Eur. Phys. J. D 60, 325–330 (2010) DOI: 10.1140/epjd/e2010-00213-6
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Propagation of magnetoacoustic surface waves along static plasma slab surrounded by moving plasma and neutral gas R. Valliammal1,a , M. Sivaraman2 , and K. Somasundaram3 1 2 3
Department of Physics, APA College (W), Palani 624615, Tamil Nadu, India Department of Physics, GRI, Gandhigram 624302, Tamil Nadu, India Department of Computer Science and Application, GRI, Gandhigram 624302, Tamil Nadu, India
Received 13 February 2010 / Received in final form 17 April 2010 c EDP Sciences, Societ` Published online 30 August 2010 – a Italiana di Fisica, Springer-Verlag 2010 Abstract. The existence and propagation of fast and slow magnetoacoustic surface waves (MASW) is investigated in our work by taking a theoretical model of a static plasma slab as the middle layer with a moving plasma region at the top and neutral gas medium as the bottom layer. Applying linear MHD, the dispersion relation is obtained and the propagation of magnetoacoustic surface waves, in the compressional limit for steady flow and for different values of dimensionless wave numbers, is analyzed. Steady flow of plasma along a structured atmosphere may cause enhancement of existing surface modes, disappearance of some modes and generation of new surface wave modes. The possible regions for the propagation of fast and slow surface and body waves for different mass density ratios and magnetic field ratios and with a small flow velocity are studied. Our discussion may help in analyzing more complicated cases.
1 Introduction The perturbations in magnetized incompressible fluid plasma propagate at the Alfv´en speed by bending the magnetic field lines in the transverse direction. But when the compressibility is taken into account, the perturbation not only bends but also compresses the magnetic field lines, resulting slow and fast wave modes. Such waves are called magnetoacoustic waves [1,2]. At a compressible plasma-plasma interface the magnetoacoustic modes can manifest as surface magnetoacoustic waves. [3–6]. Hydro magnetic surface wave propagation along an incompressible cylindrical plasma column surrounded by a neutral gas has been studied by Uberoi and Somasundaram [7]. The density and neutral temperature of magnetized helium plasma columns submerged in a cool neutral gas have been studied by Chiu and Cohen [8]. The dispersive characteristic of Alfv´en urface waves (ASW) along a moving incompressible plasma slab or cylinder has been discussed by Narayanan and Somasundaram [9], and Somasundaram and Narayanan [10]. The effect of compressibility was discussed by Naryanan [11]. The various plasma regions in the Earth’s atmosphere with different densities and temperature interact with each other. This makes the magnetosphere a very dynamic plasma system deriving energy from the sun; it a
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can support various types of plasma waves and instabilities. These waves are common feature of space plasma waves and play an important role in understanding the various phenomena like micro pulsations, sub storms, magnetosphere-ionosphere coupling. The magnetoacoustic surface wave phenomena for a plasma-plasma-neutral gas interface have not been discussed much, so far. The twodimensional model in this present work may be applied for the case of compressible plasma slab (E layer) as the middle layer which couples the magnetospheric plasma (the upper layer) to the neutral gas layer (D layer and below) in Earth’s atmosphere. The aim of this work is to study the MASW trapped within the magnetic slab allowing the presence of flow outside the slab. Flows may be important for a number of reasons: they can bring about the KelvinHelmholtz instability in magnetic structures. Our investigation is done in the compressional limit applying linear MHD and neglecting the effects of conductivity terms.
2 The geometry and formulation The study of propagation of magnetoacoustic surface waves is based on a two-dimensional model, which consists of three uniform regions. A plasma slab of density ρ01 is assumed to be embedded in a steady magnetic field B 01 (medium-1), being surrounded by a plasma moving with a velocity U (medium-2) and a neutral gas region
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x
B02 ρ02
+a 2
1
U
B01 ρ 01
Using the field components and applying the boundary conditions, we get the dispersion relation as, 2a
z -a g
B0g, ρ 0g
Fig. 1. Showing the geometry under study which has a plasma slab of thickness 2a sandwiched between a moving plasma layer above and a static neutral gas medium below along with the equilibrium density and magnetic fields in all the three regions. The flow velocity U and the magnetic field B 01,02,0g are assumed to be along the z-direction.
(medium-g) as shown in Figure 1. In all the three medium the equilibrium physical quantities: the magnetic field B, the pressure P , the density ρ and the flow velocity U of the upper plasma layer are assumed to be varying uniformly. The linearized MHD equations [3,4,11,12] for an infinitely conducting, non-viscous compressible and moving plasma slab embedded in a magnetic field B 0 are: ∂ρ ∂ρ = −ρ0 (∇ · v) − U ∂z ∂t ∂ (∇ × b) × B 0 ∂ +U v = ∇p + ρ0 ∂t ∂z μ ∂b ∂v ∂b = −U − B 0 (∇ · v) + B 0 ∂t ∂z ∂z ∇·b=0 ∂p ∂p = C 2 ρ0 (∇ · v) − U , ∂t ∂z
(1) (2) (3) (4) (5)
where v, p and b are the perturbed fluid velocity, pressure and magnetic field respectively, μ is the flow interface parameter and C is the sound speed in the plasma medium γp0 given by C = ρ0 , where γ is the flow interface pa-
rameter and p0 , ρ0 and B 0 are the unperturbed pressure, density and magnetic field respectively. Equations (1) to (5) can be Fourier transformed, assuming all perturbed quantities are proportional to f (x) exp[i(kz − ωt), i.e., ∂ ∂ ∂ ∂t = −iω, ∂x = 0 and ∂z = ik. After Fourier transforming, we can obtain the magnetic field components and velocity components. For static plasma slab also the velocity and magnetic field components can be arrived at using equations (1) to (5) by taking U = 0. The boundary conditions that are applied at x = a and x = −a [7] are:
(i) the tangential component of the electric field seen by the plasma just below the surface (static plasma – neutral gas interface) must vanish; (ii) the total pressure is continuous across the boundary (interface); (iii) the normal velocity (tangential) component is continuous.
tanh 2(m1 a) 2 2 ρ02 m21 εg k 2 vA2 − (U k − ω) coth 2(m1 a) 2 2
2 2
2 +ρ01 ρ02 m1 k vA2 − (U k − ω) k vA1 − ω 2
2 2 2 +ρ01 m1 m2 εg k 2 vA1 − ω 2 + ρ201 m2 k 2 vA1 − ω2 tanh 2(m1 a) =0 × coth 2(m1 a)
(6)
2 B0g k 2 ρ0g ω 2 where εg = , and τ = − 1 − kω2 s2 = μ kτ
1 − α2 vP2 h in which, α = vsA , is the flow interface parameter, s is the sound velocity in neutral gas medium and vph = kvωA is the normalized phase velocity of the surface waves at the interface and 2 2 2 k 2 C22 − (U k − ω) k 2 vA2 − (U k − ω) m22 = 2 ] k 2 CT2 2 − (U k − ω)2 [C22 + vA2 and
m21
2 k 2 vA1 − ω 2 k 2 C12 − ω 2 = 2 ] [k 2 C12 − ω 2 ] [C12 + vA1
where vA1,A2 =
B √ 01,02 μρ01,02
and CT2 1,T 2 =
2 2 C1,2 vA1,A2
2 (CT2 1,T 2 +vA1,A2 ) and C1,2 , vA1,A2 , CT 1,T 2 are the sound velocity, Alfv´en velocity and tube velocity in the media 1 and 2 respectively. Here m21 and m22 are functions of ω 2 and k 2 .
3 Discussion In equation (6), the term m1 a in tanh 2(m1 a) or coth 2(m1 a) can be normalized with vA1 which gives ⎞1/2 ⎛
C12 2 1 − vP2 h − v Ph ⎟ 2 ⎜ vA1 2 ⎟ m1 a = ⎜ ⎝ C2 ⎠ ka C1 T1 2 − vP h 2 2 +1 vA1 vA1
(7)
In absence of flow (U = 0) and in the limiting case when ka → ∞, the dispersion relation (6) reduces to the following equation,
2 2 m1 εg + ρ01 m2 k 2 vA1 ρ02 m1 k 2 vA2 − ω2 − ω2
2 +ρ01 m2 k 2 vA1 − ω2 = 0
(8)
and in the incompressible limit as C → ∞ results in m1 → k and m2 → k. From the first part of equation (8), we get, η1 − β12 − 1 = 0 (9) vP2 h 1 + τ as discussed by Gerwin [13] for the surface waves at the plane interface of an incompressible conducting plasma
R. Valliammal et al.: Propagation of magnetoacoustic surface waves along static plasma slab...
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Table 1. Showing the various interface parameters taken for the study and the normalized tube velocities (CT 1,2 ), sound velocities (C1,2 ) and the ratio of Alfv´en velocities (vA1,2 ). β12 = 1.5, η1 = 1.5, γ = 1.66, α = 0.2, v = 0.0 η2
CT 1 /vA1
CT 2 /vA1
C1 /vA1
C2 /vA1
vA2 /vA1
Remarks
1.0 0.707106
C2 > C1 and vA2 = vA1 C2 > C1 and vA2 < vA1
1.41421 1.0
C2 > C1 and vA2 > vA1 C2 = C1 and vA2 = vA1
β22 = 0.5 0.5 1.0
0.942247 0.942247
0.971275 0.971275
2.81336 2.81336
4.08167 2.88617 β22 = 1.0
0.5 1.0
0.942247 0.942247
0.942247 0.942247
2.81336 2.81336
3.97869 2.81336
media and a compressible gas. The second part of equation (8) is similar to the case discussed by Roberts [3,4] and in the presence of flow it becomes
m1 2 2 2 ρ01 k 2 vA1 k vA2 − (U k − ω)2 + ρ02 m2
tanh 2(m2 a) = 0 (10) −ω 2 coth 2(m2 a)
The roots of the dispersion equation (12) are obtained numerically, by taking parallel propagation and by varying the wave number ka. We do not consider the surface waves with negative phase velocity. These waves are named as backward surface waves. In a compressible interface, we get both fast and slow magnetoacoustic surface and body waves. When there is flow, the usual fast and slow magnetoacoustic surface waves are accelerated and we give the which was discussed by Nakariakov and Roberts [12] for nomenclature for accelerated fast and slow magnetoacousa plasma-plasma interface. Another limiting case can be tic surface waves as simply fast and slow magnetoacoustic applied to the dispersion relation (6) in which ka → 0 surface waves. Depending upon the condition, we get, surface waves and in the incompressible limit as C → ∞ the relation (6) when m21 > 0 or body waves when m21 < 0. Trapped waves becomes, 2 2
2 ρ02 k vA2 − kω 2 + εg = 0. (11) occur only when m2 > 0; otherwise, waves leak from the slab [12]. The roots of the dispersion relation (12) are obThe above equation can be normalized to get equation (9). tained for η = 1 and 1.5, η = 0.5 and 1, β 2 = 0.5 2 1 2 The dispersion relation (6) is a transcendental equation and 1, β 2 = 1.0 and 1.5, γ = 1.66 and α = 0.2. Here 1 and requires a numerical evaluation to find the roots. So the values of the density ratios and magnetic field dennormalizing the equation with vA1 , we get, sity ratios are randomly chosen but abiding the condi N1 N1 M1 tion that the equilibrium is subjected to stability. From N1 tan 2h [T1 T3 ] (ka) +[T3 T2 ] η1 the pressure balance conditions we have the normalized cot 2h η1 γ 2 1/2 N N2 M 2 2 2 N sound velocities as vCA11 = and N1 M1 2 N1 α2 + 2 β1 − 1 tan 2h 1/2 2 [T1 T2 ] + +η1 T2 (ka) = 0, C cot 2h C2 N2 M2 N2 = η12 v21 + γ2 1 − β22 at plasma slab-neutral v A1 (12) A1 gas interfaces and at plasma slab-moving plasma interface where, v = U/vA1 is the normalized flow velocity of the respectively. The normalized tube velocities CT 1 and CT 2 2 2 B2 2 and the ratio of Alfv´en velocities (vA2 /vA1 = β22 /η2 ) can plasma medium lying above the slab and β22 = B02 2 , β1 = 01 be determined and are tabulated as in Table 1. 2 Bg0 ρ02 are the magnetic interface parameters η = , η = One can draw the following diagram to represent the 2 2 1 ρ01 B01 ρg0 possible regions for the existence of surface and body ρ01 are the density interface parameters, and modes for the tabulated parameters above and from the 2
β2 condition for surface waves to exist m21 > 0 and for body 2 2 − v − vP h , T2 = 1 − vP h , T1 = waves m21 < 0 and m22 should be always greater than η2 2 zero. The diagram given below (Fig. 2) which gives ex
η1 C plicitly the existence of the possible regions is not drawn T3 = β12 − vP2 h , M1 = 22 − v − vP2 h T1 , τ vA1 to any scale, and we have taken the interface parameters 2 as β12 = 1.5, η1 = 1.5, β22 = 1.0, η2 = 0.5, γ = 1.66, C1 2 N1 = 2 − vP h T2 , α = 0.2 and v = 0.0. The surface waves exist only for the v 2 2 A1 condition vA2 /vA1 > 1β22 /η2 > 1) in the absence of flow.
C22 CT2 2 β22 2 The phase velocities that are obtained by solving the nor− v − vP h M2 = 2 2 + η vA1 vA1 2 malized dispersion relation are lying in the range of 1.0 to 1.3 so they are the (externally fast) fast surface waves. and 2 2 There are no body modes. Graphs are drawn with phase C1 CT 1 2 velocity verses ka as shown in Figures 3a, 3b, 4a and 4b. N2 = − v + 1 . Ph 2 2 vA1 vA1 Similar discussions can be made for the propagation of
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Fast surface Waves 1/α
5.0
C2/vA1
3.97869
C 1/ vA1
2.81336
vA2/ vA1
1.414
(Externally fast)Fast Surface Waves
1. 0
1.0
CT2/ vA1 & CT1/ vA1
0.942247
Slow Surface Waves
Fig. 2. The possible regions for the existence of MASW for the plasma interface parameters namely: flow interface parameter α = 0.2, magnetic interface parameters β22 = 1, β12 = 1.5, the density interface parameters η2 = 0.5, η1 = 1.5 and the normalized flow velocity v = 0.
(a)
(b)
Fig. 3. Shows the normalized phase velocities vs. the wave numbers. Symmetric mode (solid lines) and asymmetric mode (dotted lines) are plotted for parallel propagation for the flow interface parameter α = 0.2, magnetic interface parameters β22 = 0.5, 1, β12 = 1.5, density interface parameters η2 = 0.5, 1 η1 = 1.5 and normalized flow velocity v = 0. (b) Is like (a) but for v = 0.5.
MASW when the top layer of plasma assumes a small value of flow velocity and also for β12 = 1 and η1 = 1 both in the presence and absence of flow.
4 Surface and body waves We can note from the graphs in Figures 3a, 3b, that in the absence of flow and for β22 = 1 and η2 = 0.5, in symmetric mode as ka increases the phase velocity decreases but for asymmetric mode this behavior
is reversed. From the study of the possible regions of MASW, it is noted that as m22 > 0 the phase velocity is lying in the range as max {(CT 2 /vA1 ) + v} < vP h < min {(vA2 /vA1 ) + v}, which is nomenclatured as externally fast [14] and as m21 > 0 then the phase velocity lies in limit, [max (CT 1 /vA1 ) , 1] < vP h < min (C1 /vA1 ) which is the condition for fast surface waves. The existence of both symmetric and asymmetric modes is decided by whether vA2 > vA1 or vA2 ≤ vA1 . When β22 = 1 and η2 = 0.5 we get vA2 > vA1 , so for this condition and
R. Valliammal et al.: Propagation of magnetoacoustic surface waves along static plasma slab...
(a)
329
(b)
Fig. 4. Shows the normalized phase velocities vs. the wave numbers. Symmetric mode (solid lines) and asymmetric mode (dotted lines) are plotted for parallel propagation for flow interface parameter α = 0.2, magnetic interface parameters β22 = 0.5, 1, β12 = 1.0, density interface parameters η2 = 0.5, 1.0 η1 = 1 and normalized flow velocity v = 0. (b) Is like (a) but for v = 0.5.
when there is no flow, both the symmetric and asymmetric modes exist as (externally fast) fast surface waves only. For β22 = 0.5 there are no modes existing for both the values of η2 (η2 = 0.5, 1), which implies that in the absence of flow and as long as vA2 ≤ vA1 there are neither surface modes nor body modes. When the flow is introduced (a small value of v as v = 0.5), both the symmetric and asymmetric modes get accelerated and vP h decreases as ka increases for symmetric mode and vice versa for asymmetric mode. From the study of the possible regions of MASW in the presence of flow, it is noted that as m22 > 0 the phase velocity has the range as follows: vP h < [min {(CT 2 /vA1 ) + v} , min {(vA2 /vA1 ) + v} and min {(C2 /vA1 ) + v}], which has been nomenclatured as externally slow and as m21 > 0 then the phase velocity lies in limit, [max (CT 1 /vA1 ) , 1] < vP h < min (C1 /vA1 ) which is the condition for fast surface waves. The symmetric and asymmetric modes manifest themselves as (externally slow) fast surface waves and there are no body modes. Unlike in the case of v = 0, irrespective of whether vA2 ≤ vA1 or vA2 > vA1 we get the (externally slow) fast surface waves for v = 0 and for the values of β22 = 0.5, 1 and η2 = 0.5, 1. In order to study the effect of neutral gas-plasma interface parameters the value of β12 and η1 are varied as β22 = 1 and η2 = 1 and from the dispersion curves (Figs. 4a, 4b), we note that there is no much change in the behavior of MASW and they exist in a similar manner as in the previous case.
5 Conclusion Previously a number of works was done for plasma-plasma interface by many authors. In our work we have included the presence of neutral gas layer also and theoretically investigated the existence and propagation of fast and slow magnetoacoustic surface waves both in the presence and absence of flow for symmetric and asymmetric modes. The effect of relative mass densities, relative magnetic field densities and the presence of flow are discussed. The study shows that the effect of the presence of neutral gas and the compressibility factor is that there are no pure slow or fast magnetoacoustic surface or body waves but (externally fast) fast surface waves and (externally slow) fast surface waves alone are propagating within the possible region. In the absence of flow (externally fast) fast surface waves propagate only when vA2 > vA1 irrespective of whether the slab is cooler or warmer than the moving plasma layer. But in the presence of a small value of Alfv´en Mach number (v = 0.5) the symmetric and asymmetric modes manifest themselves as (externally slow) fast surface waves, irrespective of whether vA2 ≤ vA1 or vA2 > vA1 and irrespective of whether the slab is cooler or warmer than the moving plasma layer. Both in the absence and presence of plasma flow in medium-2, it is investigated that the magnetoacoustic surface waves propagate only when the neutral gas medium is warmer than the two plasma medium. This work is restricted to parallel propagation only. The discussion of the non-parallel propagation of magnetoacoustic surface waves will be the topic of separate publication.
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