Thermal balance of the mid-latitude F2-region at night

The thermal balance of the plasma in the night-time mid-latitude F2-region is examined using solutions of the steady-state O⁺ and electron heat balance equations. The required concentrations and field-aligned velocities are obtained from a simultaneous solution of the time-dependent O⁺ continuity and momentum equations.The results demonstrate the systematic trend for the O⁺ temperature to be 10–20 K greater than the electron temperature during the night at around 300 km, as observed at St. Santin by Bauer and Mazaudier. It is shown that frictional heating between the O⁺ and neutral gases is the cause of the O⁺ temperature being greater than the electron temperature; the greater the importance of frictional heating in the thermal balance the greater is the difference in the O⁺ and electron temperatures. A study is made of the roles played in the thermal balance of the plasma by the thermal conductivity of the O⁺ and electron gases; collisional heat transfer between O⁺ electrons and neutrals; frictional heating between the O⁺ and neutral gases; and advection and convection due to field-aligned O⁺ and electron motions. The results of the study show that, at around 300 km, electron cooling by excitation of the fine structure of the ground state of atomic oxygen plays a major role in the thermal balance of the electrons and, since the temperature of the ions is little affected by this electron cooling process, in determining the difference between the ion and electron temperatures.     

An empirically derived lunar gravity field

The heat-flow experiment is one of the Apollo Lunar Surface Experiment Package (ALSEP) instruments that was emplaced on the lunar surface on Apollo 15. This experiment is designed to make temperature and thermal property measurements in the lunar subsurface so as to determine the rate of heat loss from the lunar interior through the surface. About 45 days (1 1/2 lunations) of data has been analyzed in a preliminary way. This analysis indicates that the vertical heat flow through the regolith at one probe site is 3.3 × 10^–6 W/cm² (±15%). This value is approximately one-half the Earth's average heat flow. Further analysis of data over several lunations is required to demonstrate that this value is representative of the heat flow at the Hadley Rille site. The mean subsurface temperature at a depth of 1 m is approximately 252.4K at one probe site and 250.7K at the other. These temperatures are approximately 35K above the mean surface temperature and indicate that conductivity in the surficial layer of the Moon is highly temperature dependent. Between 1 and 1.5m, the rate of temperature increase as a function of depth is 1.75K/m (±2%) at the probe 1 site. In situ measurements indicate that the thermal conductivity of the regolith increases with depth. Thermal-conductivity values between 1.4 × 10^–4 and 2.5 × 10^–4 W/cm K were determined; these values are a factor of 7 to 10 greater than the values of the surface conductivity. If the observed heat flow at Hadley Base is representative of the moonwide rate of heat loss (an assumption which is not fully justified at this time), it would imply that overall radioactive heat production in the Moon is greater than in classes of meteorites that have formed the basis of Earth and Moon bulk composition models in the past.Lamont-Doherty Geological Observatory Contribution Number 1800.     

Thermal and transport properties of mantle rock at high pressure: Applications to super-Earths

In the present study, the temperature- and pressure-dependent transport and thermal properties, i.e., viscosity, phonon thermal conductivity, thermal expansivity and heat capacities, as well as electronic and radiative thermal conductivities, have been derived for the mantles of super-Earths. These properties are necessary to understand the interior dynamics and the thermal evolution of those planets. We assume that the mantles consist of MgSiO₃ perovskite (pv), but we discuss the effects of the post-perovskite transition, and we elaborate on an addition of periclase MgO and incorporated Fe. However, MgO is found to only significantly influence the phonon thermal conductivity – the viscosities, heat capacities and thermal expansivities of pv and MgO remain comparable. We use the Keane theory of solids, which takes into account the behavior of solid matter at the infinite pressure limit, adopt the Keane equations of state, and adjust for pv and MgO by comparison with experimental high-pressure and high-temperature data. We find the theory of the infinite pressure limit of Keane to be in excellent agreement with recent ab initio studies and experiments. To calculate the melting curve, we further use the Lindemann–Stacey scaling law and fit it to available experimental data. The best data fitting melting temperature for pv reaches 5700 K at 135 GPa and increases to 20,000 K at 1.1 TPa, corresponding to the core-mantle boundary of a 10 Earth mass super-Earth (10M_Earth). We find the pv adiabatic temperature (with a potential temperature of 1700 K) to reach 2570 K at 135 GPa and 5000 K at 1.1 TPa. To calculate the pressure-and temperature-dependent viscosity, we use the semi-empirical homologous temperature scaling to relate enthalpy change, and hence viscosity, to the melting temperature. We find that the resulting activation volume of pv decreases from 2.8 cm³/mol at 25 GPa to 1.4 cm³/mol at 1.1 TPa-resulting in a viscosity increase by ～15 orders of magnitude through the adiabatic mantle of a 10M_Earth planet. Furthermore, the thermal expansivity (of pv and MgO) decreases by a factor of eight, and the total thermal conductivity (phonon, radiative and electronic) of an Earth-like pv/MgO composite increases by a factor of seven through an adiabatic mantle of a 10M_Earth super-Earth. At higher temperatures, i.e., for super-adiabatic temperature profiles, the electronic and radiative thermal conductivities strongly increase and dominate the conductive heat transport. All findings indicate an increase of heat transfer solely by conduction in the lower mantles of super-Earths. Thus our results disagree with Earth-biased full-mantle convection assumptions made by previous models for super-Earths, and additionally raise questions about the differentiation of massive rocky exoplanets and their ability to generate magnetic fields or sustain plate tectonics.     

Thermophysical properties of Apollo 12 fines

The vacuum thermal conductivity of the Apollo 12 fines is presented as a function of temperature for densities of 1300, 1640 and 1970kg/m³. It is found to vary from about 10^?3W/m-°K at 100°K to about 3 x 10^?3W/m-°K at 400°K. The conductivity of the fines is found to be close to that of terrestrial basalt both under vacuum and at higher pressures. The thermal diffusivity is calculated from conductivity and specific heat data. Average values of the thermal conductivity, thermal diffusivity and thermal parameter are also presented.     

Ion temperature troughs induced by a meridional neutral air wind in the night-time equatorial topside ionosphere

A mathematical model has been developed to calculate consistent values for the O⁺ and H⁺ concentrations and field-aligned velocities and for the O⁺, H⁺ and electron temperatures in the night-time equatorial topside ionosphere. Using the results of the model calculations a study is made to establish the ability of F-region neutral air winds to produce observed ion temperature distributions and to investigate the characteristics of ion temperature troughs as functions of altitude, latitude and ionospheric composition. Solar activity conditions that give exospheric neutral gas temperatures 600 K, 800 K and 1000 K are considered.It is shown that the O⁺-H⁺ transition height represents an altitude limit above which ion cooling due to adiabatic expansion of the plasma is extremely small. The neutral atmosphere imposes a lower altitude limit since the neutral atmosphere quenches any ion cooling which field-aligned transport tends to produce. The northern and southern edges of the ion temperature troughs are shown to be restricted to a range of dip latitudes, the limiting dip latitudes being determined by the magnetic field line geometry and by the functional form of the F-region neutral air wind velocity. Both these parameters considerably influence the interaction between the neutral air and the plasma within magnetic flux tubes.     

Thermophysical properties of lunar material returned by Apollo missions

Data on thermophysical properties measured on lunar material returned by Apollo missions are reviewed. In particular, the effects of temperature and interstitial gaseous pressure on thermal conductivity and diffusivity have been studied. For crystalline rocks, breccias and fines, the thermal conductivity and diffusivity decrease as the interstitial gaseous pressure decreases from 1 atm to 10^–4T. Below 10^–4T, these properties become insensitive to the pressure. At a pressure of 10^–4T or below, the thermal conductivity of fines is more temperature dependent than that of crystalline rocks and breccias. The bulk density also affects the thermal conductivity of the fines. An empirical relationship between thermal conductivity, bulk density and temperature derived from the study of terrestrial material is shown to be consistent with the data on lunar samples. Measurement of specific heat shows that, regardless of the differences in mineral composition, crystalline rocks and fines have almost identical specific heat in the temperature range between 100 and 340K. The thermal parameter calculated from thermal conductivity, density and specific heat shows that the thermal properties estimated by earth-based observations are those characteristic only of lunar fines and not of crystalline rocks and breccias. The rate of radioactive heat generation calculated from the content of K, Th and U in lunar samples indicates that the surface layer of the lunar highland is more heat-producing than the lunar maria. This may suggest fundamental differences between the two regions.Now at Lamont-Doherty Geological Observatory, Columbia University, Palisades, New York, U.S.A.     

Heat blanketing envelopes and thermal radiation of strongly magnetized neutron stars

Strong (B?10⁹ G) and superstrong (B?10¹⁴ G) magnetic fields profoundly affect many thermodynamic and kinetic characteristics of dense plasmas in neutron star envelopes. In particular, they produce strongly anisotropic thermal conductivity in the neutron star crust and modify the equation of state and radiative opacities in the atmosphere, which are major ingredients of the cooling theory and spectral atmosphere models. As a result, both the radiation spectrum and the thermal luminosity of a neutron star can be affected by the magnetic field. We briefly review these effects and demonstrate the influence of magnetic field strength on the thermal structure of an isolated neutron star, putting emphasis on the differences brought about by the superstrong fields and high temperatures of magnetars. For the latter objects, it is important to take proper account of a combined effect of the magnetic field on thermal conduction and neutrino emission at densities ρ?10¹⁰ g?cm^?3. We show that the neutrino emission puts a B-dependent upper limit on the effective surface temperature of a cooling neutron star.     

A self-consistent model of a thermally balanced quiescent prominence in magnetostatic equilibrium in a uniform gravitational field

We present a theoretical model of quiescent prominences in the form of an infinite vertical sheet. Self-consistent solutions are obtained by integrating simultaneously the set of nonlinear equations of magnetostatic equilibrium and thermal balance. The basic features of the models are: (1) The prominence matter is confined to a sheet and supported against gravity by a bowed magnetic field. (2) The thermal flux is channelled along magnetic field lines. (3) The thermal flux is everywhere balanced by Low's (1975b) hypothetical heat sink which is proportional to the local density. (4) A constant component of the magnetic field along the length of the prominence shields the cool plasma from the hot surrounding. We assume that the prominence plasma emits more radiation than it absorbs from the radiation fields of the photosphere, chromosphere and corona, and we interpret the above hypothetical heat sink to represent the amount of radiative loss that must be balanced by a nonradiative energy input. Using a central density and temperature of 10¹¹ particles cm^–3 and 5000 K respectively, a magnetic field strength between 2 to 10 gauss and a thermal conductivity that varies linearly with temperature, we discuss the physical properties implied by the model. The analytic treatment can also be carried out for a class of more complex thermal conductivities. These models provide a useful starting point for investigating the combined requirements of magnetostatic equilibrium and thermal balance in the quiescent prominence.     

The thermal and physical characteristics of the Gao-Guenie (H5) meteorite

Measurements of the bulk density, grain density, porosity, and magnetic susceptibility of 19 Gao-Guenie H5 chondrite meteorite samples are presented. We find average values of bulk density 〈ρ_bulk〉=3.46±0.07 g/cm³, grain density 〈ρ_grain〉=3.53±0.08 g/cm³, porosity 〈P(%)〉=2.46±1.39, and bulk mass magnetic susceptibility 〈log χ〉=5.23±0.11. Measurements of the specific heat capacity for a 3.01-g Gao-Guenie sample, a 61.37-g Gao-Guenie sample, a 62.35-g Jilin H5 chondrite meteorite sample, and a 51.37-g Sikhote-Alin IIAB Iron meteorite sample are also presented. Temperature interpolation formula are further provided for the specific heat capacity, thermal conductivity, and thermal diffusivity of the 3.01-g Gao-Guenie sample in the temperature range 300<T (K)<800. We briefly review the possible effects of the newly deduced specific heat and thermal conductivity values on the ablation of meteoroids within the Earth's atmosphere, the modeling of asteroid interiors and the orbital evolution of meteoroids through the Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect.     

The thermal conductivity of meteorites: New measurements and analysis

We have measured the thermal conductivity at low temperatures (5-300 K) of six meteorites representing a range of compositions, including the ordinary chondrites Cronstad (H5) and Lumpkin (L6), the enstatite chondrite Abee (E4), the carbonaceous chondrites NWA 5515 (CK4 find) and Cold Bokkeveld (CM2), and the iron meteorite Campo del Cielo (IAB find). All measurements were made using a Quantum Design Physical Properties Measurement System, Thermal Transport Option (TTO) on samples cut into regular parallelepipeds of ∼2-6 mm dimension. The iron meteorite conductivity increases roughly linearly from 15 W m⁻¹ K⁻¹ at 100 K to 27 W m⁻¹ K⁻¹ at 300 K, comparable to typical values for metallic iron. By contrast, the conductivities of all the stony samples except Abee appear to be controlled by the inhomogeneous nature of the meteorite fabric, resulting in values that are much lower than those of pure minerals and which vary only slightly with temperature above 100 K. The L and CK sample conductivities above 100 K are both about 1.5 W m⁻¹ K⁻¹, that of the H is 1.9 W m⁻¹ K⁻¹, and that of the CM sample is 0.5 W m⁻¹ K⁻¹; by contrast the literature value at 300 K for serpentine is 2.5 W m⁻¹ K⁻¹ and those of enstatite and olivine range from 4.5 to 5 W m⁻¹ K⁻¹ (which is comparable to the Abee value). These measurements are among the first direct measurements of thermal conductivity for meteorites. The results compare well with previous estimates for meteorites, where conductivity was derived from diffusivity measurements and modeled heat capacities; our new values are of a higher precision and cover a wider range of temperatures and meteorite types. If the rocky material that makes up asteroids and provides the dust to comets, Kuiper Belt objects, and icy satellites has the same low thermal conductivities as the ordinary and carbonaceous chondrites measured here, this would significantly change models of their thermal evolution. These values would also lower their thermal inertia, thus affecting the Yarkovsky and YORP evolution of orbits and spin for solid objects; however, in this case the effect would not be as great, as thermal inertia only varies as the square root of the conductivity and, for most asteroids, is controlled by the dusty nature of asteroidal surfaces rather than the conductivity of the material itself.     

Thermal equilibria of isobaric coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of isobaric thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. The possible thermal equilibria can be shown to depend on two parameters L ^* p ^* and h ^*/p ^* representing the ratios of cooling (radiation) to condu and heating to cooling, respectively. Arcades can contain four types of loops: hot loops with summits hotter than 400000 K; cool loops at temperatures less than 80000 K along their lengths; hot-cool loops with cool summits and cool footpoints but hotter intermediate portions; and warm loops, cooler than 80000 K along most of their lengths but with summits as hot as 400000 K. Two possibilities for coronal heating are considered, namely a heating that is independent of magnetic field and a heating that is proportional to the square of the local magnetic field. When the arcade is sheared the thermal structure of the arcade may change, leading in some cases to non-equilibrium or in other cases to the formation of a cool core.     

The thermal structure of the magnetized solar transition region

The detailed thermal structure of the magnetized solar transition region, as measured by itsdifferential emission measure [DEM(T)], is unknown. Proposals have been made that envision a significant lower-temperature contribution to the energy balance from cross-field (ion) heat flux. In this paper, we describe a self-consistent, 2-D, MHD simulation (including the full effects of anisotropic thermal conduction) of a conceptual model due to Athay (1990). We display the detailed, irregular, thermal and magnetic structure of the transition region, and demonstrate that the predicted DEM agrees with observations, particularly in theT < 10⁵ K regime where previous theories had difficulty.     

Studies on some properties of coronal mass ejections based on angular width

The paper investigates the effects of thermal conductivity and non-uniform magnetic field on the gravitational instability of a non-uniformly rotating infinitely extending axisymmetric cylinder in a homogeneous heat conducting medium. The non-uniform rotation and magnetic field are supposed to act along θ and z directions of the cylinder. It is found that the gravitational instability of this general problem is determined by the same criterion as obtained by Dhiman and Dadwal (Astrophys. Space Sci. 325(2):195–200, 2010) for the self-gravitating isothermal medium in the presence of non-uniform rotation and magnetic field with the only difference that adiabatic sound velocity is now replaced by the isothermal sound velocity. It is found that the thermal conductivity has stabilizing effect on the onset of gravitational instability. Further, the stabilizing/destabilizing effect of the non-uniform magnetic field on the gravitational instability of heat conducting medium has been discussed and is illustrated by considering some special forms of the basic magnetic fields.     

Thermal equilibria of coronal magnetic loops

Equations of thermal equilibrium along coronal loops with footpoint temperatures of 2 × 10⁴ K are solved. Three fundamentally different categories of solution are found, namely hot loops with summit temperatures above about 4 × 10⁵ K, cool loops which are cooler than 8 × 10⁴ K along their whole length and hot-cool loops which have summit temperatures around 2 × 10⁴ K but much hotter parts at intermediate points between the summit and the footpoints. Hot loops correspond to the hot corona of the Sun. The cool loops are of relevance for fibrils, for the cool cores observed by Foukal and also for active-region prominences where the magnetic field is directed mainly along the prominence. Quiescent prominences consist of many cool threads inclined to the prominence axis, and each thread may be modelled as a hot-cool loop. In addition, it is possible for warm loops at intermediate summit temperatures (8 × 10⁴K to 4 × 10⁵ K) to exist, but the observed differential emission measure suggests that most of the plasma in the solar atmosphere is in either the hot phase or the cool phase. Thermal catastrophe may occur when the length or pressure of a loop is so small that the hot solution ceases to exist and there are only cool loop solutions. Many loops can be superimposed to form a coronal arcade which contains loops of several different types.     

Temperature behavior of elemental abundances in the atmospheres of magnetic peculiar stars

We analyze the temperature dependence of the abundances of the chemical elements Si, Ca, Cr, and Fe in the atmospheres of normal, metallic-line (Am), magnetic peculiar (Ap), and pulsating magnetic peculiar (roAp) stars in the range 6000–15000 K. The Cr and Fe abundances in the atmospheres of Ap stars increase rapidly as the temperature rises from 6000 to 9000–10000 K. Subsequently, the Cr abundance decreases to values that exceed the solar abundance by an order of magnitude, while the Fe abundance remains enhanced by approximately +1.0 dex compared to the solar value. The temperature dependence of the abundances of these elements in the atmospheres of normal and Am stars is similar in shape, but its maximum is several orders of magnitude lower than that observed for Ap stars. In the range 6000–9500 K, the observed temperature dependences for Ap stars are satisfactorily described in terms of element diffusion under the combined action of gravitational settling and radiative acceleration. It may well be that diffusion also takes place in the atmospheres of normal stars, but its efficiency is very low due to the presence of microturbulence. We show that the magnetic field has virtually no effect on the Cr and Fe diffusion in Ap stars in the range of effective temperatures 6000–9500 K. The Ca abundance and its variation in the atmospheres of Ap stars can also be explained in terms of the diffusion model if we assume the existence of a stellar wind with a variable moderate rate of ～(2–4) × 10^{? 15}M_⊙ yr^?1.     

Mapping of Io's thermal radiation by the Galileo photopolarimeter-radiometer (PPR) instrument

Between 1999 and 2002, the Galileo spacecraft made 6 close flybys of Io during which many observations of Io's thermal radiation were made with the photopolarimeter-radiometer (PPR). While the NIMS instrument could measure thermal emission from hot spots with T>200 K, PPR was the only Galileo instrument capable of mapping the lower temperatures of older, cooling lava flows, and the passive background. We tabulate all data taken by PPR of Io during these flybys and describe some scientific highlights revealed by the data. The data include almost complete coverage of Io at better than 250 km resolution, with extensive regional coverage at higher resolutions. We found a modest poleward drop in nighttime background temperatures and evidence of thermal inertia variations across the surface. Comparison of high spatial resolution temperature measurements with observed daytime SO₂ gas pressures on Io provides evidence for local cold trapping of SO₂ frost on scales smaller than the 60 km resolution of the PPR data. We also calculated the power output from several hot spots and estimated total global heat flow to be about 2.0-2.6 W m⁻². The low-latitude diurnal temperature variations for the regions between obvious hot spots are well matched by a laterally-inhomogeneous thermal model with less than 1 W m⁻² endogenic heat flow.     

Solar-interplanetary modeling: 3-D solar wind solutions in prescribed non-radial magnetic field geometries

A model is presented which describes the 3-dimensional non-radial solar wind expansion between the Sun and the Earth in a specified magnetic field configuration subject to synoptically observed plasma properties at the coronal base. In this paper, the field is taken to be potential in the inner corona based upon the Mt. Wilson magnetograph observations and radial beyond a certain chosen surface. For plasma boundary conditions at the Sun, we use deconvoluted density profiles obtained from synopticK-coronameter brightness observations. The temperature is taken to be 2 × 10⁶ K at the base of closed field lines and 1.6 x 10⁶K at the base of open field lines. For a sample calculation, we employ data taken during the period of the 12 November 1966 eclipse. Although qualitative agreement with observations at 1 AU is obtained, important discrepancies emerge which are not apparent from spherically symmetric models or those models which do not incorporate actual observations in the lower corona. These discrepancies appear to be due to two primary difficulties - the rapid geometric divergence of the open field lines in the inner corona as well as the breakdown in the validity of the Spitzer heat conduction formula even closer to the Sun than predicted by radial flow models. These two effects combine to produce conductively dominated solutions and lower velocities, densities, and field strengths at the Earth than those observed. The traditional difficulty in solar wind theory in that unrealistically small densities must be assumed at the coronal base in order to obtain observed densities at 1 AU is more than compensated for here by the rapid divergence of field lines in the inner corona. For these base conditions, the value ofβ(ratio of gas pressure to magnetic pressure) is shown to be significantly greater than one over most of the lower corona - suggesting that, for the coronal boundary conditions used here, the use of a potential or force-free magnetic field configuration may not be justified. The calculations of this paper point to the directions where future research on solar-interplanetary modelling should receive priority:

1. better models for the coronal magnetic field structure
2. improved understanding of the thermal conductivity relevant for the solar wind plasma.

     

A dynamical model of the corona

The hydrodynamic equations for an ideal, inviscid, fully ionized hydrogen gas in a gravitational, but not magnetic, field are solved by an explicit Lax-Wendroff two-step technique using a one-dimensional slab symmetry. Radiation and thermal conductivity are included. The model spans 100000 km starting from the chromosphere-corona transition region. An initially isothermal gas is seen to evolve coronal properties in 4000 s, by which time it settles into dynamic equilibrium characterized by a 2000 km transition region, a temperature maximum of 1.6 × 10⁶ K at a height of 60000 km, and a solar wind mass flux of 10^-9 g cm^-2 s^-1.     

Flare loops heated by thermal conduction

A flare observed with the Hard X-Ray Imaging Spectrometer (HXIS) was studied during its rise to maximum temperature and X-ray emission rate. Two proximate flare loops, of lengths 2.8 × 10⁹ cm and 1.1 × 10¹⁰ cm, rose to temperatures of 21.5 × 10⁶ K and 30 × 10⁶ K, respectively, in 30 s. Assuming equal heat flux F into each loop from a thermal source at the point where they met, we derive a simple relationship between temperature T and loop length , which gives a loop temperture ratio of 0.68, in close agreement with the observed ratio of 0.72. The observations imply that heating in each loop was maintained by a thermal flux of 5 × 10⁹ ergs cm^-2 s^-1. It is suggested that conductive heating adequately describes the rise and maximum phase emissions in the loops and that long flare loops reach higher temperatures than short loops during the impulsive phase because of an equipartition of energy between them at their point of interaction.     

The charge state in a highly dense molecular cloud

We have calculated the desorption rates of both physisorbed and chemisorbed ions from grain surfaces, due to the temperature increase at densities higher than 10^–13 g cm^–3. It has been found that physisorbed ions desorb from grain surfaces at neutral densities ofn>1.3×10¹¹ cm^–3, assuming that the desorption energyD is equal to 0.1 eV, while the desorption of chemisorbed ions from grain surface can only occur at neutral densities ofn>10¹⁵ cm^–3, at which point thermal ionization becomes more dominant.The electrons are assumed to be emitted from grain surfaces in a manner similar to the thermonic emission from heated solid surfaces. It was found that the temperature at which electrons are emitted from negatively charged grains depends on the value of the work function of the material of the grain.The charge state has been calculated for two limiting cases. Neglecting the grain surface reactions in case 1, the resulting relative charge density represents an upper limit, such that the electrical conductivity remains high. In this situation the magnetic flux dissipation is mainly contributed by ambipolar diffusion. In the second case, it has been assumed that the charged particles are chemically adsorbed on grain surfaces such that their desorption is negligible. In this case the charge density decreases sharply with increase of neutral density. Therefore, the electrical conductivity decreases sufficiently and Ohmic dissipation becomes effective.