ACCELERATION OF TYPE II SPICULES IN THE SOLAR CHROMOSPHERE

Type II spicules are a recently discovered phenomenon. They appear to be upward jets of heated plasma in the chromosphere. They were first observed at the limb (De Pontieu et al. 2007a, 2007b, 2007c) and later identified with so-called rapid blueshifted excursions (RBEs) on the disk (Langangen et al. 2008; Rouppe van der Voort et al. 2009; Sekse et al. 2012). They extend over height ranges ∼10³ km; have maximum vertical flow speeds ∼50–150 km s⁻¹, durations ∼10–150 s, are characteristic diameters ≲ 200 km; are heated to transition region temperatures during their acceleration; and exhibit horizontal motions, sometimes inferred to be due to superimposed Alfvén waves, with speeds ∼5–10 times less than the maximum vertical speeds.

Maximum vertical flow speeds of ∼50–150 km s⁻¹ are inferred for type II spicules from limb observations (De Pontieu et al. 2007a, 2007b, 2007c). Maximum vertical flow speeds of ∼20–70 km s⁻¹ (Rouppe van der Voort et al. 2009) and ∼10–50 km s⁻¹ (Sekse et al. 2012) are inferred for RBEs from disk observations. The smaller values of the speeds inferred from the disk observations are probably due to a combination of projection effects on the disk that tend to reduce the apparent upward speed, the fact that limb observations are biased toward detecting those type II spicules that extend farthest above the limb and so tend to have the largest upward speeds, and the tendency of the on-disk velocity observations to sample lower heights in the atmosphere where speeds tend to be smaller (Rouppe van der Voort et al. 2009).

It is speculated that type II spicules are a major source of mass and energy for the corona (De Pontieu et al. 2009, 2011). Their acceleration and heating mechanisms are unknown.

There is an opposing view that RBEs are not type II spicules and that type II spicules are not jets of plasma but instead are warping, mainly vertical, two-dimensional sheets of magnetic field lines (Judge et al. 2011). The first part of this view is based on comparing the estimate of the occurrence rate of type II spicules made by Judge & Carlsson (2010) with the estimate of the occurrence rate of RBEs made by Rouppe van der Voort et al. (2009). Those estimates suggest that type II spicules occur ∼10–100 times more frequently than RBEs. Doppler shift measurements from the upcoming IRIS mission¹ are expected to contribute to resolving this controversy.

In the magnetohydrodynamic (MHD) model of a plasma, the acceleration of bulk flow is driven by the pressure gradient and Lorentz forces. The purpose of this paper is to use a 2.5D, time-dependent MHD model to determine if the Lorentz force generated by a horizontally localized current density of finite duration can accelerate vertical flows to speeds characteristic of type II spicules within their observed lifetimes and horizontal diameters, and within ranges of density, temperature, and magnetic field strength believed to be reasonable for the middle-upper chromosphere.

Solutions to the model suggest that type II spicules can be driven by this Lorentz force under chromospheric conditions. For these solutions the Joule heating flux, when averaged over horizontal regions with diameters of 50–200 km, reaches values ∼0.1–3.7 times the characteristic time-averaged middle-upper chromospheric radiative loss rate of ∼10⁶ ergs cm⁻² s⁻¹ (Anderson & Athay 1989; Fontenla et al. 2002) for ∼10–20 s before decreasing back to background values.

The strategy is to consider a model that can be solved with high accuracy to avoid significant numerical error and that contains an essential description of a 2.5D bulk flow accelerated by a spatially and temporally localized Lorentz force. This is done by specifying a form for a spatially and temporally localized vertical current density and then deriving the corresponding radial current density, in cylindrical coordinates, under the assumption that the azimuthal current density is zero. The magnetic field is computed from the current density. Additional assumptions then allow the momentum and mass conservation equations to be solved analytically for the velocity, density, and pressure. Although the model is restricted to a specific form of the current density, and hence of the magnetic field, it is proposed that more general current densities with orders of magnitude and spatial and temporal localization scales similar to those used in the model also give rise to Lorentz forces that can accelerate chromospheric plasma to type II spicule velocities. In this sense, the model represents a special case of a more general Lorentz force acceleration mechanism that might play an important role in accelerating type II spicules.

Assume cylindrical coordinates (R, θ, z), and let t be time. All quantities are independent of θ, and z is height above some unspecified point in the middle-upper chromosphere. Assume a weakly ionized H plasma with a constant temperature T. Then p = ρk_BT/m_p = ρV²_s, where p, ρ, V_s, m_p, and k_B are the pressure, mass density, sound speed, proton mass, and Boltzmann's constant. Assume p = p₁(R, t)exp (− z/L) and ρ = ρ₁(R, t)exp (− z/L), where L = k_BT/m_pg is the ideal gas pressure scale height and g = 2.74 × 10⁴ cm s⁻².

Let V be the center-of-mass (bulk flow) velocity. Assume the magnetic field B = b(R, t)exp (− z/2L). Neglecting viscous effects in the momentum equation, it follows from the momentum and mass conservation equations that an exact solution for the height dependence of V is that it is independent of height. This form of the solution is assumed here, so V = V(R, t). The assumed form of the height dependence of B, V, and ρ implies that the flow is accelerated by a height-independent Lorentz force per unit mass along a column of gas. The current density J = j(R, t)exp (− z/2L) = c∇ × B/4π. It is assumed that j_θ = 0.

Under the preceding assumptions, the momentum and mass conservation equations are

$$\begin{equation} \dot{V}_R + V_R V_R^\prime - \frac{V_\theta ^2}{R} + V_s^2 \frac{\rho _1^\prime }{\rho _1}=- \frac{j_z b_\theta }{c \rho _1}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \dot{V}_\theta + V_R \frac{(R V_\theta)^\prime }{R} = \frac{j_z b_R - j_R b_z}{c \rho _1}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \dot{V}_z + V_R V_z^\prime = \frac{j_R b_\theta }{c \rho _1}, \end{equation} \tag{ 3 }$$

$$\begin{equation} \dot{\rho }_1 + \rho _1 \frac{(R V_R)^\prime }{R} +V_R \rho _1^\prime = \frac{\rho _1 V_z}{L}. \end{equation} \tag{ 4 }$$

Here the dot and prime denote ∂/∂t and ∂/∂R.

The assumption that T is constant and that p varies with z according to the factor exp (− z/L) causes the gravitational force term −ρg and ∂p/∂z to cancel one another in the vertical component of the momentum equation, which is Equation (3). That is why these terms do not appear in that equation. Then, the vertical pressure gradient and the gravitational force do not play any role in the vertical acceleration of the plasma for the model presented here, although they are both time dependent and increase by orders of magnitude during the acceleration process for the model solutions in Section 4.

Since j_θ = 0, it follows that (∇ × B)_θ = 0, so B_R and B_z together make up a potential field. This equation, together with the constraint ∇ · B = 0, determines this field to be a constant vertical field B₀ plus a radial and additional vertical component given by b_r = b₀J₁(R/2L) and b_z = b₀J₀(R/2L), where J_{0, 1} are Bessel functions of the first kind of order 0 and 1. Here, b₀ is chosen to be zero for simplicity. Then, B_R = 0 and B_z = B₀.

Type II spicule velocities are mainly vertical and appear to be parallel to a guide magnetic field, but they can have a significant component perpendicular to this main flow. Sekse et al. (2012) observe horizontal motions of RBEs with speeds mainly ∼5–10 km s⁻¹, with most peak speeds ∼5 km s⁻¹ and maximum speeds ∼20 km s⁻¹. These horizontal motions are superimposed on vertical flows with apparent speeds ∼10–50 km s⁻¹, noting that the actual vertical flow speeds may be significantly higher due to projection effects. De Pontieu et al. (2012) observe torsional and swaying motions of type II spicules on the limb with speeds ∼25–30 and 15–20 km s⁻¹ that are perpendicular to and superimposed on nearly vertical flows (∼20° inclination) with speeds ∼50–100 km s⁻¹. De Pontieu et al. (2012) conjecture that these transverse, mainly horizontal motions are due to Alfvén waves with periods ∼100–300 s that propagate along the spicules. Since these periods are ≳ spicule lifetimes (∼100 s), it is difficult or impossible to observe a complete oscillation. It is expected that Alfvén waves occur in type II spicules because these waves are one of the fundamental MHD wave modes that must be continually excited in a magnetized atmosphere driven by convection with a wide range of temporal and spatial scales. It is not known if these waves play an important role in accelerating or heating type II spicules.

The model used here does not model waves, and although the model solutions have horizontal flows, these flows are restricted to be sufficiently small in magnitude to simplify solving the model for the vertical flow speed. This restriction is consistent with the observation that the primary acceleration is in the vertical direction, but it results in the model generating horizontal flow speeds that are not as large a fraction of the vertical flow speeds as suggested by the observations of Sekse et al. (2012) and De Pontieu et al. (2012). The horizontal flow speeds in the model are restricted in the following way. Constraints are imposed on V_R to justify omitting the terms involving V_R in the momentum Equations (1)–(3). Inspection of these equations indicates that the constraints |V_R| ≪ |V_z| and |V_R| < V_s should be adequate. The additional constraint $|V_\theta | < V_s \bar{R}^{1/2}$ allows the V²_θ/R term in Equation (1) to be omitted in first approximation. Here $\bar{R}=R/R_0$, where R₀ is the radial scale of the current density, defined in Section 2.1. As shown in Section 3, the constraint on V_R places an upper bound on R₀, and the constraint on V_θ places an upper bound on B₀.

Despite the neglect of wave motion and the upper bounds placed on the horizontal flow speeds, the model solutions show that the vertical flow speed can be accelerated to type II spicule values, with maximum speeds comparable to slow solar wind speeds (∼ 200–400 km s⁻¹) on sub-resolution horizontal spatial scales. This suggests that the primary acceleration mechanism of type II spicules does not involve waves with periods ≲ type II spicule lifetimes, consistent with the observations and conjecture of De Pontieu et al. (2012) discussed above.

After applying the constraints on the horizontal flow speeds indicated above, the radial component of the momentum equation, which is Equation (1), states that ∂p/∂R balances the radial Lorentz force. This gives rise to radial compression of the plasma. This is a z pinch effect since the radial Lorentz force is generated by J_z and B_θ.

2.1. Current Density and Azimuthal Magnetic Field

The vertical current density j_z is assumed to be localized in time and space with characteristic time and radial scales t₀ and R₀ and to have the following functional form:

$$\begin{equation} j_z = j_0 \exp (-\bar{t})(1-\exp (-\bar{t})) \exp (-\bar{R}^2). \end{equation} \tag{ 5 }$$

Here $\bar{R} = R/R_0$ and $\bar{t} = t/t_0$.

It follows from ∇ · j = 0 that the radial current density

$$\begin{equation} j_r = \frac{j_0 R_0}{4 L} \exp (-\bar{t})(1-\exp (-\bar{t})) \frac{(1-\exp (-\bar{R}^2))}{\bar{R}}. \end{equation} \tag{ 6 }$$

From ∇ × B = 4πJ/c, it follows that

$$\begin{equation} b_\theta =b_{\theta 0} \exp (-\bar{t})(1-\exp (-\bar{t})) \frac{(1-\exp (-\bar{R}^2))}{\bar{R}}, \end{equation} \tag{ 7 }$$

where b_θ0 ≡ 2πj₀R₀/c. A plot of b_θ/b_θ0 is shown in Figure 1.

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With respect to $\bar{t}$, the maximum values of j_z, j_r, and b_θ are reached at $\bar{t}=\ln (2) \,{\sim}\, 0.693147$, for which $\exp (-\bar{t})(1-\exp (-\bar{t})) = 1/4$. With respect to $\bar{R}$, the maximum values of j_r and b_θ are reached at $\bar{R}= 1.1209$, for which $(1-\exp (-\bar{R}^2))/\bar{R} = 0.6382$. The maximum value of b_θ and hence of B_θ is then 0.6382 b_θ0/4 = 0.1596 b_θ0. Then specifying b_θ0 determines the maximum magnitude of the magnetic field and determines j₀R₀.

Omit the terms involving V_R in Equations (1)–(3), and omit the term involving V_θ in Equation (1), assuming the constraints on V_R and V_θ discussed in Section 2. Solving these equations successively for ρ₁, V_θ, and V_z gives

$$\begin{equation} \rho _1 = \rho _0 + \frac{b_{\theta 0}^2}{2 \pi V_s^2}(\exp (-\bar{t})(1-\exp (-\bar{t})))^2 (I_{\rm max} - I(\bar{R})), \end{equation} \tag{ 8 }$$

where

$$\begin{equation} I(\bar{R}) = \int _0^{\bar{R}} \frac{\exp (-x^2)(1- \exp (-x^2))}{x} \; dx \end{equation} \tag{ 9 }$$

and I_max = I(∞) = 0.3466. $I(\bar{R})$ reaches 99% of I_max at $\bar{R}=1.87$,

$$\begin{eqnarray} V_\theta &=& - \frac{b_{\theta 0} B_0 t_0}{8 \pi L \rho _0} \left(\frac{1- \exp (-\bar{R}^2)}{\bar{R}}\right) \nonumber\\ &&\times \int _{\exp (-\bar{t})}^1 \frac{(1-x)}{1+A(\bar{R})(x(1-x))^2}\; dx, \end{eqnarray} \tag{ 10 }$$

where

$$\begin{equation} A(\bar{R})=\frac{b_{\theta 0}^2}{2 \pi \rho _0 V_s^2}(I_{\rm max} - I(\bar{R})) \end{equation} \tag{ 11 }$$

$$\begin{eqnarray} V_z &=& \frac{b_{\theta 0}^2 t_0}{8 \pi L \rho _0}\left(\frac{1-\exp (-\bar{R}^2)}{\bar{R}}\right)^2 \nonumber\\ &&\times \int _{\exp (-\bar{t})}^{1} \frac{y(1-y)^2}{1+A(\bar{R})\left(y(1-y)\right)^2}\; dx. \end{eqnarray} \tag{ 12 }$$

Then, solving Equation (4) for V_R gives

$$\begin{eqnarray} &&V_R=\frac{R_0}{\rho _1 \bar{R}} \int _0^{\bar{R}} x \left(\frac{\rho _1 V_z}{L} + \frac{2 \left(\rho _1-\rho _0\right)}{t_0} \frac{1-2 \exp (-\bar{t})}{1-\exp (-\bar{t})} \right) \; dx. \nonumber\\ \end{eqnarray} \tag{ 13 }$$

ρ₁, V_z, and V_θ depend on R only through $\bar{R}$, and in this sense they are scale invariant. V_R explicitly involves a factor of R₀, so the profile of V_R scales linearly with R₀. This is where the constraint on R₀ enters the solution. For given values of the other input parameters, R₀ must be chosen sufficiently small so that |V_R| ≪ |V_z| and |V_R| < V_s. Only V_θ involves B₀. For given values of the other input parameters, B₀ must be chosen sufficiently small so that $|V_\theta | < V_s \bar{R}^{1/2}$.

The average of any quantity f(R, z, t) over the horizontal circular area πR² centered at R = 0 is denoted by 〈f〉_R and is computed as

$$\begin{equation} \langle f \rangle _R(z,t) = \frac{2}{\bar{R}^2} \int _0^{\bar{R}} f(R,z,t) \bar{R} \; d\bar{R}. \end{equation} \tag{ 14 }$$

Equation (12) shows that V_z increases with $\bar{t}$ to a finite, terminal, $\bar{R}$-dependent speed determined by the inputs to the model. As $\bar{t} \rightarrow \infty$, Equation (8) shows that $\bar{\rho } \rightarrow \rho _0$. It follows from Equation (13) that the terminal profile for V_R is

$$\begin{equation} V_R(R,\infty)= \frac{R_0 \bar{R}}{2L} \langle V_z(\bar{R},\infty)\rangle _R. \end{equation} \tag{ 15 }$$

For the particular solutions presented in Section 4, the terminal velocity profile is essentially reached for $\bar{t}\, {\sim }\,3\hbox{--}4$.

The acceleration is assumed to occur in the middle-upper chromosphere. The inputs to the model are R₀, t₀, b_θ0, ρ₀, and T₀. Two solutions are presented. For both solutions t₀ = 33.3 s, T₀ = 7000 K, B₀ = 0.5 G, and ρ₀ ≡ m_pn₀, where n₀ = 10¹² cm⁻³. Then V_s = 7.6 km s⁻¹ and L = 211 km. The choice of such a small value of B₀ is necessary to satisfy the constraint on V_θ discussed in Section 3 and is probably an artifact of the simplicity of the model. All plots in Sections 4.1 and 4.2 are at the reference height z = 0, except as otherwise indicated.

4.1. Solution 1

For this solution, R₀ = 5 km and b_θ0 = 156.7 G. Then, B_θ ⩽ 25 G and j₀ = 4.984 A m⁻². Figures 2– 8 show radial profiles of the solution for selected times.

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Figure 2 shows the number density n = ρ/m_p. During the acceleration process, the plasma is compressed to a density ∼100 times its pre-acceleration value. Maximum compression is on the axis, and the region of significantly increased density has a diameter of ∼20 km. Maximum compression is reached at t ∼ 0.693t₀ = 23 s, and the density returns to its pre-acceleration value after a duration of ∼4t₀ = 133 s.

Figure 3 shows the vertical velocity V_z. The plot extends to $\bar{R} = 20$, corresponding to R = 100 km, and hence to a diameter of 200 km. The terminal velocity profile is reached at t ∼ 3t₀ = 100 s. The maximum of V_z = 142.4 km s⁻¹ and occurs at R = 10.5 km. V_z < V_s for R > 50 km.

Figures 4 and 5 show V_R and V_θ. They are ≪V_z almost everywhere, in particular in the region where V_z is near its maximum value. Then V is essentially vertical. Figure 4 shows that at early times V_R < 0 and that V_R rapidly increases to positive values for later times.

Figure 6 shows 〈V_z〉_R, V_R, V_θ, and the horizontal flow speed V_H = (V²_R + V²_θ)^1/2 at t = 4t₀, after the acceleration is over. The figure suggests that if an observation effectively averages the vertical velocity over diameters ∼50, 100, or 200 km, the inferred terminal vertical flow speeds are ∼66, 27, and 10 km s⁻¹.

4.1.1. Joule Heating Rate

The standard expression for the Joule heating rate per unit volume due to the Spitzer resistivity η and the magnetization-induced Pedersen resistivity ηΓ is (e.g., Mitchner & Kruger 1973)

$$\begin{equation} Q_J = \eta \big(J^2 + \Gamma J_\perp ^2 \big), \end{equation} \tag{ 16 }$$

where J_⊥ is the component of J⊥B. The magnetization factor Γ = M_eM_p(ρ_n/ρ)², where M_e, M_p, and ρ_n are the electron and proton magnetizations and the neutral density, respectively. In the chromosphere ρ_n ∼ ρ. Explicitly,

$$\begin{eqnarray} \eta &=& m_e^{1/2} \left(\frac{4 (2 \pi)^{1/2} e^2 \ln (\lambda)}{3 (k_B T)^{3/2}} + \frac{\sigma \rho (k_B T)^{1/2}}{n_e m_p e^2}\right), \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \lambda &=& \frac{3(k_B T)^{3/2}}{2 e^3(\pi n_e)^{1/2}}, \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} \eta \Gamma &=& \frac{B^2 m_p^{1/2}}{c^2 \sigma \rho n_e (k_B T)^{1/2}}. \end{eqnarray} \tag{ 19 }$$

Here m_e, n_e, e, and σ(= 5 × 10⁻¹⁵ cm²) are the electron mass, density, and charge magnitude and the charged-neutral particle scattering cross section, respectively (Osterbrock 1961).² All quantities in Equations (17)–(19) are known except n_e.

An approximate expression for n_e under solar chromospheric conditions is derived in Goodman & Judge (2012). It is

$$\begin{eqnarray} n_e(\mbox{cm$^{-3}$}) &=& n_{\rm H}^{1/2}(\mbox{cm$^{-3}$}) \left(\frac{3.2 \times 10^4}{\alpha }\right)^{1/2} \nonumber\\ &&\times \left(\frac{T}{10^4}\right)^{|b|/2} \exp \left(-T_2/2T\right) \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \hspace{-3.1pc}&\equiv & n_{\rm H}^{1/2} I(T). \end{eqnarray} \tag{ 21 }$$

Here T₂ = 1.18187 × 10⁵ K, α = 3.373 × 10⁻¹³, b = −0.786, and n_H is the H i density. Equation (20) includes all lowest order non-LTE (NLTE) effects. In the present model, n_H = ρ/m_p. Equation (20) with T = T₀ is used here to estimate Q_J. The heating flux F_J ∼ ∫^∞₀Q_J dz ∼ 2LQ_J(R, t).

Figure 7 shows Q_J. Joule heating is confined within a radius of 10 km. The Pedersen current dissipation rate ΓηJ²_⊥ is ∼10²–10³ times larger than the Spitzer heating rate ηJ². This is due to the large magnetization-induced resistivity Γη ≫ η. Since Q_J(R, z, t) ∼ exp (− z/2L)Q_J(R, t), where 2L = 422 km, Q_J decreases by a factor of 10 over a height range of 10³ km.

Figure 8 shows 〈F_J〉_R. The figure shows that the maximum value 〈F_J〉_R over diameters of 50, 100, and 200 km is ∼(13, 3, 1) × 10⁵ ergs cm⁻² s⁻¹, respectively, to be compared with the observed middle-upper chromospheric heating rate of ∼10⁶ ergs cm⁻² s⁻¹. This suggests that the resistive heating rate in type II spicules can exceed that in the surrounding plasma by ∼10%–100% for ∼10–20 s.

4.2. Solution 2

This solution explores the effect of increasing the maximum value of B_θ by a factor of 2 to 50 G. Then R₀ must be reduced by a factor of 4 to 1.25 km to keep |V_R| sufficiently small. Then, b_θ0 = 313.4 G and j₀ = 39.875 A m⁻², which is a factor of 8 larger than for Solution 1.

Equation (8) shows that the maximum density increases by a factor of ∼4 over that of Solution 1, to ∼(3–4) × 10¹⁴ cm⁻³. The region of significantly increased density has a diameter of ∼5 km.

Figure 9 shows V_z. As before, the terminal velocity profile is reached at t ∼ 3t₀ = 100 s. The maximum of V_z = 456.3 km s⁻¹, which is comparable to the slow solar wind speed and occurs at R = 3 km. V_z < V_s for R > 25 km.

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Figure 10 shows 〈V_z〉_R, V_R, V_θ, and V_H at t = 4t₀. The figure suggests that if an observation effectively averages the vertical velocity over diameters of ∼50, 100, and 200 km, the inferred terminal vertical flow speeds are ∼35.5, 11.6, and 3.6 km s⁻¹, respectively. These vertical speeds are ∼2–3 times smaller than those for Solution 1, even though the maximum vertical flow speed is 3.2 times larger than for that solution. This is because the current density, and hence the acceleration, is confined within a smaller radius.

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Both solutions have an annular region defined by $1 \lesssim \bar{R} \lesssim 5$ in which the vertical flow speed is highly supersonic, with maximum values comparable to slow solar wind speeds. Since the thickness of these regions is ∼5–20 km, relatively high resolution is needed to resolve them if they exist. One test of the viability of the acceleration mechanism modeled here is to see if future, higher resolution observations reveal overall higher flow speeds as the region of maximum flow speed predicted by the model is increasingly better resolved.

Figure 11 shows Q_J. It is ∼35 times larger than Q_J for Solution 1.

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Figure 12 shows 〈F_J〉_R. The figure shows that 〈F_J〉_R over diameters of 50, 100, and 200 km is ∼(36.8, 9.2, 2.3) × 10⁵ ergs cm⁻² s⁻¹, respectively. Similar to the corresponding result for Solution 1 shown in Figure 8, this suggests that the resistive heating rate in type II spicules can exceed that in the surrounding plasma by ∼23%–368% for ∼10–20 s.

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The result for Solutions 1 and 2 that V_z reaches maximum speeds ∼150 and 460 km s⁻¹ over radially localized regions with sub-resolution diameters suggests that significant slow solar wind acceleration occurs in type II spicules on sub-resolution horizontal spatial scales.

The magnetic energy before and after the acceleration is the same. Then Poynting's theorem implies that all the energy for accelerating and heating the plasma flows into the acceleration region as a Poynting flux and is entirely converted into thermal and bulk flow kinetic energy. Explicitly, Poynting's theorem implies ∫^∞₀(Q_J + V · (J × B)/c) dt = −∫^∞₀∇ · S dt, where S is the Poynting flux. Then, the integral of Q_J + V · (J × B)/c over time and volume gives the total energy input into the plasma as the sum of the thermal energy input E_J and the work W done in accelerating the plasma. These quantities are estimated as follows:

$$\begin{eqnarray} W &=& 2 \pi \int _0^\infty dt \int _0^\infty dz \int _0^\infty \frac{({\bf J} \times {\bf B}) \cdot {\bf V}}{c} R \; {\it dR} \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} &=& \frac{2 \pi L}{c} \int _0^\infty dt \int _0^\infty \left(-j_z b_\theta V_R - 2 j_R B_z V_\theta + j_R B_\theta V_z\right) R \; {\it dR} \nonumber\\ \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} & \sim & \frac{2 \pi L R_0^2 t_0}{c} \int _0^6 d\bar{t} \int _0^{\bar{R}_{\rm max}} (-j_z b_\theta V_R - 2 j_R B_z V_\theta \nonumber\\ && + j_R B_\theta V_z) \bar{R} \; d\bar{R}, \end{eqnarray} \tag{ 24 }$$

$$\begin{equation} E_J \sim 4 \pi L R_0^2 t_0 \int _0^6 d\bar{t} \int _0^{\bar{R}_{\rm max}} Q_J(\bar{R},\bar{t}) \bar{R} \; d\bar{R}. \end{equation} \tag{ 25 }$$

For Solutions 1 and 2, let $\bar{R}_{\rm max} = 20$ and 80, corresponding to a diameter of 200 km. For Solution 1, W = 2.36 × 10²² ergs and E_J = 1.35 × 10²¹ ergs, so the total energy input is 2.5 × 10²² ergs. About 94.4% of this energy is converted into bulk flow kinetic energy. For Solution 2, W = 1.93 × 10²² ergs and E_J = 3.81 × 10²¹ ergs, so the total energy input is 2.31 × 10²² ergs. About 84% of this energy is converted into bulk flow kinetic energy. Almost all of the electromagnetic energy that heats and accelerates the plasma is converted into bulk flow kinetic energy. The total energy inputs for Solutions 1 and 2 are ∼5 times smaller than characteristic nanoflare energies of ∼10²³–10²⁴ ergs.³

Current densities localized on space and time scales of type II spicules and that generate reasonable magnetic field strengths can give rise to a Lorentz force that accelerates chromospheric plasma to terminal vertical and horizontal velocities similar to those of type II spicules. A vertical pressure gradient is not needed to generate the observed flows, although in reality it might play an important role in spicule acceleration.

The maximum vertical flow speeds occur in horizontally localized regions with sub-resolution diameters near the axis of the spicules. These speeds can reach values comparable to slow solar wind speeds ∼150–460 km s⁻¹. This suggests that the acceleration of type II spicules is an important part of the mechanism that accelerates the slow solar wind.

Almost all of the magnetic energy that accelerates and heats the plasma is converted into bulk flow kinetic energy, as opposed to thermal energy.

The radial component of the Lorentz force compresses the plasma during the acceleration process, possibly by factors as large as ∼100, although the highest densities remain within the known chromospheric density range. The Joule heating flux generated during the acceleration process is essentially entirely due to proton Pedersen current dissipation, and it can be a significant fraction of and may exceed the heating flux of ∼10⁶ ergs cm⁻² s⁻¹ associated with the relatively quasi-steady middle-upper chromospheric emission.

In the model used here, the Joule heating rate is uncoupled from the predicted V, ρ, and p in that it has no effect in determining these quantities. This is a result of the model not including an energy equation, which is redundant given the assumption of constant T. If this assumption is relaxed, an energy equation must be included in the model, which then self-consistently determines T along with V, ρ, and p. Such a model can use the approximate analytic expressions for the NLTE ionization fraction and radiative cooling rate in the chromosphere derived in Goodman & Judge (2012), in which case the electron density and radiative cooling rate are also computed self-consistently by the model.

A model that self-consistently includes an energy equation, variable T, realistic transport coefficients (most importantly the electrical conductivity tensor for a partially ionized plasma), and NLTE ionization and radiative cooling and that is solved with sufficient numerical resolution to resolve the smallest relevant length scales is needed to accurately model energy flow and transformation rates, including radiative emission. No such model exists. The smallest length scales that must be resolved to accurately describe type II spicule heating and acceleration are unknown, although the results presented here suggest that a horizontal resolution ≲ 5 km is necessary.

If magnetic reconnection plays a role in type II spicule formation, then the radiating current sheet solutions in Goodman & Judge (2012) might give an indication of the resolution needed to resolve the resistive length scales in the chromosphere. These solutions indicate that a numerical resolution ≲ 10–100 m is needed, aside from it being necessary to ensure that any artificial diffusive effects introduced into the model are much smaller than the physical diffusive effects described by the realistic transport coefficients in the model. The fact that the ionization fraction and radiative emission rates are exponentially sensitive functions of T, and that T is largely determined by diffusive transport processes such as Joule heating, emphasizes the need to use realistic transport coefficients and fully resolve their effects.

The origin of the current density in the model used here is not indicated. Processes that generate this current density, or more generally that generate a Lorentz force similar to the one modeled here, need to be identified. Magnetic reconnection might be such a process. Other processes must also be considered, such as that suggested by the simulations of Martínez-Sykora et al. (2011), which involves horizontal compression by a Lorentz force causing a vertical pressure gradient that accelerates plasma along a mainly vertical magnetic field. This horizontal compression and the radial compression in the model solutions presented here might cause a transient compressive heating rate −p∇ · V that causes an observable increase in radiative emission.

The author gratefully acknowledges support from grants ATM-0650443 and ATM-0848040 from the Solar Physics Program of the National Science Foundation to the West Virginia High Technology Consortium Foundation. The author thanks the referee for constructive remarks that led to significant improvements in the paper. This research made use of NASA's Astrophysics Data System (ADS).