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Instability at the Leading Edge of a Reconnection Jet
INSTABILITY AT THE LEADING EDGE OF A RECONNECTION JET R. TanDokorol and M. Fujimotol
1Dept. Earth Planet. Sci., Tokyo Inst. Tech., Meguro, Tokyo 152-8551, J A P A N ABSTRACT In a transient reconnection process, the leading edge of the reconnection jet pushes and compresses the plasma standing ahead of it. This causes curved magnetic field lines and a strong pressure gradient to develop at the edge. It implies that the leading edge of the jet could be unstable to the ballooning instability. We studied this situation by three-dimensional MHD simulations and clarified that the interface indeed becomes unstable. The leading edge on the current sheet plane is deformed into a wavy shape and then to a mushroom-like pattern subsequently. The growth rate of the instability is controlled by the wavelength in the current-wise direction with a shorter wavelength mode growing faster. The dispersion curve obtained from a series of simulations is given.
Introduction Magnetic reconnect ion is one of the most important processes in space physics, in which magnetic energy is converted to kinetic energy of plasma (e.g., Priest and Forbes, 2000). Ill tile process, plasma as well as magnetic field outside the current simet is convected toward the diffusion region located within the current sheet and a pair of hot jets are ejected out of the region along the current sheet. In a transient case where reconnection is initiated at a certain time, there stands ahead of the accelerated jets the pre-existing current sheet plasma at rest. The interaction between the reconnect ion jet and the standing plasma is one of the key issues in space plasma.s, whereby some visible effects of the energcti(: phenomena in the magnetotail, the solar corona, etc., are mediated (e.g., Haerendel, 1992; Masuda et al., 1994). In the interaction, curved field lines an(t a significant pressure gradient are t)ro(luced at the jet front. It is quite likely that the interface is unstable to a kind of interchange instability, or the ballooning instability to be specific, when the third-dimension along the current is taken into account. The saturated state of the two-dimensional tearing instability, in which a similar situation arises, is shown by Dahlburg et al. (1992) to bc unstable to three-dimensional disturbances. A full particle simulation of a reconnection jet has shown that a highly-localized strong plasma flows develops due to coupling to an interchange and a kink modes (Prit(:hett and Coroniti, 1997; Pritchett, 2001), whose wavelengths are order of the ion gyro-radius. Coupling to an interchange mode is reported also in hybrid simulations (Nakamura et al, 2002a, 2002b). In this paper, we show results from three-dimensional MHD simulations of reconnection to demonstrate that the instability indeed exists and show its dispersion relation within the MHD approximation obtained from a series of simulations. Simulation Model In this study, we have conducted three dimensional MHD simulations to examine the behavior of the leading edge of the reconnection jet. The well-known MHD equations with anomalous resistivity fixed in space and time are used. We use the usual coordinates system in magnetospheric physics: The x axis along the tail axis directed positive (negative) earthward (tailward), the y axis directed positive from dawn to dusk, and the z axis positive northward. The initial magnetic field in our simulation is anti-parallel B(z) = tanh(z)~'z. Here normalizations are follows: magnetic field by the the lobe magnetic field B0, plasma density by the lobe density no, spatial scale by the half thickness of the current sheet D, velocity
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Fig. 1. The Bz plots on the current sheet plane (z=0 plane) at T=60, 90, 120,and 150. The jet front becomes wavy and subsequently grows into a bubble-like pattern. by the lobe Alfv6n speed VA, and time by the Alfvdn transit time 7 = D/VA. The plasma temperature is assumed to be lmiform with the plasma beta in the lobe set to 0.5. Magnetic reconnection is initiated by the localized anomalous resistivity in the cllrrent sheet, whose spatial form is 7/ = ri0 exp [ - ( x / D ) 2 - (z/D)2]. 7]0 = 0.03 is chosen. Note that the resistivity does not change in the dawn-dusk (y) direction. The instability is seeded by adding perturbations to the initial equilibrium density and pressure. The perturbations have the form Z i a i sin(kv,iy ) exp [ - ( z / L x / 2 ) 2 - ( z / D ) 2 , that is, periodic in the y direction and spatially confined in the x and z directions (L, is the system size in the x direction). We have confirmed that only the (list~lrbances symmetric: at)~mt the z=0 plane grow significantly. The results shown in this paI)er are from the rims in which symmetric |~ollndary ~'onditiolls at ttm z=0 are imposed. Simulation Results We first show the results of a case where the system size in the y direction L u is 40 and the initial perturbations of ky,i = 2~Lv/i, i = 1 ~ 5, are considered. The amplitude of the initial perturbation of each mode is set the same. Figure 1 shows temporal development of the Bz component on the current sheet (z=0) plane. At T=60, plasma compression develops around the interface between the reconnection jet and pre-existing plasma ahead of the jet. The Bz component is piling-up there. Slight wavy deformation of the interface is visible at this time. The deformation grows to show a bubble-like configuration at T=90. Significant portion of reconnected magnetic field lines are converging into the fore-moving bubbles at T = 120 to result in a mushroom-like shape. The magnetic bubbles are accelerated faster than other parts, and the interface between the jet and the plasma ahead of it, which would be a single line without the instability, is now highly elongated in the z direction. The above resulted from an initial condition composed of a superposition of 5 modes. We can find that the shortest wavelength mode (A=8) obviously dominates throughout. To examine how the growth of the instability depends on the wavelength, we have conducted a series of runs with only a single mode included in each run. Figure 2 shows the same as in Figure 1 but from various single mode calculations at T=100. In each panel, the wavelength of the initial perturbation is the same as the system size Ly = 5, 10, 20, and 40. For shorter wavelength cases shown in Figure 2a and 2b, a mushroom-like structure has already developed by this time and the jet's leading part is becoming elongated. In contrast, the ,~=20 and A= 40 cases in Figure 2c and 2d show that the interface is deformed but only to a wavy shape by this time.
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Fig. 2. The Bz plots on the current sheet plane (z=0 plane) at T=100. The panels show the results from various runs in which wavelength are changed: (a)Ly-5, (b)Ly-10, (c)Ly=20, and (d)Ly=40. To measure the growth rates ~, quantitatively, we have measured the quantity ABz(t) -- max{Bz,maz(x, t)Bz,min(x,t)}, where Bz,maz(X, t) and Bz,min(X, t) are the maximum and the minimum value, respectively, of the Bz component along a given constant x on the z - 0 plane at time t. Having found that this ABz(t) to show linear growth in the small amplitude phase for various cases, we have decided to adopt the growth rate of this quantity as the growth rate of the instability. We note that, because the background field is changing in time at a pace comparable to tile growth rate of the instability, we would think that this experimental rather than a theoretical determination of the growth rate is a reasonable choice at present. Figure 3 shows the dispersion relation of the instability obtained in this way. It can be seen that the larger wavenumber (shorter wavelength) modes have larger growth rates. The increa.se of the growth rate is more or less saturated around ky ~ 1 presumably due to the finite width effect of the jet's leading edge. This saturation makes the growth rate range to be limited within a factor of 2 even though the wavenurnber is varied for about two-orders of magnitudes.
Discussion We have shown that, in a transient reconnection process, the interface between the jet and tile current sheet plasma standing ahead of it is unstable to an inter(:hange instability. The shorter wavelength nlo(le growing faster than the longer wavelength mode is a typical feature of interchange instabilities. The geometry of the interface in which curved field lines are collocated with a sharp pressure gradient makes us to conclude that the instability is the ballooning instability. This conclusion is supported by tile results from other runs in which tile plasma density at the current sheet (z = 0) is reduced while keeping the plasma pressure the same. The growth of the instability is faster when the initial density at z = 0 is lower. The change in the initial density profile changes the density structure at the jet leading edge but the pressure structure stays essentially the same. The density gradient at the jet leading edge develops less for lower initial z = 0 density. This implies that it is the steep pressure gradient but not a density gradient that is crucial for the instability. Unlike the Raleigh-Taylor instability, which is the most probable alternative candidate that requires Vn, only the Vp effect is crucial for the ballooning. Thus the results are in favor of the ballooning mode. Because of the velocity difference between the fore-going part of the mushroom-like structure and those left behind, the jet's leading part is elongated. When the elongation provides enough space for a velocity shear instability, one would expect this secondary instability to develop at the sides of the elongated spines. Our preliminary results show this to be true. The growth rates differ only by a factor of two when the wavenumbers are varied over the two-orders of magnitudes range. This implies that the results may depend heavily on the initial power spectrum shape of the perturbation. We have shown the shortest wavelength mode to dominate when a run is started from a -145-
Fig. 3. The dispersion relation -~ versus ky of the instability obtained from a series of single-mode simulations.
flat spectrum. The results from a run Inay change significantly when a power-law st)ectrum is considered for the initial perturbation, in which a long wavelengtll mode may dominate if the spectrum is steep enougll. Understanding how the resultant field line structure would (:hange a(:cor(ting to the initial spectrum is important, for the stru(:ture should have significant impacts on parti(:le energization processes taking t)la(:e at the site. Wc have studied only the (:ases without a guide field. The guide field effe(:ts (:oul(t be either ways, bringing more complication into the interface structure up to a moderate value but suppressing the instability beyon(t a certain value. This issue will be studied in a near filture.
Acknowledgments R. T. thanks Dr. M. S. Nakamura fi)r fruitflll discussion and usefld r A C T - J S T project 12D-1.
M. F. is a member of the
REFERENCES Dahlburg, R. B., et al., Secondary instability in three-dimensional magnetic reconnection, Phys. Fluids B. vol. 4. 3902, 1992. Hacrcndel, G., Disruption, ballooning, or auroral avalanche - on the cause of sut)stroms, Proc. ICS-1, ESA, Kiruna, 417, 1992. Masuda, S., et al., A loop-top hard X-ray source in a compact solar flare, Nature, 371, 495, 1994. Nakamura, M. S., M. Fujimoto, and H. Mastulnoto, Instability at interface between reconnection jet and pre-existing pla~sma sheet, Adv. Space Res., 29/7, 1125, 2002a. Nakamura, M. S., H. Matsumoto, and M. Fujimoto, Interchange instability at the leading part of reconnection jets, Geophys. Res. Lett., 29, 10.1029/2001GL013780, 2002b. Priest, E., and T. Forbes, Magnetic reconnection, Cambridge Univ. Press, Cambridge, 2000. Pritchett, P.L., Collisionless magnetic reconnection in a three-dimensional open system, J. Geophys. Res., 106, 25,961, 2001 Pritchett, P. L., and Coroniti, F. V., Interchange and kink modes in the near-Earth plasma sheet and their associated plasma flows, Geophys. Res. Lett., vol. 24, 1997.
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1Dept. Earth Planet. Sci., Tokyo Inst. Tech., Meguro, Tokyo 152-8551, J A P A N ABSTRACT In a transient reconnection process, the leading edge of the reconnection jet pushes and compresses the plasma standing ahead of it. This causes curved magnetic field lines and a strong pressure gradient to develop at the edge. It implies that the leading edge of the jet could be unstable to the ballooning instability. We studied this situation by three-dimensional MHD simulations and clarified that the interface indeed becomes unstable. The leading edge on the current sheet plane is deformed into a wavy shape and then to a mushroom-like pattern subsequently. The growth rate of the instability is controlled by the wavelength in the current-wise direction with a shorter wavelength mode growing faster. The dispersion curve obtained from a series of simulations is given.
Introduction Magnetic reconnect ion is one of the most important processes in space physics, in which magnetic energy is converted to kinetic energy of plasma (e.g., Priest and Forbes, 2000). Ill tile process, plasma as well as magnetic field outside the current simet is convected toward the diffusion region located within the current sheet and a pair of hot jets are ejected out of the region along the current sheet. In a transient case where reconnection is initiated at a certain time, there stands ahead of the accelerated jets the pre-existing current sheet plasma at rest. The interaction between the reconnect ion jet and the standing plasma is one of the key issues in space plasma.s, whereby some visible effects of the energcti(: phenomena in the magnetotail, the solar corona, etc., are mediated (e.g., Haerendel, 1992; Masuda et al., 1994). In the interaction, curved field lines an(t a significant pressure gradient are t)ro(luced at the jet front. It is quite likely that the interface is unstable to a kind of interchange instability, or the ballooning instability to be specific, when the third-dimension along the current is taken into account. The saturated state of the two-dimensional tearing instability, in which a similar situation arises, is shown by Dahlburg et al. (1992) to bc unstable to three-dimensional disturbances. A full particle simulation of a reconnection jet has shown that a highly-localized strong plasma flows develops due to coupling to an interchange and a kink modes (Prit(:hett and Coroniti, 1997; Pritchett, 2001), whose wavelengths are order of the ion gyro-radius. Coupling to an interchange mode is reported also in hybrid simulations (Nakamura et al, 2002a, 2002b). In this paper, we show results from three-dimensional MHD simulations of reconnection to demonstrate that the instability indeed exists and show its dispersion relation within the MHD approximation obtained from a series of simulations. Simulation Model In this study, we have conducted three dimensional MHD simulations to examine the behavior of the leading edge of the reconnection jet. The well-known MHD equations with anomalous resistivity fixed in space and time are used. We use the usual coordinates system in magnetospheric physics: The x axis along the tail axis directed positive (negative) earthward (tailward), the y axis directed positive from dawn to dusk, and the z axis positive northward. The initial magnetic field in our simulation is anti-parallel B(z) = tanh(z)~'z. Here normalizations are follows: magnetic field by the the lobe magnetic field B0, plasma density by the lobe density no, spatial scale by the half thickness of the current sheet D, velocity
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Fig. 1. The Bz plots on the current sheet plane (z=0 plane) at T=60, 90, 120,and 150. The jet front becomes wavy and subsequently grows into a bubble-like pattern. by the lobe Alfv6n speed VA, and time by the Alfvdn transit time 7 = D/VA. The plasma temperature is assumed to be lmiform with the plasma beta in the lobe set to 0.5. Magnetic reconnection is initiated by the localized anomalous resistivity in the cllrrent sheet, whose spatial form is 7/ = ri0 exp [ - ( x / D ) 2 - (z/D)2]. 7]0 = 0.03 is chosen. Note that the resistivity does not change in the dawn-dusk (y) direction. The instability is seeded by adding perturbations to the initial equilibrium density and pressure. The perturbations have the form Z i a i sin(kv,iy ) exp [ - ( z / L x / 2 ) 2 - ( z / D ) 2 , that is, periodic in the y direction and spatially confined in the x and z directions (L, is the system size in the x direction). We have confirmed that only the (list~lrbances symmetric: at)~mt the z=0 plane grow significantly. The results shown in this paI)er are from the rims in which symmetric |~ollndary ~'onditiolls at ttm z=0 are imposed. Simulation Results We first show the results of a case where the system size in the y direction L u is 40 and the initial perturbations of ky,i = 2~Lv/i, i = 1 ~ 5, are considered. The amplitude of the initial perturbation of each mode is set the same. Figure 1 shows temporal development of the Bz component on the current sheet (z=0) plane. At T=60, plasma compression develops around the interface between the reconnection jet and pre-existing plasma ahead of the jet. The Bz component is piling-up there. Slight wavy deformation of the interface is visible at this time. The deformation grows to show a bubble-like configuration at T=90. Significant portion of reconnected magnetic field lines are converging into the fore-moving bubbles at T = 120 to result in a mushroom-like shape. The magnetic bubbles are accelerated faster than other parts, and the interface between the jet and the plasma ahead of it, which would be a single line without the instability, is now highly elongated in the z direction. The above resulted from an initial condition composed of a superposition of 5 modes. We can find that the shortest wavelength mode (A=8) obviously dominates throughout. To examine how the growth of the instability depends on the wavelength, we have conducted a series of runs with only a single mode included in each run. Figure 2 shows the same as in Figure 1 but from various single mode calculations at T=100. In each panel, the wavelength of the initial perturbation is the same as the system size Ly = 5, 10, 20, and 40. For shorter wavelength cases shown in Figure 2a and 2b, a mushroom-like structure has already developed by this time and the jet's leading part is becoming elongated. In contrast, the ,~=20 and A= 40 cases in Figure 2c and 2d show that the interface is deformed but only to a wavy shape by this time.
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Fig. 2. The Bz plots on the current sheet plane (z=0 plane) at T=100. The panels show the results from various runs in which wavelength are changed: (a)Ly-5, (b)Ly-10, (c)Ly=20, and (d)Ly=40. To measure the growth rates ~, quantitatively, we have measured the quantity ABz(t) -- max{Bz,maz(x, t)Bz,min(x,t)}, where Bz,maz(X, t) and Bz,min(X, t) are the maximum and the minimum value, respectively, of the Bz component along a given constant x on the z - 0 plane at time t. Having found that this ABz(t) to show linear growth in the small amplitude phase for various cases, we have decided to adopt the growth rate of this quantity as the growth rate of the instability. We note that, because the background field is changing in time at a pace comparable to tile growth rate of the instability, we would think that this experimental rather than a theoretical determination of the growth rate is a reasonable choice at present. Figure 3 shows the dispersion relation of the instability obtained in this way. It can be seen that the larger wavenumber (shorter wavelength) modes have larger growth rates. The increa.se of the growth rate is more or less saturated around ky ~ 1 presumably due to the finite width effect of the jet's leading edge. This saturation makes the growth rate range to be limited within a factor of 2 even though the wavenurnber is varied for about two-orders of magnitudes.
Discussion We have shown that, in a transient reconnection process, the interface between the jet and tile current sheet plasma standing ahead of it is unstable to an inter(:hange instability. The shorter wavelength nlo(le growing faster than the longer wavelength mode is a typical feature of interchange instabilities. The geometry of the interface in which curved field lines are collocated with a sharp pressure gradient makes us to conclude that the instability is the ballooning instability. This conclusion is supported by tile results from other runs in which tile plasma density at the current sheet (z = 0) is reduced while keeping the plasma pressure the same. The growth of the instability is faster when the initial density at z = 0 is lower. The change in the initial density profile changes the density structure at the jet leading edge but the pressure structure stays essentially the same. The density gradient at the jet leading edge develops less for lower initial z = 0 density. This implies that it is the steep pressure gradient but not a density gradient that is crucial for the instability. Unlike the Raleigh-Taylor instability, which is the most probable alternative candidate that requires Vn, only the Vp effect is crucial for the ballooning. Thus the results are in favor of the ballooning mode. Because of the velocity difference between the fore-going part of the mushroom-like structure and those left behind, the jet's leading part is elongated. When the elongation provides enough space for a velocity shear instability, one would expect this secondary instability to develop at the sides of the elongated spines. Our preliminary results show this to be true. The growth rates differ only by a factor of two when the wavenumbers are varied over the two-orders of magnitudes range. This implies that the results may depend heavily on the initial power spectrum shape of the perturbation. We have shown the shortest wavelength mode to dominate when a run is started from a -145-
Fig. 3. The dispersion relation -~ versus ky of the instability obtained from a series of single-mode simulations.
flat spectrum. The results from a run Inay change significantly when a power-law st)ectrum is considered for the initial perturbation, in which a long wavelengtll mode may dominate if the spectrum is steep enougll. Understanding how the resultant field line structure would (:hange a(:cor(ting to the initial spectrum is important, for the stru(:ture should have significant impacts on parti(:le energization processes taking t)la(:e at the site. Wc have studied only the (:ases without a guide field. The guide field effe(:ts (:oul(t be either ways, bringing more complication into the interface structure up to a moderate value but suppressing the instability beyon(t a certain value. This issue will be studied in a near filture.
Acknowledgments R. T. thanks Dr. M. S. Nakamura fi)r fruitflll discussion and usefld r A C T - J S T project 12D-1.
M. F. is a member of the
REFERENCES Dahlburg, R. B., et al., Secondary instability in three-dimensional magnetic reconnection, Phys. Fluids B. vol. 4. 3902, 1992. Hacrcndel, G., Disruption, ballooning, or auroral avalanche - on the cause of sut)stroms, Proc. ICS-1, ESA, Kiruna, 417, 1992. Masuda, S., et al., A loop-top hard X-ray source in a compact solar flare, Nature, 371, 495, 1994. Nakamura, M. S., M. Fujimoto, and H. Mastulnoto, Instability at interface between reconnection jet and pre-existing pla~sma sheet, Adv. Space Res., 29/7, 1125, 2002a. Nakamura, M. S., H. Matsumoto, and M. Fujimoto, Interchange instability at the leading part of reconnection jets, Geophys. Res. Lett., 29, 10.1029/2001GL013780, 2002b. Priest, E., and T. Forbes, Magnetic reconnection, Cambridge Univ. Press, Cambridge, 2000. Pritchett, P.L., Collisionless magnetic reconnection in a three-dimensional open system, J. Geophys. Res., 106, 25,961, 2001 Pritchett, P. L., and Coroniti, F. V., Interchange and kink modes in the near-Earth plasma sheet and their associated plasma flows, Geophys. Res. Lett., vol. 24, 1997.
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