EXPLAINING INVERTED-TEMPERATURE LOOPS IN THE QUIET SOLAR CORONA WITH MAGNETOHYDRODYNAMIC WAVE-MODE CONVERSION

We begin with a model in which Alfvén waves propagate up from the photosphere and gradually "feed" a turbulent cascade. This can only happen when there are counterpropagating wave packets that can collide with one another (Iroshnikov 1963; Kraichnan 1965; Howes & Nielson 2013). This situation is clearly present in a coronal loop with both footpoints being jostled by convective motions. A long series of analytic models, computer simulations, laboratory experiments, and comparisons with in situ heliospheric plasma has led us to generally believe that Alfvén waves damp out due to turbulence and produce heat at roughly the following rate:

$$\begin{eqnarray}&&{Q}_{{\rm{A}}}\,\approx \,\rho \,\displaystyle \frac{{Z}_{+}^{2}{Z}_{-}+{Z}_{-}^{2}{Z}_{+}}{{\lambda }_{\perp }}\end{eqnarray} \tag{ 15 }$$

(see, e.g., Hossain et al. 1995; Zhou & Matthaeus 1990; Matthaeus et al. 1999; Dmitruk et al. 2002; Breech et al. 2008; Chandran et al. 2011; Cranmer & van Ballegooijen 2012; van der Holst et al. 2014). The quantity λ_⊥ is a transverse correlation length, or turbulent outer scale. Z_± are the Elsässer amplitudes that characterize the strength of turbulent eddies propagating in both directions along the field line. A symmetric closed loop will experience equal contributions of Alfvén waves from both directions, and we can assume Z₊ = Z₋. In this case, the transverse rms velocity amplitude is given by ${v}_{\perp }={Z}_{\pm }/\sqrt{2}$, and the basal energy density of Alfvén waves can be estimated as ${U}_{{\rm{A}},0}={\rho }_{0}{v}_{\perp 0}^{2}$.

We assume that the loss rate of waves is sufficiently weak such that the rms velocity amplitude throughout the loop can be modeled with wave flux conservation, which for hydrostatic loops is equivalent to v_⊥ ∝ ρ^−1/4. Additionally, we assume that the correlation length scales with the radius of the flux tube (Hollweg 1986) such that λ_⊥ ∝ A^1/2 ∝ B^−1/2. In terms of the free parameters f and U_tot, we can write

$$\begin{eqnarray}&&{Q}_{{\rm{A}}}=\left[\displaystyle \frac{\alpha \,{(1-f)}^{3/2}{U}_{\mathrm{tot}}^{3/2}}{{\rho }_{0}^{1/2}{\lambda }_{\perp 0}}\right]{\left(\displaystyle \frac{\rho }{{\rho }_{0}}\right)}^{1/4}{\left(\displaystyle \frac{B}{{B}_{0}}\right)}^{1/2},\end{eqnarray} \tag{ 16 }$$

where the subscript 0 refers to the value of each quantity at the footpoints of the loop in the transition region, such that the quantity in the square brackets above can be called the basal Alfvénic heating rate ${Q}_{{\rm{A}},0}$. The constants α = 0.85 and λ_⊥0 = 1.1 Mm were adopted from the semi-empirical turbulence model of Cranmer & van Ballegooijen (2012). Note that specifying Q_A(s) requires knowing the solution for the density ρ(s), which is an output of the modeling process.

Nuevo et al. (2013) suggested that some fraction of the Alfvén wave energy may be converted, via one or more nonlinear mechanisms, into compressive and longitudinal wave energy. The resulting fluctuations are similar to acoustic waves (i.e., slow-mode magnetosonic waves in plasmas with β < 1), but are not identical (see, e.g., Hollweg 1971; Nakariakov et al. 1998; Bogdan et al. 2002; Bogdan 2006; Kaghashvili 2007; Cranmer & Woolsey 2015). Still, they behave in many ways like acoustic waves, in that they exhibit oscillations in both density (δρ) and velocity parallel to the field (δv_∥).

We assume that the mode conversion occurs in the chromosphere and lower TR, and at the lower boundary of our modeled loop there is a compressive wave energy density ${U}_{{\rm{S}},0}$ given by Equation (14). The compressive waves obey a conservation equation discussed in more detail in Section 4.1 of Cranmer et al. (2007),

$$\begin{eqnarray}&&\displaystyle \frac{1}{A}\displaystyle \frac{\partial }{\partial s}({c}_{s}{{AU}}_{{\rm{S}}})=-{Q}_{{\rm{S}}}=-\displaystyle \frac{{c}_{s}{U}_{{\rm{S}}}}{{L}_{\mathrm{damp}}},\end{eqnarray} \tag{ 17 }$$

where c_s is the sound speed,

$$\begin{eqnarray}&&{c}_{s}=\sqrt{\displaystyle \frac{\gamma P}{\rho }}=\sqrt{\displaystyle \frac{\gamma {C}_{1}}{{C}_{2}}\,\displaystyle \frac{{k}_{{\rm{B}}}T}{{m}_{p}}},\end{eqnarray} \tag{ 18 }$$

where we assume γ = 5/3. For now the damping length L_damp is assumed to be constant and is treated as a free parameter. In Section 7 we compare the most realistic values of L_damp with predictions from the theory of shock steepening and conductive dissipation.

By defining the quantity $Y\equiv {c}_{s}{{AU}}_{{\rm{S}}}$, we solve Equation (17) as

$$\begin{eqnarray}&&Y(s)={Y}_{0}\,\exp \left(-\displaystyle \frac{s}{{L}_{\mathrm{damp}}}\right)\end{eqnarray} \tag{ 19 }$$

between the base (s = 0) and the apex of the loop (s = L/2). This gives an explicit form for the heating rate due to compressive waves,

$$\begin{eqnarray}&&{Q}_{{\rm{S}}}=\displaystyle \frac{{c}_{s,0}\,f\,{U}_{\mathrm{tot}}}{{L}_{\mathrm{damp}}}\,\displaystyle \frac{{A}_{0}}{A(s)}\,\exp \left(-\displaystyle \frac{s}{{L}_{\mathrm{damp}}}\right).\end{eqnarray} \tag{ 20 }$$

The heating rate Q_S(s) can be written explicitly without knowing the solutions for density or temperature along the loop. However, writing the energy density U_S requires knowing c_s(s) and thus also T(s). The velocity amplitude of the compressive waves is defined as

$$\begin{eqnarray}&&{v}_{\parallel }(s)=\sqrt{\displaystyle \frac{3\,{U}_{{\rm{S}}}}{\rho }},\end{eqnarray} \tag{ 21 }$$

where the factor of 3 contains the assumption that compressive waves rapidly steepen into a sawtooth-shaped train of shock waves (see Stein & Schwartz 1972; Cranmer et al. 2007).

Figure 4 compares the basal amplitudes and heating rates of Alfvénic and compressive waves with one another for a range of values for the mode conversion fraction f. We find it useful to describe the conversion fraction with the associated parameter $\omega \equiv -\,{\mathrm{log}}_{10}\,(1-f)$. The parameter ω increases monotonically with f, with both quantities starting at zero together. However, as $f\to 1$, $\omega \to \infty$. Integer values of ω give the number of 9's in the decimal representation of f (i.e., ω = 3 corresponds to f = 0.999). Thus, the region near f ≈ 1 is stretched out and resolved better with ω.

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Figure 4 shows how Alfvén waves dominate at low values of ω and compressive waves dominate at high values. These models assumed U_tot = 0.1 erg cm⁻³, T₀ = 10⁴ K, and P₀ = 0.05 dyn cm⁻². For these representative parameters, the velocities at the TR are of order 20–30 km s⁻¹, which is representative of observational results from nonthermal line broadening (e.g., Banerjee et al. 2011; Hahn & Savin 2014). Note that ${U}_{{\rm{A}},0}={U}_{{\rm{S}},0}$ when f = 0.5. However, for the parameters assumed here, the heating rates ${Q}_{{\rm{A}},0}$ and ${Q}_{{\rm{S}},0}$ do not become equal until f = 0.83. In other words, for low values of f it is possible for ${Q}_{{\rm{A}},0}\gg {Q}_{{\rm{S}},0}$ even when the velocity amplitudes are comparable to one another.

At the coronal base (s = 0), we specify the boundary conditions on the gas pressure (P₀) and temperature (T₀). We fix T₀ = 10⁴ K and allow P₀ to vary as an output of our method (see below). In order to determine additional boundary conditions, we find it convenient to introduce the quantity

$$\begin{eqnarray}&&{\rm{\Psi }}=A\,\kappa (T)\,\displaystyle \frac{{dT}}{{ds}}.\end{eqnarray} \tag{ 22 }$$

At the apex of the loop (s = L/2), we take advantage of the assumed symmetry as shown in Figure 1 and assume

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{dT}}{{ds}}\right|}_{s=L/2}={\left.\displaystyle \frac{d{\rm{\Psi }}}{{ds}}\right|}_{s=L/2}=0.\end{eqnarray} \tag{ 23 }$$

Determining the bottom boundary condition on the temperature gradient dT/ds (i.e., on the quantity Ψ) requires slightly more care than just choosing a representative value. This quantity depends sensitively on the modeled thermal energy balance at T₀. Following Hammer (1982) and Schrijver & van Ballegooijen (2005), we assume that in the upper chromosphere the local heating term Q_heat is relatively unimportant in comparison to the conduction and radiative cooling terms. We also simplify the equations in the vicinity of the transition region by using a generalized power-law form of the radiative loss function,

$$\begin{eqnarray}&&{\rm{\Lambda }}(T)={{\rm{\Lambda }}}_{0}{T}^{\gamma },\end{eqnarray} \tag{ 24 }$$

for constant values of Λ₀ and γ. We also use a simpler version of the conductive flux,

$$\begin{eqnarray}&&F=-{\kappa }_{0}{T}^{5/2}\displaystyle \frac{{dT}}{{dz}},\end{eqnarray} \tag{ 25 }$$

that includes only the electron conductivity. The thermal balance equation therefore becomes

$$\begin{eqnarray}&&\displaystyle \frac{{dF}}{{dz}}=-{n}_{e}^{2}\,{{\rm{\Lambda }}}_{0}{T}^{\gamma }.\end{eqnarray} \tag{ 26 }$$

Multiplying both sides of Equation (26) by F and then dividing by dT/dz gives a straightforwardly integrable form

$$\begin{eqnarray}&&F\,{dF}=\psi \,{T}^{\gamma +(1/2)}\,{dT}\end{eqnarray} \tag{ 27 }$$

where we assume that $\psi ={\kappa }_{0}{{\rm{\Lambda }}}_{0}{({n}_{e}T)}^{2}$, proportional to the pressure squared, is a constant across the chromosphere and TR. Integrating both sides yields

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}({F}_{2}^{2}-{F}_{1}^{2})=\displaystyle \frac{2\psi }{2\gamma +3}[{T}_{2}^{\gamma +(3/2)}-{T}_{1}^{\gamma +(3/2)}],\end{eqnarray} \tag{ 28 }$$

which can be simplified further by considering that T and $| F|$ are both increasing rapidly with increasing height, such that T₂ ≫ T₁ and $| {F}_{2}| \gg | {F}_{1}|$. This allows us to eliminate the "2" subscript and write

$$\begin{eqnarray}&&{F}^{2}=\displaystyle \frac{4\psi \,{T}^{\gamma +(3/2)}}{2\gamma +3}.\end{eqnarray} \tag{ 29 }$$

Using the above definitions, this gives

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dz}}=\sqrt{\left(\displaystyle \frac{4}{2\gamma +3}\right)\,\displaystyle \frac{| {Q}_{\mathrm{rad}}| \,T}{\kappa }},\end{eqnarray} \tag{ 30 }$$

and we calculate γ = 9.9 from our tabulated values of the radiative loss function at T₀ = 10⁴ K. We use this value in Equations (22) and (30) to obtain Ψ₀ at the lower boundary.

We solve for the density and temperature throughout the loop using a similar iterative method to that of Schrijver & van Ballegooijen (2005). Thermal energy conservation is a second-order differential equation, which we separated into two first-order coupled equations: Equation (22) and

$$\begin{eqnarray}&&\displaystyle \frac{d{\rm{\Psi }}}{{ds}}=-A({Q}_{\mathrm{rad}}+{Q}_{\mathrm{heat}}).\end{eqnarray} \tag{ 31 }$$

These coupled equations were integrated from the base (s = 0) to the apex (s = L/2) using first-order Euler steps to obtain T(s) and Ψ(s). However, both Q_rad and Q_heat depend on the coronal density as well as on temperature. To obtain ρ(s), we first integrated Equation (8) for P(s) and also used Equation (4). Doing this requires an initial guess for the temperature. This was given by an analytic solution to a balance between a constant coronal heating rate Q_heat and a conduction rate Q_cond dominated by the κ₀ electron term. In Cartesian geometry (A = constant), this is given by

$$\begin{eqnarray}&&{T}_{\mathrm{guess}}(s)=({T}_{\max }-{T}_{0}){\left[1-{\left(\displaystyle \frac{z}{{z}_{\max }}-1\right)}^{2}\right]}^{2/7}+\,{T}_{0}.\end{eqnarray} \tag{ 32 }$$

For this first guess, we assumed representative chromospheric and coronal values T₀ = 10⁴ K and T_max = 10⁶ K.

After creating the above initial temperature function, we also chose an initial guess for P₀ and proceeded to integrate Equations (22)–(31). In general, this initial solution does not satisfy the upper boundary conditions at the loop apex (see above), so we used a Newton–Raphson root-finding algorithm to adjust P₀ toward a value at which Equation (23) is satisfied. We used a fixed initial guess of P₀ = 0.01 dyn cm⁻² for the Newton–Raphson method. This value is noticeably lower than the typical base pressure found for the loops, but we find that values of the base pressure that are too high tend to cause divergent solutions. In rare cases when the Newton–Raphson method failed to converge to a solution, we found the root with a more stable but significantly slower bisection method (searching between bounding values of P₀ = 0.001 and 10 dyn cm⁻²).

As the numerical iteration progresses, the tabulated solutions for T(s) and P(s) are updated and the most recent values are used as guesses in successive rounds of integrating Equations (8) and (22)–(31). The finite-difference tables contain 1000 values of s between the base and apex, and they use a nonuniform grid determined by the function

$$\begin{eqnarray}&&s=\displaystyle \frac{L}{2}{\left(\displaystyle \frac{n}{999}\right)}^{4}\end{eqnarray} \tag{ 33 }$$

for 0 ≤ n ≤ 999. Grid zones are made intentionally dense near s = 0 in order to capture the rapid temperature changes seen near the base of the loop. We tested the adequacy of these discrete grid parameters (and the use of first-order Euler integration) everywhere in the interior of the loop grid by considering the sum of the three heating/cooling terms in Equation (9) divided by the absolute value of the largest of Q_heat, Q_rad, or Q_cond. When averaged over the length of the loop, this testing ratio was found to almost never exceed 1% for both short and long loops, with the average usually closer to 0.6%–0.9% for both cases. At worst, for some combinations of parameters this ratio would reach 15% at some point along the short loops and 27% at some point along the long loops. The instances with the relatively high test ratio tended to coincide with low-ω and high-U_tot loops that would have unrealistically high peak temperatures and temperature gradients near the top of a quiescent corona. The usual low values of the test ratio, especially for loops matching the observational data, indicate a comparable level of fractional accuracy in the other parameters.

Because of the possibility of large relative changes from one iteration to the next, we used an undercorrection scheme for the temperature. This technique was adopted from the ZEPHYR code (Cranmer et al. 2007), and the value at each distance s is updated using

$$\begin{eqnarray}&&T{(s)={T}_{\mathrm{old}}(s)}^{N}\,{T}_{\mathrm{new}}{(s)}^{1-N}\end{eqnarray} \tag{ 34 }$$

where T_old is the previous iteration's solution and T_new is the direct output of the numerical integration for the current iteration. The constant N is a parameter between 0 and 1 that we set to 0.65 for most cases. For loops that tend to diverge during early iterations, N is first set to 0.2 and then gradually raised to 0.5. Once T(s) is updated, the table of P(s) values is updated, and the process is repeated until the relative temperature difference between one iteration and the next is less than a thousandth of a percent for 10 sample grid points spaced evenly through the grid, from n = 99 to n = 999.

In order to determine whether the time-steady solutions are stable to small perturbations, we test for unstable gravitational stratification. According to Aschwanden et al. (2001), a hydrostatic loop becomes dynamically unstable if

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial s}\gt 0\end{eqnarray} \tag{ 35 }$$

in the region 0 ≤ s ≤ (L/2). Often, it is not necessary to apply this criterion because unstable solutions tend to be unphysical in other ways. In almost all cases, our models exhibit monotonically decreasing density with increasing s. If the coronal heating is concentrated too strongly at the base, Equation (35) may end up being satisfied, but these models tend to have insufficient heating toward the apex to maintain a time-steady hot corona. The iteration process in this case leads to negative temperatures, clearly indicating an unphysical solution.

There have been a number of different thermal instabilities proposed that could disrupt some coronal loops and give rise to prominence condensation or "coronal rain" (e.g., Priest 1978; Habbal & Rosner 1979; Gómez et al. 1990; Cally & Robb 1991; Mok et al. 2008; Sasso et al. 2012, 2015; Antolin et al. 2015). This process—in which the loop is stretched or twisted beyond a stable length and cools catastrophically to chromospheric temperatures—is just one of several proposed to explain the highly dynamic and intermittent ultraviolet and X-ray emission seen from loops. However, we defer the study of these kinds of instabilities to future work, since the large-scale quiescent loops modeled in this paper appear to have parameters quite distinct from the (mainly low-lying) unstable loops in the studies cited above.

We consider loops of two different lengths, both given in Table 1, and we refer to them as "long" and "short" loops with reference to the categories defined by Nuevo et al. (2013). The short loops are characterized by θ = 8° and L = 0.333 R_⊙, and the long loops have θ = 25° and L = 1.584 R_⊙. To span the full parameter space of possible loop solutions for each length, we fill a three-dimensional grid of values for f, U_tot, and L_damp. Twenty different values for f are used, with 10 values evenly spaced between 0 and 0.9 (i.e., 0 ≤ ω ≤ 1) and another 10 values evenly spaced in ω between 1.3 and 4.0. Ten different values are used for U_tot that are evenly spaced on a logarithmic scale bounded by 0.01 and 3.16 erg cm⁻³. For most values of f we use 13 different values of L_damp. However, when f = 0, the compressive heating term described in Equation (20) vanishes and the coronal heating term is purely Alfvénic. It is therefore only necessary to simulate the loops once for each value of U_tot when f = 0 since the damping length no longer affects the heating rate. For all other values of f, the 13 values of L_damp are evenly spaced between 0.004 and 0.04 R_⊙. In total, 4960 loops are simulated, half of which are short and half of which are long.

In order to draw as close a parallel as possible to the findings of Nuevo et al. (2013), we often average values along the loop "legs" at the heights probed by the DEMT technique. The legs are defined to span heights between 0.03 and 0.2 R_⊙, with the larger value replaced by z_max for the short loops that do not extend as high as 0.2 R_⊙. Specifically, we compute mean temperatures $\langle T\rangle$ in the legs, and we use the slope of a linear fit to T(z) in the legs to determine a mean temperature derivative $\langle {dT}/{dz}\rangle$. Models with a positive slope are characterized as "up loops," and those with a negative slope are "down loops." We note, however, that this one parameter does not always accurately characterize the full spatial dependence of T(s).

Figure 5 shows a selection of temperature and pressure profiles for long loops (L = 1.583 R_⊙) with a range of ω values and fixed values of ${U}_{\mathrm{tot}}=1$ erg cm⁻³ and L_damp = 0.025 R_⊙. Increasing the amount of mode conversion (i.e., larger values of ω) decreases the total amount of heat deposited along the loop, but it also makes the heating rate more strongly peaked at the footpoints. Aschwanden & Schrijver (2002) showed that a sufficiently steep decrease in Q(s) indeed gives rise to a "down loop" with a local minimum in temperature at the apex.

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We determined that stable down loops become possible when the mode conversion from Alfvénic to compressive waves is strong (i.e., ω ≳ 2). Down loops tend to be seen only for a finite range of U_tot values, and the bounds of this range vary inversely with L_damp. For values of U_tot smaller than this finite range, loops remain "normal" (i.e., with monotonically increasing temperatures) for all values of ω. For values of U_tot larger than this finite range, the amount of deposited energy at the base appears to be too large for stable coronal loops to exist, and the code is unable to find solutions with T > 0 everywhere.¹ Detailed maps of the parameter space are shown below.

A number of loops are poorly described as "up" or "down" when their slope $\langle {dT}/{dz}\rangle$ is only taken from a linear fit along the legs as described above. In some cases, the peak temperature occurs below the apex, but it is too high for it to be registered as a down loop. Of the 1940 short loops recorded as "up" loops, 214 (11%) have maximum temperatures before the apex, while the same is true for 189 (8.9%) of the 2115 long "up" loops. It is possible that these kinds of loops are not reported by Nuevo et al. (2013) because they only considered loops for which it was possible to create a high-quality linear fit (i.e., with R² > 0.5) from the DEMT data.

In Figure 6 we also illustrate a transitional nonmonotonic temperature profile that occurs for some long-loop models. Standard up and down loops exhibit one- and two-temperature maxima (when measured from footpoint to footpoint), respectively. However, these "hybrid" loops exhibit three temperature peaks, i.e., they have a local maximum at the apex, local minima below the apex, and additional local maxima below them on either side. Figure 6(b) illustrates how these hybrid loop cases can occur. Equation (31) indicates that if $| {Q}_{\mathrm{rad}}| \gt {Q}_{\mathrm{heat}}$, the second derivative of T is positive and the temperature curve near the peak is concave up, consistent with down loops. If $| {Q}_{\mathrm{rad}}| \lt {Q}_{\mathrm{heat}}$, the curve is concave down at the apex, consistent with up loops. However, in Figure 6(b) we see that for the hybrid loop case, the ratio $| {Q}_{\mathrm{rad}}| /{Q}_{\mathrm{heat}}$ oscillates above and below 1 a number of times between the base and the apex. Thus, despite the strong conduction that tends to smear out small-scale thermal fluctuations along the field, these rare cases do demand multiple minima and maxima in the time-steady T(s).

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Figures 7 and 8 are contour plots that indicate the mean temperature gradient $\langle {dT}/{dz}\rangle$ in units of MK ${R}_{\odot }^{-1}$. Each panel plots $\langle {dT}/{dz}\rangle$ versus ω and U_tot for a fixed value of L_damp. Figure 7 shows results for the short-loop models, and Figure 8 shows results for the long-loop models. We continue the color scheme from Figures 5 and 6 (and from Nuevo et al. 2013), in which up-loop parameters are in red and down-loop parameters are in blue.

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White areas in Figures 7 and 8 indicate combinations of parameters for which the code did not find physically realistic solutions. This tends to occur for the largest values of both ω and U_tot, i.e., very strong basal heating and a rapid drop-off of Q_heat with increasing height. All cases of "good" solutions (not in white) were found to be also gravitationally stable (∂ρ/∂s < 0) everywhere from footpoint to apex. This result disagrees somewhat with Aschwanden & Schrijver (2002), who found some cases of gravitationally unstable loops with positive temperatures at all values of s. These cases occurred for similar (i.e., rapidly decreasing) heating functions to our no-solution cases, but it should be noted that the exponential form for Q_heat(s) used by Aschwanden & Schrijver (2002) was different from what is used here.

As noted above, the sign of the observationally motivated mean gradient $\langle {dT}/{dz}\rangle$ does not always convey an accurate picture of the number of temperature minima and maxima. Specifically, it does not reveal the region of parameter space that produces "hybrid loops" (i.e., ones with three maxima in T(s) from footpoint to footpoint) at all. Thus, in Figure 9 we give an example of the same parameter space shown in Figures 7–8, but with a finer grid and a color scale that denotes the exact topological nature of each model's temperature profile. This grid contained 12,000 models that were only simulated until the peak temperature had sufficiently converged: 100 values of ω and 120 values of U_tot that span the limits shown in Figure 9. The hybrid loops (three peaks) occupy a narrow strip of parameter space between the standard regions of up loops (one peak) and down loops (two peaks), with some small-scale boundary structure that depends on the relative magnitudes of the heating terms Q_rad and Q_heat as discussed above. The purple "spurs" that extend to the right in Figure 9 correspond to loops with extremely shallow minima beneath their apex. In those cases the temperature variation at the top of the loop is almost flat, and a plot of T(s) would look nearly indistinguishable from a two-peak down loop.

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The numerical results for apex temperature T_max and base pressure P₀ can be compared straightforwardly to several existing analytical models (e.g., Rosner et al. 1978; Serio et al. 1981; Aschwanden & Schrijver 2002; Martens 2010). For simplicity's sake we compare only with Rosner et al. (1978, hereafter RTV), who assumed a constant heating rate Q versus distance along a loop with constant pressure and cross section. For standard choices of the radiative and conductive constants, the RTV laws can be written as

$$\begin{eqnarray}&&{T}_{\max }=1.302\,{\rm{MK}}{\left(\displaystyle \frac{Q}{{10}^{-4}\mathrm{erg}{\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}}\right)}^{2/7}{\left(\displaystyle \frac{L}{\text{100 Mm}}\right)}^{4/7}\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}{P}_{0} & = & 0.1467\,\mathrm{dyn}\,{\mathrm{cm}}^{-2}{\left(\displaystyle \frac{Q}{{10}^{-4}\mathrm{erg}{\mathrm{cm}}^{-3}{{\rm{s}}}^{-1}}\right)}^{6/7}\\ & & \times \,{\left(\displaystyle \frac{L}{\text{100 Mm}}\right)}^{5/7}.\end{eqnarray} \tag{ 37 }$$

Because the heating rates used in this paper are monotonically decreasing from footpoint to apex, it is not clear what value(s) of Q to use in Equations (36)–(37). After some experimentation, we found that using the geometric mean of the basal and apex heating rates (i.e., the square root of their product) produces reasonable agreement with the RTV predictions. Figure 10 shows a model-by-model comparison for the grid of short loops. A comparable plot for the long loops looks similar to this one, but with all values of T_max and P₀ shifted slightly upward. Note that the up loops (red points) tend to be more RTV-like than the down loops (blue), since the latter often have a more pronounced and complicated s-dependence to their heating rates and pressures.

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For a more direct comparison with the observational data, Figure 11(a) shows how $\langle T\rangle$ and $\langle {dT}/{dz}\rangle$ compare against one another. Polygonal outlines (adapted from Figure 10 of Nuevo et al. 2013) give an approximate indication of the observed region of parameter space for up and down loops. Out of the 4960 cases simulated in the short- and long-loop grids, only 698 of them (14%) fall inside the polygonal outlines. Figure 11(b) outlines the regions of parameter space (in U_tot and ω) corresponding to these models. There seems to be an intermediate range of values for U_tot (which changes slightly based on the mode coupling efficiency) required to match the observations. More wave power would result in too much heating (i.e., $\langle T\rangle \gt 2$ MK), and less wave power would give too little ($\langle T\rangle \lt 0.8$ MK). There are no clear limiting values of L_damp that identify the subset of models that agree with the observations.

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Figure 11 gives the general impression that the measured quiescent up loops are produced when there is a small amount of mode energy transfer from Alfvén to compressive waves at the TR, and the down loops are produced when there is large mode energy transfer. For low values of ω, the band of observationally appropriate parameters for up loops corresponds to basal values of the Alfvén velocity amplitude around 3–6 km s⁻¹. These values are quite a bit smaller than the canonical 20–30 km s⁻¹ nonthermal amplitudes often seen in the brighter and more compact active-region loops. However, the large quiescent loops traced out by the DEMT technique (Huang et al. 2012; Nuevo et al. 2013) are believed to compose a diffuse coronal background that must be heated more weakly than the small and bright loops that are easier to resolve in extreme-ultraviolet images.

Nuevo et al. (2013) measured the plasma β ratio for the observed loops, where

$$\begin{eqnarray}&&\beta =\displaystyle \frac{P}{{B}^{2}/8\pi }.\end{eqnarray} \tag{ 38 }$$

The leg-averaged value $\langle \beta \rangle$ was found to be systematically larger for down loops (median values of 1–3) than for up loops (median values of 0.5–0.9). Nuevo et al. (2013) attributed this difference mainly to the fact that down loops tend to have weaker magnetic fields than up loops. They also speculated that the larger values of $\langle \beta \rangle$ in the down loops should make them more susceptible to mode conversion between Alfvén and compressive waves.

Unfortunately, we cannot make a direct comparison with the observed trends in β because our modeled loops do not depend on an absolute normalization for B. However, we have estimated the value of β₀ at the TR under the assumption that all loops share a common value of B = 2 G at the TR. In fact, we believe that β₀ may be a more important parameter for determining the degree of basal mode conversion than $\langle \beta \rangle$ measured higher up in the corona. With the above assumption about B, the up loops within the red polygonal outline in Figure 11(a) have a median value of β₀ = 0.636, and the corresponding down loops have a median value of β₀ = 1.32. This difference is attributable solely to differences in P₀ in the two populations of models, but it does go in the same direction as the observed variation in $\langle \beta \rangle$. It should be noted that a similar trend in base pressures is also evident in the measured mean values given in Table 1 of Nuevo et al. (2013). Thus, it is unclear whether the observed differences in $\langle \beta \rangle$ between the up and down loops can be attributed primarily to differences in gas pressure or magnetic pressure.