Prominence oscillations and stability

Astron. Astrophys. 323, 969–985 (1997) ASTRONOMYAND

ASTROPHYSICS

Prominence oscillations and stability

Communicating the distant photospheric boundary

N.A.J. Schutgens

Sterrekundig Instituut Utrecht, Utrecht University, The Netherlands

Received 29 July 1996 / Accepted 16 January 1997

Abstract. The photosphere provides an important boundarycondition for prominence support. The conservation of pho-tospheric flux (sometimes called line tying) sets a serious con-straint on the evolution of coronal magnetic fields. This bound-ary condition can only be communicated to the prominence byAlfven and magneto-acoustic waves. As a result, the boundarycondition as experienced by the prominence at height h lagsbehind a time h/vA (vA: Alfvenspeed) as compared to the in-stantaneous situation at the location of the photosphere.

In this paper I study vertical oscillations and stability ofprominences, taking retardation effects into account. An equa-tion of motion for a Kuperus-Raadu prominence is derived, de-scribing the prominence as a line current and the photosphereas a perfectly conducting plate. Solving this equation of motionimplies solving the full time-dependent Maxwell equations, thusguaranteeing a realistic field evolution under the assumption ofphotospheric line tying. In terms of the currents that flow, sucha description is equivalent to the corresponding MHD picture.

The results indicate that the travel time h/vA is an impor-tant parameter of the system as it influences the decay or growthtimes of prominence oscillations greatly. A new kind of instabil-ity is found, whereby the prominence experiences oscillationsgrowing in time, even in the nonlinear regime. This instabilityoccurs when the travel time h/vA is comparable to or greaterthan the oscillation period. Also, forced oscillations can only besignificant for rather precisely matched values of h/vA and thedriving period.

Key words: Sun: prominences – Sun: oscillations – magneticfields – waves

1. Introduction

Prominence oscillations were first observed, using narrow band-pass filter-grams, in so-called ‘winking’ filaments (see Ramsey

Send offprint requests to: Nick Schutgens, Sterrekundig InstituutUtrecht, Utrecht University, postbus 80,000, 3508 TA Utrecht, TheNetherlands

& Smith 1966, Hyder 1966, Kleczek & Kuperus 1969). Thewinking of filaments is presumably triggered by nearby flares.As a result the whole (quiescent) prominence oscillates hori-zontally.

With the advance of more sophisticated spectral techniques,other oscillatory modes of prominences were detected. Theseinvolve only part of the prominence body and are not necessar-ily flare related. Most observations of quiescent prominenceshave been made on the limb. They usually reveal oscillations inDoppler shifts of spectral lines, but hardly ever changes in linewidth or line intensity. See Tsubaki (1988) for a review. Theoscillations can be grouped in short period (3–8 min.) and longperiod (40–80 min.) oscillations. The velocity amplitudes arearound 2 km/s. The absence of variations of the line width andthe line intensity suggests incompressible phenomena. Theseresults have since 1988 been reconfirmed by observations ofSuematsu et al. (1990) and Mashnich et al. (1993).

None of the above mentioned observations had the requiredspatial resolution and field of view to resolve the particu-lar prominence structures that were oscillating. Recently 2D-spectral scans of filaments on the disk were studied by Thomp-son & Schmieder (1991), Yi Zhang et al. (1991) and Yi Zhang& Engvold (1991). Due to higher spatial resolution the actualstructures that oscillate within a prominence could be resolved.It appears that the oscillations are confined to thread-like spa-tial structures. Thompson & Schmieder (1991) found periodsof 2.5–4 min. with velocity amplitudes of 0.5 km/s. Yi Zhang etal. (1991) found periods of 5–16 min. Velocity amplitudes were2 km/s. Jensen et al. (1994) proposed Alfven waves travelingalong flux-tubes to explain the observations of Yi Zhang et al.

The observed thread-like structures are probably related tothe filament fine structure (see also Schmieder 1988, 1992).These fine structures are made up of fibrils 300 to 1000 kmin diameter and contain most of the filament matter. In diskfilaments these fibrils are parallel to the photosphere, but usuallyinclined at angles of 15◦ to 30◦ to the photospheric neutral line.It is very well possible that the observed structures consist ofmore than one fibril (Mein & Mein 1991).

On the theoretical side, considerable effort has been madeto describe oscillations. The earliest attempts at describing os-

970 N.A.J. Schutgens: Prominence oscillations and stability

cillations concern ‘winking’ filaments (Hyder 1966, Kleczek &Kuperus 1969). These authors modelled the prominence slab asa rigid free harmonic oscillator. More detailed studies followedlater by Galindo-Trejo (1987), Oliver et al. (1992, 1993, 1995)and Joarder & Roberts (1992, 1993), who studied prominenceoscillations using the ideal MHD equations linearized around aprominence equilibrium model. Recently Joarder et al. (1997)studied the influence of fine structure on prominence oscilla-tions. All authors derive oscillation periods that agree with theobserved periods. There is however a large discrepancy in thepolarization of the oscillatory modes. This is probably due to dif-ferent choices of the equilibrium model. However, it should benoted that allmost all of these models have the same topology,which coincides with the topology of a Kippenhahn-Schluter(1957) prominence (hereafter KS). It is also noteworthy that alloscillatory modes thus found are marginally stable, with the ex-ception of the modes obtained by Hyder and Kleczek & Kuperuswho introduced some ad-hoc form of damping.

The stability of KS-type prominence equilibria was studiedby De Bruyne & Hood (1993), Strauss & Longcope (1994),Longbottom et al. (1994), Longbottom & Hood (1994a,b) usingan energy-method. This allows one to study how prominencemass and field strength etc. influence the stability properties. Asexpected for KS-type models, line-tying greatly stabilizes suchprominences.

For a different class of prominence models (Kuperus &Raadu, 1974, hereafter KR), it was shown that photosphericflux conservation, that leads to line-tying, can also destabilizeprominences. In these models the prominence is characterizedby a strong coronal current, whose field cannot penetrate thephotosphere. The photosphere effectively repels the prominenceand can thus support it against gravity or the downward forceof an overlying arcade. Van Tend & Kuperus (1978) showedthat such a prominence becomes unstable if the current is largerthan some critical value, resulting in a sudden upward motion ofthe filament. This instability has been used to study two ribbonflares (Kaastra 1975, Martens & Kuin 1989).

Comparative studies of the vertical stability of KS- and KR-type prominences have been made by Amari & Aly (1989),Demoulin & Priest (1988), Ridgway et al. (1991) and Demoulinet al. (1991) (see also Anzer & Ballester 1990 and Low 1993).These authors model the prominence as an infinitely thin wireor slab located in a force free field and then perform a quasi-stationary stability analysis. Demoulin & Priest and Demoulinet al. show that both types can suffer a van Tend-Kuperus kindof instability, but that KR prominences are more prone to it.

All stability studies of KR prominences mentioned sofar usea quasi-stationary approach for the field evolution. The basicforce balance is then

mz(t) = I0

[µ0

4πI0

z(t)−Bcor(z(t))

](1)

where m is the prominence mass per meter, z(t) the height attime t and I0 the prominence current. The magnetic permeabil-ity of vacuum is given by µ0. The repelling force due to thephotosphere is here modelled through a mirror current, which

Alfven wavesAlfven waves

Fast waves

Fast waves

Fig. 1. A single flux tube and its ambient field. The dashed lines outsidethe flux tube represent the ambient field as formed by surrounding fluxtubes and the coronal arcade. In this field topology, Alfven wave pack-ets transport information along the flux tube, while fast wave packetstransport it through the whole corona.

explains the first term on the right hand side. The second term isdue to an overlying coronal arcade that pushes the prominencedown. Van den Oord & Kuperus (1992) pointed out that such aquasi-stationary approach ignores a potentially very importanteffect: the delay in the repelling force. Photospheric currents setup the field that will repel the prominence. but it takes these fieldsa time z/vA to reach the prominence! The real delay is actuallytwice that value as the prominence determines the photosphericcurrents that repel it. The repelling field does not change untila magnetic wave has travelled down from prominence to pho-tosphere, the photospheric currents have changed, and anothermagnetic wave has travelled upward to the prominence. Alter-natively, one can say that field changes near the prominencespread through the corona. Those that reach the photosphereare reflected and can influence the prominence. Van den Oord& Kuperus used the following equation of motion

mz(t) = I(t)

[µ0

4π2I(t′)

z(t) + z(t′)−Bcor(z(t))

], (2)

vA(t− t′) = z(t) + z(t′), (3)

where the last equation is the so-called retardation condition.It simply states that the Alfven travel time between the promi-nence at height z(t) and the mirror current at −z(t′) is t − t′.This is a heuristical approach as the field is not computed self-consistently from the Maxwell equations.

Consider a collection of flux tubes in the corona, anchoredin the photosphere. These flux tubes can be seen as constitutinga prominence. The dominant coronal field (apart from the fluxtube) is an arcade (bipolar region). Under the usual assump-tion that the corona is a low β plasma, the influence of slowmagneto-acoustic waves can be neglected. Also, both Alfvenand fast magneto-acoustic waves travel with approximately thesame (Alfven) group speed. Changes in the fields are propagatedas wave packets. The Alfven wave packets travel only along thefield lines, while the fast magneto-acoustic wave packets travel’isotropically’ through the coronal medium. Thus photospheric

N.A.J. Schutgens: Prominence oscillations and stability 971

0

Photosphere

Flux tube

length l

heig

ht h

0

Fig. 2. Geometry of prominence (solid line) and two possible wavetravel paths (dashed resp. dotted lines). Fast wave packets first commu-nicate a local change in the prominence equilibrium to the photosphere,and consequently the local change in photospheric currents (read: re-pulsive force) to parts (in this case the top) of the prominence. At eachpart of the prominence, not a single discrete lag, but a distribution oflags is experienced.

twisting of the flux tubes is propagated through Alfven wavepackets, while the fast magneto-acoustic wave packets propa-gate global motion of the tubes. It is the fast magneto-acousticwave that establishes ‘communication’ between photosphereand prominence (see Fig. 1).

Assume a height h0 and a length l0 for the tube. As thefast magneto-acoustic waves travel isotropically, the typical lagswill range from 2h0/vA to 2

√h2

0 + l20/4/vA (see Fig. 2). Forquiescent prominences typical values are h0 = 30 000 km, l0 =100 000 km. Taking vA = 1000 km/s for the corona, lags of1–2 min. can be expected. The shortest prominence oscillationperiods are of the same magnitude.

A self-consistent way of studying the dynamics of a KRprominence under the influence of a distant photospheric bound-ary thus involves describing the magnetic waves that run be-tween the photosphere and filament. More particular, I will studythe oscillations of a system of a current wire (a prominence) sus-pended above a perfectly conducting plate (photosphere) witha background field (coronal arcade) by simultaneously solvingthe equation of motion and Maxwell’s equations. In the actualcomputation, the influence of the conducting plate is describedby a mirror current, chosen in such a way that the flux at thephotospheric level is the same.

To describe prominence oscillations, a differential equationwith delays is used (Eq. 6). Such differential equations arisenaturally in a host of research fields, like population biology(Cushing 1977, MacDonald 1978) or control and optimizationtheory (Minorsky 1962). Equation (2) is an example of a differ-ential equation with a delay. A good introduction to differentialequations with delays is found in Saaty (1981). Recently theoscillation theory for these differential equations was discussedin Gyori & Ladas (1991). Two outstanding treatments on differ-ential difference equations (differential equations with constantdelays) are by Pinney (1958) and Bellman & Cooke (1963).

The paper is organized as follows. In Sect. 2, the magneticfield of a prescribed and evolving current distribution is dis-cussed. This is the first step in describing the dynamical field ofthe moving mirror current. Next, in Sect. 3, the Lorentz forcedue to the mirror current is derived and incorporated into an

equation of motion. This equation of motion is discussed andcompared with both the van Tend-Kuperus and van den Oord-Kuperus equations. Section 4 describes numerical studies of theequation of motion and also the analytic solution of its linearizedform. Oscillations that grow or damp in time appear naturally.The next two sections explore in more depth the characteristicsof free (Sect. 5) and forced (Sect. 6) oscillations. Both sectionscompare the theory developed in this article with observations.Finally, in Sect. 7, the physical basis of the model and specifi-cally the energy partitioning are discussed. Also, the universalityof retardation in MHD configurations in general is pointed out.A summary of the model and its consequences can be found inSect. 8. All dimensional variables in this paper are in rationalMKSA units.

2. Magnetic field of a current distribution

To study the motion of current carrying filaments, the evolutionof the magnetic field of a prescribed time-dependent currentdistribution must be known. Given a current distribution J (r, t)in vacuum, the magnetic field B(r, t) can be computed fromMaxwell’s equations. Adopting the Lorentz gauge and assumingcharge neutrality, these equations reduce to the wave equationfor the vector potential A(r, t) (Jackson 1975, p. 220),

∆A− ε0µ0∂2A

∂t2= −µ0J .

The magnetic B and electric E fields follow from

B = ∇×A , E = −∂A

∂t.

In the case of an infinite space (boundary condition: A(|r| →∞, t) = 0) the formal solution is given by (Morse & Feshbach1953, p. 834)

A(r, t) = µ0

∫ t

−∞dt′∫V

dr′3 G3D(r, t|r′, t′) J (r′, t′). (4)

The solution is constructed using Green’s function for the threedimensional space

G3D(r, t|r′, t′) =δ[|r − r′|/c− (t− t′)

]4π|r − r′| ,

where c = 1/√ε0µ0 is the speed of light. According to Green’s

function, the contribution of an impulsive source propagatesoutwards on a spherical shell with the speed of light. Outsidethe sphere no knowledge at all exists of the source. Inside thesphere, the effect of the source belongs to the past: it is no longerdetectable.

Now consider the case of a two dimensional current distri-bution, assuming invariance along the x-axis. Green’s functionin two dimensions contains a Heaviside function θ, not a Diracdelta. Green’s functionG2D(r, t|r′, t′) is obtained by integratingGreen’s function for three dimensional space along the x-axis(Morse & Feshbach, p. 842) and is given by

G2D(r, t|r′, t′) =c θ[c(t− t′)−

√(y − y′)2 + (z − z′)2]

2π√c2(t− t′)2 − (y − y′)2 − (z − z′)2

.


Fig. 3. Evolution of the magnetic field (arbitrary units). Unit of time:τ0.

In two dimensions sources constitute infinite lines, parallel tothex-axis. Contributions from impulsive sources propagate out-wards within an infinitely long cylinder whose radius expandswith the speed of light. Outside the cylinder the presence of theimpulsive source is unknown. Every infinitesimal part of the linesource acts as a point source that results in an outward expand-ing spherical shell (cf. Huygens’ principle), carrying away theperturbation from its origin. Note that the singularity in Green’sfunction for two dimensions is integrable.

To illustrate the effects of retardation in two dimensions,the field of a wire current is computed at a distance r from thewire for two different current models. For an observer with afixed position with respect to the wire, the retardation conditionis easily solved. The situation is invariant for rotation aroundthe x-axis so cylindrical coordinates can be used. Let τ0 be thetravel time r/c between wire and observer.

For a stationary wire that is switched on at t = 0 for aninstant, I(t) = I0δ[t], the magnetic field is

Bφ(r, t) =

0 t < τ0

µ0I02πr

1τ0

√(tτ0

)2− 1

−3

t ≥ τ0.

The field evolution is shown in Fig. 3. Three effects are clearlyvisible. Firstly, the initial delay: until t = τ0 the field at the ob-server’s position remains zero. Secondly, the ‘wake’: after theinitial cylindrical wavefront has passed, spherical wavefrontscontinue to arrive at the observer’s position resulting in a wake.This wake is a phenomenon typical for two dimensional con-figurations; it does not occur in one or three dimensional spaces(Morse & Feshbach 1953, p. 841). Thirdly, there is a singularityat t = τ0, arising from the discontinuity of I(t). Continuouslychanging currents do not exhibit such a singularity. They doexhibit however the initial delay and the wake-phenomenon,although the latter may not be apparent.

A stationary wire with an oscillating current I(t) =I0 cos(ωt) has a field

Bφ(r, t) =µ0I0

4rωτ0

√J2

1 (ωτ0) + Y 21 (ωτ0) cos(ωt + α).

Fig. 4. Amplitude of the magnetic field oscillation (arbitrary units) asfunction of ωτ0.

where tanα = J1(ωτ0)/Y1(ωτ0) and J1(x) and Y1(x) are theBessel functions of the first resp. second kind, both of orderunity. The amplitude of the oscillation is a monotonously in-creasing function of ωτ0, while the phase is quasi-periodic inωτ0. For ωτ0 � 1 (but not necessarily τ0 � 1!), the field ap-proaches as expected µ0I(t)/2πr, the quasi-stationary solution(see also Fig. 4). For ωτ0 >∼ 1 the amplitude becomes signif-icantly larger than in the quasi-stationary approximation. Thecurrent now oscillates so fast, as compared with the wave traveltime, that subsequent phase surfaces of the vector potential arespatially at distances of the order of r or smaller. The gradi-ents in A, and hence the field amplitude, now rapidly increasewith rising ωτ0. Apparently the ratio of the lag and the inher-ent time scale of the system is a good indicator of the relativeimportance of the influence of retardation. Retardation is onlyimportant when the lag is comparable to or larger than any in-herent time scale. No initial delay is present as the current hasbeen oscillating from t = −∞ onward.

3. A dynamical Kuperus-Raadu model for prominences

To compute the repulsive force due to the photosphere, I usea mirror current. In Appendix A the field of a moving, infiniteand straight wire current is discussed. The expression obtainedis valid for a vacuum, not for a plasma. The main difference is themagnitude of the group speeds of the waves. In vacuum this is thespeed of light c. In a plasma it is vA, the Alfvenspeed. Therefor,change c to vA in Eq. (A3) to obtain a heuristic description ofthe evolution of the field in a plasma. The substitution c→ vA isstraight forward when using the MKSA system of units, as longas√ε0µ0 is interpreted as c. In the Gaussian system c sometimes

is interpreted as a normalization constant, not as a signal speed.These instances of c should then not be interpreted as vA.

The field of the mirror current Bmir(t, z) results from thesubstitutions: h(t′) → −z(t′) and I(t) = I0 → −I0 in Eq. (A3).Note that Bmir(t, z) is only the y-component of the mirror field.

The prominence is located in a coronal background field thatexerts a downward Lorentz force on the current. For this field


a simple configuration (linear force free arcade), in which thefieldstrength drops off exponentially with height, is taken

Bcor = B0e−z/H0 .

Here H0 is the scale height of the coronal field. Of course Bcor

refers only to the y-component of the field.The influence of the coronal plasma on the prominence mo-

tion is modeled by a damping term. This damping is not spec-ified further: its magnitude depends on the damping mecha-nism, precise values of plasma parameters and the geometriesinvolved. Various models have been proposed: mass friction(Hyder, 1966), emission of sound waves (Kuperus & Kleczek,1969) and emission of Alfven waves (van den Oord & Kuperus,1992). In all three cases damping is linear with the prominence’sspeed.

To obtain dimensionless equations, introduce the followingdimensionless quantities t, z, x, I0, νph, χ and τ0

t =√

µ0m4π

1B0t, I0 = 4π

µ0z0B0I0, νph =

√4πmµ0

B0νph,

z = z0z, x = z0x, χ = z0H0

τ0 = 2 z0vAB0

√4πµ0m

.(5)

The dimensionless equation of motion is (drop the tildes)

z(t) = I0{Bmir(t, z)− e−χz(t)

}− νphz(t) + f (t). (6)

The forcing term f (t) is introduced to perturb the equilibrium.Physically it represents a shock wave, triggered by a flare andhitting the prominence or a change in the coronal backgroundfield due to flux emergence at the photosphere.

The field of the mirror current is given by (cf. Eq. (A3))

Bmir(t, z) =

I0

∫ ∞

0dx′

1F 2t′R

2t′

[2(z(t) + z(t′))

Ft′Rt′+ τ0

z(t′)Ft′

−

−τ 20

(z(t) + z(t′))2z(t′)2Ft′Rt′

]t−t′=τ0Rt′/2

, (7)

with

Rt′ =√x′2 + (z(t) + z(t′))2 (8)

and

Ft′ = 1 +12τ0z(t) + z(t′)

Rt′z(t′). (9)

Because of Bmir, Eq. (6) is a nonlinear differential equationwith delays. As the equation of motion contains terms with thehighest order derivative z evaluated at both the present t and thepast t′, this is called a neutral differential equation (NDE, seeSaaty 1981, p. 214).

In the expression for Bmir, t′ is an implicit function of x′

according to the retardation condition

t− t′ =12τ0Rt′ (10)

and ranges from τ0(z(t) + z(t′))/2 to infinity. This is gener-ally called a distributed delay. The dimensionless Alfvenspeedequals 2/τ0. I show τ0 to be an important parameter of the model.The retarded time t′ is a monotonously increasing function of tfor |z(t)| < 2/τ0,

dt′

dt= 1− 1

2τ0z(t) + z(t′)

Rt′z(t).

In stationary situations Bmir will reduce to the magneto-static force between two currents and hence Eq. (6) will havethe same equilibrium as the dimensionless van Tend-Kuperusequation of motion. Let the scaling parameter z0 be the equilib-rium height. In dimensionless variables, the equilibrium currentthen becomes I0 = e−χ.

The initial conditions for Eq. (6) will be functions on theinterval t ∈ [−∞, 0], if the solution starts at t = 0. As Bmir

contains delays up to infinity, the system is influenced by itscomplete history, i.e. z(t′), z(t′), z(t′) for t′ ≤ t. The most logi-cal choice for initial conditions is then a solution to the equationof motion itself. Equilibrium is the only practical option. WithI0 = e−χ, the initial conditions are z(t) = 1, z(t) = z(t) = 0 fort ≤ 0.

For comparison, the dimensionless equation of motion usedby van den Oord & Kuperus (1992) is given

z(t) = I0

{2I0

z(t) + z(t′)− e−χz

}− νphz(t) + f (t), (11)

where

t− t′ =12τ0(z(t) + z(t′)).

As the retardation condition is now independent of x′, van denOord & Kuperus used a single discrete delay. Both Eqs. (6) and(11) reduce to the van Tend-Kuperus equation for τ0 = 0,

z(t) = I0

{I0

z(t)− e−χz(t)

}− νphz(t) + f (t).

4. Solving the equation of motion

4.1. Numerical results: sample runs

In this paragraph solutions to the dimensionless equation ofmotion, Eq. (6), for fixed ν and χ, but varying τ0 are presented.To this end, a numerical code was developed, based on oneof the techniques described in Cryer (1972). The solution wasadvanced using a second order Runge-Kutta method, while theintegral expression (7) was computed using the Gill-Miller thirdorder finite difference method (Gill & Miller 1972). As this in-tegration method allows for unequally spaced x′-values, the re-sults from previous advances could be used as t′, z(t′). To obtainthese values for x′ = 0, I used the second order Langrangianinterpolation. The overall order of this method is two.

The physical parameters for the prominence are chosen tobe z0 = 30 000 km and m = 3.1 × 104 kg/m (Hirayama 1989,Anzer 1995). For the coronal arcade I chose B0 = 0.001 T


and H0 → ∞. The latter implies that the field is constant withheight (χ = 0) and hence no van Tend-Kuperus instability (vanTend & Kuperus 1978) exists. As I am are not interested in theglobal evolution of a prominence, but only in the stability ofits equilibrium, the choice χ = 0 is not unphysical. It merelyimplies that, near the equilibrium height of the filament, thearcade field is constant.

The frequency of the undamped van Tend-Kuperus solu-tion is now

√4πB2

0/µ0m = 5.8 min, or in dimensionlessunits: 1. With a dimensionless damping of νph = 0.2, thesolution to Eq. (6) for τ0 � 1 is expected to oscillate atω0

vtk =√I2(1− χ)− ν2/2 =

√0.99 with a quality factor

Q = ω0vtk/νph ≈ 5. The delay parameter τ0 can be changed

by adjusting vA without altering the scaling.All simulations start with equilibrium. At t = 0 an unspeci-

fied driving force acts on the filament:

f (t) =

0 t < 0,

1vA

2τ0

2tc

sin2 πttc

0 ≤ t ≤ tc,

0 t > tc.

All variables are in the dimensionless units defined in the pre-vious section, except for vA which is specified in km/s. Thisdriving force is such that it would finally give the prominencean upward speed of 1 km/s if it were the only acting force. Notethat the driving force and its first derivative are continuous. Forthese simulations tc = 1.07 (dimensionless), which correspondsto 0.5 min.

For τ0 ∈ [10−2, 10] the results are presented in Fig. 5. Thesolution to Eq. (6) is a damped or growing oscillation with onedistinct period. For τ0 small, the solution approximates the vanTend-Kuperus solution, as conjectured. For τ0

>≈ 1 the dampingfactor of the oscillation depends very strongly on the delay. Forτ0 = 1.07, a strongly damped oscillation is observed. With aslightly higher delay (τ0 = 1.428) the solution becomes unsta-ble. The damping is maximal for τ0 = tc. There is however noevidence that this is related to any resonance between the driv-ing force and the field of the mirror current. Different values oftc yield qualitatively similar solutions. The last simulation wasstopped when the velocity became larger than vA. The promi-nence moved faster than the plasma waves at that moment andEq. (6) looses validity due to the resulting shock waves (seeSects. 1 and 3). The period of the oscillation is relatively con-stant and changes only by a factor of 2, as τ0 varies over severaldecades.

In Sect. 2 it was noted that retardation has the largest influ-ence when the lag is comparable to or larger than any time scaleinherent to the system (see also Fig. 4). This is also illustratedby the results of this section, as the solution remains fairly closeto the van Tend-Kuperus solution for small ω0

vtkτ0, but changesits nature drastically for ω0

vtkτ0 >∼ 1. In Sect. 7 an explanationof these results is presented.

4.2. Analytical solution

The non-linearity and the distributed time lag, depending on thehistory of z(t), make Eq. (6) very hard to solve. Yi Zhang et

al. (1991) observed line-of-sight velocities of about 2 km/s. Asthe oscillation periods are of the order of 10 min., this impliesheight variations of 300 km, the height of quiescent prominencesusually being higher than 30 000 km (Schmieder 1988). For suchsmall amplitude oscillations the linearized equation of motioncan be used, which is derived in Appendix B.1. The lag canthen be evaluated as if in equilibrium. Hence the retardationcondition becomes

t′ = t− 12τ0R0 = t− 1

2τ0

√x′2 + 4. (12)

The error made is of second order (δzδz) (see Appendix B.1.).The linearized equation of motion becomes

δz(t) + νphδz(t) + I20 (

12− χ)δz(t)+

I20

∫ ∞

0dx′

{24− 2R2

0

R50

δz(t− 12τ0R0)+

+τ012−R2

0

R40

δz(t− 12τ0R0) + τ 2

02R3

0

δz(t− 12τ0R0)

}= f (t). (13)

This is a neutral differential difference equation with a dis-tributed delay that is independent of t, t′ and z. This equationis solved using a Laplace transform L (for mathematical de-tails, see Appendix B.2). Introducing Z(s) ≡ L[δz(t)](s) andF (s) ≡ L[f (t)](s), the transformed linearized equation of mo-tion is h(s) Z(s) = F (s), with h(s) the characteristic function

h(s) = s2 + νphs + I20

{12− χ− 1

2τ 2

0 s2K ′

1(τ0s)

}. (14)

HereK1(s) is the modified Bessel function of the second kind, ofinteger order 1 (see also Abramowitz & Stegun 1968, Ch. 9). Theprime stands for differentiation with respect to the argument.

The solution has a highly oscillatory character. The zeroes ofthe characteristic function define the frequencies and dampingrates of the oscillations. Equation (14) is a trancedental functionand one has to rely on numerical methods to determine its zeroes.To distinguish between these zeroes, I can label them with asubscript k = 0, 1, 2, 3 . . ., such that Re s0 ≥ Re s1 ≥ Re s2 ≥. . . Each zero can be written as sk = νk + ωk ı, where νk ∈ IRand ωk ∈ IR can be viewed as the characteristic damping rateand frequency of a particular term in Eq. (16). If νk > 0 it ismore appropriate to call it a growth rate. Obviously, stability isdetermined by the sign of ν0.

The characteristic function has an infinite number of zeroes,with only a finite number of zeroes located in the positive half-plane (Krall 1964, see also Pontryagin 1955). Note that for τ0 >0, zeroes always come in complex conjugate pairs: ifKn(s) = 0then Kn(s∗) = 0.

For k ∈ IN sufficiently large and m = 0, 1, 2, 3, . . ., anapproximation to the zeroes sufficiently far from the origin canbe found:

sk+m = − 2τ0

log2 4√k + m

I0τ0± 2(k + m)π + π

4

τ0ı. (15)


Fig. 5. Numerical solutions of the equation of motion for different values of τ0, νph = 0.2, χ = 0.0. The dashed line is the solution of the vanTend-Kuperus equation.

Deriving this formula is quite an involved process (see eitherPontryagin 1955, Bellman & Cooke 1963 or Pinney 1958). Thereader can verify the formula by substituting it inh(s) and notingthat, as k is large, several terms can be ignored. Obviously,the zeroes appear in a chain in the complex plain. Also, forτ0 increasing, those zeroes given by the asymptotic form movetowards the origin.

It is unclear whether there exist only simple zeroes, or alsozeroes of higher multiplicity. Numerical analysis seems to sug-gest that for τ0 > 0 only simple zeroes exist. However, forτ0 = 0 and ν2 = 4I2

0 ( 12 − χ) a double zero does exist.

The solution δz(t) can be obtained by an inverse Laplacetransform, as detailed in Appendix B.3:

δz(t) = integral +∞∑k=0

1h′(sk)

∫ t

0e−skuf (u) du eskt (16)

As h(s∗) = h∗(s), the expression for δz(t) is real. Besides theseries, the expression for δz(t) contains a double integral. How-ever, its contribution is rather small and does not, in any way,alter the global behaviour of the solution (see Appendix B.3).According to the approximate expression Eq. (15) for the ze-roes, only the first few terms in Eq. (16) are important. Thezeroes associated with these first few terms are called dominat-ing zeroes.

For comparison, the characteristic function for the van Tend-Kuperus model is

htk = s2 + νphs + I20 {1− χ} , (17)

and for the van den Oord-Kuperus model:

hok = s2 + νphs + I20

{12− χ +

12

e−τ0s

}. (18)

For a study of the zeroes of functions like hok, I refer to Cooke& Grossman (1982).

For τ0 → 0, both equations (14) and (18) reduce to Eq. (17).

5. The case of free oscillations

5.1. Location of zeroes

In Sect. 4.1 numerical examples were given that clearly showthe influence of the delay τ0 on the behaviour of the solution.Also, from the previous section it is known that the zeroes ofEq. (14) fully determine this solution.

The zeroes are completely determined by specifying the pa-rameters νph, χ and τ0. Keeping νph and χ fixed, while vary-ing τ0, the zeroes of the characteristic function will describecontinuous paths in the complex plane. Determining the zeroesas a function of τ0 enables to explain the numerical results of


Fig. 6. The location of the five most dominant zeroesof the characteristic function for νph = 0.2, χ = 0.0and τ0 ∈ [10−1, 104]. Each dot is a zero. Many dots lieon the same curve, that forms the path a particular zerodescribes. For some specific values of τ0 markers havebeen put in the graph. The van Tend-Kuperus zero isthe only zero present for τ0 = 0.1. The dashed linesare lines of constant Q = |ω/2ν|.

the previous section. Only the upper half-plane is considered(Im sk ≥ 0), as the solutions of h(s) = 0 come in conjugatepairs. As in the previous section, the parameter values χ = 0.0(I0 = e−χ = 1) and ν = 0.2 are chosen.

For τ0 = 0 there is only one zero s0 = −νph/2 + ω0vtk ı

(referred to as the van Tend-Kuperus zero). The other zeroescan be thought of as located at the infinite far left in the complexplane. This is most consistent with their subsequent motion inthe complex plane and the asymptotic form of Eq. (15). In thefollowing the behaviour of the van Tend-Kuperus zero and thelocation of the dominant zeroes will be studied in more detail.The latter are interesting because they effectively determine thesolution of the linearized equation of motion, the van Tend-Kuperus being the dominant zero for ω0

vtkτ0 � 1. To this end, Ideveloped a code, called ROOTS, based on an algorithm devisedby Kuiken (1968) to locate the zeroes of an arbitrary complexequation.

Using the ROOTS code it is possible to determine the zeroesof the characteristic function in a finite domain of the complexplane. In Fig. 6 the results for 10−1 ≤ τ0 ≤ 103 are shown.With increasing τ0 they all move downward and to the rightat first, except for the van Tend-Kuperus zero. The dominantvan Tend-Kuperus zero moves to the left, which is why thenumerical solution of Sect. 4.1 shows such strong damping forτ0 ≈ 1. After that another zero, that started at the infinite farleft, becomes the dominant one. In the numbering of zeroes,the van Tend-Kuperus zero is now s1. The solution presentedin Sect. 4.1 shows an increase in the frequency over a relativelyshort range of τ0 values as ω0 > ω1 = ωvtk. At some value of τ0,the first zero crosses over into the positive real half-plane andstability is lost. It appears that once stability is lost, it remainslost. The growth rate diminishes again for τ0 very large.

Clearly, different zeroes are dominant for different valuesof τ0. This does not result in a different mode of oscillation, but

merely a different frequency. In principle it is possible that two(or more) well-defined frequencies are present in the solution.For the present choice of parameter values ν, χ this does nothappen as the damping is too strong around τ0 ≈ 1. If this werenot the case, and if at the same time ω0 ≈ ω1, one might evenexpect a beat frequency |ω0−ω1| to be present. It is possible to leta solution, which at τ0 = 0 was over critically damped, oscillatewith a substantial quality factor Q = ωk/2νk for τ0 > 0.

In Fig. 7 the real part of the dominant zero (max Re sk, k ∈IN) is plotted as a function of τ0. This zero determines the sta-bility properties and sets an upper limit on growth rates and anlower limit on damping rates. For small τ0 the van Tend-Kuperusmodel is a good approximation. For τ0 ≈ 1 the stability prop-erties of the filament change drastically. This coincides withω0

vtkτ0 ≈ 1, which in Sect. 2 was found to be an important indi-cator for the influence of retardation (see also Fig. 4). The min-imum in maxk∈IN Re sk corresponds to the τ0 value for whichthe van Tend-Kuperus zero stops being the dominant zero.

5.2. Parametric stability analysis

It is also interesting to see how certain characteristics of the so-lution change, as the parameters νph, χ and τ0 are altered. In par-ticular I like to know how damping and delay influence stabilityand the occurence of periods such as observed in prominences.Basically this comes down to determining whether zeroes areor are not present in certain areas of the complex plane. TheCONTOUR code (see Appendix D) allows one to determinethe number of zeroes of the characteristic function, within aspecified contour in the complex plane. A prominence and ar-cade as in Sect. 4.1, so z0 = 30 000 km, m = 3.1 × 104 kg/mand B0 = 0.001 T were assumed. Again, the coronal back-ground field was taken constant (χ = 0.0). The Alfvenspeedcan be safely estimated at 102 ≤ vA ≤ 104 km s−1. The damp-ing is governed by any of the three mechanisms mentioned in


Fig. 7. The dominant damping/growth rateν0 = max(Re sk) as functionof τ0, for νph = 0.2, χ = 0.0. The dominant frequency varies between0.5 and 1.5 in the same τ0 range.

Fig. 8. Stability regions in the νph, τ0-plane. The number of zeroes inthe positive real half plane is denoted by a grey scale (white: none;lightest gray: 2; black: 40 or more). The white area indicates dampedoscillations.

Sect. 3. Of these, emission of Alfven waves produces damping2 or 3 magnitudes larger than the others. A safe range would be10−4 ≤ νph ≤ 105 kg/s.

First stability properties will be discussed. For this a con-tour that spans the whole of the infinite half-plane Re s > 0is needed. This is of course impractical. However, using theasymptotic form of K ′

1(τ0s) (Watson 1962, especially §7.25 p.204) it can be shown that the magnitude of any zero with Res > 0 is of order max(ν, I0, I

40τ

30 ) or less. This permits to choose

a finite contour and still draw conclusions about the whole ofthe positive real half-plane. In Fig. 8 results for χ = 0.0 are pre-sented. Over a large range of νph, stability is lost for ω0

vtkτ0 ≈ 1.With increasing τ0 a large number of zeroes rapidly crosses theimaginary axis into the positive real half plane. The dampinghas a stabilizing influence, as expected. For other values of χthe graph hardly differs. With increasing χ instability sets in forsomewhat higher values of τ0.

Fig. 9. Occurrence of observed prominence periods. The white area isthe parameter regime that has damped oscillations as its solution. Thelight gray regions in the νph, τ0-plane show where the damped solutionhas periods in the range 1 – 20 min. and quality factors Q ≥ 10. Thedark gray regions show where solutions are unstable, black regionssuffer from unreliable results from the CONTOUR code. Right of thedashed line at νph = 2 the van Tend-Kuperus model does not permitany kind of oscillation. χ = 0.0,m = 3.1× 104 kg/m.

Secondly, and perhaps more interesting, the CONTOURcode permits to determine whether the periods observed inprominences (see Sect. 1), can be predicted by the model. Peri-ods in the range of 1–20 min. with quality factors Q = ω/2ν ≥10 were looked for. Using m = 3.1× 104 kg/m and B0 = 0.001T, one can transform these periods into dimensionless time. Therequired contour is now a halved trapezoid in the negative realhalf-plane. The size is determined by the dimensionless periodand the minimum quality factor. For χ = 0.0, there are tworegions in parameter space νph, τ0 that have solutions with therequired periods (see Fig. 9). The region in the lower left cornercontains the unretarded case τ0 ≈ 0, as for νph = 0 ω0

vtk = 5.8min. The other region lies just below the instability region andruns along the full extent of νph values. Apparently, a substan-tial delay can cause the observed oscillations, even when thephysical damping is in fact enormous. The lower left regionis probably caused by the van Tend-Kuperus zero for τ0 > 0,while the strip below the instability region is caused by otherzeroes that came in from the far left side of the complex plane(see also Fig 6). For smaller substructures within the promi-nence (e.g. m = 3.1× 102 kg/m), the van Tend-Kuperus periodfor τ0 = 0 is out of the range of required periods (if νph ≤ 2).For these substructures one still finds the observed oscillationsfor substantial delays and large damping (νph > 2, see Fig. 10,here it was assumed that Q ≥ 2.5).

Due to the high quality factors, the obtained oscillations withperiods of 1–20 min. are likely to be caused by one or more ofthe dominating zeroes of the characteristic function.

6. The case of forced oscillations

Oscillation periods of three and five minutes have often beenfound in prominences. The close resemblance to the chromo-


spheric three and photospheric five minute oscillation has ledto the believe that these prominence oscillations are forced, in-stead of free (see Balthasar et al. 1986, 1988). The oscillatingphotosphere shakes the footpoints of the prominence and canthus induce forced oscillations in the prominence body.

In the simplified model discussed in this paper, photosphericdriving can be simulated with f (t) = f0 cos(ωdt)θ[t]. Eq. (16)can of course be written down explicitly, but that would bevery cumbersome. Instead, note that switching on a harmonicforce will lead to the appearance of a lot of transient terms inEq. (16). For t → ∞ most terms will damp out, because thefrequencies ωd and Im sk are incommensurable. In the case ofa harmonic driving force f (t) = f0 cos(ωdt), one can thereforlook for harmonic solutions of the equation of motion with thesame frequency ωd. Substituting δz(t) = h0 cos(ωdt + φ) in Eq.(13), and computing the absolute value gives:

h0 =f0

| − ω2d + νphωd ı + I2

0

{12 − χ + 1

2τ20ω

2dK

′1(τ0ωd ı)

} | .The denominator equals h(ωdτ0 ı). Using (Abramowitz & Ste-gun, p. 375)

Kn(yı) =−π2ın

[Yn(y) + ıJn(y)] ,

where Jn(y) and Yn(y) are Bessel functions of first and secondkind, of order n, the expression for h0 can be rewritten as

h0 =

[(−ω2

d + I20

{12− χ +

π

4τ 2

0ω2dY

′1 (τ0ωd)

})2

+

+(νphωd +

π

4I2

0 τ20ω

2dJ

′1(τ0ωd)

)2] 1

2

f0.

Fig. 11 is a plot of h0 (f0 = 1) as a function of τ0 and ωd forν = 0.2, χ = 0.0. For τ0 = 0, h0 is the amplitude of a forcedoscillation in the van Tend-Kuperus model. For ωd = ω0

vtk =√0.99, the free oscillation period, this amplitude is maximal.

As the graph shows, relatively little changes as τ0 remains small.With increasing τ0, the maximal amplitude drops off to zero.Well-defined peaks appear at frequencies different from ω0

vtk.These peaks are of infinite amplitude.

The peaks correspond to imaginary zeroes of h(s). The lo-cation of the zeroes of h(s), for fixed νph and χ is a function ofτ0 alone. As τ0 changes, the zeroes move through the complexplane. For a certain τ0 a zero sm,m ∈ IN of h(s) crosses theimaginary axis at frequency ωm. If, for that value of τ0, onedrives the system with the frequency ωd = ωm, the term k = min Eq. (16) will be strongly excited.

A forced oscillation can only be excited efficiently for spe-cific combinations of ωd and τ0. This provides a diagnostic tool.The resonances are all caused by zeroes very close to the theimaginary axis. Observed resonances are likely to be caused bythe dominant zero, for otherwise the system is unstable. Any per-turbation apart from harmonic forcing would cause the promi-nence to disrupt.

Fig. 10. Occurrence of observed prominence periods. The white area isthe parameter regime that has damped oscillations as its solution. Thelight gray regions in the νph, τ0-plane show where the damped solutionexhibits periods of 1 – 20 min. and quality factors Q ≥ 2.5. The darkgray regions show where solutions are unstable, black regions sufferfrom unreliable results from the CONTOUR code. Right of the dashedline at νph = 2 the van Tend-Kuperus model does not permit any kindof oscillation. χ = 0.0,m = 3.1× 102 kg/m.

Fig. 11. Amplitude of the forced oscillation as a function of ωd and τ0.A logarithmic scaling of the τ0 axis is used to stress the importance ofthe delay. The ridge A is of finite height (as νph /= 0), but the peaks B& C are in fact infinite.

From Fig. 11 it follows that for νph = 0.2 the only observableresonance occurs at (ωd, τ0) = (1.53, 1.35). The product ωdτ0 isindependent of the scaling. The two most common driving peri-ods in the solar photosphere are 3 min. and 5 min. Hence forcedoscillations are expected to be most noticable for photosphere– fibril travel times of 0.5 min. resp. 0.8 min. Using an averageAlfvenspeed of 1000 km/s these fibrils are located at heights of30 0 00 resp. 48 000 km. Further numerical analysis shows thatthis product ωdτ0 ≈ 1 depends only very weakly on νph.


7. Discussion

7.1. The energy budget

In this section I discuss where the energy for the growing oscil-lations comes from. After t > tc there is no driving force andthe oscillation is free.

Consider, instead of two infinitely long wires, a single cur-rent loop, partially submerged below the photosphere. If the toppart of the loop is moved upward, the total flux through the loopincreases. This change in flux will, according to Faraday’s law,result in an electromotive force along the current (e.g. Jack-son 1975). By Lenz’s law this electromotive force will opposethe moving charges in the current. The loop’s amperage willdecrease and the motion of the top is counteracted. To keepthe amperage nevertheless constant the current sources must dowork. For this an external energy source is required. This energysource is implicitly assumed in the present model, as the currentis taken constant.

A loop current can also be viewed as an LR-circuit, withL the self-inductance of the current and R the total resistance.In such a circuit the current changes on typical time-scales ofL/R (e.g. Duffin 1965, p. 222). Martens (1986) assumed thatfor prominences this time-scale is about 75 hours. Hence foroscillation periods of the order of minutes, it seems reasonableto assume that the current is indeed constant.

As the system is basically a damped harmonic oscillator, itis possible to present an approximate description of the energypartitioning. Take the linearized equation of motion (13) andsplit the Lorentz force due to the mirror current in two parts:Bmir = Bvtk+b. The Lorentz force is considered to be the sum ofthe force in the van Tend-Kuperus model and some modificationb. Rewrite the equation of motion as

δz(t) + I20 (1− χ)δz(t) = −νphδz(t)− I0b(t, δz(t)),

then multiply both sides with δz(t) and integrate t over a periodT

∆Ekin + ∆Epot = −∫ T

0dt {νphδz(t) + I0b(t, δz(t))} δz(t).

The left-hand side of this equation is the total energy changein the van Tend-Kuperus model during one period T . The righthand side represents the possible energy sources and sinks. Incase τ0 = 0, b = 0 and there is only a sink due to some physicaldamping mechanism.

From the numerical studies it follows that the solution is, to afair approximation, a damped harmonic oscillation with nearlyconstant period: δz(t) = h0 cos(ω0

vtkt); the van Tend-Kuperuszero is the dominant zero up to ω0

vtkτ0 ≈ 1. Substitute this intothe right-hand side of the energy equation. The b-field gives riseto terms like

h20I

20ω

0vtk

∫ ∞

0dx′

24− 2R20

R50

h0 cosω0vtk

(t− 1

2τ0R0

)sin(ω0

vtkt).

Apparently the phase difference between the velocity and theb-field determines whether this term will act as an energy sourceor sink. The exact result is

Fig. 12. Retardation and phase lags. The solid line is the dominantdamping as obtained by the ROOTS code (see Sect. 5). The effectivedampingν as obtained from energy considerations is plotted as a dashedline. The two dotted lines are the minimal and maximal average phasedifferences between the resultant restoring force and the prominencespeed.

∆Ekin + ∆Epot =

−νphω0vtkπ −

(π2

)2ω0

vtk2τ 2

0 I20J

′1(ω0

vtkτ0),

where h0 is taken equal to unity.

In Fig. 12 the effective damping is plotted, obtained from theright-hand side of the last equation by dividing it by 2πω0

vtk. Alsoplotted is the dominant damping as obtained in Sect. 5 (see alsoFig. 7). As expected, the two graphs coincide for ω0

vtkτ0 <∼ 1.Both the increase in damping and the loss of stability are pre-dicted by the simple energy considerations. Quantitatively thegraphs differ forω0

vtkτ0 >∼ 1 as dominant damping and frequencyvary strongly and the assumption that δz(t) = h0 cos(ω0

vtkt) be-comes invalid.

In the case of the undamped van Tend-Kuperus model(νph = 0, τ0 = 0), there is a phase difference ∆φ = π/4 be-tween the resultant restoring force and the velocity. This is typ-ical of the classic harmonic oscillator. It is possible to estimatethe additional phase difference due to the finite travel times ofthe magnetic field, for τ0 /= 0. In Eq. (B1) five positive definitedelay kernels can be identified. The first kernel is e.g. k(v) =1.5τ 4

0 I20/v

4√v2 − τ 2

0 . One can associate an average delay withthis kernel: <τ0>=

∫∞τ0dv vk(v)/

∫∞τ0dv k(v). This delay gives

rise to an average phase difference ∆φ = π/4+ <τ0> ω0vtk/2π

between the resultant restoring force and the velocity. In Fig. 12the smallest and largest phase difference for four delay kernels(since the fourth delay kernel in Eq. (B1) has an infinitely largeaverage delay, it is ignored) are plotted as dotted lines . Oneclearly sees that maximum damping occurs when ∆φ ≈ 0.5,while maximum growth occurs when force and velocity are inphase.


7.2. Back to basics: the physical model

Realistic models for prominence oscillation always incorporateretardation, if only implicit, as it is a direct consequence of finitegroup speeds and a distant boundary condition. This holds forKuperus-Raadu prominences, but also for Kippenhahn-Schlutermodels (see e.g. Fig. 1, Priest 1990). However, the current in aKS prominence is much less than in a KR prominence. Therefor,the Lorentz force due to the mirror current is likely to play onlya minor role in the force balance of KS models. Consequently,one can expect retardation to be more important for KR modelsthan for KS prominences. This is indeed shown to be the case ina forthcoming paper (accepted by A&A). This may also explainwhy other authors (Oliver et al. 1992, 1993, 1995, Joarder &Roberts 1992, 1993) do not find a strong dependence of growthtimes on Alfvenspeed. Their global equilibria invariably are ofthe KS type.

Throughout this paper, the Alfvenspeed is considered con-stant with height. This is of course not the case, the Alfvenspeedbeing substantially lower in the chromosphere and photosphere,than in the corona. As a matter of fact, this is exactly why wavesgenerated in the corona are reflected at the photosphere and isthus crucial to the idea of a delayed boundary condition. Usingan inhomogeneous Alfvenspeed alters the retardation condi-tion, and will surely change the details of the stability analy-sis. However, the problem becomes mathematically intractable,as no standard solution method exists for an inhomogeneousmedium. The basic delay τ0 is however unchanged and a con-stant Alfvenspeed is likely a justifiable approximation. One mayconsider the Alfvenspeeds mentioned in this paper as ‘height-average’.

As the body of a prominence has a finite extension, delaysother than the distributed delay discussed in this paper can beexpected. In particular, there will be delays in the ‘communica-tion’ between top and bottom of a prominence, or between itslongitudinal ends (evolution of the current). However, no impor-tant supporting forces are associated with these delays as long asthe current is constant, and the basic delay for communicationwith the photosphere is in any case τ ≈ 2z0/vA. Moreover, theAlfvenspeed within a prominence is usually about 100 km/s.But the Alfvenspeed close to the photosphere can be as smallas 0.1 km/s (Priest 1984, fig. 2.1). Consequently, delays forprominence–photosphere communication are much longer thanthe Alfven crossing times of a prominence body. It then seems avalid approximation to retain only the largest delay and considerthe prominence as a rigid straight wire.

The approach used in this paper, a one dimensional current,instead of a three dimensional magnetized plasma, has a bigadvantage. In this model it is very easy to alter the signalspeedvA, without altering anything else. Hence retardation can bestudied in its pure form. With an MHD approach, changing theAlfvenspeed implies changing the equilibrium field or density(usually both). One then also changes the particulars of the orig-inal force balance and the effect of finite wave speeds becomesquite unclear. The disadvantage, of course, is that this modelonly describes global motions of the prominence body.

8. Summary

In this article I have studied vertical oscillations and stabil-ity of prominences, and in particular the influence of retarda-tion, caused by the distant photosphere. The model presented isequivalent to an MHD description, but focuses on the interac-tions between currents.

The prominence is assumed to have a KR configuration.Prominence and photosphere are replaced by a wire currentand a conducting plate respectively. The upward force, result-ing from the conducting plate repelling the wire current’s field, iscalculated in a self-consistent way by solving Maxwell’s equa-tions. As a consequence it is possible to investigate the effectof finite propagation speeds of the magnetic field on verticalprominence oscillations and stability. With the exception of vanden Oord & Kuperus (1992), this has never been done before.

The difference between this article and the article by vanden Oord & Kuperus is twofold. First, van den Oord & Kupe-rus assumed that the lag in ‘communication’ between promi-nence and photosphere was determined entirely by the shortestpossible distance between the two. They therefor used a singlediscrete delay 2z0/vA. In the present paper, it is argued thatisotropically travelling fast waves give rise to a distributed de-lay τ ∈ [2z0/vA,∞]. Such a distributed delay is much moredestabilizing than a single delay. Secondly, the force expressionin van den Oord & Kuperus is in essence still quasi-stationary,although it is shifted by an amount of one delay 2h0/vA back-ward in time. In the present model the force is calculated byexplicitly computing the time dependent field from the Maxwellequations.

The equation of motion Eq. (6) belongs to a special classof equations: neutral differential equations. It can be solved nu-merically and, in linearized form, analytically. In the latter case,a characteristic function h(s) is found, whose zeroes define allthe complex frequencies present in the solution. Usually one ofthese zeroes dominates the solution. Like for instance Hastings(1983), it is found that the product of dominating frequency anddelay ωτ0 is a good indicator of the influence of finite propaga-tion speeds. For ωτ0 � 1, the solutions are similar to the vanTend-Kuperus model (i.e. vA = ∞, see Eq. (1)). For ωτ0 >∼ 1retardation effects come into play and radically change the so-lution. For a broad range of τ0 values, the dominant frequencyhardly changes, while the dominant damping rate depends sen-sitively on the lag. Depending on the value of ωτ0 the systemcan become strongly damped or even unstable. This can all beexplained by considering the phase lags between the resultantforce and speed. Forced oscillations will be most prominent ifωτ0 ≈ 1.

Considering the usual heights of quiescent promi-nences (h0 ∈ [30 000, 50 000] km) and acceptable coronalAlfvenspeeds (vA ∈ [102, 103] km/s), retardation will only bea significant effect for short oscillation periods (1–16 min.). Ifωτ0 ≈ 1 the quality factor of the free oscillation is extremelysensitive to changes in ωτ0, so to changes in height. Assumingthat all other relevant physical parameters (such as flux tubemass) are more or less the same, certain periods will prevail at


certain heights. Forced oscillations by foot point shaking due tothe photospheric 3 and 5 min. oscillations will be concentrated atcertain heights. For τ0 large enough growing oscillations exist,which may be the onset for a prominence disruption.

Acknowledgements. The author is financially supported by the Nether-lands Organisation for Scientific Research (NWO) under grant nr. 781-71-047. The author gratefully acknowledges stimulating discussionswith Michael Raadu, Max Kuperus and especially Bert van den Oord.The author thanks the referee, dr J.L. Ballester, for his useful commentsand criticism.

Appendix A: magnetic field of an infinite straight wire cur-rent in vacuum

An infinite straight wire parallel to the x-axis, carrying a currentI(t), at height h(t) above the x, y-plane at time t, is describedby

Jx(r, t) = I(t) δ[y] δ[z − h(t)]. (A1)

In Eq. (4) only the x-component of the vectorpotential has asource

Ax(r, t) =µ0

2π

∫ ∞

0dx′

∫ t

−∞dt′

I(t′) δ[Rt′/c− (t− t′)]Rt′

=µ0

2π

∫ ∞

0dx′

∫ t

−∞dt′

I(t′) δ[t′ − t∗]{1− z−h(t′)

cRt′h(t′)

}Rt′

.

where Rt′ =√x′2 + y2 + (z − h(t′))2 and t∗ determined by

Rt∗ = c(t− t∗). This can be rewritten as

Ax(r, t) =µ0

2π

∫ ∞

0dx′

[I(t′)Ft′Rt′

]Rt′=c(t−t′)

(A2)

with

Ft′ =dtdt′

= 1− {z − h(t′)} h(t′)cRt′

.

Note that t′ has become an implicit function of x′. Given z andh(t) the retardation condition traces out a path in x′, t′-spacealong which the integrand of Eq. (A2) is integrated.

If the current had been absent for t < 0, the lower inte-gration bound for t′ would have been t′ = 0 in all previousexpressions. As a consequence x′ would have an upper boundof√c2t2 − y2 − (z − h(0))2 in Eq. (A2).

In Sect. 3 only the By component of the field matters:By(r, t) = ∂

∂zAx(r, t). Changing the order of differentiationand integration, the differential operator now operates directlyon the integrand. Bearing in mind that, in the integrand, t′ is animplicit function of z, this implicit dependence gives rise to anexplicit additional dependence of Ax on z. So the partial dif-ferential operator in front of the integral sign, becomes a totaldifferential operator in front of the integrand

By(r, t) =µ02π

∫∞0 dx

′[∂t′∂z

∂∂t′

(I(t′)Ft′Rt′

)+ ∂∂z

(I(t′)Ft′Rt′

)]Rt′=c(t−t′)

.

Using ∂t′∂z =

(∂z∂t′)−1

and z =√c2(t− t′)2 − x′2 + h(t′) gives

By(r, t) = −µ0

2π

∫ ∞

0dx′

1cF 2

t′R2t′×[

{z − h(t′)} I(t′) +{z−h(t′)}2

I(t′) h(t′)cFt′Rt′

+

+{z−h(t′)}2

I(t′) h(t′)Ft′R2

t′− {z−h(t′)} I(t′) h2(t′)

cFt′Rt′+

+c{z−h(t′)} I(t′)

Rt′− I(t′)h(t′)

]Rt′=c(t−t′)

.

Multiply the fifth term by Ft′Rt′/Ft′Rt′ and the sixth term bycFt′Rt′/cFt′Rt′ . Then add the third and fifth resp. fourth andsixth term. Elementary algebra finally leads to

By(r, t) = −µ0

2π

∫ ∞

0dx′

1cF 2

t′R2t′×[

{z − h(t′)} I(t′) +{z−h(t′)}2

I(t′) h(t′)cFt′Rt′

+

+c{z−h(t′)} I(t′)

Ft′Rt′− I(t′) h(t′)

Ft′

]Rt′=c(t−t′)

. (A3)

Appendix B: on solving the equation of motion

A solution to the equation of motion Eq. (6) is obtained in threedistinct steps. First the equation is linearized. This not onlysimplifies the equation itself, but also the retardation conditionEq. (10). Next a Laplace transform is performed and the so-called characteristic function h(s) is derived. The solution isthe convolution of the inverse Laplace transform of 1/h(s) andthe driving term f (t).

B.1. Linearizing the equation of motion

In this section the equation of motion (6) is linearized aroundthe equilibrium using z(t) = 1 + δz(t).

Consider the retardation condition t−t′ = 12τ0Rt′ . For equi-

librium values the delay is given by 12τ0R0 (R0 =

√x′2 + 4).

Linearize the retardation condition around this equilibrium de-lay

t− t′ ≈ 12τ0R0 +

τ0

R0(δz(t) + δz(t′)).

Hence any deviations of the delay from its equilibrium valueare of the same order as δz.

Next, approximate expressions like z(t′) in t′ = t− 12τ0R0.

Write t′ = t− 12τ0R0 + δt′,

z(t′) ≈ z(t− 12τ0R0) + z(t− 1

2τ0R0) δt′.

By evaluating the height, the speed and the acceleration at t −12τ0R0 instead of t−τ , a second order error (zδz) is introduced.

The integrand inBmir(t, z) can now be expanded in a Taylorseries of z(t)+z(t′), z(t′) and z(t′). For this purpose z(t)+z(t′),


z(t′) and z(t′) are treated as if independent. This is a commontechnique in the stability analysis of stationary points of non-linear differential equations.

Keeping only zero and first order terms and replacing z(t′)by z(t− 1

2τ0R0), the four different terms of the integrand become

2(z(t)+z(t′))I0

F 3t′R

3t′

= 4I0

R30− 12τ0I0

R40δz(t− 1

2τ0R0)+

+(2R2

0 − 24)I0

R50

(δz(t) + δ(t− 12τ0R0)),

τ0I0z(t′)F 3t′R

2t′

= τ0I0

R20δz(t− 1

2τ0R0),τ 2

0 (z(t)+z(t′))2I0z(t′)2F 3

t′R3t′

= 2τ 20 I0

R30δz(t− 1

2τ0R0).

As one might expect, only the first term of the integrand has azero-order contribution∫ ∞

0

4I0

R30

dx′ = I0.

The expansions above result in a linearized equation of motion.Note that the zero order contributions of the two Lorentz forcescancel each other. Also the term containing δz(t) in the integrandcan be integrated directly. The resulting equation of motion is alinear differential difference equation (DDE), Eq. (13). Insteadof a trancedental equation for t′, there now is a constant time-lagt− t′ = 1

2τ0R0 (constant in t′, not in x′!). There is no need foran explicit retardation condition anymore.

By introducing a new integration variable v = 12R0 the final

form of the linearized equation of motion becomes

δz(t) + νδz(t) + I20 (

12− χ)δz(t)+

+12τ 2

0 I20

∫ ∞

τ0

dv1√

v2 − τ 20

{3τ 2

0 − v2

v4δz(t− v)+

+3τ 2

0 − v2

v3δz(t− v) +

τ 20

v2δz(t− v)

}= f (t). (B1)

The integral can be seen as the sum of several separate integralsof the form of

∫∞τ0

dv k(v)z(n)(t−v) dv n = 0, 1, 2. The weightfunction k(v) is called a delay kernel. DDEs very much like theabove equation (B1) are described by Pinney (1958). In fact theonly difference is that the delay kernels in this DDE all have an(integrable) singularity (at v = τ0), while Pinney’s delay kernelshave none.

B.2. On the Laplace transform of the linearized equation ofmotion

Consider the usual Laplace transform L. Define Z(s) ≡L[δz(t)](s) and F (s) ≡ L[f (t)](s). The first three terms onthe left-side of our linearized equation of motion (13) are eas-ily transformed. The integral expression can be evaluated bychanging the order of integration. The full Laplace transformthen reads

s2Z(s) + νphZ(s) + I20 (

12− χ)Z(s) +

12I2

0Z(s)×

{3Ki4(τ0s)− Ki2(τ0s) + 3τ0sKi3(τ0s)− τ0sKi1(τ0s)+

+τ 20 s

2Ki2(τ0s)}

= F (s),

with

Kin(s) =∫ ∞

1dv

e−sv

vn√v2 − 1

.

The functions Kin(s) are known as the repeated integrals ofK0(s). This last function is known under a variety of names:modified Bessel function of the second kind of order zero, Mac-Donald’s function and Basset function are the most common.The repeated integrals Kin are defined in Abramowitz & Stegun1968 (Ch.11).

Using the recurrence relation for the repeated integrals(Abramowitz & Stegun, Ch.11), the Laplace transform Z(s)can be expressed as

Z(s) =F (s)

s2 + νphs + I20

{12 − χ− 1

2τ20 s

2K ′1(τ0s)

} .The denominator is called the characteristic function. HereK1(s) is the modified Bessel function of the second kind, ofinteger order 1 (see Abramowitz & Stegun 1968, Ch. 9). Theprime stands for differentiation with respect to the argument.

B.3. Solution of the linearized equation of motion

According to Appendix C the inverse of the characteristic equa-tion can be written as

h−1(s) =1π

∫ 0

−∞dx

Im h(x)(s− x)|h(x)|2 +

∞∑k=1

1h′(sk)

1s− sk

where it is assumed that the characteristic function has onlysimple zeroes. If one multiplies h−1(s) with F (s) and performsan inverse Laplace transform, one obtains

δz(t) =∞∑k=1

1h′(sk)

∫ t

0e−skvf (v)dv eskt +

1π

∫ 0

−∞ext

Im[h(x)]|h(x)|2

∫ t

0f (u)e−xududx. (B2)

Assuming f (t) continuous and

f (t) =

0 if t = 0f (t) if 0 < t < tc0 if t ≥ tc

one can easily demonstrate that the contribution by the doubleintegral

– is zero for t = 0– is negative definite for t > 0– has a minimum for some time 0 < t < tc– increases to zero for t > tc


The upper bound is thus zero. Numerical calculations show thatthe absolute value of the minimum is usually about 10 % ofthe maximal amplitude of the damping oscillation. This is atleast true for the numerical solutions presented in Sect. 4.1. Inthe case of a zero close to the negative real axis however, theintegral might contribute considerably.

The contribution by the double integral in Eq. (B2) is there-for quantitatively not negligible, but does not in any case changequalitatively the behaviour dictated by the series.

Clearly, the zeroes sk determine the overall behaviour ofδz(t). In particular: zeroes with positive real parts imply insta-bility.

Appendix C: on the Cauchy decomposition of meromorphicfunctions

Suppose a complex function h(s) is analytic in the principal re-gion (being the complex plane with the exception of the originand the negative real axis). Across the negative real axis h(s) isdiscontinuous. Assume this function has an infinite number ofcomplex and isolated simple zeroes sk, all located in the prin-cipal region. Then its inverse function h−1(s) is also analytic inthe principal region, except for the neighbourhoods of an infinitenumber of complex and isolated simple poles sk. The functionh−1(s) is called a meromorphic function in the principal region(Sansone & Gerretsen 1960, p. 120). Suppose that, at large dis-tances from the origin, except near the neighbourhoods of theaforementioned poles, this function goes to zero at least as fastas |s|−1. Such a function can be decomposed as

h−1(s) =1π

∫ 0

−∞dx

Imh(x)(s− x)|h(x)|2 +

∞∑k=1

1h′(sk)

1s− sk

.

Proof: h−1(s) is a function whose only singularities are simplepoles. Order the zeroes of h(s) as follows:

0 ≤ |s1| ≤ |s2| ≤ |s3| ≤ . . .

It is possible to find a q > 0 so that q < |Im sk| for every k.This follows as the isolated poles are all in the principal regionand do not have an accumulation point.

Construct a sequence of closed contours Cn in such a waythat Cn includes the poles s1, s2 ... sn, but no other poles. AlsoCn should avoid the negative real axis and the origin (see Fig.14).Consider the integral

In =1

2πi

∫Cn

1(w − s)h(w)

dw,

where s is a point inside Cn. The integrand has poles at all thesk, but also atw = s. Computation of the integral using residues(Sansone & Gerretsen 1960, p. 116) leads to

In =n∑k=1

1(sk − s)h′(sk)

+1

h(s).

Ims

3

s

s

s 1

δ 1

3

2

4

Re

C 1

C 3

R

Fig. 13. The curves Cn in the complex plane. sk, k=1,2,3,4 are zeroesofh(s). The curves have to be traversed anti-clockwise in the evaluationof In.

The contour Cn can be split in two parts: Cin and Cii

n (See Fig.14). The first part Ci

n is an arc at a characteristic distance to theorigin of order Rn. The second part Cii

n is a path around thenegative real axis at a distance of order δn. If n increasesRn →∞ and δn ↓ 0. This permits to calculate both contributions. Thelength of contour partCi

n isLn. Note thatLn = O(Rn). At largedistances from the origin h−1(s) on Ci

n goes to zero at least asfast as R−1

n . If n is increased, one can estimate its contributionIin to In to be

|Iin| ≤Ln

2π(Rn − |z|) maxCn

|h−1(w)|.

Thus for n going to infinity, Iin tends to zero. The contributionof Cii

n is computed explicitly and for n → ∞ reduces to (useh(s∗) = h∗(s))

Iiin = − 1π

∫ 0

−∞dx

Imh(x)(x− s)|h(x)|2 .

Not only has the stated decomposition been proved, but alsothat the series converges uniformly on the complex plane.

A similar proof, but for functions analytic on the wholeplane, can be found in Titchmarsh (1939 p. 110).

Appendix D: the CONTOUR code

A simple stability analysis of a solution like Eq. (16) can bemade by determining the number of zeroes lying in the righthalf-plane where Res > 0. A necessary and sufficient conditionfor stability is that this number is zero.

Consider a contour C in the complex plane like in Fig. 14.Suppose a function h(s) is analytic in a simple connected part ofthe complex plane that comprises the contour C. Suppose alsoh(s) has no zeroes on the contour C. How many zeroes doesh(s) have within C?

According to the argument principle (Sansone & Gerretsen,1960, p. 147), there are

p =1

2πı

∫C

ds1h

∂h

∂s


Table 1. Counting the encirclements of the origin.

conditionsindex anti-clockwise? encircle origin? ai bi

i = N xN > xN−1 sgn(xN ) /= sgn(xN−1) +1 1i = N xN > xN−1 sgn(xN ) = sgn(xN−1) +1 0i = N xN < xN−1 sgn(xN ) /= sgn(xN−1) -1 1i = N xN < xN−1 sgn(xN ) = sgn(xN−1) -1 0

1 < i < N sgn(xi+1 − xi) = sgn(xi−1 − xi) sgn(xi) = sgn(xi−1) +1 01 < i < N sgn(xi+1 − xi) /= sgn(xi−1 − xi) sgn(xi) = sgn(xi−1) -1 01 < i < N sgn(xi+1 − xi) = sgn(xi−1 − xi) sgn(xi) /= sgn(xi−1) +1 11 < i < N sgn(xi+1 − xi) /= sgn(xi−1 − xi) sgn(xi) /= sgn(xi−1) -1 1

zeroes within contourC. However the right hand expression hasa simple geometrical interpretation. One can use the functionh(s) to map contour C into the complex plane. The image of Cunder this mapping is called h(C) (see Fig. 14) and is a contouritself. From residue calculus it follows that this new contourencircles the origin p times in anti-clockwise fashion

p =1

2πı

∫h(C)

dh1h.

Notice that a zero of multiplicity m will be counted m times.Hence two encirclements imply two simple zeroes or one doubezero.

Now take a contour C symmetric in the real axis. For map-ping functions where h(s∗) = h∗(s), the mapped contour h(C)will be symmetric in the real axis as well. Knowing h(C) andits intersections with the real axis, one can split up the con-tour h(C) in several smaller contours by making cuts at theseintersections. Each of the now obtained contours topologicallyresembles a circle. Each circle either encircles the origin onceor never. A clockwise encirclement is counted as -1, an anti-clockwise encirclement as +1. Summing these numbers for thedifferent circles gives the number of times h(C) encircles theorigin. Suppose the N intersection points are denoted xi ∈ IR.The intersections are numbered as follows. Starting on the realaxis and following contour C, intersection i is encountered be-fore intersection i + 1, but after intersection i− 1, 1 ≤ i ≤ N .Look at the circle between intersection points xi and xi−1. Onecan associate two integer numbers ai and bi with this circle. Theai define the sense of rotation, ai = +1 if anti-clockwise. The bidefine the encirclement of the origin, bi = 0 if no encirclement(see Table 1). The number of encirclements is

p =N∑i=2

bi

N∏j=i

aj .

A short code named CONTOUR was developed to map a con-tour C, with a given transform h(s). The contour C is a userdefined polygon, symmetric with respect to the real axis. Thecode determines h(C) and its intersections with the real axis.From this the number of encirclements of the origin is found.This number equals the number of zeroes of h(s) within C. The

C

1 zero

no zero’s

h(C)

3 zero’s

h(C)

h(C)

A B

C D

Im Im

Im Im

Re

Re Re

Re

Fig. 14a–d. Contourmapping: a the contour C defines the area wherezeroes are looked for; b-d different mappings h(C) of C. The numberof encirclements of the origin equals the number of zeroes. Here it wasassumed that h(s) analytic on C.

contour C is determined either by a priori specified frequenciesand quality factors that one hopes to find for certain parametervalues τ0, νph, χ or by specified minimum values of the damp-ing rate. To determine stability one chooses a rectangle withone side on the imaginary axis and the remainder in the positivereal half plane. This was done in Sect. 5. Using the asymptoticform of K ′

1(τ0s) (Watson 1962, §7.25 p. 204), one can showthat the maximal imaginary part of any zero in the positive realhalf plane is of order max(ν, I0, I

40τ

30 ). The NAG FORTRAN

s18dcf routine for modified Bessel functions can only just han-dle arguments of order 104. For τ0 > 10 the contour neededbecomes too large. If the contour is chosen to be such that any40 zeroes in the positive real half plane will always be encom-passed by it (but not necessarily 41 or more zeroes), one canstill apply CONTOUR and get sensible results with a ‘satura-tion’ of 40. A contour with the side along the imaginary axisof length 60 turns out to be sufficient for the parameter regimeunder consideration.


One can check the results by expanding the contour by afactor of two and then compare the results. The existence ofasymptotic forms for the location of zeroes means that mostzeroes are regularly spaced.

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Prominence oscillations and stability

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