Review article: MHD wave propagation near coronal null points of magnetic fields
aa r X i v : . [ a s t r o - ph . S R ] A p r Space Science Reviews manuscript No. (will be inserted by the editor)
Review article: MHD wave propagation near coronal nullpoints of magnetic fields
J. A. McLaughlin · A. W. Hood · I.De Moortel
Received: 30 January 2010 / Accepted: 8 April 2010
Abstract
We present a comprehensive review of MHD wave behaviour in theneighbourhood of coronal null points: locations where the magnetic field, and hencethe local Alfv´en speed, is zero. The behaviour of all three MHD wave modes, i.e. theAlfv´en wave and the fast and slow magnetoacoustic waves, has been investigatedin the neighbourhood of 2D, 2.5D and (to a certain extent) 3D magnetic nullpoints, for a variety of assumptions, configurations and geometries. In general, itis found that the fast magnetoacoustic wave behaviour is dictated by the Alfv´en-speed profile. In a β = 0 plasma, the fast wave is focused towards the null point bya refraction effect and all the wave energy, and thus current density, accumulatesclose to the null point. Thus, null points will be locations for preferential heating byfast waves . Independently, the Alfv´en wave is found to propagate along magneticfieldlines and is confined to the fieldlines it is generated on. As the wave approachesthe null point, it spreads out due to the diverging fieldlines. Eventually, the Alfv´enwave accumulates along the separatrices (in 2D) or along the spine or fan-plane(in 3D). Hence, Alfv´en wave energy will be preferentially dissipated at these locations .It is clear that the magnetic field plays a fundamental role in the propagation andproperties of MHD waves in the neighbourhood of coronal null points. This topic isa fundamental plasma process and results so far have also lead to critical insightsinto reconnection, mode-coupling, quasi-periodic pulsations and phase-mixing.
Keywords
Magnetic fields, Coronal · Magnetic fields, models · Waves, Magneto-hydrodynamic · Waves, Propagation · Magnetohydrodynamics
Magnetohydrodynamic (MHD) wave motions (e.g. Roberts 2004; Nakariakov &Verwichte 2005; De Moortel 2005) are ubiquitous throughout the solar corona(Tomczyk et al. 2007). Several different types of MHD wave motions have now been
J. A. McLaughlinSchool of Computing & Engineering, Northumbria University, Newcastle Upon Tyne, NE18ST, UKE-mail: [email protected] McLaughlin et al. observed by various solar instruments: slow magnetoacoustic waves have been seenin
SOHO data (e.g. Ofman et al. et al. et al.
TRACE data (De Moortel etal.
TRACE (Aschwanden et al. et al.
Hinode (Ofman& Wang 2008). Non-thermal line narrowing / broadening due to Alfv´en waves hasbeen reported by Harrison et al. (2002) / Erd´elyi et al. (1998), Banerjee et al. (1998) and O’Shea et al. (2003). Alfv´en waves have possibly been observed in thecorona (Okamoto et al. et al. et al. et al. et al. null points - locations wherethe magnetic field, and hence the Alfv´en speed, is zero, and separatrices - topo-logical features that separate regions of different magnetic flux connectivity. Acomprehensive review can be found in Longcope (2005).This paper will provide a comprehensive literature review of the nature of MHDwave propagation in the neighbourhood of coronal null points. This topic exists atthe overlap of two important areas of solar physics: MHD wave and magnetic null-point theories. A brief introduction to both of these areas is provided in § . § .
3, including a mathematical description of magnetic null points. § § § . § § § ρ h ∂ v ∂t + ( v · ∇ ) v i = −∇ p + 1 µ ( ∇ × B ) × B + ν ∇ · π ,∂ B ∂t = ∇ × ( v × B ) + η ∇ B ,∂ρ∂t + ∇ · ( ρ v ) = 0 ,ρ h ∂ǫ∂t + ( v · ∇ ) ǫ i = − p ∇ · v + 1 σ | j | + νε ij π ij , (1)where ρ is the mass density, v is the plasma velocity, B the magnetic induction(usually called the magnetic field), p is the plasma pressure, µ = 4 π × − Hm − isthe magnetic permeability, ν is the coefficient of classical viscosity, π ij = ε ij − δ ij ∇· HD waves and coronal null points 3 v is the stress tensor, ε ij = ( ∂v i /∂x j + ∂v j /∂x i ) / σ isthe electrical conductivity, η = 1 /µσ is the magnetic diffusivity, ǫ = p/ρ ( γ −
1) isthe specific internal energy density, where γ = 5 / j = ∇ × B /µ is the electric current density. ν and η are assumed to be constants.Note that the classical viscous term used in equations (1) is in fact not themost appropriate for the solar corona since, in the presence of strong magneticfields, the viscosity takes the form of a non-isotropic tensor. However, only thepapers of Craig & Litvinenko (2007) and Craig (2008) mentioned in this reviewwill invoke the non-isotropic viscous tensor and so, for brevity, we do not provide afull description here. The mathematical details of the non-isotropic viscous tensorcan be found in Braginskii (1965) and, for example, Van der Linden et al. (1998),Ofman et al. (1994) and Erd´elyi & Goossens (1994; 1995).1.2 MHD wavesA wave is a disturbance that propagates through space and time, usually withthe transference of energy. Such a disturbance, either continuous or transient,propagates by virtue of the elastic nature of the medium. In MHD, the magnetictension provides an elastic restoring force, such that we would expect waves topropagate along uniform magnetic fieldlines with a characteristic speed: v A = | B |√ µρ , where v A is called the Alfv´en speed . Transverse waves travelling at this speed alongmagnetic fieldlines are called
Alfv´en waves .If we consider a compressible medium, then we can define the sound speed as: c s = r γpρ . When assuming a compressible medium, the Alfv´en wave still remains, but thesound and Alfv´en speed can now couple together to give magnetoacoustic waves.Two combinations arise: the higher frequency mode is known as the fast magne-toacoustic wave and the lower frequency wave is known as the slow magnetoacousticwave . These three wave types, the Alfv´en wave and the fast and slow magnetoa-coustic waves, make up the three MHD waves considered in this review paper.The fundamental properties and nature of linear MHD waves in uniform mag-netic fields have been reported in detail by several authors, for example in an un-bounded homogeneous medium (Cowling 1976), and in a bounded inhomogeneousslab / cylindrical density profile embedded in a uniform magnetic field (Roberts1981) / (Edwin & Roberts 1983; Cally 1986; Roberts & Nakariakov 2003).Finally, in MHD it is important to consider the ratio of magnetic pressure tothermal pressure. This ratio is called the plasma − β and is given by: β = 2 µp | B | = 2 γ c s v A . (2)The properties of the fast and slow magnetoacoustic waves have a strong depen-dence on the magnitude of the plasma − β , namely because it is directly propor-tional to the square of the ratio of the sound speed to the Alfv´en speed. Thus, McLaughlin et al. in a regime where β ≪
1, magnetic pressure and magnetic tension dominate thepropagation and vice versa. Table 1 lists the main properties of the three wavetypes depending upon their environment. Note that the Alfv´en wave behaviour isindependent of the plasma- β , as it is a purely magnetic wave (in the linear regime).The plasma − β parameter varies greatly with height across the layers of thesolar atmosphere (see Gary 2001 for well-constrained values). However, magneticpressure generally dominates thermal pressure in the solar corona, and thus it isusual to assume plasma − β ≪ − or zero − β environment, although there are some caveats to this ( § . plasma- β ≫ β ) plasma − β ≪ β )Alfv´en wave Transverse wave propagating at speed v A Fast MA wave Behaves like sound wave(speed c s ) Propagates roughly isotropicallyPropagates across magnetic fieldlines(speed v A )Slow MA wave Guided along B (speed v A ) Guided along B Longitudinal wave propagatingat speed c s Table 1
Properties of MHD waves depending on the plasma − β . topology nomenclature to reduce a complicated setof fieldlines to something more understandable. In 2D, a general magnetic con-figuration contains separatrix curves (separatrices) which split the magnetic planeinto topologically distinct regions, in the sense that within a specific region allthe fieldlines start at a particular source and end at a particular sink. There is asecond important topological aspect: magnetic null points (or neutral points) aresingle-point locations where the magnetic field vanishes ( B = ). There are twotypes of magnetic null point: X-type null points , commonly called X-points, whichoccur at the intersection of separatrix curves, and
O-type null points , or O-points,located at the center of magnetic islands. Magnetic topologies that contain nullpoints are common in the presence of multiple magnetic sources.A magnetic fieldline that joins two null points (itself a special type of separa-trix) is called a separator . Thus, instead of showing all the magnetic field lines ina region, we can just show the important aspects of the topology; such a pictureof the magnetic structure is called the magnetic skeleton of the field. In 3D, wehave similar properties, now with separatrix surfaces separating the volume intotopologically distinct regions, and these surfaces intercept at a separator.
HD waves and coronal null points 5 (a) (b) (c)
Fig. 1 ( a ) X-type null point for α = 1. This potential neutral point has separatrices (redlines) intersecting at an angle of π/
2. ( b ) O-type null point, with α = −
1. A blue star denotesthe null point. ( c ) Single potential magnetic null point configuration created by interaction oftwo dipoles. Here, A z = y/ (cid:2) ( x + λ ) + y (cid:3) + y/ (cid:2) ( x − λ ) + y (cid:3) , for λ = 0 .
5. Red lines denotethe separatrices.
Let us first consider null points in 2D (e.g. Dungey 1953; 1958). Following § . . B = [ B x ( x, y ) , B y ( x, y ) , . A null point occurs at the point ( x , y ) if: B x ( x , y ) = 0 and B y ( x , y ) = 0 . Expanding B x and B y in a Taylor series about ( x , y ) gives the linear approxi-mation: B x = ∂B x ∂x (cid:12)(cid:12)(cid:12) ( x ,y ) ( x − x ) + ∂B x ∂y (cid:12)(cid:12)(cid:12)(cid:12) ( x ,y ) ( y − y )= a ( x − x ) + b ( y − y ) , (3) B y = ∂B y ∂x (cid:12)(cid:12)(cid:12)(cid:12) ( x ,y ) ( x − x ) + ∂B y ∂y (cid:12)(cid:12)(cid:12)(cid:12) ( x ,y ) ( y − y )= c ( x − x ) − a ( y − y ) , (4)where the coefficients a, b, c are arbitrary.Let us now introduce the vector potential (also called the flux function), A ,such that B = ∇ × A , and in 2D we have A = (0 , , A z ). Thus, we have: B = (cid:18) ∂A z ∂y , − ∂A z ∂x , (cid:19) . (5)Integrating equations (3) and (4) gives the corresponding vector potential as: A z = a ( x − x )( y − y ) + b y − y ) − c x − x ) , McLaughlin et al. (c)(b)(a)
Fig. 2
Magnetic fields containing two null points. ( a ) Magnetic configuration containing bothX-type and O-type null points. Here A z = λ x − y − ( x − λ ) / B = (cid:2) − y, ( x − λ ) − λ (cid:3) )for λ = 0 .
5. Red lines/blue star denotes the separatrices/O-type null point. ( b ) Potentialmagnetic configuration containing two X-type null points connected by a separator. Here, A z = − x y + y / λ y , where λ = 1. ( b ) Potential magnetic configuration containing twoX-type null points not connected by a separator. Here, A z = − xy + x / − λ x , where λ = 1.Red lines denote the separatrices. where we have chosen the arbitrary constant of integration such that A z vanishes at( x , y ). Further simplification is possible by rotating the xy -axes through an angle θ to give new x ′ , y ′ -axes, and choosing the angle θ such that tan 2 θ = − a/ ( b + c ).This simplification gives the corresponding vector potential as: A z = B L h(cid:0) y ′ − y ′ (cid:1) − α (cid:0) x ′ − x ′ (cid:1) i , (6)where BL = 2 a + b − c p a + ( b + c ) , α = 4 a a + b − c − . Here, B is the characteristic strength of the magnetic field and L is the charac-teristic length-scale over which the field varies. The corresponding field componentsare: B x = BL ( y ′ − y ′ ) and B y = BL α ( x ′ − x ′ ) . (7)Magnetic field lines are defined by A z equal to a constant. For α >
0, thefieldlines are hyperbolic, giving an X-type null point. The separatrices are givenby y ′ − y ′ = ± α ( x ′ − x ′ ) and are inclined at an angle ± arctan α to the x ′ -axis.The magnetic fieldlines for α = 1, x ′ = y ′ = 0 can be seen in Figure 1a.The value of α (and thus the angle between the separatrices) is related to thecurrent density. The current density is given by: j = 1 µ ( ∇ × B ) = − µ ∇ A z ˆ z = − BµL (cid:0) − α (cid:1) ˆ z . Thus, a null point is potential if α = ± π/
2. Note that for an O-point, α is imaginary and so the current density is always non-zero. Thus, O-type neutralpoints can never be potential. HD waves and coronal null points 7 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) z zy x y x(a) (b)
Fig. 3 ( a ) Proper radial null point, described by B = ( x, y, − z ), i.e. ǫ = 1. ( b ) Improperradial null point, described by B = ( x, ǫy, − [ ǫ + 1] z ), for ǫ = 1 /
2. Note for ǫ = 1 /
2, the fieldlines rapidly curve such that they run parallel to the x − axis along y = 0. In both figures, the z − axis indicates the spine and the xy − plane at z = 0 denotes the fan-plane . The red fieldlineshave been tracked from the z = 1 plane, the blue from z = − An O-type null point magnetic configuration can be seen in Figure 1b, for α = − x ′ = y ′ = 0. However, note that the simple 2D magnetic field configuration ofequation (7) is only valid close to the null point: as x ′ and/or y ′ get very large, B becomes unphysically large. Figure 1c denotes a more realistic single magnetic nullpoint configuration created by the interaction of two dipoles. This configurationcomprises of four separatrices and an X-point, and as x ′ and/or y ′ get large, thefield strength becomes smaller (i.e. a more physical field).Magnetic configurations can also contain multiple null points, and it can beargued that null points appear in pairs; a double null point may arise as a localbifurcation of a single 2D null point (see e.g. Galsgaard et al. Magnetic null points also exist in three dimensions, but occur in a different formto those described in § B = BL ( x, ǫy, − [ ǫ + 1] z ) , (8)where the parameter ǫ is related to the predominate direction of alignment ofthe fieldlines in the fan plane. Parnell et al. (1996) investigated and classified thedifferent types of linear magnetic null points that can exist (our ǫ parameter iscalled p in their work). Topologically, this 3D null consists of two key parts: the z − axis represents a special, isolated fieldline called the spine which approaches thenull from above and below (as found by Priest & Titov 1996) and the xy − plane McLaughlin et al. through z = 0 is known as the fan-plane and consists of a surface of fieldlinesspreading out radially from the null. Figure 3 shows two examples of 3D nullpoints: ǫ = 1 (Figure 3a) and ǫ = 1 / et al. § IV ). For ǫ ≥
0, 3D nulls are describedas positive nulls, i.e. the spine points into the null and the field lines in the fan aredirected away. In addition, all potential nulls are designated radial , i.e. there is nospiral motions in the fan-plane.In this review paper, we only consider positive, potential null points, and thusthere are three cases to consider: – ǫ = 1: describes a proper null (Figure 3a). This magnetic null has cylindricalsymmetry about the spine axis (so is actually only a 2.5D null point). – ǫ > , ǫ = 1: describes an improper null (Figure 3b). Field lines rapidly curvesuch that they run parallel to the x − axis if 0 < ǫ < y − axisif ǫ > – ǫ = 0: equation (8) reduces to a simple 2D X-point potential field in the xz − plane and forms a null line along the y − axis through x = z = 0.1.4 Statistics of coronal null pointsWe have provided a mathematical description of null points, but how commonare null points in the corona? Null points are an inevitable consequence of thedistributed isolated magnetic flux sources at the photospheric surface. Using pho-tospheric magnetograms to provide the field distribution on the lower boundary,both potential and non-potential (nonlinear force-free) field extrapolations suggestthat there are always likely to be null points in the corona. The number of suchsingular points will depend upon the magnetic complexity of the photosphericflux distribution. Detailed investigations of the coronal magnetic field, using suchpotential field calculations, can be found in Beveridge et al. (2002) and Brown& Priest (2001). The properties of coronal null points have also been consideredthrough theoretical considerations (e.g. Parnell et al. et al. et al. et al. et al. et al. (2004) calculated a po-tential field extrapolation from a high resolution MDI magnetogram and found1 . × − magnetic null points per square megameter. R´egnier et al. (2008) per-formed a similar investigation using a magnetogram from the Narrowband FilterImager onboard Hinode and found 6 . × − Mm − . Longcope & Parnell (2009)investigated 562 MDI magnetograms using the Fourier spectrum of magnetogramsand found 3 . × − ± . × − coronal null points per square megameter (ataltitudes greater than 1.5 Mm). Alternatively, we can estimate the total numberof coronal null points by multiplying these results by the surface area of the Sun(i.e. to provide a crude estimate, where we assume the Sun is free of active regionsand coronal holes). This corresponds to approximately 10 ,
000 (Close et al.
HD waves and coronal null points 9 ,
000 (Longcope & Parnell 2009) or 40 ,
000 (R´egnier et al. et al. (2009) investigated the solar cycle variation of coro-nal null points using a potential field source surface model in spherical geometry,and find that there is no significant variation in the number of nulls found from therising to the declining phase (indicating that null points are present throughoutboth phases of the solar cycle).Several investigations also consider specific examples of null points in thecorona. For example, Aulanier et al. (2000) investigated a class M3 flare thatoccured on 14 July 1998 above a δ − spot. Using potential field extrapolations,the authors recreated the pre-flare magnetic topology from Kitt Peak line-of-sightmagnetograms and revealled a single coronal null point located above the δ − spot.Secondly, Ugarte-Urra et al. (2007) investigated the magnetic topology of 26 CMEevents by performing potential field extrapolations from corresponding MDI mag-netograms, and find that magnetic null points are present in a large number of thepre-CME topologies.Finally, we note that a null point plays a key role in the magnetic breakout model (e.g. Antiochos 1998; Antiochos et al. et al. et al. et al. – MHD wave propagation in inhomogeneous media is a fundamental plasma pro-cess, and the study of MHD wave behaviour in the neighbourhood of magneticnull points directly contributes to this area. – We now know that MHD wave perturbations are omnipresent in the corona.We also know that null points are an inevitable consequence of the distributedisolated magnetic flux sources at the photospheric surface (where the numberof such singular points will depend upon the magnetic complexity of the pho-tospheric flux distribution). Thus, these two areas of scientific study; MHDwaves and magnetic topology, will encounter each other at some point, i.e.MHD waves will propagate into the neighbourhood of coronal null points (e.g.blast waves from a flare will at some point encounter a null point). Thus, thestudy of MHD waves around null points is itself a fundamental coronal process. – The study of MHD wave behaviour in the neighbourhood of magnetic nullpoints is also interesting in its own right and, as we shall see, often providescritical insights into other areas of plasma physics, including: mode-conversion et al. ( § . § . § .
7) and phase-mixing( § . The first investigation into the behaviour of MHD waves in the neighbourhoodof 2D magnetic null point was performed by Bulanov & Syrovatskii (1980). Theauthors considered the linearised MHD equations (equations 1) under the coldplasma approximation ( β = 0) in a cylindrically symmetric geometry, where acircular boundary is imposed at r = 1 at which the fieldlines are held fixed. Theirinvestigation produces a detailed discussion of the propagation of fast and Alfv´enwaves in the neigbourhood of an equilibrium magnetic field: B = BL ( − y, − x ) , (9)which corresponds to the magnetic configuration in Figure 1a. The subscript 0denotes an equilibrium quantity.Bulanov & Syrovatskii also noted that in this 2D geometry, the z − componentmotions are decoupled from xy − plane motions. Thus, the Alfv´en wave, whichis governed by motions transverse to the magnetic field (i.e. v z ˆ z ) is decoupledfrom the magnetoacoustic waves (motions in the xy − plane). Hence, it is possibleto consider the Alfv´en wave and the magnetoacoustic waves separately in a 2Dgeometry.In their paper, harmonic fast waves are generated at the r = 1 boundary andthese propagate inwards towards the null point, and Bulanov & Syrovatskii findthat, in the asymptotic limit r →
0, the perturbations have azimuthal symmetry,i.e. propagate as cylindrical waves. This was the first piece of work that indicateda key relationship between fast waves and null points. However, the assumed cylin-drical symmetry means that the disturbances can only propagate either towardsor away from the null point, and are already encircling the null point. Thus, it isunclear if this is a general result.To investigate the Alfv´en wave, Bulanov & Syrovatskii make a coordinate trans-formation such that the coordinate lines coincide with the lines of force, where: ζ = 12 (cid:0) x − y (cid:1) = 12 r cos 2 θ , η = xy = 12 r sin 2 θ . (10)The authors find that Alfv´en perturbations propagate along magnetic fieldlinesat the local Alfv´en speed. The inhomogeneity of the Alfv´en speed profile leadsto an exponential increase in the gradients in the system, and in the asymptoticlimit t → ∞ , these gradients accumulate at the separatrices of the field, i.e. ζ = 0.Again, this result was the first to indicate a relationship between Alfv´en wavepropagation and the location of the separatrices.An alternative approach to the study of MHD wave behaviour in the neigh-bourhood of a 2D magnetic null point was investigated in a series of papers byCraig and co-workers (Craig & McClymont 1991; 1993; Craig & Watson 1992)in which the authors considered perturbations of the flux function A z . Here, thefocus was on applications for magnetic reconnection, specifically to investigate ifnull points (viewed as weaknesses in the magnetic field) could collapse in response HD waves and coronal null points 11 to imposed boundary motions. To clearly demonstrate their results, we repeat partof their analysis here:In polar coordinates, an X-type null point, located at the origin, can be ex-pressed as: B = BR (cid:0) r sin 2 θ ˆ r + r cos 2 θ ˆ θ (cid:1) , (11)where B denotes the equilibrium magnetic field and R represents the typical sizeof a coronal magnetic structure.The MHD equations (1) can be simplified by invoking the flux function A z (equation 5). In polar coordinates, the equilibrium flux function ( A ) for a simple2D X-point (i.e equation 6) is: A = − r cos 2 θ (12)where the evolution of A z is governed by the induction equation and the dynamicsare governed by the momentum equation: ∂A z ∂t + ( v · ∇ ) A z = η ∇ A z ,∂ v ∂t + ( v · ∇ ) v = −∇ A z · ∇ A z . These equations can be linearised about the equilibrium flux function such that A z = A + A , and combined to form a single differential equation for the perturbedflux function ( A ): ∂ A ∂t = |∇ A | ∇ A + η ∇ ∂A ∂t , (13)where, from equation (12), |∇ A | = r .We can see that the right-hand-side of equation (13) has two parts. Considera region close to the origin, r c , where r c ∼ η / is the skin depth. Far from theorigin, r ≫ r c , advection effects dominate and equation (13) reduces to a waveequation: ∂ A ∂t = r ∇ A (14)Here, the rate of propagation of information is governed by the wave speed pro-portional to r which makes the signal travel time logarithmic in r . Thus, a dis-turbance on the outer boundary ( r = 1) propagates into the diffusion region in atime δt ∼ | ln r c | ∼ ln η .Alternatively, close to the origin, r ≪ r c , and equation (13) reduces to thediffusion equation: ∂A ∂t = η ∇ A . Diffusion is ultimately responsible for dissipating the kinetic and magnetic energyin the system.Next, let us assume separation of variables such that: A ( r, θ, t ) = f ( r ) e imθ e λt , et al. where m is the azimuthal wavenumber. The eigenequation for f ( r ) is then: r h ∂∂r (cid:16) r ∂f∂r (cid:17)i = (cid:20) λ (1 + ηλ/r ) + m (cid:21) f . (15)Close to the origin ( r ≪ r c ), the radial dependence of the current j z = −∇ A = J m h ( − λ/η ) / r i . We note that only the J Bessel function is nonzero atthe origin (which is equivalent to a nonvanishing displacement in the flux function)and so the m = 0 mode is the only mode which allows topological reconnection.Assuming m = 0, the evolution of A z can then be obtained from the analytical ornumerical solution of equation (13).Craig & McClymont (1991) derived equation (13) and investigated the fieldlinespassing through the (circular) boundary in a particular manner in order to perturbthe field - shifting the footpoints so as to “close up” the angle of the X-point. Theresulting field perturbations cause the null point to collapse to form a current sheetin which reconnection can release magnetic energy. In these models the boundarymotions move the field lines but do not return them to their original positions (akinto photospheric footpoint motion). Thus, the Poynting flux induced by the imposedmotion (and then fixing the field after the motion is complete) accumulates at theresulting current sheet and provides the energy released in the reconnection.Craig & McClymont find that, as their system evolves, the fieldlines recon-nect as they pass through the null point (located at the origin), indicating thatresistivity is essential to this mode. Their initial vertical ‘current sheet’ beginsreconnecting and the inertia of the flowing plasma carries the system past theequilibrium configuration, until a weaker horizontal current sheet is formed. Then,a much weaker vertical current sheet returns at a later time (one complete cycle).After three such cycles, the system is close to its equilibrium (potential) configu-ration. The reconnection is found to be oscillatory , with inertial overshoot of theplasma carrying more magnetic flux through the neutral point than is requiredto reach a static equilibrium. The reconnection rate scales as | ln η | , and so thereconnection is described as fast . This is an important results as it shows thatthe damping of the fast wave is still highly significant even when the resistivity issmall.Craig & Watson (1992) considered the radial propagation of the m = 0 modeand solved equation (15) using a mixture of analytical and numerical solutions.They demonstrated that the propagation of the m = 0 wave towards the null pointgenerates an exponentially large increase in the current density and that magneticresistivity dissipates this current in a time related to log η , in agreement with Craig& McClymont (1991). Their initial disturbance is given as a function of radius,i.e. an internal perturbation is considered. In their investigation, the outer radialboundary is held fixed so that any outgoing waves will be reflected back towardsthe null point. This means that all the energy in the wave motions is containedwithin a fixed region.Craig & McClymont (1993) investigated the normal mode solutions for both m = 0 and m = 0 modes with resistivity included. Again they emphasised thatthe current builds up as the inverse square of the radial distance from the nullpoint. Craig & McClymont (1993) also explicitly report on the focusing of thewave (energy) onto the neutral point due to the gradient in Alfv´en speed (clearlyseen in equation 14). HD waves and coronal null points 13
Independently, Hassam (1992) also investigated the behaviour of stressed X-points in a cylindrically symmetric geometry. Hassam performed a similar deriva-tion to that of equation (13) and recognised that equation (15) can be recast as ahypergeometric equation (e.g. Oberhettinger 1990) when m = 0:( z − ddz (cid:16) z ddz f (cid:17) = (cid:16) λ (cid:17) f , using the transformation z = − r /ηλ . The relaxation time found using this for-mulation is | ln η | and so is in agreement with that found by Craig & McClymont(1991; 1993).Craig et al. (2005) comment that it is surprising that fluid viscosity has beenneglected in previous studies, and state that the leading terms in the viscous stresstensor actually dominate the plasma resistivity by many orders of magnitude fortypical coronal plasmas (as emphasised by Hollweg 1986). These authors extendthe model of Craig & McClymont (1993) to include the effect of (isotropic) scalarfluid viscosity. They find the inclusion of viscosity can have a dramatic effect: fornon-reconnective modes ( m > η > ν the problem isdominated by ln η , and ν > η it is dominated by ln ν ). For reconnective disturbances( m = 0), the oscillatory reconnection is suppressed if ν > η , whereas for η > ν , fastoscillatory reconnection is regained. However, the authors do note that althoughviscosity can dramatically influence the rate of dissipation in the system, finiteresistivity is still required for reconnection to occur. Craig (2008) has extendedthis study to include non-isotropic viscosity (Braginskii 1965) and find that themain results of Craig et al. (2005) are still valid.These papers have led the way in understanding MHD motions in the neigh-bourhood of a 2D X-point. However, the assumed cylindrical symmetry means thatthe magnetoacoustic disturbances can only propagate either towards or away fromthe null point, and attention has been restricted to a circular reflecting boundary,so all outgoing waves are reflected back into the vicinity of the null point. Thus, ina sense there is nowhere else for the wave to propagate except into the null point.In addition, in all these papers (except Bulanov & Syrovatskii 1980 whichconsidered asymptotic limits) the boundary motions move the field lines but do notreturn them to their original positions. The Poynting flux induced by this imposedmotion provides the energy released in the resultant current sheet. However, if theboundary motions are simply due to the passing of incoming waves through theboundary, then it is not clear that the null point need collapse and form a currentsheet. Furthermore, if this is the case, then it is not clear if the energy in the wave(again due to the Poynting flux through the boundary) will dissipate or simplypass through one of the other boundaries.Finally, all these papers have assumed that the system is best described interms of normal modes, where a single normal mode can be thought of as the longtime evolution of a system. However, normal mode analysis does not allow us toconcentrate on the transient features of the wave propagation.For these reasons, it is informative to specifically track the propagation ofboundary-driven (as opposed to internally generated) asymmetric disturbancesinto the domain. et al. The first investigation that specifically made the extension away from polar sym-metry was performed by Ofman (1992) and Ofman et al. (1993), who performednonlinear, resistive 2D MHD calculations of the reconnection in the stressed nullpoint and obtained the | ln η | scaling law numerically, as well as solving the lineardispersion relation (equation 15) for all azimuthally nonsymmetric perturbations( m >
0) analytically, in agreement with the results of Craig & McClymont (1991;1993) and Hassam (1992). However, the authors also stressed the significance of thechoice of boundary condition, and the strong influence they have on the permis-sible solutions. Steinolfson et al. (1995) investigated the effects of the boundaryconditions further by perfoming nonlinear, resistive MHD simulations to studystressed X-points, contrasting both rigid and open boundary conditions. The au-thors found that, for rigid boundary conditions, they could recover the results ofCraig & McClymont (1991; 1993), but found that for open boundary conditions,the X-point was instead deformed by their perturbation to form a current sheet.Shortly thereafter, Hassam & Lambert (1996) utilised a cartesian geometry toinvestigate the propagation of the Alfv´en wave in the neighbourhood of a simpleX-point (of the form seen in Figure 1a). Using the same coordinate transformations(equation 10) as in Bulanov & Syrovatskii (1980), Hassam & Lambert again findthat the Alfv´en wave propagates along magnetic fieldlines, i.e. the fluid elementsare confined to the magnetic fieldlines they are generated on. However, Hassam& Lambert also stress the distinction between two different types of boundarydriven Alfv´en wave motions. Perturbations that initially straddle the separatriceseventually accumulate along the separatrices, whereas disturbances that do notinitially straddle the separatrices do not accumulate along the separatrices, butleave the system and are eventually damped by phase-mixing (Heyvaerts & Priest1983).Hassam & Lambert investigated the propagation of the Alfv´en wave in a squarenumerical box by driving a harmonic wave train (polarised transverse to the planeof the magnetic field) on both the left and right boundaries, simulating periodicfootpoint motion. Their choice to drive wave-trains into the domain from oppositesides of the box means the forcing is antisymmetric in the horizontal direction.In addition, line-tied conditions are used on the upper and lower boundaries, i.e.both velocity and the normal derivative of magnetic field are kept zero. Thus, wavemotions are reflected back into the numerical domain, limiting the evolution.An investigation of MHD wave propagation in a β = 0 plasma in the neigh-bourhood of a 2D null point has been looked at in a series of papers by McLaughlin& Hood (2004; 2005; 2006a), where the focus was on more general disturbances,more general boundary conditions and single wave pulses (rather than harmonicwave trains). By looking at boundary-driven disturbances generated from a sin-gle boundary, coupled with non-reflecting boundary conditions, such investigationallow us to focus on the transient features on the propagation.To clearly demonstrate the results of McLaughlin & Hood, we repeat part oftheir analysis here: HD waves and coronal null points 15 v = + v (no backgroundflows). The only exception is B = B + b , where b = ( b x , b y , b z ).McLaughlin & Hood now consider a special coordinate system for v : v = v k | B | (cid:18) B | B | (cid:19) − v ⊥ | B | (cid:18) ∇ A | B | (cid:19) + v y ˆ y , where A is the equilibrium vector potential. The terms in brackets are unit vec-tors. This splits the velocity into parallel and perpendicular components. Thiswill make our MHD mode detection and interpretation easier. For example, in alow β − plasma, the slow wave is guided by the magnetic field and has a velocitycomponent that is mainly field-aligned. This makes perfect sense when β ≪ β ≫ v = ¯v v ∗ , v ⊥ =¯v B v ∗⊥ , v k = ¯v B v ∗k , B = B B ∗ , b = B b ∗ , x = Lx ∗ , z = Lz ∗ , p = p p ∗ , ∇ = ∇ ∗ /L , t = ¯ tt ∗ , A = BLA ∗ and η = η , where we let * denote a dimensionless quantityand ¯v, B , L , p , ¯ t and η are constants with the dimensions of the variable theyare scaling. We then set B/ √ µρ = ¯v and ¯v = L/ ¯ t . We also set η ¯ t/L = R − m ,where R m is the magnetic Reynolds number, and set β = 2 µp /B , where β isthe plasma- β at a distance unity from the null/origin.This process non-dimensionalises the linearised MHD equations. For the restof this section, we drop the star indices; the fact that the variables are now non-dimensionalised is understood. Thus, the linearised, non-dimensionalised equationsare: ρ ∂∂t v ⊥ = −| B | [( ∇ × b ) · ˆ y ] + β ∇ A · ∇ p ,ρ ∂∂t v k = − β B · ∇ ) p ,ρ ∂v y ∂t = ( B · ∇ ) b y ,∂b x ∂t = [( ∇ v ⊥ × ˆ y ) · ˆ x ] + 1 R m ∇ b x ,∂b y ∂t = ( B · ∇ ) v y + 1 R m ∇ b y ,∂b z ∂t = [( ∇ v ⊥ × ˆ y ) · ˆ z ] + 1 R m ∇ b z ,∂p ∂t = − γ (cid:20) ∇ · (cid:18) B v k | B | (cid:19) − ∇ · (cid:18) v ⊥ ∇ A | B | (cid:19)(cid:21) . (16)Note that the 2D geometry considered by McLaughlin & Hood is in the xz − plane.As noted first by Bulanov & Syrovatskii (1980), this means that the Alfv´en wave (inthe ˆ y − direction) is decoupled from the magnetoacoustic waves (in the xz − plane).Finally, we note that equations (16) simplify greatly if we neglect pressureperturbations (i.e. assume β = 0). Under the β = 0 assumption, the plasma pres-sure plays no part in the dynamics of the system, and so the linearised equation et al. of mass continuity has no influence on the momentum equation and so in effectthe plasma is arbitrarily compressible (Craig & Watson 1992) and we assume thebackground gas density is uniform ( ρ ). A spatial variation in ρ can cause phasemixing (Heyvaerts & Priest 1983; De Moortel et al. et al. β = 0, linearised MHD equations naturally decouple into two equa-tions for the fast magnetoacoustic wave (governed here by v ⊥ ) and for the Alfv´enwave (governed by v y ). The slow wave is absent in the β = 0 limit (v k = 0).Under these assumptions, the linearised equations for the fast magnetoacousticwave can be combined to form a single wave equation: ∂ ∂t v ⊥ = v A ∇ v ⊥ = v A (cid:18) ∂ ∂x + ∂ ∂z (cid:19) v ⊥ , (17)where v A ( x, z ) = | B | / √ ρ = p ( B x + B z ) /ρ is the equilibrium (unperturbed)Alfv´en speed.Similarly, the linearised equations for the Alfv´en wave can be combined to forma single wave equation: ∂ ∂t v y = ( B · ∇ ) v y = (cid:16) B x ∂∂x + B z ∂∂z (cid:17) v y . (18)Wave equations (17) and (18) are the primary equations governing the be-haviour of the linear, β = 0, fast and Alfv´en waves in an equilibrium magnetic field B . These equations form the basis of the investigations carried out by McLaughlin& Hood (2004; 2005; 2006a) for various magnetic configurations.3.2 Single two-dimensional null pointMcLaughlin & Hood (2004) investigated the behaviour of the fast and Alfv´en wavesabout a simple 2D X-type null point using the following equilibrium magnetic field: B = BL ( x, , − z ) . (19)This magnetic field represents a π/ a .Note that this particular choice of magnetic field is only valid in the neighbourhoodof the null point located at x = 0, z = 0.McLaughlin & Hood (2004) considered a single wave pulse coming in from thetop boundary of the form:v ⊥ ( x, z max ) = (cid:26) sin ωt for 0 ≤ t ≤ πω , (20) ∂ v ⊥ ∂x (cid:12)(cid:12)(cid:12) x = x min = 0 , ∂ v ⊥ ∂x (cid:12)(cid:12)(cid:12) x = x max = 0 , ∂ v ⊥ ∂z (cid:12)(cid:12)(cid:12) z = z min = 0 , where ω is the frequency. The authors find that the linear fast magnetoacous-tic wave travels towards the neighbourhood of the X-point and bends around it.Since the Alfv´en speed, v A ( x, z ) = | B | / √ ρ = x + z , is spatially varying, thewave travels faster the further it is away from the null point. Thus, the wavedemonstrates refraction and this can be seen in Figure 4a. A similar refractionphenomenon was found by Nakariakov & Roberts (1995). This refraction effect HD waves and coronal null points 17 (a) (b)
Fig. 4 ( a ) Contour of v ⊥ at t = 2 for a fast wave driven from the upper boundary. Theoverplotted black lines are the WKB solution, where the three lines represent the leading,middle and trailing edges of the wave pulse. The blue cross denotes the null point (locatedat the origin). ( b ) Ray paths of the WKB solution for an Alfv´en wave driven at the upperboundary, after a time t = 2, for starting points of x = 0 , . , ... . wraps the wave around the null point, and it is this that is the key feature oflinear, β = 0 fast wave propagation. This refraction effect is the (non-radial) gen-eralisation of the (purely radial) focusing effect reported by Bulanov & Syrovatskii(1980) and Craig & McClymont (1991; 1993).Note, since the Alfv´en speed drops to zero at the null point, the wave neverreaches there, but the length scales (which can be thought of as the distance be-tween the leading and trailing edges of the wave pulse) rapidly decrease, indicatingthat gradients, and hence the current density, will rapidly increase. McLaughlin& Hood (2004) show that for the simple 2D X-point, all gradients increase expo-nentially as they approach the null point. The rate of this build-up is extremelyimportant, as it implies that resistive dissipation will eventually become impor-tant, regardless of the size of η , and this will convert the wave energy into (ohmic)heat. In fact, the exponential growth of the current density indicates that the timefor magnetic diffusion to become important will depend on ln η .This is in good agreement with Craig & McClymont (1991; 1993) and Craig& Watson (1992) who had previously found that the reconnection rate scales as | ln η | . This means that wave dissipation will be very efficient, and predicts thatnull points will be the natural locations of linear fast wave energy deposition andpreferential heating.To confirm their results, McLaughlin & Hood (2004) also solved equation (17)approximately using the WKB approximation (e.g. Murray 1927; Sneddon 1957).The WKB solution of McLaughlin & Hood assumes ω ≫ et al. out, following the field lines, in agreement with Bulanov & Syrovatskii (1980).The wave is confined to the fieldlines it is excited on. As the wave approaches theseparatrix (defined by the x − axis), the pulse thins but keeps its original amplitude.The wave eventually accumulates along the separatrices. As for the fast wave, wehave decreasing length scales, and for this choice of set-up, j x grows exponentiallyin time. Hence, the authors find that the Alfv´en wave causes current density toaccumulate along the separatrices (in agreement with Hassam & Lambert 1996).Hence, all the Alfv´en wave energy will be dissipated along the separatrices and thesewill be the locations for preferential heating.A WKB solution was also obtained for the Alfv´en wave and this was in excellentagreement with the numerical results. In Figure 4b, we can see the evolution of fluidelements that begin at points x = 0 , . , ... .
5, which clearly demonstrates howthe fluid elements simply travel along the fieldlines they start on. In addition, notethat at t = 2 the fluid elements have all travelled different distances along theirrespective fieldlines, but the wave remains planar. McLaughlin & Hood explain thisand show that the leading edge has in fact reached z = z max e − t = 2 e − = 0 . § et al. B = BL (cid:0) x − z − λ , , − xz (cid:1) , (21)where 2 λ is the distance between the null points. Note that this introduces acharacteristic length scale into the system, whereas previously the single X-pointhad no charactistic length scale. The other equilibrium magnetic field consideredtakes the form: B = BL (cid:0) xz, , x − z − λ (cid:1) . (22)These equilibrium magnetic field configurations can be seen in Figures 2b and 2c.The authors considered the β = 0 linearised MHD equations and solved equations(17) and (18) for these two magnetic equilibria.McLaughlin & Hood (2005) find that for the linear fast magnetoacoustic waveapproaching the two null points from above, the wave travels down towards thenull points and begins to refract around them both. The wave the ‘breaks’ into HD waves and coronal null points 19 two along the line x = 0 (due to symmetry), with each half of the wave goingto its closest null point. Each part of the wave then continues to wrap aroundits respective null point repeatedly, eventually accumulating at that specific nullpoint.In the case of the fast wave pulse travelling in from the side boundary, we seea similar effect (i.e. a refraction effect, wave breakage and accumulation at thenulls), but in this case the wave is not equally shared between the null points.For example, for a fast wave travelling in from the left boundary, initially thepulse thins, begins to feel the effect of the left-hand-side null point and begins torefract around this null. As the ends of the wave wrap around behind the left nullpoint, they then become influenced by the right-hand-side null point. These armsof the wave then proceed to wrap around the right null point, flattening the wave.Furthermore, the two parts of the wave now travelling through the area betweenthe null points have non-zero Alfv´en speed, and so can propagate through thisarea. These parts of the wave break along x = 0 and then proceed to wrap aroundthe null point closest to them.As before, it is clear the refraction effect focuses all the energy of the incidentwave towards the null points, but McLaughlin & Hood (2005) find that the anglethat the fast wave approaches the null points from will determine what proportionof wave energy ends up at each null point (i.e. where the wave ‘breaks’ ). In thecase of the fast wave, all the wave energy is accumulating at the null points andsince we have a changing perturbed magnetic field with increasing gradients, thisis where current density will accumulate. Thus, fast wave heating will naturallyoccur at both null points.For the Alfv´en wave, the results show that the wave propagates along the fieldlines, thins but keeps its original amplitude, and eventually accumulates along theseparatrices (again, in agreement with previous studies).Thus, McLaughlin & Hood (2005) find that the key results from McLaughlin& Hood (2004) carry over from a single 2D null configuration to that of a pair of2D null points.3.4 Single null point configuration created by two magnetic dipolesMcLaughlin & Hood (2006a) again investigate the behaviour of the fast and Alfv´enwave in an ideal, β = 0 plasma, but now address a key problem with the first twopapers: that the simple null points considered (i.e. equations 19, 21, 22) are onlyvalid locally, because as x and z get very large, B also gets unphysically large.To address this issue, McLaughlin & Hood (2006a) investigate the behaviourof MHD waves near an equilibrium magnetic field created by two dipoles: B x = BL − ( x + λ ) + z (cid:2) ( x + λ ) + z (cid:3) + − ( x − λ ) + z (cid:2) ( x − λ ) + z (cid:3) ! ,B z = − BL x + λ ) z (cid:2) ( x + λ ) + z (cid:3) − x − λ ) z (cid:2) ( x − λ ) + z (cid:3) ! , (23)where 2 λ is the separation of the dipoles. This magnetic field (with 2 λ = 1) canbe seen in Figure 1c. It comprises of four separatrices and an X-point located at et al. ( x, z ) = (0 , λ ). Note that as x or z gets very large, the field strength becomes small.Hence, this is a more physical field than those previously investigated.McLaughlin & Hood (2006a) drive a fast-wave planar pulse on their lowerboundary (along z = 0). As the wave propagates upwards, the planar wave isdistorted by the two regions of high Alfv´en speed and the wave forms two peakswith maxima located over the loci of the magnetic field. Close to the null point, thefast wave begins to refract around the null point. Meanwhile, the rest of the wave(referred to as the wings ) continue to propagate upwards and spread out (sincethe fast wave propagates roughly isotropically). The wave is stretched betweenits two goals (part wrapping around the null and part travelling away from themagnetic skeleton) and this leads to the wave splitting; near the regions of highAlfv´en speed the localised high speed thins the wave and forces the split. Thus,part of the (now split) wave spirals into the null and the other part propagatesaway from the magnetic skeleton. Finally, a WKB solution demonstrates that thereis a critical radius of influence within which a fast wave will be captured by thenull point and that, for the parameters considered by McLaughlin & Hood, 40%of the wave is trapped by the null. This makes intuitive sense: if the fast wave istoo far away from the magnetic null, it will not feel its effect.McLaughlin & Hood (2006a) also looked at the behaviour of the Alfv´en waveand found that, as before, the propagation follows the magnetic fieldlines but thatnow only part of the wave accumulates along the separatrices and the other partof the wave appears to propagate away from the magnetic skeleton. However, theAlfv´en wave is actually just following the fieldlines (and these are spreading out)recovering the key results of Hassam & Lambert (1996), i.e. there is a key differencebetween the evolution of perturbations that initially straddle the separatrices andthose that do not.3.5 MHD wave behaviour at β = 0 null pointsThe key results from McLaughlin & Hood (2004; 2005; 2006a) demonstrate that thebehaviour of the fast wave in a β = 0 plasma is entirely dominated by the Alfv´en-speed profile, and since the magnetic field drops to zero at the X-point, the wavewill never reach the actual null. The next step is to extend the model to includeplasma pressure ( β = 0 plasma). The most obvious effect is the introduction of slow magnetoacoustic waves . The fast wave can now also pass through the null point(there is a non-zero sound speed there) and thus perhaps take energy away fromthat area. Such a model could also involve mode coupling in areas where the soundspeed and Alfv´en speed become comparable in magnitude.Such an investigation has been carried out into the behaviour of magnetoa-coustic waves by McLaughlin & Hood (2006b) for a single 2D null point (i.e.equilibrium magnetic field given by equation 19). Note that the plasma pressureplays no role in the propagation of the linear Alfv´en wave, and so the descriptionby McLaughlin & Hood (2004) remains valid.In most parts of the corona, the plasma- β (equation 2) is much less than unityand hence the pressure gradients in the plasma can be neglected. However, nearnull points the magnetic field vanishes and so the plasma- β can become very large.Thus, understanding the changing plasma- β is of key importance here. Considering HD waves and coronal null points 21 equilibrium quantities; β = 2 µp L B ( x + z ) ⇒ β = β x + z = β r , where r = x + z and β = 2 µp L /B .Thus, the plasma- β varies through the whole region, since magnetic field isvarying everywhere throughout our model. In fact, the plasma − β is infinite at thenull point. In particular, we note that outside a radius of unity we have a low- β environment and inside we have a high − β environment. This will have importantconsequences as fast and slow waves have differing properties depending upon theirenvironment (see Table 1).There is also coupling between the perpendicular and parallel velocity compo-nents (when β = 0) and this coupling is most effective where the sound speed andthe Alfv´en velocity are comparable in magnitude. Bogdan et al. (2003) call thiszone the magnetic canopy or the β ≈ β for the (true, varying) plasma- β and β for the (constant) value of theplasma- β at a radius of unity.The β = 1 layer occurs at radius r = √ β . However, it is not the β = 1 layerthat is most important in understanding this system, but the layer where the soundspeed is equal to the Alfv´en speed, i.e. c s = v A . Recalling that c s = p γβ / v A ,this means that the c s = v A layer (or alternatively the β = 2 /γ layer) occurs at aradius r = p γβ /
2. Of course, the difference between the β = 1 layer at r = √ β and the c s = v A layer at r = p γβ / β ≈ β = 0 system is: c = 12 (cid:0) v A + c s (cid:1) + 12 q(cid:0) v A + c s (cid:1) − v A c s cos θ which for perpendicular propagation reduces to c = v A + c s = γ β + x + z . (24)McLaughlin & Hood (2006b) solve the linearised MHD equations (16) with β = 0. Using identical boundary conditions to McLaughlin & Hood (2004), i.e.equation 20, a wave pulse is driven in the perpendicular velocity component on theupper boundary, which corresponds to driving a low- β fast wave. It is found thatthe low- β fast wave propagates into the neighbourhood of the null point and beginsto refract around it. However, as the wave crosses the c s = v A layer, the low- β fast wave transforms into a high- β fast wave (due to the change in environment)and also generates a high- β slow wave. The fraction of the incident wave convertedinto slow wave is found to be proportional to β . The magnetoacoustic propagationnow proceeds in three ways: – Firstly, the generated slow wave spreads out along the fieldlines, eventuallyaccumulating along the separatrices. – Secondly, the remaining part of the fast wave inside the c s = v A layer continuesto refract and some of it now passes across the null . We identify this wave as ahigh- β fast wave. The high- β fast wave can pass through the null point because,although v A (0 ,
0) = 0, there is now a non-zero sound speed there (clearly seen et al. in equation 24). After it has crossed the null, the high- β fast wave continuesto propagate downwards and crosses the c s = v A layer for a second time. As itemerges, the wave now becomes a low- β fast wave and spreads out isotropically. – Finally, the (low- β ) fast wave located away from the null and away from the c s = v A layer (again referred to as the ‘wings’ of the low- β wave) are notaffected by the non-zero sound speed (as v A ≫ c s ) and so here the refractioneffect dominates. In fact, as these wings wrap around below the null point,they encounter the high- β fast wave as it is emerging from the c s = v A layer.This results in a complicated interference pattern, but it appears that the twowaves pass through each other (due to the linear nature of the system).It is clear that there are two competing phenomena: ( a ) the refraction effect due to the varying Alfv´en speed and ( b ) a non-zero sound speed at the null whichallows the fast wave to pass through it. It is the value of β that dictates whicheffect dominates. Thus, two extremes can occur. The first occurs when β →
0, in which case the refraction effect dominates and we recover the results ofMcLaughlin & Hood (2004). The second occurs when β → ∞ and the systembecomes hydrodynamic. In this case, the fast wave reduces to an acoustic wave andso completely passes through the null (effectively, it does not even see the magneticfield, since v A ≪ c s ). Thus, it is possible to understand the whole spectrum ofvalues of the parameter β .McLaughlin & Hood (2006b) also noted a tendency for their system to developseveral lobe-like structures in various variables. This occurs because the choiceof magnetic equilibrium naturally leads to a sin 2 θ and sin 4 θ dependence in v k when v ⊥ is driven (this was demonstrated in polar coordinates, using equilibriummagnetic field 11). Thus, with this choice of null point, if we drive any of thevelocity variables then the system will naturally develop a θ − dependence.Finally, the authors compared their results with an analytical WKB approxi-mation. This resulted in two separate wave descriptions, corresponding to the fastor slow wave. However, the WKB solution could only reproduce the propagationof the fast wave (in both its low and high- β environments) or the slow wave butnot both together. Instead, the WKB solution broke down at the conversion layer(at c s = v A ) where the approximation becomes degenerate. Thus, the WKB solu-tion (in the form presented by McLaughlin & Hood) cannot be used to investigatemode conversion. Instead, addition terms are needed in the approximation (theauthors only consider the first-order terms in the WKB approximation). Alterna-tively, this degeneracy can be overcome by using the method developed by Cairns& Lashmore-Davies (1983) to match WKB solutions across the mode conversionlayer. This has been done in 1D (McDougall & Hood 2007) but a 2D investigationhas yet to be completed.3.6 Nonlinear simulations in the neighbourhood of a 2D X-type null pointMcLaughlin & Hood (2004; 2005; 2006a; 2006b) investigated the behaviour of thelinear fast and slow magnetoacoustic waves and Alfv´en waves in the neighbourhoodof a variety of 2D null points using equations (16). However, the validity of thelinearisation is questionable once the perturbed velocity becomes comparable tothe magnitude of the local Alfv´en speed. McLaughlin et al. (2009) extend themodels of McLaughlin & Hood (2004) and Craig & McClymont(1991) to include HD waves and coronal null points 23 nonlinear effects in a β = 0 plasma, and consider the behaviour of the nonlinearfast wave. The authors solved the nonlinear, compressible, resistive MHD equations(equations 1) using a Lagrangian-remap, shock-capturing code ( LARE2D , Arber et al. B = BL ( y, x, , which corresponds to the magnetic field seen in Figure 1a. Note that Ofman (1992),Ofman et al. (1993) and Steinolfson et al. (1995) had previously performed non-linear 2D calculations of stressed X-points.McLaughlin & Hood (2004) and previous results from investigations conductedin a cylindrical geometry ( §
2) clearly demonstrate that the Alfv´en speed (v A = B x + B y = x + y = r ) plays a vital role. Hence, it is natural to considereither a polar coordinate system or to drive a circular pulse. In addition, as com-mented by McClements et al. (2004), a disturbance initially consisting of a planewave is refracted as it approaches the null in such a way that it becomes moreazimuthally-symmetric. Thus, as a first step in investigating the nonlinear regime,it is appropriate to consider an azimuthally-symmetric initial condition in velocityof the form: v ⊥ ( x, y, t = 0) = 2 C sin [ π ( r − . . ≤ r ≤ . , (25)v k ( x, y, t = 0) = 0 , which corresponds to a circular, sinusoidal wave pulse in v ⊥ , where 2 C is the initialamplitude. When the simulation begins, this initial pulse naturally splits into twowaves, each of amplitude C : an outgoing wave and an incoming wave. The authorsfocus on the incoming wave, i.e. the wave travelling towards the null point, andset C = 1.The authors find that at early times the incoming wave (identified as a linearfast wave) propagates across the magnetic fieldlines and that the initial pulseprofile (an annulus) contracts as the wave approaches the null point. This is thesame refraction behaviour that has previously been reported. The authors alsonote that the incoming wave pulse develops an asymmetry, where in the y − / x − direction the wave peak / trailing footpoint is catching up with the leadingfootpoint / wave peak, and eventually forms discontinuities. This can be seen inFigure 5a. The asymmetry develops directly due to the choice of a velocity initialcondition. In the nonlinear regime, specifying an initial condition in velocity alsoprescribes a background velocity profile. Thus, the initial condition (equation 25)appears to excite the m = 0 mode, but this actually corresponds to the m = 2mode in cartesian components.At later times, these discontinuities develop into fast oblique magnetic shockwaves, leading to local heating of the plasma. In addition, the shocks above andbelow y = 0 began to overlap, forming a triangular ‘cusp’ (called the shock-cusp)and this leads to the development of hot jets, which again heat the local plasmaand significantly bent the local magnetic fieldlines. The hot jets (which fit thedescription of Forbes 1988) set up slow oblique magnetic shock waves emanatingfrom the shock-cusp. In addition, there is evidence of slow shocks along the sidesof the jet upstream of the tip and we see kinks in the fieldlines at the tip ofthe jet, indicative of a fast shock. Thus, the jet heating itself is accomplished et al. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) time (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (a) (b)(c)j (0,0) z time Fig. 5
Contours of v ⊥ for a fast wave pulse initially located at a radius r = 5 and its resultantpropagation at ( a ) time t = 1 and ( b ) time t = 2 .
6. The black lines denote the (changing)separatrices and the null point is located at their intersection (origin). Note in ( b ) that theseparatrices have been deformed and now form a ‘cusp-like’ magnetic field structure. Theamplitude of v ⊥ varies substantially throughout the evolution, and hence each subfigure isassigned its own colour bar. ( c ) Plot of time evolution of j z (0 ,
0) for 0 ≤ t ≤
60 . Insert showstime evolution of j z (0 ,
0) for 25 ≤ t ≤
60 (i.e. same horizontal axis, different vertical axis).Dashed lines indicate maxima (red) and minima (blue). Green line shows limiting value of j z (0 ,
0) = 0 . by a combination of slow and fast shocks. It is interesting to note that the jethas a bimodal structure consisting of a hot, narrow jet incased within a broader,lower temperature jet, which is a feature that is not predicted by steady-statereconnection theory.Eventually, the fast shocks reach the null point and the shocks have deformedthe magnetic field such that the separatrices now touch one another rather thanintersecting at a non-zero angle (called ‘cusp-like’ by Priest & Cowley 1975). Thiscan be seen in Figure 5b. However, the separatrices continue to evolve and sothis field structure is not sustained for any length of time. The (deformed) nullpoint itself continues to collapse and forms a horizontal current sheet (again, suchbehaviour was not seen in the linear systems).Subsequently, the system evolves as follows: the hot jets to the left and rightof the null continue to heat the plasma, which in turn expands. This expansionsquashes and shortens the horizontal current sheet, forcing the separatrices apart. HD waves and coronal null points 25
The (squashed) horizontal current sheet then returns to a ‘cusp-like’ null pointwhich, due to the continuing expansion from the heated plasma, in turn forms avertical current sheet. The evolution then proceeds through a series of horizontaland vertical current sheets and displays oscillatory behaviour, in a similar mannerto the linear results of Craig & McClymont (1991). The oscillatory nature of thesystem can be clearly seen from the time evolution of j z (0 ,
0) shown in Figure 5c.The red / blue lines indicate maxima / minima in the system and the green lineshows j z (0 ,
0) = 0 . t diffusion ∼ R m = 10 ).McLaughlin et al (2009) provide two pieces of evidence for reconnection in theirsystem. Qualitatively, they observe changes in fieldline connectivity and quantita-tively they look at the evolution of the vector potential at the null point. Since theyhave both oscillatory behaviour and evidence for reconnection, they conclude thatthe system displays oscillatory reconnection (as detailed by Craig & McClymont1991).The authors then extended the study to look at the effect of changing theamplitude, C , of their initial condition (equation 25). They conclude that a largerinitial amplitude results in a larger amount of current being left in the system atthe end of the simulation, i.e. in the non-potential final state.Thus, it is clear that the nonlinear behaviour is completely different to that ofthe linear regime. For example, current density now accumulates at many locations,such as along horizontal or vertical current sheets, along slow oblique magneticshocks and at the location of shock-cusps. This was not the case in the linearregime, where all the current density accumulated at the null point exponentiallyin time.The work of McLaughlin et al. (2009) provides a link between two traditionallyseparate areas of solar physics: MHD wave theory and reconnection, and is thefirst demonstration of reconnection naturally driven by MHD wave propagation.3.7 Quasi-Periodic PulsationsQuasi-Periodic Pulsations (QPPs) have been observed in radio, optical and X-rayemission of solar flares (e.g. Inglis et al. et al. et al. et al. et al. Fast magnetoacoustic wave behaviour in the neighbourhood of magnetic nullpoints provides two alternative mechanisms. Firstly, Nakariakov et al. (2006) ex-tend the model of McLaughlin & Hood (2004) to include a harmonic driver. Asbefore, the refraction effect wraps the fast wave around the null point, and largegradients and currents develop. By utilising a harmonic driver, the resultant cur-rent growth and accumulation is itself periodic, i.e. the periodicity of the incomingfast waves is efficiently transmitted into the periodic modulation of the currentdensity. In addition, Nakariakov et al. provide an explanation of where such anincoming fast wave could originate: fast waves can leak from an oscillating loopsituated near a null point but magnetically disconnected from it. Interestingly, thiswould imply that the period of the observed QPPs is linked to the properties ofthe neighbouring, oscillating loop.An alternative explanation may be provided by the work of McLaughlin etal. (2009) in which oscillatory reconnection is naturally driven by an incomingfast magnetoacoustic wave. Here, the oscillatory behaviour is due to a cycle ofhorizontal and vertical current sheets (i.e. a different physical mechanism to that ofNakariakov et al. β = 0 plasma, as well as conduct adetailed parametric study.3.8 Phase-MixingFinally, we consider a novel piece of work by Fruit & Craig (2006) whereby thephenomenon of phase-mixing (Heyvaerts & Priest 1983) is invoked in the viscousand resistive dissipation of standing Alfv´en waves within a line-tied X-point geom-etry. Here, the authors consider magnetic fieldlines anchored into a rigid, reflectiveboundary and excite the system with an initial velocity profile. During the simu-lation, each fieldline, being rigidly tied at the end, oscillates back and forth, withthe oscillation frequency depending upon the length and magnetic field strengthof that fieldline. The phase difference between neighbouring lines increases as timeevolves, leading to growing cross-gradients, and eventually leading to strong visco-resistive damping. Thus, the authors conclude that phase-mixing can provide anefficient mechanism for energy dissipation of standing Alfv´en waves in the vicinityof 2D null points.Craig & Litvinenko (2007) extend the model of Fruit & Craig (2006) to includeanisotropic (Braginskii bulk) viscosity, and find that the main results are still valid. An obvious and essential extension of the 2D work described so far is to extendthe models to full 3D simulations. However, two extension are possible that couldprovide useful signposts before the move to 3D. Firstly, it is possible to extendthe model to 2.5D with the addition of a third spatial coordinate, by taking intoaccount an extra Fourier component of the form e imy , where m is the azimuthalmode number. Secondly, we can consider the addition of a small axial magnetic HD waves and coronal null points 27 field perpendicular to the plane of the magnetic X-point, i.e. a longitudinal guide-field. Note that with the addition of a longitudinal guide-field we are no longerconsidering null points: even though the X-point geometry remains, B = 0 at theX-point (due to the cross field) and we actually have an X-line. However, since thetwo systems are so closely related we will still provide a literature review of therelevant papers.Extending the model to 2.5D and 3D will lead to coupling of all the wave modes,and thus will most likely result in energy accumulating at both the separatricesand the null points.The first work in this area was done by Bulanov et al. (1992). Bulanov et al. considered an X-point magnetic field configuration with a longitudinal (along theX-line) magnetic field B k . The authors write down, for the first time, the governinglinearised equations in the ideal β = 0 regime, and show that the fast magnetoa-coustic wave and Alfv´en wave are linearly coupled by the gradients in the field.Furthermore, the authors show that magnetoacoustic perturbations can transforminto Alfv´en waves and vice versa, thus leading to current density accumulation ateither the null point or along the separatrices. However, the rate or efficiency ofthis process is not explored in this analytical work. In the limit of B k →
0, the twomodes are decoupled and the results of 2D work are recovered.McClements et al. (2006) also investigated the coupling of Alfv´en waves andfast waves in the vicinity of a magnetic X-point with a weak longitudinal guide fieldpresent ( B k ≪ B ⊥ ), and extended the model of Bulanov et al. (1992) to includeresistive effects. These authors solve the initial value problem for a fast wave beingdriven by a harmonic Alfv´en wave train and find that energy is channelled into thefast wave, and that large gradients start to build-up near the X-point, due to theAlfv´en speed profile. The results indicate that a significant fraction of the Alfv´enwave energy is converted into fast wave energy. Ben Ayed et al. (2009) extendthe work of McClements et al. (2006) to include a strong guide-field ( B k ≫ B ⊥ ).Again, the authors found that the Alfv´en wave is coupled into the fast mode, withthe coupling strongest on the separatrices and far from the X-line.The three works (Bulanov et al. et al. etal. β = 0 plasma, and McClements et al. (2006) and BenAyed et al. (2009) concentrate on driving linear Alfv´en wave trains and observingthe coupling to the fast mode.Conversely, Landi et al. (2005) considered the nonlinear propagation of a har-monic Alfv´en wave train in a 2.5D geometry, consisting of a magnetic X-pointthreaded by an axial magnetic field. Interestingly, the authors find that the drivenAlfv´en waves couple to the fast mode through the magnetic geometry, and thatthe generated fast waves have a frequency equal to that of the driven Alfv´en wavesand an amplitude that scales linearly with the amplitude of the incoming Alfv´enwaves. This is different to nonlinear formation of fast waves from a propagatingAlfv´en wave (due to the ponderomotive force), in which the generated fast waveshave a frequency twice that of the driven Alfv´en wave (Nakariakov et al. et al. et al. (2005) propose that this indicates amechanism of mode conversion that differs from the standard nonlinear fast waveexcitation via the ponderomotive force. The authors also find that these generatedfast waves rapidly develop into fast-mode shocks, and thus the wave dissipationis concentrated into thin current structures. However, this is not surprising as the et al. authors consider very large amplitudes for their driven Alfv´en waves, preciselyso that they can observe shock formation within their simulation domain. Theauthors do not report whether the fast waves experience the refraction effect orwhether the Alfv´en waves are confined to the magnetic fieldlines that originate on.McClements et al. (2006) and Ben Ayed et al. (2009) consider radial symmetryfor their driven waves, and Landi et al. (2005) considers a large-amplitude, periodicwave train driven simultaneously on both side boundaries. Hence, it would beinteresting to see if the results of all these papers persist if one considers a singlewave pulse, as this would allow the transient behaviour to be observed, and alsoif the waves were driven on a single side of the numerical domain. Finally, let us now review the behaviour of MHD waves in the neighbourhood of3D null points. However, surprisingly few papers have been written that addressthis issue, or at least, papers that concentrate on the the transient propagation ofthe modes. Most papers have focused on the dynamics of current formation in anattempt to locate regions where reconnection is most likely to occur, rather thanon the transient propagation of the MHD waves.The first study of the dynamics of current formation at 3D null points wasperformed by Rickard & Titov (1996). These authors solved the linear, β = 0MHD equations, and studied a 3D null point that is axisymmetric about the spine(i.e. a proper 3D null, see § m , and the analysis is performed in cylindricalgeometry.Several different perturbations are considered by Rickard & Titov (see theirfigure 2) and are driven with a velocity pulse (in v r , v θ , v z ) on the boundaries(either the radial boundaries or upper/lower boundaries). Numerical simulationsshow that axisymmetric perturbations ( m = 0) lead to current accumulation alongthe spine, whilst the m = 1 mode produces currents in the fan plane and at thenull point itself. For m >
1, there is no current accumulation anywhere along theskeleton. Rickard & Titov also noted that in these axisymmetric equilibria, theazimuthal components decouple from the remaining components when m = 0, andthat in 2D only the m = 0 mode was associated with producing currents at thenull.The primary aim of this study was to investigate current accumulation, and notto investigate the behaviour of MHD waves around 3D nulls point. Of course, theperturbations investigated correspond to a combination of Alfv´en wave and fastmagnetoacoustic waves, but the authors do not use this terminology. In addition,the authors drive a combination of v r , v θ , v z which generates both wave types,and thus in some cases it is difficult to confirm that current accumulation resultsfrom a certain wave-type. However, some behaviours are clear: a pure m = 0 wavepulse corresponds to a torsional Alfv´en wave, which the authors note is channelledalong the equilibrium magnetic fieldlines. Secondly, the authors note that the m = 1 motions can propagate across magnetic fieldlines, and that a focusing effectis seen in the evolution of j r and j θ (as we would expect for the fast wave) forcertain disturbances. HD waves and coronal null points 29
Thus, Rickard & Titov (1996) give a tantalising suggestion that our under-standing of 2D behaviour of the fast and Alfv´en wave transfers to 3D. However, itis not possible to see from their paper if driving a pure Alfv´en wave results in gen-eration of a fast mode disturbance through the magnetic geometry, or vice versa,as expected from §
4. The simulations of Rickard & Titov also utilise reflectingboundary conditions in their cylindrical geometry. In addition, it can be arguedthat their investigation is actually 2.5D and not a fully 3D experiment. Never-theless, Rickard & Titov (1996) is still a landmark paper for the investigation of3D MHD wave behaviour in the neighbourhood of magnetic null points. It wouldbe interesting to repeat the work of Rickard & Titov (1996) but to try to excitepure modes and to focus on the resultant transient wave propagation and possiblemode conversion.The work of Rickard & Titov (1996) has been extended to multiple null pointtopologies in a series of papers by Galsgaard and co-workers. Galsgaard et al. (1996; 1997a) looked at shearing a 3D potential null point pair, with continuous(opposite) shear on two opposite boundaries (parallel to the separator), wherethe fieldlines were not returned to their original position. This generated a wavepulse that travelled towards the interior of the domain from both directions, andresulted in current accumulation along the separator line with maximum value atthe null points. Galsgaard et al. (1997b) looked at perturbations in 3D magneticconfigurations containing a double null point pair connected by a separator. Theboundary motions used were very similar to those described above (i.e. shear theboundary and fix). Their experiments showed that the nulls can either accumu-late current individually or act together to produce current along the separator.Galsgaard & Nordlund (1997) found that when a magnetic structure containingeight null points is perturbed, current density accumulates along separator lines.In all these Galsgaard et al. papers, the boundary conditions tried to mimicthe effect of photospheric footpoint motions by moving the boundary and holdingit fixed. As in Rickard & Titov (1996), the terminology of MHD waves was notinvoked, and the perturbations considered were a combination of MHD waves.The first investigation specifically looking at MHD wave propagation about aproper 3D null point was reported in Galsgaard et al. (2003), where the authorslooked at a particular type of wave disturbance and solved the nonlinear β = 0MHD equations. Galsgaard et al. (2003) investigated the effect of rotating the fieldlines around the spine to generate a twist wave (essentially a torsional Alfv´en wave)and followed its propagation towards the null point. Twists were imposed simul-taneously on the upper and lower boundaries, with both the same and oppositevorticities considered. The authors found that the helical Alfv´en wave spreads outas it propagates towards the null point and is confined to the magnetic fieldlines itoriginates on. As the wave approaches the fan plane, the wave spreads out alongthe diverging fieldlines and produces current accumulation in the fan plane.In addition, the authors also observe the generation of a fast-mode wave, andfind that this wave focuses and wraps around the null point. The authors suggestthat their twist wave is a pure helical Alfv´en wave, and that nonlinear effectsgenerate the fast-mode wave (using the ponderomotive mechanism detailed byNakariakov et al. et al. Galsgaard et al. (2003) confirm that several of the key properties of fast andAlfv´en waves transfer to 3D geometry. However, again their investigation is ac-tually 2.5D and not fully 3D. In addition, closed boundaries were used, and thuswaves propagate toward the boundaries and are reflected back into the domain.The authors also analyse the linear β = 0 MHD equations using the WKBmethod for their azimuthally-symmetric 3D null, and find good agreement. Theyfind that the equations for fast and Alfv´en perturbations decouple, although thisis not surprising as, due to their choice of symmetrical geometry, their resultantequations are two-dimensional (since a proper 3D null point is essentially 2D incylindrical polar coordinates).Pontin & Galsgaard (2007) and Pontin et al. (2007) have performed numericalsimulations in which the spine and fan of a proper 3D null point are subjectedto rotational and shear perturbations. They found that rotations of the fan planelead to current density accumulation about the spine, and rotations about thespine lead to current sheets in the fan plane. In addition, shearing perturbationslead to 3D localised current sheets focused at the null point itself. Again, this is ingood agreement with what we may expect for MHD wave behaviour, i.e. currentaccumulation at certain parts of the topology.The first study of MHD waves in the neighbourhood of an improper 3D nullpoint was investigated by McLaughlin et al. (2008). These authors consider thepropagation of the fast and Alfv´en waves about the magnetic equilibrium B =[ x, ǫy, − ( ǫ + 1) z ], for both proper ( ǫ = 1) and improper ( ǫ > ǫ = 1) 3D potentialnull points. The authors utilise the 3D WKB approximation of the linear β = 0MHD equations.McLaughlin et al. (2008) find that the fast magnetoacoustic wave experiencesrefraction towards the magnetic null point, and confirms that, as in 2D, the effectof this refraction is dictated by the Alfv´en speed profile. The fast wave, and thusthe wave energy, accumulates at the null point. The current build-up is shown tobe exponential and the value of the exponent depends upon ǫ . Thus, as in 2D,there is preferential heating at the null point for the fast wave.For the Alfv´en wave, the authors find that the wave propagates along theequilibrium fieldlines and that a fluid element is confined to the fieldline it startson. For an Alfv´en wave generated along the fan-plane, the wave accumulates alongthe spine. For an Alfv´en wave generated across the spine, the value of ǫ determineswhere the wave accumulation will occur: either the fan-plane ( ǫ = 1), along the x − axis (0 < ǫ <
1) or along the y − axis ( ǫ > et al. (2008) also provide a quantitative analysis of the preferentialheating/energy release process. They derive an analytical expression for currentevolution resulting from fast wave propagation along the spine (their equation 23)which is of the form | j | = ωe ( ǫ +1) t / [ z ( ǫ + 1)] (where ω is the wave frequencyand z is the starting point on the spine). Furthermore, in order to give an order-of-magnitude estimate, the authors show that for characteristic coronal values( L = 10 Mm, B = 10 G, ρ = 10 − kg m − ) a planar fast wave propagating alongthe spine will build-up a current of 0 . t = 1 seconds, and thatresistive effects become non-negligible after a time t = log ( ωλ /η ) / ≈ .
87 mA and hastravelled a distance of 7 .
13 Mm). The Alfv´en wave is degenerate with the fast
HD waves and coronal null points 31 wave along the spine in a β = 0 plasma, and so has identical estimates for thesecharacteristic conditions.However, McLaughlin et al. (2008) are unable to reach any conclusions aboutthe coupling of the fast and Alfv´en wave types due to the geometry of the magneticfield. The WKB approximation, in the form utilised by McLaughlin et al. , doesnot take into account the coupling of the fast and Alfv´en wave types due to thegeometry of the magnetic field. Under their approximation, the waves actually seethe magnetic field as locally uniform.Thus, a fully 3D model of MHD wave behaviour in the neighbourhood of ageneral 3D null point has yet to be investigated, where the investigation tracksthe propagation and evolution of each MHD wave, and determines the efficiencyof mode-conversion due to the magnetic geometry against that due to nonlineareffects. The behaviour of all three MHD wave types; Alfv´en, fast and slow wave, hasbeen investigated in the neighbourhood of 2D, 2.5D and (to a certain extent) 3Dmagnetic null points, in a variety of geometries and under a variety of assumptions.The main conclusions may be summarised as follows: – The linear, fast magnetoacoustic wave behaviour is dictated by the equilibriumfast wave speed profile (i.e. p v A + c S ), which in low- β plasmas can be thoughtof as the equilibrium Alfv´en-speed profile. The fast wave is guided towards thenull point by a refraction effect and wraps around it. The fast wave slowsas it approaches the null, leading to a decrease in length scales and thus anincrease in current density close to the null point. In a β = 0 plasma, the fastwave cannot cross the null point and the build-up of current is exponential,indicating that dissipation will occur on a timescale related to log η . Thus,linear fast wave dissipation is very efficient, and null points will be locations forpreferential heating . For β = 0, the fast wave can cross the null point, due tothe finite sound speed there, and wave energy can now escape the null point.In this case, there exists two competing phenomena and the dominate effect isdetermined by the value of the plasma- β . – The linear Alfv´en wave propagates along the equilibrium fieldlines and a fluidelement is confined to the fieldline it starts on. Since the propagation followsthe fieldlines, the Alfv´en wave spreads out as it approaches the diverging nullpoint. In 2D, all the Alfv´en wave energy accumulates along the separatricesand the current build-up is exponential in time. In 3D, for an Alfv´en wavegenerated along the fan-plane, the wave accumulates along the spine and foran Alfv´en wave generated across the spine, the value of ǫ determines where thewave accumulation will occur: fan-plane ( ǫ = 1), along the x − axis (0 < ǫ < y − axis ( ǫ > Alfv´en wave energy will be dissipatedalong the separatrices/ separatrix surfaces and these will be the locations forpreferential heating. – The behaviour of the slow wave in the neighbourhood of null points has re-ceived the least attention in the literature. The linear slow wave is found tobe wave-guided and accumulates along the separatrices. A low- β fast wave cangenerate/convert into both a high- β fast and high- β slow wave as it crosses the et al. v A = c S mode-conversion layer. Such a layer is a natural consequence for a nullpoint emersed in a β = 0 plasma. In fact, the value of β grows as r − close toa null point. – The addition of a weak guiding field leads to linear coupling between the fastand Alfv´en waves in a low- β plasma, and thus the propagation of either modecan generate the other. Such a configuration is, of course, no longer a null point,but rather an X-line. However, the nature of mode-coupling for 3D null pointsis, at this time, uncertain. Fast waves have been shown to be generated by thepropagation of the Alfv´en wave, but it is unclear if this is due to the fieldlinegeometry, nonlinear coupling or both waves being simultaneous generated bya common driver. – Results in 2D show that in the nonlinear regime, the fast magnetoacousticwave can deform the equilibrium X-point configuration, leading to a cycle ofhorizontal and vertical current sheets and associated changes in connectivity.Thus, the system exhibits oscillatory reconnection . – It is clear that the equilibrium magnetic field plays a fundamental role inthe propagation and properties of MHD waves. In general, an arbitrary dis-turbance/perturbation will generate all three wave modes and current accu-mulation could occur at all the null points, and/or along the spine, fan andseparators. Thus, the results described in this review all highlight the impor-tance of understanding the magnetic topology in determining the locations ofwave heating.However, several big questions still remain in this area: – The nature of the coupling of the three modes in 3D needs to be addressed,and the importance of coupling due to the magnetic geometry verses nonlinearcoupling should be investigated. – The theory of nonlinear fast waves driving oscillatory reconnection should beextended to study more general disturbances, and to investigate how robustthe initial findings of McLaughlin et al. (2009) are. – The key results for the linear fast and Alfv´en wave make clear predictionsas to where preferential heating can occur. It would be interesting to see thetheoretical models developed with forward modelling (see e.g. Kilmchuk &Cargill 2001; De Moortel & Bradshaw 2008) to provide tell-tale observationalsignatures, and for these synthetic results to be compared with observationaldata.In conclusion, we have seen that the study of MHD wave behaviour in theneighbourhood of magnetic null points is a fundamental plasma process, and canprovide critical insights into other areas of plasma behaviour including: mode-conversion ( § . § . § . § . § . will propagate in the neighbourhood of coronal null HD waves and coronal null points 33 points. Thus, MHD wave propagation about magnetic null points is itself - theo-retically - a fundamental coronal process.However, there is as yet no clear observational evidence for MHD wave be-haviour in the neighbourhood of coronal null points. In the lead author’s opinion,the successful detection of MHD oscillations around coronal null points will re-quire input from two areas: high-spatial/temporal resolution imaging data as wellas potential/non-potential extrapolations from co-temporal magnetograms. Twoof the instruments onboard the recently launched Solar Dynamics Observatory(SDO) may satisfy these requirements: the
Atmospheric Imaging Assembly (whichwill provide high-quality imaging data) and the
Helioseismic and Magnetic Imager (which will provide vector magnetograms). Thus, the first detection of MHD wavesin the neighbourhood of coronal null points may be reported in the near future.
Acknowledgements
JAM acknowledges financial assistance from the Leverhulme Trust.IDM is grateful for support through a Royal Society University Research Fellowship.
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