A numerical method to compute Euler potentials for non dipolar magnetic fields

. The magnetospheric magnetic ﬁeld may be conveniently described by two scalar functions ( a , b ), known as the Euler potentials. They are not uniquely deﬁned, and they may be di(cid:129)cult to derive for conﬁg-uration more complex than a simple dipole. We propose here a simple numerical method to compute one possible pair ( a , b ). In magnetospheric regions of closed ﬁeld lines, a can be chosen as a function of the tube volume of unit magnetic ﬂux. The method can be applied to a wide class of magnetic ﬁelds which describe the magneto-spheric domain of closed ﬁeld lines and the conjugated ionosphere. Here, it is used with the T87 Tsyganenko model. The results coincide with the dipolar potentials at close distances from the Earth. At larger distances, they display an increasing distortion with the radial distance (or the invariant latitude in the ionosphere) and the magnetic activity. In the magnetosphere, the contours of a and b are stretched towards the nightside. In the ionosphere, they also extend towards the nightside and present major distortions in a narrow ring at the polar cap boundary, which maps distant boundary layers in the magnetosphere.


Introduction
The large-scale convection of the magnetospheric plasma is strongly related to the topology of the magnetic ®eld B and the coupling with the ionosphere.The curvature and the gradients of the magnetic ®eld control the drift of the magnetospheric plasma.Particle ex-change, current circulation and electric ®eld transmission between the magnetosphere and the ionosphere contribute to couple the magnetospheric motion strongly to the ionospheric motion.A convenient way to solve the magnetospheric transport is the use of the Euler potentials (a, b) as system coordinates.They are de®ned as:

B raÂrb 1
The interest of this representation has been extensively discussed by Stern (1967Stern ( , 1970Stern ( , 1976Stern ( , 1994b)).Brie¯y, it includes directly the expression of the magnetic ®eld lines: a constant and b constant.Two points belong to the same ®eld line only if they have the same Euler potentials, at least for a ®eld line topology suciently regular as is the case for the planetary ®elds (Stern, 1994b).This property reveals itself to be very powerful as a means of studying the transport of the magnetospheric ¯ux tubes, and their coupling with the ionospheric plasma: the formulation of the basic transport equations is much simpli®ed by choosing the Euler potentials as a reference frame (BoÈ strom, 1975;Peymirat and Fontaine, 1994b;Stern, 1994b and references therein).This representation has been widely used in the inner magnetosphere where the magnetic ®eld is essentially dipolar and simple analytical expressions can be derived for the Euler potentials (Stern, 1976 and references therein).This method can be applied to outer regions of the magnetosphere, or to dynamical situations like substorms, where the magnetic ®eld departs signi®cantly from the dipole (see Hilmer and Voigt, 1995;Tsyganenko, 1996;Ostapenko and Maltsev, 1997;Tsyganenko, 1997 for the most recent magnetic ®eld models).
The derivation of the Euler potentials is very easy for a dipolar magnetic ®eld but is rather dicult for any type of magnetic ®eld.Equation ( 1) is a highly non linear dierential equation, and from a practical point of view, it restricts the use of Euler potentials.For current-free ®elds, they can be exactly derived for a limited class of magnetic ®elds (Stern, 1994a) or approximately obtained with perturbations methods (Stern, 1967).Stern (1987) showed that the stretching of magnetic ®eld lines in the tail produces realistic models but the Euler potentials are not in general explicitly known.Alternatively, they can be numerically computed.Cheng (1995 and references therein) solved the Grad-Shafranov equation relating the Euler potentials to the pressure of the magnetospheric plasma.Euler potentials were computed from a given pressure pro®le.Tsyganenko and Stern (1996) and Khurana (1997) modelled the Euler potentials as theoretical functions ®tted to databases, to reproduce respectively the distribution of the ®eld aligned currents and the jovian magnetic ®eld.More recently, Ho et al. (1997) developed a general numerical method to compute the Euler potentials associated with the Neptune's magnetic ®eld.This method uses the relationship between Euler potentials and the magnetic ¯ux per unit area.The Euler potentials are similar to those of a dipole in the regions where the magnetic ®eld is dipole-like.
The Euler potentials represent a powerful tool to investigate the magnetospheric plasma transport.The diculty of deriving them for any magnetic con®guration has generally reduced their use to that of the dipolar approximation.Our purpose is to compute the Euler potentials for non-dipolar magnetic con®gurations.Following Ho et al. (1997), we propose a simple numerical method which applies to regions of closed magnetic ®eld lines.Section 2 brie¯y recalls some mathematical properties of the Euler potentials and Sect. 3 describes the numerical method used.It is then tested in Sect. 4 with the magnetic ®eld model of Tsyganenko (1987).

A short review of mathematical properties of Euler potentials
The mathematical properties of these Euler potentials and their applications to physical problems have been reviewed in detail by Stern (1970Stern ( , 1976)).We will use two of their interesting mathematical properties: they are non unique and their tridimensional distribution can be derived from knowledge of how they map on a surface.

Non uniqueness of Euler potentials
Assume that one set of Euler potentials (a 1 , b 1 ) has been derived for a particular magnetic ®eld B: Consider two functions a 2 (a 1 , b 1 ) and b 2 (a 1 , b 1 ).Then and a 2 and b 2 will be a new set of Euler potentials provided that Therefore, an in®nite number of Euler potentials can be derived once one has been computed (Stern, 1970).

Computation of Euler potentials from a surface
If the topology of the magnetic ®eld B is regular enough, it is not necessary to solve Eq. ( 2) in three dimensions due to the non-divergence of B (Stern, 1970(Stern, , 1976)).We assume that Eq. ( 2) has been solved in the vicinity of one point X We consider the ®eld line l passing through this point, and label it (a 1 , b 1 ).As a 1 and b 1 are constant along l : B Ára 1 0 5 which implies a function p such that B p ra 1 Ârb 1 7 exists.From Eq. ( 2), its value is unity at point P.Then, a 1 and b 1 , which are two independent functions, are completed by a third one, c 1 , to form a set of coordinates along l generally not orthogonal.c 1 can be, for instance, the linear distance along l and allows us to label every point along l .With p considered as a function of a 1 , b 1 and c As a 1 , b 1 and c 1 are independent, the term rc 1 Á ra 1 Â rb 1 does not cancel, and dp dc 1 vanishes.
p does not depend on c 1 and is constant along l where its value is unity as for point X From Eq. ( 7) B ra 1 Ârb 1 10 everywhere along l .Therefore, it is sucient to compute the Euler potentials at every point of a surface and the mapping along magnetic ®eld lines will complete the 3D-distribution (Stern, 1970(Stern, , 1976)).
3 Numerical computation method Stern (1970) described a general way to compute the Euler potentials on a surface in generalized orthogonal or non-orthogonal coordinates.We use this formalism to derive a ®rst set of Euler potentials (a 1 , b 1 ), and from it a second one (a 2 , b 2 ) which is more convenient to use.We then describe the method to compute Euler potentials in regions of closed magnetic ®eld lines and discuss its possible extension to open magnetic ®eld lines.

Derivation of a ®rst set of Euler potentials
One solves Eq. (2) on a plane surface crossed only once by the magnetic ®eld lines.For instance, can be the equatorial plane of the magnetosphere or any other plane of physical interest.Such a plane avoids computing multiple-valued Euler potentials (Stern, 1970).
De®ne a spherical (r, h, u) system relative to , with r, h and u respectively the radius, the colatitude and the azimuth of the point considered.The surface is de®ned by h p 2 .u 0 corresponds to midnight, u p to noon, u b 0 in the dawn sector and u `0 in the dusk sector.
Equation (2) then gives a non linear system with 3 equations and 6 unknowns, the partial derivatives of a 1 and b 1 with respect to r, h and u.It can be reduced to a 3-3 non-linear system if the expression of one Euler potential is given.For example, it can be convenient to choose a 1 as a function of the volume 5 of a ¯ux tube of unit ¯ux, a 1 = a 1 (5) with where s is the distance along the closed magnetic ®eld line and f the module of B. The integration is carried on between the two ionospheric ends of the ®eld line.This particular choice is made in reference to BoÈ strom (1975), Peymirat and Fontaine (1994b) and Stern (1994b) who showed that the derivation of the magnetospheric plasma transport equations is simpli®ed if a 1 is function of 5 only.Ho et al. (1997)  with h pa2 and hence dh = 0. Along a curve of constant value of a 1 on , da 1 da 1 dr dr da 1 du du 0 13 holds and Eq. ( 12) reduces to db 1 The term in bracket is simpli®ed from Eq. ( 2) projected on the h axis where b 1 =0 for u=0 and the integration carried along a curve of constant a 1 .Equation ( 16) would be equivalent to the relationship used by Ho et al. (1997) if the unit area were the elementary area in the (r, h, u) space.a 1 ( 5) is de®ned similarly to the dipole approximation as where f 0 and e are respectively the equatorial magnetic ®eld on the Earth and the radius of the Earth.

Derivation of a second set of Euler potentials with gauge constraints
In the ideal case of the dipolar axisymmetrical magnetic ®eld, symmetrical orthogonal Euler potentials a and b can be derived (Stern, 1976).Symmetrical Euler potentials for symmetrical magnetic ®elds allow an easier interpretation of the physical properties of the studied system.Orthogonal Euler potentials, i.e. ra Á rb 0, simplify the computations when they are used as coordinates.However, it is rarely possible to ®nd orthogonal Euler potentials (Stern, 1994b) and we will limit our computation to Euler potentials complying with the symmetries of the magnetic ®eld when they exist.
On the surface , a 1 de®nes a family of curves where a 1 is constant.These curves cross the tail of the magnetosphere and encircle the Earth except at considerable distances in the noon sector where they are not de®ned due to the opening of the ®eld lines.The third spherical coordinate u varies along these curves.We note u max a 1 its extremum value and b 1Ymax a 1 the corresponding value of b 1 .In the case of a dipole, where the contours of a 1 encircle the Earth, u max a 1 p for every a 1 and b 1Ymax a 1 e u max a 1 e p.In reference to the dipole, we take the following gauge constraint for the new set of Euler potentials (a 2 , b 2 ) In the case of a symmetrical magnetic ®eld, Eqs. ( 19), ( 20) and ( 21) lead to symmetrical Euler potentials.If the magnetic ®eld is not symmetrical, Euler potentials computed with Eqs. ( 19), ( 20) and ( 21) are not symmetrical but are such that b 2 0 at midnight, and b 2 p e at noon whenever the contour of a 2 is closed.
The integration of Eq. ( 4) along a curve of constant a 1 with Eq. ( 21) leads to such that from Eqs. ( 19) and ( 20) from which one gets where a 2Ymin a 1Ymin is a constant given as in Eq. ( 18) as with 5 min , the minimal value of 5.

Computation method for closed magnetic ®eld lines
The computation of the Euler potentials proceeds in two steps.One ®rst calculates the volume 5 from which a 1 and b 1 are derived from Eqs. ( 17) and ( 18).Then, one calculates the second set of Euler potentials a 2 and b 2 from Eqs. ( 24), ( 25) and ( 26).The knowledge of the magnetic ®eld comes into play in Eqs. ( 17) and ( 18).The derivation of a 1 implies a full tridimensional model to obtain the volume 5, while the computation of b 1 makes only use of a bidimensional distribution of the f h component on a surface .This method also applies similarly for two dimensional models of the magnetic ®eld.In that case, the computation of the volume 5 is not necessary.a 1 is simply a given but single valued function that one arbitrarily chooses.b 1 is then derived from the same formula (17).In that way the method can directly be applied to a data set of magnetic ®eld measurements.But the computation of the tridimensional distribution will then require ®eld line tracing, which is possible if the quality of the data is good enough.

Extension of the method to open magnetic ®eld lines
The method developed therebefore applies to regions of closed magnetic ®eld lines.It can be extended to open magnetic ®eld lines in the following way.Equation ( 2) is solved on a surface which is no more a plane but a sphere around the planet.The sphere must be suciently close to the planet such that any ®eld line crosses the sphere at least once.Then one proceeds as in Sects.(3.1) and (3.2) and gets where f r is the r component of B, and r the radius of the sphere.
The main dierence and diculty is the calculation of a 1 .a 1 can be any single valued function except in regions of closed magnetic ®eld lines where the surface is crossed twice because a 1 must take the same value along the same ®eld line.In regions of closed magnetic ®eld lines, one must ®nd on the surface the two conjugate hemispheres, which assumes that ®eld line tracing is possible.On one hemisphere a 1 can be any single valued function with the same value for the conjugate point of the other hemisphere.a 1 must also be continuous and derivable along the equatorial boundary of the polar cap region of each hemisphere to avoid any non physical discontinuity.
4 Application of the Euler potentials to the Tsyganenko model (Tsyganenko, 1987) The modelling of the magnetospheric ®eld has developed considerably.The most recent models were proposed by Hilmer and Voigt (1995), Skone et al. (1995), Tsyganenko (1996), Ostapenko and Maltsev (1997), Tsyganenko (1997).The description of the magnetospheric currents diers from one model to the other.The input parameters are as dierent as the hst index, the up index, the ei index, the interplanetary magnetic ®eld, the solar wind pressure, the magnetopause stando distance, the midnight equatorward boundary of the diuse aurora, the amount of stretching of the ®eld lines and the orientation of the merging lines.Particular forms and versions (Donovan, 1993;Peredo et al., 1993;Tsyganenko, 1987Tsyganenko, , 1989Tsyganenko, , 1995) ) have been built to improve the consistency with the observations or for speci®c purposes.For example, Peredo et al. (1993) modi®ed existing forms of the Tsyganenko (1987) and Tsyganenko (1989) models to better reproduce the observations in the near-tail magnetosphere; Lui et al. (1994) combined the models of Tsyganenko (1987) and Tsyganenko (1989) to model the quiet time nightside magnetosphere; Pulkkinen et al. (1994) modi®ed the model of Tsyganenko (1989) to study the recovery phase of a substorm; Skone et al. (1995) improved the model of Donovan (1993) to compute the quiet time magnetic ®eld at geostationary orbits; Kullen and Blomberg (1996) added the eect of the IMF in the model of Tsyganenko (1989).It is not our intention to describe in detail the numerous existing models, and the interested reader is referred, for example, to the review by Stern (1994b).We do not intend to test our general method to all the existing models of the magnetic ®eld.For simplicity, we tested it on the model of Tsyganenko (1987) because it is simple, has been widely used in numerous developments, and contains the main features of the other models: a stretched tail, a dayside compression, open ®eld lines at high latitudes and a dependence on the magnetic activity.
The short version of the external ®eld valid up to 30 e , which is more accurate than the long version (Tsyganenko, 1987), is considered.The main ®eld is the dipole ®eld and the geodipole tilt angle is set to zero.The computations are performed between 2 e where the ®eld is very dipolar, and the region where the ®eld lines open r `30 e .We use a spatial grid in the equatorial plane of the magnetosphere, with a step of 0.01 e for radial distances r and 2pa80 for azimuthal angles u.The small radial step is chosen in order to compute with a good accuracy the radial derivatives of the unit ¯ux tube volume 5 (Eq.17).
Figure 1a, b illustrates in the GSM coordinates the ®eld lines of the Tsyganenko (1987) model in the noonmidnight meridian plane for 3À `up `3 (top) and 5À `up (bottom) corresponding respectively to moderate and disturbed magnetic activity.The bold lines at polar latitudes are connected to the poles; at lower latitudes, they represent the external boundary of the domain, i.e. the last closed ®eld lines that we consider to compute the Euler potentials.The ®eld lines become a little more compressed in the noon sector and more stretched in the midnight sector with the magnetic activity.
Figure 1c, d shows the equatorial conjugates (ZGSM = 0) of points of the ionosphere with invariant latitudes between 0 and 90 with a 0.1 step for two geomagnetic activity levels (3À `up `3 ; 5À `up).Close to the Earth, the ®eld lines are closed and map to the equatorial plane of the magnetosphere in a regular and continuous way such that the equatorial plane is full of points.Far away from the Earth, the ®eld lines open, corresponding in our de®nition to radial distances larger than 30 e , and the equatorial plane starts to empty.An interesting feature is the ®sh shape of Fig. 1d, showing the in¯uence of the magnetic activity on the ®eld lines close to the polar cap.
We use the equatorial plane of the magnetosphere to compute the Euler potentials.The results are ®rst presented as equivalent McIlwain parameters, then in the equatorial plane of the magnetosphere and ®nally in the ionosphere.

Equivalent L McIlwain parameter
The L McIlwain parameter corresponds approximately to the largest geocentric distance, expressed in earth radii, attained by a ®eld line (Stern, 1976).In the case of the dipole, v is obtained in the equatorial plane of the magnetosphere where the following relationship holds between v and a v a Àf 0 e a 28 As suggested by Stern (1976), we generalize this relation to a non dipolar magnetic ®eld.This parameter, called equivalent McIlwain parameter, allows an interpretation from a comparison to the dipole.
Figure 2 shows v as a function of XGSM in the noon-midnight meridian plane for the magnetic ®eld models of Stern (1967) and of Tsyganenko (1987).The dotted and dotted-dashed lines correspond to the magnetic ®eld model built by Stern (1967) and completely described by analytical Euler potentials.He used a perturbation method to derive them from the magnetic ®eld model of Mead (1964).The dotted line illustrates v given by Stern (1967), the dotted-dashed line the normalized v that we computed from Eqs. (24± 26) and the volume 5 associated to his magnetic ®eld model, and the bold line the dipole.Indeed, the computation of the label v of magnetic ®eld lines depends on the various choices made for the Euler potentials, and which may have no relationship between them.We note that close to the Earth, below 5 Re, the curves are identical and coincide with the dipole.When the distance increases, the two curves depart from the dipole v becoming larger than the dipole one.The two curves behave approximately similarly in the midnight sector, but dier more signi®cantly in the dayside near the magnetopause.As they represent the same magnetic ®eld model, this dierence illustrates the only eect on mapping on dierent Euler potentials.In general, it is not the signature of physical processes, such as the compression or extension of magnetic ®eld lines.
The plain and dashed lines of Fig. 2 illustrate the normalized v parameter that we computed for the Tsyganenko model from Eqs. (24±26) for moderate and disturbed magnetic conditions.For both activity levels, in the noon and midnight sectors, the equivalent v is found to be smaller than the dipole and than the previous curves computed from Stern's model (Stern, 1967).Again, this result cannot be directly related to any physical processes.However, one particular feature allows us to go a little further in the comparison of results for both activity levels from Tsyganenko's model (Tsyganenko, 1987): the equivalent parameter v maps down to the same latitude in the ionosphere.This means that the ionospheric footprint of a given ®eld line labelled by a given equivalent v does not depend on the activity.The main eect of the activity can be seen in Fig. 2 in the nightside: it increases the geocentric distance of the ®eld line in the equatorial plane, which can be interpreted this time as the signature of the tail-like extension with activity.In the dayside, the curves are very close, which tends to indicate that the activity level has only a weak in¯uence on the dayside magnetic ®eld, as illustrated in Fig. 1a, b.

Equatorial distribution of the Euler potentials
The distributions of the Euler potentials associated with the Tsyganenko model are illustrated in the equatorial plane of the magnetosphere in Fig. 3a, b for moderate (top) and disturbed magnetic activity (bottom).For 5À `up, the shape of the domain where the Euler potentials are computed looks like a spear head to avoid the opening of the ®eld lines in the 0400±0600 MLT and the 1800±2000 MLT sectors illustrated in Fig. 1d.In Fig. 3a, b, the contours of a 2 are represented by the corresponding contours of equivalent v, from v 2 and steps of one unit.Close to the Earth, they are circles around the Earth as for a dipole.The only dierence is a v-value smaller than the radial distance: for example, for disturbed magnetic activity, the contour v 6 is located approximately at r 8 e at 0000 MLT.The contours become non-concentric when the distance increases in the midnight sector.Antonova andGanushkina (1996, 1997) computed the volumes associated with the Tsyganenko model for a lower magnetic activity, up 2, and obtained the same behaviour.When the magnetic activity increases, the contours become more stretched at larger distances in the tail in the nightside.Stern (1967) used the magnetospheric model of Mead (1964), andCheng (1995) solved the Grad-Shafranov equation with a given pro®le of the magnetospheric pressure to compute the Euler potentials.They obtained similar results except at noon where their contours approach closer to the Earth with the magnetic activity.This is due to the normalization used in the computation of the Euler potentials (Eqs.24±26) as discussed in Sect.4.1.
The contours of b 2 are represented by the radial curves crossing the contours of v.When the radial distance increases, they ®rst rotate towards the dayside, then in the opposite direction towards the nightside.The bending of the contours of b 2 increases with the magnetic activity illustrating the eect of the solar wind on the magnetic ®eld and the departure from a dipole where the contours of b 2 are the radial lines u onstnt.Similar contours are computed by Stern (1967) with the magnetic ®eld model of Mead (1964) but where the only rotation is towards the nightside.The rotation towards the dayside in the Tsyganenko model is not due to the normalization as we checked it with the non normalized Euler potentials a 1 and b 1 .As b 1 is computed along the a 1 contours from f h , it depends in a complex way on the magnetic ®eld model and we did not ®nd any simple explanation.We shall return to it in the next paragraph.
Figure 3c shows the normalized Euler potentials a 2 and b 2 associated with the Stern model (1967), with the Euler potentials calculated by Stern between 1200 and 2400 MLT, and the Euler potentials that we compute between 0000 MLT and 1200 MLT from the volume 5 of the Stern model (1967).The dierence between both potentials clearly shows that several sets of Euler potentials can be found for a given magnetic ®eld depending on the computation method.Close to the Earth where the magnetic ®eld is nearly dipolar, the v contours that we computed are slightly displaced tailwards in comparison to the dipole where they are located at a distance v.This is due to Eq. ( 18) which is an estimate of the true formula for the dipole where the quadratic terms have been neglected for simplicity.Interestingly, the b contours are slightly rotated toward the dayside close to the Earth as for the Tsyganenko model.This rotation is not obtained by Stern, and depends on the choice of a.
Figure 4 illustrates in the equatorial plane of the magnetosphere for moderate magnetic activity, the contours of the north-south component f h computed (Fig. 4a) directly from the model of Tsyganenko (1987) and (Fig. 4b) from the Euler potentials that we just  Stern (1967).The dawn sector illustrates the Euler potentials that we compute from the volume 5 of the Stern (1967) magnetic ®eld model derived and displayed in the previous paragraph (see Fig. 3a).Both results coincide over the restricted domain where we computed the Euler potentials, which shows the precision of our computation method although Euler potentials are calculated with a precision of about 1.5%.Similar agreements are obtained for the case of large magnetic activity.5a) and great (Fig. 5b) magnetic activity.At lower latitudes, the contours of v are very similar to those of a dipole.At higher latitudes, they concentrate on the nightside due to the ®eld lines stretching in the nightside and the small compression of the ®eld lines in the dayside.Antonova and Ganyushkina (1995 and references therein) and Peymirat and Fontaine (1994a) computed for various magnetic activities the volume of unit magnetic ¯ux tubes using the magnetic ®eld of Tsyganenko (1987) but with the Internal Geophysical Reference Field (1985,1986) as the main ®eld.Their results are in agreement with ours.

Ionospheric projection of the Euler potentials
The contours of b 2 start as the dipole for low latitudes and become distorted at higher latitudes.The distortion increases again with the magnetic activity.Wings appear in the mapping at dawn and dusk at the polar cap boundary for large magnetic activity.They re¯ect the particular domain that we used in the equatorial plane of the magnetosphere to avoid the regions with strongly distorted ®eld lines close to the polar cap as illustrated by the points A, B, C, D and E. Their equatorial conjugates are indicated in Fig. 3b.The mapping is regular except for points A and B which map at the same MLT in the equatorial plane of the magnetosphere.The ®eld lines connecting these two points to their equatorial conjugates originate from the dayside of the ionosphere and bend towards the tail due to the solar wind.

Conclusion
The Euler potentials (aY b) provide a very useful set of coordinates for the description of the magnetic topology.They simplify the derivation of the basic equations of the plasma transport in the magnetosphere and the mapping for studies related to the coupling between the ionosphere and the magnetosphere.
We proposed a numerical method to compute the Euler potentials based on a formalism previously made by Stern (1970) and which applies to regions of closed ®eld lines.The Euler potentials are ®rst computed on a surface, and their 3-D distribution is then obtained by magnetic ®eld line mapping.We chose for this initial surface the equatorial plane of the magnetosphere, which is the simplest surface crossed only once by closed magnetic ®eld lines.The results are represented both in the equatorial plane, and in the ionosphere.
The pair (aY b) of Euler potentials describing a magnetic ®eld is not unique.Ho et al. (1997) assumed that b is given as a complex function numerically computed from which they derived a using the relationship between the Euler potentials and the magnetic ¯ux per unit area.We propose another general method that reduces to the one used by Ho et al. (1997) if the unit area considered is the elementary area in the spherical coordinates space.It assumes that a only depends on the volume 5 of a ¯ux tube of unit magnetic ¯ux with the constraint that b 0 at midnight and that the existing symmetries of the magnetic ®eld are preserved.The method is developed to recover the usual pair of Euler potentials in the case of a dipolar magnetic ®eld.In fact, the conditions mentioned are not absolutely necessary and the method developed here is much more general.For example, it can be directly applied to a set of magnetic ®eld measurements if the dependence of a on 5 is disregarded, or extended to open magnetic ®eld lines if the considered surface is a sphere around the planet.
The method is illustrated with the model of Tsyganenko (1987).In the magnetosphere, close to the Earth, the Euler potentials are similar to those of a dipole, the departure from the dipole increasing with the distance and the magnetic activity.The a contours are illustrated as contours of the equivalent McIlwain parameter v (Stern, 1976).In the nightside, the main eect of the magnetic activity is to stretch tailward the magnetic ®eld lines.On the dayside, it has practically no in¯uence.The contours of b are rotated towards the dayside close to the Earth, and are also stretched with the magnetic activity in the nightside.The dayside rotation is associated to the choice of a. Finally, the Euler potentials are mapped on the ionosphere along the magnetic ®eld.The departure from a dipole increases with the latitude and the magnetic activity.At large latitudes, the contours of b display a distortion increasing with the magnetic activity.The numerical method illustrated with the model of Tsyganenko (1987) can be applied to a wide class of magnetic ®elds in regions of closed ®eld lines for studies of the magnetospheric convection coupled to the ionosphere.In particular, we intend to implement it in our numerical model of the magnetospheric transport (Peymirat and Fontaine, 1994b) to derive the physical consequences of a magnetic ®eld distribution more realistic than the dipolar approximation.

Fig. 1a±d .
Fig. 1a±d.Field lines in the XGSM-ZGSM plane of the Tsyganenko (1987) magnetic ®eld model every 2 invariant magnetic latitude from the North Pole with the dipole ®eld as the main ®eld and zero geodipole tilt angle.The bold lines correspond to the poles at high latitudes and to the external boundary of the considered domain at low latitudes.a Corresponds to 3À `up `3 where the external

Fig. 2 .
Fig. 2. Normalized equivalent v parameter as a function of XGSM in the noon-midnight meridian plane.The plain line corresponds to 3À `up `3, the dashed line to 5À `up, the dotted-dashed line to v computed from the volume 5 of the Stern model, the dotted line to v inferred by Stern (1967) and the bold line to the dipole

Fig. 3a±c .
Fig. 3a±c.Contours of the Euler potentials in the equatorial plane of the magnetosphere between the Earth's centre and 30 e with the Magnetic Local Time indicated on the external circle a for 3À `up `3 and b for 5À `up.The a contours are represented as contours of the equivalent v parameter.They encircle the Earth while the b contours cross them.The contour interval is 1 for v with the closest contour to the Earth corresponding to 2, and is 2p e a40 for b with b 0 in the midnight meridian c Contours of the Euler potentials in a format similar to a, b.The dusk sector (1200 `MLT `2400) corresponds to the Euler potential inferred byStern (1967).The dawn sector illustrates the Euler potentials that we compute from the volume 5 of theStern (1967) magnetic ®eld model

Figure
Figure5a, b represents polar maps of the ionosphere with the pole at the centre and latitudes above 40 .They display the ionospheric projection of the contours of v and b 2 along the magnetic ®eld lines for moderate (Fig.5a) and great (Fig.5b) magnetic activity.At lower latitudes, the contours of v are very similar to those of a dipole.At higher latitudes, they concentrate on the nightside due to the ®eld lines stretching in the nightside and the small compression of the ®eld lines in the dayside.Antonova and Ganyushkina (1995 and references therein)  andPeymirat and Fontaine (1994a)

Fig
Fig. 4a, b.Contours of the f h component (in nT) in equatorial maps similar to Fig. 3a, b for 3À `up `3.The f h contours are represented as contours of the quantity Log(f h ).The contour interval is 0.25 and the closest contour to the Earth corresponds to 3.5.f h is given from the Tsyganenko (1987) model in a and computed from the Euler potentials illustrated in Fig. 3a in b.The bold line in a corresponds to the external boundary of the domain where the Euler potentials are computed