Hydrogen ionization in the solar atmosphere

Hydrogen ionization in the solar atmosphere

Jorrit Leenaarts

photosphere: 3% ionized

low chromosphere: 0.1-50%

ionized

Hα chromosphere: 10% - 99%

ionized

corona: 100% ionized

Ionization is a substantial buffer of energy

Ionization strongly affects thermal evolution H: 90% of nuclei

He:10% of nuclei

• ^{E(H→H 2 ) / E th} = 22 @ 2 kK

• E(H I→H II) / E th =17 @ 8 kK

• E(He I→He II) / E th = 3 @ 10 kK

• E(He II→He III) / E th = 2 @ 30 kK

Ionization is slow in the chromosphere

solution:

rate equations:

Dn ₁

Dt = n ₂ P ₂₁ − n ¹ P ₁₂ Dn ₂

Dt = n ₁ P ₁₂ − n ² P ₂₁ n _H = n ₁ + n ₂

n ₁ (t) = n ₁ (∞) + [n ¹ (0) − n ¹ (∞)]e ^−t(P

¹²

^+P

²¹

⁾ n ₁ (∞) = n _H P ₂₁

P ₁₂ + P ₂₁

Dn ₁

Dt = n ₂ P ₂₁ − n ¹ P ₁₂ Dn ₂

Dt = n ₁ P ₁₂ − n ² P ₂₁ n _H = n ₁ + n ₂

n ₁ (t) = n ₁ (∞) + [n ¹ (0) − n ¹ (∞)]e ^−t(P

¹²

^+P

²¹

⁾

Ionization is slow in the chromosphere

How does hydrogen ionize?

see Carlsson & Stein, 2002, ApJ, 572, 625

that when chemical rates are slow, populations do not have time to reach their equilibrium values. Indeed, some of the disagreement with observations may be due to this nonequi- librium. The solar chromosphere will experience some addi- tional heating besides that included in our model. However, as we show later, even in such hotter models the ionization/

recombination rates are slow, and the results presented here will be qualitatively correct.

3. IONIZATION AND EXCITATION PROCESSES

In the upper chromosphere and lower transition region the density is much too low to ensure LTE, and the full rate equations have to be solved to calculate the hydrogen ion- ization. Furthermore, inspection of the rates involved shows that typical ionization/recombination timescales are much longer than dynamical timescales, so that the ionization bal- ance can be expected to be out of statistical equilibrium. It is thus necessary to take into account the advection and time- derivative terms in the equations. In this section we analyze the rate equations to find what processes dominate the hydrogen ionization balance. In the next section we analyze the timescales involved.

Figure 2 shows the hydrogen ionization fraction as a function of column mass (m_c) in the initial radiative equili- brium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

The processes that are important for the ionization bal- ance are not the same in the transition region and in the chromosphere. In the solar chromosphere the hydrogen ion- ization is dominated by photoionization from the first excited state. The reason is that the much lower particle den- sity of the first excited state (#3 $ 10⁴ cm^!3) than in the ground state (#10¹¹ cm^!3) is more than compensated by the higher mean intensity at the Balmer edge at 364 nm com- pared to that of the Lyman edge at 91 nm. At a radiation temperature of 5000 K the ratio is 3 $ 10⁸. In addition, the atmosphere is optically thick to Lyman photons and opti- cally thin to Balmer photons in all of the chromosphere.

The result is that photoionization by Lyman photons plays an insignificant role in the hydrogen ionization throughout the chromosphere. Figure 3 shows the net rates between all the hydrogen energy levels at lg m_c ¼ !4, which corre- sponds to a height of 1.2 Mm in the initial atmosphere. This picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m_c ¼ !5:6. Ionization is primarily through a net rate from n ¼ 2 to the continuum (photoionization in the Balmer continuum) balanced by recombination to the higher levels cascading down to the n ¼ 2 level through bound-bound transitions with Dn ¼ 1. The reservoir of elec- trons is the ground state. There is transfer of electrons between the ground state and first excited state by collisions.

The rate of this transfer is very slow, because chromospheric temperatures are of order 1 eV and the energy jump to the first excited level is 10.2 eV.

In the very thin zone (2 km), where the ionization fraction goes from 30% to fully ionized, the situation is different.

Collisional ionization from the ground state dominates, with net photorecombination in all continua and again bound-bound transitions with Dn ¼ 1 dominating the return channel to the ground state (Fig. 4).

Figures 3–4 show only the net rates and do not answer the question of what is driving the system and what rates are just adjusting to provide a closed loop. One way to investigate this aspect is to see how the system responds to perturba- tions. We have therefore solved the equations of statistical equilibrium repeatedly perturbing the rates one by one.

Increasing the photoionization cross section in the Balmer continuum increases the ionization, while the opposite is true for the bound-free transitions from the higher levels, consistent with the rate picture where the ionization is through photoionization in the Balmer continuum and recombination to the higher levels. Even though the H!

(n ¼ 3 ! 2) transition has the largest net rate in the chro- mosphere, changing this rate has no effect on the ionization balance. The same is true for all other bound-bound transi-

Fig. 2.—Hydrogen ionization as a function of column mass in the initial radiative equilibrium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

Fig. 3.—Net rates between the hydrogen levels in the chromosphere (at lg m_c ¼ !4, which corresponds to a height of 1.2 Mm in the initial atmo- sphere, but the picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m^c ¼ !5:6). The thickness is proportional to the net rate, with arrows showing the direction of the net rate. The ionization is dominated by photoionization in the Balmer continuum balanced by photorecombination to higher levels and cascades through bound-bound transitions with Dn ¼ 1.

No. 1, 2002 DYNAMIC HYDROGEN IONIZATION 629

How does hydrogen ionize?

see Carlsson & Stein, 2002, ApJ, 572, 625

that when chemical rates are slow, populations do not have time to reach their equilibrium values. Indeed, some of the disagreement with observations may be due to this nonequi- librium. The solar chromosphere will experience some addi- tional heating besides that included in our model. However, as we show later, even in such hotter models the ionization/

recombination rates are slow, and the results presented here will be qualitatively correct.

3. IONIZATION AND EXCITATION PROCESSES

In the upper chromosphere and lower transition region the density is much too low to ensure LTE, and the full rate equations have to be solved to calculate the hydrogen ion- ization. Furthermore, inspection of the rates involved shows that typical ionization/recombination timescales are much longer than dynamical timescales, so that the ionization bal- ance can be expected to be out of statistical equilibrium. It is thus necessary to take into account the advection and time- derivative terms in the equations. In this section we analyze the rate equations to find what processes dominate the hydrogen ionization balance. In the next section we analyze the timescales involved.

Figure 2 shows the hydrogen ionization fraction as a function of column mass (m_c) in the initial radiative equili- brium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

The processes that are important for the ionization bal- ance are not the same in the transition region and in the chromosphere. In the solar chromosphere the hydrogen ion- ization is dominated by photoionization from the first excited state. The reason is that the much lower particle den- sity of the first excited state (#3 $ 10⁴ cm^!3) than in the ground state (#10¹¹ cm^!3) is more than compensated by the higher mean intensity at the Balmer edge at 364 nm com- pared to that of the Lyman edge at 91 nm. At a radiation temperature of 5000 K the ratio is 3 $ 10⁸. In addition, the atmosphere is optically thick to Lyman photons and opti- cally thin to Balmer photons in all of the chromosphere.

The result is that photoionization by Lyman photons plays an insignificant role in the hydrogen ionization throughout the chromosphere. Figure 3 shows the net rates between all the hydrogen energy levels at lg m_c ¼ !4, which corre- sponds to a height of 1.2 Mm in the initial atmosphere. This picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m_c ¼ !5:6. Ionization is primarily through a net rate from n ¼ 2 to the continuum (photoionization in the Balmer continuum) balanced by recombination to the higher levels cascading down to the n ¼ 2 level through bound-bound transitions with Dn ¼ 1. The reservoir of elec- trons is the ground state. There is transfer of electrons between the ground state and first excited state by collisions.

The rate of this transfer is very slow, because chromospheric temperatures are of order 1 eV and the energy jump to the first excited level is 10.2 eV.

In the very thin zone (2 km), where the ionization fraction goes from 30% to fully ionized, the situation is different.

Collisional ionization from the ground state dominates, with net photorecombination in all continua and again bound-bound transitions with Dn ¼ 1 dominating the return channel to the ground state (Fig. 4).

Figures 3–4 show only the net rates and do not answer the question of what is driving the system and what rates are just adjusting to provide a closed loop. One way to investigate this aspect is to see how the system responds to perturba- tions. We have therefore solved the equations of statistical equilibrium repeatedly perturbing the rates one by one.

Increasing the photoionization cross section in the Balmer continuum increases the ionization, while the opposite is true for the bound-free transitions from the higher levels, consistent with the rate picture where the ionization is through photoionization in the Balmer continuum and recombination to the higher levels. Even though the H!

(n ¼ 3 ! 2) transition has the largest net rate in the chro- mosphere, changing this rate has no effect on the ionization balance. The same is true for all other bound-bound transi-

Fig. 2.—Hydrogen ionization as a function of column mass in the initial radiative equilibrium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

Fig. 3.—Net rates between the hydrogen levels in the chromosphere (at lg m_c ¼ !4, which corresponds to a height of 1.2 Mm in the initial atmo- sphere, but the picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m^c ¼ !5:6). The thickness is proportional to the net rate, with arrows showing the direction of the net rate. The ionization is dominated by photoionization in the Balmer continuum balanced by photorecombination to higher levels and cascades through bound-bound transitions with Dn ¼ 1.

No. 1, 2002 DYNAMIC HYDROGEN IONIZATION 629

slow leakage between

n=1and n=2 because of

10.2 eV jump

How does hydrogen ionize?

see Carlsson & Stein, 2002, ApJ, 572, 625

that when chemical rates are slow, populations do not have time to reach their equilibrium values. Indeed, some of the disagreement with observations may be due to this nonequi- librium. The solar chromosphere will experience some addi- tional heating besides that included in our model. However, as we show later, even in such hotter models the ionization/

recombination rates are slow, and the results presented here will be qualitatively correct.

3. IONIZATION AND EXCITATION PROCESSES

In the upper chromosphere and lower transition region the density is much too low to ensure LTE, and the full rate equations have to be solved to calculate the hydrogen ion- ization. Furthermore, inspection of the rates involved shows that typical ionization/recombination timescales are much longer than dynamical timescales, so that the ionization bal- ance can be expected to be out of statistical equilibrium. It is thus necessary to take into account the advection and time- derivative terms in the equations. In this section we analyze the rate equations to find what processes dominate the hydrogen ionization balance. In the next section we analyze the timescales involved.

Figure 2 shows the hydrogen ionization fraction as a function of column mass (m_c) in the initial radiative equili- brium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

The processes that are important for the ionization bal- ance are not the same in the transition region and in the chromosphere. In the solar chromosphere the hydrogen ion- ization is dominated by photoionization from the first excited state. The reason is that the much lower particle den- sity of the first excited state (#3 $ 10⁴ cm^!3) than in the ground state (#10¹¹ cm^!3) is more than compensated by the higher mean intensity at the Balmer edge at 364 nm com- pared to that of the Lyman edge at 91 nm. At a radiation temperature of 5000 K the ratio is 3 $ 10⁸. In addition, the atmosphere is optically thick to Lyman photons and opti- cally thin to Balmer photons in all of the chromosphere.

The result is that photoionization by Lyman photons plays an insignificant role in the hydrogen ionization throughout the chromosphere. Figure 3 shows the net rates between all the hydrogen energy levels at lg m_c ¼ !4, which corre- sponds to a height of 1.2 Mm in the initial atmosphere. This picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m_c ¼ !5:6. Ionization is primarily through a net rate from n ¼ 2 to the continuum (photoionization in the Balmer continuum) balanced by recombination to the higher levels cascading down to the n ¼ 2 level through bound-bound transitions with Dn ¼ 1. The reservoir of elec- trons is the ground state. There is transfer of electrons between the ground state and first excited state by collisions.

The rate of this transfer is very slow, because chromospheric temperatures are of order 1 eV and the energy jump to the first excited level is 10.2 eV.

In the very thin zone (2 km), where the ionization fraction goes from 30% to fully ionized, the situation is different.

Collisional ionization from the ground state dominates, with net photorecombination in all continua and again bound-bound transitions with Dn ¼ 1 dominating the return channel to the ground state (Fig. 4).

Figures 3–4 show only the net rates and do not answer the question of what is driving the system and what rates are just adjusting to provide a closed loop. One way to investigate this aspect is to see how the system responds to perturba- tions. We have therefore solved the equations of statistical equilibrium repeatedly perturbing the rates one by one.

Increasing the photoionization cross section in the Balmer continuum increases the ionization, while the opposite is true for the bound-free transitions from the higher levels, consistent with the rate picture where the ionization is through photoionization in the Balmer continuum and recombination to the higher levels. Even though the H!

(n ¼ 3 ! 2) transition has the largest net rate in the chro- mosphere, changing this rate has no effect on the ionization balance. The same is true for all other bound-bound transi-

Fig. 2.—Hydrogen ionization as a function of column mass in the initial radiative equilibrium atmosphere. The ionization fraction rises from about 10^!5 at the classical temperature minimum at lg m_c ¼ !1 (height of 0.5 Mm) to about 30% at the base of the transition region 1 Mm further up.

Fig. 3.—Net rates between the hydrogen levels in the chromosphere (at lg m_c ¼ !4, which corresponds to a height of 1.2 Mm in the initial atmo- sphere, but the picture is almost identical in the whole chromosphere from lg m_c ¼ !1 up to the base of the transition region at lg m^c ¼ !5:6). The thickness is proportional to the net rate, with arrows showing the direction of the net rate. The ionization is dominated by photoionization in the Balmer continuum balanced by photorecombination to higher levels and cascades through bound-bound transitions with Dn ¼ 1.

No. 1, 2002 DYNAMIC HYDROGEN IONIZATION 629

fast equilibration of excited states and continuum

slow leakage between

n=1and n=2 because of

10.2 eV jump

Bifrost

see Gudiksen et al.,2011, A&A,531,154

3D MHD code, MPI parallelized

photospheric radiation

thermal conduction

non-equilibrium ionization

etc...

• 9 extra variables + 8 MHD variables = 17 variables

• 7 hydrogen populations are advected

• ⁿ

^e

^{and T}

^g

are not advected

• close system with energy, particle and charge conservation

How to deal with H ionization

J. Leenaarts et al.: On the minimum temperature of the quiet solar chromosphere.

tion into account using approximations for the radiation field based on Sollum (1999). This simulation, however, still com- puted the radiative losses based on Saha equilibrium, and re- quired an ad-hoc heating term that prevented the temperature to drop below 2400 K.

All of the above simulations were limited in such a way that an accurate prediction of the minimum temperature occurring in the quiet solar chromosphere could not be made.

In this paper we discuss the temperature structure in a simu- lation that tries to remedy the various limitations of the previous models: it is two-dimensional, includes a non-equilibrium equa- tion of state, includes radiative losses without the assumption of Saha equilibrium and includes heating through absorption of coronal radiation. We do not include a significant magnetic field, so that our simulation serves as a baseline against which simula- tions with stronger field can be compared.

In Sec. 2 we discuss the RMHD model assumptions in de- tail and discuss some of the basic physical processes that occur in the quiet chromosphere. In Sec. 3 we discuss the results of our run, especially paying attention to the accuracy of the radia- tive heating. We finish with a discussion and our conclusions in Sec. 4.

2. The model

We model the chromosphere using the RMHD code Bifrost (Gudiksen et al. 2011).

MHD equations. The Bifrost code solves the equations of resis- tive MHD including the effects of heat conduction and radiation:

∂ρ

∂t + ∇ · (ρu) = 0, (1)

∂ρu

∂t + ∇ · (ρuu − τ) = −∇P + j × B + ρg, (2)

∂e

∂t + ∇ · (eu) = −P(∇ · u) − ∇ · F ^c − ∇ · F ^r + Q _visc + Q _Joule , (3)

µ ₀ j = ∇ × B, (4)

∂B

∂t = ∇ × (u × B) − ∇ × η j, (5)

with ρ the mass density, u the velocity, τ the viscous stress ten- sor, P the gas pressure, j the current density, B the magnetic field, g the gravitational acceleration, e the internal energy per volume, F _c the energy flux due to heat conduction, F _r the radia- tive energy flux, Q _visc the viscous energy dissipation, Q _Joule the Joule heating, µ ₀ the permeability of free space and η the electri- cal resistivity tensor. Bifrost employs a staggered grid and uses sixth-order operators to compute spatial derivatives. The time- stepping is done using a third-order predictor-corrector scheme by Hyman (1979), modified for variable timestep.

Equation of state. Conservation of internal energy is expressed as

e = 3kT 2

 

n ^e + n _H2 +

n

_l

�

i=1

n _i + n _o

 

 + n ^H2 (e _H2 + χ _H2 )

+

n

_l

�

i=1

n _i χ _i + e _o (n _e , T ). (6)

Here, k, T , n _e , n _H2 , n _l , n _i , n _o are Boltzmann’s constant, the gas temperature, the electron density, the H2 molecule density, the number of levels in the hydrogen model atom, the density of atomic hydrogen in excitation or ionization state i and the num- ber density of all other atoms and molecules that are not, or do not contain, hydrogen, respectively. The rotational and vi- brational energy per H ₂ molecule is e _H2 , χ _H2 is the dissociation energy of H ₂ , χ _i is the excitation or ionization energy of atomic hydrogen and e _o is the internal energy of all other atoms and molecules that are not, or do not contain, hydrogen. The last quantity depends on the temperature and electron density and is computed in LTE.

Eq. 6 is solved together with the equations for charge conser- vation, hydrogen nucleus conservation, evolution equations for the atomic hydrogen level populations n _i (5 bound levels plus the continuum):

∂n _i

∂t + ∇ · (n ⁱ u) =

n

_l

�

j, j �i

n _j P _ji − n ⁱ

n

_l

�

j, j �i

P _{i j} , (7)

and an equation for non-equilibrium formation of H ₂ molecules:

∂n _H2

∂t + ∇ · (n ^H2 u) = R _f n ³ ₁ − R ^b n ₁ n _H2 , (8) where n ₁ is the ground state population of atomic hydrogen.

The radiative part of the rate coefficients P _{i j} is computed us- ing the approximations given by Sollum (1999), the temperature- dependent rate coefficients R _f and R _b are taken from the UMIST database (Woodall et al. 2007, www.udfa.net).

These equations yield values for T , n _e , n _H2 , and n _i . The gas pressure is then given by

P = kT

 

n ^e + n _H2 +

n

_l

�

i=1

n _i + n _o

 

 , (9)

and used in the momentum and energy equations (Eqs. 2–3).

This non-equilibrium equation-of-state requires advection of 6 atomic and 1 molecular hydrogen population in addition to the 8 MHD variables, for a total of 15 advected quantities. See Leenaarts et al. (2007), Golding (2010) and Gudiksen et al.

(2011) for more details.

Radiative heating. Formally, the radiative flux divergence is given by

∇ · F ^r =

� _∞

0

�

Ω

α(ν, ˆn) (S (ν, ˆn) − I(ν, ˆn)) dΩ dν, (10) with S , α and I the source function, opacity and intensity at frequency ν in direction ˆn into the solid angle Ω. In practice, this double integral is too computationally expensive to com- pute, and various approximations are made.

The integral over frequency is replaced by four representa- tive radiation bins, with their own associated bin-averaged opac- ity per mass unit κ _i , photon destruction probability � _i and thermal emission E _i , assuming isotropic scattering and isotropic source function and opacity (Nordlund 1982; Skartlien 2000). The ra- diative heating of such a so-called multi-group scheme is then given by

Q _rad = 4πρ

� 4 i=1

κ _i (� _i J _i − E ⁱ ) (11)

The implementation of this scheme in the Bifrost code is dis- cussed in detail by Hayek et al. (2010). The quantities κ _i , � _i and

E _i are precomputed and tabulated assuming LTE populations

2

J. Leenaarts et al.: On the minimum temperature of the quiet solar chromosphere.

tion into account using approximations for the radiation field based on Sollum (1999). This simulation, however, still com- puted the radiative losses based on Saha equilibrium, and re- quired an ad-hoc heating term that prevented the temperature to drop below 2400 K.

All of the above simulations were limited in such a way that an accurate prediction of the minimum temperature occurring in the quiet solar chromosphere could not be made.

In this paper we discuss the temperature structure in a simu- lation that tries to remedy the various limitations of the previous models: it is two-dimensional, includes a non-equilibrium equa- tion of state, includes radiative losses without the assumption of Saha equilibrium and includes heating through absorption of coronal radiation. We do not include a significant magnetic field, so that our simulation serves as a baseline against which simula- tions with stronger field can be compared.

In Sec. 2 we discuss the RMHD model assumptions in de- tail and discuss some of the basic physical processes that occur in the quiet chromosphere. In Sec. 3 we discuss the results of our run, especially paying attention to the accuracy of the radia- tive heating. We finish with a discussion and our conclusions in Sec. 4.

2. The model

We model the chromosphere using the RMHD code Bifrost (Gudiksen et al. 2011).

MHD equations. The Bifrost code solves the equations of resis- tive MHD including the effects of heat conduction and radiation:

∂ρ

∂t + ∇ · (ρu) = 0, (1)

∂ρu

∂t + ∇ · (ρuu − τ) = −∇P + j × B + ρg, (2)

∂e

∂t + ∇ · (eu) = −P(∇ · u) − ∇ · F ^c − ∇ · F ^r + Q _visc + Q _Joule , (3)

µ ₀ j = ∇ × B, (4)

∂B

∂t = ∇ × (u × B) − ∇ × η j, (5)

with ρ the mass density, u the velocity, τ the viscous stress ten- sor, P the gas pressure, j the current density, B the magnetic field, g the gravitational acceleration, e the internal energy per volume, F _c the energy flux due to heat conduction, F _r the radia- tive energy flux, Q _visc the viscous energy dissipation, Q _Joule the Joule heating, µ ₀ the permeability of free space and η the electri- cal resistivity tensor. Bifrost employs a staggered grid and uses sixth-order operators to compute spatial derivatives. The time- stepping is done using a third-order predictor-corrector scheme by Hyman (1979), modified for variable timestep.

Equation of state. Conservation of internal energy is expressed as

e = 3kT 2

 

n ^e + n _H2 +

n

_l

�

i=1

n _i + n _o

 

 + n ^H2 (e _H2 + χ _H2 )

+

n

_l

�

i=1

n _i χ _i + e _o (n _e , T ). (6)

Here, k, T , n _e , n _H2 , n _l , n _i , n _o are Boltzmann’s constant, the gas temperature, the electron density, the H2 molecule density, the number of levels in the hydrogen model atom, the density of atomic hydrogen in excitation or ionization state i and the num- ber density of all other atoms and molecules that are not, or do not contain, hydrogen, respectively. The rotational and vi- brational energy per H ₂ molecule is e _H2 , χ _H2 is the dissociation energy of H ₂ , χ _i is the excitation or ionization energy of atomic hydrogen and e _o is the internal energy of all other atoms and molecules that are not, or do not contain, hydrogen. The last quantity depends on the temperature and electron density and is computed in LTE.

Eq. 6 is solved together with the equations for charge conser- vation, hydrogen nucleus conservation, evolution equations for the atomic hydrogen level populations n _i (5 bound levels plus the continuum):

∂n _i

∂t + ∇ · (n ⁱ u) =

n

_l

�

j, j �i

n _j P _ji − n ⁱ

n

_l

�

j, j �i

P _{i j} , (7)

and an equation for non-equilibrium formation of H ₂ molecules:

∂n _H2

∂t + ∇ · (n ^H2 u) = R _f n ³ ₁ − R ^b n ₁ n _H2 , (8) where n ₁ is the ground state population of atomic hydrogen.

The radiative part of the rate coefficients P _{i j} is computed us- ing the approximations given by Sollum (1999), the temperature- dependent rate coefficients R _f and R _b are taken from the UMIST database (Woodall et al. 2007, www.udfa.net).

These equations yield values for T , n _e , n _H2 , and n _i . The gas pressure is then given by

P = kT

 

n ^e + n _H2 +

n

_l

�

i=1

n _i + n _o

 

 , (9)

and used in the momentum and energy equations (Eqs. 2–3).

This non-equilibrium equation-of-state requires advection of 6 atomic and 1 molecular hydrogen population in addition to the 8 MHD variables, for a total of 15 advected quantities. See Leenaarts et al. (2007), Golding (2010) and Gudiksen et al.

(2011) for more details.

Radiative heating. Formally, the radiative flux divergence is given by

∇ · F ^r =

� _∞

0

�

Ω

α(ν, ˆn) (S (ν, ˆn) − I(ν, ˆn)) dΩ dν, (10) with S , α and I the source function, opacity and intensity at frequency ν in direction ˆn into the solid angle Ω. In practice, this double integral is too computationally expensive to com- pute, and various approximations are made.

The integral over frequency is replaced by four representa- tive radiation bins, with their own associated bin-averaged opac- ity per mass unit κ _i , photon destruction probability � _i and thermal emission E _i , assuming isotropic scattering and isotropic source function and opacity (Nordlund 1982; Skartlien 2000). The ra- diative heating of such a so-called multi-group scheme is then given by

Q _rad = 4πρ

� 4 i=1

κ _i (� _i J _i − E ⁱ ) (11)

The implementation of this scheme in the Bifrost code is dis- cussed in detail by Hayek et al. (2010). The quantities κ _i , � _i and E _i are precomputed and tabulated assuming LTE populations

2

How to deal with H ionization

J. Leenaarts et al.: Non-equilibrium hydrogen ionization in 2D simulations of the solar atmosphere. 9

is e _H2 . The latter is computed using polynomial approximations by Vardya (1965)

Chemical equilibrium. The equation of chemical equilibrium between neutral atomic hydrogen and molecular hydrogen is given by

n _H2 =

! " ₅

i=1 n _i # 2

K , (A.2)

where K is the chemical equilibrium constant whose value and its temperature derivative are given by polynomial approxima- tions as function of T by Tsuji (1973). The molecular hydrogen density depends on T and n _i (i = 1, . . . , 5). The derivatives are

∂ n _H2

∂ T = −



 

 



" ₅

i=1 n _i K



 

 



2 ∂ K

∂ T , (A.3)

∂ n _H2

∂ n _i = 2 !

" ₅

i=1 n _i #

K . (A.4)

Charge conservation. The electron density is given by

n _e = n ₆ + n ^tot _H n ^noH _e , (A.5)

which we rewrite in a form suitable for Newton-Raphson itera- tion:

F ₁ = 1 − 1 n _e

! n ₆ + n ^tot _H n ^noH _e #

= 0. (A.6)

The functional F ₁ depends on n _e , n ₆ and T . The partial deriva- tives are

∂ F ₁

∂ n _e = n ₆ + n ^tot _H n ^noH _e

n ² _e − n ^tot _H n _e

∂ n ^noH _e

∂ n _e , (A.7)

∂ F ₁

∂ n ₆ = − 1

n _e , (A.8)

∂ F ₁

∂ T = − n ^tot _H n _e

∂ n ^noH _e

∂ T . (A.9)

Internal energy. The second functional specifies the distribu- tion of the internal energy e _i over the various contributions:

F ₂ = 1 − 1 e _i



 

 

 



3kT 2



 

 

 



n _e + n _noH + n _H2 +

- 6 i=1

n _i



 

 

 

 + n ^tot _H e _noH + n _H2 e _H2 +

- 6 i=1

n _i χ _i



 

 

 



= 0, (A.10)

with χ _i the energy of hydrogen level i. This functional depends on n _e , n _i and T . The partial derivatives are

∂ F ₂

∂ n _e = − 1 e _i

1 3kT

2 + n ^tot _H ∂ e _noH

∂ n _e 2

, (A.11)

∂ F ₂

∂ n _i = − 1 e _i

1 3kT 2

3 ∂n _H2

∂ n _i + 1 4

+ e _H2 ∂ n _H2

∂ n _i + χ _i 2

, (A.12)

∂ F ₂

∂ T = − 1 e _i



 

 

 



3k 2



 

 

 



n _e + n _noH + n _H2 + T ∂ n _H2

∂ T +

- 6 i=1

n _i



 

 

 

 + n ^tot _H ∂ e _noH

∂ T + n _H2 ∂ e _H2

∂ T + e _H2 ∂ n _H2

∂ T 2

. (A.13)

Particle conservation. The conservation of the total number of hydrogen nuclei n ^tot _H is given by:

F ₃ = 1 − 1 n ^tot _H



 

 

 



- 6 i=1

n _i − 2n ^H2



 

 

 



= 0. (A.14)

This functional depends on n _i and T . The partial derivatives are:

∂ F ₃

∂ n _i = − 1 n ^tot _H

1

1 + 2 ∂ n _H2

∂ n _i 2

, i = 1, . . . , 5, (A.15)

∂ F ₃

∂ n ₆ = − 1

n ^tot _H , (A.16)

∂ F ₃

∂ T = − 2 n ^tot _H

∂ n _H2

∂ T . (A.17)

LTE populations. LTE is imposed to start up a new computation and at the lower boundary. The LTE hydrogen populations then obey the Saha-Boltzmann equations (i = 1, . . . , 5)

F _3+i = 1 − 1 n _e

n _i n ₆

2g ₆ g _i

1 2πm _e kT h ²

2 3/2

e ^−(χ

⁶

^−χ

ⁱ

^)/kT = 0, (A.18) where g _i is the statistical weight of level i. These equations de- pend on n ₆ , n _i , n _e and T . Defining

K _i = 1 n _e

n _i n ₆

2g ₆ g _i

1 2πm _e kT h ²

2 3/2

e ^−(χ

⁶

^−χ

ⁱ

^)/kT , (A.19) the partial derivatives are given by

∂ F _3+i

∂ n ₆ = K _i

n ₆ , (A.20)

∂ F _3+i

∂ n _i = − K _i

n _i , (A.21)

∂ F _3+i

∂ n _e = K _i

n _e , (A.22)

∂ F _3+i

∂ T = −K ⁱ 1 χ ₆ − χ ⁱ

kT ² + 3 2T

2

. (A.23)

Rate equations. Outside LTE, the change of the hydrogen pop- ulations n ₁ . . . n ₆ over a timestep ∆t can be expressed as

n _i (t ₀ + ∆ t) − n ⁱ (t ₀ ) =

5 _t

₀

+∆ t t

₀

- 6 j, j!i

n _j !

R _ji + C _ji #

−n ⁱ

- 6 j, j!i

! R _{i j} + C _{i j} #

dt, (A.24)

with R _{i j} and C _{i j} the radiative and collisional rate coefficients from level i to level j. Defining

P _{i j} = R _{i j} + C _{i j} , (A.25)

P _ii = −

- 6 j, j!i

! R _{i j} + C _{i j} #

(A.26) and writing n _i = n _i (t ₀ +∆ t) and n ^t _i

⁰

= n _i (t ₀ ) yields the discretized, implicit form of Eq. A.24

F _3+i = n _i

n ^t _i

⁰

− ∆ t n ^t _i

⁰



 

 

 



- 6 j=1

n _j P _ji



 

 

 

 − 1 = 0. (A.27)

The equations are dependent on n _i , n _j ( j ! i), n _e and T . The partial derivatives are

∂ F _3+i

∂ n _i = 1 − ∆t P ⁱⁱ

n ^t _i

⁰

, (A.28)

energy conservation:

6 Gudiksen et al.: The numerical code Bifrost

Proper modeling of this ionization requires that non-equilibrium effects be taken into account.

The hydrogen ionization module solves the time-dependent rate equations for the atomic hydrogen level populations n _i

∂ n _i

∂ t + ∇ · (n ⁱ u) =

n

_l

!

j, j!i

n _j P _ji − n ⁱ

n

_l

!

j, j!i

P _{i j} . (31)

together with an equation for time-dependent H ₂ molecule for- mation and equations for energy and charge conservation. Here P _{i j} is the rate coefficient for the transition from level i to j and n _l is the number of energy levels in the hydrogen atom (normally set to 6 in Bifrost ). Solution of this system of equations yields the gas temperature, electron density and the atomic and molec- ular hydrogen populations (n _H2 ). The radiative transition rate co- efficients in the rate equations are prescribed as determined by Sollum (1999). This removes the effect of the global coupling of the radiation and makes the problem computationally tractable, but still computationally demanding.

The gas pressure is computed as P = kT



 

 

  n _e + n _H2 +

n

_l

!

i=1

n _i + n _o



 

 

  , (32)

with n _o the number density of all other atoms and molecules that are not, or do not contain, hydrogen.

Full non-LTE radiation tables as functions of n _e and T can be used to replace the radiation tables as functions of e and ρ, which assume hydrogen ionization based on Saha ionization equilib- rium.

The method is described in detail in Leenaarts et al. (2007), the additional equation for time-dependent H ₂ formation is given in Appendix B.

10. Thermal Conduction

As the plasma temperature rises towards one million degrees in the tenuous transition region and corona, thermal conduction be- comes one of the major terms in the energy equation, and mod- eling of this term is vital if this portion of the atmosphere is to be simulated with any fidelity. The implemented form of the ther- mal conduction takes the form (Spitzer 1956) :

F _c = −κ ⁰ T ^5/2 ∇ # T (33)

where the gradient of T is taken only along the magnetic field (∇ # ) and κ ₀ is the thermal conduction coefficient. The conduc- tion across the field is significantly smaller under the conditions present in the solar atmosphere and is smaller than the numer- ical diffusion, so it is ignored. Since thermal conduction is de- scribed by a second order diffusion operator, this introduces sev- eral difficulties: The Courant condition for a diffusive operator such as that scales with the grid size ∆x ² instead of with ∆x for the magneto-hydrodynamic operators. This severely limits the time step ∆t the code can be stably run at. We have implemented two solutions to this problem.

In the first, the thermal flux is calculated and if the diver- gence of the thermal flux sets severe restraints on the timestep, it is throttled back by locally lowering the thermal conduction coefficient κ ₀ . This method is only acceptable when the number of points where the conduction has to be throttled back is very small and the results must be analyzed carefully.

A second solution is to proceed by operator splitting, such that the operator advancing the variables in time is L = L _hydro +

L _conduction . The conductive part of the energy equation is handled by discretizing

∂ e

∂ t = −∇ · F ^c = −∇ _# · (

κ ₀ T ^5/2 ∇ _# T )

(34) and solving the resulting problem implicitly.

We discretize the L _conduction operator using the Crank- Nicholson (or ‘θ’) method and thus write

(e ⁿ⁺¹ − e ^∗ )/dt = θ∇ # · F ⁿ⁺¹ c + (1 − θ)∇ # · F ⁿ c ) (35) where the quantities with superscript n are computed before the hydrodynamic timestep, the quantities with superscript ∗ are computed after the hydrodynamic timestep and the quantities with superscript n + 1 are the temperatures deduced implicitly.

The variable θ is set to a value between 0.5 and 1. The implicit part of the problem is in our case computed using a multi-grid solver (A concise introduction to multi-grid methods may be found in chapter 19 of Press et al. (1992)).

In implementing the multi-grid solver, we use the same do- main decomposition as in the magneto-hydrodynamic part of the code. The small scale residuals in Eq. 35 are smoothed using a few Gauss-Seidel sweeps at high resolution, before the result- ing temporary solution is injected onto a coarser grid on which smoothing is again performed. The process continues by Gauss- Seidel smoothing sweeps on steadily coarser grids until the size of the problem on individual processors is small enough to be communicated and stored on each and every processor. At this point the partial solutions are spread to all processors which con- tinue the Gauss-Seidel smoothing, now on the global problem, on steadily coarser grids. Finally, when the coarsest grid size is reached, the solution is driven to convergence by performing a great number of Gauss-Seidel sweeps. Subsequently, the coarser solutions are corrected and then prolonged on succesively finer grids, first globally, but after a certain grid size is reached the temporary solution is spread and the problem is again solved lo- cally, using the same domain decomposition as in the magneto- hydrodynamic part of the code. After each prolongation, the so- lution is smoothed using a few Gauss-Seidel sweeps. The entire cycle can be repeated as many times as desired, until a converged solution is found.

11. Tests and Validation

Bifrost has been extensively tested in order to validate the results it provides. When possible, the tests are taken from the literature to validate the test and allow comparison with previous work.

These tests were selected for their simplicity, their utility and their challenge to the different terms in the algorithms.

11.1. The Sod Shock Tube Test

The Sod Shock tube test (Sod 1978) has become a standard test in computational HD and MHD. It consists of a one-dimensional flow discontinuity problem which provides a good test of a com- pressible code’s ability to capture shocks and contact discontinu- ities within a small number of zones, and to produce the correct density profile in a rarefaction wave. The test can also be used to check if the code is able to satisfy the Rankine-Hugoniot shock jump conditions, since this test has an analytical solution.

The code is set to run a 1D problem and using the initial conditions for the Sod problem. The fluid is initially at rest on

J. Leenaarts et al.: Non-equilibrium hydrogen ionization in 2D simulations of the solar atmosphere. 9

is e _H2 . The latter is computed using polynomial approximations by Vardya (1965)

Chemical equilibrium. The equation of chemical equilibrium between neutral atomic hydrogen and molecular hydrogen is given by

n _H2 =

! " ₅

i=1 n _i # 2

K , (A.2)

where K is the chemical equilibrium constant whose value and its temperature derivative are given by polynomial approxima- tions as function of T by Tsuji (1973). The molecular hydrogen density depends on T and n _i (i = 1, . . . , 5). The derivatives are

∂ n _H2

∂ T = −



 

 



" ₅

i=1 n _i K



 

 



2 ∂ K

∂ T , (A.3)

∂ n _H2

∂ n _i = 2 ! " ₅

i=1 n _i #

K . (A.4)

Charge conservation. The electron density is given by

n _e = n ₆ + n ^tot _H n ^noH _e , (A.5)

which we rewrite in a form suitable for Newton-Raphson itera- tion:

F ₁ = 1 − 1 n _e

! n ₆ + n ^tot _H n ^noH _e #

= 0. (A.6)

The functional F ₁ depends on n _e , n ₆ and T . The partial deriva- tives are

∂ F ₁

∂ n _e = n ₆ + n ^tot _H n ^noH _e

n ² _e − n ^tot _H n _e

∂ n ^noH _e

∂ n _e , (A.7)

∂ F ₁

∂ n ₆ = − 1

n _e , (A.8)

∂ F ₁

∂T = − n ^tot _H n _e

∂ n ^noH _e

∂T . (A.9)

Internal energy. The second functional specifies the distribu- tion of the internal energy e _i over the various contributions:

F ₂ = 1 − 1 e _i



 

 

 



3kT 2



 

 

 



n _e + n _noH + n _H2 +

- 6 i=1

n _i



 

 

 

 + n ^tot _H e _noH + n _H2 e _H2 +

- 6 i=1

n _i χ _i



 

 

 



= 0, (A.10)

with χ _i the energy of hydrogen level i. This functional depends on n _e , n _i and T . The partial derivatives are

∂ F ₂

∂n _e = − 1 e _i

1 3kT

2 + n ^tot _H ∂ e _noH

∂n _e

2

, (A.11)

∂ F ₂

∂n _i = − 1 e _i

1 3kT 2

3 ∂n _H2

∂n _i + 1 4

+ e _H2 ∂ n _H2

∂n _i + χ _i 2

, (A.12)

∂ F ₂

∂ T = − 1 e _i



 

 

 



3k 2



 

 

 



n _e + n _noH + n _H2 + T ∂ n _H2

∂ T +

- 6 i=1

n _i



 

 

 

 + n ^tot _H ∂ e _noH

∂ T + n _H2 ∂ e _H2

∂ T + e _H2 ∂ n _H2

∂ T

2

. (A.13)

Particle conservation. The conservation of the total number of hydrogen nuclei n ^tot _H is given by:

F ₃ = 1 − 1 n ^tot _H



 

 

 



- 6 i=1

n _i − 2n ^H2



 

 

 



= 0. (A.14)

This functional depends on n _i and T . The partial derivatives are:

∂ F ₃

∂ n _i = − 1 n ^tot _H

1

1 + 2 ∂ n _H2

∂ n _i 2

, i = 1, . . . , 5, (A.15)

∂ F ₃

∂ n ₆ = − 1

n ^tot _H , (A.16)

∂ F ₃

∂T = − 2 n ^tot _H

∂ n _H2

∂T . (A.17)

LTE populations. LTE is imposed to start up a new computation and at the lower boundary. The LTE hydrogen populations then obey the Saha-Boltzmann equations (i = 1, . . . , 5)

F _3+i = 1 − 1 n _e

n _i n ₆

2g ₆ g _i

1 2πm _e kT h ²

2 3/2

e ^−(χ

⁶

^−χ

ⁱ

^)/kT = 0, (A.18) where g _i is the statistical weight of level i. These equations de- pend on n ₆ , n _i , n _e and T . Defining

K _i = 1 n _e

n _i n ₆

2g ₆ g _i

1 2πm _e kT h ²

2 3/2

e ^−(χ

⁶

^−χ

ⁱ

^)/kT , (A.19) the partial derivatives are given by

∂ F _3+i

∂n ₆ = K _i

n ₆ , (A.20)

∂ F _3+i

∂ n _i = − K _i

n _i , (A.21)

∂ F _3+i

∂ n _e = K _i

n _e , (A.22)

∂ F _3+i

∂T = −K ⁱ 1 χ ₆ − χ ⁱ

kT ² + 3 2T

2

. (A.23)

Rate equations. Outside LTE, the change of the hydrogen pop- ulations n ₁ . . . n ₆ over a timestep ∆t can be expressed as

n _i (t ₀ + ∆ t) − n ⁱ (t ₀ ) =

5 _t

₀

+∆ t t

₀

- 6 j, j!i

n _j !

R _ji + C _ji #

−n ⁱ

- 6 j, j!i

! R _{i j} + C _{i j} #

dt, (A.24)

with R _{i j} and C _{i j} the radiative and collisional rate coefficients from level i to level j. Defining

P _{i j} = R _{i j} + C _{i j} , (A.25)

P _ii = −

- 6 j, j!i

! R _{i j} + C _{i j} #

(A.26) and writing n _i = n _i (t ₀ +∆ t) and n ^t _i

⁰

= n _i (t ₀ ) yields the discretized, implicit form of Eq. A.24

F _3+i = n _i

n ^t _i

⁰

− ∆ t n ^t _i

⁰



 

 

 



- 6 j=1

n _j P _ji



 

 

 

 − 1 = 0. (A.27)

The equations are dependent on n _i , n _j ( j ! i), n _e and T . The partial derivatives are

∂ F _3+i

∂ n _i = 1 − ∆t P ⁱⁱ

n ^t _i

⁰

, (A.28)

charge conservation:

pressure follows:

How to deal with H ionization

H ionization acts as an equation-of-state.

Radiation field

• detailed evaluation of radiation field computationally expensive

• but needed for rate equations:

• Sollum (1999): fixed radiative rates can be

used as confirmed in detailed 1D simulations P _ij = C _ij (n _e , T _e ) + R _ij (J _ν , n _e )

J _ν

• specify in each transition

• rate coefficients are local

T _rad

P _ij = C _ij (T _e , n _e ) + R _ij (T _rad , n _e )

Radiation field

Lyα is special

see Carlsson & Leenaarts, 2012, A&A

• Use detailed computations to compute the Lyα downward rate:

• Solve the transfer equation using rate and n

1

populations to provide source function and (fudged, poor manʼs scattering) opacity

• Obtain upward rate from J

• Rates not implicit, possible source of problems

R ₂₁ = L(T ) E(τ ) n _e n ₁

Lyα is special

Ionization degree

Temperature

Increased temperature fluctuations

HION LTE

Unrealistic clustering at temperatures where He

ionizes

No such clustering for hydrogen

H 2 molecules provide lower limit in LTE

Low temperature end

Time-dependent chemistry removes lower limit below ρ=10 ^-9

Low temperature end

Conclusions (1)

• proper modeling of ionization essential for chromospheric temperature structure

Conclusions (1)

• proper modeling of ionization essential for chromospheric temperature structure

• clumping of gas at 10 kK and 30kK is an artifact of LTE Helium ionization

Conclusions (1)

• proper modeling of ionization essential for chromospheric temperature structure

• clumping of gas at 10 kK and 30kK is an artifact of LTE Helium ionization

• real temperature structure in upper chromosphere unknown, He ionization should be modeled better

Conclusions (1)

• proper modeling of ionization essential for chromospheric temperature structure

• clumping of gas at 10 kK and 30kK is an artifact of LTE Helium ionization

• real temperature structure in upper chromosphere unknown, He ionization should be modeled better

• time-dependent H

2

chemistry removes lower limit of temperature in expansion bubbles

Conclusions (1)

Conclusions (2)

• Conclusions (1) are generally valid

• Magnetized chromosphere influenced by ionization- dependence of neutral-ion effects

• See Juanʼs (and others) talk for details