EFECTIVIDAD DE LA PRÁCTICA RECONSTRUIDA

In this c h ap ter, w e consider a sim ple th eo retical model of p rom in en ce oscillations in te rm s of the m agnetoacoustic modes of oscillation of a th in plasm a slab. The slab is e m b e d d ed in a ho tte r and r a re r coronal plasm a. The effects of g rav ity are ignored. We assum e th a t both the prom in en ce slab and its coronal e n v iro nm e n t are p e rm e a te d by a uniform m agnetic field th a t is aligned along the long axis of the slab. Such an assu m p tio n reg ard in g th e m agnetic field of a prom inence was, in fact, co nsid ered by Kleczek and K uperus (1969) to explain the so-called w inking phenom eno n ' ob se rv ed in q u iescen t prom inences. A sim ilar model of p rom in en ces has also b e e n p ro p osed re c e n tly by Roberts (1991c; see also J o a rd e r and Roberts, 1992a) to explain the ob se rv ed long period (of abou t an hou r) oscillations in quiescent prom inences. The models by Kleczek and K uperus

( 1969) and Roberts (1991c) are described in Chapter I.

The model w e exam ine in this ch ap ter form s a n a tu ral extension of the

earlier models of Kleczek and K uperus (1969) and Roberts (1991c). Here,

we exam ine th e effect of a finite thickness of prom inence slab, and also the effect of the e x te rn al coronal m edium on the oscillation freq u e n cie s and

modes of th e model prom inence.

In re ality , a p rom in e n ce m agnetic field is no t lo n gitu d in al, b u t is inclined at a fine angle of a p p ro xim ately 2 0 ° with th e long axis of the p rom in en ce; see the ob se rv atio n al discussion in Section 1.1.2. With an av erag e m agnetic field stre n g th of 12 G in prom inences (see Table 1.1 in C h ap ter I), this p rod u ces a dom in a n t lo n gitu d in al com po ne n t of field stre n g th 10-11 G and a w e ak e r tran sv erse com ponent of stre n g th 3-5 G. In this c h a p te r, w e co nsid er the dom in a n t lo n gitu d in al field com po nen t, ignoring th e tra n s v e rs e com po ne n t altogether. The w e a k e r tr a n s v e r s e com po nen t of the m agnetic field ignored in this discussion, is trad itio n ally b eliev e d to pe rta in to the coronal m agnetic arcad e th a t th r e a d s th e prom inence plasm a tran sv ersely , providing it w ith the n ece ssary su p po rt against gravity; see, for exam ple, Kippenhahn-S chluter model (Fig. 1.1(a)) in C hapter I for a traditional description of prom inences. The footpoints of

po larity in the photosphere, thus producing a line-tying effect (Section

1.4.3). We ignore this photospheric line-tying effect in the p re se n t model

and consider the coronal plasm a to be of infinite extent on e ith er side of the prom inence slab.

W ith these simplifying assum ptions regarding the prom inence and its

coronal en v iro nm en t, the equilibrium th at we exam ine in this c h a p te r is th e n easily co nstru cted from Fig.1.5 (see Section 1.5, C hapter I) by substituting (|)=0 and C Here (t>is the angle of inclination of the m agnetic field to the slab axis and 2t is the anchor point separation distance of the m a g n e tic field lines. For co n ven ien ce, the modified e q u ilib riu m configuration is sketched in Fig. 2.1.

The model th a t w e are considering falls far short of satisfying th e situ atio n actually observed in quiescent prom inences. The no rm al modes co ns id e re d h e re , can no t the re fo re be com p a red d ire c tly with th e ob s e rv a tio ns of prom in en ce oscillations. Nonetheless, it is u seful to consider the modes of oscillation for this model as it provides a basis for

^ X

Fig. 2.1. A schematic diagram showing the model prominence slab with a longitudinal magnetic field.

th e u n d e rs ta n d in g of th e modes of oscillation th a t arise in th e more complex models investigated in later chapters.

The no rm al modes w e h ere describe are in fact the th re e d im ensional c ou n te r p a r ts of th e h y d rom a g n e tic modes of oscillatio n of a two d im e n sio n al p lasm a slab discussed in Roberts (1 9 8 1 b ) and Edwin and Roberts (1982), and in this context are of some in te re s t for v ariou s solar and astrophysical phenom ena. The modes are also sim ilar in n a tu re to the h y d rom ag n etic modes of a straight, uniform cylindrical p lasm a tu b e w ith a m agnetic field th a t is aligned to the axis of the tube; see Edwin and Roberts (1983), Roberts (1985). The stu d y of such modes has a w id e ran g e of ap plicatio ns in th e e n e rg y tra n s po rt m echanism s in su n spo ts and th in p ho tosp h eric flux tubes, as well as in coronal loops; see th e discussion in Edwin and Roberts (1983), Roberts et al. (1984), Roberts, (1991 a,b,c, 1992).

The eq u ilib riu m state in th e p articular case of a p u rely lo n gitu d in al (<j)=0) m agnetic field is described by the condition of continuity of th e gas p ressu re across the plasm a slab, nam ely, p^T^ = p^T^ , w h ere p^, p^ and T^, Tç are the equilibrium densities and te m p e ra tu re s inside and outside the slab. The m ean molecular w eight (p.= 1 / 2 ) is assum ed to be th e sam e inside and outside the prom inence slab.

The c h a ra c te ristic MHD speeds th a t e n te r in our d e scrip tio n of the modes of oscillation of the plasm a slab are the Alfven speeds Vy^^ and v^^ in sid e and ou tsid e th e slab, th e sound sp eed s Cg^ and Cg^, and th e m agnetoacoustic cusp speeds Cy^ and Cy^. Here the suffix o' stan d s for the p ro perties inside the slab (the object), w h ereas the suffix e' (the exterior) is used for the p ro perties of the slab's en v iro nm en t. We also define two critical sp eed s for n o n -p ara llel propagation in each of the uniform m edia inside and outside the plasm a slab. These are

V_CO A o jse c2 0: f 2_■'so" V 2 Ao^^sec"^6-4v^^CgQsec^8 y 1/2 1 1 / 2 (2.1a)

21 ^se'^^Ae Sec^Gi- / ^se'*'^Ae2 2 LV

sec^0-4v^gCggSec20 1/2 1 1 / 2 _(2.1b)

w h e re 0 is the angle of inclination b e tw ee n the direction of propagation of the w a v e mode and th e m agnetic field B. The sp eed s v^^ and v^g are sim ply th e m agnetoacoustic speeds along the m agnetic field B, w ith

refe rrin g to the fast w av e and ' referrin g to the slow w a ve (see Section

1.3). Note th a t as 8-> 0, v^^ min(v^Q.Cgg) and v tnin(v^g,Cgg), w h e re a s

n

^co and ^ max(Vy^g.Cgg). Also, as 9 -> 0^^, v ^

c^g, w h e re a s v^^ -> oo a nd v^g ->oo. These lim its are co nsisten t w ith th e

phase speeds of the m agnetoacoustic modes given in Section 1.3.2 (see Fig.

1.2a, b). For in te rm e d iate values of the angle of propagation, we h av e Vgg > Vco > CTo and > c^e.

2.2 Basic equations and the dispersion relation

We now c o nsid er sm all p ertu r b a tio ns of th e fo rm (1.9) a bou t th e e q u ilib riu m co nfig u ratio n d isplay ed in Fig. 2.1. In each of th e th re e uniform m edia inside and outside the slab, the linearized MHD eq u atio ns governing sm all p ertu rb a tio ns can be obtained from Eqs. (1.10)-(1.14) of C hapter I a fte r su b stitu tin g By = B = |B| and B^= B^= 0. Elim ination of the p ressu re p ertu rb a tio n p and the density p ertu rb atio n p from the linearized MHD eq u atio ns leads to th re e coupled o rd in a ry d ifferen tial e qu atio ns in the com ponents of the velocity p e rtu rb atio n (v^,Vy,v^). These eq u atio ns are

(see Roberts, 1981a)

^d^v ^

(® - k y V A ^ + "dv.,\ k_{k y \ dx}o ^d ^ A 0, (2.2a)

, , 2 2 2\ 2^dVy^

(kyCg'Co )Vy + ikyCg J+ k y k _{= 0}

and

(2.2b)

co^-(ky + kg)v^lvg - 0)2A z l _''' . 2, _{z^dx J}= 0 (2.2c)

differential equation (Roberts, 1981a; Rae and Roberts, 1983)

(kyV* - - (m 2 + k ^ )v j = 0, (2.3a)

w h e re

(kyCg - o)2)(kyVA - 0)2

= ___ _ (2.3b)

The first b rac k ete d () term in Eq.(2.3a) sim ply re p re se n ts an Alfven w ave pro p a g a tin g along the field and it is com pletely d e cou pled from the m ag n eto acou stic w av es. The second te rm in p a re n th e s is (), gives the

governing differential equation for the m agnetoacoustic w aves.

In th e following, w e do not discuss fu rth e r the Alfven modes of the slab. A p art from the triv ial n a tu re of th ese modes in th e p a rticular case of a p u rely longitudinal m agnetic field, th ese modes are highly anisotropic and incom pressible in ch aracter.(see Section 1.3.2) and are th ere fo re not likely to play any significant role in th e global motion of the prom inence slab, but in stea d m ay produce localized d istu rb a n c e s in th e in d iv id u al m agnetic field lines. In this c h ap ter, w e confine a tten tio n to the m agnetoacoustic

modes of th e slab.

+

It is convenient to rew rite Eq. (2.3a) in term s of the critical speeds v as_c defined in Eq. (2.1). For the m agnetoacoustic w aves, w e ob tain (Rae and

Roberts, 1983; Joarder and Roberts, 1992a)

d2v^ 2 ^ - k y M ^ v , , , (2.4a) w h e re + 2 o V -2

,

(vc - - c^) m2 ^ " - ( 2 --- ^ ---A - ^ . tan^e (2.4b)

The lin earized MHD e q u atio ns also give expressions for the v a riou s i m po r t a n t pe r tu r b a tio ns in te r m s of th e com po ne n t of v elo city p e rtu rb a tio n v^. These expressions are:

. 2 ic„ V y ( x ) / 2 _ 2 V , Y, z z 2\ ^ 2 2 ,jk y C T - C O .^ jk y dx (2.5) ik. v^(x) _ kyM^ dx (2.6) 2/, _fkyV^-co'^2 2 2^ p(x) = and r 2 2 V . Y , 2z z 2\ ^2 . kyCT-“

K

A / (2.7) ip_(o,e)^kyVA-co2^ p.j.(x) = fdv. cokyM^ dx (2.8)

Here p(x) and p^(x) are the p ertu rb a tio n am plitudes in gas p re ssu re and in

the total (gas+ magnetic) pressure, respectively.

Eqs. (2.2)-(2.8) apply inside (|x|< a) the slab w ith Cg = Cg^, v ^ = v^^, c^ = Cfo- Vg = VgQ, m^ = nig, M^ = Mg, and outside (|x| > a) the slab w h e re Cg = Cgg,

VA = VAe> " ^Te> ai^ = m^, M ^ = M

In th e ir model of prom inence oscillations, Kleczek and K uperus (1969)

In document Cuatro pasos para aprender la matemática (página 44-50)