General Solution of Two-Dimensional Chapman-Ferraro Problem of Magnetospheric Cavity

Source spectra of S waves were determined using records of eighteenearthquakes occurring in the Chia-Yi and Tai-Nan area with local magnitudesof 2.8≤M(subscript L)≤5.8 as obtained from a rock-site station.In addition tothe correction of geometrical spreading,elimination of the anelastic attenuationeffect from the observed spectra was carefully examined to measurethe high-frequency spectral levels of seismic sources.As to the source spectra,two types of spectral shapes may be observed.For earthquakes of M(subscript L)＜5.4,the spectra obey the ω-squared model with asingle corner frequency.However,this observation cannot provide an adequaterepresentation for earthquakes of M(subscript L)≥5.4,since they clearly demonstratethe existence of two corner frequencies on the spectrum.The differencein spectral shapes may reveal that the rupture of larger earthquakesproceeded as a series of multiple events while a single fault patch results insmaller earthquakes.This explanation is supported by both spectral shapesand waveform characteristics,and may disclose the complexity of earthquakesources of larger magnitude.The seismic moment of M0 measured from spectral level at low frequencyrange satisfies a relation with lower corner frequency of f0 inM0∞f^(-3)0.For the set of earthquakes,the average stress drop is 125 bars.Nonetheless,this model is a poor fit to the shapes of source spectra forevents of M(subscript L)≥5.4.The source spectra obtained by the two greater events,the 1991 Chiali(M(subscript L)=5.7)and 1993 Tapu(M(subscript L)=5.8)earthquakes,werediscussed in this subject.In describing these spectra,a stress drop of about60 bars was estimated from the spectral level in a lower frequency range,while 600 bars was required to interpret the high-frequency amplitudes.By applying the Sato and Hirasawa(1973)source model,the average scalelength of the fault heterogeneities inferred from the higher corner frequencyof f0 is about 300 meters,and this is almost identical to the source radius ofthe Brune(1970,1971)model for small events with a magnitude of around3.Based on the seismic moments taken from the Harvard centroid-momenttensor(CMT)solution and this study,for the Tapu earthquake theestimated values of local stress drop obtained using the specific barrier model(Papageorgiou and Aki,1983)are about 700 and 516 bars.The high stressdrop of 600 bars for our result,as observed from high-frequency sourcespectra,lies in between,and its validity is also confirmed by the agreementof total seismic energy between the results obtained from specific barriermodel and those from the Gutenberg-Richter relation(1956).


INTRODUCTION
The pioneering work of Chapman and Ferraro on geomagnetic storms in 1930' s started modem research on the earth's magnetosphere, although the latter name was coined by Gold much later in the 1950s, at the dawn of space age. The 'cavity', as it was called by Chapman and Ferraro (1931), for the void carved out of the streams of solar corpuscles is a magneto spheric region not reached by the solar wind, in today's nomenclature. Before the ultimate acceptance of the suggestion by Dungey ( 1961 a) that field line reconnection between inter planetary and terrestrial magnetic fields takes place to link the earth to interplanetary space magnetically, the cavity was thought to be covered on its entire surface by a sheet-like current to form a closed magnetosphere. This Chapman-Ferraro current on the magnetopause is in duced by the impingement of the solar wind. It produces a magnetic field that adds to the earth's dipolar field so that the resultant magnetic field has a magnetic pressure to balance the dynamic pressure exerted by the solar wind.
Instead of determining the distribution of the dynamic pressure of the impinging solar wind from magnetohydrodynamic consideration of the interaction between the solar wind and the earth, Ferraro (1960) adopted the approximation of specular reflection for the incident corpuscular particles. Such a simplification reduces the mathematical determination of the magnetospheric cavity to a problem of free boundary on which pressure balance is to be satis fied. This constitutes the famous Chapman-Ferraro problem. To date, the Chapman-Ferraro problem has been solved only for its two-dimensional ver sion to the extent of fully revealing the solution's dependence on the earth's magnetic moment and the solar wind's mass density and flow velocity. Based on the insights provided by this two-dimensional solution, approximate solutions of the three-dimensional Chapman-Ferraro problem have been attempted, using various numerical schemes for some values of the param eters (e.g., Mead and Beard, 1964). These exemplary solutions have served as the framework for building reference models to organize observational data on the magnetosphere (see Siscoe, 1988). Strictly speaking, solutions for open magnetosphere should be used to build the refer ence models because the earth's magnetosphere is now regarded as open, with interconnective field lines. But, until now, the corresponding ChapmanFerraro-Dungey problem for a partially open magnetosphere has not yet been formulated. Recently, Yeh (1997) elucidated the mag netic topology of the magnetosphere by a model of partially open magnetosphere, which has a front part much like a closed magnetosphere and a rear part like a fully open magnetosphere. Thus, a re-examination of the Chapman-Ferraro problem is desirable to provide some useful clues for the formulation of the ChapmanFerraro-Dungey problem. For this purpose, we shall present the general solution of the two-dimensional Chapman-Ferraro problem.
The exact solution of the two-dimensional Chapman-Ferraro problem with Ferrao's ap proximation of pressure balance as the boundary condition was given by Dungey (1961b) and Hurley (1961) in different mathematical forms. They applied different transformations to the Laplace equation that governs the scalar potential and flux function of the magnetic field. They gave the solutions explicitly and showed that the given solutions indeed satisfy the pre scribed boundary conditions. However, neither of them revealed how the explicit solutions were obtained.
In this paper, we shall elucidate the methodologies for obtaining the general solution for arbitrary boundary conditions. The general solution amounts to Poisson's integral formula for a Dirichlet problem with a half-plane or a circle domain. Conformal transformations, such as the Schwarz-Christoffel transformation, are utilized in the mathematical process. This general solution enables us to obtain desirable solutions with other boundary conditions.

CHAPMAN�FERRARO PRO B LEM IN TWO DIMENSIONS
We use rectangular coordinates (x, y) for the noon-midnight meridional plane, with the origin at the earth's center, the x-axis sunward and the y-axis northward. Each coordinate point can be represented by a complex number z=x+iy. The magnetic field 1 B +1 B can be represented by a complex function B(z)=B Y (x, y)+iB/x, y). For a current-free magnetic field, we can write B=-d <l>/dz in terms of a complex potential c;l>(z)='l'(x, y)+iQ(x, y), with'¥ being a flux function and Q a scalar potential. The relations a'I' an

399
(2) indicate that'¥ and n as functions of x and y satisfy Cauchy-Riemann equations. Hence, each of 'P and Q satisfies Laplace equation Accordingly, <I> is an analytic function of the complex variable z. In our usage of complex variables the earth's dipole field has the complex potential MJz, with M0 being the magnitude of the earth's southward dipole moment. The problem at hand is to find a desirable <I> that behaves asymptotically like a dipole near the origin (viz., <l>�MJz as z�O) and also satisfies the boundary condition specified on the free boundary for the magnetopause. We may choose'¥ to have a value of 0 on the field lines that contain the north/south neutral points and n to have a value of 0 on the equipotential lines that contain the subsolar/antisolar stagnation points. Conceptually, the solution should have field lines and equipotential lines like what is depicted in Figure 1 as far as topology is concerned.
In Ferraro's (1960) approximation of specular reflection for incident particles, the solar wind's dynamic pressure on the magnetopause is taken as p1cos2x, with p1=2(m + +m_)n 1 U/ being the dynamical pressure at the subsolar point (m+ being proton mass, m_ electron mass, n1 number density and U1 velocity of the solar wind) and X the incident angle of the sun-to-earth velocity with respective to the outward normal to the magnetopause. The pressure balance lµ0 1 1Bl2=p1cos2X (with µ0 being the magnetic permeability) takes the form of 2 (5 ) by virtue of IBl=dntd f, and cosx=dy/d f, (hence cosx9BI dy/d.Q) on the meridional trace of the magnetopause. Here y 0 denotes the y-coordinate of the north neutral point and 00 the magnetic potential at the south neutral point. These two positive-valued parameters, related by 0/ y 0=V 2µop1 , are to be determined in terms of the two known parameters M0 and p1 which are indicative of the strengths of the geomagnetic field and incident solar wind. The crux of the problem is the free boundary on which the y versus Q condition of equa tion (5) is to be satisfied. There are two different methodologies to handle the free boundary. Both of them make use of a complex-valued intermediary, W=U+iV or p:::: r exp i0, to trans form the free boundary in the z-plane into a fixed straight line in the W-plane or a fixed circle in the p-plane. In the first approach, extending what was done by Dungey (196lb), the inter- mediary variable Wis related to <I> by a conformal mapping between two 'hodograph' planes.
In the second approach, used by Hurley (1961), the intermediary variable� is related to z by a conformal mapping between two coordinate planes.

A SCHWARZ-CHRISTOFFEL TRANSFORMATION
First, we consider the methodology that uses W=U+iV as the intermediary variable, Let x('I', Q) and y('I', .Q) denote the inverses of the pair of functions 'l'(x, y) and .Q(x, y). Their a'l'2 an2 In other words, the inverse function z(<I>) of the analytic function <l>(z) is analytic too (a well known theorem in the theory of analytic functions). The function z(<I>) has a branch cut, from i0.0 to -i00 on the 0-axis, which interfaces the images of the front and rear segments of mag netopause. See Figure 2. The front segment, which has the subsolar point as its midpoint, and the rear segment, which has the antisolar point as its midpoint, join at the north and south neutral points. The Q versus y condition on the magnetopause now appears as {y0 (n1n0 + 2) if 'I'= -0 and 0 < n < Q 0, Without revealing the methodology, Dungey (1961b) gave the following analytic function and showed its satisfaction of the condition across the branch cut.
We shall explore a methodology to obtain Dungey's solution by means of a Schwarz Christoffel transformation (e.g., see Greenberg, 1978). The transformation deforms the whole <1>-plane conformally so that the vertical branch cut in the <l>-plane becomes a horizontal branch  w dW c wno)1 1 2cw + '2 0 ) 1 12 ·. ---- --------: : which has the asymptotic behavior of <ll --7 W in the limit W--700 near the dipole. The inverse mapping is 404 TAO, Vol. 10, No. 2, June 1999 The mappings of field lines and equipotential lines are orthogonal straight lines in the <l>-plane (see Figure 2) and are orthogonal curves in the W-plane (see Figure 3). The Cauchy-Riemann equations (7) become au a v a v au and the Laplace equations (8) and (9) by Poisson's integral formula (e.g., see Greenberg, 1978). Its partial derivatives ()y/au and aytav provide the values for -axtav and dxldV, respectively, by virtue of equations (15). From the combination cax1aU)dU+(ax/dV)dV for dx, we obtain l n o .

A CANONICAL CONFIGURATION
Next, we consider the methodology that uses s=P exp ie as the intermediary variable. The complex potential <; describes a magnetic field B=(MJi;2+Bc) dSfdz, depending on how the intermediary variable sis related to the coordinate variable z. Its flux function 'P=(MJp-Bcp)cose has a value of 0 on the circle p=p0 with p0:::: (M0/Bc)112 and also on the polar axis e=±l1t in the s-plane. Its scalar potential !2=-(M0/p+Bcp) sine has a sinusoidal variation 2 !2=-!lo sine, (26) with 00=2M0/p0, on that circular boundary. Figure 4 shows the mappi�gs of field lines and equipotential lines, with the mappings of the two neutral points at s= P 0 e ± m12. In the pedagogic case of s=z mapping, the magnetic field M0/z2+Bcrepresents the noon-meridional configura tion of a cavity with a spherical magnetopause (cf. Yeh,19 97). This canonical configuration (see Figure 4) has a northward uniform field lyBc to account for the part of magnetic field due to the magnetopause current on the spherical cavity. It is equivalent to the magnetic field due to a pair of line currents at infinity on opposite sides of the dipole, carrying oppositely directed currents of infinite strength proportional to their distance of separation.
A conformal transformation between the s-plane and the z.:.plane will have z as an ana lytic function of S· The two functions x(p, e) and y(p, e) satisfy Laplace equations: a 2 1 a 1 a 2 ( -+--+---)x= O , ap 2 P ap P 2 a0 2 a 2 a a 2 (-+ 1-+_ l -)y= O . ap 2 P ap P 2 ae 2 Moreover, the y versus Q relation (10) on the magnetopause takes the form y0(-2-sin 0) It serves as the boundary condition for equation (28). This Dirichlet problem for a circular domain has the solution by Poisson's integral formula (e.g., see Greenberg, 1978). The corresponding x(p, 0) can be found from the Cauchy-Riemann equations It is TAO, Vol. JO, No. 2, June 1999 a 1 a -x=-y, ap Pae 1 a a --x=-y. P ae ap i n .
which defines an analytic function of s for z. When y(0) is odd-symmetric, equation (33) becomes The asymptotic behavior z�(4yJnp 0 )s+O(s2) indicates that we may choose p0-4y 0 to make 1t z�s near the dipole. Accordingly, Bc= ( K)2Miy 0 2in terms of M0 and y0• 4 To evaluate the integral for z(s), we split the odd function y(0) into a constant part and a sinsusoidal part . The former yields (- .

DISCUSSION
has dy/d.Q equal to .!.. x c 2 !('/Mo+vMc )2 at the subsolar point, equal to ooat the neutral points, 2 equal to 0 at the peak points, and equal to .!_ X c 2/(V Mc-1' Mo )2 at the antisolar point. The dy/ 2 d.Q slope at the subsolar point is smal�er than y i.0.0, which is � xc 2 /(M0 + Mc), whereas the slope at the antisolar point is larger. Pressure balance on this cavity requires a non-zero gas pressure at the neutral points on the earthward face to match the dynamic pressure of the solar wind. Moreover, the magnetic pressure on the earthward face of antisolar segment of the bound ary is to be matched by that of interplanetary magnetic field because the solar wind does not impinge directly to exert a dynamic pressure there. Accordingly, the interplanetary magnetic field should have an effective value of 2(1/ Mc-1/ Mo )2/x c 2 at the antisolar point.