ON VISCOUS AND RESISTIVE DISSIPATION OF ALFVN WAVES IN AN ISOTHERMAL ATMOSPHERE

In this article we will investigate reflection and dissipation of Alfven waves, resulting 
from a uniform vertical magnetic field, in a viscous, resistive and isothermal atmosphere. It is shown that 
the atmosphere may be divided into two distinct regions connected by an absorbing and reflecting 
transition layer. In the transition layer the reflection, dissipation and absorption of the magnetic energy of 
the waves take place and in it the kinematic viscosity changes from small to large values. In the lower 
region the effect of the resistive diffusivity and kinematic viscosity changes from small to large values. In 
the lower region the effect of the resistive diffusivity and kinematic viscosity is negligible and in it the 
solution can be represented as a linear combination of two, incident and reflected, propagating waves 
with different wavelengths and different dissipative factors. In the upper region the effect of the resistive 
diffusivity and kinematic viscosity is large and the solution, which satisfies the prescribed boundary 
conditions, will behave as a constant. The reflection coefficient, the dissipative factors are determined 
and the conclusions are discussed in connection with solar heating.


INTRODUCTION
It is well known that the solar corona is extremely hot, the typical temperature is 2 x 106K, at heights in excess of 2000 Km, compared with 5 x 10SK, temperature minimum, in the atmosphere (see Priest  [3]   for references).As a result, thermal energy must be continually supplied to maintain this temperature against radiative cooling.The old ideas for coronal heating were that of souM waves being generated in the convection zone that could propagate through the solar chromosphere, steeping into shocks and to give global heating.Sound waves have been ruled out in connection with coronal heating, however, because their low group velocity means that they cannot supply the necessary energy However, the remnants of this idea remain.Recent theories of the solar heating invoke, strongly, the magnetic energy dissipation as a source of thermal energy.In particular, recent investigations are focusing on the dissipation of magnetic energy, resulting from a vertical magnetic field, and its role in the heating process of the solar atmosphere.
The aim of this paper is to investigate the combined effect of the electrical conductivity and the viscosity on the reflection and dissipation of Alfvtn waves, resulting from a uniform vertical magnetic field, in an isothermal atmosphere.It is shown that if the effect of the viscosity is small compared to that of the resistive diffusivity the atmosphere can be divided into two different regions.In the lower region the influence of the resistive diffusivity and kinematic viscosity is negligible and the solution can be written as a linear combination of incident and reflected waves with different wavelengths and different dissipating factors.In fact the wavelength factor of the reflected wave is smaller than that of the incident wave.This indicates that the magnetic energy which dissipates from the incident wave is larger than that of the reflected wave and the difference in the magnetic energy of the reflected and the incident waves represents the dissipated energy in the transition region.Moreover, the dissipative factors are a function of the electrical resistivity and the dynamic viscosity.In the upper region the effect of the resistive diffusivity and the kinematic viscosity is large.As a result, the solution, which satisfies the dissipation condition, behaves as a constant.The lower and the upper regions are connected by a transition layer in which the kinematic viscosity changes from small to large values because of the exponential decrease of the density with height.Moreover, in the transition region the reflection and dissipation of Alfv6n waves take place.The differences in both, wavelengths and dissipating factors, result from the absorption and dissipation process of the magnetic energy in the transition layer, the reflection coefficient and the dissipative factors are determined.The conclusions are discussed in connection with the process of the heating of the solar atmosphere.

MATHEMATICAL FORMULATION OF THE PROBLEM
We will consider an isothermal atmosphere, which is viscous and resistive, and occupies the upper half-space z > 0. It will be assumed that the gas is thermally non-conducting and under the influence of a uniform vertical magnetic field.We will investigate the problem of small oscillations about equilibrium, i.e. oscillations which depend only on time 4, on the vertical coordinate z and on the horizontal coordinate z.
Let the equilibrium pressure, density, temperature and magnetic field strength be denoted by P0(z), p0(x), To, and Bo(x), where P0(z), po(Z) and To satisfy the gas law P0(x)= RToPo(x) and the hydrostatic equation P(x) / gpo(z) 0.Here R is the gas constant, g is the gravitational acceleration and the prime denotes differentiation of the pressure with respect to x.The equilibrium pressure and density, P0(z) P0(0)exp(-z/K), p0(:r) p0(0)exp(-z/K), (2.1) where K RTo/g is the density scale height.Let p(z,z,t), p(z,z,t), V(z,z,t), and h(x,z,t) be the perturbations quantities in the pressure, density, velocity, and the magnetic field strength.
Alfv6n waves are incompressible because they have motions transverse to the magnetic field, i.e. they do not couple to slow or fast magnetohydrodynamics waves in a homogeneous medium, Priest [3].As a result, they can be described only by the induction and momentum equations and dissipation of linear waves is not affected by thermal conduction or radiation.Thus, the equations of motion are c9-' V(x)= v(z,z,t)ey and is the where n(x, z, t) B(x) + h(x, z, t), V ex + % + , pbi of the maetic field, kby [3,8].Here v d v-e, respeively, the incompressible d compressible scosities.Moreover, c denotes the spd of light in valuta d is the Oc electfic conducti.The induction equation (2.2) bces maetic field oscillation, veloci trspon ong the maetic field lines d compressibili agnst resiive dissipation by O effect, the HI effe bng oned.The momentum uation (2.3) bces the ineia force d pressure gradient against weight, maetic d scous forces.
In this article we will consider the case of a uniform vertical magnetic field B0(x) B0, as well as r/= and v, i.e. they are constants.It follows from equation (2.1) that the kinematic viscosity and Alfv6n speed can be written in the following form (z) (U/po(O))e'/K, (2.4) ) where CA V/#/4rp0(0) Bo.Moreover, the linear forms of equations (2.2) and (2.3) are: Dh(x, z, t) rl[D,h(x, z, t) + Dh(x, z, t)] BoDv(x, z, t), (2.6) Dv(x,z,t) (x)[Dv(x,z,t) + Dv(x,z,t)] (c/Bo)Dh(x,z,t). (2.7) In addition, the veloci v(x, z, t) c be efited to obtn equation for h(x, z, , oy Ts c be aompBshed by deremiag uafions (2.6) d (2.7) th resp to t d using uation (2.7).BOARY CONDONS.To mple the problem foulafion cenn bound conditions must be impos to ensure a que solmion.Since the g is scous md resistive the dissipation condition l be neces md mfficiem, as m upper bound ndition, to ensure a uque solution.
the &ssipafion condition rues the fiteness of the rate of the ener dissipation in m ite colu offld of a ut cross-section.Ts implies, < , (2.9) lR(,k,)ld where R(x, k, ) dotes the c field sputum of a wave th equen w, wavelenh k d at the position x (e equation 3.5).Ts bound confion 11 not be applible if u a 0, bm it 11 be applicable if u 0 or a 0.Moreover, a bound ndifion is so rr at x 0, d we shl set h(0, , ) , (2.0) by suitably nong h(x, k, ).It 11 be sn that the bound conditions 11 ensure a uque lution to tn a multipficative nstt.

SOLOMON OF E PROBLEM
In s ion we 11 invegate e bettor ofthe lmion ofthe differemi equation (2.8) d we s dee a fies solution d obtn the ptotic behaor of the solution.We use Fourier represmtion in z d t for the mastic field prbation h(x, z, ) bemuse the propeRies of the atmosphe depend on x oy.In other words the maetic field h(x, z, ) is ven by the foilong Fourier representation h(x,,t) d R(x,k,t)exp[i(k -)]dk, (3.1)where R(, , t) denotes the maic field perbatin spmmm fr a wave of equency , trsverse wavenumber k at a psition .Moreover, we 11 introduce the fllong dimensiNess peters, '=/, =, =/(0, =/(0, =/, =/.(3.whe the prime on the dimensioNess vable is for simplici.er the subgitutin of equation (3.1) d the densioNess peters, defin in equation ( 32), into equation (2.), we have the follong erenfi uation [I +e-=/(]D2R(x,k,w) [/32(1-i2)+(f/o-iol/O2)e-=]R(x,k,w)=0, (3 3)   where d d/dx.Since the solution will depend on p/a, it will be convenient to introduce a new dimensionless variable X defined by then the operator D goes over into For fixed value a > 0 the point X 0 corresponds to x oo, the point X0 1/a to x 0 and the segment connecting these points in the complex x-plane to x > 0. Moreover let R(X, .,,,.,) X'(X), (3.5)   then differential equation (3.3) can be written in the following form where the prime denotes the derivative of with respect to X, provided that the parameter r satisfies the following relation r fV/1 ic2.
with c=l-2r, a+b=2r, ab=i(a2/c3-c2). (3.9) Moreover, equation (3.6) has three regular singular points, X 0, X 1, and X c.The imermediate regular singular point X 1 corresponds to the reflecting layer.Solving for the dimensionless parameters a and b we have a=/3[V/1-ic2 + V/1 ic/13,2c3], ( For IX[ < 1, the hypergeometric equation (3.6) has two linearly independent solutions of the following form: (I)I(X) F(a,b, 2,X), where F(a,b, 2,X) .=o(c).r() r( + )r( + ) x__i r()r() z_..=o r( + ) !For IX[ > 1 and [arg( X)] < r, the solution of equation (3.6) can be written in the following form: .(X)=(-X)-aF(a,l-c+a,l-b+a,x-'), ( (x) (-x)-'F(,, a + , ,, + ,x-'). (3.16) The second solution 2(X) will be eliminated by the boundary condition (2.9) because it increases to infinity as x--oo.As a result, the solution of the differential equation (3.6), which satisfies the dissipation condition, is a multiple of (X), i.e.O(X) 001 (X) (TF(a, b, cX), (3.17)where (7 is a constant which can be determined from the boundary condition (2.10).For IXl > 1, larg( X)I < r, the analytic continuation of the solution of the differemial equation (3.6) carl be written like: ) ,(x) c r()r(c ) r()r( ) + r()r(-) X)-"F(a, 1 c + a, 1 b + a,x-1) x)-F(, For [X[ > 1 and ]arg( X)[ < r the asymptotic behavior of the solution, defined in equation (3.8), as a 0 can be obtained by retaining the most significant terms in equation (3.18), the resulting equation is 4. MAGNITUDE OF THE REFLECTION COEFFICIENT AND CONCLUSIONS In this section we will investigate the behavior of the solution of the differential equation (3.6) defined by the hypergeometric function (3.17) and its asymptotic expansion defined by equation (3.19).In particular, we will determine the reflection coefficient and the dissipative factors To do this we have to make use ofthe dimensionless parameters a and b.It is easy to see that a= [V/1-io2 + V/1 ivt2/2sl=dl-id, It is clear that dl> a and da > rib.Reintroducing the original variable x via (3.4), the equation (3.19) can be written in the following form c [r()r( b)exp(-d2 + idb)loga (x) r()r( ) [exp[( d2 + idb)X] + R exp[(dl id,)x]], (4.3)where R denotes the reflection coefficient obtained from the ratio of the amplitudes of the reflected and incident waves and defined by The constant C can be determined from the boundary condition (2.10).As a result,  and equation (4.3) can be written in the following form (x) As a consequence ofthe above results and discussion we have the following conclusions. ( [A] Equation (4.3) represents the behavior of the solution of the differential equation (3.6), which satisfies the prescribed boundary conditions, in the lower region.It indicates that the solution can be written as a linear combination of an incident and a reflected wave.In the upper region the solution, which satisfies the dissipation condition, should behave like a constant.These two regions are connected by a transition region.
[B] In the transition region the reflection, dissipation and modification of the waves takes place.Moreover, the kinematic viscosity changes from small to large values because ofthe exponential decrease ofthe density with x.
[C] The incident wave decays exponentially like exp( dx), while the reflected wave decays like exp( dlx).Since dl > d2, the dissipative factor of the incident wave is larger than that of the reflected wave.On the contrary, the wavelength of the reflected wave A, 27r/da is smaller than that of the incident wave A, 27r/rib, because da > rib.This indicates that the larger pan of the dissipated magnetic energy comes from the incident wave.
[D] The dissipative factors are functions ofthe dynamic viscosity and electrical conductivity and they behave like exponentially decaying waves.This indicates that the magnetic energy of the wave dissipates as the wave propagates, upward and downward, but still most of the dissipation of the magnetic energy takes place in the transition region.