% \documentstyle[12pt]{article} \addtolength{\textheight}{1.5in} \addtolength{\topmargin}{-0.5in} \addtolength{\evensidemargin}{-0.5in} \addtolength{\textwidth}{0.5in} \raggedbottom % GENERAL DEFINITIONS \newcommand{\be}{\begin{eqnarray}} \newcommand{\ee}{\end{eqnarray}} %\newcommand\boxx {\thinspace\setbox0\hbox{A}\maketypebox\thinspace} \def\ni{\noindent} \def\ul{\underline} \def\cl{\centerline} \newcommand\ie {{\it i.e.}} \newcommand\eg {{\it e.g.}} \newcommand\etc{{\it etc.}} \newcommand\cf {{\it cf. }}\newcommand\grad{\nabla} \begin{document} \setlength{\baselineskip}{20.5 pt} \thispagestyle{empty} %\thispagestyle{empty} \pagenumbering{arabic} \begin {center} \ni {\Large\bf The role of space-time dependent ionospheric conductivity in the evolution of field line resonances: Relation to auroral arc} \end {center} \vskip 10mm \begin {center} {\large Manju Prakash \\ \\ Department of Physics and Astronomy, \\ SUNY at Stony Brook, \\ Stony Brook, New York 11794, USA}\\ \vskip 3mm {\large and\\ \\ Robert Rankin \\ \\ Canadian Network for Space Research,\\ Department of Physics, University of Alberta,\\ Edmonton, T6G 2J1 Canada}\\ \end {center} \vskip 5mm {\bf Abstract:} { The present work examines the nonlinear evolution of field line resonances (FLRs) in the presence of electron inertia and the space-time dependent ionospheric conductivity. The nonlinear space-time evolution results from the ponderomotive forces that are exerted by the large amplitude FLRs that are excited during magnetospheric substorms. The nonlinear equations are derived using the MHD equations and the generalized Ohm's law. The effects arising from the non-uniform and time-dependent Pedersen conductivity are incorporated through the boundary conditions on field -aligned currents (FAC) that close the ionospheric currents. \ni The space-time dependent ionospheric conductivity can increase the FAC associated with FLRs. This increase leads to the intensification of the pre-existing auroral arc observed during substorms. The increase in the FAC can also enhance the nonlinear effects arising from the ponderomotive forces. These effects can result in the density perturbations which can structure the FLRs. The structuring of the FLRs can lead to the restructuring of the auroral arc observed during substorms. These studies indicate that the ionospheric manifestation of the substorms is modified by the space-time dependent ionospheric conductivity. \newpage \section {Introduction} \vskip 4mm \ni The effect of the ionosphere on the detectability, polarization structure, and decay of ultra low frequency (ULF) magnetospheric waves (frequency range 1mHz--1 Hz) has been a topic of interest for more than thirty years ({\it Allan and Knox,} 1979; {\it Southwood and Hughes,} 1983). The degree of ionospheric conductivity controls the amplitude of the ULF signal, its reflection, and its subsequent decay on the ground. The increasing use of the ground based networks such as CANOPUS (to study auroral substorms) provide further impetus to understand the modification of the ULF signals between their point of generation in space and their detection on the ground. The finite conductivity of the ionosphere leads to partial reflections of the shear Alfv\'{e}n waves (SAWs). The SAW is a transverse ULF wave which is guided along the geomagnetic field lines. It carries field-aligned current, momentum, and electromagnetic energy into and out of the Earth's magnetosphere. This energy is converted into the kinetic energy of the auroral particles and is also dissipated in the form of Joule heating of the particles at the polar ends of the auroral flux tubes ({\it Louarn et al.,} 1994; {\it Lysak,} 1990;{\it Wahlund et al.}, 1994). During substorms, the dynamic changes in the plasma sheet are communicated to the ionosphere by the field-aligned currents (FAC) associated with the SAWs. On reflection from the ionosphere the SAWs transfer ionospheric drag back to the magnetosphere. Thus Alfv\'{e}n waves play an important role in coupling the dynamics of the inter-regions of the Earth's magnetosphere ({\it Louarn et al.,} 1994; {\it Lysak and Dum,} 1983; {\it Lysak,} 1990; {\it Prakash and Lysak,} 1992; {\it Wahlund et al.,} 1994). The degree of space-time dependent ionospheric conductivity can alter the FACs and restructure the auroral arc ({\it Ellis and Southwood,} 1983; {\it Glassmeier,} 1983). The non-uniform conductivity can also modify the transport of energy and momentum within the magnetosphere ({\it Ellis and Southwood,} 1983). Therefore, it is important to examine the role of space-time dependent ionospheric conductivity in magnetosphere-ionosphere coupling. The SAWs play an important role in the magnetosphere-ionosphere coupling through the excitation of FLRs. The FLRs are standing SAW structures (frequency range 1-10mHz) which are formed along the closed geomagnetic field lines when the SAWs are reflected at the conjugate ends of the ionosphere. The FACs associated with FLRs can intensify the pre-existing auroral arc. This intensification is an important signature of the onset of substorms ({\it Samson et al., } 1991, 1992a;b). Near the ionosphere, the FLRs have a latitudinal scale size of 50 km which is comparable to the latitudinal size of the auroral arc ({\it Rankin et al.,} 1994). These observations indicate that the FLRs play an important role in the dynamics of the auroral region during substorms. The observations of the vortices in the auroral zone that are caused by the coupling of the shear flows (resulting from the $\bf E\times B$ drift in FLRs) with the pressure gradients ({\it Rankin et al.,} 1997; {\it Samson et al.,}1996; {\it Voronkov et al.,} 1997) further support the role of FLRs in the auroral dynamics during substorms. During substorms, a significant amount of energy is pumped into the excitation of the FLRs. This leads to excitation of the large amplitude FLRs that exert ponderomotive forces on the ambient plasma. The ponderomotive forces can narrow the meridional scale size of the FLRs by various physical effects listed in section 2.1 of this paper ({\it Rankin et al.,} 1994; 1995). When the scale size is comparable to the electron inertial length ($\sim 1 \rm km$) in the auroral zone, the parallel electric fields $\rm E_z$ develop ({\it Hasegawa, } 1976; {\it Wei et al., }1994). The parallel electric fields can accelerate electrons to several hundred eV and lead to spatial modulation of the auroral arc (\it Wei et al., } 1994). The nonlinear effects resulting from the ponderomotive forces can narrow the auroral arcs to a scale size less than 5 km ({\it Bellan and Stasiewicz,} 1998; {\it Rankin et al., } 1995; {\it Trondsen et al.,} 1997; {\it Wei et al.,} 1994). The present work will focus on the structuring of the auroral arc related to substorms only. It will not address various mechanisms which lead to the formation of arc sizes down to 100m ({\it Stasiewicz et al.,} 1997). It has been reported that some discrete auroral arcs may be spatially modulated with FLR frequencies in the range 1-4mHz ({\it Samson et al., } 1996). The characteristic features of the auroral arc as well as the dynamics of the other observable physical effects can be understood theoretically by examining the nonlinear evolution of the FLRs. The current studies on the nonlinear evolution of FLRs, however, assume that the ionospheric conductivity is constant and uniform ({\it Rankin et al., } 1993). On the dayside, ionosphere conductivity is uniform and constant, because it is primarily produced by the impact of the solar UV radiation. On the nightside, the ionospheric conductivity is caused by the precipitating particles and energy flux from the magnetosphere. This results in a non-uniform and time dependent conductivity. The FAC resulting from the conductivity can lead to feedback interactions which can intensify the auroral arc ({\it Lysak,} 1991; {\it Trakhtengertz and Feldstein,} 1984) observed during substorms. Therefore, the current theoretical studies on the evolution of FLRs should be extended to include effects arising from the inhomogeneous and time-dependent conductivity ({\it Ellis and Southwood, } 1983; {\it Glassmeier,} 1983). The present work is an effort in this direction. In section 2 of the paper, we discuss the mathematical and the physical aspects of the FLR wave model. In section 3, the ionospheric model and the FAC observed during substorms will be studied. In section 4, the equations describing the nonlinear evolution of FLRs in the presence of electron inertia and space-time dependent conductivity will be derived. The significant results and directions for the future research will be summarized in section 5 of the paper. \vskip 5mm \section {The Field Line Resonance Wave Model}\\ \subsection {Physical Model} The formation of FLRs along the closed field lines of the terrestrial magnetosphere can be understood using magnetohydrodynamic (MHD) equations ({\it Southwood and Hughes,} 1983). The MHD equations, when coupled with the boundary of the magnetopause, yield compressional mode solutions with discrete frequencies. The discrete modes are formed when the compressional Alfv\'{e}n modes (normal modes of the magnetosphere) are reflected at the magnetopause boundary (at one end) and at the wave turning points (on the other end). These modes constitute a discrete spectrum of the compressional waves in the Earth's magnetospheric cavity ({\it Kivelson and Southwood,} 1986). The energy to excite these cavity modes may be derived from pressure pulses in the solar wind, transient dayside reconnection, Kelvin Helmholtz instability, or other substorm-related instabilities. These cavity modes can couple their energy with the SAWs at the resonance points. The coupling results in the formation of standing wave structures (along the closed geomagnetic field lines) when the SAWs reflect at the northern and southern ionospheres. These standing wave structures are know as FLRs. There are observations based on the ground based magnetometer, radar, and optical data which support that FLRs are standing SAWs that are formed along the closed field lines ({\it Greenwald and Walker,} 1980; {\it Samson et al.,} 1991; {\it Walker et al.,} 1992). We remark that the present theoretical formalism underlying the formation of FLRs is based on ideal MHD equations which neglect the parallel electric field. The parallel electric field effects are incorporated through the Hall term in the generalized Ohm's law. This approach has been criticized by {\it Bellan} (1994; 1996) who claims that when the parallel electric field is incorporated into the fluid equations the FLRs do not exist. Therefore, FLR is an artifact of ideal MHD plasma and do not exist in reality. In view of this, all observations pertaining to FLRs should be re-evaluated. We believe that MHD formalism with Hall effect is a suitable approach at wavelengths of hundreds of kilometers of FLRs. It is meaningless to apply fluid models at such large scales. During substorms, large amount of energy is pumped into the excitation of the FLRs. The resulting large amplitude FLRs can exert ponderomotive force defined as: \be F = - \nabla(\frac {}{2\mu_{0}}). \ee Here, the average is taken over the SAW period. The force moves the ambient magnetospheric plasma along the field lines, away from the ionosphere and towards the equatorial region. This redistribution and the spatial movement of the plasma can lead to following effects:\\ \ni {\it Changing the frequency SAWs:} The change in plasma density can result in frequency shifts of FLRs that are time-dependent. These shifts are different for different field lines. This leads to spatial variation in the phases of the FLRs. The temporal dephasing on adjacent field lines can narrow the FLRs ({\it Rankin et al., } 1995). When the width of the resonances is comparable to electron inertial length $\lambda_{e}$, the mode conversion of SAW into inertial Alfv\'{e}n wave can take place. The sharp density gradient due to ponderomotive forces can facilitate the mode conversion process. The electron inertial wave propagates the energy away from the resonance and saturates the amplitudes of the FLRs. The dispersion relation of the inertial wave is given as: \be \omega^{2}= \frac {k_{z}^{2}v_{z}^{2}}{1+k_{\perp}^2\lambda_{e}^{2}}. \ee Here, $\lambda_e= \frac{c} {\omega_pe}$ is the electron inertial length. One of the 3D computer simulations including nonlinear effects and electron inertia were carried out by {\it Wei et al. } (1994). \\ \ni {\it Generation of higher harmonics:} The density perturbations due to the ponderomotive forces result in the excitation of slow magnetosonic wave (SMW) along the ambient magnetic field. The nonlinear interaction between the SMW and FLRs results in the generation of higher harmonics. These higher harmonics can restructure the magnetic field of the pre-existing auroral arc and can lead to the filamentation of the FAC ({\it Rankin et al.,} 1995). Auroral ion acceleration and the formation of density cavities (in the ionosphere) by the ponderomotive forces has been studied in the past ( {\it Bellan and Stasiewicz,} 1998; {\it Li and Temerin, }1993; {\it Rankin et al., } 1999). The goal of the present work is to examine the nonlinear structuring of the auroral arc (by ponderomotive forces) in the presence of electron inertia and the space-time dependent conductivity. In contrast to linear theory the nonlinear effects can lead to rapid structuring and mode conversion of the FLRs ({\it Hasegawa, } 1976). These nonlinear effects due to the ponderomotive forces occur on the time scale $t_{nl}$ ({\it Rankin et al, } 1994). The observed SAW fields are generally of the order of 300-400 nT near the ionosphere. The FLR amplitudes (in the frequency range 1-1.3 mHz) can result in the density perturbation of order one in a time interval of of 5-10 SAW period ({\it Rankin et al.,} 1994; 1999). The density perturbations can lead to structuring of the auroral arc due to the physical effects listed above. \subsection {Mathematical Model} \ni To examine mathematically the nonlinear evolution of FLRs (based on the above FLR wave model), Earth's magnetosphere is approximated to a rectangular ``box" with straight magnetic lines ({\it Rankin et al., } 1994). The density and the field gradients are assumed in the $x$ direction (radial direction). The $y$ axis is along the azimuthal direction and the $z$ axis is along the ambient magnetic field. The MHD equations with the generalized Ohm's law are: \be \frac {\partial \rho} {\partial t}+ \nabla\cdot (\rho{\bf V}) &=& 0 \\ \rho \frac { d {\bf V}} {d t} &=& -\nabla P +\bf J\times B \\ \frac {\partial\bf B}{\partial t} &=& -\nabla\times \bf E \\ \nabla\times\bf B &=& \mu_{0}\bf J \\ \bf E+ \bf V\times B &=& \eta \bf J +\lambda_{e}^2\mu_{0}\frac {\partial J}{\partial t}\,, \ee \ni where $\rho (x)$ is the plasma density, $\bf v$ is the flow velocity, $P$ is the plasma pressure and $\bf J$ is the current density. The quantity $\bf B (x)$ is the magnetic field; $\bf E$ is the electric field; $\eta$ is the resistivity in the magnetosphere due to the electron conductivity and $m_{e}$ is the electron mass. The last term in Eq. (7) describes the electron inertial effects ({\it Lysak,} 1990). These effects play an important role when the characteristic length scale is comparable to the electron inertial length $\lambda_e$. The electron inertial effects lead to the development of a parallel electric field $\rm E_z$. The $\rm E_z$ field can accelerate auroral particles up to hundreds of eV in energies during substorms.\\ In the next two sections we will study the nonlinear evolution of FLRs in the presence of space-time dependent ionospheric conductivity.\\ \vskip 5mm \section { The role of space-time dependent ionospheric conductivity during substorms:} \subsection {Plane Polar Ionosphere} We model the Earth's ionosphere as a slab with height integrated Pedersen and Hall conductivities $\Sigma_{P}$ and $\Sigma_{H}$, respectively. The sheet ionosphere approach is justified because the extension of the ionosphere is a few hundred kilometers while the wavelength of the typical ULF pulsation is of the order of tens of thousands of kilometers. We define the $z$ axis as the positive upward direction. It should be noted that the conclusions of our studies are not significantly changed by assuming that the geomagnetic field lines are straight. The effects of the dipolar geometry can be incorporated by including an appropriate cosine factor.\\ \subsection {Field-aligned Current During Substorms:} During substorms, an enhanced magnetospheric convection sets up an electric field E in the ionosphere. The electric field drives a large ionospheric current. In the presence of non-uniform Pedersen and Hall conductivities, the ionospheric current is closed through the FAC in the magnetosphere ({\it Lysak,} 1990). The upward FAC is given as: \be J = \Sigma_{p}\nabla_{\perp}^2\phi+\nabla_{\perp}\Sigma_{p}\cdot\nabla_{\perp} \phi-\nabla_{\perp}\Sigma_{H}\times\nabla_{\perp}\phi\cdot\bf z. \ee Here, the conductivities $\Sigma_{p}$ and $\Sigma_{H}$ are space-time dependent. The FAC ($J_{alf}$) associated with FLRs is given as: \be J_{alf}= \frac {1}{\mu_0}\frac {\partial B_y}{\partial x}. \ee Using dispersion relation for Alfv\'{e}n waves: \be E_x= V_{A} B_{y}. \ee \be J_{alf}= \frac {1}{\mu_0 V_A}\frac {\partial E_x}{\partial x}. \ee The Alfv\'{e}n travel time between the two ends of the ionosphere is 30 s to few minutes. Therefore, in the steady state the current continuity equation for the auroral plasma implies that the FAC ($J_{alf}$) of the FLRs is equal to the ionospheric current given in Eq. (8). During substorms, the space-time dependent conductivities can lead to enhanced FAC due to space-time dependent conductivities. The enhanced FAC can intensify the pre-existing auroral arc. In a steady state, the Hall current closes within the ionospheric E-region and does not contribute to the FAC. However, when the steady state assumption is relaxed, the Hall current driven by the inductive electric field becomes divergent and closes through the magnetospheric FAC. The effects can become important at high frequencies or at hundreds of kilometers (the scale size of FLRs). The magnetospheric component of the Hall current can significantly modify the dynamic evolution of the auroral arc during substorms ({\it Buchert and Budnik, } 1997; {\it Yoshikawa and Itonaga,} 1996). The role of the divergent Hall current in the evolution of the auroral arc is a complex problem by itself. The present work will not address these studies further. \\ We will further assume either that Hall conductivity is uniform or that the gradient in the Hall conductivity is parallel to the applied electric field. Therefore, the contribution of the Hall term in Eq. (8) is negligible. For uniform ionospheric conductivity, each reflection of Alfv\'{e}n waves at the ionospheric end can enhance the FAC by a factor of two ({\it Kan,} 1991). The enhanced FAC can intensify the pre-existing auroral arc. The ponderomotive forces due to the enhanced magnetic field (associated with the current) increase and lead to restructuring of the auroral arc due to the effects listed in section 2.1 of the paper. The typical height integrated conductivity of the ionosphere is 3-10 mho. The conductivity gradient occurs on the length scale of 20 km. This length scale is much larger than the electron inertial length (a few km) in the auroral zone. Hence, the conductivity gradient does not introduce a new length scale which will lead to fine scale structuring of the auroral arc.\\ \vskip 5mm \section {Nonlinear evolution of FLRs in the presence of inhomogeneous and time-dependent conductivity} Following the approach developed in the literature, we derive amplitude equations describing the nonlinear evolution of FLRs in the presence of electron inertia and the space-time dependent ionospheric conductivity ({\it Frycz et al., } 1998; {\it Rankin et al.,} 1995). Since the magnetospheric plasma is coupled to the auroral region by FAC, the effects due to the space-time dependent conductivity are incorporated through the boundary conditions on the FAC.\\ We make following assumptions:\\ (1) The variation in the field quantities and ionospheric conductivity is along the $x$ direction only, and (2) the nonlinear effects enter through the azimuthal component $B_y$ of the magnetic field ({\it Rankin et al., } 1995). We have assumed $B_z\sim B_0$ and $B_y$ is much less than $B_0$. \\ \vskip 5mm \ni {\bf Nonlinear evolution of the FLRs in the presence of non-uniform Pedersen Conductivity:} \ni The FAC associated with FLRs is given as: \be J_{alf} = \Sigma_{p}\nabla_{\perp}^2\phi+\nabla\Sigma_{p}\cdot\nabla_{\perp}. \phi \ee Here, $E_{x}=- \nabla_{\perp}\phi$. \be J_{alf}= -\Sigma_{p}\frac {\partial E_x}{\partial x}-\frac {\partial \Sigma_p}{\partial x}E_x. \ee Using Faraday's law: \be \frac {\partial B_y}{\partial t}=-(\nabla\times\bf E)_{y}. \ee \be \frac {\partial B_{y}}{\partial t}=\frac {\partial E_z}{\partial x}-\frac {\partial E_x}{\partial z}. \ee \be \frac {\partial ^2B_y}{\partial t^2}= \frac {\partial ^2E_z}{\partial x\partial t}-\frac {\partial ^2 E_x}{\partial z\partial t}, \ee where the parallel electric field is given as: \be E_{z}=\lambda_{e}^2\mu_{0}\frac {\partial J_{z}}{\partial t}. \ee From Ampere's law: \be J_{z}= \frac {1}{\mu_0}\frac {\partial B_y}{\partial x}. \ee This leads to : \be E_z= \lambda_e^2\frac {\partial ^2 B_y}{\partial x\partial t}. \ee \be \frac {\partial ^2 E_z}{\partial x\partial t}= \lambda_{e}^2\frac {\partial ^4 B_y}{\partial x^2 \partial t^2}. \ee Next, we proceed to determine $\frac {\partial ^2 E_x}{\partial z\partial t}$ Taking the $x$ component of the Eq. (7): \be E_x+{({\bf V\times B})}_{x}=\eta J_x. \ee Using Ampere's law and momentum equation: \be J_x= -\frac {1}{\mu_0}\frac {\partial B_y}{\partial z}. \ee \be \frac {\partial V_y}{\partial t}= \frac {B_0}{\mu_0\rho_0}\frac {\partial B_y}{\partial z}- \frac {B_0}{\mu_0\rho_0}\frac {\partial B_z}{\partial y}. \ee Combining Eqs. (21-23), we obtain: \be \frac {\partial ^2 E_x}{\partial z \partial t}= 2\Gamma_{A}\frac {\partial B_y}{\partial t}- V_{A}^2 \frac {\partial ^2B_y}{\partial z^2}+V_{A}^2\frac {\partial ^2 B_z}{\partial y\partial z}. \ee Combining Eqs. (10; 16; 24) we obtain: \be \frac {\partial ^2B_y}{\partial t^2} +\Gamma_{A} \frac {\partial B_y}{\partial t}-V_{A}^2 \frac{\partial ^2B_y}{\partial z^2} = \lambda_{e}^2\frac {\partial ^4 B_y}{\partial z^2 \partial t^2}- V_{A}^2\frac {\partial ^2B_z}{\partial y\partial z}. \ee We use envelope approximation ({\it Rankin et al., } 1995): \be A(x,y,z,t)&= &Re[A(x,t) \exp(i\omega t-ik_{y}y)] \sin (k_z z) \,, \ee and neglect the higher derivatives of the slowly varying component $b(x,t)$. Using Eq.(26) and keeping terms to lowest order only we obtain an equation describing nonlinear evolution of the perturbed magnetic field b. \be \dot{b}+ 2 \Gamma_{A}b = -i\left (\Delta \omega -\frac {1}{4}\omega n \right) b- i\frac {\omega}{2}\lambda_{e}^2\frac {\partial ^2 b}{\partial x^2} + \frac {1}{2} R \omega. \ee The amplitude equation for the perturbed density n is given as ({\it Rankin et al., } 1994): \be \ddot{n} +\Omega^{2} n &=&-\frac {1}{2}\omega^{2}b^2. \ee \ni Here, R is the strength of the driver: \be R &=& \frac {k_yb_c}{k_z B_0}, \ee \ni where $b_c$ is a constant ({\it Rankin et al., } 1995) and the other symbols are as follows: \be b(t) = \frac {b_y}{B_0} , \quad {\rm and} \quad \frac {\delta \rho}{\rho_0} = n \cos(2k_z z),\quad \Omega = 2k_{z}C_{s}, \ee \be \Gamma_{A} =-\frac {1}{2} \frac {\eta}{\mu_0}\nabla^2, \quad {\rm and}\quad \Delta \omega (x) = \omega (x_{c}-x)/2L. \ee \ni The Eq. (27) assumes an Alfv\'{e}n wave profile of the following form, \be \frac {1}{V_A^2}\sim \frac {k_z^2}{\omega^2} \left( 1- \frac {x-x_c}{L}+ \frac {\delta\rho}{\rho_0}\right). \ee Here, L is the length scale characterizing the radial variation of the Alfv\'{e}n speed ($V_{A}$) around $x_{c}$. The point $x_c$ is determined by the resonance condition $V_{A}^2(x_{c})= \omega^2/k_z^2$. The quantity $\delta\rho$ (Eq. 32) denotes the density fluctuations about the density $\rho_{0}$.\\ It is clear from Eq. (27) that the ponderomotive forces and electron inertia result in the frequency shifts of the SAWs (which form FLRs) by $\frac {1}{4}\omega n$ and by $\frac {\omega}{2}\lambda_{e}^2 \frac {\partial^ 2 b}{\partial x^2}$, respectively ({\it Rankin et al., } 1994). The frequency shift (due to the ponderomotive forces) is proportional to the density perturbation (n) of the magnetosonic wave of frequency $\Omega$. The density perturbation (Eq. 28) is driven by the ponderomotive forces resulting from the large amplitude of the FLRs. In order to study the nonlinear evolution of FLRs in the presence of space-dependent Pedersen conductivity, the above equations should be are coupled with the boundary conditions on the FAC associated with FLRs. The boundary conditions at the two ionospheric ends are obtained by equating the FAC of FLRs with the FAC which closes the Pedersen current. \be \frac {1}{\mu_0}\frac {\partial B_y}{\partial x}= -\Sigma_{p}V_{A}\frac {\partial B_y}{\partial x}-\frac {\partial \Sigma_p}{\partial x}V_{A} B_{y}. \ee The Eqs (27,28) along with the boundary condition (Eq. 33) can be studied numerically to examine the nonlinear evolution of FLRs in the presence of space-dependent ionospheric conductivity. \\ {\bf Nonlinear evolution of FLRs in the presence of time-dependent Pedersen conductivity} \\ In this section, we study the case when the Pedersen conductivity is independent of ``x'' but is a function of the ``t'' co-ordinate only. We note that the time scale of variation of the ionospheric conductivity is much larger than Alfv\'{e}n bounce period. Therefore, during the time scale of variation in the conductivity, the steady state is not reached. Assuming that the time-dependent conductivity does not provide additional sources of charges we can use current continuity equation to yield boundary conditions at the two ionospheric ends: \be \frac {1}{\mu_0}\frac {\partial B_y}{\partial x}= -\Sigma_{p}V_{A}\frac {\partial B_y}{\partial x} \ee \ni Here, $\Sigma_p$ is time-dependent. The boundary condition (Eq. 34) is coupled with Eqs (27,28) to study the nonlinear evolution of FLRs in the presence of time-dependent ionospheric conductivity. Efforts are in progress to solve equations (28-29) numerically using the boundary conditions given by equations (33) and (34) with appropriate profiles for the ionospheric conductivity ({\it Richmond,} 1995). The time-dependent conductivity can modify the dynamical evolution of auroral arc observed during the substorms.\\ We list the plasma sheet parameters ({\it Hughes,} 1995) as follows:: $\rm T_e$ =1keV; $\rm T_i$ =0.5keV; $\rho_i$ =100km; $\rm B_{0}$ =20$\gamma$; $\rm n_{0}$ = $1\rm cm^{-3}$& ; $\beta$ =0.2. The wave and plasma parameters ({\it Wahlund et al, } 1994) in the auroral zone at the Freja altitude are give as,$\rm T_e$ =5eV; $\rm T_i$ =1eV; $\rm B_{0}$ = $3\times 10^4$ nT; $\rm n_{0}$ =100 $\rm cm^{-3}$; $\rm C_{s}$ =16 km/s; $\rm V_{A}$ = $15 \times 10^{3}$ km/s. Here, $\gamma$= $10^{-5}$ Gauss. In addition to the cold ion component, the ions with energies in the range 5-30 eV have also been reported at the Freja altitudes ({\it Wahlund et al., } 1998). In the cold plasma with low plasma $\beta$ value, the absence of thermal pressure fails to prevent the concentration of plasma into the nodes of the Alfv\'{e}n waves. Therefore, large magnitudes of density perturbations are possible, which can lead to structuring of the auroral arc. It has been reported that the nonlinear effects resulting from the the ponderomotive forces are strong in cold plasma, $\beta << b^2$ ({\it Rankin et al., } 1994). Therefore, with b of one per cent, we conclude (based on the above parameters) that the nonlinear effects can play a significant role in structuring the auroral arc in the cold component of the ions observed at the Freja altitudes. The effects due to the ponderomotive forces will be less prominent in the hot component of the auroral plasma observed at the Freja altitudes as well as near the plasma sheet. \section {Summary and Conclusions} We have derived amplitude equations and boundary conditions to study the nonlinear evolution of FLRs in the presence of electron inertia and the space-time dependent ionospheric conductivity. Efforts are in progress to solve these equations numerically to study the fine scale structuring of the FLRs and the auroral arc. The outcome of these studies will be compared with data available from the CANOPUS ground based network. The theoretical and data analysis studies will shed some light on the mechanism of the small scale structures of the auroral arcs observed during substorms. We note that the space-time dependent conductivity can modify the phase mixing length of the FLRs. These effects and the feedback of the FLR currents on the ionospheric conductivity will be examined in future. The Eqs.( 27-28;33; 34) provide the starting point to carry out extensive studies relating to these effects.\\ Our studies ignore the role of space-time dependent conductivity on the magnetosonic wave describing the density perturbations. These effects are essentially collisional and will result in the linear damping of the wave. These effects will not significantly modify the nonlinear evolution of the FLRs. We have also ignored the effects arising from the non-uniform Alfv\'{e}n speed and the temperature changes due to the inhomogeneous conductivity ({\it Lysak,} 1990). The space-time dependent conductivity can modify the growth rate of the feedback instability as well as the reflection of Alfv\'{e}n waves from the auroral acceleration region ({\it Lysak, } 1991; Trakhengertz and Feldstein, } 1984). The effects of the Hall current are significant in the dipole geometry and lead to the coupling between the toroidal and poloidal modes ({\it Allan and Knox,} 1979). These studies on the Hall current are also important in interpreting the ground signals of FLRs. 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