max, F 1/R F P. The hoop force is difficult to evaluate, even in axisymmetry. In J ϕ B ˆR 20

21 (1/R 2 )RJ ϕ ( ψ/ R), the poloidal field (1/R)( ψ/ R) nearly cancels across plasma, except for the higher order effects of 1/R and the outward Shafranov shift of the magnetic axis due to pressure. The current density J ϕ (1/R) ψ is often peaked off axis at R > R o. The local force on a toroidal segment remains the same order as for the axisymmetric, circular cross section torus [12], F H ( ϕ/2π)(ip/2)( / R 2 o )(L e + L i ) ϵ 2, in terms of the loop current I p, external inductance L e = µ o R o (ln(r o /a) ), and internal inductance L i = (2/Ip) 2 d 3 x Bθ 2/2µ o R o (2/Ip) 2 Bθ 2 da. Since I p B θ ϵb o, F H ϵ 2, one order smaller than F P and F 1/R at high beta and comparable at low beta. Gravity ρĝ( ˆR sin ϕ + ˆϕ cos ϕ) leads to a radial force F G = ϕ R o Ĝ sin ϕ ρ da. It, like the vertical field, does not depend on large- and small-r geometric differences, so gravity appears in the momentum equation at one order in ϵ higher than J B or p to exert a comparable force on the loop. V.3. Maximum stable height At finite pressure β o ϵ 2, gravitational nonaxisymmetry has far-reaching effects. Although steady states exist for smaller gravity, the maximal gravity Ĝ ϵβ o is expected to determine the maximum stable loop height. As a loop expands away from the photosphere, n o and B generally decrease. The gravity parameter Ĝ n o/bo 2 will tend to grow, at a faster rate than β o if the loop temperature decreases with the expansion in volume. Eventually Ĝ will exceed the stable value ϵβ o. At this point the strong downward acceleration on the plasma along the loop overcomes the temperature buoyancy and the density at the top of the loop falls below the level where the local gravitational force ρĝ can counteract the radial expansion. Based on the thin loop parameters, the maximum stable height appears to be relatively low, on the order of 10 8 m compared to the solar radius R = m. Loops above this height will expand, under weak gravitational restraint. This is consistent with the observation that most sufficiently large loops tend to expand. Factors external to the loop that may not be easily observable can also influence the existence of a steady state. Even if the loop parameters fall in the steady state range, a mismatch in the required degree of nonaxisymmetry at the photosphere or in the surrounding corona can prevent the existence of a steady state. 21

22 V.4. Extensions The motivation for the analysis is to find numerical solutions for loop steady states. The results demonstrate the necessity of the analytical study. Nonaxisymmetric toroidal steady states described by multiple parameters and free functions are extremely difficult to find by trial and error. The lack of good observations of loop parameters, shape, and profiles increases the difficulty. Numerical solutions could be computed using existing nonlinear fusion simulation codes such as M3D[21, 22] that embed a toroidal plasma in an MHD resistive vacuum. They handle nonaxisymmetry and require relatively small modification for gravity. Nonlinear simulation may be necessary to find nonaxisymmetric steady states and fully investigate radial stability. VI. SUMMARY A consistent ideal MHD model for the steady states of a simple coronal loop with gravity, modeled as a partial-torus magnetic flux rope that confines a plasma with density and pressure higher than its surroundings, is developed through expansion in small inverse aspect ratio ϵ = a/r o. Steady state requires stabilization of the radial (major radius) expansion instability of a curved, current-carrying plasma. Three parameters at the top of the loop, ϵ, the plasma beta β o, and a gravity Ĝ = ga/v2 A, where g is the acceleration due to gravity, a the average minor radius, and v A the shear Alfvén velocity, provide a unified picture of the steady states of simple loops. The solar gravity has important effects linked to nonaxisymmetry, previously neglected. In the high beta β o ϵ 1 and low beta β o ϵ 2 tokamak orderings, steady states exist at Ĝ ϵβ o and O(ϵ 2 )β o. Larger Ĝ generally corresponds to longer loops. At the maximal Ĝ ϵβ o, loops are nonaxisymmetric and radially stabilized by gravity acting on the upper part of the loop. Loops with Ĝ O(ϵ2 )β o are more axisymmetric and structurally resemble high- or low-beta tokamaks. Radial stability is provided primarily by the fixed loop footpoints combined with leading order axisymmetry. Very low beta steady states, β o = O(ϵ 3 ), may also exist, but are sensitive to the higher order force balance. Sufficiently long loops tend to have Ĝ > ϵβ o (B o and n o decrease with increasing height above the photosphere, while the loop temperature decreases with increasing loop volume). This is unstable to expansion because the strong gravity reduces the loop-top mass density 22

23 below the level at which the gravitational force ρĝ ˆR can counteract expansion, consistent with the observation that most sufficiently long loops expand. Thin coronal loops commonly observed in solar active regions have typical ϵ 0.02 and fit the high beta ordering. Gravity Ĝ orders the loops by height and the cases Ĝ/β o ϵ and ϵ 2 span the range of heights. Fatter loops with ϵ may satisfy the low beta ordering, including the loops that appear along magnetic neutral lines in active regions. After a flare or CME, some thick loops are observed to reach steady states at heights around < R o 10 8 m, consistent with rough estimates of the maximum stable height. Complete solution requires numerical simulation, but it is nontrivial in nonaxisymmetry and needs better observational constraints on the loop structure. ACKNOWLEDGMENTS Work partially supported by U.S. DOE OFES contract DE-SC [1] R. Rosner, W. H. Tucker, and G. S. Vaiana, Astrophys. J (1978). [2] Issue on Self-organization in magnetic flux ropes, Plasma Phys. Controll. Fusion 56, (2014), and references therein. [3] P. F. Chen, Living Solar Phys. 8, 1 (2011). [4] Issue on Flux Rope Structure of Coronal Mass Ejections, Solar Phys. 284:1 (2013), and references therein. [5] A. Vourlidas, Plasma Phys. Controll. Fusion 56, 1 (2014). [6] j. V. Hollweg, Solar Phys (1981). [7] J. A. Klimchuk, J. T. Karpen, and S. K. Antiochus, Astrophys. J. 714, 1239 (2010). [8] C. J. Schrijver, M. L. DeRosa, T. Metcalf, G. Barnes, B. Lites, T. Tarbell, J. McTiernan, G. Valori, T. Wiegelmann, M. S. Wheatland, T. Amari, G. Aulanier, P. Dmoulin, M. Fuhrmann, K. Kusano, S. Régnier, J. K. Thalmann, Astrophys. J (2008). [9] A. A. van Ballegooijen,Astrophys. J (2004). [10] M. G. Bobra, A. A. van Ballegooijen, and E. E. DeLuca, Astrophys. J (2008). [11] Y. Su, V. Surges, A. van Ballegooijen, E. DeLuca, and L. Golub, Astrophys. J (2011). 23

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25 [32] V. S. Titov and P. Démoulin, Astron. Astrophys (1999) [33] T. Amari, J. F. Luciani, Z. Mikić, J. Linker, Astrophys. J. Lett. 529 L49 (2000). [34] Y. Yan, Y. Deng, M. Karlický, Q. Fu, S. Wang, and Y. Liu, Astrophys. J. Lett. 551 L115 (2001). [35] T. Torök and B. Kliem, Astrophys. J. 630 L97 (2005). [36] M. J. Aschwanden, Physics of the Solar Corona, Praxis Publishing, Chichester UK (2006). Appendix A: ϵ 2 equations At order ϵ 2, momentum equation (11) becomes ˆϕ (1) : 0 = (1/2)ρ 0c Ĝ 2 + (1/ ˆR 2 ) Ĩ 0 ψ 0 ˆϕ (A1) +(1/2 ˆR 2 )[ Ĩ 0c ψ 0c ˆϕ + Ĩ 0s ψ 0s ˆϕ] ˆϕ cos : 0 = (ρ 0 + ρ 0c2 /2)Ĝ2 p 1s / ˆR (A2) +(1/ ˆR 2 )[ (Ĩ0 + Ĩ0c2/2) ψ 0c ˆϕ + Ĩ 0c ψ 0 ˆϕ] ˆϕ sin : 0 = p 1c / ˆR + (1/ ˆR 2 )[ Ĩ 0s ψ 0 ˆϕ (A3) + (Ĩ0 Ĩ0c2/2) ψ 0s ˆϕ] ˆR (1) : 0 = (1/ ˆR 2 ) 2 ψ 0 ( ψ 0 / R) (A4) ( ˆR o I 0 / ˆR 2 )( Ĩ1/ R) ( p 2 / R) (1/2)ρ 0s Ĝ 2 (1/2 ˆR 2 )[ 2 ψ 0c ( ψ 0c / R) + 2 ψ 0s ( ψ 0s / R)] (1/2 ˆR 2 )( / R)(Ĩ2 0c + Ĩ2 0s + Ĩ2 0c2) ˆR sin : 0 = (ρ 0 ρ 0c2 /2)Ĝ2 ( p 2s / R) (A5) ( ˆR o I 0 / ˆR 2 )[( Ĩ1s/ R) + (1/ ˆR)( ψ 0c / Z)] (1/ ˆR 2 )[ 2 ψ 0s ( ψ 0 / R) + 2 ψ 0 ( ψ 0s / R)] (1/ ˆR 2 )( / R)(Ĩ0Ĩ0s Ĩ0c2Ĩ0s/2) ˆR cos : 0 = ( p 2c / R) (A6) ( ˆR o I 0 / ˆR 2 )[( Ĩ1c/ R) (1/ ˆR)( ψ 0s / Z)] (1/ ˆR 2 )[ ψ 2 0c ( ψ 0 / R) + ψ 2 0 ( ψ 0c / R)] (1/ ˆR 2 )( / R)(Ĩ0Ĩ0c + Ĩ0c2Ĩ0c/2) 25

26 ˆϕ sin2 : 0 = (1/2)ρ 0s Ĝ 2 + 2p 1c2 / ˆR (A7) +(1/2 ˆR 2 )[ Ĩ 0c ψ 0s ˆϕ Ĩ 0s ψ 0c ˆϕ] ˆϕ cos2 : 0 = (1/2)ρ 0c Ĝ 2 (A8) +(1/2 ˆR 2 )[ Ĩ 0c ψ 0c ˆϕ Ĩ 0s ψ 0s ˆϕ] +(1/2 ˆR 2 ) Ĩ 0c2 ψ 0 ˆϕ ˆR sin2 : 0 = (1/2)ρ 0c Ĝ 2 ( ˆR o I 0 / ˆR 2 )( Ĩ1s2/ R) (A9) (1/2 ˆR 2 )[ 2 ψ 0s ( ψ 0c / R) + 2 ψ 0c ( ψ 0s / R)] (1/2 ˆR 2 )( / R)(Ĩ0cĨ0s) p 2s2 / R ˆR cos2 : 0 = (1/2)ρ 0s Ĝ 2 ( ˆR o I 0 / ˆR 2 )( Ĩ1c2/ R) (A10) (1/2 ˆR 2 )[ ψ 2 0c ( ψ 0c / R) ψ 2 0s ( ψ 0s / R)] (1/2 ˆR 2 )( / R)[Ĩ0Ĩ0c2 + (Ĩ2 0c Ĩ2 0s)/2] ( p 2c2 / R). The hydrostatic cos 2ϕ harmonics Ĩ0c2,p 1c2,ρ 0c2 are retained, as discussed in Sec. III.3.1. Factors (1/2) come from cos 2 ϕ = (1/2)(1 + cos 2ϕ), sin 2 ϕ = (1/2)(1 cos 2ϕ) and cos 2ϕ sin ϕ = (1/2)(sin 3ϕ sin ϕ), cos 2ϕ cos ϕ = (1/2)(cos 3ϕ + cos ϕ), and in the sin 2ϕ equations from (1/2) sin 2ϕ = sin ϕ cos ϕ. The Ẑ equations corresponding to the ˆR ones are not shown. Appendix B: ϵ 3 equations for Ĝ ϵ3 At ϵ 3, the equations for Ĝ3 at high or low beta are ˆϕ (1) : 0 = (1/2)ρ 0c Ĝ 3 (B1) + (1/ ˆR 2 )[ Ĩ 1 ψ 0 ˆϕ + Ĩ 0 ψ 1 ˆϕ +(1/2)( Ĩ 1c ψ 0c ˆϕ + Ĩ 1s ψ 0s ˆϕ + Ĩ 0c ψ 1c ˆϕ + Ĩ 0s ψ 1s ˆϕ)] ˆϕ cos : 0 = (ρ 0 + ρ 0c2 /2)Ĝ3 p 2s / ˆR (B2) + (1/ ˆR 2 )[ (Ĩ1 + Ĩ1c2/2) ψ 0c ˆϕ + Ĩ 1c ψ 0 ˆϕ + Ĩ 0 ψ 1c ˆϕ + Ĩ 0c (ψ 1 + ψ 1c2 /2) ˆϕ +(1/2) Ĩ 0s ψ 1s2 ˆϕ] ˆϕ sin : 0 = p 2c / ˆR + (1/ ˆR 2 )[ Ĩ 1s ψ 0 ˆϕ (B3) 26