Ionospheric disturbance magnetic field continuation from the ground to the ionosphere using spherical elementary current systems

Transcription

1 Eath Planets Space, 51, , 1999 Ionospheic distubance magnetic field continuation fom the gound to the ionosphee using spheical elementay cuent systems O. Amm and A. Viljanen Finnish Meteoological Institute, Geophysical Reseach Division, P.O. Box 503, FIN Helsinki, Finland (Received Mach 1, 1999; Revised June 23, 1999; Accepted June 24, 1999) A new technique fo continuation of the gound magnetic field caused by ionospheic cuents to the ionosphee in spheical geomety is pesented that makes use of elementay ionospheic cuent systems, which wee intoduced by Amm (1997) in extension of an ealie wok by Fukushima (1976). The measued gound magnetic distubance is expanded in tems of the gound magnetic effect of a spatial distibution of such elementay cuent systems. Using a matix invesion technique, the scaling factos fo each elementay cuent system, and theefom the ionospheic equivalent cuents ae calculated. The technique can be applied to both global and local scales. Its advantages compaed to the common field continuation techniques with Fouie (local scale), spheical cap (local to medium scale), o spheical (global scale) hamonic expansions ae: 1) No fixed limitation of the spectal content has to be given fo the whole analysis aea, as it has to be done fo the othe techniques by tuncation of a seies expansion. 2) The locations of the elementay cuent systems can be chosen feely, such that they ae most suitable with espect to the available measuement sites o the type of cuent system to be analysed. Results of the new technique ae discussed in compaison to esults of the spheical cap hamonic expansion method fo a model of a Cowling channel. 1. Intoduction Being intoduced as ealy as by Gauß (1836), continuation of the magnetic field distubance due to extenal souces fom the gound to the ionosphee emains a cucial step in using gound magnetomete data in ionospheic-magnetospheic eseach (e.g., Untiedt and Baumjohann, 1993, and efeences theein). While the gound magnetic field o the gound equivalent cuents by themselves can only be used qualitatively fo the estimation of ionospheic electodynamic paametes, the height-continued equivalent cuents at the ionospheic level can be combined with infomation of the ionospheic electic field via Ohm s law to spatially obtain quantitative esults on ionospheic conductances, tue ionospheic cuents, and field-aligned cuents (e.g., Richmond and Baumjohann, 1983; Inheste et al., 1992; Amm, 1998). Field continuation in geneal (not only fom the gound to the ionosphee) is also an impotant tool fo the constuction of thee-dimension geomagnetic efeence models (e.g., Haines, 1985a, Tota et al., 1992, De Santis et al., 1997), o fo geological applications such as to study custal magnetic anomalies by means of satellite data (e.g., De Santis et al., 1989), o the Eath s conductivity stuctue (e.g., Tota and De Santis, 1996). As we shall see, the technique descibed in this pape can be applied to such poblems as well with small modifications. The common technique fo field continuation in spheical geomety is the expansion of a magnetic potential into a seies of spheical hamonics, each of which is a solution Copy ight c The Society of Geomagnetism and Eath, Planetay and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society fo Planetay Sciences. of Laplace s equation = 0 that holds in aeas fee of cuents (e.g., Chapman and Batels, 1940): [ Ni (,ϑ,ϕ) = R E n n=1 m=0 {gn m,i N e ( RE n=1 m=0 ) n+1 Pn m (cos ϑ) cos(mϕ) + h m,i n sin(mϕ)} n ( ) n + Pn m (cos ϑ) {g m,e n R E cos(mϕ) + h m,e n sin(mϕ)} whee R E is the Eath s adius, n and m the intege degee and ode of the associated Legende function Pn m,(,ϑ,ϕ) the coodinates of a spheical coodinate system, and gn m and h m n the spheical hamonic coefficients to be detemined fom the measuements, usually by means of a least-squae eos method. The fist sum in (1) coesponds to the pat of the magnetic potential caused by intenal souces, the second one to that caused by extenal cuents (maked with supescipts i and e fo the coefficients, espectively). Once the hamonic coeffients ae detemined, the magnetic potential can be continued to any adius R C inside the cuent-fee aea by setting = R C in (1). While this taditional spheical hamonic analysis (SHA) is well suited fo global studies, poblems appea if the aea of inteest and of measuements is confined to a pat of the Eath s suface only: The SHA coefficients will then be pooly defined, o vitual data points have to be added. A way out of these poblems is povided by spheical cap hamonic analysis (SCHA; Haines, 1985b). Let us assume ] (1) 431

2 432 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION that a given data set can be coveed by a spheical cap with midpoint (ϑ p,ϕ p ) (in geogaphical coodinates) and a halfangle ϑ 0 of the cap. The SCHA expansion of the magnetic potential in the cuent-fee egion in the spheical coodinate system with the midpoint of the cap as the nothen pole is (Haines, 1985b): [ Ki (,ϑ,ϕ) = R E k k=0 m=0 {g m, j k K e ( ) nk (m)+1 RE Pn m k (m)(cos ϑ) cos(mϕ) + h m, j k sin(mϕ)} k ( ) nk (m) + k=1 m=0 {g m,e k R E Pn m k (m)(cos ϑ) cos(mϕ) + h m,e k sin(mϕ)} ]. (2) The stuctue of Eq. (2) is simila to Eq. (1) of SHA, but to yield appopiate basis functions on the cap, the SHA integal degee n has to be eplaced by a SCHA non-integal degee n k (m) whee k is an intege index. The n k (m) ae detemined by the bounday conditions fo the associated Legende functions P m n k (m) (cos ϑ) at ϑ = ϑ 0 dpn m k (m) (cos ϑ 0) = 0 fo k m even, dϑ Pn m k (m) (cos ϑ 0) = 0 fo k m odd, i.e., fo a given m and ϑ 0, those Legende functions which fulfill Eq. (2) ae seached with inceasing n k (m) and ae indexed by k. Accodingly, a definition of the Legende functions is needed that does not ely on an intege degee n (Hobson, 1931; Haines, 1985b): P m n (cos ϑ) = Kn m sinm ϑ ( F m n; n + m + 1; 1 + m; 1 cos ϑ ) (4) 2 whee F(α; β; γ ; x) is the hypegeometic function and Kn m ae nomalisation factos (in geophysics usually Schmidt nomalisation, e.g., Chapman and Batels, 1940). Since n k (m) k, with the same numbe of coefficients, SCHA uses highe degee Legende functions than SHA, and theefoe obtains a bette spectal esolution. SCHA is suitable fo local to medium scale studies, with scale lengths in the ode of magnitude of 1000 km 1000 km. Anothe method fo field continuation on local scales when the cuvatue of the Eath s suface can be neglected will biefly be mentioned: Be (k x, k y ) the Fouie tansfom of a magnetic potential (x, y), with x and y denoting catesian coodinate axes, and k x and k y the coesponding wave numbes. Then, fo extenal souces (as we ae mainly concened with in this pape), the Fouie tansfomed potential on the gound at z = 0 elates to the Fouie tansfomed potential on anothe height z = z C as (k x, k y, z C ) = (k x, k y, 0) e k zz C (5) with k z = kx 2 + k2 y and z positive downwads (cf., e.g., Untiedt and Baumjohann, 1993). (3) Fo all of the methods discussed, a minimum wavelength that can be esolved in the analysis has to be chosen globally, i.e., fo the whole analysis aea. Fo SHA, this wavelength is λ min = (2π R E )/n max, fo SCHA it is λ min = (2π R E )/n k (m) max, and fo the Fouie tansfom method it becomes λ min = 2π/k z,max, with the subscipt max denotes the highest value of n, n k (m), ok z used in the analysis, espectively. As shown in the application example below, the duty to choose this esolution bounday globally can lead to poblems in the field continuation pocedue if the spectal content of the field to be analysed is highly vaying, o if the density of the measuement sites is. If the esolution bounday is adopted to the pat of the analysis aea with the highest spectal content of the field o the most dense measuements, the highe spheical hamonic o Fouie coefficients might become eatic due to lack of data in the emaining pats. If it is set lowe, the analysis might be unable to epoduce details of the field in the fome aea. In this pape, we pesent a method fo field continution that is not based on spectal decomposition as the ones above. It expands the measued gound magnetic field into a sum of the magnetic field effect of spheical elementay cuent systems (SECS) placed in the ionosphee. The centes of these elementay cuent systems (called poles hee) can be placed feely, such that thei locations ae most suitable with espect to the density of the measuement sites o to the type of the magnetic field distubance to be analysed. 2. Spheical Elementay Cuent Systems Two types of spheical elementay (sheet) cuent systems have been defined by Amm (1997), one ( J df,el ) being divegence-fee and the othe ( J cf,el ) cul-fee. Witten in a spheical coodinate system (,ϑ,ϕ ) with unit vectos (e, e ϑ, e ϕ ) that has its nothen pole in the cente (o the pole ) of the elementay system (compae the sketch in Fig. 1), thei definition is J df,el ( ) = I 0,df cot(ϑ /2)e 4π R ϕ (6) I and J cf,el ( ) = I 0,cf cot(ϑ /2)e 4π R ϑ (7) I whee is the adius of the ionosphee, assumed to be an infinitely thin laye at 100 km above the Eath s suface thoughout this pape, and I { 0, df } ae called the scaling factos of the elementay systems. The cul-fee elementay cf system (7) is associated with a field-aligned cuent (i.e., a divegence of J cf,el ) of magnitude I 0,cf at its pole, and unifom, oppositely diected FACs of magnitude - I 0,cf /4π R 2 I on the est of the ionosphee, so that the net FAC ove the whole ionosphee is zeo. This cul-fee elementay system is the same that Fukushima (1976) attibuted to Pedesen cuents in an unifomly conducting ionosphee. Howeve, as shown by Amm (1997), fo the following it is not necessay to conside how the cuent was actually poduced. By using Helmholtz s theoem, Amm (1997) showed that any ionospheic cuent density J can be uniquely constucted by a supeposition of (6) and (7), placing poles of elementay systems all ove the ionosphee. In fact, the elementay systems as defined above can be used to expand any

3 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION 433 Fig. 1. Sketch of spheical elementay cuent systems (SECS); Left side: cul-fee elementay system associated with a field-aligned cuent (FAC) at its pole at ϑ = 0, and oppositely diected small FACs of constant magnitude on the est of the sphee. This system does not poduce any magnetic effect below the ionosphee and is not needed to expand equivalent cuents; Right side: Divegence-fee elementay system. continuosly diffeentiable vecto field on a sphee. When we ae dealing with the continuation of the gound magnetic field distubance to the ionosphee, we need to have in mind that we cannot econstuct the total o actual cuent density J fom it. We only obtain the equivalent cuents J eq,ion which ae those hoizontal cuents flowing in the ionosphee that poduce the same magnetic effect below the ionosphee than the actual, thee-dimensional cuent system consisting of hoizontal cuents and FACs (e.g., Untiedt and Baumjohann, 1993). J eq,ion is divegence-fee because the cul-fee pat of the actual cuent system, like the cul-fee elementay cuent system (7), does not poduce any magnetic effect below the ionosphee, as has also been shown by Fukushima (1976). Hence, fo the pupose of ou pape, i.e., the upwad continuation of the gound magnetic field to the ionosphee and its epesentation in tems of J eq,ion, we only need the divegence-fee elementay system (6) to expand J eq,ion ( ) = Ionosph. [cul J( )] 4π cot( ϑ/2)e ϕ d 2 (8) whee ϑ and ϕ denote the coodinates of in the spheical coodinate system with its pole at, and e ϑ and e ϕ ae the unit vectos accoding to this coodinate system. The ϕ = 0 diection fo each coodinate system can be defined feely, but fixed. Fo (8), we made use of Stokes laws to yield K, 0 [cul J( )] d 2 = I 0,df ( ) (whee K is a cicula ionospheic aea with adius aound, and I 0,df ( ) denotes the scaling facto of the elementay cuent systems with pole at ). In case of a discete gid epesentation, the integal in (8) divides into a sum ove discete (gid) points. Assuming [cul J( )] = C G to be constant ove a gidpoint aea F G, the scaling facto of the elementay cuent system at each gidpoint G is appoximated to I 0,df ( G ) = C G F G. We biefly note that a simila set of elementay cuent systems fo the plana geomety has been given in Amm (1997), equation (4). Due to a mispint in that equation, 4π has to be eplaced by 2π in the denominato. The pocedues descibed in this pape can be used analogously in the plana geomety by using the divegence-fee one of these elementay systems instead of (6). 3. Gound Magnetic Field Distubance of Divegence-fee Spheical Elementay Cuent System As mentioned above, the magnetic field effect of the culfee elementay cuent system (7) vanishes below the ionosphee (cf. Fukushima, 1976). To calculate the gound magnetic field effect of J df,el (Eq. (6)), we deived its vecto potential A( ) below the ionosphee by expanding the distance between the souce cuent filaments in the ionosphee and the point whee A is evaluated in spheical hamonics. Using the geneating function fo the associated Legende functions, we can wite A in a closed-fom expession and finally obtain B = ot A. Details of this calculation ae given in the appendix. Fo a point with adius < and pole angle ϑ fom the pole of the elementay cuent system, the magnetic field effect of a divegence-fee elementay cuent system J df,el with scaling facto I 0, flowing at =,is B (,ϑ ) = µ 0I 0 4π cos ϑ + ( ) 1 2 (9)

4 434 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION Fig. 2. Gound magnetic effect of a divegence-fee elementay cuent system with a scaling facto A, in the spheical coodinate system with the pole of the elementay system as the noth pole, as a function of the gound distance fom this pole; B ϕ is zeo. and B ϑ (,ϑ ) = µ 0I 0 4π sin ϑ 1 cos ϑ 2 cos ϑ + ( ) + cos ϑ 2. (10) whee Z = Z 1,ϑ Z 1,ϕ Z 2,ϑ Z 2,ϕ. Z nobs,ϑ (12) B ϕ is zeo and thee is no ϕ dependence, as can be immediately seen fom the symmety. Figue 2 shows the esulting magnetic distubances B and B ϑ at = R E, fo an elementay cuent system with a scaling facto of I 0,df = A, as a function of the gound distance fom the pole of the elementay cuent system. As expected, B eaches a maximum (of exactly 10 nt fo = 100 km) below the pole and deceases apidly with inceasing distance fom it. B ϑ is zeo diectly below the pole, then shows a minimum at about 127 km distance fom it, and deceases slowly with futhe inceasing distance. The equations fo the magnetic effect of J df,el at > ae given in the appendix. 4. Matix Fomulation of Expansion of Gound Magnetic Distubance into Distubances of Elementay Cuent Systems With the pevious, the matix fomulation of the field continuation is staightfowad: Let us assume that the gound magnetic field distubance Z k has been measued at points k,obs = (ϑ k,obs,ϕ k,obs ), k = 1,...,n obs which ae usually iegulaly spaced. Poles of elementay cuent systems ae placed at points l,el = (ϑ l,el,ϕ l,el ), l = 1,...,n el, which may be but do not have to be located on a egula gid. We can then detemine the scaling factos of the elementay cuent systems that fit best to the gound magnetic field obsevation by solving T I = Z (11) is the vecto of obsevations, I = Z nobs,ϕ I 0,df,1 I 0,df,2. I 0,df,nel (13) is the vecto of the scaling factos fo the elementay cuent systems, and T 11,ϑ T 12,ϑ T 1nel,ϑ T 11,ϕ T 12,ϕ T 1nel,ϕ T 21,ϑ T 22,ϑ T 2nel,ϑ T = T 21,ϕ T 22,ϕ T. (14) 2nel,ϕ.. T nobs 1,ϕ T nobs 2,ϕ T nobs n el,ϕ Hee, the T { k,l, ϑ } denote the ϑ o ϕ component of the gound ϕ magnetic effect of an elementay cuent system with a scaling facto of 1 A and its pole at l at the obsevation point k, expessed in the same coodinate system as the obsevations (usually geogaphic). In pactice, they ae easily obtained by calculating the gound magnetic effect of the elementay system using the equations as given in the Appendix, and

5 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION 435 (a) (b) (c) Fig. 3. Model of a Cowling channel (details see text and Amm (1997)); (a) Actual ionospheic cuents and FACs (cossed cicles mak downwad, dotted ones upwad flowing FACs).; (b) Tue equivalent cuents; (c) By 90 degees clockwise otated gound magnetic distubance. conveting it fom the spheical coodinate system of the elementay system into that of the measuements. As mentioned above, the expansion of any given equivalent cuent systems into a supeposition of spheical elementay cuent systems as given in Eq. (6) is analytically unique. Since the elation between the elementay cuent systems and thei gound magnetic effect is bijective, ou expansion of the gound magnetic distubance in tems of the elementay cuent system distubances is also unique. The esulting scaling factos I ae identical with those of the expansion of J eq,ion in the discete fom of Eq. (8), thus allowing to calculate J eq,ion. 5. Matix Equation Solving Technique To solve Eq. (11) fo I, in pinciple any matix solving technique could be used. Howeve, in pactise ou poblem will most often be highly undedetemined, because the amount of data points n obs whee magnetometes have measued the gound magnetic distubance is typically much smalle than the desied numbe of elementay cuent sys-

6 436 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION tems n el to obtain a good epesentation of J eq,ion in the ionospheic plane. In this case, T will be badly conditioned, and the solution space of (11) will contain a nullspace. A ecommendable solving technique fo such a case is the singula value decomposition method (SVD, e.g., Pess et al., 1992): The matix T is decomposed into T = U w V T (15) whee U and V T ae othogonal matices, i.e. the columns of U and the ows of V T ae bases fo the n el -dimensional space of all possible combinations of the n el elementay systems. w is diagonal, and its diagonal elements w mm, m = 1,...,n el, ae called the singula values of T. If n el > n obs, those n el n obs singula values that ae connected with the basis of the nullspace of T in U and V T ae zeo. The decisive point in SVD is now that it allows to sepaate out othe badly conditioned pats of T as well, since they ae connected with nonzeo, but small singula values. This is done by setting all w mm with w mm ε Max.{w mm } to zeo. Typical values fo ε ange between 0.01 and 0.1. Afte that, (11) can be solved by I = V (diag( w mm ))U T Z (16) whee { wmm 1 fo w mm 0, w mm = (17) 0 fo w mm = 0. Fo an undedetemined system of equations, this pocedue will pick the solution with minimum I 2 fom the total solution space (Pess et al., 1992). With the solution fo I, the ionospheic equivalent cuents J eq,ion can be obtained at any ionospheic point using (8) and the emaks fo the discete fomulation afte that equation. The lage ε is chosen, the smoothe the solution fo J eq,ion will be in geneal. Howeve, it should be noted that the choice of ε does not invoke a spectal esolution bounday, but a sepaation with espect to well and badly conditioned pats of ou linea system of equations. 6. Application Example Finally, we test the SECS method of upwad field continuation fom the gound to the ionosphee in a model example of a Cowling channel and compae its esults to those of the spheical cap hamonic analysis (SCHA). We choose SCHA fo the compaison since on the scales of typically 1000 km 1000 km that ae most inteesting and applicable fo upwad continuation in ionospheic-magnetospheic physics (e.g., Richmond and Baumjohann, 1983; Walke et al., 1997; Amm, 1998), it is pobably the most advanced existing technique (compae, e.g., Haines, 1990). A Cowling channel is a confined ionospheic aea with an enhanced conductance, leading to hoizontal ionospheic cuents with magnitudes that ae enhanced compaed to the backgound. The channel usually has a lage extent in the diection of the cuent flow (called channel diection ) than pependicula to it. At the edges of the Cowling channel in the channel diection, field-aligned cuents (FACs) ae pesent to feed o divege the enhanced hoizontal cuents, espectively (e.g., Bostöm, 1974, Baumjohann et al., 1981). We will not discuss the details of this model hee, since we (a) (b) Fig. 4. Spheical cap hamonic analysis (SCHA) esults of upwad continuation; (a) With K e = 10; (b) Inceasing to K e = 12 leads to eatic vectos. use the same model that has been descibed in detail by Amm (1997), and since fo the pupose of this pape only the total cuent flow as sketched in Fig. 3(a) is elevant. In ou model, the channel is limited to the aea between 66 and 70 degees of latitude and 20 to 22 degees of longitude. The channel cuents ae flowing southwad, being fed at the nothen and diveged at the southen end of the channel via FACs. Note that the longitude on which we place ou model has no special physical meaning. It is meely fo convenience adjusted to

7 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION 437 (a) (b) (c) Fig. 5. Results of the spheical elementay cuent system (SECS) method; (a) with poles of elementay systems on the gid points shown only; (b) Refined gid in the aea of loops of the by 90 degees otated gound magnetic field; (c) Result with poles of elementay systems on the efined gid. Nothen Fennoscandia. The ionospheic equivalent cuents J eq,ion that cause the same magnetic effect below the ionosphee as the theedimensional cuents of Fig. 3(a) ae shown in Fig. 3(b). As can be seen, besides the dominating southwad cuents in the channel aea, the effect of the FACs esults into two equivalent cuent loops at both flanks of the Cowling channel pependicula to the channel diection. The same stuctue, somewhat smoothed out due to the distance fom the souce, can still be seen in the by 90 degees clockwise otated gound magnetic distubance as calculated with Biot-Savat s law fom ou model cuents (Fig. 3(c)). This gound magnetic distubance is taken as the input fo the field continuation methods, and the SECS method as well as SCHA will be tested in how well they can epoduce the model equivalent cuent patten as pesented in Fig. 3(b).

8 438 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION Fig. 6. Longitudinal pofile of the noth component of the ionospheic equivalent cuents; Solid line: tue pofile (see Fig. 3(b)); Dotted line: SCHA upwad continuation esult with K e = 10 (see Fig. 4(a)); Boken line: SECS method upwad continuation esult with poles of elementay systems on efined gid (see Fig. 5(c)). The SCHA algoithm used is the one of Haines (1988), with slight modifications by the authos. The equations fo the equivalent cuents in tems of the SCHA expansion can be found in Haines and Tota (1994). In Fig. 4(a), the SCHA esult is shown fo a spheical cap half angle of ϑ 0 = 12, and K e = 10 in Eq. (2) (intenal coefficients ae ielevant in ou model case). Since the SCHA fit is subject to eos at the bounday of the spheical cap which may continue inwads duing the upwad continuation, it is necessay to select a somewhat lage ϑ 0 than would be equied solely by the aea of data coveage (e.g., Tota and De Santis, 1996). Howeve, inceasing ϑ 0 to lage values than the one selected hee does not anymoe impove the esults. Clealy, SCHA is able to epoduce the geneal patten of the tue ionospheic equivalent cuents well. Howeve, if the esult is examined in moe detail, it tuns out that the cuents in the cente of the channel ae undeestimated by about 20 ma/m (i.e., by about 20%), wheeas the equivalent etun flow at the flanks pependicula to the channel diection is lage than in the model (compae Fig. 3(b)). The smoothing of the eal J eq,ion distibution that is pesent in the SCHA esult is most obvious fom the compaison of the noth component of J eq,ion on a longitudinal pofile at 68 degees of latitude (Fig. 6): The SCHA esult (dotted line) is not able to follow the elatively naow peak of the model ( tue ) cuents (solid line) in the cente of the channel. The natual emedy of this is to incease K e fo the SCHA expansion. As the esult with K e = 12 shows (Fig. 4(b)), this leads to a bette epesentation of the cuents in the cente of the channel, but now a population of vectos at the flanks become eatic (the same happens with K e = 11, but we show the K e = 12 case fo cleae illustation). The same effect is obseved if ϑ 0 is deceased. The eason fo this behaviou is that the spectal components that ae needed to epesent the peak in the cente of the channel, ae ill-detemined at its flanks and lead to a swinging effect thee. The esults of the SECS method is shown in Fig. 5. The value of ε used is which was obtained in an optimisation pocess as the K e fo SCHA. Fist, we allow poles of spheical elementay systems on the gid points of ou model only. The upwad continued J eq,ion essentially shows the same poblems as dicussed above fo SCHA in this case, as can be seen in Fig. 5(a). Howeve, fom the geomety of the gound magnetic distubance (see Fig. 3(c)), it is obvious that the cul of the equivalent cuent system and theefoe also the magnitude of the scaling factos of ou elementay cuent systems peaks nea the flanks of the channel pependicula to the channel diection. Hence, in these aeas we efine the gid fo the poles of the elementay cuent systems as shown in Fig. 5(b). While no geneal optimum ule fo such a efinement exists, fom ou expeience a good choice is to decease the gid spacing fo the poles of the elementay cuent systems in those aeas to about one thid of the aveage spacing of the input data, i.e., hee of the gid spacing of ou model. A efinement in aeas whee the cul of the measued equivalent cuents is small will not impove the esults. The esult of the SECS method using the efined gid is shown in Fig. 5(c). Although the impovement is not easily visible on the fist view fom this vecto plot, it becomes obvious by a compaison of the J eq,ion,noth

9 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION 439 pofiles at 68 degees of latitude (Fig. 6). The solution based on elementay cuent (boken line) systems follows the tue pofile much close than the SCHA esult (dotted line). Especially, it epoduces the minimum in the cente of the channel nealy exactly, and follows the naow shape of the tue pofile thee closely. Such on the fist view small diffeences may become impotant if the upwad continued equivalent cuents ae used quantitatively in combination with, e.g., the ionospheic electic field to detemine othe ionospheic electodynamic paametes (e.g., Inheste et al., 1992; Amm, 1998). This application example shows that the SECS method is able to obtain at least as good, and with an appopiate gid efinement even bette esults in upwad continuation than the aleady quite advanced SCHA method. In paticula, a efinement of the gid does not lead to numeical poblems as does the incease of spectal content by inceasing K e in SCHA. Additionally, the SECS method can even povide some infomation of equivalent cuent souces outside the aea of measuements by placing poles of elementay cuents thee and fitting thei magnetic effect on the measuements. 7. Conclusions and Outlook We have intoduced a new method fo field continuation, woked out in this pape fo the pupose of upwad continuation of the gound magnetic field distubance to the ionosphee, and its epesentation as ionospheic equivalent cuents. In contast to existing methods that ely on a spectal decomposition of the magnetic potential, this method is based on spheical elementay cuent systems (SECS) as intoduced in Amm (1997). Thus, no fixed bounday fo the minimum wavelength to be esolved has to be given ove the total analysis aea like in the existing methods by teminating a seies, but poles of elementay systems can feely be placed as they ae most suitable with espect to the density of mesuements o the type of distubance to be analysed. The value of this advantage was demonstated in the model example of a Cowling channel whee the new method, with appopiately placed elementay systems, gained a bette econstuction of the tue ionospheic equivalent cuents than the spheical cap hamonic analysis (SCHA). We have selected the Cowling channel model with its elatively simple geomety fo this pape in ode to illustate the effects of the diffeent upwad continuation methods most clealy. The authos have tested the SECS method also fo cuent systems of moe complex geomety, and in all cases gained upwad continuation esults of simila quality as shown in this pape. The method pesented in this pape can easily be adapted to othe field continuation poblems than fom the gound to the ionosphee, just by placing simila spheical elementay cuents like in (6) on the sphee(s) whee the cuents that cause the field ae assumed to flow and calculating the magnetic field effect of the elementay cuent system with scaling facto of 1 A on the sphee(s) whee the measuements wee taken (of couse, in Eq. (6) has to be eplaced by the espective adius of the sphee(s)). Likewise, sepaation of the pats of a field caused by extenal and intenal souces can be caied out with the SECS method by placing elementay cuent systems on two sphees, one epesenting the extenal, the othe the intenal pat. Typically, the oute shell would be the ionosphee, wheeas the inne one might be placed with espect to a pefect conducto that is used to appoximately eplace the eal conductivity stuctue of the Eath (e.g., Baumjohann et al., 1981; Gustafsson et al., 1981, Viljanen et al., 1995). It should, howeve, be noted that the intoduction of moe elementay cuent systems fo a given amount of data will incease the undedetemination of the system of linea equations in (11). Futhemoe, to pefom the field sepaation, measuements of B and coesponding tansfe functions have to be included into Eqs. (12) and (14), espectively. Finally, all of the upwad continuation methods mentioned in this pape can also be used to deive models of the measued quantities, and theeby to inte- o extapolate J eq,ion, o the measued gound magnetic field distubance, since the detemination of the SECS, SCHA, SHA, o Fouie coefficients allows in pinciple to calculate these quantities at any place in the espective shells. Not supisingly, the quality of the esult will depend on the data coveage nea the point whee the quantities ae to be intepolated. Also fo this pupose, the SECS method has shown to be moe obust than the spectal methods. Acknowledgments. The authos like to thank K. H. Glaßmeie (Baunschweig) fo valuable discussions, and P. Janhunen (Helsinki) fo his comments on the manuscipt. The wok of O. A. was suppoted by a DAAD-fellowship HSP III, financed by the Geman Ministey fo Reseach and Technology. The Edito thanks J. M. Tota and anothe efeee fo thei assistance in evaluating this pape. Appendix In this appendix, we outline the calculation of the magnetic field distubance of a divegence-fee elementay cuent system J df,el flowing in the ionosphee at adius. All calculations ae done in the coodinate system with the pole of the elementay system at ϑ = 0, i.e., at the noth pole. Fo simplicity, we omit the quotes used in the main text to sepaate this coodinate system fom the geogaphical one hee. We fist show the deivation of Eqs. (9) and (10) fo <, and then give simila equations fo >. Fist we inset Eq. (6) into the expession of the vecto potential A of the desied magnetic field distubance to obtain A( ) = µ 0I 0 16π 2 d 3 δ( ) cot ϑ /2 e ϕ. (A.1) By expanding line 1 into spheical hamonics and using thei othogonality elations (e.g., Afken, 1985) we aive at A(,ϑ)= µ 0I 0 4π n=1 ( ) 1 n Pn 1 n(n + 1) R (cos ϑ)e ϕ. I (A.2) With the definitions λ := / x := cos ϑ, and and consequently f (λ) := n=1 f (λ) = 1 λ 1 n(n + 1) λn+1 P 1 n (x) λ n Pn 1 (x) n=1 (A.3) (A.4)

10 440 O. AMM AND A. VILJANEN: DISTURBANCE MAGNETIC FIELD CONTINUATION we can use the geneating function fo the associated Legende polynomials Pn m (x) fo the special case m = 1 (e.g., Afken, 1985) λ n Pn+1 1 (x) = 1 λ n Pn 1 λ (x) n=0 n=1 1 x 2 = (A.5) (1 2xλ + λ 2 ) 3/2 to obtain a closed-fom expession fo A afte solving the diffeential equation fo f (λ): A(,ϑ) = µ 0I 0 4π sin ϑ ( ) 2 cos ϑ cos ϑ 1 e R ϕ. I (A.6) Fom (A.6), we can diectly deive (9) and (10) by using B = ot A. The equations fo the magnetic effect of J df,el fo > can be deived analogously to the above. The esults ae and B (,ϑ)= µ 0I 0 4π 2 B ϑ (,ϑ) = µ 0I 0 4π cos ϑ 1 sin ϑ cos ϑ 2 2 cos ϑ + R 2 I + ( ) (A.7) (A.8) Of couse, the ionospheic adius can be eplaced in (A.7) and (A.8) as well as in (9) and (10) by any othe adius whee the elementay cuent system is assumed to flow. Refeences Amm, O., Ionospheic elementay cuent systems in spheical coodinates and thei application, J. Geomag. Geoelect., 49, , Amm, O., Method of chaacteistics in spheical geomety applied to a Haang discontinuity situation, Ann. Geophys., 16, , Afken, G., Mathematical Methods fo Physicists, Academic Pess, pp. 985, San Diego, U.S.A., Baumjohann, W., R. J. Pellinen, H. J. Opgenooth, and E. Nielsen, Joint twodimensional obsevations of gound magnetic and ionospheic electic fields associated with auoal zone cuents: Cuent system associated with local auoal beak-ups, Planet. Space Sci., 29, , Bostöm, R., Ionosphee-magnetosphee coupling, in Magnetospheic Physics, edited by B. M. McComack, p. 45, D. Reidel, Nowell, Mass., Chapman, S. and J. Batels, Geomagnetism, vol. II, pp. 1049, Oxfod Univesity Pess, New Yok, De Santis, A., D. J. Keidge, and D. R. Baaclough, A spheical cap hamonic model of the custal magnetic anomaly field in Euope obseved by MAGSAT, in Geomagnetism and Palaeomagnetism, edited by F. J. Lowes et al., pp. 1 17, De Santis, A., C. Falcone, and J. M. Tota, SHA vs. SCHA fo modelling secula vaiation in a small egion such as Italy, J. Geomag. Geoelect., 49, , Fukushima, N., Genealized theoem fo no gound magnetic effect of vetical cuents connected with Pedesen cuents in the unifom-conductivity ionosphee, Rep. Ionos. Space Res. Japan., 30, 35 40, Gauß, C. F., Edmagnetismus und Edmagnetomete, in: Gauß, C. F., Weke, hsg. von de Königlichen Gesellschaft de Wissenschaften, Göttingen, , Gustafsson, G., W. Baumjohann, and I. Ivesen, Multi-method obsevations and modeling of the thee-dimensional cuents associates with a vey stong Ps 6 event, J. Geophys., 49, , Haines, G. V., Spheical cap hamonic analysis of geomagnetic secula vaiation ove Canada , J. Geophys. Res., 90, , 1985a. Haines, G. V., Spheical cap hamonic analysis, J. Geophys. Res., 90, , 1985b. Haines, G. V., Compute pogams fo spheical cap hamonic analysis of potential and geneal fields, Comput. Geosci., 14, , Haines, G. V., Regional magnetic field modelling: a eview, J. Geomag. Geoelect., 42, , Haines, G. V. and J. M. Tota, Detemination of equivalent cuent souces fom spheical cap hamonic models of geomagnetic field vaiations, Geophys. J. Int., 118, , Hobson, E. W., The Theoy of Spheical and Ellipsoidal Hamonics, pp. 500, Cambidge Univesity Pess, New Yok, Inheste, B., J. Untiedt, M. Segatz, and M. Küschne, Diect detemination of the local ionospheic Hall conductance distibution fom twodimensional electic and magnetic field data, J. Geophys. Res., 97, , Pess, W. H., B. P. Flanney, S. A. Teukolsky, and W. T. Vetteling, Numeical Recipes, 2nd ed., pp. 973, Cambidge Univesity Pess, Cambidge, Richmond, A. D. and W. Baumjohann, Thee-dimensional analysis of magnetomete aay data, J. Geophys., 54, , Tota, J. M. and A. De Santis, On the deivation of the Eath s conductivity stuctue by means of spheical cap hamonic analysis, Geophys. J. Int., 127, , Tota, J. M., A. Gacia, J. J. Cuto, and A. De Santis, New epesentation of geomagnetic secula vaiation ove esticted egions by means of spheical cap hamonic analysis: application to the case of Spain, Phys. Eath Planet. Inte., 74, , Untiedt, J. and W. Baumjohann, Studies of pola cuent systems using the IMS Scandinavian magnetomete aay, Space Sci. Rev., 63, , Viljanen, A., K. Kauistie, and K. Pajunpää, On induction effects at EIS- CAT and IMAGE magnetomete stations, Geophys. J. Int., 121, , Walke, J. K., V. Y. Semenov, and T. L. Hansen, Synoptic models of high latitude magnetic activity and equivalent ionospheic and induced cuents, J. Atmos. Te. Phys., 59, , O. Amm ( Olaf.Amm@fmi.fi) and A. Viljanen