Slow magnetohydrodynamic waves in stratified and ...

PHYSICS OF PLASMAS 13, 042108 共2006兲

Slow magnetohydrodynamic waves in stratified and viscous plasmas Istvan Ballai,a兲 Robert Erdélyi, and James Hargreaves Solar Physics & upper-Atmosphere Research Group (SPARG), Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, United Kingdom

共Received 31 January 2006; accepted 21 March 2006; published online 27 April 2006兲 The propagation of slow magnetohydrodynamic waves in vertical thin flux tubes embedded in a vertically stratified plasma in the presence of viscosity is shown here to be governed by the Klein-Gordon-Burgers 共KGB兲 equation, which is solved in two limiting cases assuming an isothermal medium in hydrostatic equilibrium surrounded by a quiescent environment. The results presented here can be applied to, e.g., study the propagation of slow magnetohydrodynamic waves generated by the granular buffeting motion in thin magnetic photospheric tubes. When the variation in the reduced velocity occurs over typical lengths much larger than the gravitational scale height, the KGB equation can be reduced to a Klein-Gordon equation describing the propagation of an impulse followed by a wake oscillating with the frequency reduced by viscosity and the solution has no spatial or temporal decay. However, in the other limiting case, i.e., typical variations in the reduced velocity occur over characteristic lengths much smaller than the gravitational scale height, waves have a temporal and spatial decay. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2194847兴

␻ = ± 冑k2cT2 + ␻2b .

I. INTRODUCTION

The magnetic field in solar and stellar atmospheres tends not to be diffuse but, instead, to concentrate in small cylindrical entities called magnetic flux tubes or magnetic loops. In these tubes the magnetic field can be, as a first approximation, considered to be axial. Such structures are a perfect medium for magnetohydrodynamic 共MHD兲 wave propagation, playing the role of waveguides. We consider a vertical magnetic flux tube expanding through a vertically stratified plasma. If the tube is sufficiently thin so that the radial variations can be neglected to leading order, we can apply the so-called thin flux tube approximation resulting in a much simpler mathematical analysis.1,2 We study the propagation of linear longitudinal waves in a nonideal 共viscous兲 plasma. The dynamics of linear longitudinal waves in thin stratified and ideal waveguides is governed by the Klein-Gordon equation 共see, e.g., Refs. 1 and 3兲 2 ⳵ 2Q 2⳵ Q + ␻2bQ = 0, 2 − cT ⳵t ⳵␹2

共1兲

where Q is a physical quantity 共e.g., density, speed, etc., reduced by an exponential term兲, ␹ is a coordinate parallel to the gravitational acceleration vector, cT is the propagation speed of slow magnetoacoustic waves along the tube, and ␻b is a cutoff frequency and is a constant quantity for an isothermal medium, depending upon the sound speed, Alfvén speed, and gravitational acceleration. Employing a normal mode analysis 共Q ⬃ ei共␻t−k␹兲兲, the dispersion relation of these linear waves is given as a兲

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共2兲

Due to the given k dependence of the dispersion relation 关Eq. 共2兲兴, waves are dispersive; i.e., waves with smaller wavelength 共larger k兲 propagate faster. Calculating the group speed kc2 ⳵␻ = ± 2 2T 2 冑k cT + ␻b ⳵k waves with smaller wave number will have smaller group speed, the maximum of the group speed 共at k → ⬁兲 being cT. Equation 共2兲 has been studied in the context of pulse propagation in the solar photosphere and chromosphere 共see, e.g., Refs. 1 and 3–7兲. The impulsive excitation of waves in a flux tube leads to the formation of a pulse that propagates away with the speed cT, followed by a wake in which the flux tube oscillates with the frequency ␻b. All previous studies assumed waves propagating in an ideal medium, though it is widely recognized that the medium in which these waves can propagate is dissipative. The effect of nonadiabacity on the propagation of waves in stratified media solar plasmas was explored by Ref. 8. Roberts argues that the cutoff frequencies actually are changed in the photosphere due to the stratification, and the modes observed by, e.g., Refs. 9 and 10 are, in fact, nonadiabatic acoustic 共-gravity兲 modes propagating along a slender magnetic flux tube. This idea was further exploited in detail by Refs. 11 and 12. Here we investigate the effect of dissipation 共e.g., viscosity兲 on the propagation of waves in a thin magnetic flux tube filled with a vertically stratified plasma. Throughout the derivation we will assume, for the sake of simplicity, that the medium, in hydrostatic equilibrium, is isothermal and the temporal variation in the pressure outside the tube is the largest temporal scale in the problem. Here we present the

13, 042108-1

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042108-2

Phys. Plasmas 13, 042108 共2006兲

Ballai, Erdélyi, and Hargreaves

mathematical derivation and the solution of the equation describing the propagation of longitudinal magnetoacoustic waves in details. A further study on how MHD slow waves under the combined conditions of stratification and dissipation will propagate in specific solar or astrophysical plasmas 共e.g., observed drivers, constraints about magnetic and geometrical structure, characteristic values, etc.兲 will complement the present study in the near future. II. GOVERNING EQUATIONS

We consider an isolated magnetic tube embedded in a magnetic free compressible and isothermal medium. We suppose that the coordinate system is such that the x axis is parallel to the gravitational acceleration vector; i.e., g = −gxˆ. The tube is considered to be thin, with circular cross section A共x兲 and in thermal equilibrium with its environment. Waves propagate in a plasma described by the pressure p共x兲 and density ␳共x兲, and is permeated by a longitudinal magnetic field B共x兲. The equilibrium of the tube is prescribed by the hydrostatic equilibrium dp0 = − ␳0g, dx

共3兲

where the quantities with an index “0” describe the equilibrium state and the lateral pressure balance p0 + B20 / 2␮ = pe, with pe being the kinetic 共thermal兲 pressure of the environment. We suppose oscillations with wavelengths comparable to the gravitational scale height H, which for an isothermal medium is a constant quantity. Since the equilibrium configuration is isothermal, the scale height outside the tube will also be H. We suppose the x dependences of the equilibrium quantities are given as p0共x兲 = p0共0兲e−x/H, B0共z兲 = B0共0兲e−x/2H,

␳0共x兲 = ␳0共0兲e−x/H , A0共x兲 = A0共0兲ex/2H ,

共4兲

describing a barometric atmosphere. With these particular choices, the sound speed 共␥ p0 / ␳0兲1/2, the Alfvén speed B0 / 共␮␳0兲1/2, and the cusp speed cT = cSvA / 共c2S + vA2 兲1/2 are all constant quantities. Equations 共4兲 also verify that the dependence of the magnetic field and the cross section of the tube are such that the magnetic flux in the tube is always conserved. The linear perturbation of the equilibrium state is governed by the system of equations

⳵ ⳵ 共␳0A + ␳A0兲 + 共␳0A0v兲 = 0, ⳵t ⳵x ␳0

⳵v ⳵p ⳵ 2v + + ␳g = ␯␳0 2 , ⳵t ⳵x ⳵x

冉

共5兲

共6兲

冊

⳵␳0 ⳵p ⳵␳ dp0 = c20 +v +v , dx ⳵x ⳵t ⳵t

共7兲

where v is the component of the velocity along the x axis. These equations must be supplemented by two equations that describe the conservation of the magnetic flux and the con-

servation of total pressure 共the sum of the kinetic and magnetic pressure兲 inside the tube is balanced at the boundary by the external kinetic pressure pe. These two conditions are expressed as B0A + BA0 = 0,

p+

B0 B = pe . ␮

共8兲

In Eq. 共6兲 ␯ is the coefficient of kinematic viscosity 共here considered as a constant quantity兲. The choice of viscosity as a dissipative mechanism was arbitrary; it may be possible that for particular applications 共e.g., flux tubes in stellar atmospheres兲 magnetic diffusivity or thermal conduction would be more appropriate dissipative mechanism. It can be easily shown that the final form of the equation describing the dynamics of slow waves does not change; only the magnitude of the dissipative term would be changed. After a lengthy but still straightforward calculus, we find that the dynamics of a linear perturbation propagating along the magnetic field in a magnetically isolated, stratified, and viscous expanding tube is described by the inhomogeneous Klein-Gordon-Burgers 共KGB兲 equation

冉

冊

2 ⳵ 1 ⳵ ⳵ 2Q 1 ⳵2 2⳵ Q 2 + − c + ␻ Q − ␯ + Q T b ⳵ t 16H2 2H ⳵ x ⳵ x2 ⳵ t2 ⳵ x2

=−

冉冊冉

e−x/4H cT ␳0共x兲 vA

2

冊

⳵ ⳵ pe g + pe , ⳵ t ⳵ x c2S

共9兲

where Q共x , t兲 = v共x , t兲exp共−x / 4H兲 is the reduced speed and ␻2b is given by

␻2b =

冉冊冉冊 9 2 2 3 2 − − ␻a − 4 ␥ 2 ␥

2

␤␥ 2 ␻ , ␤␥ + 2 a

共10兲

where ␻a = cS / 2H is the acoustic cutoff frequency, and ␤ = 2␮ p0 / B20 is the plasma-beta. It is interesting to note that for ␥ = 5 / 3, the magnetic field increases the cutoff frequency 共compared to the acoustic cutoff frequency 兲 if ␤ 艋 1.5. If the last term in the left-hand side of Eq. 共9兲 is neglected 共ideal medium兲, we recover the governing equation given by Ref. 3. If, in addition, we suppose that waves propagate in an ideal medium surrounded by a quiescent environment 共i.e., the temporal variation of the external pressure is the longest temporal scale in the problem兲, we obtain the KG equation 2 ⳵ 2Q 2⳵ Q − c + ␻2bQ = 0. T ⳵ t2 ⳵ x2

共11兲

The subject of the present study is the propagation of slow MHD waves in a viscous plasma embedded in a quiescent environment. For this model, the KGB equation reduces to

冉

冊

2 ⳵ 1 ⳵ ⳵ 2Q 1 ⳵2 2⳵ Q 2 + 2 Q = 0. 2 − cT 2 + ␻ bQ − ␯ 2 + ⳵ t 16H ⳵t ⳵x 2H ⳵ x ⳵ x

共12兲 Before embarking on to solve the KGB equation let us remind some of the essential properties and solutions of the KG equation to be used later.


042108-3


Slow magnetohydrodynamic waves¼

III. PROPERTIES AND SOLUTIONS OF THE KLEIN-GORDON EQUATION

⌿共x,s兲 =

冕

⬁

Q共x,t兲e−stdt,

q2 =

s2 + ␻2b cT2

0

Probably the simplest nontrivial solution of the KG equation is the uniform wave train solution of the form Q = A cos k␪, where ␪ = x ± c0t with the speed c0 given by the dispersion relation 共关Eq. 共2兲兴 c20 = cT2 +

␻2b 2.

⌿共x,s兲 =

k

1 2cT2 q

冕

+

The KG equation is a particular case of the so-called telegrapher’s equation 共see, e.g., Ref. 13兲. For any function u共x , t兲 satisfying the telegrapher’s equation a

with the solution

⳵ 2u ⳵u ⳵ 2u + cu = + 2b , ⳵ t2 ⳵t ⳵ x2

u = u共x,t兲,

共13兲

with a ⬎ 0, b, and c constants, we can introduce the a new function Q共x , t兲 = u共x , t兲e共b/a兲t, which transforms the telegrapher’s equation into the KG equation 2 ⳵ 2Q 2⳵ Q − m + n2Q = 0, ⳵ t2 ⳵ x2

共14兲

再冕

x

e−q共x−␩兲关sf共␩兲 + g共␩兲兴d␩

冎

−⬁

⬁

e−q共␩−x兲关sf共␩兲 + g共␩兲兴d␩ .

x

共18兲

In order to obtain a solution for the KG equation 共11兲, we have to invert Eq. 共18兲 using an inverse Laplace transform 共Bromwich integral兲 in the form Q共x,t兲 =

1 2␲i

冕

␥+i⬁

␥−i⬁

est⌿共x,s兲ds,

where ␥ is chosen such that all singular points of ⌿共x , s兲 lie on the left of the line Re共s兲 = ␥ in the complex s plane. Calculating the inverse Laplace transform of Eq. 共18兲, we obtain the solution of the Klein-Gordon equation

where 1 Q共x,t兲 = 关f共x − cTt兲 + f共x + cTt兲兴 2

ac − b2 n = . a2

1 m = , a 2

2

In Appendix A we show that the KG equation can be further reduced to a Bessel equation of order zero through a series of variable changes, so that the solution of the KG equation in the original variables has the form Q = AJ0

冋

册

␻b 冑 2 c 共t − t0兲2 − 共x − x0兲2 . cT T

冏冏 ⳵Q ⳵t

= g共x兲,

− ⬁ ⬍x⬍ ⬁,

共16兲

t=0

supplemented by the requirement that at ±⬁, the function Q共x , t兲 vanishes 共boundary condition兲. Initially, the tube is set to motion with the shape f共x兲 and is also accelerating with the given shape g共x兲. Using a Laplace transform, the KG equation can be reduced to an ordinary differential equation of the form d2⌿共x,s兲 f共x兲 g共x兲 − q2⌿共x,s兲 = − s 2 − 2 , 2 dx cT cT where

共17兲

1 2cT

冕

x+cTt

x−cTt

冦

g共␩兲J0

␻bcTtJ1 − f共␩兲

共15兲

This result has been obtained without considering any initial and/or boundary conditions, nevertheless this solution serves as an initial guess of the form of the final solution. In order to find the value of A, initial and/or boundary conditions have to be taken into account. Depending on the type of the initial condition 共spatial or temporal兲 the form of the solution will differ. Let us first suppose that the KG equation is subject to the conditions Q兩t=0 = f共x兲,

+

冋

冋

␻b 冑 2 2 c t − 共x − ␩兲2 cT T

␻b 冑 2 2 c t − 共x − ␩兲2 cT T

冑cT2 t2 − 共x − ␩兲2

册

冧

册共19兲

.

This solution can be interpreted as follows. There are two propagating waves each half of the original displacement moving in opposite directions, followed by an oscillating tail with decaying amplitude. In obtaining Eq. 共19兲, we have used the properties of the Laplace transform14

冋冉

L c TJ 0

冤

冉冊

L c T␦ t −

=

冊冉冊册

␻b 冑 2 2 2 x c t −x H t− cT T cT

se−qx , q

x − cT

=

e−qx , q

x ⬎ 0,

冉冑冊冉冊冑

cT␻btJ1 ␻b

t2 −

x2 t2 − 2 cT

x2 cT2

H t−

x cT

冥

x ⬎ 0,

where ␦共x兲 and H共x兲 are the Dirac-delta and Heaviside functions, respectively. The nature of the solution is changed when we suppose an initial condition such as


042108-4



Q兩t=0 = 0,

冏冏 ⳵Q ⳵t

= 0,

Q兩x=0 = A0共t兲,

0⬍x⬍⬁

These initial conditions suggest that the entire tube is at rest at t = 0 and no part is accelerating, however, the footpoint of the tube 共x = 0兲 is driven by a velocity perturbation A0共t兲. In this case, the solution of the KG equation is15

冉冊冉冊冕 x x H t− + cT cT

t

A0共t − ␶兲W共x, ␶兲d␶ ,

0

共21兲 where

冋冑冉冊册冉冊冑

␻bxJ1 ␻b W共x, ␶兲 = −

␻2 − ␻2b cT2

t=0

共20兲

Q共x,t兲 = A0 t −

k2 =

␶2 −

x cT

2

cT2 ␶2 − x2

x H ␶− . cT

Now let us turn our attention to the KGB equation. In what follows, we are going to solve Eq. 共12兲 in two limiting cases.

In this first case, the KGB equation describing longitudinal wave propagation in a viscous thin flux tube in a quiescent environment can be reduced to 2 ⳵ 2Q ␯ ⳵Q 2⳵ Q = 0. + ␻2bQ − 2 − cT ⳵t ⳵ x2 16H2 ⳵ t

␯␻

ki ⬇

共22兲

共24兲

32H cT冑␻2 − ␻2b 2

共25兲

,

which is clearly a positive quantity; therefore, we conclude that in the limit when the characteristic variation in Q共x , t兲 is larger than the gravitational scale height waves will have neither spatial damping nor temporal one. Employing the aforementioned normal mode analysis, we can derive the characteristic speeds of these waves; i.e., phase and group speed. Using Eq. 共23兲, the phase speed is found as

␻ i␯ =− ± k 32H2k

冑

cT2 +

冉

冊

1 ␯2 2 ␻ − , 共26兲 k2 b 4 ⫻ 162H4

where ± denotes upward/downward propagating waves. Assuming a real wave number, waves with larger wavelengths will propagate faster provided ␯ ⬍ 32H2␻b. This means that there is an interval 共though possibly less applicable to realistic photospheric and/or chromospheric conditions兲 where waves with longer wavelengths will propagate more slowly. The group speed can be obtained as vg =

A. The variation in Q occurs over typical length scales much larger than the gravitational scale height

␯␻ , 16H2cT2

and waves will decay if the imaginary part of k is negative. Assuming mainly propagating waves 共兩kr 兩 Ⰷ 兩ki 兩兲, we obtain

vph =

IV. PROPERTIES AND SOLUTIONS OF THE KLEIN-GORDON-BURGERS EQUATION

+i

⳵␻ = ⳵k

冑

kcT2 k2cT2

+

␻2b

␯2 − 4 ⫻ 322H4

,

共27兲

which, for real or complex wave number approaching infinity, will tend to cT, similar to the ideal case 共see, for example, Ref. 3兲. Equation 共27兲 shows that the more viscous the medium, the larger the group speed. Equation 共22兲 is again a “telegrapher-type” equation and it can be reduced to a KG equation by introducing the transformation 2

In order to have a first glimpse of the behavior of the solution of Eq. 共22兲, let us carry out a normal mode analysis. For the function Q共x , t兲, we suppose a dependence of the form exp关i共␻t − kx兲兴. This leads us to the dispersion relation

␻2 +

i␯␻ − k2cT2 − ␻2b = 0, 16H2

共23兲

where the second term describes the contribution due to viscosity. Supposing a temporarily decaying wave, waves will damp provided Im共␻兲 is a positive quantity. Solving the dispersion relation 共23兲 for ␻, we find that waves cannot damp temporally; moreover, waves will propagate provided the condition

␯ ⬍ 32H2冑k2cT2 + ␻2b is satisfied 共easily satisfied under solar photospheric or chromospheric conditions兲. Supposing a spatially damped wave 共i.e., complex wave vector, but now ␻ is real兲, the dispersion relation can be recast as

Q共x,t兲 = q共x,t兲e␯t/32H . 关This transformation in essence is similar to the introduction of a transformation of the type Q共x , t兲 = q共x , t兲exp共␭t兲 in Eq. 共22兲, and, the parameter ␭ is chosen such that the resulting equation does not contain first-order derivatives in time.兴 With this transformation, the equation of wave propagation in a viscous plasma is given by

冉

冊

2 ⳵ 2q ␯2 2⳵ q 2 − c + ␻ − q = 0, T b ⳵ t2 ⳵ x2 322H4

共28兲

and describes a wave propagating with the group speed cT followed by a wake whose frequency is reduced by viscosity 共when compared to the ideal counterpart兲. The solution of this equation 关written for the original function Q共x , t兲兴 can be given according to the type of the initial and/or boundary condition in one of the forms given by Eqs. 共19兲 and 共21兲 with ␻b replaced by 冑␻2b − ␯2 / 共322H4兲 and the solution is multiplied by exp关␯t / 32H2兴. The solution of this equation shows a decaying behavior for t ⬍ ta = 32H2 / ␯. For photospheric conditions, the solution will decay for times less than


042108-5



6.4⫻ 108 s since the launch of the wave, i.e., for a slow wave propagating with a speed of 6 km s−1, waves will amplify for distances larger than 17 a.u. In reality, the time dependence of the Bessel functions in the solutions ensures the temporal decay of the solution. In fact, an asymptotic analysis of the solution given by Eq. 共19兲共i.e., t Ⰷ x / cT兲 shows that the solution is proportional to exp关t / ta兴 / t1/2 which, for scale heights of the order of ⬃100 km shows a decaying behavior for large t.

d2⌿共x,s兲 ␯ d2 f共x兲 g共x兲 − q21⌿共x,s兲 = 2 2 − 2 2 dx dx cT + ␯s cT + ␯s sf共x兲 , cT2 + ␯s

− where ⌿共x,s兲 =

冕

⬁

q21 =

Q共x,t兲e−stdt,

0

B. The variation in Q occurs over typical length scales much smaller than the gravitational scale height

⌿共x,s兲 =

␻= ±

冑

k2cT2

+

␻2b

␯ 2k 4 i ␯ k 2 − + , 4 2

共30兲

where the second term in the dispersion relation describes the damping of these modes. Since the imaginary part of the frequency is positive, waves in this limit will have a temporal decay. Supposing a complex wave number, for mainly propagating waves, the imaginary part is given by

␯␻ ki ⬇ − 2cT

冑

␻2 − ␻2b cT4 + ␯2␻2

,

which is a negative quantity; therefore, waves in this limit will have spatial damping. Waves will propagate if the viscosity coefficient satisfies the condition

␯⬍

.

冋冕冉冕冉

2q1共cT2 + s␯兲

g共␩兲 + sf共␩兲 − ␯

−⬁

⬁

d2 f共␩兲 d␩2

g共␩兲 + sf共␩兲 − ␯

x

册

冊冊

d2 f共␩兲 d␩2

⫻e−q1共␩−x兲d␩ .

共32兲

Taking into account the expression of q1, we have to find the inverse Laplace transform of the following quantities: I1 =

1

冑共s2 + ␻2b兲共cT2 + s␯兲

冉冑冊

I2 =

s

冑共s2 + ␻2b兲共cT2 + s␯兲

s2 + ␻2b

exp − x

and

cT2 + s␯

冉冑冊 s2 + ␻2b

exp − x

cT2 + s␯

,

共33兲

.

共34兲

The detailed derivation of these transforms are given in Appendix B. Taking the two values of the inverse Laplace transform, the solution of the KGB equation is Q共x,t兲 =

2 2 2 冑k cT + ␻2b . k2

Comparing to the case presented in the previous section, the damping rate now depends on k 共waves with larger wavelength will damp stronger than those with smaller wavelength兲 and a positive imaginary frequency means that waves have temporal damping. Computing the phase and group speed of the damped modes 共the real part of the dispersion relation兲, the phase speed of waves is larger for smaller wavelengths; the maximum group speed is cT 共as in the ideal case兲, and is reached for finite k. When k → ⬁, the real part of the group speed tends to −⬁. Here we have neglected the imaginary part of the group speed since for a wave number of ⬃共10−10 , 10−4兲 m−1, the ratio between the real part and the imaginary part of the group speed is of the order of 105 − 106. Let us now consider the KGB equation subject to the same boundary conditions as given by Eq. 共16兲. After applying a Laplace transform, the ordinary differential equation describing the propagation of waves is given by

cT2 + s␯

x

1

⫻e−q1共x−␩兲d␩ +

共29兲

Normal mode analysis is carried out by supposing the ansatz Q共x , t兲 ⬃ exp关i共␻t − kx兲兴. Equation 共29兲 yields the dispersion relation

s2 + ␻2b

The solution of this differential equation, which is bounded at ±⬁, is

In this second case, the KGB equation is reduced to 2 ⳵ 3Q ⳵ 2Q 2⳵ Q 2 − c + ␻ Q − ␯ = 0. T b ⳵ t ⳵ x2 ⳵ t2 ⳵ x2

共31兲

1 2 −

冕冉冕 ⬁

g共x⬘兲 − ␯

−⬁

⬁

1 2

1

␲冑␯

冕

1

⬁

cT2 /␯

␲冑␯

共35兲

冉冑冑

e−t␩ cos

and

L2 = −

f共x⬘兲L2dx⬘ ,

−⬁

where

L1 =

冊

d2 f共x⬘兲 L1dx⬘ dx⬘2

冕

⬁

c2/␯

x − x⬘

␯

␩2 + ␻2b ␩ − cT2 /␯

冑共␩2 + ␻2b兲共␩ − cT2 /␯兲

冉冑冑

␩e−t␩ cos

x − x⬘

␯

冊

d␩ ,

␩2 + ␻2b ␩ − cT2 /␯

冑共␩2 + ␻2b兲共␩ − cT2 /␯兲

冊

共36兲

d␩ , 共37兲

where we have used the properties cos共−x兲 = cos共x兲 and sinh共−x兲 = −sinh共x兲. If the initial conditions are given as temporal function, i.e., we prescribe the form of Q共x , t兲 when x = 0, then the


042108-6



solution of the KGB equation is changed. Supposing that when x = 0, Q共0 , t兲 = A共t兲, the solution of the KGB equation can be expressed as Q共x,t兲 =

1

␲冑␯

冕

t

A共t − ␶兲W共x, ␶兲d␶ ,

where W共x, ␶兲 = −

1 ␲

−

2 ␲

共38兲

0

冕冕

⬁

cT2 /␯

␻b

冉冑冑冊冉冑冑冊

e−␶␩ sin

e−i␶␩ sinh

−␻b

␩2 + ␻2b d␩ ␩ − cT2 /␯

x

␯

x

␯

APPENDIX A: THE CONNECTION BETWEEN THE KLEIN-GORDON AND THE BESSEL EQUATIONS

In this section we will explore the connection between the two equations, and we will show that the KG equation can be reduced to a Bessel equation of order zero. As shown in Sec. II, by a simple change of the dependent variable, the telegrapher equation can be reduced to KG equation 共see Eq. 共14兲 with m being the cusp speed, cT, and n the frequency of the wake ␻b when comparing 共14兲 to 共1兲兲. Let us introduce the new independent variables

␩2 + ␻2b d␩ . 共39兲 cT2 /␯ − i␩

␰=

n 共mt + x兲, m

␨=

n 共mt − x兲, m

This equation can be solved for different type of sources; e.g., monochromatic source 关A共t兲 = V0ei⍀t兴, delta-function pulse 关A共t兲 = V0␦共␻bt / 2␲兲兴, etc.

Eq. 共14兲 can be reduced to the canonical form

V. CONCLUSIONS

which is a linear hyperbolic differential equation. Introducing the new variable

We have developed a model for analyzing the propagation of slow MHD modes in a vertical thin flux tube with a vertical stratification in the presence of viscous dissipation. The evolution of these waves is described by the KleinGordon-Burgers equation written for the reduced velocity Q. Restricting ourselves to a simplified model in which the temporal response of the environment is the largest temporal scale in the problem 共in other words the wave guide is rigid兲, we solved the KGB equation in two limiting cases: 共a兲 when spatial variation in the reduced velocity occur over typical length scales much larger than the gravitational scale height and 共b兲 when the changes in Q occur over distances much shorter than the gravitational scale height. According to the variation scale of Q, the slow wave propagating in the dissipative plasma has different attenuation, in case 共a兲 we obtained that slow waves do not have spatial or temporal damping, while in the latter case we found a temporal and spatial decay. The impulsive excitation of slow waves in case 共a兲 results in a wave front propagating in opposite directions with half of the amplitude of original displacement followed by a wake that oscillates with a frequency decreased by the viscosity; i.e., in this case the viscosity acts on the wake only. In case 共b兲 it is not so straightforward to decompose the motion into a leading front and a wake. The solutions presented here are given in a general form, integrals should be solved for particular forms of the initial driver 共e.g., sinusoidal, pulse-like, etc.兲. An analysis of the results presented here for particular solar drivers in the solar photosphere, coronal plumes or solar wind spaghetti structures will be the subject of future studies.

⳵ 2Q Q + = 0, ⳵␰ ⳵ ␨ 4

w = 共␰ − ␰0兲共␨ − ␨0兲, then the differential equation transforms into an ordinary differential equation w

d2Q dQ 1 + + Q = 0. dw2 dw 4

共A2兲

Changing the variable w = ␣2, we obtain d2Q 1 dQ + + Q = 0, d␣2 ␣ d␣

共A3兲

which is a Bessel equation of order zero with solutions in form of AJ0共␣兲 + BY 0共␣兲, where J0 and Y 0 are the zeroth order Bessel functions of first and second kinds, respectively, and A and B are constants. The solution can be further simplified by neglecting the solution containing Y 0共␣兲 since it is divergent at zero. Therefore, the solution of the differential equation 共14兲 is Q = AJ0关冑共␰ − ␰0兲共␨ − ␨0兲兴.

共A4兲

The solution of the KG equation in form of Bessel functions will be used throughout further derivations. APPENDIX B: THE INVERSE LAPLACE TRANSFORM OF THE INTEGRALS I1 AND I2 GIVEN BY EQS. „33… and „34…

Considering the complex z plane, the transforms we have to calculate are given by the Bromwich integrals

ACKNOWLEDGMENTS

I. B. acknowledges the financial support by Nuffield Foundation 共NAL/00746/G兲. R. E. acknowledges M. Kéray for patient encouragement. I. B. and R. E. also acknowledge the financial support obtained from the NFS Hungary 共OTKA, T043741兲.

共A1兲

I1 =

冑冕

1

2␲i ␯

␥+i⬁

␥−i⬁

冑共z2 +

冉冑冑

⫻exp −

etz

␻2b兲共cT2 /␯ ␻2b

x

z2 +

␯

cT2 /␯ + z

冊

dz,

+ z兲共B1兲

and


042108-7



冕

F共z兲etz = − i

冕

cT2 /␯

⬁

C2

e−t␩

冑共␩2 + ␻2b兲共␩ − cT2 /␯兲

冉冑冑 x

⫻exp − i

␯

冊

␩2 + ␻2b d␩ . ␩ − cT2 /␯

共B3兲

Ⲑ

Ⲑ

On C3, the new variable is z = ␩e−i␲ with z + cT2 ␯ = 共 cT2 ␯ − ␩兲, so that the integral becomes

冕

F共z兲etz = −

C3

冕

e−t␩

0

冑共␩2 + ␻2b兲共cT2 /␯ − ␩兲

cT2 /␯

冉冑冑 x

⫻exp −

␯

冊

␩2 + ␻2b d␩ . cT2 /␯ − ␩

共B4兲

In a similar way, we can define a new variable for all remaining integration paths. Along C4: z = ␩e−i␲/2, z − i␻b = 共␻b + ␩兲e3i␲/2, z + i␻b = 共␻b − ␩兲ei␲/2; FIG. 1. The integration contour and the branch cuts for the inverse Laplace transforms of the quantities 共33兲 and 共34兲. The branch cuts are shown by crosses.

along C5: z = ␩e−i␲/2, z − i␻b = 共␻b + ␩兲ei␲/2, z + i␻b = 共␻b − ␩兲e−i␲/2; along C6:

I2 =

冑冕

1

2␲i ␯

␥+i⬁

␥−i⬁

冑共z2 + ␻2b兲共cT2 /␯ + z兲

冉冑冑

⫻exp −

z = ␩ei␲/2, z − i␻b = 共␻b − ␩兲e−i␲/2, z + i␻b = 共␻b + ␩兲ei␲/2;

zetz

x

z2 + ␻2b

␯

cT2 /␯ + z

冊

dz.

along C7: 共B2兲

Let us consider the first transform, the second one can be calculated in an analog way as presented here. The integrand in I1 contains two branch cuts: one is running on the real axis from z = −cT2 / ␯ to −⬁ and the other one is running on the imaginary axis in between ±i␻b 关in principle, we could have the symmetric branch cuts, but it can be shown that in this case the inverse Laplace transform cannot be defined as there will be impossible to choose such a Re共s兲 = ␥ in the complex s plane, so that all singularities lie on the left of the line at Re共s兲 = ␥兴. The integration contour is deformed as in Fig. 1, and the Bromwich integral transforms into a contour integral over the closed contour C. According to Fig. 1, the contour integral comprises the line integrals C1 , C2 , . . . , C10, plus the integration around the infinitesimally small circles around ±i␻b and −cT2 / ␯. First to note is that for large z, the integrand behaves as 1 / z5. Consequently, the contributions along the arcs C1 and C10 vanish as the radius of the arcs 共R兲 tends to infinity. This result is a direct consequence of Jordan’s lemma. On C2 we change the variable such that z = ␩e−i␲; i.e., z+

cT2

␯

冉冊

= ␩−

cT2

␯

e−i␲ .

The integral transforms into

z = ␩ei␲/2, z − i␻b = 共␻b − ␩兲e3i␲/2, z + i␻b = 共␻b + ␩兲ei␲/2; along C8: z = ␩ e i␲,

z + cT2 /␯ = c2/␯ − ␩;

along C9: z = ␩ e i␲,

z + cT2 /␯ = 共␩ − cT2 /␯兲ei␲ .

Upon substituting these relations into the integrals and after simplifying, one obtains

冕冕 =−

C3

,

C8

冕冕 +

C4

=

C5

冑冕

1

␲ ␯

␻b

0

+

C6

C7

=

冑冕

1

␲ ␯

冑共␻2b − ␩2兲共cT2 /␯ − i␩兲

冉冑冑

⫻cosh

冕冕

e−it␩

x

␻2b − ␩2

␯

cT2 /␯ − i␩

␻b

0

d␩ ,

eit␩

冑共␻2b − ␩2兲共cT2 /␯ + i␩兲

冉冑冑

⫻cosh

冊

x

␻2b − ␩2

␯

cT2 /␯ + i␩

冊

d␩ ,


042108-8



冕冕 =i

C9

⬁

e−t␩

cT2 /␯

冑共␻2b + ␩2兲共␩ − cT2 /␯兲

冉冑冑

⫻exp − i

x

␻2b + ␩2

␯

␩ − cT2 /␯

冊

I1 =

d␩ .

共B5兲

冕

C2艛C9

=

冑冕

1

␲ ␯

冉冑冑

e−t␩ cos

⬁

cT2 /␯

冑共␩

2

+

␩ + ␻2b x 2 ␯ ␩ − c T/ ␯ ␻2b兲共␩ − cT2 /␯兲

冊

冕

C4艛C5艛C6艛C7

⳵ = ␲冑␯ ⳵ x

冕

␻b

−␻b

e−i␩t ␻2b − ␩2

冉冑冑

⫻sinh

x

␻2b − ␩2

␯

cT2 /␯ − i␩

e−t␩

冑共␩2 + ␻2b兲共␩ − cT2 /␯兲

冉冑冑冉冑冑

冊

␩2 + ␻2b ⳵ d␩ + 2 ⳵x ␩ − c T/ ␯

x

␯

x

␯

␻2b − ␩2 cT2 /␯ − i␩

冊册

冕

␻b

−␻b

e−i␩t ␻2b − ␩2

d␩ .

共B8兲

Applying the described method for I2, we obtain that I2 = −

d␩ . 共B6兲

冑冋冕

1

␲ ␯

⬁

cT2 /␯

␩e−t␩

冑共␩2 + ␻2b兲共␩ − cT2 /␯兲

冉冑冑冉冑冑

⫻cos

⫻sinh

冊

⬁

cT2 /␯

⫻sinh

Summing up the integrals along the branch cut on the imaginary axis, we obtain that 1

␲ ␯

⫻cos

Adding all these integrals together, the integrals along the arcs C3 and C8 cancel, while the integrals along C2 and C9 will result in 2

冑冋冕

1

x

␯

冊

␩2 + ␻2b ⳵ d␩ − 2 ⳵x ␩ − c T/ ␯

x

␻2b − ␩2

␯

cT2 /␯ − i␩

冊册

d␩ .

冕

␻b

␩e−i␩t 2 2 −␻b ␻ b − ␩ 共B9兲

B. Roberts and A. R. Webb, Sol. Phys. 56, 5 共1978兲. A. Ferriz-Mas, M. Schuessler, and V. Anton, Astron. Astrophys. 210, 425 共1989兲. 3 I. C. Rae and B. Roberts, Astrophys. J. 256, 761 共1982兲. 4 W. Kalkofen, P. Rossi, G. Bodo, and S. Massaglia, Astron. Astrophys. 284, 976 共1994兲. 5 S. S. Hasan and W. Kalkofen, Astrophys. J. 519, 899 共1999兲. 6 Z. E. Musielak and P. Ulmschneider, Astron. Astrophys. 370, 541 共2001兲. 7 Z. E. Musielak and P. Ulmschneider, Astron. Astrophys. 400, 1057 共2003兲. 8 B. Roberts, Sol. Phys. 87, 77 共1983兲. 9 R. G. Giovanelli, W. C. Livingstone, and J. V. Harvey, Sol. Phys. 59, 40 共1978兲. 10 B. De Pontieu, T. Tarbell, and R. Erdélyi, Astrophys. J. 590, 502 共2003兲. 11 B. De Pontieu, R. Erdélyi, and S. P. James, Nature 共London兲 430, 536 共2004兲. 12 B. De Pontieu, R. Erdélyi, and I. De Moortel, Astrophys. J. 624L, 21 共2005兲. 13 C. A. Coulson, Waves 共Wiley Interscience, New York, 1955兲, p. 15. 14 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Table of Integral Transforms 共McGraw-Hill New-York, 1954兲, Vol. 1, p. 102. 15 G. Sutmann, Z. E. Musielak, and P. Ulmschneider, Astron. Astrophys. 340, 556 共1998兲. 1

d␩ .

共B7兲

Next we have to calculate the value of the integral around the singular points ±i␻b and −cT2 / ␯. When performing the integrals around the singular points on the imaginary axis, we introduce a new variable such that z = ± i␻b + ⑀ei␪ and the variable ␪ varies from −␲ / 2 to 3␲ / 2 for z = i␻b and −3␲ / 2 to ␲ / 2 at z = −i␻b. In both cases, the radius of the circle 共⑀兲 tends to zero. It can be shown that both integrals tend to zero when the radius of the circles surrounding the singular points tend to zero. At the singularity on the real axis, we introduce a new variable such that z = −cT2 / ␯ + ⑀ei␪, and the integrals are evaluated such that ␪ varies from −␲ to zero for the arc below the singularity and zero to ␲ for the arc above the singularity. Taking the radius of the circle infinitesimally small, both integrals tend to zero. Finally, the inverse Laplace transform of I1 takes the form

2
