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Numerical studies of vertically propagating acoustic and magnetoacoustic waves in an isothermal atmosphere (III)
Journal of Molecular Structure (Theochem) 640 (2003) 39–47 www.elsevier.com/locate/theochem
Numerical studies of vertically propagating acoustic and magnetoacoustic waves in an isothermal atmosphere (III) A.F. Jalbouta,*, H. Alkahbyb, A. Talmadgeb, P. Frempong-Mirekub a
Department of Chemistry, University New Orleans, New Orleans, LA 70112, USA b Department of Mathematics, Dillard University, New Orleans, LA 70122, USA Received 11 February 2002; accepted 5 May 2003
Abstract In this paper we investigate, numerically, the generation and propagation of acoustic and magnetoacoustic waves and their roles in the heating process of the chromosphere. The combined effects of viscosity, thermal conduction and a uniform horizontal magnetic field on an upward and a downward propagating wave in an isothermal atmosphere are considered. It is shown that when the heating mechanisms are dominated by the effects of thermal conduction. The chromosphere atmosphere can be divided into three distinct regions, low, middle and high. The middle region acts like a semitransparent transition layer and it is produced by thermal conduction and connects middle and high chromosphere. In the transition region part of the energy transmitted upward, part is dissipated and the other part is reflected downward. Moreover, viscosity creates an absorbing and reflecting layer and the magnetic field forms a totally reflecting barrier because of its dissipationless nature. When the combined effects of the viscosity and magnetic field dominated the oscillatory process, thermal conduction can be eliminated because the solution decays exponentially with altitude before the effects of thermal conduction take place. The formulation of the model leads to a system of differential equations of the velocity and temperature and it will be used for the numerical solutions, and for the analytical solutions we have a fourth order differential equation. The differential equations in both cases are linear but with exponential coefficients. Approximate and exact solutions of the mathematical model are studied, in low, middle, and high chromospheres, both numerically and analytically. The analysis of both studies is in complete agreement with previously observed and reported results and conclusions about the heating process of the chromosphere. The results of the numerical solutions are discussed in connection with the heating mechanisms of the three regions of the chromosphere. Finally, the case where the values of thermal conduction, viscosity and magnetic field are arbitrary is considered. q 2003 Published by Elsevier B.V. Keywords: Magnetoacoustic waves; Transistion regions; Alfven waves; Chromosphere; Reflection coefficient
1. Introduction * Corresponding author. E-mail addresses: [email protected] (A.F. Jalbout), [email protected] (H. Alkahby), [email protected] (A. Talmadge), [email protected] (P. Frempong-Mireku). 0166-1280/03/$ - see front matter q 2003 Published by Elsevier B.V. doi:10.1016/S0166-1280(03)00377-4
Traditionally, the sun and its activities have been divided into two different classes: quiet and active. The quiet sun is viewed as a static, spherically symmetric ball of plasma, whose properties depend
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mainly, to first approximation, on radial distance from the core (center) and whose magnetic field is negligible. On the other hand, the active sun consists of transient phenomena, such as sun spot and flares which are superimposed on the quiet sun and most of which owe their existence and activities to the presence and structure of the magnetic field. For the understanding of the activities of the quiet and active suns, their mechanism and influences on the universe, many approaches were suggested. One of these approaches is the study of the generation and propagation of the magnetoacoustic waves and their effect on the heating process of the solar atmosphere. More importantly, the existence and structure of the magnetic field in the sun and its influences on the activities of the solar atmosphere must be understood in order to have the basic facts about the active sun. It is well known that the solar corona is extremely hot, typical temperatures are 106 K compared with 5 £ 103 K in the photosphere. Consequently, thermal energy must be continually supplied to maintain this temperature against radiative cooling. As a result, these two questions must be answered: how is the energy supplied to the corona, and how is it dissipated? To answer these questions, many mathematical models and dissipative mechanisms are suggested see Alkahby [1 – 3], Alkahby and Yanowitch [5], Yanowitch [10 – 12], Lyons and Yanowitch [8]. In this paper we study numerically the heating process of the chromosphere by the energy that is dissipated from upward and downward propagating magnetoacoustic waves in an isothermal atmosphere. The combined effects of viscosity, thermal conduction and a uniform horizontal magnetic field are considered in two cases. In the first case, the effects of thermal conduction dominates the oscillatory process. As a result, the atmosphere of the chromosphere can be divided into three distinct regions low, middle, and high. In the low chromosphere, near the photosphere, the motion is adiabatic and the solution of the problem that satisfies the prescribed boundary conditions. This solution can be written as a linear combination of an upward and a downward propagating wave with equal and adiabatic wavelengths. In the high chromosphere, the motion is an isothermal one because the increase of the thermal diffusivity with altitude transforms
the motion from the adiabatic form below the transition region to an isothermal one above it. Once again, in the high chromosphere, the solution can be written as a linear combination of an upward and a downward propagating wave with equal and isothermal wavelengths. The difference in wavelengths will be accounted for by reflection. The low and high chromospheres are connected by the middle chromosphere, which is a semitransparent transition region in this region wave modification takes place and part of the energy is reflected downward, part is dissipated (this may explain the sharp increase of the temperature), and the other part transmitted upward. In the high chromosphere, the transmitted waves will be reflected downward by the second transition region that is created by the combined effects of the viscosity and magnetic field. The second transition region connects the high chromosphere and the corona. In the corona, the solutions that satisfy the prescribed boundary conditions decay with altitude representing the dissipation of the energy in the atmosphere after being emitting from the sun. The middle chromosphere and corona are connected by a transition region. This region’s properties depend on the relative strength of the dynamic viscosity with respect to Alfven speed. When the viscosity dominates the activities, the transition region acts like a reflecting and absorbing layer. On the other hand if the magnetic field effect is dominant, the upper transition behaves like a totally reflecting barrier. In fact the physical nature of these transition regions is not fully understood because of their complicated magnetic structure. In the transition regions, the multiple wave reflection, energy absorption and dissipation will contribute greatly to the heating process of the chromosphere. This mathematical model leads to a fourth order singular perturbation problem which is solved in two cases. In the first case thermal conduction dominates the motion and the problem is solved by matching inner and outer solutions in an overlapping domain. In the second case the combined effects of the viscosity and magnetic field dominate the motion and the differential equation is solved in the same method. Also it is shown that when the combined effects of the viscosity and magnetic field are dominant the effect of thermal conduction can be eliminated. This result is expected
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because of the physical nature of the transition region that is formed by them. For arbitrary values of the kinematic viscosity, thermal diffusivity, and Alfven speed the solution is similar to that of the second case. In all cases the reflection coefficient is determined and the conclusions are discussed in connection with the heating process of the chromosphere and the numerical and analytical results are in complete agreement. Finally, it has to be indicated that 1 is obtained from Priest [9] and added to the body of the paper for completeness and to show the value of the model. It is clear that the solution of the mathematical model represents exactly what was predicted in 1 about the temperature distribution in the atmosphere of the chromosphere.
2. Sources of energy in the sun and physical structure of the chromosphere In the core of the sun, He nuclei are being produced from H nuclei, mainly by proton– proton cycle but partly by the C –N cycle. At the end of each cycle, which lasts 107 years. Groups of four protons 1H have been converted into nuclei (4He), according to the following equation 41 H ! 4 He þ 2e þ 2n þ 26:7 MeV
41
chromosphere to the high chromosphere. More specifically, the temperature increases rapidly in the boundary between the middle and high chromosphere and at the transition region between the high chromosphere and the corona. In this model we assume that the heating source is the dissipation of the energy from magnetoacoustic waves, which were generated in different regions in the sun. More specifically, waves generated in the photosphere of the sun and modified greatly by the effect of the magnetic field of the sun. It will be seen that, when thermal conduction dominates the oscillatory process, a semitransparent reflecting layer is created. Consequently, part of the energy will be transmitted upward and the other part will be reflected downward. On the other hand the magnetic field produces a totally reflecting layer if the viscous effect on the motion in the second region transition is negligible. Also the viscosity forms an absorbing and reflecting layer if the effect of the magnetic field in the second transition layer is negligible. In addition, resonance may occur in the second transition region, when the wavelength is matched with the strength of the magnetic field, and this may contribute for the understanding of the nature of the activities in the active sun.
ð2:1Þ 3. Mathematical formulation of the problem
It is clear that the net charge on each side of this equation is þ 4 because each proton has a charge of þ 1 and each nucleus has a charge of þ 2. It is known that each He nucleus is smaller in mass by 3% than the sum of the masses of the original protons with this difference being converted to energy. According to Einstein’s equation E ¼ mc2 ; each kilogram of mass is equivalent to 9 £ 106 J. In the above equation the energy is released in the form of g-rays (26.7 MeV) and two neutrinos (0.5 MeV) denoted by n: This is one known source of energy of the solar atmosphere and the second suggested source is the energy that comes from the dissipation of different types of waves generated in the sun, which is the subject of this paper. Moreover, as observed and modeled the chromosphere can be divided, according to its temperature, into three chromospheres, low chromosphere, middle chromosphere and high chromosphere (see Priest [9]). Also the temperature rises monotonically from the low
Suppose an isothermal atmosphere, which is viscous and thermally conducting, occupies the upper half-space z . 0: It is assumed that the gas is under the influence of a uniform horizontal magnetic field and it has an infinite electrical conductivity. We will investigate small oscillations about equilibrium which depend only on the time and on the vertical coordinate z: Let P; r; W; B; and T be the perturbations in the pressure, density, vertical velocity, horizontal magnetic field strength and temperature. Let P0 ; r0 ; W0 ; B0 ; and T0 be their corresponding equilibrium quantities. Equilibrium pressure, temperature, and density satisfy the gas law P0 ¼ Rr0 T0 and the hydrostatic equation P00 þ gr0 ¼ 0; here g is the gravitational acceleration and R is the gas constant, and the prime denotes the derivative with respect to z. Consequently, the pressure and density can be written
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in the following form: z P0 ðzÞ ¼ P0 ð0Þexp 2 ; H z r0 ðzÞ ¼ r0 ð0Þexp 2 H
the temperature: ð3:1Þ
ð3:2Þ
! gv2 WðzÞ þ gh expðzÞD2 WðzÞ D 2Dþ 4 2
þ igðD 2 1ÞTðzÞ ¼ 0
ð3:8Þ
and
where H is the density scale height defined by H ¼ RT0 =g: Moreover, the linearized equations of motion (conservation of momentum and mass, magnetic induction and heat equations and the gas law) can be written as: B0 4 r0 W t þ Pz þ rg þ ð3:3Þ B ¼ mWzz 3 4p z
rt þ ðr0 WÞz ¼ 0
ð3:4Þ
Bt þ B0 W z ¼ 0
ð3:5Þ
r0 ðcV Tt þ gHWz Þ ¼ kTzz
ð3:6Þ
P ¼ Rðr0 T þ T0 rÞ
ð3:7Þ
Here cV is the specific heat at constant volume, m is the dynamic viscosity coefficient, which is assumed to be constant, and k is the thermal conductivity coefficient. The subscripts z and t denote the differentiation of the independent variables z and t, respectively. To simplify the problem formulation we will consider only solutions which are harmonic in time t, i.e. Wðz; tÞ ¼ W p ðzÞexpð2istÞ and Tðz; tÞ ¼ T p ðzÞ expð2istÞ where s denotes the frequency of the wave. It is more convenient to rewrite the differential equations of motion in dimensionless form: zp ¼ z=H; sa ¼ c=2H is the adiabatic cut off frequency of the wave, where c2 ¼ gRT0 ¼ ggH is the adiabatic sound speed and g denotes the ratio of the specific heats c2A ¼ B20 =4pr0 ð0Þ; dp ¼ c2A =c2 ; h ¼ dp 2 isp mp ; W p ¼ W=c; mp ¼ 2m=3r0 cH; sp ¼ s=sa ; tp ¼ tsa ; T p ¼ T=2gT0 ; and kp ¼ 2k=cV cH r0 ð0Þ: The star can be omitted, since all variables are written in dimensionless form from this point forward. Substituting all these quantities in the differential equations (3.3) – (3.7) and eliminating the pressure and density we obtain the following system of differential equations for the velocity and
ðg 2 1ÞDWðzÞ ¼ gðivÞTðzÞ þ gk expðzÞD2 TðzÞ; ð3:9Þ where D denotes the derivative of the indicated variable with respect to z. In addition, we introduce the dimensionless parameter Pr ¼ cP m=k which denotes the Prandtl number and cP is the specific heat at constant pressure. Letting a ¼ h=k it is clear that Prandtl number measures the relative strength of the viscosity with respect to the thermal conduction and we have: " ! ! v2 gv2 2 2 2 D 2Dþ 2 k expðzÞD D 2 D þ 4 4 2 gsak expðzÞDðD þ 1Þ 2 igaðk expðzÞÞ2 D2 # ðD þ 1ÞðD þ 2Þ TðzÞ ¼ 0:
ð3:10Þ
3.1. Boundary conditions To obtain a unique Solution for the differential equation (3.10), physically relevant conditions must be imposed. The first is the dissipation condition, which requires that the rate of change of the energy dissipation in an infinite column of fluid of unit crosssection to be finite. Since the dissipation function depends on the squares of the velocity gradients, the dissipation condition is equivalent to ð1 m lWz l2 dz , 1 ð3:11Þ 0
The second one is the entropy condition, which is determined by the equation for the rate of the entropy, from which it follows that ð1 k lTz l2 dz , 1 ð3:12Þ 0
Boundary conditions are also required at z ¼ 0; and we shall adopt the lower boundary condition: in a fixed interval 0 , z , z0 ; the solution of the differential equation should approach some solution of
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the limiting differential equation as k ! 0 and m ! 0; i.e. " # s2 2 TðzÞ ¼ 0 ð3:13Þ D 2Dþ 4 For numerical computations, which are the aim of this paper, we assume as lower boundary conditions that Tð0Þ ¼ 1
and Wð0Þ ¼ 1
ð3:14Þ
The upper boundary conditions (3.11) and (3.12) and the lower boundary conditions (3.13) and (3.14) are necessary and sufficient, to ensure unique analytical and numerical solutions. The upper boundary conditions are only applicable if the atmosphere is viscous and thermally conducting. If the atmosphere is not viscous the magnetic energy condition will be used as an upper boundary condition to ensure a unique solution. If the wave propagation is affected only by thermal properties, or if the effects of all the dimensionless parameters vanish, a radiation condition must be imposed to ensure a unique solution. Finally, the wave speed and temperature have the same Solution behavior in all regions of the atmosphere of the chromosphere.
4. Behavior of the solution in the region of the low chromosphere (for small values of lal) In this section we study the behavior of the solutions of the differential equation (3.10), in the region of the low chromosphere for small values of lal; (when thermal conduction effects dominate the motion), subjected to the upper (3.11) and (3.12) and lower boundary conditions (3.13) and (3.14). As we stated in the formulation of the problem the parameters d; m; and k are dimensionless, sufficiently small, and proportional to the Alfven speed, kinematic viscosity and thermal conductivity at z ¼ 0: To understand the physical nature of the behavior of the solution of the differential equation (3.10) we should investigate the relationships among these dimensionless parameters because the nature of the solution, the representation of the three regions of the chromosphere, the magnitude of the reflection coefficients, the nature of the transition regions, and the properties of the reflecting layer depend mainly on relationships
43
among these parameters. The parameter lal can be written in the following form d 2 im h d 2 i m r0 ¼ lal ¼ ¼ k k k r0
:
ð4:1Þ
Moreover, we call ld 2 im=r0 l the magnetoviscous diffusivity and lk=r0 l thermal diffusivity. Thus lal measures the relative strength of the magnetoviscous diffusivity with respect to thermal diffusivity. In addition, for small lal the atmosphere of the chromosphere can be divided into three different regions: low, middle, and high. The low and high chromospheres are connected by a semitransparent reflecting layer, while the high chromosphere and corona are joined by a transition region whose properties depend mainly on the relative strength of the kinematic viscosity with respect to Alfven speed. It has been indicated that the chromosphere is divided into three distinct regions according to its temperature (Priest [9]). The differential equation (3.10) is solved asymptotically by matching inner and outer approximation in an overlapping domain. Similar procedure was used in Alkahby [3] and the computations need not be repeated but we indicate only results for completeness. In the region of low chromosphere, the solution of the differential equation (3.10), which satisfies the prescribed boundary conditions can written in the following form:
1 1 TðzÞ , C exp þ ib z þ RL exp 2 ib z ; 2 2 ð4:2Þ where Cis a constant and RL denotes the reflection coefficient, which is defined by RL ¼ expð2pbÞ{cosð2i½u2 2 bp logðgskÞÞ þ i sinð2i½u2 2 bp logðgskÞÞ}Ra
ð4:3Þ
with
b¼
ffi 1 pffiffiffiffiffiffiffiffi s 2 2 1; 2
u1 ¼ argðhÞ;
bp ¼
qffiffiffiffiffiffiffiffiffiffiffi 1 gs2 2 1; 2
ð4:4Þ
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2
6G u2 ¼ arg6 4
2
3 1 2 ib Gð2ibÞGðibp 2 ibÞ 7 2 7; 5 Gðibp þ ibÞ
2 3 1 p 2 p þ i 2 b G ð2i b Þ 6 7 2 7 u3 ¼ arg6 4 Gðibp þ ibÞGðibp 2 ibÞ 5;
Ra ¼
ð4:5Þ
ð4:6Þ
c1 2 c2 expðFÞ ; c3 2 c4 expðFÞ
c1 ¼ expð2pbp Þ 2 expð2pbÞ;
ð4:7Þ
c1 þ c2 ¼ c3 þ c4 ¼ 2 sinhð2pbp Þ c3 2 c1 ¼ c4 2 c2 ¼ 2 sinhð2pbÞ; p
ð4:8Þ
p
and F ¼ b ðp þ 2u1 Þ þ 2iðu3 2 b loglalÞ: Here b and bp are the dimensionless wave numbers corresponding to the adiabatic and Newtonian sound speed, respectively. Eq. (4.2) represents the behavior of the solution in the low chromosphere. It is a linear combination of an upward and a downward propagating wave with equal wavelengths lL ¼ 2p=b: In the low chromosphere the reflection process is more complicated because of the existence of two reflecting layers one produced by thermal conduction and the second one created by the combined effect of the viscosity and magnetic field. The magnitude of the reflection coefficient depends on the relative strength of thermal diffusivity with respect of magnetoviscous diffusivity and it will be seen in Section 5 that if the magnetoviscous diffusivity dominates the oscillatory process there will one reflecting layer attributable to the combined effects of the viscosity and magnetic field. Also the magnitude of the reflection coefficient depends on the relative locations of the reflecting layers and its value less is than expð2pbÞ: In fact the magnitude of the reflection coefficient is periodic, which is clear from its definition in Eq. (4.3). More conclusions about the nature of the solution in the low chromosphere shall be summarized in Section 7.
5. Behavior of the solution in the region of the middle and high chromosphere (for small values of lal) Before discussing the nature of the solutions in the high chromosphere, it is desirable and important to explain the behavior of the solution in the middle chromosphere. The region of middle chromosphere is represented by a transition region that connects low and high chromospheres in which heat dissipation and wave modification take place. This region is semitransparent, where part of the energy is transmitted upward and part is reflected downward. In this region temperature is increased substantially because the waves dump part of their energies. Propagation, transmission, and reflection will continue until all the energy of the waves is dissipated. This analysis is consistent with widely reported and observed results (Priest [9] and the references therein). In the region of high chromosphere, the solution of the differential equation (3.10), that satisfies the prescribed boundary condition, can be written in the following form:
1 1 p TðzÞ , C exp þ ib z þ Rh exp 2 ib z ; 2 2 ð5:1Þ where Rh denotes the reflection coefficient in the high chromosphere which is given by: Rh ¼ exp½2bp u1 þ 2iðu 2 bp loglghlÞ; and
u ¼ 2 arg
1 3 2 ibp Gð2ibp ÞG2 2 ibp : 2 2
ð5:2Þ
ð5:3Þ
It is clear that Eq. (4.2) shows that the Solution of differential equation (3.10) that satisfies the prescribed boundary conditions can be written as a linear combination of an upward and a downward propagating wave with equal wave number bp : Also the magnitude of the reflection coefficient lRh l ¼ expð2bp u1 Þ where u1 ¼ argðhÞ: As a result, if the viscosity dominates the oscillatory process in the region of high chromosphere, h ! ism and argðhÞ ! 2p=2; which leads to lRh l ! expð2pbp Þ: On the contrary, if the magnetic field dominates the motion in the high chromosphere h ! d and argðhÞ ! 0; consequently lRh l ! 1: It follows that the viscosity creates an absorbing and reflecting layer
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and the magnetic field produces a totally reflecting barrier. This result is expected for the magnetic field because of its dissipationless nature. Such situations occur in the region of high chromosphere, especially in sunspots where the magnetic field is very strong. In addition, when the magnetic field dominates the oscillatory process, resonance may occur in the region of high chromosphere for infinitely many values of the magnetic field and infinitely many values of the frequency of the wave if the magnetic field strength is matched with the wavelength of the wave (see Alkahby [1 –3], Alkahby and Mahrous [4,6]). At the resonance frequencies the kinetic and magnetic energies shall increase to very large values and this may explain the monotonic increase in the temperature in the high chromosphere as well as some activities in sunspots. We hope this will stimulate more interest in the study of the structure of the sun’s magnetic field and its influences on the activities of the solar atmosphere in general and its heating mechanisms in particular.
6. Behavior of the solution for large values of lal In this section we investigate the behavior of the solution of the differential equation (3.10), when the motion is dominated by the combined effects of the viscosity and magnetic field. In this case the differential equation is similar to the differential equation given in Alkahby, Mahrous and Vatsala [7] (see Eq. (12)) but the physical conclusions are different. The Solution that satisfies the prescribed boundary conditions can be obtained in the same way and the details of the computations are omitted. Consequently, the solution of the differential equation (3.10) that satisfies the prescribed boundary conditions can be written like: where denotes the reflection coefficient and defined by:
1 1 TðzÞ , C exp þ ib z þ Rg exp 2 ib z ; 2 2 ð6:1Þ where Rg denotes the reflection coefficient defined by Rg ¼ F exp½2bu1 þ 2iðu 2 loglghlÞ;
ð6:2Þ
F ¼ 2 arg½Gð2ibÞG2 ðbÞGðibp 2 ibÞGðibp þ ibÞ
45
ð6:3Þ
Once again the solution of the differential equation (3.10) can be written as a linear combination of an upward and a downward propagating wave. In this case solutions are influenced by the effect of thermal diffusivity because the reflecting barrier either a totally reflecting or reflecting and absorbing one. Consequently, the waves will reflected downward before the effect of thermal diffusivity takes place. In fact the problem can be approximated by letting the effect of thermal conduction be negligible ðk ¼ 0Þ in the differential equation (3.10). Once again the resonance will occur for infinitely many values of the frequency and the magnetic field (see Alkahby [1 – 3]. More conclusions will be summarized in Section 7.
7. Computing scheme and results of computations The results of the previous sections are asymptotically valid as lal ! 0: In order to determine the range of lal in which they are reasonably accurate, the boundary-value problem (defined by the system of Eqs. (3.8) and (3.9)) was solved numerically for several values of lal and b: For the numerical integration it is convenient to deal with the system (3.8) and (3.9) with the prescribed boundary conditions at z ¼ 0: The upper boundary conditions are replaced by these two conditions: DW ! 0; DT ! 0 as z ! 1: Using centered differences, we can replace the system of differential equations (3.8) and (3.9) by a set of difference equations that will be solved numerically, An Fnþ1 þ Bn Fn þ Cn Fn21 ¼ 0
ð7:1Þ
where Fn is the column vector with components iTðzn Þ and Wðzn Þ and An ; Bn ; Cn are 2 £ 2 matrices. The system was solved in an interval 0 # z # L sufficiently large to allow T and W to reach their limiting values, and the boundary condition DW ! 0; DT ! 0 as z ! 1, was replaced by FN ¼ FN21 ; where N is the index of the point z ¼ L: The problem was solved by the standard method in which the Fn are computed from a linear equation by backward integration. On the other hand,
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the coefficient matrices
an ¼ 2½Bn þ Cn an21 21 An
ð7:2Þ
and the vectors
bn ¼ 2½Bn þ Cn an21 21 Cn bn21
ð7:3Þ
are computed by forward integration. Moreover, let M and m denote the maximum and minimum pffiffiffiffiffiffi values of the oscillation amplitude and d ¼ M=m; then the magnitude of the reflection coefficient can be computed from the formula lRC l ¼
d21 dþ1
ð7:4Þ
where RC denotes any of the reflection coefficients defined in the body of the paper. The computations were done different values of the dimensionless parameters d; m; and k: 7.1. Observations and conclusions (1) When the effect of thermal conduction dominates the oscillatory process, the atmosphere of the chromosphere can be divided into three regions with different temperatures:the low, middle, and high chromosphere. The low and the high regions are connected by the middle chromosphere, which is a semitransparent transition region, and in this region wave modification takes place. Here part of the energy propagated upward, part is dissipated, and another part is reflected downward. In addition, the upward propagating waves have different wavelength because the exponential increase of thermal diffusivity changes the motion from the adiabatic form, below the reflecting layer to an isothermal form above it (2). The change in the wavelength accounts for the reflection process and the magnitude of the reflection coefficient is 4.3. Moreover, the solution that satisfies the prescribed boundary conditions can write as a linear combination of an upward and a downward propagating waves with equal and adiabatic wavelengths (4.2). (2) The high chromosphere and corona are connected by a second transition region, its properties depend on the relative strength of the viscosity with respect to the magnetic field. When the magnetic field dominates the motion, it will
creates a totally reflecting transition region and the magnitude of the reflection coefficient is one (5). The dominance of the viscosity effect produces an absorbing and reflecting transition region, the magnitude of the reflection coefficient, in the second region is given in Eq. (5.2). (3) The upward propagating waves in the high chromosphere will be reflected downward by the second reflecting layer. In this case the magnitude of the reflection coefficient depends on the relative strength of the viscosity with respect to the magnetic field as indicated in the previous observation. The energy absorption and multiple reflection will continue until all the energy of the dissipated and this may contribute to the heating process of the chromospheres. (4) It is clear from the above remarks that the reflection process in the lower region is very complicated and the magnitude of the reflection coefficient depends mainly of relative strength of the competing factors: viscosity, thermal conduction, and magnetic field strength. In fact, the relative location of the transition region is the main factor effecting the magnitude of the reflection coefficient. For example, if the combined effects of the viscosity and magnetic field are dominant, the effect of thermal conduction will be eliminated, because the upward propagating waves will be reflected downward before the effect of thermal conduction takes. Moreover, the differential equation (3.10) can be approximated by setting and we obtain a second order differential equation that can be transformed to a hypergeometric differential equation (see Alkahby [1 –3] and Yanowitch [10 – 12]). (5) When the magnetic field dominates the motion in the high chromosphere, the resonance may occur for infinitely many values of the magnetic field and frequency of the wave if the magnetic field strength is matched with the wavelength of the wave, see Alkahby [1 –3], Alkahby and Yanowitch [5]. In this case, the magnetic and kinetic energies of the waves will be increased greatly and this may contribute to the heating process of solar atmosphere in general and the chromosphere in particular. In addition, resonance occurrence may explain some of the activities on the surface of the active sun, for example: sunspots.
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(6) The increase of the temperature in the middle chromosphere is due to the dissipation of the magnetic and kinetic energies in this region and to the increase of thermal diffusivity with altitude. The rapid increase of the temperature in the transition region between the high chromosphere and the corona is due to the heat ejection from the surface of the sun toward the corona. This phenomena usually takes place in region where the structure of the magnetic field is very complicated. The regions where these activities are taking place, are the sunspots because of the strength and complicated nature of their magnetic field. As soon as the heat is ejected from the high chromosphere to the corona, a strong current of particles is coming from the corona toward the surface because of the rapid decrease in the pressure.
References [1] H. Alkahby, On the coronal heating mechanism by the resonant absorption of Alfven waves, Int. J. Math. Math. Sci. 16 (4) (1993) 811–816. [2] H. Alkahby, On the heating of the solar corona by the resonant absorption of Alfven waves, Appl. Math. Lett. 6 (6) (1993) 59–64.
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[3] H. Alkahby, Reflection and dissipation on hydromagnetic waves in a viscous and thermally conducting isothermal atmosphere, Geophys. Astrophys. Fluid Dynam. 72 (1993) 197 –207. [4] H. Alkahby, M. Mahrous, On the reflection and dissipation of oblique Alfven waves in an isothermal atmosphere, Int. J. Math. Math. Sci. 22 (1) (1999) 161–169. [5] H. Alkahby, M. Yanowitch, The effect of Newtonian cooling on the reflection of vertically propagating acoustic waves in an isothermal atmosphere, Wave Motion 11 (1989) 419 –426. [6] M. Mahrous, H. Alkahby, F. Jalbout, Numerical studies of vertically propagating acoustic and magnetoacoustic waves in an isothermal atmosphere (i), Proc. Fifteenth Annu. Conf. Appl. Math. (1999) 1 –13. [7] M. Mahrous, H. Alkahby, A.S. Vatsala, Reflection and dissipation of acoustic waves in an isothermal atmosphere, Proc Dynam. Syst. Applicat. 2 (1996) 45 –52. [8] P. Lyons, M. Yanowitch, Vertical oscillations in a viscous and thermal conducting isothermal atmosphere, J. Fluid Mech. 66 (1974) 273 –288. [9] E.R. Priest, Solar Magnetohydrodynamics, Reidel, New York, 1984. [10] M. Yanowitch, Effect of viscosity on vertical oscillations of an isothermal atmosphere, Can. J. Phys. 45 (1967) 2003–2008. [11] M. Yanowitch, Effect of viscosity on gravity waves and upper boundary conditions, J. Fluid Dynam. 29 (1967) 209 –231. [12] M. Yanowitch, Vertically propagating hydromagnetic waves in an isothermal atmosphere with a horizontal magnetic field, Wave Motion 1 (1979) 123–125.
Numerical studies of vertically propagating acoustic and magnetoacoustic waves in an isothermal atmosphere (III) A.F. Jalbouta,*, H. Alkahbyb, A. Talmadgeb, P. Frempong-Mirekub a
Department of Chemistry, University New Orleans, New Orleans, LA 70112, USA b Department of Mathematics, Dillard University, New Orleans, LA 70122, USA Received 11 February 2002; accepted 5 May 2003
Abstract In this paper we investigate, numerically, the generation and propagation of acoustic and magnetoacoustic waves and their roles in the heating process of the chromosphere. The combined effects of viscosity, thermal conduction and a uniform horizontal magnetic field on an upward and a downward propagating wave in an isothermal atmosphere are considered. It is shown that when the heating mechanisms are dominated by the effects of thermal conduction. The chromosphere atmosphere can be divided into three distinct regions, low, middle and high. The middle region acts like a semitransparent transition layer and it is produced by thermal conduction and connects middle and high chromosphere. In the transition region part of the energy transmitted upward, part is dissipated and the other part is reflected downward. Moreover, viscosity creates an absorbing and reflecting layer and the magnetic field forms a totally reflecting barrier because of its dissipationless nature. When the combined effects of the viscosity and magnetic field dominated the oscillatory process, thermal conduction can be eliminated because the solution decays exponentially with altitude before the effects of thermal conduction take place. The formulation of the model leads to a system of differential equations of the velocity and temperature and it will be used for the numerical solutions, and for the analytical solutions we have a fourth order differential equation. The differential equations in both cases are linear but with exponential coefficients. Approximate and exact solutions of the mathematical model are studied, in low, middle, and high chromospheres, both numerically and analytically. The analysis of both studies is in complete agreement with previously observed and reported results and conclusions about the heating process of the chromosphere. The results of the numerical solutions are discussed in connection with the heating mechanisms of the three regions of the chromosphere. Finally, the case where the values of thermal conduction, viscosity and magnetic field are arbitrary is considered. q 2003 Published by Elsevier B.V. Keywords: Magnetoacoustic waves; Transistion regions; Alfven waves; Chromosphere; Reflection coefficient
1. Introduction * Corresponding author. E-mail addresses: [email protected] (A.F. Jalbout), [email protected] (H. Alkahby), [email protected] (A. Talmadge), [email protected] (P. Frempong-Mireku). 0166-1280/03/$ - see front matter q 2003 Published by Elsevier B.V. doi:10.1016/S0166-1280(03)00377-4
Traditionally, the sun and its activities have been divided into two different classes: quiet and active. The quiet sun is viewed as a static, spherically symmetric ball of plasma, whose properties depend
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mainly, to first approximation, on radial distance from the core (center) and whose magnetic field is negligible. On the other hand, the active sun consists of transient phenomena, such as sun spot and flares which are superimposed on the quiet sun and most of which owe their existence and activities to the presence and structure of the magnetic field. For the understanding of the activities of the quiet and active suns, their mechanism and influences on the universe, many approaches were suggested. One of these approaches is the study of the generation and propagation of the magnetoacoustic waves and their effect on the heating process of the solar atmosphere. More importantly, the existence and structure of the magnetic field in the sun and its influences on the activities of the solar atmosphere must be understood in order to have the basic facts about the active sun. It is well known that the solar corona is extremely hot, typical temperatures are 106 K compared with 5 £ 103 K in the photosphere. Consequently, thermal energy must be continually supplied to maintain this temperature against radiative cooling. As a result, these two questions must be answered: how is the energy supplied to the corona, and how is it dissipated? To answer these questions, many mathematical models and dissipative mechanisms are suggested see Alkahby [1 – 3], Alkahby and Yanowitch [5], Yanowitch [10 – 12], Lyons and Yanowitch [8]. In this paper we study numerically the heating process of the chromosphere by the energy that is dissipated from upward and downward propagating magnetoacoustic waves in an isothermal atmosphere. The combined effects of viscosity, thermal conduction and a uniform horizontal magnetic field are considered in two cases. In the first case, the effects of thermal conduction dominates the oscillatory process. As a result, the atmosphere of the chromosphere can be divided into three distinct regions low, middle, and high. In the low chromosphere, near the photosphere, the motion is adiabatic and the solution of the problem that satisfies the prescribed boundary conditions. This solution can be written as a linear combination of an upward and a downward propagating wave with equal and adiabatic wavelengths. In the high chromosphere, the motion is an isothermal one because the increase of the thermal diffusivity with altitude transforms
the motion from the adiabatic form below the transition region to an isothermal one above it. Once again, in the high chromosphere, the solution can be written as a linear combination of an upward and a downward propagating wave with equal and isothermal wavelengths. The difference in wavelengths will be accounted for by reflection. The low and high chromospheres are connected by the middle chromosphere, which is a semitransparent transition region in this region wave modification takes place and part of the energy is reflected downward, part is dissipated (this may explain the sharp increase of the temperature), and the other part transmitted upward. In the high chromosphere, the transmitted waves will be reflected downward by the second transition region that is created by the combined effects of the viscosity and magnetic field. The second transition region connects the high chromosphere and the corona. In the corona, the solutions that satisfy the prescribed boundary conditions decay with altitude representing the dissipation of the energy in the atmosphere after being emitting from the sun. The middle chromosphere and corona are connected by a transition region. This region’s properties depend on the relative strength of the dynamic viscosity with respect to Alfven speed. When the viscosity dominates the activities, the transition region acts like a reflecting and absorbing layer. On the other hand if the magnetic field effect is dominant, the upper transition behaves like a totally reflecting barrier. In fact the physical nature of these transition regions is not fully understood because of their complicated magnetic structure. In the transition regions, the multiple wave reflection, energy absorption and dissipation will contribute greatly to the heating process of the chromosphere. This mathematical model leads to a fourth order singular perturbation problem which is solved in two cases. In the first case thermal conduction dominates the motion and the problem is solved by matching inner and outer solutions in an overlapping domain. In the second case the combined effects of the viscosity and magnetic field dominate the motion and the differential equation is solved in the same method. Also it is shown that when the combined effects of the viscosity and magnetic field are dominant the effect of thermal conduction can be eliminated. This result is expected
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because of the physical nature of the transition region that is formed by them. For arbitrary values of the kinematic viscosity, thermal diffusivity, and Alfven speed the solution is similar to that of the second case. In all cases the reflection coefficient is determined and the conclusions are discussed in connection with the heating process of the chromosphere and the numerical and analytical results are in complete agreement. Finally, it has to be indicated that 1 is obtained from Priest [9] and added to the body of the paper for completeness and to show the value of the model. It is clear that the solution of the mathematical model represents exactly what was predicted in 1 about the temperature distribution in the atmosphere of the chromosphere.
2. Sources of energy in the sun and physical structure of the chromosphere In the core of the sun, He nuclei are being produced from H nuclei, mainly by proton– proton cycle but partly by the C –N cycle. At the end of each cycle, which lasts 107 years. Groups of four protons 1H have been converted into nuclei (4He), according to the following equation 41 H ! 4 He þ 2e þ 2n þ 26:7 MeV
41
chromosphere to the high chromosphere. More specifically, the temperature increases rapidly in the boundary between the middle and high chromosphere and at the transition region between the high chromosphere and the corona. In this model we assume that the heating source is the dissipation of the energy from magnetoacoustic waves, which were generated in different regions in the sun. More specifically, waves generated in the photosphere of the sun and modified greatly by the effect of the magnetic field of the sun. It will be seen that, when thermal conduction dominates the oscillatory process, a semitransparent reflecting layer is created. Consequently, part of the energy will be transmitted upward and the other part will be reflected downward. On the other hand the magnetic field produces a totally reflecting layer if the viscous effect on the motion in the second region transition is negligible. Also the viscosity forms an absorbing and reflecting layer if the effect of the magnetic field in the second transition layer is negligible. In addition, resonance may occur in the second transition region, when the wavelength is matched with the strength of the magnetic field, and this may contribute for the understanding of the nature of the activities in the active sun.
ð2:1Þ 3. Mathematical formulation of the problem
It is clear that the net charge on each side of this equation is þ 4 because each proton has a charge of þ 1 and each nucleus has a charge of þ 2. It is known that each He nucleus is smaller in mass by 3% than the sum of the masses of the original protons with this difference being converted to energy. According to Einstein’s equation E ¼ mc2 ; each kilogram of mass is equivalent to 9 £ 106 J. In the above equation the energy is released in the form of g-rays (26.7 MeV) and two neutrinos (0.5 MeV) denoted by n: This is one known source of energy of the solar atmosphere and the second suggested source is the energy that comes from the dissipation of different types of waves generated in the sun, which is the subject of this paper. Moreover, as observed and modeled the chromosphere can be divided, according to its temperature, into three chromospheres, low chromosphere, middle chromosphere and high chromosphere (see Priest [9]). Also the temperature rises monotonically from the low
Suppose an isothermal atmosphere, which is viscous and thermally conducting, occupies the upper half-space z . 0: It is assumed that the gas is under the influence of a uniform horizontal magnetic field and it has an infinite electrical conductivity. We will investigate small oscillations about equilibrium which depend only on the time and on the vertical coordinate z: Let P; r; W; B; and T be the perturbations in the pressure, density, vertical velocity, horizontal magnetic field strength and temperature. Let P0 ; r0 ; W0 ; B0 ; and T0 be their corresponding equilibrium quantities. Equilibrium pressure, temperature, and density satisfy the gas law P0 ¼ Rr0 T0 and the hydrostatic equation P00 þ gr0 ¼ 0; here g is the gravitational acceleration and R is the gas constant, and the prime denotes the derivative with respect to z. Consequently, the pressure and density can be written
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in the following form: z P0 ðzÞ ¼ P0 ð0Þexp 2 ; H z r0 ðzÞ ¼ r0 ð0Þexp 2 H
the temperature: ð3:1Þ
ð3:2Þ
! gv2 WðzÞ þ gh expðzÞD2 WðzÞ D 2Dþ 4 2
þ igðD 2 1ÞTðzÞ ¼ 0
ð3:8Þ
and
where H is the density scale height defined by H ¼ RT0 =g: Moreover, the linearized equations of motion (conservation of momentum and mass, magnetic induction and heat equations and the gas law) can be written as: B0 4 r0 W t þ Pz þ rg þ ð3:3Þ B ¼ mWzz 3 4p z
rt þ ðr0 WÞz ¼ 0
ð3:4Þ
Bt þ B0 W z ¼ 0
ð3:5Þ
r0 ðcV Tt þ gHWz Þ ¼ kTzz
ð3:6Þ
P ¼ Rðr0 T þ T0 rÞ
ð3:7Þ
Here cV is the specific heat at constant volume, m is the dynamic viscosity coefficient, which is assumed to be constant, and k is the thermal conductivity coefficient. The subscripts z and t denote the differentiation of the independent variables z and t, respectively. To simplify the problem formulation we will consider only solutions which are harmonic in time t, i.e. Wðz; tÞ ¼ W p ðzÞexpð2istÞ and Tðz; tÞ ¼ T p ðzÞ expð2istÞ where s denotes the frequency of the wave. It is more convenient to rewrite the differential equations of motion in dimensionless form: zp ¼ z=H; sa ¼ c=2H is the adiabatic cut off frequency of the wave, where c2 ¼ gRT0 ¼ ggH is the adiabatic sound speed and g denotes the ratio of the specific heats c2A ¼ B20 =4pr0 ð0Þ; dp ¼ c2A =c2 ; h ¼ dp 2 isp mp ; W p ¼ W=c; mp ¼ 2m=3r0 cH; sp ¼ s=sa ; tp ¼ tsa ; T p ¼ T=2gT0 ; and kp ¼ 2k=cV cH r0 ð0Þ: The star can be omitted, since all variables are written in dimensionless form from this point forward. Substituting all these quantities in the differential equations (3.3) – (3.7) and eliminating the pressure and density we obtain the following system of differential equations for the velocity and
ðg 2 1ÞDWðzÞ ¼ gðivÞTðzÞ þ gk expðzÞD2 TðzÞ; ð3:9Þ where D denotes the derivative of the indicated variable with respect to z. In addition, we introduce the dimensionless parameter Pr ¼ cP m=k which denotes the Prandtl number and cP is the specific heat at constant pressure. Letting a ¼ h=k it is clear that Prandtl number measures the relative strength of the viscosity with respect to the thermal conduction and we have: " ! ! v2 gv2 2 2 2 D 2Dþ 2 k expðzÞD D 2 D þ 4 4 2 gsak expðzÞDðD þ 1Þ 2 igaðk expðzÞÞ2 D2 # ðD þ 1ÞðD þ 2Þ TðzÞ ¼ 0:
ð3:10Þ
3.1. Boundary conditions To obtain a unique Solution for the differential equation (3.10), physically relevant conditions must be imposed. The first is the dissipation condition, which requires that the rate of change of the energy dissipation in an infinite column of fluid of unit crosssection to be finite. Since the dissipation function depends on the squares of the velocity gradients, the dissipation condition is equivalent to ð1 m lWz l2 dz , 1 ð3:11Þ 0
The second one is the entropy condition, which is determined by the equation for the rate of the entropy, from which it follows that ð1 k lTz l2 dz , 1 ð3:12Þ 0
Boundary conditions are also required at z ¼ 0; and we shall adopt the lower boundary condition: in a fixed interval 0 , z , z0 ; the solution of the differential equation should approach some solution of
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the limiting differential equation as k ! 0 and m ! 0; i.e. " # s2 2 TðzÞ ¼ 0 ð3:13Þ D 2Dþ 4 For numerical computations, which are the aim of this paper, we assume as lower boundary conditions that Tð0Þ ¼ 1
and Wð0Þ ¼ 1
ð3:14Þ
The upper boundary conditions (3.11) and (3.12) and the lower boundary conditions (3.13) and (3.14) are necessary and sufficient, to ensure unique analytical and numerical solutions. The upper boundary conditions are only applicable if the atmosphere is viscous and thermally conducting. If the atmosphere is not viscous the magnetic energy condition will be used as an upper boundary condition to ensure a unique solution. If the wave propagation is affected only by thermal properties, or if the effects of all the dimensionless parameters vanish, a radiation condition must be imposed to ensure a unique solution. Finally, the wave speed and temperature have the same Solution behavior in all regions of the atmosphere of the chromosphere.
4. Behavior of the solution in the region of the low chromosphere (for small values of lal) In this section we study the behavior of the solutions of the differential equation (3.10), in the region of the low chromosphere for small values of lal; (when thermal conduction effects dominate the motion), subjected to the upper (3.11) and (3.12) and lower boundary conditions (3.13) and (3.14). As we stated in the formulation of the problem the parameters d; m; and k are dimensionless, sufficiently small, and proportional to the Alfven speed, kinematic viscosity and thermal conductivity at z ¼ 0: To understand the physical nature of the behavior of the solution of the differential equation (3.10) we should investigate the relationships among these dimensionless parameters because the nature of the solution, the representation of the three regions of the chromosphere, the magnitude of the reflection coefficients, the nature of the transition regions, and the properties of the reflecting layer depend mainly on relationships
43
among these parameters. The parameter lal can be written in the following form d 2 im h d 2 i m r0 ¼ lal ¼ ¼ k k k r0
:
ð4:1Þ
Moreover, we call ld 2 im=r0 l the magnetoviscous diffusivity and lk=r0 l thermal diffusivity. Thus lal measures the relative strength of the magnetoviscous diffusivity with respect to thermal diffusivity. In addition, for small lal the atmosphere of the chromosphere can be divided into three different regions: low, middle, and high. The low and high chromospheres are connected by a semitransparent reflecting layer, while the high chromosphere and corona are joined by a transition region whose properties depend mainly on the relative strength of the kinematic viscosity with respect to Alfven speed. It has been indicated that the chromosphere is divided into three distinct regions according to its temperature (Priest [9]). The differential equation (3.10) is solved asymptotically by matching inner and outer approximation in an overlapping domain. Similar procedure was used in Alkahby [3] and the computations need not be repeated but we indicate only results for completeness. In the region of low chromosphere, the solution of the differential equation (3.10), which satisfies the prescribed boundary conditions can written in the following form:
1 1 TðzÞ , C exp þ ib z þ RL exp 2 ib z ; 2 2 ð4:2Þ where Cis a constant and RL denotes the reflection coefficient, which is defined by RL ¼ expð2pbÞ{cosð2i½u2 2 bp logðgskÞÞ þ i sinð2i½u2 2 bp logðgskÞÞ}Ra
ð4:3Þ
with
b¼
ffi 1 pffiffiffiffiffiffiffiffi s 2 2 1; 2
u1 ¼ argðhÞ;
bp ¼
qffiffiffiffiffiffiffiffiffiffiffi 1 gs2 2 1; 2
ð4:4Þ
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2
6G u2 ¼ arg6 4
2
3 1 2 ib Gð2ibÞGðibp 2 ibÞ 7 2 7; 5 Gðibp þ ibÞ
2 3 1 p 2 p þ i 2 b G ð2i b Þ 6 7 2 7 u3 ¼ arg6 4 Gðibp þ ibÞGðibp 2 ibÞ 5;
Ra ¼
ð4:5Þ
ð4:6Þ
c1 2 c2 expðFÞ ; c3 2 c4 expðFÞ
c1 ¼ expð2pbp Þ 2 expð2pbÞ;
ð4:7Þ
c1 þ c2 ¼ c3 þ c4 ¼ 2 sinhð2pbp Þ c3 2 c1 ¼ c4 2 c2 ¼ 2 sinhð2pbÞ; p
ð4:8Þ
p
and F ¼ b ðp þ 2u1 Þ þ 2iðu3 2 b loglalÞ: Here b and bp are the dimensionless wave numbers corresponding to the adiabatic and Newtonian sound speed, respectively. Eq. (4.2) represents the behavior of the solution in the low chromosphere. It is a linear combination of an upward and a downward propagating wave with equal wavelengths lL ¼ 2p=b: In the low chromosphere the reflection process is more complicated because of the existence of two reflecting layers one produced by thermal conduction and the second one created by the combined effect of the viscosity and magnetic field. The magnitude of the reflection coefficient depends on the relative strength of thermal diffusivity with respect of magnetoviscous diffusivity and it will be seen in Section 5 that if the magnetoviscous diffusivity dominates the oscillatory process there will one reflecting layer attributable to the combined effects of the viscosity and magnetic field. Also the magnitude of the reflection coefficient depends on the relative locations of the reflecting layers and its value less is than expð2pbÞ: In fact the magnitude of the reflection coefficient is periodic, which is clear from its definition in Eq. (4.3). More conclusions about the nature of the solution in the low chromosphere shall be summarized in Section 7.
5. Behavior of the solution in the region of the middle and high chromosphere (for small values of lal) Before discussing the nature of the solutions in the high chromosphere, it is desirable and important to explain the behavior of the solution in the middle chromosphere. The region of middle chromosphere is represented by a transition region that connects low and high chromospheres in which heat dissipation and wave modification take place. This region is semitransparent, where part of the energy is transmitted upward and part is reflected downward. In this region temperature is increased substantially because the waves dump part of their energies. Propagation, transmission, and reflection will continue until all the energy of the waves is dissipated. This analysis is consistent with widely reported and observed results (Priest [9] and the references therein). In the region of high chromosphere, the solution of the differential equation (3.10), that satisfies the prescribed boundary condition, can be written in the following form:
1 1 p TðzÞ , C exp þ ib z þ Rh exp 2 ib z ; 2 2 ð5:1Þ where Rh denotes the reflection coefficient in the high chromosphere which is given by: Rh ¼ exp½2bp u1 þ 2iðu 2 bp loglghlÞ; and
u ¼ 2 arg
1 3 2 ibp Gð2ibp ÞG2 2 ibp : 2 2
ð5:2Þ
ð5:3Þ
It is clear that Eq. (4.2) shows that the Solution of differential equation (3.10) that satisfies the prescribed boundary conditions can be written as a linear combination of an upward and a downward propagating wave with equal wave number bp : Also the magnitude of the reflection coefficient lRh l ¼ expð2bp u1 Þ where u1 ¼ argðhÞ: As a result, if the viscosity dominates the oscillatory process in the region of high chromosphere, h ! ism and argðhÞ ! 2p=2; which leads to lRh l ! expð2pbp Þ: On the contrary, if the magnetic field dominates the motion in the high chromosphere h ! d and argðhÞ ! 0; consequently lRh l ! 1: It follows that the viscosity creates an absorbing and reflecting layer
A.F. Jalbout et al. / Journal of Molecular Structure (Theochem) 640 (2003) 39–47
and the magnetic field produces a totally reflecting barrier. This result is expected for the magnetic field because of its dissipationless nature. Such situations occur in the region of high chromosphere, especially in sunspots where the magnetic field is very strong. In addition, when the magnetic field dominates the oscillatory process, resonance may occur in the region of high chromosphere for infinitely many values of the magnetic field and infinitely many values of the frequency of the wave if the magnetic field strength is matched with the wavelength of the wave (see Alkahby [1 –3], Alkahby and Mahrous [4,6]). At the resonance frequencies the kinetic and magnetic energies shall increase to very large values and this may explain the monotonic increase in the temperature in the high chromosphere as well as some activities in sunspots. We hope this will stimulate more interest in the study of the structure of the sun’s magnetic field and its influences on the activities of the solar atmosphere in general and its heating mechanisms in particular.
6. Behavior of the solution for large values of lal In this section we investigate the behavior of the solution of the differential equation (3.10), when the motion is dominated by the combined effects of the viscosity and magnetic field. In this case the differential equation is similar to the differential equation given in Alkahby, Mahrous and Vatsala [7] (see Eq. (12)) but the physical conclusions are different. The Solution that satisfies the prescribed boundary conditions can be obtained in the same way and the details of the computations are omitted. Consequently, the solution of the differential equation (3.10) that satisfies the prescribed boundary conditions can be written like: where denotes the reflection coefficient and defined by:
1 1 TðzÞ , C exp þ ib z þ Rg exp 2 ib z ; 2 2 ð6:1Þ where Rg denotes the reflection coefficient defined by Rg ¼ F exp½2bu1 þ 2iðu 2 loglghlÞ;
ð6:2Þ
F ¼ 2 arg½Gð2ibÞG2 ðbÞGðibp 2 ibÞGðibp þ ibÞ
45
ð6:3Þ
Once again the solution of the differential equation (3.10) can be written as a linear combination of an upward and a downward propagating wave. In this case solutions are influenced by the effect of thermal diffusivity because the reflecting barrier either a totally reflecting or reflecting and absorbing one. Consequently, the waves will reflected downward before the effect of thermal diffusivity takes place. In fact the problem can be approximated by letting the effect of thermal conduction be negligible ðk ¼ 0Þ in the differential equation (3.10). Once again the resonance will occur for infinitely many values of the frequency and the magnetic field (see Alkahby [1 – 3]. More conclusions will be summarized in Section 7.
7. Computing scheme and results of computations The results of the previous sections are asymptotically valid as lal ! 0: In order to determine the range of lal in which they are reasonably accurate, the boundary-value problem (defined by the system of Eqs. (3.8) and (3.9)) was solved numerically for several values of lal and b: For the numerical integration it is convenient to deal with the system (3.8) and (3.9) with the prescribed boundary conditions at z ¼ 0: The upper boundary conditions are replaced by these two conditions: DW ! 0; DT ! 0 as z ! 1: Using centered differences, we can replace the system of differential equations (3.8) and (3.9) by a set of difference equations that will be solved numerically, An Fnþ1 þ Bn Fn þ Cn Fn21 ¼ 0
ð7:1Þ
where Fn is the column vector with components iTðzn Þ and Wðzn Þ and An ; Bn ; Cn are 2 £ 2 matrices. The system was solved in an interval 0 # z # L sufficiently large to allow T and W to reach their limiting values, and the boundary condition DW ! 0; DT ! 0 as z ! 1, was replaced by FN ¼ FN21 ; where N is the index of the point z ¼ L: The problem was solved by the standard method in which the Fn are computed from a linear equation by backward integration. On the other hand,
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A.F. Jalbout et al. / Journal of Molecular Structure (Theochem) 640 (2003) 39–47
the coefficient matrices
an ¼ 2½Bn þ Cn an21 21 An
ð7:2Þ
and the vectors
bn ¼ 2½Bn þ Cn an21 21 Cn bn21
ð7:3Þ
are computed by forward integration. Moreover, let M and m denote the maximum and minimum pffiffiffiffiffiffi values of the oscillation amplitude and d ¼ M=m; then the magnitude of the reflection coefficient can be computed from the formula lRC l ¼
d21 dþ1
ð7:4Þ
where RC denotes any of the reflection coefficients defined in the body of the paper. The computations were done different values of the dimensionless parameters d; m; and k: 7.1. Observations and conclusions (1) When the effect of thermal conduction dominates the oscillatory process, the atmosphere of the chromosphere can be divided into three regions with different temperatures:the low, middle, and high chromosphere. The low and the high regions are connected by the middle chromosphere, which is a semitransparent transition region, and in this region wave modification takes place. Here part of the energy propagated upward, part is dissipated, and another part is reflected downward. In addition, the upward propagating waves have different wavelength because the exponential increase of thermal diffusivity changes the motion from the adiabatic form, below the reflecting layer to an isothermal form above it (2). The change in the wavelength accounts for the reflection process and the magnitude of the reflection coefficient is 4.3. Moreover, the solution that satisfies the prescribed boundary conditions can write as a linear combination of an upward and a downward propagating waves with equal and adiabatic wavelengths (4.2). (2) The high chromosphere and corona are connected by a second transition region, its properties depend on the relative strength of the viscosity with respect to the magnetic field. When the magnetic field dominates the motion, it will
creates a totally reflecting transition region and the magnitude of the reflection coefficient is one (5). The dominance of the viscosity effect produces an absorbing and reflecting transition region, the magnitude of the reflection coefficient, in the second region is given in Eq. (5.2). (3) The upward propagating waves in the high chromosphere will be reflected downward by the second reflecting layer. In this case the magnitude of the reflection coefficient depends on the relative strength of the viscosity with respect to the magnetic field as indicated in the previous observation. The energy absorption and multiple reflection will continue until all the energy of the dissipated and this may contribute to the heating process of the chromospheres. (4) It is clear from the above remarks that the reflection process in the lower region is very complicated and the magnitude of the reflection coefficient depends mainly of relative strength of the competing factors: viscosity, thermal conduction, and magnetic field strength. In fact, the relative location of the transition region is the main factor effecting the magnitude of the reflection coefficient. For example, if the combined effects of the viscosity and magnetic field are dominant, the effect of thermal conduction will be eliminated, because the upward propagating waves will be reflected downward before the effect of thermal conduction takes. Moreover, the differential equation (3.10) can be approximated by setting and we obtain a second order differential equation that can be transformed to a hypergeometric differential equation (see Alkahby [1 –3] and Yanowitch [10 – 12]). (5) When the magnetic field dominates the motion in the high chromosphere, the resonance may occur for infinitely many values of the magnetic field and frequency of the wave if the magnetic field strength is matched with the wavelength of the wave, see Alkahby [1 –3], Alkahby and Yanowitch [5]. In this case, the magnetic and kinetic energies of the waves will be increased greatly and this may contribute to the heating process of solar atmosphere in general and the chromosphere in particular. In addition, resonance occurrence may explain some of the activities on the surface of the active sun, for example: sunspots.
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(6) The increase of the temperature in the middle chromosphere is due to the dissipation of the magnetic and kinetic energies in this region and to the increase of thermal diffusivity with altitude. The rapid increase of the temperature in the transition region between the high chromosphere and the corona is due to the heat ejection from the surface of the sun toward the corona. This phenomena usually takes place in region where the structure of the magnetic field is very complicated. The regions where these activities are taking place, are the sunspots because of the strength and complicated nature of their magnetic field. As soon as the heat is ejected from the high chromosphere to the corona, a strong current of particles is coming from the corona toward the surface because of the rapid decrease in the pressure.
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