Notes on PFSS Extrapolation

Transcription

1 Notes on PFSS Extapolation Xudong Sun Abstact This is a documentation on the Stanfod PFSS model. Detailed mathematical deductions ae povided fo the use of the model. Some bief documentation on PFSS-like models SCS, HCCSSS, etc.) is also povided. 1. Basic Equation The most common vesion of PFSS model Hoeksema, 1984; Wang and Sheeley J., 1992) cuently in use takes global adial Caington synoptic maps as input. In these maps, photospheic fields ae sampled on a heliogaphic coodinate, evenly spaced eithe in latitude o sine-latitude steps. If the field is puely potential, we have B = Ψ, 1) whee 2 Ψ =. 2) We assume the existence of a spheical souce suface at a adius of usually at 2.5R ), beyond which all field lines ae open and adius. The potential aises fom both inside the inne bounday R photsphee, o R ) and outside the oute bounday, o the souce suface: Ψ = Ψ I + Ψ O, 3) with Ψ I = l+1) l= m= l f Ilm Y lm θ, φ), 4) Ψ O = l= l m= l f Olm Y lm θ, φ). 5) Scale in tems R fo Ψ I and in tems of fo Ψ O. Use the fact that Y lm θ, φ) = k lm P m l cos θ)e imφ. 6) The eal pat of Ψ can be genealized fom Equation 3) though 6): { [ R ) l+1 Ψ = R Pl m cos θ) g lm cos mφ + R ) l s c lm] l= m= R

2 2 [ R ) l+1 + h lm sin mφ + R ) l s d lm]}, 7) R whee g lm, h lm, c lm and d lm ae the unknown coefficients. Note that the nomalization of the spheical hamonics and associated Legende fuctions can be ticky. We will simply pesent the nomalization we adopted hee and leave the detailed desciption to Section 2. By definition, the field lines tun adial at the souce suface. This means the field vecto is puely adial at, o athe, the potential is a constant on the souce suface. Set this potential to, we then have c lm = d lm = R ) l+2 = c l. 8) Now ou sole task is to detemine g lm and h lm, using the inne bounday condition photospheic field). Wite B fom Equation 1) specifically: B, θ, φ) = Ψ = lm At inne bounday, we have B R, θ, φ) = l= m= Pl m cos θ)g lm cos mφ + h lm sin mφ) [ l + 1) R ) l+2 ) ] l 1 l c l. 9) P m l cos θ)g lm cos mφ + h lm sin mφ), 1) whee [ ) ] 2l+1 g lm = g lm R, 11) h lm = h lm [ R ) 2l+1 ]. 12) Now we make use of the othogonal popety of the associated Legende function in ou convention) 2π sin θdθ Pl m cos θ) cos sin mφ P m l cos θ)cos sin m φ = 4π 2l + 1 δ ll δ mm. 13) Note when m = Equation 13) holds fo the cos mφ case, while the sin mφ case simply yields. An integation of Equation 1) then shows us how to obtain g

3 PFSS 3 and h. Hee, h l is obviously. 2π sin θdθ B R, θ, φ)pl m cos θ) cos sin mφ = 4π g lm. 14) 2l + 1 h lm Fo a synoptic map X Y ) in sine-latitude fomat, the Equation 14) becomes glm h lm ) = 2l + 1 XY X Y i=1 j=1 B R, θ i, φ j )Pl m cos θ i ) cos sin mφ j. 15) Thus the potential is solved. Thee ae othe ways to compute g and h, as we will see in Section 3. To conclude, we have the solutions in the following fom. B, θ, φ) = Ψ = R l= m= P m l cos θ)g lm cos mφ + h lm sin mφ) ) [ l+2 ) ] /[ 2l+1 R ) 2l+1 ], 16) B θ, θ, φ) = 1 Ψ θ = Pl m cos θ) g lm cos mφ + h lm sin mφ) θ l= m= ) [ l+2 ) ] /[ 2l+1 ) ] 2l+1 R R 1,17) B φ, θ, φ) = 1 Ψ sin θ φ = l= m= ) [ l+2 R mpl m cos θ) g lm sin mφ h lm cos mφ) sin θ ) ] /[ 2l+1 ) ] 2l+1 R 1 18), whee g and h ae detemined by Equation 15). At the poles θ =, π), only some l, m) tems contibute to the solution. This is mainly because Pl m cos θ) sin m θ. Fo B, only the m = tems contibute. Fo B θ and B φ, only the m = 1 tems contibute. 2. Issues on Nomalization The associated Legende functions may have diffeent nomalization conventions in diffeent cases. Fom Equation 13), in ou scheme we have 1 1 P m l x) 2 dx = 2 2l d ), d = { 1, m =, m. 19)

4 4 A moe widely used fom is 1 1 with the following popeties P l m x) 2 2l + m)! dx =, m l, 2) 2l + 1)l m)! P m m l m)! l x) = 1) l + m)! P l m x), 21) P l l x) = 1) l 2l 1)!!1 x 2 ) l/2, 22) l m + 1) P l+1x) m = 2l + 1)x P l m x) l + m) P l 1x). m 23) Two sets of Legende functions ae elated by Pl m x) = 1) m l m)! 2 m d P l + m)! l x). 24) So in ou convention Equation 21)-23) become P l l x) = P m l x) = 1) m Pl m x), 25) 2l 1)!! 2 d 1 x 2 ) l/2, 26) 2l)!! l + 1)2 m 2 Pl+1x) m = 2l + 1)xPl m x) l 2 m 2 Pl 1x). m 27) Equation 24) can be used to convet standad Legende functions fo ou use. Equations 25)-27) can be used ecusively to geneate ou own set of Legende functions. In ou implementation Zhao and Hoeksema, 1994), we stat by calculating Pm m using Equation 26). Then we calculate all tems fo fixed m using Equation 27), utilizing the fact that Pm 1x) m =. We also need to calculate Pl m cos θ)/ θ. In ou implementation, we have l + 1)2 m dp m 2 l+1 x) = 2l + 1) dx [x dp l m x) dx ] 1 x 2 Pl m x) l 2 m 2 dp m l 1 x) dx. 28) 3. Altenative Method fo Computing g and h We may altenatively utilize the spheical hamonic expansion esult fom helioseismology to obtain the g and h coefficients. If the signal on the photosphee

5 PFSS 5 at a paticula moment is fθ, φ), then fθ, φ) = l= m= l f m l Y m l θ, φ), f m l C, 29) with the following nomalization 2π In the most common vesion, sin θdθyl m θ, φ)y m l θ, φ) = 4πδ ll δ mm. 3) Y m l = 1) m Y m l. 31) Conside Equation 6), 2), 3) and 31), we have Yl m l m)! θ, φ) = 2l + 1) l + m)! P l m cos θ)e imφ, 32) whee P m l is defined in Equation 2). So we have the following expansion: f m l = 1 4π = 1 4π 2π 2π 2π = 1)m 2l + 1 4π 2 d sin θfθ, φ)y m l θ, φ) l m)! sin θfθ, φ) 2l + 1) l + m)! P l m cos θ)e imφ Compae this with Equation 15), we find the connection sin θfθ, φ)p m l cos θ)e imφ. 33) g lm = 1) m 2l + 1)2 d ) Rf m l ), 34) h lm = 1) m 2l + 1)2 d ) If m l ). 35) 4. Two Othe Vesions: Global Helioseismology and Solasoft As diffeent codes may use diffeent nomalizations and algoithms, we need to be caeful when using coefficient sets. Hee ae two othe codes the community is using. Schou s Schou and Bown, 1994) global helioseismology code is adapted to compute the hamonic expansion coefficient. The synoptic map is consideed

6 6 as a single point time seies. FFT is fist applied on each ow of points and a mask dot poduct is used to get the coefficient. In this vesion, fl m is given fo non-negative m s. The following quations link the esult to ou g s and h s. 2l + 1)2 d ) g lm = Rfl m ), 36) 2 2l + 1)2 d ) h lm = Ifl m ). 2 37) The PFSS package in Sola Soft takes anothe oad. The map is fist esampled to the optimized Gauss-Legende gid befoe computes f Ilm and f Olm in Equation 4) and 5). The esult satisfies g lm R lf Ilm + l + 1)f Olm ), 38) h lm I lf Ilm + l + 1)f Olm ). 39) The esult might be sensitive to the esampling. The constant facto hee is yet to be established. 5. PFSS-Like Models The effect of cuents ae included in seveal othe PFSS-based models. Schatten Schatten, 1971) poposed a model called potential-field-cuent-sheet PFCS) that includes the effect of the cuent sheets in steames. Zhao and Hoeksema Zhao and Hoeksema, 1994) povided a modified hoizontal-cuent-cuentsheet model HCCS) that intoduces the effect of hoizontal cuent, based on Bogdan and Low s Bogdan and Low, 1986) magnetic-static solution. Late, Zhao and Hoeksema Zhao and Hoeksema, 1995) evised the HCCS model, adding a souce suface at a highe altitude. This new model is called hoizontal-cuentcuent-sheet-souce-suface HCCSSS) model. In bief, these PFSS-like models divide the sola atmosphee into diffeent egions. Below a spheical suface cusp suface) the field is moe potential-like, without the effect of cuent sheet. Above this suface the cuent sheet comes in. HCCSSS model an exta souce suface at highe up opens all field to be adial. Both the HCCS and HCCSSS models have a hoizontal cuent flowing eveywhee. We genealize the algoithm as following Zhao and Hoeksema, 1994; Zhao and Hoeksema, 1995). Fist, the field evey whee can be computed fom a potential function Ψ: Ψ, θ, φ) = l= m= and fields ae computed by R l )P m l cos θ)g lm cos mφ + h lm sin mφ), 4) B = η) Ψ ˆ 1 Ψ θ ˆθ 1 Ψ sin θ φ ˆφ, 41)

7 PFSS 7 Table 1. Vaious fom of R l ) in Diffeent Models fo Equation 4) Model Lowe Region Uppe Region PFCS R R ) l+1 [1 ) 2l+1 ] R ) 2l+1 Rc 2 R c + a) l l + 1) + a) l+1 HCCS R 2 R + a) l l + 1) + a) l+1 Rc 2 R c + a) l l + 1) + a) l+1 HCCSSS R 2 R + a) l l + 1) + a) l+1 Rc 2 1 +a) l+1 +a)l +a) 2l+1 l+1 R c+a) l + lrc+a)l+1 Rs+a) 2l+1 PFSS R R ) l+1 [1 ) 2l+1 ] R ) 2l+1 whee η) = 1 + a )2, 42) in which a paameteizes the length scale of hoizontal eletic cuent in the coona. Note when a = we simply etun to models without hoizontal cuents PFSS). The potential is detemined by bounday conditions on the imaginay spheical sufaces that we use to divide sola atmosphee. 1) To get the spheical hamonics g lm and h lm in Equation 4)), photospheic field is decomposed fo the lowe egion below cusp suface). Field value is then computed at the uppe bounday of the lowe egion cusp suface), whee Schatten s least-squae technique Schatten, 1971) is used to get the coefficient fo the highe egion. 2) The adial pat Φ in Equation 4)) is detemined by the assumption we make of the field on the boudaies: whethe it is adial, etc. To summeize, we put the fom of Φ function in Table 1. Refeences Bogdan, T.J., Low, B.C.: 1986, Astophys. J. 361), 271. doi:1.186/ Hoeksema, J.T.: 1984, Stuctue and evolution of the lage scale sola and heliospheic magnetic fields. PhD thesis, Stanfod Univ.. Schatten, K.H.: 1971, Cosmic Electodyn. 2, 232.

8 8 Schou, J., Bown, T.M.: 1994, Aston. Astophys. Suppl. 17, 541. Wang, Y.M., Sheeley J., N.R.: 1992, Astophys. J. 392, 31. doi:1.186/ Zhao, X., Hoeksema, J.T.: 1994, Sol. Phys. 1511), 91. Zhao, X., Hoeksema, J.T.: 1995, J. Geophys. Res. 1A1), 19. Othe Refeences Spheical Hamonics: hamonics Associated Legende Functions: function Nomalized Associated Legende Functions: