Alfvén Waves in the Structured Solar Corona
MMNRAS in press, 1– ?? (2015) Preprint 12 November 2018 Compiled using MNRAS L A TEX style file v3.0
Alfv´en Waves in the Structured Solar Corona
Paul S. Cally (cid:63)
School of Mathematical Sciences and Monash Centre for Astrophysics, Monash University, Clayton, Victoria 3800, Australia
ABSTRACT
A simple model of a periodic ensemble of closely packed flux tubes sitting atop avertically stratified layer reveals that an incident fast wave from below preferentiallyconverts almost immediately to Alfv´en waves in the flux tubes, with kink waves re-stricted to at most a very few Fourier modes. This suggests that observations of coronalkink modes in such structured systems may greatly underestimate the net wave en-ergy flux being transported into and through the corona, much of which may reside inharder-to-observe Alfv´en waves. The processes of mode conversion/resonant absorp-tion and Alfv´en phase mixing are implicated. It is suggested that the Sun’s internal p -mode field – the 5-minute oscillations – may contribute substantially to the processby supplying incident fast waves in the chromosphere that scatter and mode-convertin the tube ensemble. Key words:
Sun: oscillations – magnetohydrodynamics – Sun: helioseismology –Sun: corona
Since Alfv´en waves were first shown to be incompressive so-lutions of the magnetohydrodynamic (MHD) equations in aconducting fluid (Alfv´en 1942), they have been postulatedto contribute to many phenomena on the Sun. In moderntimes, their role from photosphere to solar wind has beenwidely invoked (Cranmer & van Ballegooijen 2005), includ-ing as a source of coronal heating via a turbulent cascadeand of solar wind acceleration (Cranmer et al. 2007).The first direct detection of coronal Alfv´en waves wasreported a decade ago by Tomczyk et al. (2007) in an articlepointedly entitled
Alfv´en Waves in the Solar Corona : “Wereport the detection of Alfv´en waves in intensity, line-of-sightvelocity, and linear polarization images of the solar coronataken using the FeXIII 1074.7-nanometer coronal emissionline with the Coronal Multi-Channel Polarimeter (CoMP)instrument at the National Solar Observatory” . At only 0.3km s − rms amplitude though, these Alfv´en waves are tooweak to be energetically important. The observed velocityfluctuations show a distinct signature of the solar p -mode 5-minute oscillations, suggesting a link with the Sun’s internalseismology. This is absent in intensity though, indicatingthat the helioseismology-related power is predominantly inincompressive or near-incompressive modes, and specificallynot in slow modes.This was quickly followed by Hinode Solar Optical Tele-scope (SOT) CaII H-line observations of swaying chromo-spheric spicules by De Pontieu et al. (2007), who judgedthese Alfv´en waves “strong enough to power the solar wind” . (cid:63) E-mail: [email protected]
Their sophisticated 3D radiative MHD simulations con-firmed the interpretation. They “(did) not see evidence forstable waveguides or MHD kink-mode waves” in the sim-ulations. Importantly, transmission coefficients across thechromosphere-corona transition region (TR) of 3–15% werecalculated (compared to 5% reported by Cranmer & van Bal-legooijen 2005), suggesting that the corona and solar windmay indeed benefit from a substantial Alfv´en energy injec-tion.Cirtain et al. (2007) found similar Alfv´en waves inlonger-lived solar X-ray jets, suggesting a ubiquitous phe-nomenon.However, Erd´elyi & Fedun (2007) in the same Spe-cial Section of
Science raised the question of whether these“Alfv´en waves” were instead kink waves. This was on thebasis that the only allowable polarization of Alfv´en waves ina radially structured cylindrical flux tube is torsional, andnot transverse as observed. Torsional Alfv´en waves on suchtubes would show up only as line broadening. Van Doorsse-laere et al. (2008) also argued that these waves are “guidedkink magnetoacoustic waves”, not Alfv´en waves.Jess et al. (2009) note the kink/Alfv´en uncertainty, butconcentrate specifically on pure torsional oscillations in ax-isymmetric flux tubes that are undoubtedly Alfv´en (see alsoKudoh & Shibata 1999). Mathioudakis et al. (2013) arguedthat, although fundamentally magnetoacoustic, transversalkink waves are only very weakly compressional, and so sharemany characteristics with true Alfv´en waves.Coincidently, the discrete flare-induced coronal loop os-cillations observed with the Transition Region and CoronalExplorer (TRACE) were interpreted as kink waves from thefirst (Aschwanden et al. 1999, 2002; Nakariakov et al. 1999), c (cid:13) a r X i v : . [ a s t r o - ph . S R ] D ec P.S. Cally though with their observed decay soon being ascribed to res-onant absorption, i.e., conversion to Alfv´en waves at reso-nant surfaces (Goossens et al. 2002). These and other coronalwave types (e.g., EIT waves, compressible waves in plumesand loops, and various types of standing loop waves) were in-terpreted as MHD waves of one type or another (a detaileddiscussion distinguishing the various MHD wave types inmagnetic flux tubes is presented by Goossens et al. 2009).These observations and their interpretations are discussedin historical and scientific detail in the
Living Review byNakariakov & Verwichte (2005), and need not be revisitedhere.Though fully aware of the plethora of observations ofcoronal waves identified as magneto-acoustic (the fast andslow waves, including their kink and sausage manifestationsin flux tubes), Tomczyk & McIntosh (2009) made the casethat the “spatially and temporally ubiquitous” waves seen inthe CoMP Doppler time series by Tomczyk et al. (2007) arenovel in character. This is because (i) they lack apprecia-ble intensity fluctuations (suggesting Alfv´en or near-Alfv´encharacter), making them invisible to intensity imaging in-struments; (ii) their displacement amplitudes are nearlyan order of magnitude below what could be observed byTRACE, and two orders of magnitude below SOHO/EITcapabilities; and (iii) existing Doppler imaging instrumentslacked the required sensitivity, spatial extent, and cadence.They claimed the CoMP observations could widen the remitof coronal seismology.These ubiquitous waves in structured ensembles of fluxtubes will be modelled in subsequent sections, with particu-lar attention to irrotational and incompressive parts, whichdisplay respectively kink-like and Alfv´en-like characteristics.McIntosh et al. (2011) report far higher coronal Alfv´enwave amplitudes than Tomczyk et al. (2007), 20 km s − rather than 0.3 km s − , sufficient to power the quiet so-lar corona and fast solar wind. Higher spatial and tempo-ral resolution available using the He II 304-˚A and Fe IX171-˚A channels of the Atmospheric Imaging Assembly (AIA)aboard the Solar Dynamics Observatory (SDO) allowed di-rect imaging of swaying motions rather than predominantlyline broadening seen with CoMP, reducing the effect of high-optical-depth averaging. By this stage, the waves were beingreferred to as “Alfv´enic” rather than “Alfv´en”, allowing forthem being hybrid Alfv´en/kink in nature.There are differences between Alfv´en and kink wavesthat may have practical consequences. Although only weaklycompressional, kink waves are to some extent subject todissipation mechanisms associated with compressibility, andAlfv´en waves are not. The primary source of kink wave dis-sipation in structured flux tubes though is believed to beresonant absorption, which is simply mode conversion toAlfv´en waves (Cally & Andries 2010) tightly bound to reso-nant surfaces (for a single frequency). In the radially strati-fied flux tube context, these Alfv´en waves are predominantlytorsional and have far reduced observational signatures, soby this stage the kink wave may appear to have died out(Goossens et al. 2002). The Alfv´en waves generated in thisway are subject to phase mixing (cascade to smaller scales)and therefore ultimately to dissipation via non-ideal pro-cesses.Goossens et al. (2014) make the point that the identifi-cation of kink and Alfv´en waves in structured plasmas is not absolute or global. With particular reference to nonuniformcircular-cross-section flux tubes, they argue that the oscilla-tions are hybrid in nature, taking on either classic kink-likeor torsional-Alfv´en characteristics in different spatial regionsor time periods (in an initial value problem). In particular,the oscillations become more Alfv´en-like (Alfv´enic) near theAlfv´en resonance layers (in agreement with the modelling ofCally & Andries 2010 and Hanson & Cally 2011). This viewis supported by the results presented herein.Photosphere-to-heliosphere Alfv´en models commonlyinvoke the photospheric convective power spectrum as thesource of Alfv´en waves. However, with the very low ioniza-tion fraction of the temperature minimum region taken intoaccount, Vranjes et al. (2008) conclude that the generatedAlfv´en flux is reduced by orders of magnitude, which is po-tentially fatal for such atmospheric Alfv´en wave models.An alternate source of Alfv´en waves that does not suf-fer this weakness relies on fast-to-Alfv´en mode conversionin the chromosphere, where the ionization fraction is muchhigher and standard MHD more applicable. With a givenhorizontal wavenumber, an upward travelling fast wave in avertically stratified atmosphere reflects where its horizontalphase speed matches the Alfv´en speed (assuming a zero- β (cold) plasma), and partially converts to upward or down-ward travelling Alfv´en waves (Cally & Hansen 2011). Thesame behaviour is seen in the warm plasma model of Cally &Goossens (2008), and verified in simulations by Khomenko& Cally (2011, 2012), and Felipe (2012).In this scenario internal solar p -modes (partially) con-vert to fast waves at the Alfv´en acoustic equipartition sur-face (Schunker & Cally 2006), and thence to Alfv´en waveshigher up in the low- β upper chromosphere. Hansen & Cally(2012) found that these newly created Alfv´en waves werebetter able to penetrate the transition region than thoseoriginating from the photosphere, with transmission coeffi-cients up to 30%, depending on field inclination and waveattack angle. This fast-to-Alfv´en conversion process is es-sentially the same as the resonant absorption mechanism influx tubes mentioned above, but with the required Alfv´enspeed gradient being a consequence of gravitational stratifi-cation instead of radial tube structure. For inclined magneticfield though, the generated Alfv´en waves are spatially dis-tributed rather than confined to discrete resonant surfaces(Cally & Hansen 2011). Linking atmospheric Alfv´en wavesto internal seismology is in line with the identification ofpeak power with the 5-minute p -mode spectrum by Tom-czyk et al. (2007).These mode conversion analyses though assume eithervertical or predominantly vertical Alfv´en speed stratifica-tion with little or no cross-field structure. The recent ob-servations on the other hand have been in the context of avery cross-field-structured corona. In the photosphere andlow chromosphere, the density scale height is typically 100–150 km, which is much shorter than most relevant horizontallength scales, so vertically stratified models are appropriate.On the other hand, moving higher in the atmosphere sees themagnetic field take over from gravity as the primary causeof inhomogeneity, either directly or by allowing different fluxtubes to contain different temperature and density plasma.Both scenarios present opportunities for mode coupling. Itis the purpose of this article to explore a model in which theatmosphere transitions smoothly from vertical to cross-field MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona inhomogeneities with increasing height z , and to calculatethe relative wave energy fluxes carried by fast and Alfv´enwaves as a function of z .Although Goossens et al. (2009) specifically reject call-ing kink waves “fast”, De Moortel & Nakariakov (2012) say “the kink mode is locally a fast magnetoacoustic wave, prop-agating obliquely to the magnetic field and guided along thefield-aligned plasma structure (a waveguide) by reflection orrefraction” . That this is indeed the case is made clear byPascoe et al. (2010, 2011); the reader is particularly referredto the cartoon Figure 1 of the latter article. This view of thekink (or sausage, or fluting) mode as a fast wave trapped ina low Alfv´en speed waveguide will inform the interpretationof the numerical solutions obtained in the following sections.So, the precise definition of “fast” and “Alfv´en” in aninhomogeneous plasma is not clear-cut. In general it is noteven possible to separate them unambiguously, as the twomode types are inextricably coupled (though if the couplingis weak, they may be separated as zeroth order modes in aperturbation expansion). Nevertheless, some quantificationcan be given in terms of irrotational and incompressive partsof the displacement vector, that to some extent may be asso-ciated primarily with the fast and Alfv´en parts. To separatethem unambiguously, a uniform region will be appended atthe top of the computational region so that the modes de-couple.It will be shown that the easily-observed kink-like irro-tational part typically carries far less wave-energy flux up-ward than does the difficult-to-observe Alfv´en-like incom-pressible part. This suggests that observations may signifi-cantly understate the true MHD wave flux into the corona. For the purposes of this study, it will be sufficient to adoptthe zero- β approximation (cold plasma model), in which thesound speed is assumed negligible compared to the Alfv´enspeed. This is a reasonable description of the regime of in-terest in the solar corona. Consider a cold ideal MHD plasma with uniform magneticfield B , supporting both fast and Alfv´en waves. The slowwave has been frozen out by the cold plasma assumption c/a →
0, where c and a are the sound and Alfv´en speeds re-spectively. As shown by Cally & Hansen (2011), the plasmadisplacement ξ obeys the linearized wave equation (cid:18) ∂ (cid:107) − a ∂ t (cid:19) ξ = − ∇ p χ, (1)where χ = ∇· ξ is the dilatation, ∂ t is the time deriva-tive, ∂ (cid:107) = ˆ e (cid:107) ·∇ is the field-aligned directional derivative,ˆ e (cid:107) = ˆ B is the unit vector in the direction of the magneticfield, and ∇ p = ∇ − ˆ e (cid:107) ∂ (cid:107) is the complementary perpendicu-lar component of the gradient. Note that there is no displace-ment along field lines B · ξ = 0. Even though the magneticfield B is assumed uniform, the square of the Alfv´en speed a = B / µ ρ is an arbitrary function of position throughits dependence on the density ρ ( x ). The displacement may be Helmholtz-decomposed intoirrotational and incompressive parts using potentials, ξ = ∇ p Φ − ∇× Ψˆ e (cid:107) = ξ f + ξ A = ( ∂ ⊥ Φ − ∂ y Ψ) ˆ e ⊥ + ( ∂ y Φ + ∂ ⊥ Ψ) ˆ e y , (2)whence χ = ∇ Φ and ζ = ∇ Ψ , (3)where ζ = ˆ e (cid:107) ·∇× ξ = ∇· ( ξ × ˆ e (cid:107) ). The potential Φ repre-sents the fast wave, and Ψ characterizes the Alfv´en wave ina uniform medium.Equation (1) then becomes (cid:18) ∇ − a ∂ t (cid:19) ∂ ⊥ Φ = (cid:18) ∂ (cid:107) − a ∂ t (cid:19) ∂ y Ψ (4a) (cid:18) ∂ (cid:107) − a ∂ t (cid:19) ∂ ⊥ Ψ = − (cid:18) ∇ − a ∂ t (cid:19) ∂ y Φ , (4b)neatly separating the two components in terms of the fastand Alfv´en operators F = ∇ − a − ∂ t and A = ∂ (cid:107) − a − ∂ t ,showing how they couple. As previously noted by Cally &Hansen (2011), the fast and Alfv´en waves decouple in thetwo-dimensional (2D) case ∂ y ≡
0. They also decouple ifthe Alfv´en speed a is uniform, for which case F χ = 0 and A ζ = 0 result.Boundary conditions are chosen to model mode con-version from fast waves injected at the bottom, z bot , withno incoming Alfv´en waves there, and with only evanescentor escaping waves at the top, z top . Details are presented inSection 2.4.1.Of course, there is a gauge ambiguity about the po-tentials Φ and Ψ: the physical displacement ξ is invariantunder the mapping Φ → Φ + ∂ y Υ, Ψ → Ψ + ∂ ⊥ Υ for ar-bitrary perpendicular-harmonic function Υ (i.e., ∇ Υ = 0).This produces an uncertainty in partitioning the displace-ment into fast and Alfv´en parts, but none in χ or ζ . The common understanding of fast and Alfv´en waves derivesfrom the well-known case of a uniform atmosphere, wherethey may be unambiguously distinguished. Specifically, inthat circumstance,(i) They each have their own dispersion relations, ω = a k and ω = a k (cid:107) respectively (in a zero- β plasma);(ii) The group velocity (energy propagation vector) of thefast wave is aligned with the wave vector whilst that of theAlfv´en wave is identically along the magnetic field direction;(iii) The fast wave is irrotational and the Alfv´en wave isincompressive; and(iv) Their displacements are orthogonal to each other.Which of these characteristics is essential, and which is inci-dental and specific to the uniform plasma case? This goes tothe definition of the fast and Alfv´en waves in a structuredmedium.The distinction adopted here via Equation (2) is basedon the fast wave being irrotational and the Alfv´en wave in-compressive. This makes χ and ζ unambiguously character-istic of the fast and Alfv´en wave respectively. When there isan ignorable direction, ∂ y ≡
0, Equations (4) confirm that
MNRAS in press, 1– ?? (2015) P.S. Cally this definition also recovers the expected “dispersion rela-tions” that Φ is associated exclusively with the fast waveoperator F and Ψ with the Alfv´en operator A . In this case,the polarizations (in the ⊥ and y directions respectively) areorthogonal too.Ultimately, in the general case without an ignorable di-rection, it is orthogonality that is jettisoned. This is crucial,as it allows the two modes to interact and exchange energy.In that sense, they are no longer distinct, and alternate par-titions are feasible. Strictly, there are no pure fast and Alfv´enwaves anymore, though as a matter of definition, the terms“fast” and “Alfv´en” will continue to be applied to the Φ andΨ solutions respectively. Especially where the coupling isweak, this is a useful convention.However, the convention adopted is irrelevant in a uni-form region appended at the top of the computational box,since there the attributions are unambiguous. When orthog-onality is re-established, so is the expected partition of thefast and Alfv´en wave energy flux directions (see Section 2.5later). Irrespective of the assumed partition in the interven-ing region, the final fast and Alfv´en fluxes at the top aredefinitive. The calculations presented here may therefore beregarded as scattering experiments: for a given incident fastwave at the bottom, how much energy emerges at the topand what form does it take? Consider a uniform magnetic field inclined at angle θ fromthe vertical in the x - z plane and embedded in an atmospherewith density (and hence Alfv´en speed) that is predominantlyvertically stratified below z ≈
0, becoming gradually morestructured by field line above this level, representing a peri-odic ensemble of inclined “flux tubes”. A convenient form is1 a = 1 a ( z ) − (cid:15) ( z ) [cos( x − z tan θ ) + cos y ] , (5)where a < (cid:54) (cid:15) ( z ) < . Thespecific choice a ( z ) = 1 + δ + (1 − δ ) tanh( z/h )2(e − z/h + 1) , (6)with “chromospheric” scale height h >
0, and transitionregion thickness h . Note that a ∼ δ e z/h as z → −∞ and a → z → + ∞ . The parameter δ < π of the flux tubes in x and y de-fines the unit of length, and the normalization of Equation(5) defines the unit of time. This structure smoothly transi-tions from an exponentially decreasing behaviour with z anddensity scale height h in z < z > π in dimensionless units), reasonableparameter values for the solar chromospheric scale heightand transition region thickness are h = h = 1, with δ =0 .
02. This case is illustrated in Figure 1.The flux tubes are made to fade out over dis-tance W above z = L by prescribing (cid:15) ( z ) = (cid:15) (1 − tanh[( z − L ) /W ]).What should be expected from such a model? At large - �� - � � � �� �� �������������������� � � � Figure 1.
The vertically stratified basic Alfv´en speed a as afunction of height z for the case h = h = 1, δ = 0 . negative z , the atmosphere is effectively plane parallel, withAlfv´en speed increasing exponentially with height. Thiscauses an upgoing fast wave to refract, and indeed reflectif its frequency ω is low enough that it does not reach z ≈ k y (cid:54) = 0, there isfast-to-Alfv´en conversion associated with the vertical strat-ification (see Cally & Hansen 2011 for the case of Alfv´enspeed increasing exponentially with height, and Hansen &Cally 2012 for when there is a steep “transition region” aswell). Additionally, the horizontal structuring that becomesapparent at z (cid:38) ExpandΦ( x , t ) = ∞ (cid:88) m = −∞ ∞ (cid:88) n = −∞ φ mn ( z, t ) e i[( m + r ) x +( n + s ) y ] , (7a)Ψ( x , t ) = ∞ (cid:88) m = −∞ ∞ (cid:88) n = −∞ ψ mn ( z, t ) e i[( m + r ) x +( n + s ) y ] , (7b)where r = m /M and s = n /N are rational numbers (with m and M relatively prime, and similarly for n and N )characterizing the (2 M π , N π ) periodicities of the initial orboundary conditions. Steady oscillations are sought, withexp( − i ω t ) time dependence.The various operations in Equation (4) are easily ren-dered in Fourier space. For example, the Laplacian ∇ isequivalent to ∂ z − ( m + r ) − ( n + s ) . Similarly, ∂ y ≡ i( n + s ), ∂ (cid:107) ≡ cos θ ∂ z +i( m + r ) sin θ , and ∂ ⊥ ≡ i( m + r ) cos θ − sin θ ∂ z .The effect of multiplying by a − , as given by Equation(5), is to scatter in m and n space by ± k x = r and k y = s , makes the square structure lessspecial.It is convenient to define Y = (Φ , Ψ , P, V ) T , where P = MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona ∂ ⊥ Φ and V = ∂ ⊥ Ψ. Then equations (4) may be representedin matrix operator form as M mn Y mn = R n Y m +1 n + L n Y m − n + U n +1 Y m n +1 + U n − Y m n − . (8)Here M mn = − i ( n + s ) A F
0i ( n + s ) F A− ∂ ⊥ I − ∂ ⊥ I , (9)where F = ∇ + ω /a is the (zero (cid:15) ) fast wave operator, A = ∂ (cid:107) + ω /a is the Alfv´en operator, and I is the identity.On the right hand side U n = ω (cid:15) − i( n + s ) I n + s ) 0 0 I , (10) R n = e i z tan θ U n , and L n = e − i z tan θ U n .In practice, the equations are solved in finite differenceform, with N grid points z = ( z , . . . , z N ) T vertically. Thuseach of the entries in the above matrices become N × N sub-matrices, with M mn , R n , L n , and U n becoming 4 N × N .Off-diagonal (band) terms in the submatrices of M are gen-erated by z -derivatives in F , A and ∂ ⊥ . The width of thebands depends on the order of the finite difference deriva-tives employed (arbitrary order is coded, with 12 th ordergenerally used).The various functions of z appearing in these matrices(specifically a , (cid:15) , and e ± i z tan θ ) manifest as diagonal ma-trices. Thus for example, (cid:15) ( z ) becomes diag[ (cid:15) ( z ) , . . . , (cid:15) ( z N )]in each N × N submatrix.Boundary conditions are required to complete the spec-ification of the problem. The top and bottom of the computational region, z top and z bot , are placed in regions where the horizontal structuringis negligible: z top − L (cid:29) W and ( − z bot ) (cid:29) h . At theseextremes, it is imposed that there are no incoming waves,save for the m = n = 0 fast wave at z bot .The dispersion relation resulting from Equations (4) is( k ⊥ + k y )( ω − a k )( ω − a k (cid:107) ) = 0 , (11)where k = | k | = k x + k y + k z = k ⊥ + k y + k (cid:107) .The pure fast wave and Alfv´en wave dispersion rela-tions, ω = a k and ω = a k (cid:107) respectively, may be solvedfor the z -component of k . For the fast wave k z = ± (cid:114) ω a − ( m + r ) − ( n + s ) = ± K mn (12a)with eigenvector (cid:18) ΦΨ (cid:19) = (cid:18) (cid:19) . (12b)For the Alfv´en wave k z = ± ωa sec θ − ( m + r ) tan θ = κ ± mn (13a)with eigenvector (cid:18) ΦΨ (cid:19) = (cid:18) (cid:19) . (13b) In each case, the positive root is upgoing.There is though another spurious “mode”, independentof frequency, with dispersion relation k ⊥ + k y = 0, i.e., k ⊥ = ∓ i k y , or k z = ( m + r ) cot θ ± i ( n + s ) csc θ = σ ± mn , (14a)with eigenvector (cid:18) ΦΨ (cid:19) = (cid:18) ∓ i (cid:19) . (14b)Note that exp[i σ ± mn z ] are the two linearly independent so-lutions of ∇ Υ = 0. These solutions represent the gaugefreedom discussed in Section 2.1.The spurious mode results from the system being 6 th order (for each m, n pair). It needs to be removed by ap-propriate application of auxiliary conditions. In light of itseigenvectors, this may be accomplished by usingΛ mn = Φ (cid:48)(cid:48) mn + K mn Φ mn (15)to filter out the fast wave, and then applying (optionally)zero or evanescence conditions at the boundaries. This pre-scription works by removing the pure oscillatory fast modeexp( ± i K mn z ), leaving an arbitrary linear combination ofexp(i σ ± mn z ).Based on these behaviours, the six boundary conditionsapplied are,at z bot :Λ mn = 0 or Λ (cid:48) mn − i σ ∓ mn Λ mn = 0 (16.I) χ (cid:48) mn + i K mn χ mn = − k δ m δ n (16.II) ζ (cid:48) mn − i κ − mn ζ mn = 0; (16.III)at z top :Λ mn = 0 or Λ (cid:48) mn − i σ ± mn Λ mn = 0 (16.IV) χ (cid:48) mn − i K mn χ mn = 0 (16.V) ζ (cid:48) mn − i κ + mn ζ mn = 0 . (16.VI)The upper sign is taken in Equations (16.I) and (16.IV) if( n + s ) csc θ >
0, and the lower sign otherwise.The radiation conditions (16.II), (16.III), (16.V) and(16.VI) are applied to χ and ζ rather than the potentials,as they unambiguously represent the fast and Alfv´en wavesrespectively. The solutions obtained are therefore robust ir-respective of gauge, though their attributions via the poten-tials to mode type depend on gauge choice. The prescriptions(16.I) and (16.IV) perform well in that regard, subject to theaccuracy of the WKB approximation, with little practicaldifference between the solutions found with zero or evanes-cence boundary conditions. The former are therefore usedfor simplicity.The six boundary conditions (16) are inserted intothe 4 N × N matrix equation (8), replacing respectivelyrows 1, N + 1, 3 N + 1, N , 2 N , and 4 N , i.e., the topand bottom Φ, Ψ, and V equations. The injection equa-tion (16.II) contributes an inhomogeneous source term S = In that regard, an arbitrary gauge could be applied to the so-lution as a post-process without invalidating it. This choice was arrived at by experiment. Solutions obtainedwith it coincide very well with shooting method solutions, whichdo not need to make such substitutions.MNRAS in press, 1– ?? (2015) P.S. Cally (0 , , . . . , , − k , , . . . , T , where the sole nonzero term isat entry N + 1. The full equation then takes the form M mn Y mn − R n Y m +1 n − L n Y m − n − U n +1 Y m n +1 − U n − Y m n − = S mn , (17)where it is to be understood that the six boundary replace-ments in M mn etc. have been made, as described. The quadratic wave energy equation in conservation formmay be constructed directly from Eqn (1) by contractionwith v = ∂ t ξ . After some algebra, it follows that ∂ t E + ∇· F = 0 , (18)where E = 12 ρ | v | + B µ (cid:0) χ + | ∂ (cid:107) ξ | (cid:1) (19)is the wave energy density and F = − B µ (cid:0) χ v + ( v · ∂ (cid:107) ξ )ˆ e (cid:107) (cid:1) (20)is the wave energy flux. The field-directed unit vector is de-noted by ˆ e (cid:107) . The perpendicular-to-the-field term in F pro-portional to χ v is just the rate of working of the fast wave’smagnetic pressure perturbation. The field-aligned tension-related term ( v · ∂ (cid:107) ξ )ˆ e (cid:107) on the other hand includes both fastand Alfv´en contributions.In terms of the complex solutions of Equations (4),the vertical component to wave-energy flux associated withFourier numbers m and n and averaged over a period in both x and y is F mn = F Im (cid:8) χξ ∗ z + ( ξ ∗ · ∂ (cid:107) ξ ) cos θ (cid:9) , (21)where F = ωB / µ . With the amplitude of the incident fastpotential set to | Φ inc00 | = 1 / (2 | k z | ) from boundary condition(16.II), the incident flux may be normalized to unity by set-ting F = 4 | k z | Re { k z } k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z bot . (22)The fast wave vertical wavenumber k z = K is evaluatedat the base, with k x = r , k y = s , and k = k ⊥ + k y =( k x cos θ − k z sin θ ) + k y . It is assumed that K is real,i.e., that the fast wave is travelling, not evanescent, at theinjection height.Total x - y -averaged flux (cid:80) m (cid:80) n F mn is independent ofheight, but individual components F mn are not in general,since energy cascades in Fourier space. Energy is injected into the system through z bot in mode m = n = 0 only. It is distributed between modes via thecouplings to its North, South, East and West neighbours inFourier space, i.e., from (0 ,
0) to (0 , ±
1) and ( ± , ω real), total flux (cid:80) m (cid:80) n F mn summed over allmodes remains independent of z . Strictly, Equations (17) should be solved simultaneouslyfor all modes ( m, n ), but the computational expense is pro-hibitive.Several solution strategies present themselves. Block-Jacobi iteration M mn Y ( j +1) mn = R n Y ( j ) m +1 n + L n Y ( j ) m − n + U n +1 Y ( j ) m n +1 + U n − Y ( j ) m n − + S mn , (23)is cheap, even with several thousand grid points in z , since M mn may be LU-decomposed or similar once for each ( m, n )pair, resulting in each iteration requiring only inexpensiveback-substitution. It is also readily parallelized. This schemedoes not strictly conserve flux, but once converged does so tothe required tolerance. Convergence is contingent on the ap-propriate spectral radius being less than unity. The spectralradius is not determined explicitly, but experience indicatesthat the scheme is convergent in some cases and divergentin others. It is certainly convergent for small enough (cid:15) .Convergence may be improved by taking advantage ofblock-tridiagonal structure generated by adopting an im-plicit formulation in one direction. For example, implicitcoupling in the horizontal ( m ) direction yields M mn Y ( j +1) mn − R n Y ( j +1) m +1 n − L n Y ( j +1) m − n = U n +1 Y ( j ) m n +1 + U n − Y ( j ) m n − + S mn . (24)A vertical formulation is defined analogously: M mn Y ( j +1) mn − U n +1 Y ( j +1) m n +1 − U n − Y ( j +1) m n − = R n Y ( j ) m +1 n + L n Y ( j ) m − n + S mn . (25)Both of these schemes conserve flux along their implicitdirection, but not perpendicularly. Again, convergence re-dresess this. Experience suggests that these line-implicitschemes converge very rapidly, though the decompositions ofthe left hand sides are very expensive and memory-intensive.Once calculated though, it is comparatively cheap to be ap-ply them recursively till convergence. Nevertheless, simpleJacobi iteration is far less memory-intensive and far quicker,provided it converges.The adopted finite difference scheme in z typically uses12 th order finite differences on a regular grid of about 8000–12000 points. The code though is written to accommodatearbitrary order and an optionally stretched grid. The variousJacobi or line-iterations are performed in parallel on multi-core machines. Iteration is continued untilmax m,n,z (cid:12)(cid:12)(cid:12) Y ( j +1) mn − Y ( j ) mn (cid:12)(cid:12)(cid:12) < − . (26)Memory is the determining limitation in the line-implicitmethods. The coupled ordinary differential equations (17) are solvednumerically in an atmosphere structured as displayed in Fig-ure 2. (cid:15) = 0For purposes of comparison, it is of interest to first solvethe problem for the case (cid:15) = 0, where there are no “flux MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona Figure 2.
Representative image of the flux tubes, as represented by the Alfv´en speed a ( x, y, z ), in the vertical x - z plane at y = 0 (left)and the horizontal x - y plane at z = 6 (right) for the case θ = 50 ◦ , (cid:15) = 0 . h = h = 1, δ = 0 . L = 20, W = 1. With M = N = 5(corresponding to r = 1 / s = 4 / x - y plot represents the periodicity in both directions. tubes”. Only m = n = 0 need be considered, as there isno coupling to the other Fourier modes. This is the caseextensively explored by Cally & Hansen (2011), though witha different Alfv´en speed profile.The maximum Alfv´en speed attained is 1, so any m = n = 0 fast wave with frequency ω will reflect if ω < ω =( r + s ) / . If ω (cid:62) ω there is no (total) reflection, and thefast wave propagates (partially) to infinity. It is convenientto define the parameter α such that ω = α ω , so the incidentwave reflects if α < α = 0 .
85, so it is trapped. Figures 3 and 4 showthe coupling effect of nonzero k y = s , as anticipated fromEquations (4). The incident fast mode (unit flux) from belowmostly reflects, but with 39% escaping the top as an Alfv´enwave. There is no fast wave flux at the top, as expected.Outgoing flux at the bottom consists of 56% in the fast waveand 5% in the Alfv´en wave.The cross-flux, shown dotted in Figure 4, consists ofthose terms in the quadratic formula (4) containing both Φand Ψ. It is therefore a direct measure of the fast/Alfv´encoupling. Once it dies out with increasing z , the decouplingof the fast and Alfv´en components is essentially complete.For the second case, α = 1 . (cid:15) = 0 case presented hereis for purposes of comparison. Atmospheres with periodictube structures will be the subject of investigation from nowon. - �� - � � � �������� � Φ �� - �� - � � � �� - �� - ������� � Ψ �� Figure 3. Φ and Ψ as functions of height z for the case ofFig. 2 with r = 1 / s = 4 / ω = 0 . ω (reflecting fast wave),and (cid:15) = 0 (no tubes). The evanescence of the fast wave (Φ) andthe upward travelling nature of the Alfv´en wave (Ψ) are apparent.The vertical line indicates the position of the fast wave reflectionpoint. Real and imaginary parts are shown as full and dashedcurves respectively. < (cid:15) < . MNRAS in press, 1– ????
85, so it is trapped. Figures 3 and 4 showthe coupling effect of nonzero k y = s , as anticipated fromEquations (4). The incident fast mode (unit flux) from belowmostly reflects, but with 39% escaping the top as an Alfv´enwave. There is no fast wave flux at the top, as expected.Outgoing flux at the bottom consists of 56% in the fast waveand 5% in the Alfv´en wave.The cross-flux, shown dotted in Figure 4, consists ofthose terms in the quadratic formula (4) containing both Φand Ψ. It is therefore a direct measure of the fast/Alfv´encoupling. Once it dies out with increasing z , the decouplingof the fast and Alfv´en components is essentially complete.For the second case, α = 1 . (cid:15) = 0 case presented hereis for purposes of comparison. Atmospheres with periodictube structures will be the subject of investigation from nowon. - �� - � � � �������� � Φ �� - �� - � � � �� - �� - ������� � Ψ �� Figure 3. Φ and Ψ as functions of height z for the case ofFig. 2 with r = 1 / s = 4 / ω = 0 . ω (reflecting fast wave),and (cid:15) = 0 (no tubes). The evanescence of the fast wave (Φ) andthe upward travelling nature of the Alfv´en wave (Ψ) are apparent.The vertical line indicates the position of the fast wave reflectionpoint. Real and imaginary parts are shown as full and dashedcurves respectively. < (cid:15) < . MNRAS in press, 1– ???? (2015) P.S. Cally - �� - � � � �� - ������������ � � � � � � � � � � = �� � = � Figure 4.
The fast (full black), Alfv´en (dashed), and cross(dotted) fluxes against z for the case of Fig. 3. The total flux F = F f + F A + F c = 0 .
39 (horizontal red line) is constant, asrequired since there is no lateral energy loss. - �� - � � � �� - �� - �� - ������� � Φ �� - �� - � � � �� - � - � - ������ � Ψ �� Figure 5. Φ and Ψ as functions of height z for the case ofFig. 2 with r = 1 / s = 4 / ω = 1 . ω (transmitting fast wave),and (cid:15) = 0 (no tubes). The travelling natures of the fast wave (Φ)and Alfv´en wave (Ψ) is apparent. Alfv´en speed (high density) tubes. This requires 1 − (cid:15) <α < < α < (cid:15) in whichthese tubes merge into a contiguous “Swiss cheese” is alsobriefly addressed (Section 4.2.2). α = 0 . − (cid:15) , (cid:15) ) restricts the x - y surface area where thefast wave may propagate to discrete tubes, as commonlyenvisaged for kink waves. For example, with (cid:15) = 0 . α = 0 .
85, those tubes occupy 31% of the area. With α = 1 it - �� - � � � ����������� � � � � � � � � � � = �� � = � Figure 6.
The fast (full black), Alfv´en (dashed), and cross(dotted) fluxes against z for the case of Fig. 3. The total flux F = F f + F A + F c = 0 .
998 is plotted in red. is 50%, but the “tubes” become contiguous. For the moment,only the discrete-tube case α < r = 1 / s = 3 / θ = 50 ◦ , (cid:15) = 0 . α = 0 .
85 , L = 12, W = 1, z bot = − z top = 14with − (cid:54) m, n (cid:54) − (cid:54) m, n (cid:54)
3. It is clear that energy has not propagatedvery far in mode number by this level ( z top = 14). Thisverifies that enough Fourier modes have been used. Largermode sets are easily handled with the Jacobi process, but arecomputationally more problematic for line-implicit calcula-tions. Increasing the number of modes beyond the currentlevel does not seem to adversely affect stability for the casesexamined.The highest flux is Alfv´enic for m = n = 0, theonly mode at which energy is injected (as a fast wave at z bot = − m = 0, n = − k h = (cid:112) ( m + r ) + ( n + s ) is a minimum there ( k h = 0 . m = 0, n = − αω √ (cid:15) = 0 . αω √ − (cid:15) = 0 . m = 0, n = − αω √ − (cid:15) < k h < αω √ (cid:15) ; these will be true propa-gating tube waves.This illustrates how the scattering in Fourier space canopen up one or more channels for fast mode propagation de-spite the original incident fast wave being evanescent. How-ever, the Alfv´en wave propagates in all channels; it is simplya matter of how much scatters into them from the incidentwave.Figure 8 shows these fluxes in detail as functions ofheight for − (cid:54) m, n (cid:54)
1. It is apparent that the injectedfast flux in the central mode, m = n = 0, quickly converts MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona - � - � - � � � � � - � - � - ����� � � � � ( ��� ) ��� � � ( ��� ) ������������� Figure 7.
Top Alfv´en (outer annulus) and fast (inner disk) fluxesin the Fourier modes m , n for the case r = 1 / s = 3 / θ = 50 ◦ , (cid:15) = 0 . α = 0 .
85 , L = 12, W = 1, z bot = − z top = 14 with − (cid:54) m, n (cid:54) to Alfv´en flux near the reflection height, and then that thatAlfv´en flux slowly decays with height as it is transferred tosurrounding Fourier modes by phase mixing generated bythe tube structure.For the most part, the lost energy reappears as Alfv´enflux in the surrounding modes, increasing in magnitude withincreasing z , and levelling off only where the tubes fade outaround z = 12. However, the propagating fast mode in m =0, n = −
1, is very striking. It grows rapidly over 0 (cid:46) z (cid:46)
7, before itself becoming subject to phase mixing decay. Itagain levels off as the tubes fade out, but in a more realisticmodel with much longer tubes, would decay away almostcompletely, leaving only Alfv´en energy in the system. (Jacobiiterations do not converge if the tubes are much longer, sothey are restricted here for computational convenience.)The total flux, summed over all modes, is depicted inFigure 9. Two points to note are that the total flux is indeedindependent of height (this is a sign of convergence of theJacobi iterations), and that the overall flux is predominantlyAlfv´enic.Figures 10 and 11 display the central nine Fourier com-ponents for each of the fast and Alfv´en potentials Φ and Ψrespectively. Again it is clear that only the m = 0, n = − m = n = 0.Figure 12 and 13 show snapshots of the displacementvector ξ associated with the Alfv´en and fast waves indepen-dently. As is to be expected in light of the flux comparisons,the Alfv´en displacements are significantly larger than thoseassociated with the fast wave. Watching a movie of these dis-placements confirms that the Alfv´en displacements predom-inantly rotate, whereas the fast displacements are approxi-mately linear and therefore to be identified with kink-type(transverse) waves. Figure 13 in particular confirms the ear-lier conclusion that the sole propagating fast wave for this case ( m = 0, n = −
1) is space-filling, and not restricted totubes.Figure 14 shows the spectral fluxes for flux tubes in-clined only 20 ◦ from the vertical, rather than 50 ◦ as before.Correspondingly, Figure 15 indicates that there is much-reduced (though not zero) power in the kink mode for less-inclined flux tubes. Figure 16 presents the spectral fluxes,again for θ = 20 ◦ , but with a larger driving wavevector r = 1 / s = 5 /
6, exhibiting differences in detail, but thesame overall conclusions. α = 1 . Swiss Cheese
Now consider a case (1 < α < (cid:15) ) where the region inwhich fast waves may propagate becomes contiguous, notrestricted to discrete flux tubes. With α = 1 .
15 and (cid:15) = 0 . α = 1 .
15 case,with all other parameters as for Figure 7. Though fast modeflux now dominates, there is still considerable Alfv´en power(27%). In any case, the fast wave can no longer be describedas a kink wave, as there are no longer discrete wave guides.It is more a case of a propagating bulk fast wave with ex-cisions. The modes m = 0 with n = 0 and − out of the propagat-ing regime, though it has left two strong channels for fastpropagation. α = 0 . α < − (cid:15) , there is essentially no fastwave power. This case has been checked numerically for thesame model as for Figure 17 but with α = 0 .
6; it supportsonly Alfv´en waves beyond about z = 1. No graphs are pre-sented here for that case, as there is nothing surprising toreport. Despite the simplicity of the model, the results presentedhere are instructive, and illustrate a number of featuresthat might be expected in complex mixed vertical/cross-fieldstructured atmospheres.The following lessons may be drawn.(i) A major effect of the packed flux tube structure isto scatter in Fourier space. This can partially scatter anevanescent fast wave into travelling fast waves, primarilymanifesting as kink waves, but also possibly as space-fillingfast waves. Conversely, it can scatter travelling fast waves tohigher wave number where they are evanescent.(ii) For the most part, it can be expected that the bulkof seismically generated fast wave flux incident from belowreflects before it reaches the TR, so the process of scatter into travelling fast/kink modes provides a mechanism forcarrying fast waves upward that would not be available inan unstructured corona. The m = 0, n = − MNRAS in press, 1– ?? (2015) P.S. Cally - � � � ���������������������� � � � � � � � � � � = - �� � = � - � � � ���������������������� � � � � � � � � � � = �� � = � - � � � ���������������������� � � � � � � � � � � = �� � = � - � � � ������������������������������ � � � � � � � � � � = - �� � = � - � � � �� - ������������ � � � � � � � � � � = �� � = � - � � � �� - ����� - ������������������������������ � � � � � � � � � � = �� � = � - � � � ��������������������������� � � � � � � � � � � = - �� � = - � - � � � �� - ���������������������������� � � � � � � � � � � = �� � = - � - � � � �� - ������������������������������������������������ � � � � � � � � � � = �� � = - � Figure 8.
Alfv´en (green dashed), fast (black full), cross (dotted blue), and total (red full) fluxes in the Fourier modes − (cid:54) m, n (cid:54) z for the case of Figure 7. Note the different flux scales in each panel. - � � � �� - ������������ � � � � � � � � � | � | ≤ �� | � | ≤ � Figure 9.
Alfv´en (green dashed), fast (black full), cross (dottedblue), and total (red full) fluxes summed over all Fourier modes − (cid:54) m, n (cid:54) z for the case of Figure 7. panel of Figure 9 illustrates this well, with a general build-upof fast wave flux over 2 (cid:46) z (cid:46)
7. The cross-flux is relativelysmall in this region, so an interpretation in terms of fast kinkwaves is justified.(iii) However, the kink wave eventually starts to decay viaresonant coupling to the Alfv´en wave (see the same panel for8 (cid:46) z (cid:46) m, n ).(v) Alfv´en waves themselves scatter into higher modenumbers, representing the process for mode mixing. The cen-tral panel of Figure 9 is a good example of this.(vi) Alfv´en energy will be spread more widely in Fourierspace if the flux tubes are allowed to extend much higherthan numerical constraints have permitted here. This willsee the oscillations disappear from view in practice.(vii) High transmissions through the transition region at z = 0 are easily attained (recalling that the original incidentwave carried unit flux). This is encouraging from the pointof view of coronal heating and solar wind acceleration.(viii) Only incident fast waves have been considered here,in line with the supposition that these waves originate fromthe Sun’s internal seismology. Direct injection of Alfv´enwaves at the base may be of interest, but is likely to beless realistic in the solar context because of the difficulty ofgenerating Alfv´en waves at the weakly ionized photosphere.In summary, the observable kink-like oscillations pre-sumably responsible for the various CoMP, AIA, and SOTobservations may represent only a small part of the total up-ward wave flux in coronal flux tube ensembles. It is notablethat net upward flux in all cases explored is a significantfraction of the injected flux, so wave energies, both fast andAlfv´en, may in combination provide ample energy to supplythe corona. MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona - � � � �� - ����� - ����� - ������������������������� � Φ - �� - � � � �������������� � Φ �� - � � � �� - ����� - �������������������� � Φ �� - � � � �� - ���� - ���� - ���� - ������������ � Φ - �� - � � � �� - �� - � - � - � - �� � Φ �� - � � � �� - ���� - ���� - ���������������� � Φ �� - � � � �� - ���� - �������������������� � Φ - � - � - � � � �� - � - ����� � Φ � - � - � � � �� - ���� - ���� - �������������������� � Φ � - � Figure 10.
Fast wave potential Fourier coefficients Φ mn for − (cid:54) m, n (cid:54) m = − , , n = − , , - � � � �� - ��� - ��� - ������������������ � Ψ - �� - � � � �� - ��� - ��� - ��������������� � Ψ �� - � � � �� - ��� - ��������������� � Ψ �� - � � � �� - ��� - ������������ � Ψ - �� - � � � �� - ����� � Ψ �� - � � � �� - ��� - ��� - ������������������ � Ψ �� - � � � �� - ��� - ������������ � Ψ - � - � - � � � �� - ��� - ��� - ������������ � Ψ � - � - � � � �� - ��� - ������������ � Ψ � - � Figure 11.
Alfv´en wave potential Fourier coefficients Ψ mn for − (cid:54) m, n (cid:54) ????
Alfv´en wave potential Fourier coefficients Ψ mn for − (cid:54) m, n (cid:54) ???? (2015) P.S. Cally
Figure 12.
Displacement vectors at a particular time for the caseof Figure 7, corresponding to the Alfv´en component (upper panel)and the fast component (lower panel). The y -component of eachplotted vector is indeed ξ y , but the x component is ξ ⊥ ratherthan ξ x for purposes of display. That is, these are the displace-ments seen along the line of the magnetic field, though at fixedheight z = 8 .
5. Arrow lengths represent the true displacementcomparisons. Two animations, of the Alfv´en and fast displace-ments respectively, accompany this paper.
Figure 13.
Zoomed version of Figure 12, showing Alfv´en (left)and fast (right) displacements.
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Top Alfv´en (outer annulus) and fast (inner disk)fluxes in the Fourier modes m , n for the same case as in Figure7, but with less inclined magnetic field, θ = 20 ◦ . - � � � ����������� � � � � � � � � � | � | ≤ �� | � | ≤ � Figure 15.
Alfv´en (green dashed), fast (black full), cross (dottedblue), and total (red full) fluxes summed over all Fourier modes − (cid:54) m, n (cid:54) z for the case of Figure 14,i.e., with field inclination θ = 20 ◦ .De Moortel I., Nakariakov V. M., 2012, Royal Society of LondonPhilosophical Transactions Series A, 370, 3193De Pontieu B., et al., 2007, Science, 318, 1574Erd´elyi R., Fedun V., 2007, Science, 318, 1572Felipe T., 2012, ApJ, 758, 96Goossens M., Andries J., Aschwanden M. J., 2002, A&A, 394,L39Goossens M., Terradas J., Andries J., Arregui I., Ballester J. L.,2009, A&A, 503, 213Goossens M., Soler R., Terradas J., Van Doorsselaere T., VerthG., 2014, ApJ, 788, 9Hansen S. C., Cally P. S., 2012, ApJ, 751, 31Hanson C. S., Cally P. S., 2011, Sol. Phys., 269, 105Jess D. B., Mathioudakis M., Erd´elyi R., Crockett P. J., KeenanF. P., Christian D. J., 2009, Science, 323, 1582Khomenko E., Cally P. S., 2011, J. Phys.: Conf. Ser., 271, 012042Khomenko E., Cally P. S., 2012, ApJ, 746, 68Kudoh T., Shibata K., 1999, ApJ, 514, 493MNRAS in press, 1– ?? (2015) lfv´en Waves in the Structured Solar Corona - � - � - � � � � � - � - � - ����� � � � � ( ��� ) ��� � � ( ��� ) ������������� Figure 16.
Top Alfv´en (outer annulus) and fast (inner disk)fluxes in the Fourier modes m , n for the same case as in Figure14 (i.e., θ = 20 ◦ ), but with r = 1 / s = 5 / - � - � - � � � � � - � - � - ����� � � � � ( ��� ) ��� � � ( ��� ) ���������������� Figure 17.
Top Alfv´en (outer annulus) and fast (inner disk)fluxes in the Fourier modes m , n for the case r = 1 / s = 3 / θ = 50 ◦ , (cid:15) = 0 . α = 1 .
15 , L = 12, W = 1, z bot = − z top = 14 with − (cid:54) m, n (cid:54)
3. This is the same as for the case ofFigure 7, except for the higher frequency ( α = 1 .
15 rather than α = 0 . ????