The one-dimensional oblique propagation of magnetohydrodynamic waves with arbitrary amplitudes in a Hall plasma with isotropic pressure is studied under assumption that the plasma [beta] is large. It is shown that the wave evolution is described by the derivative nonlinear Schrödinger equation (DNLS)

Ruderman, M.S.

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This is a repository copy of DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-[beta] Hall plasma.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1576/Article:Ruderman, M.S. (2002) DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-[beta] Hall plasma. Journal of Plasma Physics, 67 (4). pp. 271-276. ISSN 0022-3778 https://doi.org/10.1017/S002237780200168Xeprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. J. Plasma Physics (2002), vol. 67, part 4, pp. 271–276.  2002 Cambridge University PressDOI: 10.1017/S002237780200168X Printed in the United Kingdom271DNLS equation for large-amplitude solitonspropagating in an arbitrary direction in ahigh-β Hall plasmaMICHAEL S. RUDERMANDepartment of Applied Mathematics, University of Sheffield, Hicks Building,Hounsfield Road, Sheffield S3 7RH, UK(Received 14 February 2002)Abstract. The one-dimensional oblique propagation of magnetohydrodynamicwaves with arbitrary amplitudes in a Hall plasma with isotropic pressure is studiedunder assumption that the plasma β is large. It is shown that the wave evolutionis described by the derivative nonlinear Schrödinger equation (DNLS).1. IntroductionIt has been known for almost three decades that the evolution of small-amplitudenonlinear Alfvén waves propagating quasiparallel with respect to the backgroundmagnetic field is governed by the derivative nonlinear Schrödinger equation (DNLS).This equation was first derived by Rogister (1971) starting with a Vlasov kinetic de-scription for the particle species. Later it was derived by Mjølhus (1976) and Mio etal. (1976) on the basis of Hall magnetohydrodynamics (MHD) for cold plasmas, andby Spangler and Sheerin (1982) and Sakai and Sonnerup (1983) from warm-fluidmodels. An excellent review of theory of quasiparallel small-amplitude nonlinearMHD waves based on the use of the DNLS equation and its generalizations hasbeen given by Mjølhus and Hada (1997). Application of this theory to observationof nonlinear MHD waves at the Earth’s bow shock has been discussed by Spangler(1997).This study was motivated by a recent result by Baumgärtel (1999). He numeri-cally solved the non-stationary system of Hall MHD equations using a dark DNLSsoliton as an initial condition. In this numerical study, the propagation angle was80◦ and the plasma β was 5. The numerical solution has shown that the solitonpractically does not evolve with time. This implies that the dark DNLS solitonspropagating at large angles with respect to the background magnetic field are sta-tionary (or, at least, quasistationary) solutions to the system of Hall MHD equa-tions. This result supports the suggestion by Kennel et al. (1988) that the DNLSequation might decribe nonlinear wave propagation in an arbitrary direction withrespect to the background magnetic field in high-β plasmas.The aim of this paper is to show that the numerical result obtained by Baumgärtel(1999) is not a coincidence, and that the suggestion by Kennel et al. (1988) isperfectly correct. The paper is organized as follows. In the next section, we presentthe governing equations and discuss main assumptions. In Sec. 3, we derive the272 M. S. RudermanDNLS equation for large-amplitude waves propagate in an arbitrary direction in ahigh-β plasma. We discuss our results in Sec. 4.2. Governing equationsThe starting point of our analysis is the system of Hall MHD equations describingadiabatic motions of infinitely conducting plasmas with isotropic pressure. Thissystem can be written in the form∂ρ∂t+∇ · (ρv) = 0, (2.1a)ρ(∂v∂t+ (v ·∇)v)= −∇p +1µ(∇× B)× B, (2.1b)∂B∂t=∇× (v× B) + l(ρ0µ)1/2∇×1ρ(B×∇× B), (2.1c)pργ= const. (2.1d)Here p is the pressure, ρ the density, v the velocity, B the magnetic field, γ the adia-batic exponent, and µ the magnetic permeability of empty space, and the subscript‘0’ indicates an unperturbed quantity. The ion inertia length l is given byl =(m2iµe2ρ0)1/2, (2.2)where mi is the ion mass and e the elemental electric charge.In what follows, we consider one-dimensional perturbations that only depend onthe x coordinate in the Cartesian coordinates (x, y, z). Let us introduce the squareof the sound speed, the square of the Alfvén speed in the x direction, and the plasmaβ asc2S =γp0ρ0, V 2 =B2xµρ0, β =c2SV 2, (2.3)where Bx = const is the x components of the magnetic field. We assume that β  1and use ǫ = β−1 as a small parameter. In accordance with this, we introduce thescaled sound speed, c̃S = ǫcS , and the scaled equilibrium pressure p̃0 = ǫp0.3. Derivation of DNLS equationTo derive the DNLS equation, we use the reductive perturbation method (e.g.Taniuti and Wei 1968; Kakutani et al. 1968; Engelbreht et al. 1988). We intro-duce the new variables ξ = ǫ(x − V t) and τ = ǫ2t. In these variables, (2.1) arerewritten asǫ∂ρ∂τ− V∂ρ∂ξ+∂(ρu)∂ξ= 0, (3.1a)ρ(ǫ∂u∂τ− V∂u∂ξ+ u∂u∂ξ)= −∂∂ξ(p +|B⊥|22µ), (3.1b)ρ(ǫ∂v⊥∂τ− V∂v⊥∂ξ+ u∂v⊥∂ξ)=Bxµ∂B⊥∂ξ, (3.1c)Large-amplitude DNLS solitons 273ǫ∂B⊥∂τ− V∂B⊥∂ξ= Bx∂v⊥∂ξ−∂(uB⊥)∂ξ+ ǫχV l ê×∂∂ξ(ρ0ρ∂B⊥∂ξ), (3.1d)pργ= const, (3.1e)where χ = sign(Bx), ê is the unit vector in the x direction, u is the x compo-nent of the velocity, and v⊥ and B⊥ are the components of the velocity and themagnetic field perpendicular to the x direction. In what follows, we only considerperturbations vanishing at infinity and assume that ρ → ρ0, p → p0, u → 0,v⊥ → 0, and B⊥ → B⊥0 as |ξ| → ∞. We can consider the magnetic field atinfinity, B0 = Bxêx + B⊥0, as the equilibrium magnetic field, so that perturba-tions propagate at an angle θ = arctan(|B⊥0|/|Bx|) with respect to the equilibriummagnetic field.Now we look for the solution to (3.1) in the form of expansions in power serieswith respect to ǫ. We write these expansions in the formp = ǫ−1p̃0 + p1 + ǫp2 + . . . (3.2)for the pressure, and in the formf = f1 + ǫf2 + . . . (3.3)for all other variables. In the first-order approximation, we collect terms of theorder of unity in (3.1) to obtain the system of equationsV∂ρ1∂ξ−∂(ρ1u1)∂ξ= 0, (3.4a)ρ1(V∂u1∂ξ− u1∂u1∂ξ)=∂∂ξ(p1 +|B⊥1|22µ), (3.4b)ρ1(V∂v⊥1∂ξ− u1∂v⊥1∂ξ)= −Bxµ∂B⊥1∂ξ, (3.4c)V∂B⊥1∂ξ= −Bx∂v⊥1∂ξ+∂(u1B⊥1)∂ξ, (3.4d)p0ργ1=p0ργ0. (3.4e)In deriving (3.4e), we have used the boundary conditions at infinity.It follows from (3.4e) that ρ1 = ρ0. Then, using the boundary conditions atinfinity, we immediately find from (3.4) thatu1 = 0, (3.5a)v⊥1 = −VBx(B⊥1 − B⊥0), (3.5b)p1 = −12µ(|B⊥1|2 − |B⊥0|2). (3.5c)We see that only the perturbations of v⊥, B⊥ and p are of the order of unity, whilethe perturbations of ρ and u are of the order of ǫ.In the second-order approximation, we collect terms of the order of ǫ in (3.1). As274 M. S. Rudermana result, we obtain the system of equationsV∂ρ2∂ξ− ρ0∂u2∂ξ= 0, (3.6a)ρ0V∂u2∂ξ=∂∂ξ(p2 +B⊥1 · B⊥2µ), (3.6b)ρ0(∂v⊥1∂τ− V∂v⊥2∂ξ)=Bxµ∂B⊥2∂ξ, (3.6c)∂B⊥1∂τ− V∂B⊥2∂ξ=∂∂ξ(Bxv⊥2 − u2B⊥1) + χV l ê×∂2B⊥1∂ξ2, (3.6d)p1 = c̃2Sρ2. (3.6e)In deriving (3.6c) we have used (3.6a). Using (3.5c), (3.6a), and (3.6e), we obtainu2 = −V2µρ0c̃2S(|B⊥1|2 − |B⊥0|2). (3.7)With the aid of this result and (3.5b), we rewrite (3.6c, d) asV∂B⊥2∂ξ+Bx∂v⊥2∂ξ= −∂B⊥1∂τ, (3.8a)V∂B⊥2∂ξ+Bx∂v⊥2∂ξ=∂B⊥1∂τ+ χV l ê×∂2B⊥1∂ξ2−V2µρ0c̃2S∂∂ξ[B⊥1(|B⊥1|2 − |B⊥0|2)]. (3.8b)The compatibility condition for this system of two equations for B⊥2 and v⊥2 isthat their right-hand sides have to be equal. As a result, we obtain∂B⊥1∂τ−V4µρ0c̃2S∂∂ξ[B⊥1(|B⊥1|2 − |B⊥0|2)] + 12χV l ê×∂2B⊥1∂ξ2= 0. (3.9)Returning to the original independent variables and introducing b = By1+ iBz1, weeventually arrive at the DNLS equation∂b∂t+ V∂b∂x−V4βB2x∂∂x[b(|b|2 − |b0|2)] +i2χV l∂2b∂x2= 0, (3.10)where b0 = lim|x|→∞ b. Hence we have shown that, in high-β Hall plasmas, theDNLS equation does describe oblique propagation of MHD waves with arbitraryamplitudes.4. DiscussionThe results of the previous section clearly show that, in contrast to plasmas with lowand moderate β, in high-β plasmas, the DNLS equation describes nonlinear waveswith arbitrary amplitude of the magnetic field perturbation and propagating in anarbitrary direction.However, one has to be very cautious when applying this result to real problemsin plasma physics (e.g. to problems related to space plasmas). The point of concernis that the system of Hall MHD equations is often used for the description ofLarge-amplitude DNLS solitons 275motions of collisionless plasmas. This is only justified when the plasma β is small. Inplasmas with moderate and large β, kinetic effects become important (e.g. Mjølhusand Willer 1986, 1988; Mjølhus 1988; Mjølhus and Hada 1997). Hence, both thesystem of Hall MHD equations and the DNLS equation (3.10) can be used for thedescription of high-β plasmas only if these plasmas are collisional.A plasma can be considered as collisional if the characteristic spatial scale, whichis βl, is much larger than the mean free path of the charged particles. This conditioncan be easily reduced to Ωiτ  β, where Ωi = eB0/mi is the ion gyrofrequency andτ if the mean collision time. In order that we can neglect the plasma resistivity andconsider it as infinitely conducting, the condition Ωeτ  1 must be satisfied, whereΩe = eB0/me is the electron gyrofrequency, and me is the electron mass.An example of a plasma where both inequalities are satisfied is the solar photo-spheric plasma. In this plasma, we can use the approximation τ ≈ τei, where τeiis the electron–ion collision time (Priest 1982). If we take B0 ≈ 0.1T (= 1000G),then Ωeτei varies from 2 in the lower part of the photosphere to 2 × 103 in theupper part of the photosphere (Ruderman et al. 1997). Hence, we can neglect theresistive term in Ohm’s equation in comparison with the Hall term in the middleand upper part of the photosphere. Since Ωi ≈ 0.5 × 10−3Ωe, Ωiτei 6 1  β inthe solar photosphere. Therefore, (3.10) can be used for the description of nonlinearMHD waves (e.g. solitons) in the middle and upper part of the solar photosphere.The characteristic length of these solitons is βl ≈ 0.01β cm.AcknowledgementThe author gratefully acknowledges financial support from PPARC.ReferencesBaumgärtel, K. 1999 Soliton approach to magnetic holes. J. Geophys. Res. 104, 28295–28308.Engelbreht, J. K., Fridman, V. E. and Pelinovskii, E. N. 1988 Nonlinear Evolution Equations.London: Longman.Kakutani, T., Ono, H., Taniuti, T. and Wei, C.-C. 1968 Reductive perturbation method innonlinear wave propagation. II. Application to hydromagnetic waves in cold plasmas.J. Phys. Soc. Jpn 24, 1159–1166.Kennel, C. F., Buti, B., Hada, T. and Pellat, R. 1988 Nonlinear, dispersive, elliptically po-larized Alfvén waves. Phys. Fluids 31, 1949–1961.Mio, K., Ogino, T., Minami, K. and Takeda, S. 1976 Modified nonlinear Schrödinger equationfor Alfvén waves propagating along the magnetic field in cold plasmas. J. Phys. Soc. Jpn41, 265–271.Mjølhus, E. 1976 On the modulational instability of hydromagnetic waves parallel to themagnetic field. J. Plasma Phys. 16, 321–334.Mjølhus, E. 1988 The onset of Alfvénic turbulence – Comment. Phys. Fluids 31, 2723–2724.Mjølhus, E. and Hada, T. 1997 Soliton theory of quasi-parallel MHD waves. In: NonlinearWaves and Chaos in Space Plasmas (ed. T. Hada and H. Matsumoto). Tokyo: TERRA-PUB, pp. 121–169.Mjølhus, E. and Willer, J. 1986 Alfvén solitons. Physica Scripta 33, 442–451.Mjølhus, E. and Willer, J. 1988 Nonlinear Alfvén waves in a finite-beta plasma. J. PlasmaPhys. 40, 299–318.Priest, E. R. 1982 Solar Magnetohydrodynamics. Dordrecht: Reidel.Rogister, A. 1971 Parallel propagation of nonlinear low-frequency waves in high β plasma.Phys. Fluids 14, 2733–2743.Ruderman, M. S., Hollweg, J. V. and Goossens, M. 1997 Nonlinear theory of resonant slowwaves in dissipative layers. Phys. Plasmas 4, 75–90.276 M. S. RudermanSakai, J. I. and Sonnerup, B. U. Ö. 1983 Modulational instability of finite-amplitude disper-sive Alfvén waves. J. Geophys. Res. 88, 9069–9079.Spangler, S. R. 1997 Nonlinear evolution of MHD waves at the Earth’s bow shock: Opinionson the confrontation between theory, simulations, and measurements. In: NonlinearWaves and Chaos in Space Plasmas (ed. T. Hada and H. Matsumoto). Tokyo: TERRA-PUB, pp. 171–223.Spangler, S. R. and Sheerin, J. P. 1982 Properties of Alfvén solitons in a finite-beta plasma.J. Plasma Phys. 27, 193–198.Taniuti, T. and Wei, C.-C. 1968 Reductive perturbation method in nonlinear wave propaga-tion. I. J. Phys. Soc. Jpn 24, 941–946.

DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-[beta] Hall plasma

MICHAEL S. RUDERMAN

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DNLS equation for large-amplitude solitons 
propagating in an arbitrary direction in a 
high-β Hall plasma

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DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-[beta] Hall plasma

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