RADIATING CURRENT SHEETS IN THE SOLAR CHROMOSPHERE

The Ohm's law is a reduced form of the exact generalized Ohm's law for a three species plasma derived by Mitchner & Kruger (1973). This reduced Ohm's law results from imposing long timescale approximations on the system of particle momentum equations, neglecting viscous effects, and assuming the net ionization rate of H i is zero. Under typical chromospheric conditions, the long timescale approximations translate into the constraint that the characteristic minimum timescale t_c over which a significant change in the macroscopic state of the plasma occurs satisfies t⁻¹_c ≪ a few hundred Hz.⁴ This constraint is satisfied for the current sheets generated here. The Ohm's law used here is

$$\begin{eqnarray} {\bf E}_{\rm{CM}} & = & -\frac{\nabla p_e}{e n_e} +\eta \left\lbrace {\bf J} +M_e {\bf J} \times \hat{\bf B} + \Gamma \hat{\bf B} \times ({\bf J} \times \hat{\bf B} )\right.\nonumber\\ &&\left. - \frac{\Gamma c}{B \xi _H} \hat{\bf B} \times \left(\xi _H \nabla p - \nabla p_H \right) \right\rbrace. \end{eqnarray} \tag{ 4 }$$

Here E_CM ≡ E + (V × B)/c is the CM electric field, E is the electric field, ξ_H = ρ_H/ρ is the ratio of neutral to total mass densities, $\hat{\bf B}$ is a unit vector in the direction of B, p_H = n_Hk_BT is the H i pressure, η is the parallel (Spitzer) resistivity, and Γ = ξ²_HM_eM_p, where M_e and M_p are the electron and proton magnetizations. The magnetization of a particle species s is M_s = ω_s/ν_s, where ν_s and ω_s are the total momentum transfer collision frequency and the cyclotron frequency of species s.

The term $(\Gamma c/B \xi _H) \hat{\bf B} \times (\xi _H \nabla p - \nabla p_H)$ in Equation (4) is sometimes neglected under the assumption that ξ_H ∼ 1, and nearly uniform. If ξ_H ∼ 1, this term reduces to $(2 \Gamma c/B) \hat{\bf B} \times \nabla p_e$, since p = p_H + 2p_e. Comparing this term with the Hall term $M_e {\bf J} \times \hat{\bf B}$ in the Ohm's law, and assuming that (J × B)/c ∼ ∇p based on force balance, it follows that $(2 \Gamma c/B) \hat{\bf B} \times \nabla p_e$ may be neglected relative to the Hall term if the condition |2M_p∇_⊥p_e| ≪ |∇_⊥p| is satisfied. Here ∇_⊥ denotes the gradient ⊥B. Since p_e → 0 as ξ_H → 1, it follows that this condition is satisfied for sufficiently low ionization since M_p is bounded. No assumptions are made about ξ_H in this paper. The full effect of the term $(\Gamma c/B \xi _H) \hat{\bf B} \times (\xi _H \nabla p - \nabla p_H)$ is included. It is found to have a small-to-moderate thermoelectric effect that reduces the Joule heating rate in the current sheet solutions presented in Section 4.

For the model considered here the x, y, and z components of the Ohm's law reduce to E_x = 0,

$$\begin{eqnarray} E_y + \frac{V_z B_x}{c} &=& \frac{c \eta }{4 \pi } \left((1+ \Gamma) B_x^\prime + \frac{4 \pi \Gamma }{B_x \xi _H} (\xi _H p^\prime - p_H^\prime)\right),\\ \ms E_z -\frac{V_y B_x}{c} &=& -\frac{p_e^\prime }{e n_e} - \frac{\eta c M_e B_x B_x^\prime }{4 \pi |B_x|}. \end{eqnarray} \tag{ 5 }$$

For a hydrogen plasma out of LTE, the ionization balance (equivalent to the LTE Saha equation) determining n_H, n_p, and n_e(= n_p) can be approximated from earlier work (e.g., Hartmann & MacGregor 1980) by considering a simplified hydrogen atom sitting above the solar photosphere in chromospheric plasma. To a first approximation, we can assume radiative detailed balance in the Lyman lines and continuum, as the bulk of the chromosphere is effectively thick in these transitions. Ionization of hydrogen occurs through collisions with electrons, but has a strong contribution from the flux of photospheric light in the Balmer continuum. Recombination effectively occurs to the n = 2 levels (with n being the principal quantum number) and higher because the Lyman continuum is in detailed balance (photoionization balances radiative recombination). Under these conditions, photoionization in the Balmer continuum, in cm⁻³ s⁻¹, occurs at a rate of 8000 s⁻¹ for each atom in the n = 2 quantum levels (Vernazza et al. 1981), or

$$\begin{equation} R_i= 3.2 \times 10^4 n_{H} \exp (-T_2/T). \end{equation} \tag{ 7 }$$

Here T₂ = eE_n2/k_B, where E_n2 = (3/4) × 13.58 eV is the n = 2 level energy, and T₂ ∼ 1.18187 × 10⁵ K.

The radiative recombination rate in the same units is

$$\begin{equation} R_r = \alpha n_e n_p \left(\frac{T}{10^4}\right)^b. \end{equation} \tag{ 8 }$$

We have computed recombination only to the n = 2 levels and higher which yields α = 3.373 × 10⁻¹³ and b = −0.786.

Assuming statistical equilibrium implies R_i = R_r. This equality yields the following equations for n_e as a function of n_H and T, noting that n_e = n_p:

$$\begin{eqnarray} n_e &=& n_{H}^{1/2} \left(\frac{3.2 \times 10^4}{\alpha }\right)^{1/2} \left(\frac{T}{10^4}\right)^{|b|/2} \exp (-T_2/2T)\\ \ms &\equiv & n_{H}^{1/2} I(T). \end{eqnarray} \tag{ 9 }$$

Equation (9) includes all lowest order NLTE effects under solar chromospheric conditions.

Combining Equation (10) with Equations (2) and (3) for n_e(p, T) and n_H(p, T) gives the following equation for ρ(p, T), which is used in the equations in Section 3 that are solved numerically to determine the state of the plasma:

$$\begin{equation} \rho (p,T) = m_p\left(I^2(T) + \frac{p}{k_B T}-I(T) \left(I^2(T)+ \frac{p}{k_B T}\right)^{1/2}\right). \end{equation} \tag{ 11 }$$

Radiation losses represent a dominant sink of energy from chromospheric plasma. Decades of radiative transport calculations enable us to treat, at a similar level of approximation to the ionization balance, these losses as if they are "effectively thin": the chromosphere has strong contributions from trace elements in spite of their lower abundances because lines of H and He are effectively thick in the deeper chromospheric layers and also they require enormous energies to excite them (T₂ ≫ T, for example). Anderson & Athay (1989) found that an effectively thin approximation works to within a factor of two in comparison with detailed transfer calculations because lines of Fe ii, Ca ii, and Mg ii separately tend to dominate the radiation losses each under effectively thin conditions at the different heights where they are formed. Therefore, we have computed the radiation losses under effectively thin conditions, assuming again the Lyman lines contribute nothing to the losses, using data from the CHIANTI database (see the Appendix). Below T = 1.5 × 10⁴ K, the effectively thin radiative loss rate, in erg cm⁻³ s⁻¹, is represented by

$$\begin{eqnarray} Q_R &=& 8.63 \times 10^{-6} C_E \frac{n_e (n_{H} + n_p)}{T^{1/2}} \Sigma _{i=1}^2 E_i \Upsilon _i\nonumber\\ &&\times \exp (-e E_i/k_B T)\\ \ms\ms &\equiv & R(T) n_e(n_{H} + n_p). \end{eqnarray} \tag{ 12 }$$

The form of this solution is a physical fit to a three-level generic "hydrogen atom" with two excited states at energies E_{1, 2} in eV, excited by collisions from a ground state. The quantities ϒ_i are in the form of Maxwellian-averaged collision strengths. To match the radiation losses from all the elements their values were found to be E₁ = 3.54 eV, E₂ = 8.28 eV, ϒ₁ = 0.15 × 10⁻³, and ϒ₂ = 0.065. The temperature T is in K, C_E = 1.6022 × 10⁻¹² erg eV⁻¹, and n_e, n_p, and n_H are in cm⁻³.

As shown in the Appendix, below 1.5 × 10⁴ K, this three-level mean H atom radiates the same energy as a solar plasma with the solar photospheric trace element abundances. Equation (12) for Q_R is accurate to within ∼10% for T ⩽ 1.5 × 10⁴ K.

The thermal and ionization energy per unit volume is 3p/2 + χ_Hn_e, where χ_H is the ionization energy of H.⁵ The thermal energy equation is

$$\begin{equation} \left(\left(\frac{3 p}{2} + \chi _H n_e\right) V_z\right)^\prime = - p V_z^\prime + \frac{c}{4 \pi } B_x^\prime \left(E_y + \frac{V_z B_x}{c}\right) - Q_R. \end{equation} \tag{ 14 }$$

The terms on the right-hand side (RHS) of this equation are the compressive and Joule heating rates, and the radiative cooling rate. The first two rates can be positive or negative. Joule dissipation and compressive heating are the local thermal energy generation mechanisms included in the model, respectively converting electromagnetic and CM kinetic energy into thermal energy and vice versa. Viscous heating and diffusive thermal energy flux are omitted.

The Joule heating rate Q_J and its components are obtained by multiplying Equation (16) by J_y. The equation becomes Q_J = Q_S + Q_P + Q_T, where Q_S, Q_P, and Q_T are the Spitzer, Pedersen, and thermoelectric heating rates defined below.⁶

The first term on the RHS of Equation (16) gives rise to a resistive heating rate Q_S + Q_P ≡ (η + Γη)J²_y > 0, where Q_P∝ the magnetization-induced resistivity Γη. For the current sheet solutions presented in Section 4, Q_P is the main driver of the radiative cooling rate.

The second term on the RHS of Equation (16) is the term $-(\eta\Gamma c/B \xi _H) \hat{\bf B} \times (\xi _H \nabla p - \nabla p_H)$ in Equation (4) in Section 1.1. This term corresponds to a thermoelectric contribution Q_T to Q, since it is ∝T', given by

$$\begin{equation} Q_T = \frac{\Gamma c^2 \eta B_x^\prime }{4 \pi B_x} \left(\frac{p}{k_B T} - \frac{\rho }{m_p} \right) \left(1 + |b| + \frac{T_2}{T}\right) k_B T^\prime. \end{equation} \tag{ 22 }$$

Insight into the sign of Q_T is obtained as follows. Define $\bar{\Gamma }$ by $\Gamma = \bar{\Gamma } B_x^2$. Then $\bar{\Gamma } > 0$, and since p/(k_BT) − ρ/m_p = n_e, Q_T may be written as

$$\begin{equation} Q_T = \frac{\bar{\Gamma } c^2 \eta n_e}{8 \pi }\big(B_x^2\big)^\prime \left(1 + |b| + \frac{T_2}{T}\right) k_B T^\prime. \end{equation} \tag{ 23 }$$

The sign(Q_T) = sign((B²_x)'T'), so Q_T may be positive or negative.

Figure 1 shows B_x (left panel) and J_y (right panel) versus height for the four solutions. The profiles of B_x are close to Harris sheet profiles. There is nothing deliberately built into the model or its inputs to make B_x have these profiles.

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Not all solutions to the model contain current sheets. Smooth solutions for chromospheric atmospheres extending over a 260 km height range that do not contain current sheets are also found, but not presented here. These atmospheres are weakly heated and weakly radiating compared to the current sheet solutions, and to the energy requirements of the chromosphere.⁷

From the Ohm's law equation (5) the resistive diffusivity is c²(1 + Γ)η/(4π). Dividing this by V_z gives the resistive length scale

$$\begin{equation} L_r = \frac{c^2(1+ \Gamma)\eta }{4 \pi V_z}. \end{equation} \tag{ 24 }$$

Figure 2 (left panel) shows L_r versus height for each solution. It is found that L_r characterizes the thickness of the current sheets in that essentially all of the radiative emission is from a region with a thickness equal to a few times the minimum value of L_r for a given sheet. This region is centered around the height at which Q_R is a maximum and L_r is a minimum.

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Figure 2 (right panel) shows Q_R. The radiative flux F_R is the integral of Q_R over the height range of the solutions. For the four sheets, in order of increasing n(0), F_R = (0.488, 1.55, 1.03, 4.45) × 10⁶ erg cm⁻² s⁻¹. Although the maximum peak values of Q_R occur for the smallest values of n(0), the overall trend is for F_R to increase as n(0) increases. This is due to the increase of L_r as n(0) increases. Although the peak values of Q_R tend to decrease with increasing n(0), L_r increases enough with increasing n(0) to make F_R follow the same trend. The sheets are strongly radiating in that each sheet radiates ∼5%–45% of the observed chromospheric radiative loss, with the percentage increasing with increasing n(0), interpreted as decreasing height in the chromosphere.

Magnification of the profiles for Q_R show that essentially all of F_R is due to emission from a region with a thickness ∼2–4 times the minimum value of L_r for a given solution. These regions are centered around the height where Q_R is a maximum. These thicknesses are ∼0.5, 0.7, 2.7, and 13 km. It is also found that for the four sheets ∼50%, 50%, 40%, and 70% of F_R is due to emission from a sub-region 10 times thinner.

The sheets are fully resolved since the minimum allowed step size of the adaptive step code used to solve Equations (15)–(21) is many orders of magnitude smaller than the smallest values of L_r.

These results suggest that simulations and observations need a resolution ∼5–100 m to accurately resolve and compute the emission from strongly radiating current sheets in the most strongly radiating regions of the chromosphere.

Figures 3 and 4 show the temperature, ionization fraction n_e/(n_e + n_H), and electron and H i densities. Figure 5 shows the pressure. The temperature in the sheets is ∼2000–3000 K higher than in the surrounding plasma. The maximum sheet temperatures are 5800–9000 K. The ionization fraction in the sheets is ∼2–20 times higher than in the surrounding plasma. The maximum values of the ionization fraction are ∼2.7 × 10⁻⁴–8.4 × 10⁻². The electron density in the sheets is ∼10–100 times higher than in the surrounding plasma. The maximum neutral density in the sheets is ∼1.2–3 times larger than in the surrounding plasma. The pressure experiences a similar increase since the plasma is mainly neutral and since the maximum temperature in the sheets is ∼1.5 times higher than the temperature in the surrounding plasma.

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The results of Sections 4.1–4.3 show that the current sheet solutions are characterized by plasma conditions believed to exist in the lower and middle chromosphere. As n(0) increases, the conditions within and near the corresponding sheet vary from those characteristic of the middle chromosphere to those characteristic of the lower chromosphere.

The sheets are so strongly radiating that, for increasing n(0), the number of sheets needed to emit the total net radiative flux from the chromosphere is ∼20, 7, 10, and 2. This result is discussed further in Section 5 in the context of the constraints that observations place upon the existence and filling factor of such strongly radiating sheets.

The thermal energy equation (14) may be written symbolically as Q_R = Q_J + Q_c − Q_ce, which indicates the energy sources that determine the radiative emission. Here Q_c ≡ −p∇ · V = −pV'_z is the compressive heating rate, and Q_ce ≡ ∇ · ((3p/2 + χ_Hn_e)V) = ((3p/2 + χ_Hn_e)V_z)' is the heating rate due to the convective energy (ce) flux of thermal and ionization energy, which describes the net flow of this energy into a region.

Figures 6 and 7 show Q_J, Q_R, and Q_c − Q_ce for each of the solutions. Q_c − Q_ce does not make a significant contribution to F_R, but plays an important role in energy balance outside the sheets where Q_R is not significant.

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Define the Joule, compressive, and convective energy fluxes F_J, F_c, and F_ce as the integrals of Q_J, Q_c, and Q_ce over the height range of a solution. Then F_R = F_J + F_c − F_ce.

In order of decreasing n(0) the values of these fluxes for each of the four sheets, in units of 10⁶ erg cm⁻² s⁻¹, are (F_R, F_J, F_c, F_ce) = (4.4507, 5.9742, −1.3915, −0.13206), (1.0253, 1.0845, −0.0561, −0.00311), (1.5499, 1.6035, −0.0444, −0.00914), and(0.4875, 0.5560, −0.0786, 0.00999).

In the same order, F_J/F_R = (1.34, 1.06, 1.04, 1.14), so Joule heating is more than sufficient to balance radiative cooling. The excess is balanced mainly by the negative contribution of F_c.

Joule heating is the primary source of the thermal energy that drives the radiative emission. As discussed in the next section, the primary driver of F_R is the static electric field E_y, and most of F_R is due to proton Pedersen current dissipation.

The Joule heating rate Q_J = J_y(E_y + V_zB_x/c) = Q_P + Q_S + Q_T. The first equality shows that Q_J is driven by the CM electric field, which is the sum of the static electric field E_y and the convection electric field V_zB_x/c. The relative importance of these two drivers is discussed in Section 4.5.1. The second equality shows that Q_J has three components due to different physical processes. The contributions of each one are discussed in Section 4.5.2, and the related magnetization-induced resistivity is discussed in Section 4.5.3.

4.5.1. Convection versus Static Electric Field Driven Joule Heating

Let Q_{J, m} be the maximum value of Q_J for a given sheet. It is found that for all solutions the ratio |V_zB_x/cE_y| ∼ 10⁻⁴–10⁻¹ in the region where Q_J ⩾ 10⁻²Q_{J, m}, and is ∼10⁻³–10⁻¹ in the region where Q_J ⩾ 10⁻¹Q_{J, m}. For all sheets, the smallest values of the ratio are in the regions where Q_J is a maximum. Then it is E_y that directly drives Q_J. Part of the reason for this is the relative smallness of V_z. As shown in Figure 8, V_z ∼ 3–10 m s⁻¹ in the regions that emit essentially all of F_R. This is 2–3 orders of magnitude smaller than characteristic sound and MHD wave speeds, and expected field-aligned flow speeds in the lower and middle chromosphere.

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4.5.2. Components of Q_J

Figures 9 and 10 show the components of Q_J along with Q_R for comparison. Define the Spitzer, Pedersen, and thermoelectric heat fluxes F_S, F_P, and F_T as the integrals of Q_S, Q_P, and Q_T over z. Then F_J = F_S + F_P + F_T.

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In order of decreasing n(0) the values of these fluxes for each of the four sheets, in units of 10⁶ ergs cm⁻² s⁻¹, are (F_R, F_J, F_P, F_S, F_T) =(4.451, 5.974, 3.694, 2.282, −0.0012), (1.025, 1.085, 0.779, 0.316, −0.0105), (1.55, 1.604, 1.209, 0.486, −0.0913), and(0.488, 0.556, 0.502, 0.153, −0.099).

In the same order, the fraction of the resistive heating due to Pedersen current dissipation is F_P/(F_P + F_S) = (0.6181, 0.7113, 0.7131, 0.7662). This shows that Pedersen current dissipation makes the dominant positive contribution to F_J (F_T makes a negative contribution), and hence is the main source of the thermal energy that drives radiative loss. The variation of F_P/(F_P + F_S) with n(0) suggests that Pedersen current dissipation becomes increasingly dominant with increasing height in the chromosphere, consistent with the expected increase of the magnetization-induced resistivity Γη with increasing height.

F_T makes a small-to-moderate negative contribution to F_J that in magnitude is ∼0.02%–15% of the positive contribution of F_P + F_S to F_J. This thermoelectric cooling effect is largest in the lowest density current sheet. This sheet has the largest temperature gradient.

4.5.3. Magnetization Induced Resistivity

The total resistivity (1 + Γ)η, and the amplification factor 1  +  Γ are shown in Figure 11. In the thin regions that emit almost all of F_R, the resistivity decreases by 2–3 orders of magnitude from the exterior to the interior of these regions. Since ρ_H/ρ ∼ 1, and Γ = (ρ_H/ρ)²M_eM_p, the figure shows that this variation is entirely due to the decrease in magnetization. As the neutral sheet is approached 1 + Γ → 1 since the magnetization factor M_eM_p tends to zero. Since, as indicated in Section 4.5.2, Q_P(= ΓηJ²) comprises 62%–77% of the positive contribution to Q_J, it follows that the magnetization-induced resistivity Γη plays a major role in generating Q_J, which drives the radiative loss. If Γ is set equal to zero, the current sheet solutions presented here do not exist.

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4.5.4. Total Energy Conservation and Net Energy Fluxes

The input parameter space for the model presented here is five dimensional. There does not appear to be any way to know a priori what sets of input parameter values lead to physically reasonable solutions. After choosing values for p(0), B_x(0), and T(0) based on current knowledge of chromospheric conditions, the following three heuristic guidelines are used to constrain the search for E_y and V_z(0), resulting in the discovery of the current sheet solutions presented here, and a set of solutions that do not contain current sheets:

• 1.  
  If B_x(0) > 0(< 0), a necessary condition for the solution to contain a current sheet is that the constant field E_y be <0(> 0). The reason is that this is the condition for the vertical Poynting flux to point into the current sheet on both sides. This ensures that electromagnetic energy flows into the sheet to provide the energy for the Joule heating that drives radiative emission and ionization.
• 2.  
  Although E_y is constant in the model, it is assumed that it represents a quasi-static field that is essentially constant over an observed characteristic period t_c ∼ 600 s of Doppler intensity oscillations in the chromospheric network (Lites et al. 1993: the full range of observed periods is ∼5–20 minutes). The x component of Faraday's law gives $t_c |\dot{B_x}|\sim c |E_y| t_c/L$, where $\dot{B_x} \equiv \partial B_x/\partial t$, and L is the scale height of E_y. It is required that L ≫ the height range of the solutions to the model since E_y is assumed constant over this range. It is also required that $t_c |\dot{B_x}| \ll B_x(0)$ in order that the model assumption $\dot{B_x}=0$ be justified. Choose L to be twice the ideal gas pressure scale height for T = 7000 K. Then L = 422.2 km, which is ≫ the 50 km height range over which the solutions are generated. The magnetic field variation $t_c |\dot{B_x}|$ is estimated as follows. Observed temporal variations in the intensity of the emission from the lower chromospheric network are typically ∼20% (Banerjee et al. 1999). It is assumed that this emission is mainly due to the Pedersen current dissipation rate ηΓJ²_⊥. This scales with magnetic field strength B as B⁴, where a factor B² comes from the magnetization-induced resistivity factor Γ, and a factor B² comes from J²_⊥. The fractional change in intensity during the time t_c is then assumed to be ${\sim} (t_c/B^4) \partial B^4/\partial t= 4 t_c \dot{B}/B$. Setting this equal to 20% gives $t_c \dot{B}/B \sim 0.05$. Based on this estimate choose $t_c |\dot{B_x}| \sim 0.05 B_x(0)$. Then the preceding Faraday law estimate for |E_y| gives |E_y| ∼ 0.05B_x(0)L/ct_c ∼ 3.52 × 10⁻³B_x(0) V m⁻¹. Then for B_x(0) = 25–100 G, |E_y| ∼ 0.09–0.35 V m⁻¹, which is, remarkably and without our prior knowledge, close to the values of 0.12–0.25 V m⁻¹ found for the current sheet solutions.
• 3.  
  Given the preceding estimate of E_y, an estimate for V_z(0) is obtained by assuming there is no significant Joule heating at the lower boundary point z = 0, in which case the Ohm's law is approximately the ideal MHD Ohm's law, giving V_z(0) ∼ −cE_y/B_x(0) ∼ −0.05L/t_c ∼ 35.2 m s⁻¹, using the preceding estimate of E_y. This value is comparable to the values of 4–46 m s⁻¹ found for the current sheet solutions.

These guidelines approximately define the region of the input space where the search for solutions was performed, and where current sheet solutions were found to exist. No solutions were found with V_z(0) = 0 or E_y = 0.⁹

The constant guide field B_z is set equal to zero to simplify the model. If B_z ≠ 0 the model becomes significantly more complicated in the following way. If B_z = 0, the x component of the Ohm's law reduces to E_x = 0, consistent with the fact that ∇ × E = 0 implies that E_x is constant. If B_z ≠ 0, the x component of the Ohm's law becomes an additional differential equation of the model. Then the model is overdetermined unless it is assumed that B_y ≠ 0. If B_y ≠ 0, it follows from the x and y components of the momentum equation that V_x and V_y are not constant. These components of the momentum equation can be analytically integrated to obtain V_x(z) and V_y(z) in terms of B_x, B_y, B_z, and boundary conditions at z = 0. The number of input parameters required to determine the solution increases from five to nine if B_z ≠ 0, since B_z, B_y(0), V_x(0), and V_y(0) must now be specified. Although a nonzero B_z does not directly affect the current density since B_z is constant, the fact that B_y ≠ 0 implies that J_x ≠ 0, which directly affects the Joule heating rate. Since there is variation only with respect to z, J_z remains zero.

B_y cannot be a guide field. The reason is as follows. Assume B_y is constant. Allow the constant B_z to have any value. The x component of the Ohm's law, which is E_x = 0 for the model used in the paper, is now E_x + (V_yB_z − V_zB_y)/c = ηJ_y(M_eB_z/B − ΓB_xB_y/B²), where E_x and V_y are constant. This z-dependent equation is an additional equation that must be satisfied. This equation over-determines the solution to the model, so in general there are no non-trivial solutions. Then a nonzero B_y must depend on z, and so cannot be a guide field.

A guide field prevents the magnetization, and hence the magnetization-induced resistivity Γη, from decreasing to zero at the neutral sheet. An important question is whether a guide field has a significant effect on the radiative emission flux F_R. Since F_R is driven mainly by resistive heating, which is partly determined by the resistivity (1 + Γ)η, and since the presence of a guide field increases Γ, it is expected that the guide field has a direct effect on F_R. Although the model outlined in the first paragraph of this section must be solved to obtain a reliable estimate of the effect of a guide field, a simple though crude estimate of the effect of a small guide field on F_R is obtained as follows. Modify Γ(∝B² = B²_x) by replacing B²_x by B²_x + (0.1B_x(0))², where B_x(0) is the given boundary value of B_x. This ad hoc modification mimics the effect on the resistivity of a guide magnetic field that is one order of magnitude smaller than B_x(0). Then as the neutral sheet is approached, Γ → a nonzero value instead of zero. Running the code with this modification leads to a set of current sheets as before and to corresponding values of F_R that differ from those for the solutions discussed in the paper by (− 0.39%, −0.56%, 0.73%, −0.13%), in order of increasing number density. These differences are insignificant and are consistent with the values of Γ at the neutral sheets, which are ∼10⁻⁷–10⁻⁴. Since Γ ≪ 1, the resistivity in the neighborhood of the neutral sheets is ∼η, as in the absence of a guide field. Since η is mainly a function of T, and since T in the neighborhood of the neutral sheets for the modified solutions differs by ≲ 1.36% from its values for the solutions discussed in the paper, it follows that the value of η in this neighborhood does not change. These results suggest that the addition of a guide field that is a factor of 10 or more smaller than the asymptotic value of the field component that generates the current density does not have a significant effect on the emission rate.

Pedersen currents are mainly ion currents that flow along and are driven by E_CM⊥ ≡ E_⊥ + (V × B)/c, where E_⊥ is the component of E⊥B (e.g., Mitchner & Kruger 1973).

Osterbrock (1961), following similar work by Piddington (1956), presents an analysis of damping rates of linear MHD waves in the chromosphere due to charged-particle–neutral collisions,¹⁰ and a scalar viscosity, and concludes that damping of these waves is not important for chromospheric heating. Due to observational limitations at that time, Osterbrock (1961) uses magnetic field strengths orders of magnitude smaller than those now known to exist, causing the heating rate due to charged-particle–neutral collisions to be under estimated by orders of magnitude. Advances in observations since 1961, combined with the largely complete development of the theory of collision-dominated transport in variably ionized plasmas (Braginskii 1965), allowed for the recognition through theoretical modeling that Pedersen current dissipation may be an important chromospheric heating mechanism.

There is a long-standing proposition that Pedersen current dissipation provides essentially all the thermal energy that drives chromospheric radiative emission, and that it is the increase in the magnetization of the plasma with height from the photosphere that triggers the onset of heating by Pedersen current dissipation sufficient to cause and maintain the chromospheric temperature inversion (Goodman 2000, 2004a).

Heating by Pedersen current dissipation is not effective in the weakly ionized, weakly magnetized photosphere because the magnetization-induced resistivity Γη(= M_eM_i(ρ_H/ρ)²η) < η due to weak magnetization (M_eM_i < 1), and it is not effective in the strongly ionized, strongly magnetized corona because Γ ≪ 1 due to strong ionization ((ρ_H/ρ)² ≪ 1). This heating mechanism can only be effective in the weakly ionized, strongly magnetized chromosphere, between the photosphere and corona, since it is only there where Γ can be orders of magnitude greater than unity, causing the plasma to be highly resistive with respect to the flow of Pedersen currents, allowing for, but not guaranteeing significant resistive heating.

Although Γη can take on values orders of magnitude larger than η under chromospheric conditions (e.g., Figure 11, and references cited in this section), a large resistivity does not by itself imply a significant heating rate. An outstanding question is what are the primary drivers of Pedersen current dissipation in the chromosphere, and are they sufficiently strong to cause significant chromospheric heating? Any process that generates an E_CM⊥ can in principle drive a Pedersen current, and hence some level of heating by Pedersen current dissipation. The initial modeling of driving by convection electric fields, generated by the steady bulk flow of plasma ⊥B, is presented in Goodman (1997a, 1997b) and extended in Goodman (2004b). The initial modeling of driving by linear Alfvén waves is presented in De Pontieu & Haerendel (1998) and De Pontieu et al. (2001). Their basic model is extended in Goodman (2011) to include the full effects of the inhomogeneous background atmosphere on the waves from the photosphere to the lower corona. The initial modeling of driving by slow magnetoacoustic waves and of Pedersen current dissipation in the shock layers of fast magnetoacoustic shock waves are presented in Goodman (2000, 2001) and Goodman & Kazeminezhad (2010), respectively.

Much additional work has been done in this area. Overall, the models indicate that, under conditions consistent with current understanding of the chromosphere, Pedersen current dissipation driven by several different drivers can provide all the thermal energy needed to balance the total net radiative loss from the chromosphere. What is needed most to determine the effectiveness of this heating mechanism are calculations of J based on observations of B in the chromosphere. Calculations of this type have been done for sunspot chromospheres (Socas-Navarro 2005a, 2005b; also see comments on those calculations in Section 6.1.2 of Goodman 2011). Such calculations are especially challenging since J(∝∇ × B) involves differences of derivatives of components of B, so any error in the measurement of B is multiply compounded in the calculation of J.