EQUILIBRIUM MODELS OF CORONAL LOOPS THAT INVOLVE CURVATURE AND BUOYANCY

The forces acting on an isolated, thin, magnetic fibril embedded in a field-free atmosphere have been previously derived by a variety of authors (Spruit 1981; Choudhuri 1990; Cheng 1992). Here we rederive the equilibrium forces in the presence of a magnetized external atmosphere. Following Spruit (1981), we average the MHD momentum equation over the cross-section of the fibril, with the goal of deriving an equation which describes the force per unit length along the fibril. In equilibrium, there will be a balance of these forces, and this balance will specify the shape of the fibril and its thermodynamic properties.

We start by defining what we mean by a cross-sectional average. Consider a magnetized corona that is in static equilibrium. Select any compact bundle or sheaf of field lines that does not contain any separatrices that may exist. Let A be a cross-sectional surface that is everywhere perpendicular to the magnetic field (i.e., perpendicular to ${{{\boldsymbol {\hat{s}}}}}$) and let da be a differential area of this surface. We define the cross-sectional average of a general quantity f in the natural way,

$$\begin{equation} \bar{f} \equiv \frac{1}{A} \int \limits _{ A}\!\!\!\!\int f \, da. \end{equation} \tag{ 7 }$$

We write the MHD momentum equation using a formulation of the Lorentz force that directly acknowledges that the magnetic force is transverse to the field itself,

$$\begin{equation} \rho \frac{D{{{\boldsymbol {v}}}}}{Dt} = -{{{\boldsymbol {\nabla }}}}{P} - {{{\boldsymbol {\nabla }}}}_\perp \left(\frac{B^2}{8\pi }\right) + \frac{B^2}{4\pi R}{{{\boldsymbol {\hat{k}}}}} + \rho {{{\boldsymbol {g}}}}, \end{equation} \tag{ 8 }$$

where

$$\begin{equation} {{{\boldsymbol {\nabla }}}}_\perp \equiv {{{\boldsymbol {\hat{k}}}}}({{{\boldsymbol {\hat{k}}}}} \cdot {{{\boldsymbol {\nabla }}}}) + {{{\boldsymbol {\hat{q}}}}}({{{\boldsymbol {\hat{q}}}}} \cdot {{{\boldsymbol {\nabla }}}}). \end{equation} \tag{ 9 }$$

In Equation (8), the gas pressure, magnetic-field strength, and mass density are P, B, and ρ, respectively, and D/Dt denotes the Lagrangian time derivative. When in static balance, the acceleration is identically zero and the longitudinal and transverse components of this equation have the form,

$$\begin{equation} - \frac{\partial {P}}{\partial {s}} + \rho g_\parallel = 0, \end{equation} \tag{ 10 }$$

$$\begin{equation} -{{{\boldsymbol {\nabla }}}}_\perp \left(P + \frac{B^2}{8\pi }\right) + \frac{B^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \rho {{{\boldsymbol {g}}}}_\perp = 0. \end{equation} \tag{ 11 }$$

where $g_\parallel \equiv {{{\boldsymbol {g}}}} \cdot {{{\boldsymbol {\hat{s}}}}}$ and ${{{\boldsymbol {g}}}}_\perp \equiv {{{\boldsymbol {g}}}} - g_\parallel {{{\boldsymbol {\hat{s}}}}}$.

We will average these two components separately. The average of the longitudinal equation,

$$\begin{equation} - \frac{\partial {\overline{P}}}{\partial {s}} + \overline{\rho g_\parallel } = 0, \end{equation} \tag{ 12 }$$

simply expresses hydrostatic balance along field lines. The average of the transverse equation is more interesting,

$$\begin{equation} \frac{1}{A} \int \limits _{ A}\!\!\!\!\int {{{\boldsymbol {\nabla }}}}_\perp \left(P + \frac{B^2}{8\pi } \right) \, da = \frac{1}{A} \int \limits _{ A}\!\!\!\!\int \left(\frac{B^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \rho {{{\boldsymbol {g}}}}_\perp \right) da. \end{equation} \tag{ 13 }$$

We can now invoke the two-dimensional (2D) form of the divergence theorem to convert the surface integral over A on the left-hand side into a contour integral around ∂A, the boundary of A,

$$\begin{equation} \frac{1}{A} \oint \limits _{\partial A} \left(P + \frac{B^2}{8\pi } \right) {{{\boldsymbol {\hat{n}}}}} \, dl = \frac{1}{A} \int \limits _{ A}\!\!\!\!\int \left(\frac{B^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \rho {{{\boldsymbol {g}}}}_\perp \right) da. \end{equation} \tag{ 14 }$$

The unit vector ${{{\boldsymbol {\hat{n}}}}}$ is the outward normal to the tube's bounding surface. The differential dl is the differential path length around ∂A. The use of the divergence theorem here expresses the well-known property that the net pressure force acting on a parcel of fluid is the integral of the normal pressure acting on the fluid parcel's outer surface. This equation must hold for any bundle of field lines that we might choose. Now for clarity, select a specific bundle of field lines and label the resulting balance equation with the subscript "e,"

$$\begin{equation} \frac{1}{A} \oint \limits _{\partial A} \left(P_{\rm e}+ \frac{B_{\rm e}^2}{8\pi } \right) {{{\boldsymbol {\hat{n}}}}} \, dl = \frac{1}{A} \int \limits _{ A}\!\!\!\!\int \left(\frac{B_{\rm e}^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \rho _{\rm e}{{{\boldsymbol {g}}}}_\perp \right) da. \end{equation} \tag{ 15 }$$

Note that the left hand side of Equation (15) only depends on the value of the total pressure around the outer surface of the bundle.

Now, we construct an embedded magnetic fibril in the following manner. Take the same bundle of field lines as before and increase or decrease the field strength within the bundle. Simultaneously change the mass density, gas pressure, and temperature within the bundle such that the total pressure remains continuous across the outer surface of the bundle and the bundle remains in equilibrium. Since the fibril is in pressure and force balance, the atmosphere outside the fibril remains unchanged. Therefore, Equation (15) is still valid with the forces on the right-hand side interpreted as the forces that would have existed in the absence of the fibril.

During the generation of this fibril, the geometry of the field has not changed and the variables within the fibril are still continuous (but not continuous across the outer surface). Therefore, Equation (14) must still hold for the fibril. We label this new equation with a subscript "0,"

$$\begin{equation} \frac{1}{A} \oint \limits _{\partial A} \left(P_0 + \frac{B_0^2}{8\pi } \right) {{{\boldsymbol {\hat{n}}}}} \, dl = \frac{1}{A} \int \limits _{ A}\!\!\!\!\int \left(\frac{B_0^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \rho _0 {{{\boldsymbol {g}}}}_\perp \right) da. \end{equation} \tag{ 16 }$$

Since the total pressure is continuous across the outer interface of the fibril, the pressure integrals on the left-hand sides of Equations (15) and (16) are equivalent. Therefore, if we subtract Equations (15) from Equations (16), the left-hand sides cancel,

$$\begin{equation} \frac{1}{A} \int \limits _{ A}\!\!\!\!\int \left[ \frac{B_0^2-B_{\rm e}^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \left(\rho _0 -\rho _{\rm e}\right) {{{\boldsymbol {g}}}}_\perp \right] da = 0. \end{equation} \tag{ 17 }$$

This equation expresses the force balance in terms of the cross-sectional averages and is an extension of Archimedes' principle valid for a magnetized atmosphere. The first term is the combination of the internal tension and the net contribution of the external magnetic pressure acting on the outside of the fibril. The second term is buoyancy, the combination of gravity and the external gas pressure acting on the fibril's surface.

We have not yet made any assumptions about the thinness of the magnetic fibril. We will do so now, by assuming that the magnetic field strengths, densities, and coordinate unit vectors are constant across the cross section. In this case, we can replace the cross-sectional averages by their axial values,

$$\begin{equation} \frac{\partial {P_0}}{\partial {s}} = \rho _0 g_\parallel, \end{equation} \tag{ 18 }$$

$$\begin{equation} \frac{B_0^2-B_{\rm e}^2}{4\pi R} {{{\boldsymbol {\hat{k}}}}} + \left(\rho _0 -\rho _{\rm e}\right) {{{\boldsymbol {g}}}}_\perp = 0, \end{equation} \tag{ 19 }$$

where we now use the shorthand that all variables will be evaluated on the axis of the fibril unless otherwise stated. If one prefers, one could expand all of the variables in Taylor series in the radial distance from the axis. After integration, the thin-tube approximation dictates that only the terms that are of leading order in the cross-sectional area A are retained. This more formal approach produces the same set of equations.

These two force balance equations are very general; we did not assume that either of the internal or external fields were force-free, nor did we assume that the loop is torsionless. Therefore, these equations are valid for a fibril that describes a fully 3D curve through an equilibrium atmosphere filled with a general 3D external magnetic field. The longitudinal component simply represents hydrostatic balance along field lines. The transverse component is a balance between magnetic and buoyancy forces. Note that both the magnetic and buoyancy forces are normal to the surface of the tube. Our intuition may tell us that buoyancy is aligned with gravity; however, this is only true for bodies with closed symmetric surfaces, such as a sphere. Further, just as buoyancy can point upward or downward depending on whether the fibril is over- or underdense, the net magnetic force can point up or down depending on the relative strength of the magnetic field inside and outside the fibril.

We now invoke the assumption that the fibril is torsionless (τ = 0) and vertically oriented, therefore lacking forces in the binormal direction. Under this assumption, the longitudinal and transverse components of gravity in Equations (18) and (19) can be expressed in terms of the vertical unit vector, ${{{\boldsymbol {\hat{z}}}}}$,

$$\begin{equation} \frac{\partial {P_0}}{\partial {s}} = - g \rho _0 \left({{{\boldsymbol {\hat{z}}}}} \cdot {{{\boldsymbol {\hat{s}}}}} \right), \end{equation} \tag{ 20 }$$

$$\begin{equation} \frac{B_0^2 - B_{\rm e}^2}{4\pi } R^{-1} = g \left(\rho _0 - \rho _{\rm e}\right) ({{{\boldsymbol {\hat{z}}}}} \cdot {{{\boldsymbol {\hat{k}}}}}), \end{equation} \tag{ 21 }$$

where we have used ${{{\boldsymbol {g}}}}_\perp = ({{{\boldsymbol {g}}}} \cdot {{{\boldsymbol {\hat{k}}}}}) {{{\boldsymbol {\hat{k}}}}}$, which holds in equilibrium when the fibril is confined to the x–z plane.

Equation (20) expresses hydrostatic balance in the tangential or axial direction ${{{\boldsymbol {\hat{s}}}}}$, whereas Equation (21) describes the balance in the transverse direction of the principal normal ${{{\boldsymbol {\hat{k}}}}}$. The transverse equation constrains the density contrast, ρ₀ − ρ_e, through the balance of magnetic and buoyancy forces. This balance can be characterized by a magnetic Bond number (Jain & Hindman 2012),

$$\begin{equation} \varepsilon \equiv 4\pi g X \frac{\rho _0 - \rho _{\rm e}}{B_0^2 - B_{\rm e}^2} \, \end{equation} \tag{ 22 }$$

which is a nondimensional ratio of the buoyancy to the magnetic forces. The length scale that we have used in this definition, 2X, is the footpoint separation, where x = ±X is the location of each of the fibril's footpoints in the photosphere. The magnetic Bond number appears in calculations of the Rayleigh–Taylor instability when one or more of the fluid layers are filled with a horizontal field, providing the critical wavenumber below which instability ensues (Chandrasekhar 1961).

Equation (21) expresses a condition on the radius of curvature R of the fibril's axis in terms of the magnetic Bond number ε and the geometry of the fibril (i.e., the direction of the principal normal relative to gravity ${{{\boldsymbol {\hat{z}}}}} \cdot {{{\boldsymbol {\hat{k}}}}}$),

$$\begin{equation} R^{-1} = \frac{\varepsilon }{X} ({{{\boldsymbol {\hat{z}}}}} \cdot {{{\boldsymbol {\hat{k}}}}}). \end{equation} \tag{ 23 }$$

To proceed we need expressions for the Frenet unit vectors, ${{{\boldsymbol {\hat{s}}}}}$ and ${{{\boldsymbol {\hat{k}}}}}$, the radius of curvature, R, and the arclength, s. If z₀(x) is the height of the fibril above the photosphere, given as a function of the horizontal coordinate x, and the fibril's axis is traced by the position vector ${{{\boldsymbol {r}}}}_0(x) = x \, {{{\boldsymbol {\hat{x}}}}} + z_0(x) \, {{{\boldsymbol {\hat{z}}}}}$, these quantities can be easily derived from Equations (2) and (3),

$$\begin{equation} {{{\boldsymbol {\hat{s}}}}} = \frac{{{{\boldsymbol {\hat{x}}}}} + z_0^\prime (x){{{\boldsymbol {\hat{z}}}}}}{s^\prime (x)}, \end{equation} \tag{ 24 }$$

$$\begin{equation} {{{\boldsymbol {k}}}} = -z_0^{\prime \prime }(x) \, \frac{z_0^\prime (x) {{{\boldsymbol {\hat{x}}}}} - {{{\boldsymbol {\hat{z}}}}}}{s^\prime (x)^4}, \end{equation} \tag{ 25 }$$

$$\begin{equation} R(x) \equiv \left|{{{\boldsymbol {k}}}}\right|^{-1} = \frac{s^\prime (x)^3}{\left|z_0^{\prime \prime }(x)\right|}, \end{equation} \tag{ 26 }$$

$$\begin{equation} s^\prime (x) = \sqrt{1 + z_0^{\prime }(x)^2}, \end{equation} \tag{ 27 }$$

where primes denote differentiation with respect to the photospheric coordinate x. If we insert Equations (25) and (26) into Equation (23), we obtain a nonlinear ordinary differential equation for the height of the fibril z₀(x),

$$\begin{equation} \frac{z_0^{\prime \prime }(x)}{1+z_0^\prime (x)^2} = \frac{\varepsilon }{X}. \end{equation} \tag{ 28 }$$

A quick examination of this equation reveals that the fibril will be locally concave or convex depending on the sign of the magnetic Bond number and that points of inflection correspond to vanishing magnetic Bond number. The magnetic Bond number is proportional to the signed curvature. A stable fibril with a single concave arch will have a negative Bond number everywhere, whereas a multi-arched structure will have a magnetic Bond number that changes sign.

The temperature, density, and gas pressure within the loop can all differ from the surrounding corona. We define the contrasts for these properties in the following manner:

$$\begin{equation} \Delta \rho (s) \equiv \rho _0(s)-\rho _{\rm e}(s), \end{equation} \tag{ 29 }$$

$$\begin{equation} \Delta P(s) \equiv P_0(s)-P_{\rm e}(s), \end{equation} \tag{ 30 }$$

$$\begin{equation} \Delta T(s) \equiv T_0(s)-T_{\rm e}(s). \end{equation} \tag{ 31 }$$

Technically the external variables are all functions of two spatial coordinates—e.g., x and z. In all of these definitions, however, the external variable is evaluated at the location of the fibril. We reinforce this notation by expressing these variables as a function of the path length alone. Only the contrasts in the mass density Δρ, gas pressure ΔP, and magnetic pressure ΔB²/8π play an active role in the force balance. The temperature contrast ΔT is a dependent variable that can be derived from these active variables post facto. There are four primary equations that provide all of the inter-relations between these properties of the loop:

$$\begin{eqnarray} {\rm Pressure\ Continuity} \quad \Delta P &=& -\frac{\Delta B^2}{8\pi }, \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} {\rm Hydrostatic\ Balance} \quad \; \frac{d{\Delta P}}{d{z_0}} &=& -g \Delta \rho, \end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray} {\rm Transverse\ Force\ Balance} \quad \frac{\varepsilon }{X} &\,{=}\,& \,{-}\,\frac{g}{2} \, \frac{\Delta \rho }{\Delta P} \,{=}\, \frac{z_0^{\prime \prime }}{1+(z_0^{\prime })^2}, \qquad \end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray} {\rm Field\ Strength\ Deviation} \quad \; \frac{\Delta B^2}{8\pi } &=& \alpha \frac{B_{\rm e}^2}{8\pi }. \end{eqnarray} \tag{ 35 }$$

The first of these is a direct consequence of total pressure continuity between the internal and external fluids. The second results from transforming Equation (20) from the path length variable s to the fibril height z₀. The third relation, Equation (34), arises from a combination of the definition of the magnetic Bond number (22) and the equation of transverse force balance (28), where the magnetic-pressure contrast has been replaced through use of pressure continuity, Equation (32). The last equation is a restatement of Equation (1) which ensures that the loop and the external magnetic field share a common bounding flux surface. The temperature contrast ΔT can be found by using the ideal gas law (P = R_gasρT),

$$\begin{equation} \Delta T = \frac{g(h-H_{\rm e})}{R_{\rm gas}} \frac{\Delta \rho }{\rho _0}, \end{equation} \tag{ 36 }$$

where H_e is the external corona's pressure scale height, H_e = R_gasT_e/g, and h is the scale height of the gas-pressure contrast,

$$\begin{equation} h^{-1} \equiv -\frac{2\varepsilon }{X} = - \frac{1}{\Delta P} \frac{d{\Delta P}}{d{z_0}}. \end{equation} \tag{ 37 }$$

Conveniently, these equations can be solved analytically to obtain the contrast variables as a function of the fibril's shape. If we combine Equations (33) and (34), we obtain a differential equation that relates the pressure contrast to the derivative of the fibril's height,

$$\begin{equation} \frac{1}{\Delta P} \frac{d{\Delta P}}{d{z_0}} = \frac{2 z_0^{\prime \prime }(x)}{1+z_0^{\prime }(x)^2}. \end{equation} \tag{ 38 }$$

This equation can be directly integrated to obtain,

$$\begin{equation} \Delta P = -\alpha C \left[1+(z_0^{\prime })^2\right], \end{equation} \tag{ 39 }$$

where C is a positive integration constant that can be fixed in a variety of ways. One could apply a photospheric boundary condition or one could use spectroscopic observations to fix the temperature or density contrast at the apex of the loop. Strictly speaking, the integration constant is the product αC. We have included the field-strength deviation in this product for later convenience and in order to remind the reader that the thermodynamic contrasts arise from a small deviation from the force-free equilibrium. If we subsequently use the equations of pressure continuity (32) and hydrostatic balance (33), we can derive equations for the mass density and magnetic-pressure contrasts,

$$\begin{eqnarray} \Delta \rho &=& \frac{2\alpha C}{g} z_0^{\prime \prime }, \end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray} \frac{\Delta B^2}{8\pi } &=& \alpha C \left[1+(z_0^{\prime })^2\right]. \end{eqnarray} \tag{ 41 }$$

Thus, if we are provided the shape of the loop, we can derive all of the loop's contrast variables. Different values of the integration constant αC result in different temperature profiles along the loop. Equation (36) can be rewritten to show how the temperature contrast depends on the geometry of the loop and the constant αC,

$$\begin{equation} \Delta T = - \alpha C T_{\rm e}\, \frac{\left[1 + (z_0^\prime)^2 + 2 H_{\rm e}z_0^{\prime \prime }\right]}{P_{\rm e}+ 2 \alpha C H_{\rm e}z_0^{\prime \prime }}. \end{equation} \tag{ 42 }$$

Note that the integration constant C was chosen such that it appears only in the combination of αC in the equations above. Thus, one needs only a single constraint to define the thermodynamic variables.

Given the pressure contrast ΔP, we can use Equations (32), (35), and (39) to express the external field strength at the location of the fibril in terms of the geometry of the fibril,

$$\begin{equation} \frac{B_{\rm e}^2(s)}{8\pi } = C \left[1+(z_0^{\prime })^2\right]. \end{equation} \tag{ 43 }$$

This last equation shows that the integration constant C is really just the value of the external magnetic pressure at the apex of the loop (hence C is positive). Further, the equation demonstrates that not all external fields are self-consistent with an embedded fibril. If we select an arbitrary field line within a general magnetic field, Equation (43) may not be satisfied for a constant C. This becomes readily apparent when we express the direction of the tangent vector in terms of the angle θ that it makes with the x-axis, ${{{\boldsymbol {\hat{s}}}}} \equiv \cos \theta \, {{{\boldsymbol {\hat{x}}}}} + \sin \theta \, {{{\boldsymbol {\hat{z}}}}}$. With a little basic manipulation, we find the following relation between the angle θ and the height function z₀:

$$\begin{equation} \sec ^2\theta = 1 + (z_0^\prime)^2. \end{equation} \tag{ 44 }$$

Subsequently, one can easily show that Equation (43) is equivalent to the statement that the x-component of the external magnetic field is constant along the length of the fibril,

$$\begin{equation} {{{\boldsymbol {\hat{x}}}}}\cdot {{{\boldsymbol {B}}}}_{\rm e}= B_{\rm e}\cos \theta = \sqrt{8\pi C}. \end{equation} \tag{ 45 }$$

Of course, not all force-free fields will have this property.

If needed, a self-consistent external magnetic field can be derived by noting that the shape of the fibril and the strength of the magnetic field at the location of the fibril provide a boundary condition for the coronal magnetic field that occupies the bulk of the domain. For simplicity, we assume that the external magnetic field is potential and 2D. Therefore, since the field is also solenoidal it can be described by a flux function Ψ_e,

$$\begin{equation} {{{\boldsymbol {B}}}}_{\rm e}= {{{\boldsymbol {\nabla }}}}\times \left(\Psi _{\rm e}{{{\boldsymbol {\hat{y}}}}}\right) = - \frac{\partial {\Psi _{\rm e}}}{\partial {z}} \, {{{\boldsymbol {\hat{x}}}}} + \frac{\partial {\Psi _{\rm e}}}{\partial {x}} \, {{{\boldsymbol {\hat{z}}}}}. \end{equation} \tag{ 46 }$$

Since the field is potential the flux function must be harmonic,

$$\begin{equation} \nabla ^2\Psi _{\rm e}= 0. \end{equation} \tag{ 47 }$$

Therefore, we can find a self-consistent external field solution by solving Equation (47) under the boundary condition

$$\begin{equation*} {{{\boldsymbol {\nabla }}}}\Psi _{\rm e}= B_{\rm e}(s) \, {{{\boldsymbol {\hat{k}}}}}(s), \end{equation*}$$

applied at the location of the fibril with B_e(s) fixed by Equation (43). This ensures that the external field is both parallel to the fibril and the field strength is given by B_e(s). There is an infinity of such solutions, all with different photospheric boundary conditions.

The contrast variables have a common geometrical factor, $1+(z_0^\prime)^2$, appearing in their functional forms. For the model with uniform magnetic Bond number this factor takes on a simple form. Using Equations (49) and (50), we find

$$\begin{equation} 1 + (z_0^\prime)^2 = \sec ^2\left(\varepsilon x/X\right) = \sec ^2\varepsilon \; \exp \left(-z_0/h\right). \end{equation} \tag{ 55 }$$

Using this expression and the equations derived in Section 2.3 we can solve for all of the contrast variables and the external magnetic pressure in terms of the scale height h that depends on the Bond number, h = −X/2ε,

$$\begin{eqnarray} \Delta P &=& -\frac{\Delta B^2}{8\pi } = -\alpha C \, \exp \left(\frac{z_{\rm apex}-z_0}{h}\right), \end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray} \Delta \rho &=& -\frac{\alpha C}{gh} \, \exp \left(\frac{z_{\rm apex}-z_0}{h}\right), \end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray} \Delta T &=& T_{\rm e}\left(\frac{h}{H_{\rm e}} - 1\right) \frac{\Delta \rho }{\rho _{\rm e}+\Delta \rho _{\rm e}}, \end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray} \frac{B_{\rm e}^2}{8\pi } &=& C \, \exp \left(\frac{z_{\rm apex}-z_0}{h}\right). \end{eqnarray} \tag{ 59 }$$

The density and pressure contrasts, as well as the external magnetic pressure, are revealed to be exponential functions of height, all with the same constant scale height h. The density and temperature contrast are illustrated in Figure 3, for an isothermal exterior with a scale height of H_e = 75 Mm. The constant αC has been chosen such that the fractional density contrast at the apex of the loop is Δρ/ρ_e = ±0.01. The solid curves correspond to overdense loops and the dotted curves to underdense loops, with the different colors corresponding to loops with different magnetic Bond numbers. We will see in the discussion, Section 5, that overdense loops have been heated relative to the surrounding corona and underdense loops have been cooled. Therefore, overdense loops are probably more physically relevant.

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For an isothermal exterior, where H_e is a constant, one could choose the magnetic Bond number such that h = H_e. For such a model, the temperature contrast would be identically zero all along the length of the loop and the plasma parameter β = 8πP/B² would be a constant both inside and outside the loop, while the density contrast remains nonzero. The condition H_e = h marks a transition between disparate behavior. Loops with H_e < h have fractional density and temperature contrasts that decrease with height, whereas loops with H_e > h have fractional density and temperature contrasts that increase with height.

In the previous subsection, we discovered that a fibril with constant magnetic Bond number requires that the external magnetic pressure have a constant scale height as we slide along the fibril. This suggests that we seek a solution in which the magnetic pressure is only a function of height and varies exponentially. A simple harmonic solution that satisfies this requirement is a sinusoid in the horizontal x-direction multiplied by a decaying exponential in height z,

$$\begin{equation} \Psi _{\rm e}(x,z) = \kappa ^{-1} \, \tilde{B}_{\rm e}\cos (\kappa x) \, e^{-\kappa z}. \end{equation} \tag{ 60 }$$

In this solution, 2π/κ is the horizontal repetition length of the field (Ψ_e changes sign every π/κ). With such a flux function, the magnetic field is as follows:

$$\begin{equation} {{{\boldsymbol {B}}}}_{\rm e}(x,z) = \tilde{B}_{\rm e}\left[\cos (\kappa x) {{{\boldsymbol {\hat{x}}}}} - \sin (\kappa x) {{{\boldsymbol {\hat{z}}}}} \right] \, e^{-\kappa z}. \end{equation} \tag{ 61 }$$

Since the magnetic field is periodic in the x-direction, we will only consider the arcade that is symmetric about the origin and exists in the range x ∈ (− D, D), where D = π/(2κ). The magnetic pressure for such a field depends only on height and falls off exponentially with a scale height of (2κ)⁻¹,

$$\begin{equation} \frac{B_{\rm e}^2}{8\pi } = \frac{\tilde{B}_{\rm e}^2}{8\pi } \, e^{-2\kappa z}. \end{equation} \tag{ 62 }$$

In order to satisfy Equation (59), we must insist that the external field's horizontal wavenumber is proportional to the magnetic Bond number and that the photospheric value of the field strength depends on C,

$$\begin{eqnarray} \kappa &=& -\frac{\varepsilon }{X} = \frac{2}{h}, \end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray} \frac{\tilde{B}_{\rm e}}{8\pi } &=& C \exp \left(\frac{z_{\rm apex}}{h}\right). \end{eqnarray} \tag{ 64 }$$

Lines of constant Ψ_e in the x–z plane denote field lines. The height above the photosphere, z_e, of a field line with a specified value of the flux function is easily obtained by inverting Equation (60),

$$\begin{equation} z_{\rm e}(x; \Psi _{\rm e}) = \kappa ^{-1} \ln \left(\frac{\cos (\kappa x)}{\cos (\kappa x_\Psi)}\right), \end{equation} \tag{ 65 }$$

where the footpoints intersect the photosphere at $x = \pm x_\Psi$,

$$\begin{equation} x_\Psi \equiv \kappa ^{-1} \cos ^{-1}\left(\frac{\kappa \Psi _{\rm e}}{\tilde{B}_{\rm e}}\right). \end{equation} \tag{ 66 }$$

We can clearly see from Equation (49) that this potential field solution for the external corona is consistent with the loop model, if the magnetic Bond number is inversely proportional to the repetition length of the external magnetic field, ε = −κX, which is a condition we have already imposed to ensure that the external field strength has the appropriate value along the fibril.

The field lines of this potential field are illustrated in Figure 4. In the beginning of this section we discovered that stability requires that the magnetic Bond number be bounded ε ∈ (− π/2, 0). In the context of this specific model, we can see that the magnetic Bond number selects the field line within the coronal field that corresponds to the axis of the fibril. Fibrils with different footpoint separations and magnetic Bond numbers are shown as various colored field lines in Figure 4. A small value of the magnetic Bond number corresponds to a short, flat, low-lying loop confined very near the origin, while a magnetic Bond number near the lower limit is a tall, long loop, with foot-points approaching the outer edge of the arcade. In fact, in the limit ε → −π/2, the loop is infinitely tall with vertical legs located at x = ±X = ±D. Therefore, there is a direct mapping between the magnetic Bond number and the value of the flux function that corresponds to the loop's axis,

$$\begin{equation} \Psi _{\rm fibril} = (2D/\pi)\, \tilde{B}_{\rm e}\cos \varepsilon = h \sqrt{\frac{\pi C}{2}}, \end{equation} \tag{ 67 }$$

and one can choose to place the loop along any field line in the coronal model by dialing the magnetic Bond number between −π/2 and 0.

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The shape of a coronal loop is determined by the force-free equilibrium of the surrounding corona. Given this shape, the mass along the loop must redistribute itself such that the buoyancy forces exactly oppose the Lorentz forces acting on the loop. The resulting balance of forces can be succinctly characterized by the magnetic Bond number which is fully specified by the height of the loop above the photosphere as a function of position. In equilibrium, the magnetic Bond number is proportional to the signed curvature of the loop, and therefore concave curvature results in a negative magnetic Bond number and convex curvature in a positive Bond number. Of course, inflection points correspond to locations where the Bond number vanishes. Therefore, a loop comprised of a single concave arch must have negative magnetic Bond number everywhere if in equilibrium. Equilibrium loops that possess multiple arches will have negative Bond number in the crests and positive Bond number in the dips or troughs.

All loops with the same geometric shape have exactly the same profile of the magnetic Bond number. However, these loops can be achieved in a continuum of different ways, all varying in the distribution of mass and field strength. This continuum is represented by different values of an integration constant αC which is a direct measure of the magnetic-pressure contrast at the apex of the loop. For example, an underdense loop with enhanced magnetic-field strength can have exactly the same Bond number as an overdense loop with reduced field strength. While the shape of these two loops will be the same, the thermodynamic and magnetic properties of these two loop might be very different.

For a loop with enhanced field strength, the mass density is proportional to the second derivative of the height function with respect to the horizontal photospheric coordinate, Equation (40). Thus, we can immediately deduce that peaks or crests in a loop will have a mass deficit while dips or troughs have mass accumulation. Furthermore, if we define the total mass deviation between two points as the integrated density contrast,

$$\begin{equation} \Delta M(s_1,s_2) \equiv \int _{s_1}^{s_2} \Delta \rho (s) \, A(s) \, ds, \end{equation} \tag{ 76 }$$

then the mass deviation vanishes if we choose two points that correspond to extrema in height. This can be revealed by noting that flux conservation within the loop requires the following:

$$\begin{equation} A(s) = \frac{A_{\rm apex}}{\left[1+(z_0^\prime)^2\right]^{1/2}}, \end{equation} \tag{ 77 }$$

where A_apex is the cross-sectional area of the loop at the apex. If we change variable from the path length to the photospheric x coordinate and replace the density contrast using Equation (40) the integral can be evaluated trivially,

$$\begin{equation} \Delta M(x_1,x_2) = \frac{2\alpha C A_{\rm apex}}{g} \, \left[z_0^\prime (x_2) - z_0^\prime (x_1)\right]. \end{equation} \tag{ 78 }$$

Thus, the net mass deviation between a crest and a neighboring trough is zero and the crest drains mass into the dips. For a loop comprised of a single concave arch, the mass deviation between the two footpoints is negative. Mass must drain out of the loop through the photosphere into the solar interior. In the opposite case, a loop with a reduced field strength, the mass density is proportional to the negative of the second derivative. The peaks in such a loop must have enhanced density so that buoyancy can oppose the upward Lorentz force. This obviously implies that the gas pressure must increase throughout the loop to increase the hydrostatic support. The additional mass is lifted from the photosphere into the loop.

Observations of the temperature along the length of coronal loops have suggested that some loops might be nearly isothermal (Aschwanden et al. 2001). For the relatively small density contrasts that have been explored here, the temperature contrast is also relatively small and the temperature is perforce roughly isothermal as long as the external corona is isothermal. However, we wish to point out that the temperature contrast itself is markedly constant over height for a significant fraction of the models we have illustrated. Three of the models with constant magnetic Bond number, appearing in Figure 3, have a temperature variation of less than a 30% percent from footpoint to apex. These three correspond to the models with the squattest loops.

Given a specific shape for a coronal loop there exists a family of possible solutions, each characterized by a different value of αC, the integration constant—see Equations (39)–(42). When illustrating our models, we chose to fix this parameter by specifying the fractional density contrast at the apex of the loop. We wish to point out, however, that the parameter αC is really a measure of the total heat that has been input into the coronal loop. The excess heat contained by the loop per unit length is given by

$$\begin{eqnarray} E(s) &=& c_v A(s) \left[\rho _0(s)T_0(s) - \rho _{\rm e}(s) T_{\rm e}(s)\right] \nonumber\\ &=& (\gamma -1)^{-1} A(s) \Delta P(s), \end{eqnarray} \tag{ 79 }$$

where c_v is the specific heat at constant volume and γ is the adiabatic exponent. We can obtain the total excess heat contained by the loop by integrating E(s) over the loop's length. If we eliminate the gas-pressure contrast by using Equation (39), we reduce the integral to a constant factor times a positive definite integral that depends only on the geometry of the loop,

$$\begin{equation} \int _0^L E(s) \, {ds} = -\frac{\alpha C A_{\rm apex}}{\gamma -1} \, \int _{-X}^X \left[ 1+\left(z_0^\prime \right)^2\right] \, {dx}. \end{equation} \tag{ 80 }$$

Since the constant C and the integral appearing on the right-hand side of the previous equation are always positive, we can immediately discern two facts. (1) Heated loops correspond to those with a negative field-strength deviation α and cooled loops to those with a positive value. Thus, heated loops are those that are undermagnetized and overdense and cooled loops are overmagnetized and underdense. (2) The integration constant C is a measure of the magnitude of the heat that has been injected into or extracted from the loop. Obviously, heat deposition is more likely to be physically relevant. The family of solutions defined by different values of C form a continuum of loop models with different amounts of stored heat. For example, after a large impulsive heating event, the heat content of the loop jumps but force balance is quickly re-established. This requires a redistribution of mass (and heat) along the loop and the resulting balance is characterized by a nonzero value of C. Then as radiative cooling and conduction slowly dissipate the loop's excess heat, the loop responds by moving through a sequence of equilibria while simultaneously maintaining force balance. This sequence is represented by slow temporal attenuation of the parameter C.

Seismology of a coronal loop is only sensitive to the distribution of wave speed along the loop (e.g., Jain & Hindman 2012). If we assume that the observed waves are kink oscillations and that they propagate at the kink speed,

$$\begin{equation} c_K^2 = \frac{B_0^2 + B_{\rm e}^2}{4\pi (\rho _0 + \rho _{\rm e})}, \end{equation} \tag{ 81 }$$

then, at best, a seismic analysis can provide information about the ratio of the field strength to the density. It is impossible from only a seismic analysis to measure the field strength profile independently of the density profile. However, with additional constraints, provided either from observations or from basic physical principles, one might be able to disentangle field-strength and density variations along the loop. The force-balance model constructed here provides just such a constraint. The density contrast and magnetic-pressure contrast are related to the geometry of the loop through the magnetic Bond number,

$$\begin{equation} \frac{\Delta B^2}{4\pi } = \frac{gX}{\varepsilon } \Delta \rho = gX \frac{1+(z_0^\prime)^2}{z_0^{\prime \prime }} \Delta \rho. \end{equation} \tag{ 82 }$$

If we assume that the interior magnetic field is nearly the same as the external field (hence the field-strength deviation is small), we can use this previous equation to express the kink speed solely in terms of the density contrast, the exterior density and the loop's geometry,

$$\begin{equation} c_K^2 = 2C \frac{1 + (z_0^\prime)^2}{\rho _{\rm e}+ (\alpha C/g) z_0^{\prime \prime } }. \end{equation} \tag{ 83 }$$

If we further assume that the exterior corona is isothermal, $\rho _{\rm e}= \tilde{\rho }_{\rm e}\exp (-z/H_{\rm e})$, this expression has only four free parameters: the integration constant C, the field-strength deviation α, the coronal scale height H_e and photospheric value of the exterior density $\tilde{\rho }_{\rm e}$. Thus, even the measurement of only a few mode frequencies that provide different spatial averages of the kink speed should be sufficient to either fix these parameters or to verify a choice made using other information.

We have developed a model of curved coronal loops that self-consistently couples deviations from the force-free state to the thermodynamic properties. The coupling is accomplished by requiring that the loop is in equilibrium and the Lorentz force arising from the deviation from a force-free field is balanced by buoyancy. This links the curvature of the loop to the loop's mass-density contrast and hence to the pressure and temperature through hydrostatic balance and the ideal gas law. The end result is that specification of the loop's geometry is sufficient to derive the temperature and density profiles to within a single integration constant. This integration constant can be selected in a variety of ways, either seismically through estimates of the mean kink-wave speed or spectroscopically through estimates of the density or temperature contrast at one position along the loop. Further, this integration constant is a direct assessment of the thermal history of the loop providing an estimate of the excess heat contained by the loop relative to the surrounding corona.