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Stationary MHD–equilibria of the heliotail flow
Stationary MHD-equilibria of the heliotail flow Dieter Nickeler & Hans Fahr ~ ~Institut fiir Astrophysik und extraterrestrische Forschung Auf dem Hiigel 71, 53121 Bonn, Germany
The solar wind, passing through the termination shock, is declined into the heliospheric tail and leaves the solar system as a subalfv~nic tail flow. In this article we present an analytical method to calculate nonlinear and selfconsistent stationary ideal MHD-equilibria with incompressible, field aligned magnetized plasma flow in the heliotail. In this case within the framework of a one fluid approximation it is possible to reduce the governing MHD-equations of the problem for stationary flow states to static MHDequilibria. In magnetohydrostatics (MHS), assuming quasi-independence with respect to one coordinate (-direction) in the plane perpendicular to the tail axis, the equations reduce then to one single nonlinear, elliptic partial differential equation, the Grad-Shafranovequation (GSE) (see e.g. Grad & Rubin [1]). First we calculate static equilibria to determine the geometry and topology of permitted field configurations and map this onto corresponding stationary equilibria with plasma flow to obtain the structure of the electric currents and the magnetic field.
1. I N T R O D U C T I O N Beyond the solar wind termination shock (TS) the plasma of the solar wind is decelerated and the magnetic field is amplified, so that there exists a subalfv~nic plasma flow in the downstream direction. Scherer et al. [2] showed that for small Mach numbers the bulk flow can be assumed to behave incompressible, which even more holds for field aligned flows, where the field lines are acting as quasi-isothermals. As the decelerated solar wind has to adapt to the outer magnetized VLISM (Very Local InterStellar Medium) conditions (e.g. thermal, ram and magnetic pressure) a contact or tangential discontinuity forms, the heliopause (HP), which is stretched into downwind direction. Between the TS and the HP there exists an inner heliosheath extending into the heliotail, similar to the Earth's magnetotail. In the magnetotail the MHD quantities mainly depend on one direction perpendicular to the tail axis, connected with the theta-shaped current system (e.g. Birn [3]). In view of evident similarities we here apply a similar description to the heliotail.
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D. Nickeler and H.J. Fahr 2. S T A T I O N A R Y S T A T E S I N I D E A L M H D The equations for ideal MHD are the conservation of mass, conservation of momentum with isotropic pressure P, Amp~re's law, and the induction equation reading -
•
o
.o5'
- 5" •
- p
e)
(1)
e •
(2) --+
..+
where p is the mass density, B is the magnetic field, g the velocity field, and j the electric current density vector. For incompressible flows with V. g - 0 we can define a flow vector ~ "- v ~ g" These two conditions imply g. V p - 0 and V . ~ - 0. We also introduce the 1--,2 Bernoulli-pressure II "- P + gw . .-+
2.1. Field aligned flows These stationary equilibria should later be tested with respect to their stability and be used to derive dynamical time scales, which may be important with respect to the VLISM-solar wind interaction. In the inconsistent picture given by Suess & Nerney ([4] and references therein) only a kinematic treatment (frozen-in test field approximation) is used. These authors find strong amplification of convected magnetic fields. In addition they identify a cone of 30 degrees, in which their kinematical approach is invalidated. Such orthogonal velocity fields (with respect to components of the magnetic field) have a saddle point structure in linear stability analyses (see Hameiri [5]) and are therefore not suitable to test stability. From this point of view it may be motivated to make the simplifying assumption of a field aligned flow, i.e 9 g - P xMA /~ ' with automatic fulfillment /-fi~ of the induction equation. With all these assumptions, made in section 2., the Euler equation reads VII-
1 (1- M~)(Vx/3)x/3#o
1 / 3 2 V ( 1 - M A 2)
(3)
2/-to
allowing to conclude, that the density p and the Alfv~n Mach number MA on field lines, however, varying perpendicular to them.
are
constant
2.2. Representation by Euler potentials In 1984 Zwingmann [6] had shown the similarity between MHS-equilibria and stationary MHD-equilibria with incompressible, field aligned flows. Later the theory was improved by Gebhardt & Kiessling [7], and used for modelling sunspot magnetic fields with plasma flow by Petrie & Neukirch [8]. In general, magnetic fields can be represented by using Euler potentials. The magnetic fields of MHS-equilibria used in this case are written as BMHS -- V f • Vg, where f and g are scalar functions (Euler potentials) of x, y, z. The equations below show how the magnetohydrostatic fields are mapped to the stationary fields, by performing the noncanonical transformation f = f(a,/5) and g = g(a, ~) with the Poisson bracket 0 < ([f, g],,Z)2 ._ 1 - M~ (valid for subalfv~nic flows).
BMUS
-58-
(4)
Stationary MHD-equilibria of the heliotailflow PMHS - PMHS(f, g)
~
1
P - PMHS- _~pg2
(5)
with the velocity field g - MA BMHS/V/#op (1 -- M~). In these equations f and g are the static potentials, while c~ and/5 are the stationary potentials. Since the post-shock plasma is very hot (see Jgger & Fahr [9]), we have high values of ~-~2, i.e. of the plasma/5, and of the sound velocity c~2 > v~ > V 2 , therefore we are allowed to use these transformations.
2.3. Two-dimensional equilibria Let f = A(x, y); g = z and the transformation c~ = ~(A);/3 = g, where the GSE is written as AA = - # 0 dPMHs/dA , valid under the assumptions adapted here, then the electric current of the stationary equilibrium is calculated by A a - d--~ + ~-~ -#o jz. In the first step we assume, that AA = 0 (potential field), which means that we can find suitable solutions by using a two-dimensional multipole representation. This enables us to find equilibria, e.g. with magnetic neutral points, to get stagnation points in front of the heliosphere, which can be used to receive a Parker-like configuration of flow trajectories/magnetic field lines. We use an extended form of a Laurent series (more in a forthcoming paper). For this general kind of a conformal mapping we exclude the domain around the singularity at (x, y) = (0, 0 ) , representing the inner region of the heliosphere. Symmetric systems have two kinds of boundaries: the axis of symmetry, here the x axis, for which we assume A = 0 and the other three boundaries for which we assume asymptotic boundary conditions, namely lime_~ A = B~ y, where ~ = ~/x2+ y2 and B ~ e~ is the asymptotic VLISM magnetic field, representing the influence of the VLISM. This ensures the homogeneity of the unperturbed VLISM magnetic field and the VLISM plasma flow. Therefore within a second step we have to use suitable transformations. To get three expected current sheets we use the mapping x
oz(A) - 6' A + ~ ak In cosh k=~
/
(6)
dk
where Yk are the locations of the current sheets, i.e. y~ and ya are the heliopause current sheets, and Y2 could be the continuation of the heliospheric or heliotail current sheet (with finite thickness). The other values are normalization constants (dk = thickness of current sheets, C ensures that doz/dA > 1, guaranteeing that the flow is subalfv~nic). 3. D I S C U S S I O N
AND CONCLUSIONS
In the following figures there are shown some representative MHD-values. In MHS, isocontours of the thermal pressure and of the current density are field lines. In Figs. 1 and 2 we can see that this is violated in the case of MA ~ O. If we calculate symmetric potential fields it is also possible to choose transformations to get equilibria with asymmetric field strengths, although the trajectories are symmetric. It is also possible to model a global heliosplaere with asymmetric trajectories, so the stagnation point can lie off the x-axis. The coefficients in equation (6) could be chosen in this way that different current directions are possible for the different current sheets. Therefore this technique permits a great flexibility to construct MHD-flows for stellar magnetospheres far away from the central star, moving through the VLISM.
-59-
D. Nickeler and H.J. Fahr
Figure 1. Symmetrical field/stream lines. The length scale is 100 AU, so the standoff distance of the heliopause is -1.5 (= 150AU).
Figure 2. Isocontours of the current density for a certain transformation, which corresponds to the symmetric field lines of the mapped MHS-equilibrium. The tail field lines are open, while the isocontours are closed. The sun is located in (x, y) = (0, 0).
A c k n o w l e d g e m e n t : We are grateful for financial support granted by the Deutsche Forschungsgemeinschaft within the project Fa 97/23-2. REFERENCES
1. H. Grad, H. Rubin, Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva 1958, Vol. 31, p.190 ft. 2. K. Scherer, H. Fahr, R. Ratkiewicz, A&A, 287 (1994), p. 219 3. J. Birn, JGR Vol. 92 (1987), p. 11101 4. S.T. Suess, S. Nerney, JGR, Vol. 100 (1995), p. 3463 5. E. Hameiri, Physics of Plasmas, Vol. 5 (1998), p. 3270 6. W. Zwingmann, PhD-thesis, Ruhr-Universit/it-Bochum (1984), Germany, p. 53 ft. 7. U. Gebhardt, M. Kiessling, Physics of Fluids, B 4(7) (1992), p. 1689 8. G. Petrie, T. Neukirch, Geophys. Astrophys. Fluid Dynamics, Vol. 91 (1999), p. 269 9. S. J/iger, H. Fahr, Solar Physics, Vol. 178 (1998), p. 631
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The solar wind, passing through the termination shock, is declined into the heliospheric tail and leaves the solar system as a subalfv~nic tail flow. In this article we present an analytical method to calculate nonlinear and selfconsistent stationary ideal MHD-equilibria with incompressible, field aligned magnetized plasma flow in the heliotail. In this case within the framework of a one fluid approximation it is possible to reduce the governing MHD-equations of the problem for stationary flow states to static MHDequilibria. In magnetohydrostatics (MHS), assuming quasi-independence with respect to one coordinate (-direction) in the plane perpendicular to the tail axis, the equations reduce then to one single nonlinear, elliptic partial differential equation, the Grad-Shafranovequation (GSE) (see e.g. Grad & Rubin [1]). First we calculate static equilibria to determine the geometry and topology of permitted field configurations and map this onto corresponding stationary equilibria with plasma flow to obtain the structure of the electric currents and the magnetic field.
1. I N T R O D U C T I O N Beyond the solar wind termination shock (TS) the plasma of the solar wind is decelerated and the magnetic field is amplified, so that there exists a subalfv~nic plasma flow in the downstream direction. Scherer et al. [2] showed that for small Mach numbers the bulk flow can be assumed to behave incompressible, which even more holds for field aligned flows, where the field lines are acting as quasi-isothermals. As the decelerated solar wind has to adapt to the outer magnetized VLISM (Very Local InterStellar Medium) conditions (e.g. thermal, ram and magnetic pressure) a contact or tangential discontinuity forms, the heliopause (HP), which is stretched into downwind direction. Between the TS and the HP there exists an inner heliosheath extending into the heliotail, similar to the Earth's magnetotail. In the magnetotail the MHD quantities mainly depend on one direction perpendicular to the tail axis, connected with the theta-shaped current system (e.g. Birn [3]). In view of evident similarities we here apply a similar description to the heliotail.
-57-
D. Nickeler and H.J. Fahr 2. S T A T I O N A R Y S T A T E S I N I D E A L M H D The equations for ideal MHD are the conservation of mass, conservation of momentum with isotropic pressure P, Amp~re's law, and the induction equation reading -
•
o
.o5'
- 5" •
- p
e)
(1)
e •
(2) --+
..+
where p is the mass density, B is the magnetic field, g the velocity field, and j the electric current density vector. For incompressible flows with V. g - 0 we can define a flow vector ~ "- v ~ g" These two conditions imply g. V p - 0 and V . ~ - 0. We also introduce the 1--,2 Bernoulli-pressure II "- P + gw . .-+
2.1. Field aligned flows These stationary equilibria should later be tested with respect to their stability and be used to derive dynamical time scales, which may be important with respect to the VLISM-solar wind interaction. In the inconsistent picture given by Suess & Nerney ([4] and references therein) only a kinematic treatment (frozen-in test field approximation) is used. These authors find strong amplification of convected magnetic fields. In addition they identify a cone of 30 degrees, in which their kinematical approach is invalidated. Such orthogonal velocity fields (with respect to components of the magnetic field) have a saddle point structure in linear stability analyses (see Hameiri [5]) and are therefore not suitable to test stability. From this point of view it may be motivated to make the simplifying assumption of a field aligned flow, i.e 9 g - P xMA /~ ' with automatic fulfillment /-fi~ of the induction equation. With all these assumptions, made in section 2., the Euler equation reads VII-
1 (1- M~)(Vx/3)x/3#o
1 / 3 2 V ( 1 - M A 2)
(3)
2/-to
allowing to conclude, that the density p and the Alfv~n Mach number MA on field lines, however, varying perpendicular to them.
are
constant
2.2. Representation by Euler potentials In 1984 Zwingmann [6] had shown the similarity between MHS-equilibria and stationary MHD-equilibria with incompressible, field aligned flows. Later the theory was improved by Gebhardt & Kiessling [7], and used for modelling sunspot magnetic fields with plasma flow by Petrie & Neukirch [8]. In general, magnetic fields can be represented by using Euler potentials. The magnetic fields of MHS-equilibria used in this case are written as BMHS -- V f • Vg, where f and g are scalar functions (Euler potentials) of x, y, z. The equations below show how the magnetohydrostatic fields are mapped to the stationary fields, by performing the noncanonical transformation f = f(a,/5) and g = g(a, ~) with the Poisson bracket 0 < ([f, g],,Z)2 ._ 1 - M~ (valid for subalfv~nic flows).
BMUS
-58-
(4)
Stationary MHD-equilibria of the heliotailflow PMHS - PMHS(f, g)
~
1
P - PMHS- _~pg2
(5)
with the velocity field g - MA BMHS/V/#op (1 -- M~). In these equations f and g are the static potentials, while c~ and/5 are the stationary potentials. Since the post-shock plasma is very hot (see Jgger & Fahr [9]), we have high values of ~-~2, i.e. of the plasma/5, and of the sound velocity c~2 > v~ > V 2 , therefore we are allowed to use these transformations.
2.3. Two-dimensional equilibria Let f = A(x, y); g = z and the transformation c~ = ~(A);/3 = g, where the GSE is written as AA = - # 0 dPMHs/dA , valid under the assumptions adapted here, then the electric current of the stationary equilibrium is calculated by A a - d--~ + ~-~ -#o jz. In the first step we assume, that AA = 0 (potential field), which means that we can find suitable solutions by using a two-dimensional multipole representation. This enables us to find equilibria, e.g. with magnetic neutral points, to get stagnation points in front of the heliosphere, which can be used to receive a Parker-like configuration of flow trajectories/magnetic field lines. We use an extended form of a Laurent series (more in a forthcoming paper). For this general kind of a conformal mapping we exclude the domain around the singularity at (x, y) = (0, 0 ) , representing the inner region of the heliosphere. Symmetric systems have two kinds of boundaries: the axis of symmetry, here the x axis, for which we assume A = 0 and the other three boundaries for which we assume asymptotic boundary conditions, namely lime_~ A = B~ y, where ~ = ~/x2+ y2 and B ~ e~ is the asymptotic VLISM magnetic field, representing the influence of the VLISM. This ensures the homogeneity of the unperturbed VLISM magnetic field and the VLISM plasma flow. Therefore within a second step we have to use suitable transformations. To get three expected current sheets we use the mapping x
oz(A) - 6' A + ~ ak In cosh k=~
/
(6)
dk
where Yk are the locations of the current sheets, i.e. y~ and ya are the heliopause current sheets, and Y2 could be the continuation of the heliospheric or heliotail current sheet (with finite thickness). The other values are normalization constants (dk = thickness of current sheets, C ensures that doz/dA > 1, guaranteeing that the flow is subalfv~nic). 3. D I S C U S S I O N
AND CONCLUSIONS
In the following figures there are shown some representative MHD-values. In MHS, isocontours of the thermal pressure and of the current density are field lines. In Figs. 1 and 2 we can see that this is violated in the case of MA ~ O. If we calculate symmetric potential fields it is also possible to choose transformations to get equilibria with asymmetric field strengths, although the trajectories are symmetric. It is also possible to model a global heliosplaere with asymmetric trajectories, so the stagnation point can lie off the x-axis. The coefficients in equation (6) could be chosen in this way that different current directions are possible for the different current sheets. Therefore this technique permits a great flexibility to construct MHD-flows for stellar magnetospheres far away from the central star, moving through the VLISM.
-59-
D. Nickeler and H.J. Fahr
Figure 1. Symmetrical field/stream lines. The length scale is 100 AU, so the standoff distance of the heliopause is -1.5 (= 150AU).
Figure 2. Isocontours of the current density for a certain transformation, which corresponds to the symmetric field lines of the mapped MHS-equilibrium. The tail field lines are open, while the isocontours are closed. The sun is located in (x, y) = (0, 0).
A c k n o w l e d g e m e n t : We are grateful for financial support granted by the Deutsche Forschungsgemeinschaft within the project Fa 97/23-2. REFERENCES
1. H. Grad, H. Rubin, Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva 1958, Vol. 31, p.190 ft. 2. K. Scherer, H. Fahr, R. Ratkiewicz, A&A, 287 (1994), p. 219 3. J. Birn, JGR Vol. 92 (1987), p. 11101 4. S.T. Suess, S. Nerney, JGR, Vol. 100 (1995), p. 3463 5. E. Hameiri, Physics of Plasmas, Vol. 5 (1998), p. 3270 6. W. Zwingmann, PhD-thesis, Ruhr-Universit/it-Bochum (1984), Germany, p. 53 ft. 7. U. Gebhardt, M. Kiessling, Physics of Fluids, B 4(7) (1992), p. 1689 8. G. Petrie, T. Neukirch, Geophys. Astrophys. Fluid Dynamics, Vol. 91 (1999), p. 269 9. S. J/iger, H. Fahr, Solar Physics, Vol. 178 (1998), p. 631
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