frequency dependence of resonant absorption and over-reflection of magnetosonic waves in nonuniform...

Astron. Astrophys. 358, 1090–1096 (2000) ASTRONOMYAND

ASTROPHYSICS

Frequency dependence of resonant absorption and over-reflectionof magnetosonic waves in nonuniform structures with shear mass flow

A.T. Csık1, V.M. Cadez2, and M. Goossens1

1 Center for Plasma-Astrophysics, K.U. Leuven, Celestijnenlaan 200 B, 3001 Heverlee, Belgium2 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, 9000 Gent, Belgium

Received 28 October 1999 / Accepted 16 March 2000

Abstract. Processes of resonant absorption and over-reflectionof magnetosonic modes may occur in nonuniform plasmas witha shear mass flow along the magnetic field. In this work, westudy how these two phenomena depend on the frequencyω ofwaves propagating in a given direction prescribed by polar an-glesθ andφ of wave vectork. Plasma is assumed nonuniformwithin a transitional layer that separates two semi-infinite uni-form regions. The considered wave propagates only in one ofthe two regions and is reflected from the nonuniform transitionallayer either totally or partially depending on the efficiency ofresonant processes.

The resonant absorption and over-reflection are strongly de-pendent on wave frequency for both the slow and fast magne-tosonic mode as shown in corresponding figures.

Under conditions relevant to the solar atmosphere, we foundthat typical MHD waves with periods from few seconds up tomuch larger values of several minutes and more, can be reso-nantly absorbed and also over-reflected. This important fact hasto be taken into account in making models of energy transportsin the corona.

Key words: Magnetohydrodynamics (MHD) – methods: nu-merical – Sun: corona – Sun: oscillations

1. Introduction

In this paper we continue our study (Csık et al. 1998) of reso-nant absorption and over reflection of magnetoacoustic wavesin a nonuniform plasma layer with a mass flow that may occurin solar atmosphere. We focus our attention on how these twophenomena depend on the frequency of waves and on param-eters describing basic states relevant to conditions in the solarcorona. The two models discussed in our paper, are not meantto be related to a specific feature in the solar atmosphere likehelmet streamers, arcades, etc., instead they can be consideredas constituent elements in modeling such features. Concerningthe earlier results in this field we refer to the number of paperscited in the above reference.

Send offprint requests to: A.T. Csık

The details of the theoretical and computational proceduresare described in Csık et al. (1998). Thus, we consider slow andfast magnetosonic waves with a given frequencyω and wavevectork propagating through a uniform semi infinite half-spacez ≥ L (Region 2) and encountering a nonuniform layer atL ≥z ≥ 0 where resonant absorption and over reflection of wavesmay occur (see Fig. 1). The rest of the space (z ≤ 0, Region1) is of uniform composition but assumed nontransparent forwaves we study. For this reason, the difference in amplitudes ofthe reflected and the incident wave results only from processesrelated to the resonances: The amplitude of the reflected wavemay be either larger than that of the incoming wave, in whichcase we have over-reflection, or it may be smaller, meaning thatresonant absorption takes place.

The initial unperturbed state is as considered in our pre-vious paper (Csık et al. 1998): The plasma configuration con-sists of two semi-infinite homogeneous regions, separated by anonuniform layer. Thez− axis is perpendicular to the bound-aries of the nonuniform plasma layer atL ≥ z ≥ 0 and all basicstate quantities depend on the variablez only. The magneticfield B0 = (B0, 0, 0) is constant throughout the space whilea uniform plasma flow with velocityU0 = (U0, 0, 0) alongthe magnetic field, exists atL > z only, i.e.U0 = constant atL > z andU0 = 0 elsewhere. Due to the stationary equilib-rium of the basic state, the thermal pressurep0 and the plasmaβ = 2p0 µ0/B2

0are constant, while the densityρ0(z) and the

temperatureT0(z) have smooth profiles within the layer. Theratio of the temperatures of the two uniform domains is definedby rT = T0(L)/T0(0), which is a key parameter in the rest ofthis paper.

Concerning the governing equations, the linearized idealMHD equations are reduced to two coupled ordinary differ-ential equations which we solve by Runge-Kutta method in thenonuniform layer. Singularities are treated by using the jump re-lation technique (Goossens & Ruderman, 1996). At the bound-aries of the nonuniform layer the numerical solution is con-nected to the analytic solution in the uniform domains. For moredetails regarding the applied mathematical procedure and defi-nitions we refer to our previous article by Csık et al. (1998).

The paper is organized as follows: Introductory statementsabout the equilibrium configuration and governing equations

A.T. Csık et al.: Frequency dependence of resonant absorption and over-reflection 1091

z=L

ω

ωωω

ωωω

ω

II

A

I

I

A

II

-

-

-

-

z=0

c

c

z

ω

1

2

3

4

5

6

7

8

9

10

slow wave

fast wave

Nonuniform LayerRegion 1 Region 2

Fig. 1. Schematic profiles of characteristic wave frequencies in the con-sidered model:ωA(z), ωc(z), ωI(z) andωII(z) are the local Alfven,cusp, lower and upper cutoff frequency respectively. Atz = L, the fre-quency of the incoming (fast/slow) wave formally changes fromω toΩlocated within one of ten indicated intervals. The solid and the dashedparts of horizontal arrows indicate propagating and evanescent wavesrespectively. Locations of Alfven and cusp resonances are marked bysolid and empty squares respectively.

for linear magnetoacoustic waves are described in Sect. 1. Thecomputation of the absorption coefficient and numerical resultsare given in Sect. 2 while the discussion and conclusions arefound in Sect. 3.

2. Coefficient of absorption and numerical results

Numerical calculations of the absorption coefficientA(ω, U0)are performed with normalized physical quantities: Velocitiesare scaled to the Alfven speedvA1≡ vA(0) in Region 1, thelengths are in units ofL while the density and the temperatureare normalized toρ0(0) andT0(0), respectively. Consequently,time t and wave frequencyω are scaled toL/vA1 andvA1/L,respectively.

Depending onβ, we shall consider two groups of basic statesthat are relevant to conditions in the solar atmosphere. First, wetakeβ = 0.1 a value adequate to lower parts of the solar at-mosphere and closer to the photosphere. The second choice isβ = 0.01 which is more appropriate to higher regions of thecorona. For each of these cases, we shall consider two possibil-ities for the ratio of temperatures across the transitional layer:rT = 0.1 and 0.5, i.e. region 1 is assumed warmer than region 2.

The incident wave generated in the cooler region 2, prop-agates in the negativez− direction towards the warmer region1. Inside the non-uniform layer between the two regions, thewave is partially reflected back into region 2. If its amplituderemains unchanged, we have the total wave reflection with thecoefficient of absorptionA = 0. If the amplitude of the reflectedwave is changed it can be either decreased or increased whichindicates the resonant absorption with1 ≥ A > 0, and resonantover-reflection withA < 0 respectively.

The considered waves can either be slow or fast MHD modeswith arbitrary frequencies. Propagation angles of these wavesare kept fixed toθ = 45o andφ = 30o, the values that yieldthe largest resonant absorption in a static model (Cadez et al.1997). The wavelengthλ = 2π/k of these modes is obtainedin the following form:

(

λ

L

)

f

≈ 6.3r1/2

T

ωf,

and(

λ

L

)

s

≈ 3.5βr

1/2

T

ωs,

in the lowβ approximation. Parametersβ andrT may have anyprescribed values while the flow speeds are taken sub-Alfvenicin the uniform region 1, i.e. computations were performed for|U0| ≤ 1 only.

The coefficient of absorptionA(ω, U0) is calculated by ap-plying the method described in our previous paper. However,as stated before, the resonant processes are considered only forthose incident waves that can not propagate through the wholenonuniform layer, i.e. the region 1 is taken opaque in our study.The plotted values of the absorption coefficientA are thereforeeither equal to zero as in the case of total reflection (domainslabeled by “T.R.” in figures), or they are set to zero as in thecase when region 1 becomes transparent. The correspondingdomains are labeled in Figs. 2-9 by “T.R.” (“Total Reflection”)and “N.C.” (“Not Computed”) respectively.

2.1. Model I: the basic state with β=0.1

We first give the results concerning the basic state with a rel-atively weaker magnetic field, withβ = 0.1, that can modellower parts of the solar atmosphere.

The upper figures in Figs. 2 and 4 show the dependence of theabsorption coefficientA on normalized wave frequencyω andnormalized flow speedU0 for the temperature ratiorT = 0.1 ifthe incident waves are the slow and fast mode respectively. Theleft bottom plots in Figs. 2 and 4 show top views ofA(ω, U0)while the right bottom figures give cross-sections ofA(ω, U0)at two fixed frequenciesω = 0.5 (solid curves) andω = 1.5(dotted curves) for each of the two modes.

The behaviour ofA at a larger temperature ratio ofrT = 0.5,i.e. for a smaller change of temperature across the layer, is seenin Figs. 3 and 5. The left plots of these two figures show the topview of the 3D-plotsA(ω, U0) for rT = 0.1andrT = 0.5 for the

1092 A.T. Csık et al.: Frequency dependence of resonant absorption and over-reflection

-400

-300

-200

-100

0

100

abso

rptio

n %

0

0.5

1

1.5

2frequency

-1

-0.5

0

0.5

1

flow

X Y

Z

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

N.C.

T.R.

-1 -0.5 0 0.5 1flow

-150

-100

-50

0

50

abso

rptio

n %

Fig. 2. Slow MHD waves forβ=0.1 andrT =0.1. Top: The absorp-tion coefficientA(ω, U0). Bottom left: A top view of the above 3Dplot. Labels “N.C.” and “T.R.” stand for “Not Computed” and “TotalReflection” respectively.Bottom right: Variation ofA(ω, U0) at twofixed frequenciesω= 1.5 and 0.5 indicated by a dotted and solid curverespectively.

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

T.R.

N.C.

-1 -0.5 0 0.5 1flow

-150

-100

-50

0

50

abso

rptio

n %

Fig. 3. Left: The absorption coefficient of slow MHD waves atβ=0.1and rT =0.5 (top view). Labels “N.C.” and “T.R.” stand for “NotComputed” and “Total Reflection” respectively.Right: Variation ofA(ω, U0) at two fixed frequenciesω= 1.5 and 0.5 indicated by a dot-ted and solid curve respectively.

slow and fast MHD mode respectively. On the right of the plotswe give cross-sections ofA(ω, U0) at two fixed frequenciesω = 0.5 (solid curves) andω = 1.5 (dotted curves) for each ofthe two modes.

-40

-20

0

20

40

abso

rptio

n %

0

0.5

1

1.5

2frequency

-1

-0.5

0

0.5

1

flow

X Y

Z

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

T.R.

N.C.

-1 -0.5 0 0.5 1flow

-40

-30

-20

-10

0

10

20

30

abso

rptio

n %

Fig. 4. Fast MHD waves forβ=0.1 andrT =0.1. Top: The absorp-tion coefficientA(ω, U0). Bottom left: A top view of the above 3Dplot. Labels “N.C.” and “T.R.” stand for “Not Computed” and “TotalReflection” respectively.Bottom right: Variation ofA(ω, U0) at twofixed frequenciesω= 1.5 and 0.5 indicated by a dotted and solid curverespectively.

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

-1 -0.5 0 0.5 1flow

0

5

10

15

20

25

abso

rptio

n %

Fig. 5. Left: The absorption coefficient of fast MHD waves atβ=0.1 andrT =0.5 (top view). Labels “N.C.” and “T.R.” stand for “Not Computed”and “Total Reflection” respectively.Right: Variation ofA(ω, U0)at twofixed frequenciesω= 1.5 and 0.5 indicated by a dotted and solid curverespectively.

2.2. Model II: the basic state with β=0.01

A smaller valueβ = 0.01 in the basic state would be more ap-propriate to conditions in the upper part of the solar atmosphere.Results showing functional dependence of the absorption coef-ficientA(ω, U0) on the wave frequency and the flow speed are


-300

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-100

0

abso

rptio

n %

0

0.5

1

1.5

2frequency

-1

-0.5

0

0.5

1

flow

X Y

Z

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

T.R.

N.C.

-1 -0.5 0 0.5 1flow

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

abso

rptio

n %

Fig. 6. Slow MHD waves forβ=0.01 andrT =0.1.Top: The absorp-tion coefficientA(ω, U0). Bottom left: A top view of the above 3Dplot. Labels “N.C.” and “T.R.” stand for “Not Computed” and “TotalReflection” respectively.Bottom right: Variation ofA(ω, U0) at twofixed frequenciesω= 1.5 and 0.5 indicated by a dotted and solid curverespectively.

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

T.R.

N.C.

-1 -0.5 0 0.5 1flow

-30

-20

-10

0

10

20

30

40

50

abso

rptio

n

Fig. 7. Left: The absorption coefficient of slow MHD waves atβ=0.01and rT =0.5 (top view). Labels “N.C.” and “T.R.” stand for “NotComputed” and “Total Reflection” respectively.Right: Variation ofA(ω, U0) at two fixed frequenciesω= 1.5 and 0.5 indicated by a dot-ted and solid curve respectively.

given in Figs. 6, 7, 8 and 9. All the related descriptions of plotswith β = 0.01 are now analogous to those for Figs. 2, 3, 4 and5 respectively, whereβ = 0.1.

It is straightforward to check that the values of the plasmaβ and the temperature ratiorT taken in our models yield equi-librium states in which the cusp speed in the warmer region isalways smaller than the Alfven speed in the cooler region. Thisthen implies that either the cusp or Alfven resonance can occur

-40

-20

0

20

40

abso

rptio

n %

0

0.5

1

1.5

2frequency

-1

-0.5

0

0.5

1

flow

X Y

Z

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

T.R.

N.C.

-1 -0.5 0 0.5 1flow

-40

-30

-20

-10

0

10

20

30

abso

rptio

n %

Fig. 8. Fast MHD waves forβ=0.01 andrT =0.1. Top: The absorp-tion coefficientA(ω, U0). Bottom left: A top view of the above 3Dplot. Labels “N.C.” and “T.R.” stand for “Not Computed” and “TotalReflection” respectively.Bottom right: Variation ofA(ω, U0) at twofixed frequenciesω= 1.5 and 0.5 indicated by a dotted and solid curverespectively.

-1

-0.5

0

0.5

1

flow

0 0.5 1 1.5 2

frequency

N.C.

-1 -0.5 0 0.5 1flow

0

5

10

15

20

25

30ab

sorp

tion

%

Fig. 9. Left: The absorption coefficient of fast MHD waves atβ=0.01and rT =0.5 (top view). Labels “N.C.” and “T.R.” stand for “NotComputed” and “Total Reflection” respectively.Right: Variation ofA(ω, U0) at two fixed frequenciesω= 1.5 and 0.5 indicated by a dot-ted and solid curve respectively.

individually, i.e. they can not occur together simultaneously inour models. Consequently, the waves approach the cusp reso-nance always directly while the Alfven resonance is accessedonly by the tunneling.

In Figs. 2-9 we see the domains of positive and negativeabsorption coefficient related to resonant absorption and over-reflection, respectively. We also see that the resonant over-reflection (A < 0) of both modes occurs only if the flow is


positive:U0 > 0, i.e. if the wave propagates along the flow. Thephenomenon of over-reflection cannot occur in a static mediumas the increase of amplitudes of reflected waves requires an en-ergy extraction from the flow. As to the resonant absorption(1 ≥ A > 0), it can take place also in a static medium whilethe presence of a flow will only modify it.

It can be shown that a wave with frequencyω and wavevectork is over-reflected if

ω < kxU0. (1)

This means that thex−component of the wave phase velocityhas to be smaller than the flow speed.

3. Discussion and conclusions

Based on Figs. 2-9, we can now summarize the following prop-erties of absorption and over-reflection of slow and fast magne-tosonic waves that propagate along the direction withφ = 30o

andθ = 45o:

Slow MHD waves. The resonant absorption of the slow MHDmode takes place when the wave propagates against the flow(U0 < 0) and it arises either from the Alfven or from the cuspresonance depending on the speed of flow. The Alfven reso-nance is effective at relatively higher speedsU0 and waves canreach it by tunneling only. The related coefficient of absorp-tionA has a pronounced periodic frequency dependence whosemaxima decrease withω as seen in Figs. 2 and 6 for both casesrT =0.1 andrT =0.5. The cusp resonance, however, causes theabsorption at lower speedsU0 and its efficiency grows slowlywith ω within the range of frequencies in Figs. 2 and 6. The in-cident slow MHD waves reach this resonance directly i.e. theirfrequencies fall within the slow continuum. Comparing the ab-sorption coefficients related to the two resonances, we see thatthe Alfven resonance yields higher values forA than the cuspresonance if the wave frequencies are lower (ω ≤ 0.5). Thereverse is true at larger frequencies (ω ≥ 1.5) when the cuspresonance results into a more pronounced absorption of the slowmode (Figs. 2 and 6).

The resonant over-reflection occurs provided condition (1)is satisfied. The over-reflection is far most effective at smallerfrequenciesω ≤ 0.5 when the Alfven resonance is involved.Waves in this case propagate along the flow (U0 > 0) and gainenergy from it. The absolute value|A| is usually much largerfor over-reflection than for absorption and it decreases with fre-quencyω in both cases as seen in Figs. 2 and 6.

A rippled oscillatory character of|A| is noticeable both withrespect to variations of the speed of flowU0 and wave frequencyω. For the applied scale and range of wave parameters and forthe considered choice of the basic state, the oscillatory variationof |A| is more frequent along theU0-axis than along theω-axis.This comes from the fact that the value of|A| is highly sensitiveto variations inU0 for slow waves propagating in a lowβ plasmathat we are considering. In that case,ω ∼ βk meaning that thedimensionless wavenumberk ∼ ω/β has to be sufficiently largeif the frequencyω is to be within the range used in Figs. 2 and

6. If k is large, then already small changes inU0 will cause alarge variation of the productkU0. This explains the noticeabledifference in variation of|A| taken along theω− andU0− axesin Figs. 2 and 6. The same explanation holds also for the factthat ripples in plots of|A| are getting more condensed ifβdecreases as seen in corresponding figures forβ = 0.1 andβ = 0.01 respectively. We also see lower values of|A| at thesmallerβ.

The rippling seen in plots of|A| also depends on the tem-perature ratiorT : It disappears in the domain of the Alfvenresonance ifrT is changed from 0.1 to 0.5 as seen in Figs. 3and 7.

Fast MHD waves. In the case of fast MHD waves, there areagain two pronounced domains with enhanced values of thecoefficient|A| as seen in Figs. 4 and 8. However, contrary tothe slow MHD mode, the domain of enhanced wave absorption(1 ≥ A > 0) due to the Alfven resonance now occurs at slowflow speeds, includingU0 = 0, while the cusp resonance iseffective only if the speed of flowU0 is positive and sufficientlylarge. The same holds for over-reflection (A < 0) except that itcan occur only ifU0 /= 0. The absolute value of the absorptioncoefficient|A| continuously increases with the wave frequencyω for both resonances.

In the case whereβ = 0.1, the resonant absorption and over-reflection are slightly smaller for the cusp resonance than for theAlfv en resonance. However, ifβ is reduced toβ = 0.01, thecoefficientA becomes negligible for the cusp resonance whileit remains practically unchanged for the Alfven resonance. Thissuggests that the resonant absorption and over-reflection of fastMHD waves at the Alfven resonance are not too sensitive tothe actual value of the plasmaβ for the considered range ofparameters in our models.

Another difference with respect to the slow MHD mode isthat the fast MHD mode exhibits no rippled structure in plots ofA as seen in Figs. 4 and 8.

The dependence ofA on temperature ratiorT is stronger forthe fast mode. Thus, forβ = 0.01, a change fromrT = 0.1 to0.5 removes the absorption at small flow speeds aroundU0 = 0and also the complete over-reflection vanishes in the consideredinterval of parameters. All that is left of the resonant effects isa relatively narrow domain of wave absorption for0.5 ≥ U0 ≥0.2 for β = 0.01 and two narrow domains ifβ = 0.1 (Figs. 5and 9).

3.1. Numerical examples

To study the importance of obtained results to solar conditionswe shall discuss some numerical examples for each of the twomodels from the previous section that are related toβ = 0.1 andβ = 0.01 respectively. To do this, we shall use plotsA(U0) attwo fixed frequenciesω = 0.5 andω = 1.5 shown in Figs. 2-9.The dimensional parameters relevant to the solar atmosphereare taken from the book by Cravens (1997).


To evaluate resonant effects at these two frequencies un-der conditions in the solar atmosphere, we switch back to di-mensional quantities. In a given model described by parame-tersβ andrT , we are free to prescribe the scaling quantitiesvA1≡ vA(0), L, andT0(0). In this sense, we first write thespeed of sound in the following form:

vs =

(

γp0

ρ0

)1/2

= (γRT0(0))1/2

= (0.141T0(0))1/2 km s−1 (2)

where:γ ≡ Cp/Cv = 5/3, R ≡ R/M and the molar gasconstant isR = 8.314 J mol−1 K−1. The mean molar massMfor the ionized solar atmosphere composed mostly of hydrogenand helium ions and free electrons, is takenM ≈ 7 × 10−4 kgmol−1.

Using the speed of sound as given by Eq.(2), we can writethe scaling speedvA1 as

vA1 = vs

(

2

γβ

)1/2

≈ 0.16

(

T0(0)

β

)1/2

km s−1, (3)

for initially prescribed values ofβ and temperatureT0(0) inunits of K.

The dimensional expression for the speed of flowu0 is then

u0 ≡ U0vA1 ≈ 0.16U0

(

T0(0)

β

)1/2

km s−1, (4)

whereU0 is the dimensionless speed of flow as it appears infigures with plotted results.

The dimensional wave oscillation periodτ takes the follow-ing form

τ ≡2π

ω

L

vA1

≈ 40L

ω

(

β

T0(0)

)1/2

s, (5)

whereω is the dimensionless wave frequency as it shows up inplotted results, while the thickness of the layerL that scales thelength, is given in kilometers.

Model I: β = 0.1 . In the case withβ =0.1, we take an examplewhere the temperature of the hotter region 1 isT0 = 4 × 104 Kwhich is an acceptable value for the lower part of the transitionallayer that separates the solar chromosphere from the corona.Eqs.(3)-(5) then yield the following values for the scaling ve-locity vA1, dimensional speed of flow (in km s−1) and the oscil-lation periodτ (in milliseconds) of the wave with dimensionlessfrequencyω:

vA1 ≈ 100km s−1,

u0 ≈ 100 U0km s−1,

τ ≈ 63L

ωms,

(6)

whereL is in km.A pronounced phenomenon of resonant absorption and over-

reflection (Fig. 2) takes place for a givenω depending on the

speed of flowU0 and on the mode of the wave. For a slow modewith ω ≈ 0.5 this occurs at flow speedsU0 ≈ ∓0.5 or in thedimensional form, according to Eq.(6):

τ ≈ 0.126 L s, u0 ≈ ∓50km s−1.

Thus, if the thickness of the layer isL = 1000 km, the mostabsorbed waves would have periods aroundτ ≈ 2 min if aflow speed ofu0 ≈ 50 km s−1 is present. The same flow speedbut in the opposite direction, will cause a pronounced over-reflection on this frequency. A smaller thickness of the layer,sayL = 10 km, will cause the same absorption of waves butwith a shorter period ofτ ≈ 1.2 s.

There is also a smaller and sharper peak in resonant absorp-tion reaching 50% (Figs. 2 and 3) that appears at very slow flowspeedsu0 ≈0. The peak already disappears atU0 =-0.1 i.e. ifthe speed of flow exceedsu0 = 5 km s−1 when the wave be-comes totally reflected. A similar peak occurs for over-reflectionatU0 ≈ 0.15 oru0 ≈ 7 km s−1. Thus, relatively slow flows cansignificantly influence the efficiency of considered resonances.

Looking now at the fast mode, we see from Figs. 4 and 5 thatthey are less affected by resonances than the slow mode so thatthe absorption coefficient does not exceed 30% in the domainof frequencies and flow speeds considered in figures. As to theover-reflection, it appears in Fig. 4 atU0 ≈0.6 if the temperatureratio isrT =0.1 and it does not occur at all for frequenciesω ≤ 2if the temperature ratio isrT =0.5.

Contrary to the slow mode, the fast mode is more absorbedat higher frequencies (Figs. 4 and 5) but the coefficient of ab-sorption does not exceed 30%.

Model II: β = 0.01 . If β = 0.01 is taken in Eqs.(3)-(5) thenwe have:

vA1 ≈ 1.6 T0(0)1/2km s−1,

u0 ≈ 1.6 U0T0(0)1/2km s−1,

τ ≈4L

ωT1/2

0(0)

s.

(7)

Comparing these values with(6) in model I, we see the scalingspeedvA1 is now larger and oscillation periods are shorter atthe same values of temperature andL. However, ifβ = 0.01 as-sumes applications to the corona, then we have to takeT0(0) =106 K. This results into even higher speeds of flow that can in-fluence the absorption and cause over-reflection. For example,a slow mode with any frequencyω ≤ 2 can be over-reflectedfrom the layer withrT =0.5 if the flow speed isU0 ≈0.1 oru0 ≈160 km s−1 as seen in Fig. 7. Time periods of these wavesthen range in the intervalτ ≥ L/500 s where the thicknessof the layerL is given in km. Thus, whatever the value takenfor L, there will be a broad range of slow modes that can beover-reflected at flow speedsu0 ≈160 km s−1. However, forthe same model, fast waves can be already absorbed for smallerflow speedsU0 ≈0.06 oru0 ≈95 km s−1 (see Fig. 7).

To conclude, resonant processes due to the Alfven andcusp resonance influence MHD waves at frequencies typical


for the solar atmosphere. Thus the resonant absorption and over-reflection can easily occur for waves with time periods from fewseconds and up. This means that resonances in local nonunifor-mities may play an important role in energetics of the solaratmosphere.

Acknowledgements. A. Csık is grateful to the Onderzoeksfonds of theKatholic University of Leuven, Belgium. V. M.Cadez acknowledgesa sabbatical visitor grant obtained by the Bijzonder Onderzoeksfondsof the Universiteit Gent, Belgium.

References

Cadez V.M., Csık A., Erdelyi R., Goossens M., 1997, A&A 324, 1170Csık A., Cadez V. M., Goossens M., 1998, A&A 339, 215Cravens T.E., 1997, Physics of Solar System Plasmas. Cambridge At-

mospheric and Space Science Series, Cambridge University Press,Cambridge, UK

Goossens M., Ruderman M.S., 1996, Physica Scripta T60, 171

frequency dependence of resonant absorption and over-reflection of magnetosonic waves in nonuniform...

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