A&A 424, 691-712 (2004)
DOI: 10.1051/0004-6361:20040310
M. L. Goodman^{}
Institute for Scientific Research, 2500 Fairmont Avenue - Suite 734, Fairmont, WV 26555-2720, USA
Received 20 February 2004 / Accepted 19 April 2004
Abstract
A mechanism that creates the chromospheres of solar type stars everywhere outside of flaring
regions is proposed. The identification of the mechanism is based on previous work and on the
results of a model presented here that computes the electric current, its driving electric
field, the heating rate due to resistive dissipation, and the flow velocity in a specified class of horizontally
localized, two dimensional magnetic structures in the steady state approximation. The model
is applied to the Sun over the height range from the photosphere to the upper chromosphere.
Although the model does not contain time explicitly, it contains information about the
dynamics of the atmosphere through inputs from the FAL CM solar atmosphere model, which is
based on time averages of spectroscopic data. The model is proposed to describe the time
averaged properties of the heating mechanism that creates the chromosphere. The model
magnetic structure is horizontally localized, but describes heating of the global
chromosphere in the following way. Recent observations indicate that kilogauss strength
magnetic structures exist in the photospheric internetwork with a filling factor ,
and characteristic diameters <180 km. Assuming
and a maximum field strength of
10^{3} G for the model magnetic structure, and assuming that the chromospheric heating rate
predicted by FAL CM represents a horizontal spatial average over such magnetic structures, it
is found that the model magnetic structures that best reproduce the FAL CM heating rate as a
function of height have characteristic diameters in the range of
98 - 161 km, consistent
with the upper bound inferred from observation. Based on model solutions and previous work it
is proposed that essentially all chromospheric heating occurs in magnetic structures
with sub-resolution horizontal spatial scales
,
that
the heating is due to dissipation of
Pedersen currents driven by a convection electric field, and that it is the increase in the
magnetization of particles with height in a magnetic structure from values 1 in the
lower photosphere to values 1 near the height of the temperature minimum
in the magnetic structure that causes the Pedersen current dissipation
rate to increase to a value large enough to cause a temperature inversion. The magnetization
of a particle is the ratio of its cyclotron frequency to its total collision frequency with
unlike particle species.
Key words: Sun: chromosphere - Sun: photosphere - magnetohydrodynamics (MHD) - Sun: magnetic fields - stars: chromospheres - magnetic fields
Convection of plasma exists everywhere and always over a wide range of temporal and spatial scales in the solar atmosphere, and presumably in the atmospheres of all solar type stars. A magnetic field in the presence of convection provides a mechanism for the conversion of kinetic energy of convection into the electrical energy of a convection electric field that drives Pedersen current that is dissipated by collisions, generating thermal energy (Goodman 2000, 2001). where is the center of mass (CM) velocity of the plasma and c is the speed of light. The atmosphere is a magnetohydrodynamic (MHD) power generator, converting mechanical energy of convection into electrical energy that is dissipated by the resistive load of the plasma. The magnetic field enables and mediates this energy conversion through the convection electric field. This heating mechanism operates to some degree across the full range of length and time scales that characterize the magnetic field and convection in the atmosphere.
The specific convective processes that are the primary drivers of this heating mechanism need to be identified. The work presented here and by Goodman (2000, 2001) indicates that in order for this heating mechanism to balance the net radiative loss from the chromosphere it must be concentrated in horizontally localized magnetic structures with sub-resolution horizontal scales. A sub-resolution structure is defined as one with a characteristic diameter km. This definition follows from the highest current observational resolution of bright points associated with kilogauss magnetic field concentrations in the photosphere (Lites et al. 1999). Goodman (2000, 2001) identifies MHD wave driven convection in such magnetic structures as a primary driver. All convection is time dependent, so a determination of the importance of other convective processes in driving the heating mechanism requires time dependent models, which are not considered here. Here only a few general remarks are made about driving mechanisms other than MHD waves. (i) Granular convection on space and time scales of the granular diameter of km, and the granular turnover time of min generates a convection electric field in magnetic structures concentrated in the boundary regions of the granulation. Here it is proposed that chromospheric heating occurs in magnetic structures with horizontal diameters km. If most of these structures have photospheric field strengths G, and a filling factor , then the average separation between these structures is km, which is the characteristic diameter of a granule. (ii) Horizontally localized magnetic structures in the solar atmosphere can exist only in the presence of a localized current density that contributes to heating through resistive dissipation. This current is independent of any MHD wave driven current that is generated in these structures, and might be a major source of heating. The current may be driven by a convection electric field.
Here a model is used to describe the time averaged effects of resistive dissipation in magnetic structures in the photosphere and chromosphere due to the collective effect of whatever mechanisms drive it. The model shows how the increase with height of the magnetization of charged particles, beginning in the photosphere, can cause a temperature inversion and create the chromospheres of the Sun and other solar type stars everywhere outside of flaring regions. The magnetization M_{s} of a particle species s is the ratio of its cyclotron frequency to its total collision frequency with unlike particle species.
The model is a two dimensional, steady state MHD model of a horizontally localized magnetic structure that extends vertically from the photosphere through the upper chromosphere, and that includes an expression for the electrical conductivity tensor of the multi-species, variably ionized atmosphere. A specified photospheric filling factor is used to estimate the global resistive heating rate due to a uniform, horizontal distribution of these structures over the photosphere. The heating rate is due to dissipation of Pedersen and magnetic field aligned (parallel) currents. In Sect. 4 the heating rate is required to optimally match that of model CM of Fontenla, Avrett & Loeser (communicated by Avrett & Loeser (2001), and henceforth FAL CM) over the height range of the chromosphere (FAL CM is not to be confused with CM - center of mass). The resulting magnetic structures are found to have sub-resolution horizontal length scales, and the heating is found to be due almost entirely to Pedersen current dissipation driven by a convection electric field. The model is steady state in that time does not appear explicitly in the model. Information about the dynamics of the atmosphere enters the model implicitly through density and temperature profiles generated by FAL CM. These profiles are computed using time averages of spectroscopic data, and are inputs to the model used here. FAL CM is essentially the model used by Fontenla et al. (2002) to describe energy balance, bulk flow, and diffusion in the solar transition region. Here only that part of the model that describes the photosphere and chromosphere is used.
CGS units are used throughout the paper unless indicated otherwise. Cylindrical coordinates are used. All quantities are assumed to be independent of and time. z labels height. z=0 labels optical depth unity in the FAL CM photosphere. The magnitude of any vector function is denoted by F.
A specified two dimensional magnetic field is combined with an Ohm's law and data from FAL CM to compute the properties of anisotropic resistive heating in the photosphere and chromosphere. The electrical conductivity tensor, heating rates per unit volume due to parallel and Pedersen current dissipation, center of mass (CM) electric field that drives the current, and particle current densities and magnetizations are computed as functions of R and z from z=0 to the upper chromospheric height of km where the FAL CM temperature T = 9983 K.
The magnetization M_{s} of a particle species s is defined as
The model is not extended above for the following reason. The assumed magnetic field, defined in Sect. 2.1, has the following properties. It is closed in that all field lines return to the photosphere. It is strongly localized in the radial direction at all heights. Its z dependence is given by the factor , where L_{B} is an assumed constant scale height. This z dependence is assumed to be a sufficiently good approximation below the transition region for making order of magnitude estimates of heating rates. The assumed height dependence is unacceptable at greater heights where large temperature gradients in the transition region, and temperatures K in the transition region and corona are expected to rapidly and greatly increase the characteristic magnetic field scale height with increasing height. A magnetic field model that is also valid above the chromosphere must describe fields that undergo horizontal expansion with increasing height, that may have open field lines, and that have a magnitude that decreases with height in a way that is not reasonably well described by an exponential with a constant scale height.
The photosphere and chromosphere consist almost entirely of electrons, protons, singly ionized metallic ions, H, He I, He II, and He III. An approximate conductivity tensor of this multi-species plasma is computed in Sect. 2.5 by treating the plasma as a three species plasma of electrons, a singly charged proxy ion that collectively represents protons and metallic ions, and a proxy neutral species that collectively represents H and He I.
The z component of the field is chosen to be
For this choice of B_{z} the magnetic flux through the any surface is zero since . All field lines are closed. The field is unipolar for R < R_{0}, and for R > R_{0} with the opposite sign. The magnetic flux through the core region is , where is the area of the core region. The flux through the area with R=2R_{0} is 0.2 F_{0}, so at this radius of the core flux has returned to the photosphere.
Then
implies that
This choice for
is made more realistic by requiring that
The assumed form of the magnetic field is used to represent horizontally localized, closed field line structures in the photosphere and chromosphere. It is shown in Sect. 4 that for a filling factor of 0.02 the resistive heating rate in these structures can balance the net radiative loss from the solar chromosphere if their characteristic diameters are sub-resolution, and if the magnetic field is highly twisted in that . The form of sub-resolution magnetic structures in solar type stars may be very different from the one assumed here. However, it is essentially the high degree of horizontal localization of the magnetic field that gives rise to heating rates sufficiently large to balance the net radiative loss. The reason is that a higher degree of localization implies larger horizontal derivatives of the magnetic field components which implies larger current densities which implies larger heating rates for a given conductivity tensor. This suggests that magnetic fields that are highly localized in that their field strengths decrease by orders of magnitude from G over horizontal distances have resistive heating rates that may be predicted with reasonable accuracy by any smoothly varying mathematical solution for such a localized field. It is proposed that if such localized magnetic fields exist in the photospheres and chromospheres of solar type stars, as they appear to in the solar photosphere, the model presented here may be used to predict their resistive heating rates with order of magnitude accuracy.
Ohm's law may be written as
The Pedersen and Hall current densities are
The densities of electrons, protons, H, He I, He II, He III, C, Si, Al, Mg, Fe, Na, and Ca are given by FAL CM.
The several species of singly charged metallic ions are collectively represented by a proxy ion,
denoted by ,
with a number density
and mass
defined by the quasi-neutrality
condition
The protons and proxy ions are collectively represented by an effective ion denoted by i, eff.
Only its magnetization
and mass
enter the model. They are defined by
H and He I are collectively represented by a single neutral component with a mass
The magnetizations of electrons, protons, and proxy ions are defined by
The collision frequencies are given by standard expressions listed in Goodman (2004). They are functions of the temperature, and of the particle number densities and masses. The temperature and densities used here are from FAL CM. Since the FAL CM density and temperature profiles depend only on z, the model used here uses the approximation that these profiles are independent of R although the magnetic field depends on R and z. Then the conductivity tensors depend on R only through the magnetic field.
Figure 1: FAL heating rates per unit volume, and per unit mass as a function of z. The dashed line denotes a negative heating rate. The units of the heating rate per unit mass are 10^{9} erg g^{-1} s^{-1}. | |
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The elements of the total conductivity tensor are
The elements of the electron conductivity tensor are
In principle, the relations , and should hold. Due to the approximate nature of the conductivity model these relations hold to within terms of order .
These solutions are obtained by choosing the model parameters to minimize the least squares difference between horizontal spatial averages of the model heating rate q(R,z), and the total net radiative loss computed by FAL CM and communicated by Avrett & Loeser (2001). The procedure for doing this is as follows.
Figure 2: Electron, proton, proxy ion, H, and He I number densities, temperature, and proxy ion mass as functions of z. The temperature is given in units of 500 K. The proxy ion mass is given in units of three times the proton mass. | |
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A solution to the model is determined by specifying the parameters subject to the constraint Eq. (8). The temperature minimum of 4500 K occurs at km. The height ranges of the photosphere and chromosphere are defined as and .
is computed for km. It is shown in Fig. 1. The dashed section of the curve extends from z=150km to z=600 km, and indicates where . The first height above 600 km where FAL CM predicts a positive heating rate is 650 km.
For a given radius ,
define the horizontal average of q(R,z) over an area
of the magnetic structure
by
For each of the four choices for ,
some or all of the parameters
,
and
the filling factor f are to be varied to minimize the least squares difference
Domínguez Cerdeña et al. (2003) detect a kilogauss strength magnetic field structure in the photospheric internetwork that is inferred to have D < 180 km, and to belong to a class of similar structures having . Supporting observations of Sánchez Almeida et al. (2003) (also see Socas-Navarro & Sánchez Almeida 2003) indicate that most of the unsigned magnetic flux in the photospheric internetwork is in the form of kilogauss strength magnetic field concentrations, and that fields with strengths G and G have and .
It is shown in this section that if B_{0} and f are set equal to 1000 G and 0.02 as suggested by observation, and if L_{B} is chosen to be 600 km, then the values of that minimize yield values for the characteristic diameter for the four choices of defined after Eq. (48) that are in the range of 98 - 161 km. This range of D is consistent with the observational estimate that D < 180 km.
L_{B} is fixed at the value of 600 km since this implies that if B(0,0) = 1000 G, then G, which is assumed to be a reasonable field strength in the upper chromosphere. Larger (smaller) values of L_{B} lead to larger (smaller) heating rates.
The four choices for are , and . The solution corresponding to is presented in detail in this section. The other three solutions have similar properties, and are briefly discussed in Sect. 4.2.
Let . Then the value of that minimizes is 59.5, and D = 161.32 km. The magnetic structure is highly twisted in that , which is the case for the other three solutions discussed in Sect. 4.2.
Figure 1 shows the FAL CM heating rates per unit volume, and per unit mass from the photosphere to the lower corona.
Figure 3: Area averaged heating rates per unit volume, weighted by filling factor, and the FAL heating rate per unit volume as functions of z. | |
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Figure 3 shows the result of minimizing the difference between and over the height range of . for km, and is for km. The larger differences at greater heights indicate that the model rapidly breaks down with increasing height near the base of the transition region. It is shown in Goodman (2004) that Pedersen current dissipation is not important in the transition region and lower corona on the vertical spatial scales of the one dimensional FAL CM model. Its importance on smaller spatial scales remains to be determined. The photospheric heating rate for the model solution is several orders of magnitude smaller than the FAL CM photospheric heating rate in Fig. 1. Goodman (2004) shows that Pedersen current dissipation is not important below since in that region , so there is no qualitative difference between the mechanisms of Pedersen and parallel current dissipation. There . Figure 3 also shows the core heating rate .
Define the horizontally averaged chromospheric heating flux by
Figure 4: Area averaged heating rates per unit mass, weighted by filling factor, and the FAL heating rate per unit mass as functions of z. | |
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Figure 5: Electron, proton, proxy ion, and effective ion magnetizations averaged over the areas defined by (denoted by R_{0}) and (denoted by R_{0} - 2 R_{0}) as functions of z. | |
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Figure 4 shows the result of dividing the profiles in Fig. 3 by to obtain the model chromospheric heating rates per unit mass of , and . and reach 10^{9} erg g^{-1} s^{-1} at km and km. They reach maximum values of erg g^{-1} s^{-1} and erg g^{-1} s^{-1}. Their range of values from km up to the transition region is close to the constant value of erg g^{-1} s^{-1} that the one dimensional, semi-empirical model of Anderson & Athay (1989a,b) predicts for the height range of 1000 - 2000 km.
Define the volume averaged heating rate per unit mass by
The conditions under which Pedersen current dissipation can be important as a heating mechanism from the photosphere to the lower corona are determined by Goodman (2004). Some of the main conclusions of Goodman (2004) are illustrated in Figs. 5-7. Together with Figs. 8-10 they show that the region of the atmosphere in which Pedersen current dissipation is highly efficient and qualitatively different from parallel current dissipation is also the region where the associated heating rates per unit volume and mass are largest. This region is the chromosphere, and these heating rates are one or more orders of magnitude larger in the chromosphere than in the photosphere.
Figure 5 shows the horizontally averaged magnetizations of electrons, protons, proxy ions, and effective ions as a function of height. and are averages of M_{s}(R,z) over the areas defined by and . The averages of and are essentially equal. When M_{s} increases through unity, species s is said to become magnetized. for electrons, protons, and proxy ions at , and 700 km. for electrons, protons, and proxy ions at , and 885 km.
Figure 6: Efficiency of Pedersen current dissipation as a function of R and z. | |
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Figure 7: Ratio of parallel to Pedersen conductivity as a function of R and z. | |
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Figure 8: Heating rate per unit volume as a function of R and z. | |
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Figure 9: Heating rate per unit mass as a function of R and z, multiplied by the filling factor f=0.02. | |
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Figure 10: Ratio of Pedersen current dissipation rate to parallel current dissipation rate, averaged over areas with radii of R_{0} and 2 R_{0}, as functions of z. | |
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Figure 6 shows the efficiency Q(R,z) of Pedersen current dissipation, discussed in detail in Goodman (2004). By definition, , where is the maximum possible value of for a given value of . . Q is the efficiency with which the driving electric field transfers energy to the current density that is dissipated by collisions. From Goodman (2004), . Then Pedersen current dissipation is efficient in that . Figure 7 shows . Pedersen current dissipation is a qualitatively different heating mechanism from parallel current dissipation . Then Pedersen current dissipation is efficient and qualitatively different from parallel current dissipation, and hence can be a distinct and important heating mechanism . Figures 5-7 show that this condition does not hold in the photosphere but does hold in the chromosphere, and that there is a transition beginning near the temperature minimum between the photosphere where the condition is not satisfied, and the chromosphere where it is satisfied. This transition is caused by the magnetization of the ions and protons that occurs near the temperature minimum. This region of the atmosphere is weakly ionized in that collisional momentum transfer to the charged particles is due almost entirely to collisions with H. Then . This shows that, since T^{1/2} varies slowly, it is the steady decrease of relative to B with height that causes the charged particles to become magnetized.
Figure 8 shows . In the core region, where almost all of the heating occurs, increases with height by one order of magnitude from a local minimum at km to its global maximum at km, and then decreases relatively slowly with increasing height. The height range km is centered at the height of the temperature minimum. There is a direct correlation between the chromospheric temperature inversion and the initial rapid increase in with height in the sub-resolution core region of the magnetic structure.
Figures 6 and 7 show that the inequalities first become simultaneously satisfied in . Goodman (2004) shows that these inequalities are simultaneously satisfied and . In the core region, and increase from and to and over . In this region, at km, where first becomes positive above . From Goodman (2004), since in . Then in , so Pedersen current dissipation is significantly different from parallel current dissipation throughout . Again from Goodman (2004), , which in the core region ranges from at z=400 km to at z = 700 km, corresponding to an increase of Q from its minimum of to . As the protons become magnetized in , Pedersen current dissipation becomes an efficient channel for dissipating the energy in . This process is directly correlated with the increase of with height by an order of magnitude in , and with the chromospheric temperature inversion.
Figure 9 shows . Its values in the core region are erg g^{-1} s^{-1} for km. Horizontal averaging of these values occurs during observations with a spatial resolution greater than the core diameter D_{0}= 2 R_{0}. For the four cases considered here . This range of D_{0} is currently sub-resolution for detecting magnetic structures. At a resolution much larger than D_{0}, heating in an approximately uniform distribution of similar magnetic structures with filling factor fappears as a heating rate per unit mass that varies weakly with horizontal position, and has a height dependence similar to that predicted by Anderson & Athay (1989a,b) above the height of the temperature minimum. The temperature minimum results from the horizontal average over the heights of the temperature minima of the individual structures.
Figure 10 shows for , and 2 R_{0}. In the core region, parallel current dissipation dominates in the photosphere, below . Beginning near Pedersen current dissipation rapidly dominates parallel current dissipation in the core region, exceeding it by orders of magnitude for km. For , Pedersen current dissipation exceeds parallel current dissipation everywhere by one or more orders of magnitude.
Figure 11: Magnitude of the component of the CM electric field that is perpendicular to as a function of R and z. | |
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Figure 12: Magnitude of the component of the electric field that is parallel to as a function of R and z. | |
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Figures 11 and 12 show the magnitudes of the components and of the driving electric field . These components drive the Pedersen and parallel currents. in some regions of the photosphere below . Beginning near , rapidly increases, and exceeds by orders of magnitude for .
Figure 13: Parallel conductivity as a function of z. | |
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Figure 14: Pedersen conductivity as a function of R and z. | |
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Figure 15: Magnitude of the Hall conductivity as a function of R and z. | |
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Figure 16: Ratio of the magnitudes of the effective ion and electron Pedersen current densities as a function of R and z. | |
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Figures 13-15 show the total parallel, Pedersen, and Hall conductivities. They vary by orders of magnitude, and can differ by orders of magnitude throughout the magnetic structure. This behavior has a strong effect in determining the resistive heating rate, and underscores the importance of using realistic representations of transport processes in models of the solar atmosphere. The behavior of in Fig. 14, and in Fig. 11 are inversely correlated. This suggests that the decrease of with increasing height allows to build up to sufficiently large values to cause to become significant, meaning comparable to the chromospheric net radiative loss. The particle magnetizations are so small in the photosphere that is close to its maximum possible value of . These relatively large values of prevent , and hence from being significant. There is a similar correlation between the behavior of in Fig. 13, and in Fig. 12, suggesting that the increase of with height prevents from building up to sufficiently large values to cause to be significant in the chromosphere.
Figure 17: Ratio of the magnitudes of the proton and proxy ion Pedersen current densities as a function of R and z. | |
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Figure 16 shows the relative contribution of effective ions and electrons to the Pedersen current. The effective ions dominate the Pedersen current for km. The effective ions are a composite of protons and proxy ions. Figure 17 shows the relative contribution of protons and proxy ions to the effective ion Pedersen current. Except in the height range of km, . at km, and at km. Since, as discussed in the context of Fig. 8, increases rapidly from a local minimum to its global maximum over the height range km, the dissipation of metallic ion Pedersen current makes an important contribution to chromospheric heating up to km. Goodman (2000) proposes that it is initially metallic ion Pedersen current dissipation below the height of the temperature minimum in a magnetic structure that starts the heating process leading to the chromospheric temperature inversion that is then maintained by proton Pedersen current dissipation into the upper chromosphere. The results presented here support this proposition.
Figure 18: Current density parallel to as a function of R and z. | |
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Figure 19: Magnitude of Pedersen current density as a function of R and z. | |
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Figure 20: Magnitude of Hall current density as a function of R and z. | |
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Figures 18-20 show the parallel, Pedersen, and Hall current densities. for , and for . The core region, where almost all of the heating occurs, is largely force-free. The plasma rapidly becomes non-force-free outside the core region.
Although
in the core region, the heating is due almost entirely to
Pedersen current dissipation. This may be understood in the following way. From Eqs. (18), (32), (33), (35), and
it follows that
Figure 21: Pressure and number density as functions of R at z=0. The pressure is given in units of 10^{5} dynes cm^{-2}. The number density is given in units of 10^{17} cm^{-3}. | |
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The solutions for , and are similar to the solution for presented in Sect. 4.1. Some properties of these solutions are briefly discussed here.
For it is found that km, and . The average heating rates are erg cm^{-2} s^{-1}, erg g^{-1} s^{-1}, erg cm^{-2} s^{-1}, and erg g^{-1} s^{-1}. The fraction of and due to parallel current dissipation is . The fraction of and due to parallel current dissipation is . ranges from to for , and from to for .
For it is found that km, and . The average heating rates are erg cm^{-2} s^{-1}, erg g^{-1} s^{-1}, erg cm^{-2} s^{-1}, and erg g^{-1} s^{-1}. The fraction of and due to parallel current dissipation is . The fraction of and due to parallel current dissipation is . ranges from to for , and from to for .
For it is found that km, and . The average heating rates are erg cm^{-2} s^{-1}, erg g^{-1} s^{-1}, erg cm^{-2} s^{-1}, and erg g^{-1} s^{-1}. The fraction of and due to parallel current dissipation is . The fraction of and due to parallel current dissipation is . ranges from to for , and from to for .
As in the case of the solution in Sect. 4.1, the core region for these solutions is largely force-free, the plasma rapidly becomes non-force-free outside this region, and the magnetic field is highly twisted in that . The heating rate in the core region is times larger than outside this region. Almost all of the heating occurs in the largely force-free core plasma, and is due to Pedersen current dissipation. The range of diameters of the core regions for the four solutions is km, so essentially all of the heating occurs on sub-resolution horizontal scales.
The model presented in Sect. 2 does not predict or . It is necessary to know these quantities in order to determine the contributions of and to .
Here the model in Sect. 2 is extended to compute and , and estimate the R dependence of the total pressure and density p and , which in FAL CM only depend on z.
Assuming a steady state that only depends on R and z, and ignoring viscosity and the
convection term
,
the momentum equation is
The component of Eq. (53) is identical to Eq. (5) that is used to determine . This equation connects the two parts of the model.
Using
from Sect. 2, the R and z components of Eq. (53) are solved for pand ,
giving
Figure 21 shows p(R,0) and n(R,0). The pressure and density in the core region are larger than in the surrounding relatively field free region. For they are essentially equal to their asymptotic values and . The pressure and density on the axis R=0 are only larger than their values in the field free plasma.
The mass conservation equation is
is known from Sect. 2.2, and
since
.
Then the
component of
gives
Figure 22: Magnitude of the flow velocity perpendicular to as a function of R and z. | |
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Figure 23: Magnitude of the flow velocity parallel to as a function of R and z. | |
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The equation
reduces to
and
Multiplying the R component of
by B_{R}, the z component by B_{z}, adding the two resulting equations, and using
Eq. (61) to
express E_{R} in terms of
gives the following equation for E_{z}.
The z component of
then determines by
Figures 22 and 23 show and . throughout most of the photosphere and chromosphere. exceeds the sound speed by up to a factor 7 in the chromosphere. km s^{-1} in the photosphere, and is 700 m s^{-1} in the chromosphere. generates the convection electric field that drives the Pedersen current.
Figure 24: Magnitude of the convection electric field as a function of R and z. | |
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Figure 25: Magnitude of the component of perpendicular to as a function of R and z. | |
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Figures 24-26 show , and . essentially everywhere in the chromosphere except for . rapidly dominates beginning near . and largely cancel in the photosphere in that . The Pedersen current and the associated resistive heating in the chromosphere are driven by the convection electric field.
Figure 26: Ratio of the magnitudes of the convection electric field and as a function of R and z. | |
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The rate at which the electric field transfers energy to the fluid is
In the MHD wave heating model of Goodman (2000) it is found that , V m^{-1}, and m s^{-1}at km. That model does not take account of the filling factor of the magnetic flux tubes in which the heating is found to occur. The effect of including a filling factor is estimated as follows. If is driven by a convection electric field then . Then for given , and are . Then for f=0.02 the values of and just stated are changed to V m^{-1}, and m s^{-1}. These values are comparable to the values of and at z=750 km in Figs. 11 and 22. In the model in Goodman (2000) there is no flow in the equilibrium state, and the equilibrium magnetic field is constant, and hence unbounded and current free. Then and are entirely due to the MHD wave. By contrast, the model considered here assumes a steady state with flow, and a localized, non-potential magnetic field. The fact that estimates of and are similar for these two models at km for the same value of f, and for the requirement that the resistive heating rate, which is essentially , match that predicted by FAL CM follows from the fact that , and have similar values in both models at that height.
As observational resolution increases, the observed filling factor of magnetic field structures in the photosphere increases, including structures with kilogauss magnetic field strengths. This trend, together with the model results reported here and in Goodman (2000, 2001, 2004) lead to the proposition that the chromospheres of the Sun and other solar type stars are created by Pedersen current dissipation driven by a convection electric field in magnetic structures with horizontal spatial dimensions that are currently sub-resolution. This is an MHD power generation mechanism. Mechanical work is done to drive a CM flow of plasma across magnetic field lines. The resulting convection electric field drives current against the resistive load of the plasma, generating thermal energy. Observationally, resolution of the photospheric magnetic field on spatial scales is necessary in order to resolve the core regions of the magnetic structures in which almost all of the heating occurs. The essential difference between network and internetwork chromospheric heating is that the filling factor of the sub-resolution magnetic structures in which the heating occurs is larger in the network than in the internetwork.
The proposed heating mechanism is consistent with observational and semi-empirical model evidence that most of the magnetic flux through the photosphere is in the form of magnetic field concentrations with sub-resolution diameters as small as a few kilometers, and field strengths G (Sánchez Almeida & Landi degl'Innocenti 1996; Sánchez Almeida et al. 1996; Sánchez Almeida 1997, 2000; Sánchez Almeida & Lites 2000; Sánchez Almeida et al. 2001; Socas-Navarro & Sánchez Almeida 2002; Domínguez Cerdeña et al. 2003). A common view is that the internetwork chromosphere is heated by acoustic wave dissipation, and that magnetic field related heating mechanisms may only be important in the network. Judge et al. (2003) question this view, and present observational and modeling evidence that the solar upper chromospheric internetwork is heated by a magnetic field related mechanism. This is consistent with the proposition that the entire chromosphere outside of flaring regions is heated by an MHD mechanism of the type described in this paper.
It is the increase of the particle magnetizations with height from values 1 to values >1 in the weakly ionized photospheric plasma that transforms Pedersen current dissipation
into a qualitatively new, and highly efficient heating mechanism, and, in the presence of the
CM convection across magnetic field lines that exists everywhere and always, simultaneously
causes the buildup of the magnitude of the convection electric field to a value large enough to
increase
to a value sufficiently large to cause a temperature inversion and create the
chromosphere. The main features of this process are summarized as follows.
It is proposed that this is the basic process by which the temperature inversion and associated chromosphere is created in the atmospheres of solar type stars.
A stellar atmosphere in which a time dependent transformation takes place between a state in which , in which case there is no chromosphere, and a state in which , in which case a chromosphere is generated, may have an intermittent chromosphere. The time variation of the intermittency may be controlled by the time variation of the dynamo generated magnetic field, and atmospheric convection.
Acknowledgements
This work was supported by NSF grant ATM 0242820 to the Institute for Scientific Research. The author thanks Eugene Avrett and Rudolf Loeser of the Harvard-Smithsonian Center for Astrophysics for providing numerical solutions to the FAL models, and Juan Fontenla of the Laboratory for Space and Atmospheric Physics at the University of Colorado, Boulder for discussions about these models.