%\input JGR2.sty     		% for final version
\input JGR3.sty    		% for draft
%\input EGS3.sty               	% for draft

\def\Del(#1){{{\partial}\over{\partial #1}}}
\def\DelLine(#1){\partial/\partial#1}
\def\DDel(#1){{{\partial^2}\over{\partial #1^2}}}
\def\Dsq(#1,#2){{{\partial^2#1}\over{\partial#2^2}}}
\def\DD(#1,#2,#3){{{\partial^2#1}\over{\partial#2 \partial#3}}}
\def\DoverD(#1,#2){{{\partial#1}\over{\partial#2}}}
\def\DoverDLine(#1,#2){\partial#1/\partial#2}

\def\cdotc{\!\cdot\!}
\def\Con(#1){{({\bf #1} \cdotc \nabla )}}
\def\Div(#1){{\nabla \cdotc {\bf #1} }}
\def\nablap{\nabla_{\perp}}

\def\Nabla{{\nabla_{\perp}\!\cdot }}
\def\Conp(#1){{({\bf #1}_{\perp} \cdotc \nablap )}}
\def\Up(#1){{{\bf #1}_{\perp}}}
\def\Divp(#1){{\nablap \cdotc {\bf #1}_{\perp} }}
\def\Teq(#1){{(t\! =\! #1)}}

\def\Nablaxy{{\nabla_{x,y}\!\cdot }}
\def\Conxy(#1){{({\bf #1}_{x,y} \cdotc \nabla_{x,y} )}}
\def\Uxy(#1){{{\bf #1}_{x,y}}}
\def\Divxy(#1){{\nabla_{x,y} \cdotc {\bf #1}_{x,y}}}

\def\Nablaless{{\nabla^*_{\perp}\!\cdot }}
\def\Conless(#1){{({\bf #1}^*_{\perp} \cdotc \nabla^*_{\perp} )}}
\def\Uxyless(#1){{{\bf #1}^*_{\perp}}}
\def\Divless(#1){{\nabla^*_{\perp} \cdotc {\bf #1}^*_{\perp}}}
\def\Zeq(#1){{(z\! =\! #1)}}

\def\FAC{field-aligned current }
\def\FACs{field-aligned currents }
\def\Alfven{Alfv\'en }


\title
\centerline{Numerical Simulation of Large-Scale Field-Aligned Currents} \centerline{Generation From Finite-Amplitude Magnetosonic Waves}

\authorgroup
\centerline{M. Yamauchi}\par

\addressgroup
\centerline{Geophysical Institute, University of Alaska, Fairbanks, AK 99775-0800}

\centerline{*Swedish Institute of Space Physics, Box 812, S-98128 Kiruna, Sweden}


\abstract
%{\bf Abstract}
Two-dimensional numerical simulation of finite-amplitude magnetohydrodynamics (MHD) magnetosonic wave is performed under finite-velocity background convection condition.
Isothermal cases are considered for simplicity.
External dissipation is introduced by assuming that the \FACs are generated proportional to the accumulated charges.
The simulation results are as follows:
Paired \FACs are found on the simulated waves.
The flow directions of these \FACs depend on the angle between the background convection and the wave normal, and hence two pairs of \FACs are found from a bowed wave as an overall structure.
The majority of these \FACs are closed within each pair rather than between two wings. 
These features are not observed under slow background convection.
The result could be applied to cusp current system and the substorm current system.

\vfootnote*{present\ address}

\vskip 13 pt	%only for photo-ready copy
(accepted manuscript of  https://doi.org/10.1029/94GL00883)  


{\bf Geophys. Res. Lett.}, 21, 851-854, 1994, 
(\copyright
1994 by the American Geophysical Union.\par

%\vskip 12pt
%{\parindent=0em  Paper number 94GL00883.\par  doi:10.1029/94GL00883\par}
%\bigskip
\receivedacceptance 
(Received September 27, 1993;
revised March 3, 1994;
accepted March 25, 1994.)
\bigskip


\vfill\eject	%only for draft
\vskip 13 pt	%only for photo-ready copy
\levelonehead
\centerline{1. Introduction}
\begintext

The magnetohydrodynamics (MHD) magnetosonic waves have been barely considered as the source of large-scale field-aligned currents (FACs) because they are independent of the \Alfven mode in a linear homogeneous limit.
However, in small scales, twin vortices (i.e., a pair of \FACs) are often observed in the polar region travelling tailward [{\it Friis-Christensen, et al.}, 1988], indicating that the magnetosonic mode can be coupled with the \Alfven mode [{\it Southwood}, 1987].
Furthermore, the wave coupling has been intensively investigated in studies of magnetic pulsations [e.g., {\it Zhu and Kivelson}, 1988, and references therein].
It is known that inhomogenuity such as pressure gradient can cause a coupling between MHD \Alfven waves and magnetosonic waves.  
However, those results are obtained in a linear harmonic limit.
Studies for highly nonlinear applications, e.g., large-scale phenomena remain rather poor.

Recently, {\it Yamauchi et al.}\ [1993, hereafter referred to as paper 1] showed analytically that a large-amplitude magnetosonic wave or shock in homogenous plasma can produce paired \FACs when the background convection is sufficiently large or the plasma beta ($\beta$) is sufficiently high, i.e., when $\kappa = ( 1 + 0.5 \beta) M > 1$ where $M$ is the Mach number [paper 1].
This is in nonlinear MHD, and is appropriate for large-scale phenomena.
The kinetic energy of convection is directly converted to the electrostatic energy that drives the current system, and we call this mechanism \lq\lq magnetosonic dynamo\rq\rq\ hereafter. 

The polarity of the paired \FACs depends on the angle between the wave normal and the background convection.
If the wave is bowed as shown in Figure 1a, the polarities of the \FACs are different on each wing of the wave.
In paper 1, two magnetospheric applications (Figures 1b and 1c) are suggested for such a configuration: 
1.\ the cusp current system [{\it Iijima and Potemra}, 1976] because the magnetosonic shock is found in the exterior cusp [{\it Lundin}, 1985]; 
2.\ the substorm current wedge travelling tailward during substorm expansive phase [{\it Akasofu}, 1964; {\it McPherron, et al.}, 1973] because paired \FAC structures are found inside the poleward-most part of the surge [{\it Yamauchi et al.}, 1992].

However, one problem is unsolved in paper 1.  
The current closer direction is unanswered in the bowed configuration in Figure 1.  
The \FACs can be closed (1) by longitudinal polarisation current ($J_P$) between the upstream side and downstream side within each wind as well as (2) by the transversal surface current ($J_S$) between two wings.
The latter is the ordinary dynamo circuit whereas the former represents the magnetosonic dynamo.
To know the relative importance of these current systems, we need two dimensional (2-D) modelling.
This is not achieved by the plane wave approximation in paper 1.

Here we study this by means of 2-D MHD simulation. 
Since the purpose of this paper is thus limited, we may employ several simplifying assumptions.
First we simplify the basic equations to 2-D by neglecting the curvature effect (see paper 1 for more discussion) and the field-aligned flow.
Other assumptions are found in paper 1 and in the rest of this paper.


\levelonehead
\centerline{2. Numerical Model}
\begintext


The double \FAC solution is analytically derived from the generalized MHD vorticity equation (equation (10) of paper 1), and this can be traced back to a combination of continuity equation and momentum equation, a polytropic relation, frozen-in condition, the Lorentz transform, and a linear current-charge relation (see also {\it Sato and Iijima} [1979]).
Their 2-D forms (under ${\bf u}_{\parallel} = 0$ and ${\bf B \cdotc \nabla} = 0$) are
$$\eqalignno{
  \Del(t)\rho  &= - \nablap \cdotc (\rho{\bf u})  &(1a) \cr
  \rho\Del(t)\Up(u)
  &= - \rho\Con(u)\Up(u) - \nablap P + \Up(J) \times {\bf B} &(1b) \cr
  \Del(t)P  &= - \nablap \cdotc (P{\bf u}) &(1c) \cr
  \Del(t)B  &= - \nablap \cdotc ( B {\bf u} ) &(1d) \cr
  \Up(E) &= {\bf B\times u} &(1e) \cr
  \Divp(J) &= - \sigma \Divp(E) &(1f) \cr
}$$
respectively, where we assumed an isothermal case in (1c) and a uniform conductivity ($\sigma$) in (1f), and equation (1f) comes from equation (9) of paper 1.

Equation (1f) is not an explicit form for $\Up(J)$, and we have to integrate it for the numerical calculation.
Before that, we introduce another assumption: we neglect the magnetic curvature effect and assume ${\bf B} = B{\hat z}$ (see Figure 1a for the co-ordinates).
This means that the perpendicular components of any vector (e.g., $\Up(u)$, $\Up(J)$ and $\Up(E)$) lie on the ($x$, $y$) plane.
With this 2-D assumption, the integral form of equation (1f) is written as
$$\eqalignno{\Up(J) - {\bf J}_i 
  &= \sigma ( \Up(u) \times {\bf B} - {\bf U}_0 \times {\bf B}_0 ) \cr
  &= \sigma ( B \Up(u) \times {\hat z} - B_0 {\bf U}_0 \times {\hat z} )
 &(2) \cr }$$
where ${\bf J}_i$ is a divergence-free constant vector, and hence can be expressed in a form of 
${\bf J}_i = \sigma \nablap \Psi \times {\hat z}$.
The reference values $U_0$ and $B_0$ (later we also have $\rho_0$ and $P_0$) are upstream values of $\vert u_{\perp}\vert$, $B$, $\rho$, and $P$ without disturbances.
Equation (2) is the same as equation (15) of paper 1.
The simplest form of the conductivity is a scalar, but in general it is a tensor because of the Hall effect.
As long as the non-diagonal component is relatively small, the tensor form does not make the simulation very complicated, and we use a tensor form to have a little more generality.
Then, equation (2) becomes
$$\eqalignno{\Up(J) 
  &= (\sigma + \sigma_{\perp} {\hat z} \times )
    ( B \Up(u) \times {\hat z} - B_0 {\bf U}_0 \times {\hat z} 
    + \nablap \Psi \times {\hat z} )  \quad &(3)  \cr
%  &=  ( \sigma_{\perp} - \sigma {\hat z} \times )
%    ( B \Up(u) - B_0 {\bf U}_0 + \nablap \Psi )  &(3) \cr
}$$

Now we have to solve $\Psi$.
To do so, we neglect eddy currents which close inside the source region; i.e., we assume that only the potential electric field contributes to $\Up(J)$.  
That means ${\bf J}_i$ gathers all the non-potential part.
By taking the total electric field in (3) (i.e., inside the second parenthesis) to be rotation-free, we obtain
$$\eqalignno{ \nabla^2_{\perp} \Psi
  &= - \nablap \cdotc ( B \Up(u) ) &(4)\cr }$$
This assumption implicitly removes the magnetic stress force from the momentum equation, and the magnetosonic speed becomes the same as the sound speed in this case.
External dissipation is introduced into the system instead of magnetic restore forces.
This simplification should not prevent the double \FACs solution from the magnetosonic wave according to the analytical study [paper 1], and is not an inadequate assumption for the present purpose.


For numerical simulation, we need dimensionless forms of equations (1a)-(1d), (3), and (4).
Let $L_0$, a reference value for $x$ and $y$, be the length of area under consideration.
Together with $U_0$, $B_0$, $\rho_0$, and $P_0$ (already defined above), this consists the basic set of reference values for normalization.
For example, the dimensionless time is defined as $t^*$ = $(U_0/L_0)t$.
Other reference values are obtained as
 $\Psi_0 = U_0 B_0$, $J_0 = B_0 \mu^{-1} h^{-1}$,
 $\sigma_0 = \Sigma_{A0} h^{-1} = h^{-1} \mu^{-1} V_{A0}^{-1} $,
% $\Psi^*$ = $\Psi / (U_0 B_0)$,
% ${\bf J}^*$ = ${\bf J}/(B_0/\mu_0 h)$,
% $V_{A0}$ = $B_0/\sqrt{\mu_0\rho_0}$,
% $C_{S0}$ = $\sqrt{P_0/\rho_0}$,
 where $h$ is the extent of source region along the magnetic field [paper 1].
In the dimensionless forms, two dimensionless parameters, 
$M_A = U_0 V_{A0}^{-1}$ and $M_S = U_0 C_{S0}^{-1}$,
appear in equations (1b) and (3):
$$\eqalignno{
  \Del(t^*)(\rho^* u^*_x)\! + \!\Nablaless (\rho^* u^*_x\Uxyless(u))
  \! &= \! - M_S^{-2} \DoverD(P^*,x^*)
  \! + \! {L_0\over h} M_A^{-2} J^*_y B^* \qquad &(5a) \cr
  \Del(t^*)(\rho^* u^*_y)\! + \!\Nablaless (\rho^* u^*_y\Uxyless(u))
  \! &= \! - M_S^{-2} \DoverD(P^*,y^*)
  \! - \! {L_0\over h} M_A^{-2} J^*_x B^* \qquad &(5b) \cr
  \Uxyless(J) = M_A ( \sigma^*_{\perp} - \sigma^* {\hat z} \times & )
   ( B^* \Uxyless(u) + {\hat x} + \nabla^*_{\perp} \Psi^* ) &(5c) \cr
}$$
whereas no parameter appears in the dimensionless forms of other equations (1a), (1c), (1d), and (4). 
Note that the $J_z \Uxy(B)$ term is neglected in equations (5b) and (5b) because we do not include the magnetic curvature effect and because inclusion of these small terms does not alter the solution significantly [{\it Yamauchi}, 1990].
The \FAC $J^*_{\parallel} = J^*_z\Zeq(h)$ is then obtained by height-integrating $\Divp(J)$ inside the source region:
$$\eqalignno{
  J^*_{\parallel} %&= J^*_z\Zeq(h)  \cr
  &= - {h \over L_0} \Divless(J) &(6) \cr
}$$

The initial conditions are simple: we consider only a uniform convection without any disturbances corresponding to the upstream of Figure 1a.
Therefore, we have $\rho^*$ = $P^*$ = $B^*$ = 1, $u^*_x$ = $-1$, and $u^*_y$ = $\Psi^*$ = $J^*_x$ = $J^*_y$ = 0.
For the boundary conditions, we give a gradual pressure increase on the downstream boundary, the highest pressure enhancement in the centre, and no enhancement in the side edges of the outflow boundary.
In this way we may have a bowed compressional wave propagating against the uniform convection.
The other three boundaries are free boundaries, i.e., the first spatial derivative is set zero in the normal direction to the boundary surface.
The free boundary are inelastic for waves (no reflection at the boundary).


Time evolutionary equations (1a)-(1d) are integrated using the Two Step Lax Wendroff method [{\it MacCormack}, 1969].
%Poisson's equation (5f) is solved by the method of Boisvert [1984].
The meshes of the simulation box are $40 \times 40$, and the box size is $L_0 \times L_0$.
Each time step is determined as 
$\Delta t^* = 0.25 \cdotc Min(\Delta x^*, \Delta y^*) \!\cdot\! 
 [ C^*_S+ Max(\vert V^* \vert) ]^{-1} $
so as to satisfy the Courant's stability condition [{\it Roache}, 1976].
%The typical values for the plasma sheet [{\it Parks et al.}, 1979;
% {\it Fairfield}, 1987; see also {\it Moore et al.}, 1987] are
% $L_0$ = 5 R$_E$, $U_0$ = 30--100 km$\,$s$^{-1}$,
% $\rho_0 = 1.5 \times 10^{-20}\,$kg$\,$m$^{-3}$,
% and $P_0 = 10^{-9}$Pa.
For the parameter values, we adopt $M_A$ = 0.45, $M_S$ = 0.71 (i.e., $\beta$ = 0.8), $\sigma^*$ = 0.45, $\sigma_{\perp}^*$ = 0.22, and $L_0 h^{-1} = 5$.
Such values of $M_A$ and $M_S$ are not unrealistic in the central plasma sheet if the magnetospheric convection is as fast as 30--100 km$\,$s$^{-1}$ [{\it Moore et al.}, 1987].
%In these regions, $\Sigma_A$ is high and the value
% of $\sigma^*$ (must be less than unity) is also reasonable.
With these values, we have $\kappa = 0.99$ (see introduction for $\kappa$).


\levelonehead
\centerline{3. Simulation Results}
\begintext


Figures 2{\it a} and 2{\it b} show the numerical results.  
It is clear from the figures that a finite-amplitude compressional wave is propagating against the convection as is expected from the initial and boundary conditions (subsonic convection plus pressure increase at the downstream boundary).
The wave front is characterized by a sudden increase of both the pressure and the magnetic field, indicating that this belongs to the MHD fast mode.
The propagation velocity is 0.4$U_0$ against the convection, i.e., 1.4$U_0$ relative propagation velocity.
This value is the same as the sound speed 
($C_S$ = $M_{S}^{-1} U_0$ = 1.4$U_0$).
Since the magnetosonic speed is the same as the sound speed in the present formulation (see discussion at equation (4)), this result supports that the simulated wave is the MHD fast wave.
The wave is steepened to form a shock in Figure 2{\it b}.
Since we used the conservation forms in the basic equations, the conservation relations are satisfied across the shock.

Intense \FACs are recognized on the wave front.
At first (Figure 2{\it a}) the \FACs are not in pairs: only one sheet of \FAC appears on each wing (top and bottom) of the bowed wave. Later (Figure 2{\it b}), however, oppositely-directed \FACs appear downstream side of the first ones on both sides, respectively.
The current distribution agrees with that of Figure 1a.
One important finding is that the current system is mostly closed between the upstream-side and downstream-side rather than between both wings of the bowed wave.
Hence, we have intense enough current to explain the large-scale \FACs in the ionosphere.
The intensity of those \FACs is more than 0.2$B_0 \mu^{-1} h^{-1}$, i.e., more than $B_0 \mu^{-1} L_0^{-1}$.

The downstream-side \FAC is always less intense than the upstream-side one.
The asymmetry comes from the Hall term ($\sigma_{\perp}$).
The result becomes symmetric if we set $\sigma_{\perp} = 0$.
In other words, the expected four-current structure is modified but not totally lost by including the Hall effect [{\it Yamauchi}, 1990].

The simulation is done with different sets of parameter values [{\it Yamauchi}, 1990], but the main results (wave steepening and formation of paired \FAC system) are the same unless $\kappa < 0.8$, and hence we present only this case here.  
For small $\kappa$, i.e., for slow background convection, we do not observe steepened waves nor the paired current system.
Finite background convection (finite inflow kinetic energy) is thus crucial for the magnetosonic dynamo to work.
The $\kappa$ dependence also indicates that the wave steepening is not the artifact of dispersion which is inherent to two-step Lax-Wendroff method.




\levelonehead
\centerline{4. Conclusion}
\begintext

It is numerically shown that a bowed large-amplitude magnetosonic wave propagating against the background convection can generate two pairs of \FACs as shown in Figure 1 as is predicted in paper 1.  
This solution, however, is not obtained for slow background convection.
The majority of these \FACs are found to be closed by longitudinal polarization currents within each wing of the bowed wave rather than by the transversal surface current of the wave. 
Thus, the \FAC due to magnetosonic dynamo can be stronger than that by the traditional dynamo, where the \FACs closed by the transversal current. 
We also confirmed that the solution is valid with finite Hall effect. 

In the introduction, we proposed two magnetospheric phenomena for possible applications.  
However, before we could apply the above result to the substorm current wedge, we have to consider, e.g., the effect of the magnetic curvature and the mechanism to enhance the near-Earth pressure (this generates a magnetosonic wave).
Similarly, we have to obtain a standing wave solution before we could apply to the cusp current system.
Yet, the magnetosonic mechanism in bowed magnetosonic waves can be still useful in understanding these magnetospheric phenomena.




\endtext %just before the acknowledgement
\font\SmallIt=cmti8
\def\It(#1){{\SmallIt #1\/}}
\acknowledgement 
%\It(Acknowledgments.)				%for final
\noindent {\it Acknowledgments.} \qquad	%for draft
The majority of this work has been done as a part of dissertation at the Geophysical Institute, University of Alaska Fairbanks, and has been supported in part by the NASA Grant NAGW-1520, the NSF Grant ATM 88-03133, and the Air Force Contract No. F19628-86-K-0030.
The author is grateful to S.-I., Akasofu, J. R. Kan, L. C. Lee, R. Lundin, D. W. Swift, and M. Tamao for their helpful discussions and suggestions.


\vfill\eject


\hyphenation{field-aligned}
\levelonehead
\centerline{References}

\beginreferences
%\tenpoint
%\eightpoint
\vskip 6pt

Akasofu, S.-I., The development of the auroral substorm,
 {\it Planet. Space Sci., 12}, 273, 1964.

%Boisvert, R., A fourth order accurate fast direct method for
% the Helmholtz equation, in {\it Elliptic Problem Solvers 2},
% edited by G. Birkhoff and A. Schoenstadt, pp. 35--44.,
% Academic Press, Orlando, Florida, 1984.

%Fairfield, D. H., Structure of the geomagnetic tail,
% in {\it Magnetotail physics}, edited by A. T. Y. Lui,
% Johns Hopkins University press, Baltimore, 1987.

Friis-Christensen, E., M. A. McHenry, C. R. Clauer, and S. Vennerstrom,
 Ionospheric traveling convection vortices observed near the polar
 cleft: a triggered response to sudden changes in the solar wind,
 {\it Geophys. Res. Lett., 15}, 253--256, 1988.

%Hasegawa, A., and C. G. Maclennan, Field-aligned electric field 
% accompanied by drift Alfven waves in an inhomogeneous plasma,
% {\it Geophys. Res. Lett., 17}, 1605--1608, 1990.
%
Iijima, T., and T. A. Potemra, Field--aligned currents in the dayside cusp
 observed by Triad, {\it J. Geophys. Res., 81}, 5971--5979, 1976.

Lundin, R., Plasma composition and flow characteristics in the
 magnetospheric boundary layers connected to the polar cusp,
 in {\it The Polar Cusp}, edit by J. A. Holtet and A. Egeland,
 pp. 9--32, D. Reidel, Hingham, Mass., 1985.

MacCormack, R. W., The effect of viscosity in hypervelocity impact
 cratering, {\it AIAA} paper No. 69--354, 1969.

McPherron, R. L., C. T. Russell, and M. P. Aubry,
 Satellite studies of magnetospheric substorms on August 15, 1968,
 9, Phenomenological model for substorms,
 {\it J. Geophys. Res., 78}, 3131--3149, 1973.

Moore, T. E., D. L. Gallagher, J. L. Horwits, and R. H. Comfort,
 MHD wave breaking in the out plasmashere,
 {\it Geophys. Res. Lett., 14}, 1007--1010, 1987.

%Parks, G. K., C. S. Lin, K. A. Anderson, R. P. Lin, and H. Reme,
% ISEE 1 and 2 particle observation of outer plasma sheet boundary,
% {\it J. Geophys. Res., 84}, 6471--6476, 1979.

Roache, P. J., {\it Computational Fluid Dynamics}, Hermosa Pub., 1979.

Sato, T., and T. Iijima, Primary sources of large-scale Birkeland current,
 {\it Space Sci. Rev., 24}, 347--366, 1979.

Southwood, D. J., The ionospheric signature of flux transfer events, 
 {\it J. Geophys. Res., 92}, 3207--3213, 1987.

%Wright, A. N., Asymptotic and time-dependent solutions of
% magnetic pulsations in realistic magnetic field geometries,
% {\it J. Geophys. Res., 97}, 6439--6450, 1992.
%
Yamauchi, M., A theory of field-aligned current generation from
 the plasma sheet and the poleward expansive of aurora
 substorms, Ph.D. thesis, 201 pp., Univ. of Alaska, Fairbanks,
 May 1990.

Yamauchi, M., R. Lundin, and B. Aparicio, Viking observation
 of the substorm current wedge, 
 {\it Eur. Space Agency Spec. Publ., ESA SP-335},
 495--497, 1992.

Yamauchi, M., R. Lundin, and A. T. Y. Lui, Vorticity equation for 
 MHD fast waves in geospace environment, 
 {\it J. Geophys. Res., 98}, 13523--13528, 1993.

Zhu, X., and M. Kivelson, Analytic formulation and quantitative 
 solitions of the coupled ULF wave problem, 
 {\it J. Geophys. Res., 93}, 8602--8612, 1988.



%% Elphinstone, R. D., D. Hearn, J. S. Murphree, and L. L. Cogger,
% Mapping using Tsyganenko long magnetospheric model and its relationship
% to Viking auroral images, {\it J. Geophys. Res., 96}, 1467--1480 1991.

%% Gelpi, C., H. J. Singer, and W. J. Hughes, A comparison of magnetic
% signatures and DMSP auroral images at substorm onset:
% Three case studies, {\it J. Geophys. Res., 92}, 2447--2460, 1987.

%% Lopez, R. E., and A. T. Y. Lui, A multisatellite case study of
% the expansion of a substorm current wedge in the near-Earth
% magnetotail, {\it J. Geophys. Res., 95}, 8009--8017, 1990.

%% McIlwain, C. E., Substorm injection boundaries, in
% {\it Magnetospheric Physics}, edited by B. M. McCormac, p. 143,
% D. Reidel, Hingham, Mass, 1974.

%% Nagai, T., Field-aligned currents associated with substorms in
% the vicinity of synchronous orbit 2. GOES 2 and GOES 3 observations,
% {\it J. Geophys. Res., 92}, 2432--2446, 1987.

%% Tsyganenko, N. A., On the re-distribution of the magnetic field and
% plasma in the near nightside magnetosphere during a substorm growth
% phase, {\it Planet. Space Sci., 37}, 183--192, 1989.



\endreferences


\authoraddress
M. Yamauchi, Swedish Institute of Space Physics, Box 812, 98128 Kiruna, Sweden.
\bigskip




\vfill\eject
%\pageno = 20
\caption
%\hsize=8.5 cm
Fig. 1. \hskip 1 em
A predicted distribution of the \FACs on a bowed compressional wave propagating against the finite-velocity background convection.
({\it a}) General configuration.
The transversal surface current ($J_S$) generally exists for ordinary MHD fast waves but does not diverge in 1-D configuration.
The longitudinal current ($J_P$) is responsible for the paired structure.
({\it b}) and ({\it c}) Possible magnetospheric regions (the exterior cusp and the substorm current wedge expanding against convection) where the magnetospheric dynamo can be applied (after paper 1).


%\vskip 2pc \noindent
\vskip -1pc \noindent
%\hsize=8.5 cm
Fig. 2. \hskip 1em
Numerical results at (a) 250 time steps ($t^*=1.03$) and (b) 500 time steps ($t^*=2.02$).
The top left panel shows the pressure $P^*$,
 the top right panel shows the velocity $\Uxyless(u)$,
 the bottom left panel shows the \FAC $J^*_z$, and
 the bottom right panel shows the perpendicular current $\Uxyless(J)$.
The distributions of $B_z > 0$ and $\rho$ (not shown here) are similar to the pressure distribution.
The excited large-scale MHD fast mode is accompanied by the \FACs as is expected.



\bye
\end

$$\eqalignno{
  \Del(t^*)(\rho^* u^*_x) + \Nablaless (\rho^* u^*_x\Uxyless(u)) &= 
  - M_S^{-2} \DoverD(P^*,x^*) + {L_0\over h} M_A^{-2} J^*_y B^* &(5a) \cr
  \Del(t^*)(\rho^* u^*_y) + \Nablaless (\rho^* u^*_y\Uxyless(u)) &= 
  - M_S^{-2} \DoverD(P^*,y^*) - {L_0\over h} M_A^{-2} J^*_x B^* &(5b) \cr
  \Uxyless(J) = M_A ( \sigma^*_{\perp} - \sigma^* {\hat z} \times & )
   ( B^* \Uxyless(u) + {\hat x} + \nabla^*_{\perp} \Psi^* ) &(5c) \cr
}$$
After a normalization of
 $x^* = x/L_0$, $B^* = B/B_0$, etc.,
two dimensionless parameters: 
$M_A = U_0 V_{A0}^{-1}$ and $M_S = U_0 C_{S0}^{-1}$
appear in equations (1b) and (3).
A $M_S^{-2}$ is multiplied to the $\nabla P$ term in equation (1b), a $M_A$ is multiplies to entire right hand side of (3), and a multiplier $L_0 h^{-1} M_A^{-2}$ appears at the $\Up(J) \times {\bf B}$ term in equation (1b). 

  \Del(t^*)\rho^*  + \Nablaless (\rho^*\Uxyless(u)) &= 0  &(5a) \cr
  \Del(t^*)P^*  + \Nablaless (P^*\Uxyless(u)) &= 0 &(5b) \cr
  \Del(t^*)B^*  + \Nablaless (B^*\Uxyless(u)) &=0 &(5c) \cr
   \nabla^{*2}_{x,y} \Psi^* &= - \Nablaless (B^* \Uxyless(u)) &(5g) \cr
 $\beta = P_0/P_{B0}^{-1} = 2M_A^2 M_S^{-2}$.

The finite background convection provides the energy: its kinetic energy is converted to the electromagnetic energy of the current system when it is decelerated across the wave.

