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Modeling HEDLA magnetic field generation experiments on laser facilities
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Modeling HEDLA magnetic field generation experiments on laser facilities Q4
M. Fatenejad a, *, A.R. Bell b, A. Benuzzi-Mounaix e, R. Crowston g, R.P. Drake f, N. Flocke a, G. Gregori b, M. Koenig c, C. Krauland f, D. Lamb a, D. Lee a, J.R. Marques c, J. Meinecke b, F. Miniati e, C.D. Murphy b, H.-S. Park d, A. Pelka c, A. Ravasio b, B. Remington d, B. Reville b, A. Scopatz a, P. Tzeferacos a, K. Weide a, N. Woolsey g, R. Young f, R. Yurchak c a
Flash Center for Computational Science, University of Chicago, Chicago, IL, USA Department of Physics, University of Oxford, UK LULI, Ecole Polytechnique, France d Lawrence Livermore National Laboratory, USA e ETH Zurich, Switzerland f Center for Radiative Shock Hydrodynamics, University of Michigan, USA g The University of York, UK b c
Q1
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 June 2012 Received in revised form 27 September 2012 Accepted 20 November 2012 Available online xxx
The Flash Center is engaged in a collaboration to simulate laser driven experiments aimed at understanding the generation and amplification of cosmological magnetic fields using the FLASH code. In these experiments a laser illuminates a solid plastic or graphite target launching an asymmetric blast wave into a chamber which contains either Helium or Argon at millibar pressures. Induction coils placed several centimeters away from the target detect large scale magnetic fields on the order of tens to hundreds of Gauss. The time dependence of the magnetic field is consistent with generation via the Biermann battery mechanism near the blast wave. Attempts to perform simulations of these experiments using the FLASH code have uncovered previously unreported numerical difficulties in modeling the Biermann battery mechanism near shock waves which can lead to the production of large non-physical magnetic fields. We report on these difficulties and offer a potential solution. Ó 2012 Published by Elsevier B.V.
Keywords: Magnetohydrodynamics Biermann battery FLASH
1. Introduction The universe is permeated by magnetic fields, with strengths ranging from a femtogauss in the voids between filaments, several microgauss in galaxies and galaxy clusters, and tens of kilogauss in ordinary stars to many teragauss in the vicinity of some black holes and neutron stars. These magnetic fields play a crucial role in astrophysical and cosmological phenomena. However, the origin and strength of these fields are not fully understood. The problem of understanding magnetic fields is two-fold: (1) understanding the generation of the seed fields which magnetic dynamos require in order to operate, and (2) understanding the dynamo process itself, which is generally thought to involve the interaction of turbulent flows with these seed fields. A very promising mechanism for the origin of seed magnetic fields in the universe is the asymmetric shocks that occur in
* Corresponding author. E-mail addresses: [email protected], milad@flash.uchicago.edu (M. Fatenejad).
hierarchical structure formation when smaller halos merge to form galaxies and galaxies merge to form clusters of galaxies [1e4]; for recent reviews, see Refs. [5,6]. By asymmetric, we mean shocks, or blast waves, that lack any 1D symmetry (either planar or spherical). These shocks can cause the generation of magnetic fields by thermoionic currents e the so-called Biermann battery mechanism (BBM) [7]. They may then be amplified to observed values by the turbulence in galaxies [4,8] and in galaxy clusters [8e10]. For recent reviews, see Refs. [5,6,11]. High-power lasers offer the possibility of generating asymmetric shocks in the laboratory and directly measuring magnetic fields generated by them. Many previous experiments [13e15] have measured magnetic fields generated in ablating plasmas produced by high-power lasers. These magnetic fields are not relevant to magnetic fields generated by cosmological shocks because they originate in regions of smooth flow that are supported by laser illumination of a target rather than originating near freely expanding asymmetric shocks. Recent experiments performed at the Laboratoire pour l’Utilisation des Lasers Intenses (LULI) facility use high-power lasers to generate asymmetric shock waves in gas
1574-1818/$ e see front matter Ó 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.hedp.2012.11.002
Please cite this article in press as: M. Fatenejad, et al., Modeling HEDLA magnetic field generation experiments on laser facilities, High Energy Density Physics (2012), http://dx.doi.org/10.1016/j.hedp.2012.11.002
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filled chambers, and measure the time-dependent magnetic fields produced near the shock by the BBM [16]. Using dimensionless scaling parameters, it was demonstrated that the fields measured in these experiments have strengths consistent with the seed fields in the intergalactic medium (IGM). With the experiments in-hand, the possibility now exists to validate simulation codes which are used to directly model astrophysical phenomena. Recent attempts to simulate these experiments using the multidimensional radiation magnetohydrodynamics (MHD) code FLASH [17] have uncovered previously unreported, significant challenges in modeling the BBM source term near shocks. These issues can lead to incorrect predictions of magnetic field strength. We believe that these challenges affect many widely used simulation codes and report on these challenges here. We then offer a potential partial solution that offers the possibility of accurately simulating the magnetic field generation experiments. Finally, we report some preliminary simulation results using it (Fig. 1).
2. Experiment description Fig. 2 shows a schematic of the magnetic field generation experiments performed at the LULI laser facility. To date, over 275 separate experiments have been performed where the laser properties, materials, and geometry have been altered. Here a single experimental configuration is described. A 500 mm diameter solid polystyrene sphere mounted on a thin glass stalk is placed in a chamber filled with Argon gas at 0.5 mbar pressure. The sphere is illuminated from one side with a single 400 J, 527 nm beam using a 1.5 ns square pulse. Polystyrene ablating off of the sphere expands outward, launching an asymmetric blast wave. The asymmetries are largely caused by the one-sided drive. Approximately 600 ns after the start of the laser pulse, the blast wave travels past the first of two 3-axis induction coils [18] placed 3.2 cm and 4 cm away from the target center. The induction coils measure the strength of the magnetic field as a function of time.
Fig. 1. Radio contours of the A255 cluster of galaxies overlaid on the ROSAT X-ray image of the cluster. The white þ and symbols show the positions of the centroid and peak of the cluster X-ray emission. The images at the corners of the figure show the Faraday rotation measure (FRM) for four radio galaxies in the cluster. The FRM measurements of the radio halo at the cluster center and these four radio galaxies imply a magnetic field strength that declines from the cluster center outward, with a field strength of 2.6 mG at the cluster center and an average magnetic field strength of w1.2 mG in the cluster core. Image source Ref. [12]
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3
vr þ V$ðrvÞ ¼ 0; vt
(1)
v B2 1 ðrvÞ þ V$ rvv þ P þ I BB ¼ 0; vt 4p 8p
(2)
h i v c Etot þ V$ ðEtot þ PÞv þ ðE BÞ ¼ 0; vt 4p
(3)
where r is the mass density, v is the velocity, P is the total plasma pressure, B is the magnetic field, E is the electric field, and Etot is the total energy density:
Etot ¼
Fig. 2. Schematic of an experiment to measure shock-generated magnetic fields [16].
Fig. 3 shows the magnetic field strength as measured in three shots with similar configurations. The peak in the magnetic field strength corresponds to the time at which the blast wave passes the coil location. The magnetic field strength is not large enough to be dynamically important. Thus, while measurable, the presence of a field does not significantly influence the motion of the plasma. The gradual rise in the field strength (from 300 to 600 ns in Fig. 3) is caused by the significant magnetic diffusivity present in the experiment. The magnetic Reynolds number is estimated to be approximately 16 [16]. This indicates that modeling resistivity will be important in MHD simulations. 3. Background The FLASH code, like many MHD codes, evolves equations representing the conservation of mass, momentum, and total energy in a plasma:
1 2 B2 rv þ e þ ; 2 8p
(4)
where e is the plasma internal energy density. The pressure is related to the density and internal energy through the equation of state:
P ¼ EOSðr; eÞ
(5)
The evolution of the magnetic field is governed by Faraday’s law:
vB ¼ cV E: vt
(6)
The electric field is set by the generalized Ohm’s law:
uB VPe þ hj E ¼ ; c ene
(7)
where j is the current density, h is the resistivity, Pe is the electron pressure, and ne is the electron number density. Equations (1e7) are standard representations of MHD and their derivations can be found in several texts such as Refs. [19] and [20]. Several approximations have been made including the assumption of charge neutrality and the neglect of the displacement current in Ampere’s law [20,21] so that j ¼ ðc=4pÞV B. The first term in Equation (7) is the dynamo term representing the generation of electric field due to the bulk motion of a magnetized plasma. The second term introduces resistive effects which allows the plasma to move relative to the magnetic field. The final term is the Biermann battery source which is responsible for the generation of fields in an unmagnetized plasma. The Biermann battery term arises from the mass difference between ions and electrons. Forces acting on the plasma accelerate electrons more quickly than ions, resulting in a net current and hence generation of magnetic fields. There is a corresponding decrease in plasma internal energy, e, to conserve total energy as implied by Equation (4). Inserting Equation (7) in Faraday’s law yields the induction equation
vB VPe ¼ V ðu BÞ cV ðhjÞ þ cV ; vt ene
(8)
where the final term is often expanded:
vB VPe Vne ¼ V ðu BÞ cV ðhjÞ þ c : vt en2e
Fig. 3. Magnetic field measurements from 3-axis coils. The peak in the field strength is inferred to correspond with the time at which the blast wave passes the coils.
(9)
In this form, it is clear that the Biermann battery term (BBT) is active where ever the electron pressure and density gradients are not aligned. Thus, B will remain zero in an initially unmagnetized plasma with a purely 1D (symmetric) flow, such as a perfectly spherical blast wave or a planar shock. In contrast, asymmetric shocks generate magnetic fields via the BBM downstream because
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behind the shock Vne and VPe are not parallel to the direction of the shock velocity. As a result, the BBT is non-zero, and therefore vB/vt is non-zero, leading to the generation of a magnetic field where none existed before.
3.1. The vorticity equation One common method [5] of estimating the magnetic field strength generated via the BBM is to take advantage of the similarity between the induction equation and the vorticity equation:
vu VP Vr ¼ V ðu uÞ ; vt r2
(10)
where u is the vorticity, P is the total plasma pressure, and r is the mass density. For a fully-ionized, single species, single temperature (Te ¼ Ti where Te is the electron temperature and Ti is the ion temperature) plasma, the electron number density can be written as ne ¼ zr/mi where z is the atomic number and mi is the ion mass. Assuming an ideal-gas equation of state, we can write the electron pressure as Pe ¼ zP/(zþ1). Applying these assumptions to the induction equation, and assuming no resistivity, Equation (9) yields:
vB cmi 1 VP Vr ¼ V ðu BÞ þ : vt e 1þz r2
(11)
Thus, the quantities (1 þ z)u and ui ¼ eB/cmi obey the same equations. Consequently, under the many constraints listed above, the vorticity can be used as a stand-in for the magnetic field, assuming the initial vorticity and magnetic field strength are zero. It is well known that asymmetric shocks (for example, standing bow-shocks) do generate vorticity [22,23] and hence asymmetric shocks will generate magnetic fields due to the similarity between the two quantities.
The use of the similarity to vorticity does not fully address this issue. The most obvious reason is that the assumptions that lead to the similarities in Equations (10) and (11) are not valid for many HEDP experiments, which often have high magnetic resistivity and multiple, partially-ionized species [26]. Furthermore, the evaluation of the vorticity itself requires computing numerical derivatives of a different discontinuous function, in this case the velocity, which will also lead to non-physical magnetic fields. The challenges described here have been discussed with the developers of a number of codes including CRASH [27], ENZO [28], and RAGE/CASSIO [29], which are Eulerian codes; and HYDRA [30], which is an Arbitrary LagrangianeEulerian code. We have determined that at the present none of them accurately treat the BBM near shocks.1 The artificial fields generated by direct discretization of Equation (8) or Equation (9) can be large. This is especially true for simulations of the experiments described in Section 2, where the artificial fields can dominate any physical result. It is therefore critical to address this challenge for performing realistic simulations of these experiments. A potential solution, which has been implemented in the FLASH code for 2D cylindrical simulations, attempts to circumvent this issue by detecting shocks. This can easily be accomplished by identifying regions where large velocity gradients exist. The Biermann battery source term is not computed in cells that are near shock waves and large artificial fields are therefore not produced. This allows the code to generate physically relevant magnetic fields in the downstream region, but not within the shock region itself. Physical magnetic fields can be generated within the shock region, and a limitation of the approach is that it ignores this source of the magnetic field which can be important in certain situations. Below we present the results of preliminary verification simulations that demonstrate how the shock detection method can be used to avoid the calculation of non-physical magnetic fields. 5. Verification simulation results
4. Simulation of the Biermann battery mechanism Some MHD simulation codes attempt to model the BBM using a discretization of Equation (9). This approach is reasonable in smooth flows (see for example Refs. [14,15]), but does not work near shock waves. The reason is that the battery source term contains gradients of Pe and ne e both of which may be (and usually are) discontinuous at shocks. Lagrangian [24] and Eulerian [25] simulation codes spread shocks over a small number of computational cells. The specific number of computational cells and details concerning the exact transition from the upstream to downstream state depend on the details of the specific numerical algorithm used. Mathematically, the electron pressure and density are discontinuous at a shock wave, but simulation codes treat them as quantities that vary quickly in the vicinity of the shock and have very large, purely numeric gradients. Thus, direct discretization of the electron pressure and number density gradients, using finite differences for example, near shock waves will lead to the generation of large, magnetic fields that are purely a numerical artifact. Moreover, these artificial fields do not converge and will increase in strength as the computation mesh becomes finer since the numerical gradients in Pe and ne increase with higher resolution. Many Eulerian finite-volume codes solve Equation (8) instead of Equation (9), where the former includes the source term as the curl of a flux, which has the units of an electric field. This approach does not cure the pathological behavior of the BBM at shocks, since, unlike ordinary hydrodynamic fluxes, the Biermann flux itself contains a gradient of Pe.
The challenges described in Section 4 can be demonstrated using a simple verification test. In the case of a purely 1D shock wave, the Biermann battery source term should be zero and no magnetic fields should be present if they are not introduced through initial or boundary conditions. We performed MHD simulations of a circularly expanding shock wave in 2D Cartesian (x y) geometry using FLASH. The simulations include the BBM, but did not include resistive effects. In these simulations, the flow is radial while the mesh is aligned with the x and y directions. The initialpconditions are defined such that there is only variation in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ x2 þ y2 . Initially B ¼ 0 throughout the domain and the velocity is zero. A fully-ionized ideal gas equation of state is used with Helium as the single material in the domain. The initial density is 0.0001 g/cc. The initial temperature is set to 0.5 eV for R > 0.01 microns and 14.66 eV for R < 0.01 microns. The simulation was performed for 1.8 ns with the Biermann battery source term active. Fig. 4 shows the magnetic field strength (in the z direction) at this time when no shock detection is used to suppress field generation near the shock. A large artificial field is generated in this
1 CRASH and ENZO accurately treat the BBM in smooth flow but do not do so at shocks (B. van der Holst and G. Toth, private communication, 2012; H. Xu, H. Li, and S. Li, private communication, 2011); RAGE/CASSIO do not currently have an MHD capability, though there are plans to add one (C. Fryer and J. Wohlbier, private communication, 2011); HYDRA accurately treats the BBM in smooth flow but not at shocks for which the flow is significantly misaligned with the computational mesh, though work is underway to address this (M.M. Marinak, private communication, 2012).
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Fig. 4. Density and magnetic field strength from a 2D Cartesian FLASH MHD simulation modeling a circularly expanding shock wave. The flow is not aligned with the mesh and no shock detection is used. Large, artificial magnetic fields are generated in this case.
case. The magnetic field strength does not diminish with increasing resolution. The overall field is still relatively small, with b > 104 where b represents the ratio of plasma pressure to magnetic pressure. These artificial fields are not dynamically important as is the case in the experimentally measured fields. Thus, the artificial fields will not substantially influence the motion of the plasma but must suppressed in order to obtain agreement with the experiment. Fig. 5 shows the same simulation with the Biermann battery source term suppressed. The white lines bound the region where the code detects a shock. The bounded area is approximately 5-cells
Fig. 5. Similar simulation as in Fig. 4, but shock detection is used to eliminate magnetic field generation in the 5 cells bounding the shock wave. The overall field strength is substantially reduced.
5
Fig. 6. Total magnetic energy as a function of cell size for the circularly expanding shock verification simulations. When shock detection is used, the energy decreases as the cell size is reduced.
across in the radial direction. The field strength is substantially reduced. The magnetic fields that are generated when shock detection is active are produced as a result of discretization error in the smooth flow regions. Because the shock is not aligned with the flow, the electron pressure and density gradients are not perfectly
Fig. 7. A laser illuminates a 0.1 mm thick Carbon foil located at z ¼ 0 with 450 J of 527 nm light in a 1 ns square pulse with a 0.4 mm FWHM Gaussian intensity profile. The laser enters from the bottom of the image. A shock breaks out of the foil after 2 ns into an Argon filled chamber. The image shows the density (g/cc) plot from a 2D cylindrical FLASH simulation at 50 ns. The blast wave is highly asymmetric and extends to 7 mm in the z direction and 3.5 mm in the R direction. The magnetic field strength (Gauss) is shown as an inset, colored red, white and blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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aligned. However, in this case, the error is convergent e as the mesh resolution is increased the total magnetic energy decreases, as shown in Fig. 6. This is not the case without shock detection, i.e. when simple differencing of the Biermann battery source term is used everywhere. 6. Preliminary validation simulation results The shock detection techniques have been used in preliminary 2D cylindrical MHD simulations of laser driven experiments. These simulations have a different geometry than those described in Section 2. Rather than a spherical target, a laser illuminates 0.1 mm thick Carbon foil, generating a shock which breaks out of the foil into an Argon filled chamber. Fig. 7 shows the mass density and magnetic field strength in a simulation after 50 ns. No magnetic resistivity is used in these simulations. The shock has expanded to cover a large fraction of the simulation domain. Shock detection is being used to suppress generation of unphysical magnetic fields at the shock wave. Behind the shock, the magnetic field strength is approximately 103 G. Future simulations will include resistivity and this will act to significantly reduce the peak field strength, but the results so far are promising. 7. Conclusions Recent experiments have demonstrated that astrophysically relevant magnetic fields can be produced via the Biermann battery mechanism acting downstream of asymmetric shocks. These experiments use high-power lasers to generate asymmetric blast waves in gas filled chambers and are promising validation experiments for MHD codes. Attempts to simulate these experiments using the multidimensional MHD code FLASH have revealed serious numerical challenges in modeling the Biermann battery source term near shock waves where electron pressure and number densities are discontinuous. These challenges affect many widely used simulation codes and have gone largely unnoticed, but we demonstrate them here using verification tests. A partial solution which uses shock detection to suppress the Biermann battery source in cells near discontinuities has been implemented in FLASH. This modification prevents the generation of large, artificial magnetic fields near shock waves in simulations and is being tested in full scale laser driven simulations. Acknowledgments This work was supported in part at the University of Chicago by the US Department of Energy NNSA ASC through the Argonne Institute for Computing in Science under field work proposal 57789; and the US National Science Foundation under grant PHY0903997. References [1] R.M. Kulsrud, R. Cen, J.P. Ostriker, D. Ryu, The protogalactic origin for cosmic magnetic fields, The Astrophysical Journal 480 (1997) 481. [2] K. Roettiger, J.M. Stone, J.O. Burns, Magnetic field evolution in merging clusters of galaxies, The Astrophysical Journal 518 (1999) 594. [3] G. Davies, L.M. Widrow, A possible mechanism for generating galactic magnetic fields, The Astrophysical Journal 540 (2000) 755.
[4] H. Xu, B.W. O’Shea, D.C. Collins, M.L. Norman, H. Li, S. Li, The Biermann battery in cosmological MHD simulations of population III star formation, The Astrophysical Journal Letters 688 (2008) L57. [5] R.M. Kulsrud, E.G. Zweibel, On the origin of cosmic magnetic fields, Reports on Progress in Physics 71 (2008) 046901. [6] L.M. Widrow, D. Ryu, D.R.G. Schleicher, K. Subramanian, C.G. Tsagas, R.A. Treumann, The first magnetic fields, Space Science Reviews 166 (2012) 37e70. [7] L. Biermann, Über den Ursprung der Magnetfelder auf Sternen und im interstellaren Raum (mit einem Anhang von A. Schlüter), Zeitschrift Naturforschung Teil A 5 (1950) 65. [8] D. Ryu, H. Kang, J. Cho, S. Das, Turbulence and magnetic fields in the largescale structure of the universe, Science 320 (2008) 909e912. [9] A.A. Schekochihin, S.C. Cowley, R.M. Kulsrud, G.W. Hammett, P. Sharma, Plasma instabilities and magnetic field growth in clusters of galaxies, The Astrophysical Journal 629 (2005) 139e142. [10] A.A. Schekochihin, S.C. Cowley, Turbulence, magnetic fields, and plasma physics in clusters of galaxies, Physics of Plasmas 13 (2006) 056501. [11] D. Ryu, D.R.G. Schleicher, R.A. Treumann, C.G. Tsagas, L.M. Widrow, Magnetic fields in the large-scale structure of the universe, Space Science Reviews 166 (2012) 1e35. [12] F. Govoni, M. Murgia, L. Feretti, G. Giovannini, K. Dolag, G.B. Taylor, The intracluster magnetic field power spectrum in Abell 2255, Astronomy & Astrophysics 460 (2006) 425e438. [13] J.A. Stamper, B.H. Ripin, Faraday-rotation measurements of megagauss magnetic fields in laser-produced plasmas, Physical Review Letters 34 (1975) 138. [14] C.K. Li, F.H. Séguin, J.A. Frenje, J.R. Rygg, R.D. Petrasso, R.P.J. Town, P.A. Amendt, S.P. Hatchett, O.L. Landen, A.J. Mackinnon, P.K. Patel, V.A. Smalyuk, T.C. Sangster, J.P. Knauer, Measuring E and B fields in laserproduced plasmas with monoenergetic proton radiography, Physical Review Letters 97 (2006) 135003. [15] C.K. Li, F.H. Séguin, J.A. Frenje, J.R. Rygg, R.D. Petrasso, R.P.J. Town, O.L. Landen, J.P. Knauer, V.A. Smalyuk, Observation of megagauss-field topology changes due to magnetic reconnection in laser-produced plasmas, Physical Review Letters 99 (2007) 055001. [16] G. Gregori, et al., Generation of scaled protogalactic seed magnetic fields in laser-produced shock waves, Nature 481 (2012) 480e483. [17] B. Fryxell, K. Olson, P. Ricker, F.X. Timmes, M. Zingale, D.Q. Lamb, P. MacNeice, R. Rosner, J.W. Truran, H. Tufo, FLASH: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes, The Astrophysical Journal Supplement Series 131 (2000) 273e334. [18] E. Everson, P. Pribyl, C. Constantin, A. Zylstra, D. Schaeffer, N. Kugland, C. Niemann, Design, construction, and calibration of a three-axis, highfrequency magnetic probe as a diagnostic for exploding plasmas, Review of Scientific Instruments 80 (2009). 113505e113505. Q3 [19] F.F. Chen, Introduction to Plasma Physics and Controlled Fusion, in: Plasma Physics, second ed., vol. 1, Plenum Press, 1984. [20] M. Goossens, An Introduction to Plasma Astrophysics and Magnetohydrodynamics, Kluwer Acadmeic Publishers, 2003. [21] K. Tsinganos, Theory of mhd jets and outflows, in: J. Ferreira, C. Dougados, E. Whelan (Eds.), Jets from Young Stars, Lecture Notes in Physics, vol. 723, Springer, Berlin/Heidelberg, 2007, pp. 117e159. [22] W.D. Hayes, The vorticity jump across a gasdynamic discontinuity, Journal of Fluid Mechanics 2 (1957) 595. [23] N.K.-R. Kevlahan, The vorticity jump across a shock in a non-uniform flow, Journal of Fluid Mechanics 341 (1997) 371. [24] J. von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics 21 (1950) 232e237. [25] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, third ed., Springer, 2009. [26] R.P. Drake, Perspectives on high-energy-density physics, Physics of Plasmas 16 (2009) 055501. [27] B. van der Holst, G. Tóth, I.V. Sokolov, K.G. Powell, J.P. Holloway, E.S. Myra, Q. Stout, M.L. Adams, J.E. Morel, S. Karni, B. Fryxell, R.P. Drake, CRASH: a blockadaptive-mesh code for radiative shock hydrodynamics implementation and verification, The Astrophysical Journal Supplement Series 194 (2011) 23. [28] D.C. Collins, H. Xu, M.L. Norman, H. Li, S. Li, Cosmological adaptive mesh refinement magnetohydrodynamics with Enzo, The Astrophysical Journal Supplement Series 186 (2010) 308. [29] M. Gittings, R. Weaver, M. Clover, T. Betlach, N. Byrne, R. Coker, E. Dendy, R. Hueckstaedt, K. New, W.R. Oakes, D. Ranta, R. Stefan, The RAGE radiationhydrodynamic code, Computational Science & Discovery 1 (2008) 015005. [30] M.M. Marinak, G.D. Kerbel, N.A. Gentile, O. Jones, D. Munro, S. Pollaine, T.R. Dittrich, S.W. Haan, Three-dimensional hydra simulations of national ignition facility targets, Physics of Plasmas 8 (2001) 2275e2280.
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Contents lists available at SciVerse ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Modeling HEDLA magnetic field generation experiments on laser facilities Q4
M. Fatenejad a, *, A.R. Bell b, A. Benuzzi-Mounaix e, R. Crowston g, R.P. Drake f, N. Flocke a, G. Gregori b, M. Koenig c, C. Krauland f, D. Lamb a, D. Lee a, J.R. Marques c, J. Meinecke b, F. Miniati e, C.D. Murphy b, H.-S. Park d, A. Pelka c, A. Ravasio b, B. Remington d, B. Reville b, A. Scopatz a, P. Tzeferacos a, K. Weide a, N. Woolsey g, R. Young f, R. Yurchak c a
Flash Center for Computational Science, University of Chicago, Chicago, IL, USA Department of Physics, University of Oxford, UK LULI, Ecole Polytechnique, France d Lawrence Livermore National Laboratory, USA e ETH Zurich, Switzerland f Center for Radiative Shock Hydrodynamics, University of Michigan, USA g The University of York, UK b c
Q1
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 June 2012 Received in revised form 27 September 2012 Accepted 20 November 2012 Available online xxx
The Flash Center is engaged in a collaboration to simulate laser driven experiments aimed at understanding the generation and amplification of cosmological magnetic fields using the FLASH code. In these experiments a laser illuminates a solid plastic or graphite target launching an asymmetric blast wave into a chamber which contains either Helium or Argon at millibar pressures. Induction coils placed several centimeters away from the target detect large scale magnetic fields on the order of tens to hundreds of Gauss. The time dependence of the magnetic field is consistent with generation via the Biermann battery mechanism near the blast wave. Attempts to perform simulations of these experiments using the FLASH code have uncovered previously unreported numerical difficulties in modeling the Biermann battery mechanism near shock waves which can lead to the production of large non-physical magnetic fields. We report on these difficulties and offer a potential solution. Ó 2012 Published by Elsevier B.V.
Keywords: Magnetohydrodynamics Biermann battery FLASH
1. Introduction The universe is permeated by magnetic fields, with strengths ranging from a femtogauss in the voids between filaments, several microgauss in galaxies and galaxy clusters, and tens of kilogauss in ordinary stars to many teragauss in the vicinity of some black holes and neutron stars. These magnetic fields play a crucial role in astrophysical and cosmological phenomena. However, the origin and strength of these fields are not fully understood. The problem of understanding magnetic fields is two-fold: (1) understanding the generation of the seed fields which magnetic dynamos require in order to operate, and (2) understanding the dynamo process itself, which is generally thought to involve the interaction of turbulent flows with these seed fields. A very promising mechanism for the origin of seed magnetic fields in the universe is the asymmetric shocks that occur in
* Corresponding author. E-mail addresses: [email protected], milad@flash.uchicago.edu (M. Fatenejad).
hierarchical structure formation when smaller halos merge to form galaxies and galaxies merge to form clusters of galaxies [1e4]; for recent reviews, see Refs. [5,6]. By asymmetric, we mean shocks, or blast waves, that lack any 1D symmetry (either planar or spherical). These shocks can cause the generation of magnetic fields by thermoionic currents e the so-called Biermann battery mechanism (BBM) [7]. They may then be amplified to observed values by the turbulence in galaxies [4,8] and in galaxy clusters [8e10]. For recent reviews, see Refs. [5,6,11]. High-power lasers offer the possibility of generating asymmetric shocks in the laboratory and directly measuring magnetic fields generated by them. Many previous experiments [13e15] have measured magnetic fields generated in ablating plasmas produced by high-power lasers. These magnetic fields are not relevant to magnetic fields generated by cosmological shocks because they originate in regions of smooth flow that are supported by laser illumination of a target rather than originating near freely expanding asymmetric shocks. Recent experiments performed at the Laboratoire pour l’Utilisation des Lasers Intenses (LULI) facility use high-power lasers to generate asymmetric shock waves in gas
1574-1818/$ e see front matter Ó 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.hedp.2012.11.002
Please cite this article in press as: M. Fatenejad, et al., Modeling HEDLA magnetic field generation experiments on laser facilities, High Energy Density Physics (2012), http://dx.doi.org/10.1016/j.hedp.2012.11.002
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filled chambers, and measure the time-dependent magnetic fields produced near the shock by the BBM [16]. Using dimensionless scaling parameters, it was demonstrated that the fields measured in these experiments have strengths consistent with the seed fields in the intergalactic medium (IGM). With the experiments in-hand, the possibility now exists to validate simulation codes which are used to directly model astrophysical phenomena. Recent attempts to simulate these experiments using the multidimensional radiation magnetohydrodynamics (MHD) code FLASH [17] have uncovered previously unreported, significant challenges in modeling the BBM source term near shocks. These issues can lead to incorrect predictions of magnetic field strength. We believe that these challenges affect many widely used simulation codes and report on these challenges here. We then offer a potential partial solution that offers the possibility of accurately simulating the magnetic field generation experiments. Finally, we report some preliminary simulation results using it (Fig. 1).
2. Experiment description Fig. 2 shows a schematic of the magnetic field generation experiments performed at the LULI laser facility. To date, over 275 separate experiments have been performed where the laser properties, materials, and geometry have been altered. Here a single experimental configuration is described. A 500 mm diameter solid polystyrene sphere mounted on a thin glass stalk is placed in a chamber filled with Argon gas at 0.5 mbar pressure. The sphere is illuminated from one side with a single 400 J, 527 nm beam using a 1.5 ns square pulse. Polystyrene ablating off of the sphere expands outward, launching an asymmetric blast wave. The asymmetries are largely caused by the one-sided drive. Approximately 600 ns after the start of the laser pulse, the blast wave travels past the first of two 3-axis induction coils [18] placed 3.2 cm and 4 cm away from the target center. The induction coils measure the strength of the magnetic field as a function of time.
Fig. 1. Radio contours of the A255 cluster of galaxies overlaid on the ROSAT X-ray image of the cluster. The white þ and symbols show the positions of the centroid and peak of the cluster X-ray emission. The images at the corners of the figure show the Faraday rotation measure (FRM) for four radio galaxies in the cluster. The FRM measurements of the radio halo at the cluster center and these four radio galaxies imply a magnetic field strength that declines from the cluster center outward, with a field strength of 2.6 mG at the cluster center and an average magnetic field strength of w1.2 mG in the cluster core. Image source Ref. [12]
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3
vr þ V$ðrvÞ ¼ 0; vt
(1)
v B2 1 ðrvÞ þ V$ rvv þ P þ I BB ¼ 0; vt 4p 8p
(2)
h i v c Etot þ V$ ðEtot þ PÞv þ ðE BÞ ¼ 0; vt 4p
(3)
where r is the mass density, v is the velocity, P is the total plasma pressure, B is the magnetic field, E is the electric field, and Etot is the total energy density:
Etot ¼
Fig. 2. Schematic of an experiment to measure shock-generated magnetic fields [16].
Fig. 3 shows the magnetic field strength as measured in three shots with similar configurations. The peak in the magnetic field strength corresponds to the time at which the blast wave passes the coil location. The magnetic field strength is not large enough to be dynamically important. Thus, while measurable, the presence of a field does not significantly influence the motion of the plasma. The gradual rise in the field strength (from 300 to 600 ns in Fig. 3) is caused by the significant magnetic diffusivity present in the experiment. The magnetic Reynolds number is estimated to be approximately 16 [16]. This indicates that modeling resistivity will be important in MHD simulations. 3. Background The FLASH code, like many MHD codes, evolves equations representing the conservation of mass, momentum, and total energy in a plasma:
1 2 B2 rv þ e þ ; 2 8p
(4)
where e is the plasma internal energy density. The pressure is related to the density and internal energy through the equation of state:
P ¼ EOSðr; eÞ
(5)
The evolution of the magnetic field is governed by Faraday’s law:
vB ¼ cV E: vt
(6)
The electric field is set by the generalized Ohm’s law:
uB VPe þ hj E ¼ ; c ene
(7)
where j is the current density, h is the resistivity, Pe is the electron pressure, and ne is the electron number density. Equations (1e7) are standard representations of MHD and their derivations can be found in several texts such as Refs. [19] and [20]. Several approximations have been made including the assumption of charge neutrality and the neglect of the displacement current in Ampere’s law [20,21] so that j ¼ ðc=4pÞV B. The first term in Equation (7) is the dynamo term representing the generation of electric field due to the bulk motion of a magnetized plasma. The second term introduces resistive effects which allows the plasma to move relative to the magnetic field. The final term is the Biermann battery source which is responsible for the generation of fields in an unmagnetized plasma. The Biermann battery term arises from the mass difference between ions and electrons. Forces acting on the plasma accelerate electrons more quickly than ions, resulting in a net current and hence generation of magnetic fields. There is a corresponding decrease in plasma internal energy, e, to conserve total energy as implied by Equation (4). Inserting Equation (7) in Faraday’s law yields the induction equation
vB VPe ¼ V ðu BÞ cV ðhjÞ þ cV ; vt ene
(8)
where the final term is often expanded:
vB VPe Vne ¼ V ðu BÞ cV ðhjÞ þ c : vt en2e
Fig. 3. Magnetic field measurements from 3-axis coils. The peak in the field strength is inferred to correspond with the time at which the blast wave passes the coils.
(9)
In this form, it is clear that the Biermann battery term (BBT) is active where ever the electron pressure and density gradients are not aligned. Thus, B will remain zero in an initially unmagnetized plasma with a purely 1D (symmetric) flow, such as a perfectly spherical blast wave or a planar shock. In contrast, asymmetric shocks generate magnetic fields via the BBM downstream because
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behind the shock Vne and VPe are not parallel to the direction of the shock velocity. As a result, the BBT is non-zero, and therefore vB/vt is non-zero, leading to the generation of a magnetic field where none existed before.
3.1. The vorticity equation One common method [5] of estimating the magnetic field strength generated via the BBM is to take advantage of the similarity between the induction equation and the vorticity equation:
vu VP Vr ¼ V ðu uÞ ; vt r2
(10)
where u is the vorticity, P is the total plasma pressure, and r is the mass density. For a fully-ionized, single species, single temperature (Te ¼ Ti where Te is the electron temperature and Ti is the ion temperature) plasma, the electron number density can be written as ne ¼ zr/mi where z is the atomic number and mi is the ion mass. Assuming an ideal-gas equation of state, we can write the electron pressure as Pe ¼ zP/(zþ1). Applying these assumptions to the induction equation, and assuming no resistivity, Equation (9) yields:
vB cmi 1 VP Vr ¼ V ðu BÞ þ : vt e 1þz r2
(11)
Thus, the quantities (1 þ z)u and ui ¼ eB/cmi obey the same equations. Consequently, under the many constraints listed above, the vorticity can be used as a stand-in for the magnetic field, assuming the initial vorticity and magnetic field strength are zero. It is well known that asymmetric shocks (for example, standing bow-shocks) do generate vorticity [22,23] and hence asymmetric shocks will generate magnetic fields due to the similarity between the two quantities.
The use of the similarity to vorticity does not fully address this issue. The most obvious reason is that the assumptions that lead to the similarities in Equations (10) and (11) are not valid for many HEDP experiments, which often have high magnetic resistivity and multiple, partially-ionized species [26]. Furthermore, the evaluation of the vorticity itself requires computing numerical derivatives of a different discontinuous function, in this case the velocity, which will also lead to non-physical magnetic fields. The challenges described here have been discussed with the developers of a number of codes including CRASH [27], ENZO [28], and RAGE/CASSIO [29], which are Eulerian codes; and HYDRA [30], which is an Arbitrary LagrangianeEulerian code. We have determined that at the present none of them accurately treat the BBM near shocks.1 The artificial fields generated by direct discretization of Equation (8) or Equation (9) can be large. This is especially true for simulations of the experiments described in Section 2, where the artificial fields can dominate any physical result. It is therefore critical to address this challenge for performing realistic simulations of these experiments. A potential solution, which has been implemented in the FLASH code for 2D cylindrical simulations, attempts to circumvent this issue by detecting shocks. This can easily be accomplished by identifying regions where large velocity gradients exist. The Biermann battery source term is not computed in cells that are near shock waves and large artificial fields are therefore not produced. This allows the code to generate physically relevant magnetic fields in the downstream region, but not within the shock region itself. Physical magnetic fields can be generated within the shock region, and a limitation of the approach is that it ignores this source of the magnetic field which can be important in certain situations. Below we present the results of preliminary verification simulations that demonstrate how the shock detection method can be used to avoid the calculation of non-physical magnetic fields. 5. Verification simulation results
4. Simulation of the Biermann battery mechanism Some MHD simulation codes attempt to model the BBM using a discretization of Equation (9). This approach is reasonable in smooth flows (see for example Refs. [14,15]), but does not work near shock waves. The reason is that the battery source term contains gradients of Pe and ne e both of which may be (and usually are) discontinuous at shocks. Lagrangian [24] and Eulerian [25] simulation codes spread shocks over a small number of computational cells. The specific number of computational cells and details concerning the exact transition from the upstream to downstream state depend on the details of the specific numerical algorithm used. Mathematically, the electron pressure and density are discontinuous at a shock wave, but simulation codes treat them as quantities that vary quickly in the vicinity of the shock and have very large, purely numeric gradients. Thus, direct discretization of the electron pressure and number density gradients, using finite differences for example, near shock waves will lead to the generation of large, magnetic fields that are purely a numerical artifact. Moreover, these artificial fields do not converge and will increase in strength as the computation mesh becomes finer since the numerical gradients in Pe and ne increase with higher resolution. Many Eulerian finite-volume codes solve Equation (8) instead of Equation (9), where the former includes the source term as the curl of a flux, which has the units of an electric field. This approach does not cure the pathological behavior of the BBM at shocks, since, unlike ordinary hydrodynamic fluxes, the Biermann flux itself contains a gradient of Pe.
The challenges described in Section 4 can be demonstrated using a simple verification test. In the case of a purely 1D shock wave, the Biermann battery source term should be zero and no magnetic fields should be present if they are not introduced through initial or boundary conditions. We performed MHD simulations of a circularly expanding shock wave in 2D Cartesian (x y) geometry using FLASH. The simulations include the BBM, but did not include resistive effects. In these simulations, the flow is radial while the mesh is aligned with the x and y directions. The initialpconditions are defined such that there is only variation in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ x2 þ y2 . Initially B ¼ 0 throughout the domain and the velocity is zero. A fully-ionized ideal gas equation of state is used with Helium as the single material in the domain. The initial density is 0.0001 g/cc. The initial temperature is set to 0.5 eV for R > 0.01 microns and 14.66 eV for R < 0.01 microns. The simulation was performed for 1.8 ns with the Biermann battery source term active. Fig. 4 shows the magnetic field strength (in the z direction) at this time when no shock detection is used to suppress field generation near the shock. A large artificial field is generated in this
1 CRASH and ENZO accurately treat the BBM in smooth flow but do not do so at shocks (B. van der Holst and G. Toth, private communication, 2012; H. Xu, H. Li, and S. Li, private communication, 2011); RAGE/CASSIO do not currently have an MHD capability, though there are plans to add one (C. Fryer and J. Wohlbier, private communication, 2011); HYDRA accurately treats the BBM in smooth flow but not at shocks for which the flow is significantly misaligned with the computational mesh, though work is underway to address this (M.M. Marinak, private communication, 2012).
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Fig. 4. Density and magnetic field strength from a 2D Cartesian FLASH MHD simulation modeling a circularly expanding shock wave. The flow is not aligned with the mesh and no shock detection is used. Large, artificial magnetic fields are generated in this case.
case. The magnetic field strength does not diminish with increasing resolution. The overall field is still relatively small, with b > 104 where b represents the ratio of plasma pressure to magnetic pressure. These artificial fields are not dynamically important as is the case in the experimentally measured fields. Thus, the artificial fields will not substantially influence the motion of the plasma but must suppressed in order to obtain agreement with the experiment. Fig. 5 shows the same simulation with the Biermann battery source term suppressed. The white lines bound the region where the code detects a shock. The bounded area is approximately 5-cells
Fig. 5. Similar simulation as in Fig. 4, but shock detection is used to eliminate magnetic field generation in the 5 cells bounding the shock wave. The overall field strength is substantially reduced.
5
Fig. 6. Total magnetic energy as a function of cell size for the circularly expanding shock verification simulations. When shock detection is used, the energy decreases as the cell size is reduced.
across in the radial direction. The field strength is substantially reduced. The magnetic fields that are generated when shock detection is active are produced as a result of discretization error in the smooth flow regions. Because the shock is not aligned with the flow, the electron pressure and density gradients are not perfectly
Fig. 7. A laser illuminates a 0.1 mm thick Carbon foil located at z ¼ 0 with 450 J of 527 nm light in a 1 ns square pulse with a 0.4 mm FWHM Gaussian intensity profile. The laser enters from the bottom of the image. A shock breaks out of the foil after 2 ns into an Argon filled chamber. The image shows the density (g/cc) plot from a 2D cylindrical FLASH simulation at 50 ns. The blast wave is highly asymmetric and extends to 7 mm in the z direction and 3.5 mm in the R direction. The magnetic field strength (Gauss) is shown as an inset, colored red, white and blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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aligned. However, in this case, the error is convergent e as the mesh resolution is increased the total magnetic energy decreases, as shown in Fig. 6. This is not the case without shock detection, i.e. when simple differencing of the Biermann battery source term is used everywhere. 6. Preliminary validation simulation results The shock detection techniques have been used in preliminary 2D cylindrical MHD simulations of laser driven experiments. These simulations have a different geometry than those described in Section 2. Rather than a spherical target, a laser illuminates 0.1 mm thick Carbon foil, generating a shock which breaks out of the foil into an Argon filled chamber. Fig. 7 shows the mass density and magnetic field strength in a simulation after 50 ns. No magnetic resistivity is used in these simulations. The shock has expanded to cover a large fraction of the simulation domain. Shock detection is being used to suppress generation of unphysical magnetic fields at the shock wave. Behind the shock, the magnetic field strength is approximately 103 G. Future simulations will include resistivity and this will act to significantly reduce the peak field strength, but the results so far are promising. 7. Conclusions Recent experiments have demonstrated that astrophysically relevant magnetic fields can be produced via the Biermann battery mechanism acting downstream of asymmetric shocks. These experiments use high-power lasers to generate asymmetric blast waves in gas filled chambers and are promising validation experiments for MHD codes. Attempts to simulate these experiments using the multidimensional MHD code FLASH have revealed serious numerical challenges in modeling the Biermann battery source term near shock waves where electron pressure and number densities are discontinuous. These challenges affect many widely used simulation codes and have gone largely unnoticed, but we demonstrate them here using verification tests. A partial solution which uses shock detection to suppress the Biermann battery source in cells near discontinuities has been implemented in FLASH. This modification prevents the generation of large, artificial magnetic fields near shock waves in simulations and is being tested in full scale laser driven simulations. Acknowledgments This work was supported in part at the University of Chicago by the US Department of Energy NNSA ASC through the Argonne Institute for Computing in Science under field work proposal 57789; and the US National Science Foundation under grant PHY0903997. References [1] R.M. Kulsrud, R. Cen, J.P. Ostriker, D. Ryu, The protogalactic origin for cosmic magnetic fields, The Astrophysical Journal 480 (1997) 481. [2] K. Roettiger, J.M. Stone, J.O. Burns, Magnetic field evolution in merging clusters of galaxies, The Astrophysical Journal 518 (1999) 594. [3] G. Davies, L.M. Widrow, A possible mechanism for generating galactic magnetic fields, The Astrophysical Journal 540 (2000) 755.
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Please cite this article in press as: M. Fatenejad, et al., Modeling HEDLA magnetic field generation experiments on laser facilities, High Energy Density Physics (2012), http://dx.doi.org/10.1016/j.hedp.2012.11.002
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