Anomalous resistivity and the non-linear evolution of the ion-acoustic instability

1. Anomalous resistivity and the non-linear evolution of the ion-acoustic instability Panagiota Petkaki British Antarctic Survey, Cambridge Paper ID: 57-OSP-A1276b Paper Title: (SP5/SP19) Anomalous resistivity and the non-linear evolution of the ion-acoustic instability Presentation Mode: Oral Date/Time: 6 JUL / 1430pm to 1445pm Magnetic reconnection requires the violation of ideal MHD by various kinetic-scale effects whose relative importance is uncertain. Recent research at the British Antarctic Survey has highlighted the potential importance of wave-particle interactions by showing that Vlasov simulations of unstable ion-acoustic waves predict an anomalous resistivity that is three orders of magnitude higher than a popular analytical quasi-linear estimate. Here, we investigate the non-linear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations. We compare the evolution to recent predictions and discuss how non-linearity may aid magnetic reconnection. Paper ID: 57-OSP-A1276b Paper Title: (SP5/SP19) Anomalous resistivity and the non-linear evolution of the ion-acoustic instability Presentation Mode: Oral Date/Time: 6 JUL / 1430pm to 1445pm Magnetic reconnection requires the violation of ideal MHD by various kinetic-scale effects whose relative importance is uncertain. Recent research at the British Antarctic Survey has highlighted the potential importance of wave-particle interactions by showing that Vlasov simulations of unstable ion-acoustic waves predict an anomalous resistivity that is three orders of magnitude higher than a popular analytical quasi-linear estimate. Here, we investigate the non-linear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations. We compare the evolution to recent predictions and discuss how non-linearity may aid magnetic reconnection.

2. Change in Electron inertia from wave-particle interactions In the presence of electrostatic or electromagnetic growing waves, the energy and momentum of the Particles can be changed by scattering by waves. Scattering involves energy and momentum exchange between waves and particles, which can change the direction and velocity of the particles. This process is analogous to the particle-particle collisions.In the presence of electrostatic or electromagnetic growing waves, the energy and momentum of the Particles can be changed by scattering by waves. Scattering involves energy and momentum exchange between waves and particles, which can change the direction and velocity of the particles. This process is analogous to the particle-particle collisions.

3. Anomalous Resistivity due to Ion-Acoustic Waves Resistivity from Wave-Particle interactions is important in Collisionless plasmas (Watt et al., GRL, 2002) We have studied resistivity from Current Driven Ion-Acoustic Waves (CDIAW) Used 1D Electrostatic Vlasov Simulations Realistic plasma conditions i.e. Te~Ti� Maxwellian and Lorentzian distribution function (Petkaki et al., JGR, 2003) Found substantial resistivity at quasi-linear saturation What happens after quasi-linear saturation Study resistivity from the nonlinear evolution of CDIAW We investigate the non-linear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations.

4. Evolution of Vlasov Simulation The simulation code used to study the ion-acoustic waves is a one-dimensional electrostatic Vlasov simulation code. Both electrons and protons are described using their distribution functions. The Vlasov equation is used to integrate forward the distribution functions, and Ampere�s Law is used to integrate forward the electric field. There is no magnetic field in the simulation. The use of Ampere�s Law avoids the need to apply explicit boundary conditions to the electric field. The boundary conditions for the distribution functions are periodic. The ion-acoustic waves are generated by giving the electrons a drift relative to the ions. This results in a finite current present in the system. Note that the electric field perturbations are calculated using the perturbations in the current, the spatially-averaged current is treated separately. This is an explicit finite-difference method which uses a predictor-corrector algorithm. It has the advantage of using first-order derivatives to calculate a second-order accurate solution. In order to evolve the electric field, a central difference method is used. It too is second-order, and tests of the simulation have shown that it gives identical results to using the MacCormack method for the electric field. The central difference method has the advantage of only involving one calculation per time step. The method is implemented as follows, remembering that equations involving the distribution function must be performed separately for each plasma species. Thus by using only first-order derivatives, a second-order accurate solution can be obtained. Tests of the code showed that accuracy is improved if this algorithm is applied in a ``flip-flop'' fashion: if the time step is even, then the predictor step is calculated using forward finite differences for the spatial and velocity derivatives and the corrector step using backward finite differences; if the time step is odd, then backward finite differences are used for the predictor step and forward finite differences for the corrector step. The simulation code used to study the ion-acoustic waves is a one-dimensional electrostatic Vlasov simulation code. Both electrons and protons are described using their distribution functions. The Vlasov equation is used to integrate forward the distribution functions, and Ampere�s Law is used to integrate forward the electric field. There is no magnetic field in the simulation. The use of Ampere�s Law avoids the need to apply explicit boundary conditions to the electric field. The boundary conditions for the distribution functions are periodic. The ion-acoustic waves are generated by giving the electrons a drift relative to the ions. This results in a finite current present in the system. Note that the electric field perturbations are calculated using the perturbations in the current, the spatially-averaged current is treated separately. This is an explicit finite-difference method which uses a predictor-corrector algorithm. It has the advantage of using first-order derivatives to calculate a second-order accurate solution. In order to evolve the electric field, a central difference method is used. It too is second-order, and tests of the simulation have shown that it gives identical results to using the MacCormack method for the electric field. The central difference method has the advantage of only involving one calculation per time step. The method is implemented as follows, remembering that equations involving the distribution function must be performed separately for each plasma species. Thus by using only first-order derivatives, a second-order accurate solution can be obtained. Tests of the code showed that accuracy is improved if this algorithm is applied in a ``flip-flop'' fashion: if the time step is even, then the predictor step is calculated using forward finite differences for the spatial and velocity derivatives and the corrector step using backward finite differences; if the time step is odd, then backward finite differences are used for the predictor step and forward finite differences for the corrector step.

5. Vlasov Simulation Initial Conditions CDIAW- drifting electron and ion distributions � Natural Modes in Unmagnetised Plasmas driven unstable in no magnetic field and in uniform magnetic field Centre of Current Sheet - driven unstable by current Apply white noise Electric field f? close to zero at the edges Maxwellian Drift Velocity - Vde = 1.2 x (2T/m)1/2 Mi=25 me, Ti=1 eV, Te = 2 eV ni=ne = 7 x 106 /m3

7. Time-Sequence of Full Electron Distribution Function Top figure : Anomalous resistivity Lower figure : Electron DF

8. Time-Sequence of Full Ion Distribution Function Top figure : Anomalous resistivity Lower figure : Ion DF

9. Ion-Acoustic Resistivity Post-Quasilinear Saturation Resistivity at saturation of fastest growing mode Resistivity after saturation also important Behaviour of resistivity highly variable Ensemble of simulation runs � probability distribution of resistivity values, study its evolution in time Evolution of the nonlinear regime is very sensitive to initial noise field Require Statistical Approach 104 ensemble run on High Performance Computing (HPCx) Edinburgh (1280 IBM POWER4 processors)

11. time 251.924 188.256 112.295 we 2.42415e-18 5.73702e-19 time 252.846 188.664 115.640 we 2.52313e-18 6.13928e-19 time 253.769 188.157 114.735 we 2.62207e-18 6.54139e-19 time 254.692 187.461 113.402 we 2.72037e-18 6.94223e-19 time 300.832 115.476 204.373 we 4.71294e-18 1.53517e-18time 251.924 188.256 112.295 we 2.42415e-18 5.73702e-19 time 252.846 188.664 115.640 we 2.52313e-18 6.13928e-19 time 253.769 188.157 114.735 we 2.62207e-18 6.54139e-19 time 254.692 187.461 113.402 we 2.72037e-18 6.94223e-19 time 300.832 115.476 204.373 we 4.71294e-18 1.53517e-18

14. Histogram of Anomalous resistivity values

15. Skewness and kurtosis of probability distribution of resistivity valuesskewness = 0kurtosis = 3for a Gaussian

16. Discussion Ensemble of 104 Vlasov Simulations of the current driven ion-acoustic instability with identical initial conditions except for the initial phase of noise field Variations of the resistivity value in the quasilinear and nonlinear phase The probability distribution of resistivity values Gaussian in Linear, Quasilinear, Non-linear phase A well-bounded uncertainty on any single estimate of resistivity. Estimation of resistivity at quasi-linear saturation is an underestimate. May affect likehood of magnetic reconnection and current sheet structure First, the anomalous resistivity values at any given time in the quasi-linear and non-linear phases are Gaussian distributed. This is a surprise to me, as I would have anticipated non-Gaussian distributions during this non-linear, out-of-equilibrium phase. However, we have noted Gaussian distributions in non-linear turbulent processes before (Watkins, Oughton, and Freeman, Planet, Space Sci, 49, 1233, 2001). This result is very useful to the Watt et al and Petkaki et al papers and the parameter study. It says that a well-bounded uncertainty can be placed on any single estimate of resistivity, e.g., at quasi-linear saturation. Secondly, the non-linear evolution of the instability eventually ceases (as you have observed before), the resistivity goes to zero and a well-defined distribution function results. What determines the form of this df should be very important for understanding non-linear plasmas, i.e., those in the real world ;-). Although I'm confident of the general result, it is not conclusively shown yet because of the difficulty of being sure that there is no aliasing, which has been a struggle to compete with throughout, but it's very nearly there now. First, the anomalous resistivity values at any given time in the quasi-linear and non-linear phases are Gaussian distributed. This is a surprise to me, as I would have anticipated non-Gaussian distributions during this non-linear, out-of-equilibrium phase. However, we have noted Gaussian distributions in non-linear turbulent processes before (Watkins, Oughton, and Freeman, Planet, Space Sci, 49, 1233, 2001). This result is very useful to the Watt et al and Petkaki et al papers and the parameter study. It says that a well-bounded uncertainty can be placed on any single estimate of resistivity, e.g., at quasi-linear saturation. Secondly, the non-linear evolution of the instability eventually ceases (as you have observed before), the resistivity goes to zero and a well-defined distribution function results. What determines the form of this df should be very important for understanding non-linear plasmas, i.e., those in the real world ;-). Although I'm confident of the general result, it is not conclusively shown yet because of the difficulty of being sure that there is no aliasing, which has been a struggle to compete with throughout, but it's very nearly there now.

17. References Petkaki P., Watt C.E.J., Horne R., Freeman M., 108, A12, 1442, 10.1029/2003JA010092, JGR, 2003 Watt C.E.J., Horne R. Freeman M., Geoph. Res. Lett., 29, 10.1029/2001GL013451, 2002 Petkaki P., Kirk T., Watt C.E.J., Horne R., Freeman M., in preparation Find the referenceFind the reference

18. Conclusions Ion-Acoustic Resistivity can be high enough to break MHD frozen-in condition Form of the distribution function of ions and electrons is important Gaussian statistics describes variation in ion-acoustic resistivity values Estimation of ion-acoustic resistivity can be used as input by other type of simulations

22. Grid of Vlasov Simulation Significant feature of the Code : Number of grid points to reflect expected growing wavenumbers - ranges of resonant velocities Spatial Grid : Nz=Lz/?z Largest Wavelength (Lz) ?z is 1/12 or 1/14 of smallest wavelength Velocity Grid Nv{e,i} =2 X (vcut/?v{e,i}) +1 vcut > than the highest phase velocity Vcut,e = 6 ? + drift velocity or 12 ? + drift velocity Vcut,i = 10 ? or 10 maximum phase velocity Time resolution Courant number One velocity grid cell per timestep Integration by MacCormack method, explicit finite difference equ., second order, in-pairs calculation of plasma momentsIntegration by MacCormack method, explicit finite difference equ., second order, in-pairs calculation of plasma moments

25. k = 2Te/Ti = 1.0Mi/Me = 25 Vde = 1.2 x qe

26. Initial ion temperature 1eV.Initial ion temperature 1eV.

27. Compare Anomalous Resistivity from Three Simulations S1 - Maxwellian - Vde = 1.35 x ? (? = (2T/m)1/2 ) Nz=547, Nve=1893, Nvi=227 S2 - Lorentzian - Vde = 1.35 x ? (? = [(2 ?-3)/2?]1/2 (2T/m)1/2 ) Nz=593, Nve=2667, Nvi=213 S3 - Lorentzian - Vde = 2.0 x ? (? = [(2 ?-3)/2?]1/2 (2T/m)1/2 ) Nz=625, Nve=2777, Nvi=215 Mi=25 me Ti=Te = 1 eV ni=ne = 7 x 106 /m3 Equal velocity grid resolution ?=2

28. Initial ion temperature 1eV. The stability analysis was performed for the real ion to electron mass ratio, $m_i/m_e=1836$. However, in the simulation work presented in the next section a reduced mass ratio $m_i/m_e = 25$ is used. The effect of the reduced mass ratio on the stability curves is shown in Figure~\ref{massratio}. Here we have plotted the absolute critical drift velocity as a function of $\kappa$ for $m_i/m_e=1836$ and $m_i/m_e=25$ both for $T_e/T_i=1$. The Maxwellian case is plotted as $\kappa=80$ for illustration purposes. CDIAW are excited for lower drift velocities in the real world than in the simulation one. Initial ion temperature 1eV. The stability analysis was performed for the real ion to electron mass ratio, $m_i/m_e=1836$. However, in the simulation work presented in the next section a reduced mass ratio $m_i/m_e = 25$ is used. The effect of the reduced mass ratio on the stability curves is shown in Figure~\ref{massratio}. Here we have plotted the absolute critical drift velocity as a function of $\kappa$ for $m_i/m_e=1836$ and $m_i/m_e=25$ both for $T_e/T_i=1$. The Maxwellian case is plotted as $\kappa=80$ for illustration purposes. CDIAW are excited for lower drift velocities in the real world than in the simulation one.

29. The reconnecting universe Most of the universe is a plasma. Most plasmas generate magnetic fields.

30. Hall MHD reconnection Physics Hall effect separates ion and electron length scales. Whistler waves important (not Alfven waves) Consequences fast reconnection insensitive to mechanism which breaks frozen-in Evidence Generates quadrupolar out-of-plane magnetic field. Observed in geospace[Ueno et al., J. Geophys. Res., 2003]

31. SOC Reconnection? Distributions of areas and durations of auroral bright spots are power law (scale-free) from kinetic to system scales [Uritsky et al., JGR, 2002; Borelov and Uritsky, private communication] Could this be associated with multi-scale reconnection in the magnetotail? Self-organisation of reconnection to critical state (SOC) [e.g., Chang, Phys. Plasmas, 1999] cf SOC in the solar corona [Lu, Phys. Rev. Lett., 1995]

32. Previous analytical work Analytical estimates of the resistivity due to ion-acoustic waves: Sagdeev [1967]: Labelle and Treumann [1988]: Both estimates assume Te � Ti which is not the case for most space plasma regions of interest (e.g. magnetopause). We�ve been studying resistivity due to wave-particle interactions in collisionless plasmas. Resistivity is important in collisionless plasmas as a mechanism which can violate the ideal MHD condition. This violation can result in magnetic reconnection or the generation of field-aligned electric fields. Previous analytical estimates and simulations of the resistivity due to current-driven ion-acoustic waves have concentrated on the regime where electron temperature far exceeds ion temperature. This is not always the case in space plasmas, and so it is necessary to use simulations to investigate. In a plasma with similar electron and ion temperatures, there needs to be a large current to excite unstable ion-acoustic waves. Observations do not always support the existence of large currents in regions where ion-acoustic waves are measured. However, the fact that these waves are measured in many regions of space plasma, and in laboratory plasma experiments indicates the need to study them in more detail for a range of plasma parameters. We�ve been studying resistivity due to wave-particle interactions in collisionless plasmas. Resistivity is important in collisionless plasmas as a mechanism which can violate the ideal MHD condition. This violation can result in magnetic reconnection or the generation of field-aligned electric fields. Previous analytical estimates and simulations of the resistivity due to current-driven ion-acoustic waves have concentrated on the regime where electron temperature far exceeds ion temperature. This is not always the case in space plasmas, and so it is necessary to use simulations to investigate. In a plasma with similar electron and ion temperatures, there needs to be a large current to excite unstable ion-acoustic waves. Observations do not always support the existence of large currents in regions where ion-acoustic waves are measured. However, the fact that these waves are measured in many regions of space plasma, and in laboratory plasma experiments indicates the need to study them in more detail for a range of plasma parameters.

33. Ion-Acoustic Waves in Space Plasmas Ionosphere, Solar Wind, Earth�s Magnetosphere Ion-Acoustic Waves � Natural Modes in Unmagnetised Plasmas driven unstable in no magnetic field and in uniform magnetic field Not affected by the magnetic field orientation (under certain conditions) Centre of Current Sheet - driven unstable by current Source of diffusion in Reconnection Region Current-driven Ion-Acoustic Waves � finite drift between electrons and ions Observations of Ion-Acoustic Waves, Ionosphere (Foster et al 1988, Wahlund et al, 1992, Forme et al 1995) Solar Wind (associated with suprathermal ions, Anderson et al 1981, Fuselier et al 1987 In plasma region where the frequency of the ion-acoustic waves (~=to the ion plasma frequency) is much higher Than the electron gyrofrequency, ion-acoustic waves are not affected by the orientation of the magnetic field.Observations of Ion-Acoustic Waves, Ionosphere (Foster et al 1988, Wahlund et al, 1992, Forme et al 1995) Solar Wind (associated with suprathermal ions, Anderson et al 1981, Fuselier et al 1987 In plasma region where the frequency of the ion-acoustic waves (~=to the ion plasma frequency) is much higher Than the electron gyrofrequency, ion-acoustic waves are not affected by the orientation of the magnetic field.

34. Reconnection and Geospace Courtesy of Mervyn Freeman Courtesy of Mervyn Freeman

35. Anomalous Resistivity due to Ion-Acoustic Waves 1-D electrostatic Vlasov simulation of resistivity due to ion-acoustic waves. Resistivity is 1000 times greater than Labelle and Treumann [1988] theoretical (quasi-linear) estimate (depending on realistic mass ratio) must take into account the changes in form of the distribution function. Consistent with observations in reconnection layer [Bale et al., Geophys. Res. Lett., 2002] Resistivity in non-Maxwellian and non-linear regimes.

36. Reconnection in Collisionless Plasmas Magnetosphere Magnetopause Magnetotail Solar Wind Solar Corona Stellar Accretion Disks Planetary Magnetospheres Pulsar Magnetospheres Direct in situ observations of magnetic reconnection are difficult to makeDirect in situ observations of magnetic reconnection are difficult to make

37. ?? ?? ? ?i - ?i+1

38. Important Conclusions on The Ion-Acoustic Resistivity