A 3D Magnetohydrodynamic simulation for the propagation of plasma plume transverse to applied magnetic field
Bhavesh G. Patel, Narayan Behera, R. K. Singh, Ajai Kumar, Amita Das
AA 3D Magnetohydrodynamic simulation for the propagation ofplasma plume transverse to applied magnetic field
Bhavesh G. Patel ∗ and Narayan Behera, R. K. Singh, Ajai Kumar Institute for Plasma Research, Bhat, Gandhinagar 382 428, India
Amita Das † Physics Department, Indian Institute of Technology,Delhi, Hauz Khas New Delhi -110016, India (Dated: February 26, 2021)
Abstract
We have carried out a 3D ideal-MHD (Magnetohydrodynamic) simulation to study the evolutionof laser generated plasma plume in a moderate external magnetic field (0.13 T) oriented perpen-dicular to the flow direction of the plasma plume. The outcomes of the simulation show that theplasma plume pushes the external magnetic field lines as it expands, thereby creating a cavity inplasma density as well as the external magnetic field. Formation of this cavity is supported andsustained by the plasma pressure. As the plasma pressure drops due to expansion, the imbalancebetween the magnetic energy and the internal energy results in the collapse of the cavity. Theseobservations have striking similarities with the observations of the experiments [Phys. Plasmas24, 033511 (2017)] performed recently to study the plasma plume expansion in the presence of anexternal transverse magnetic field. This similarity indicates that the physical mechanisms domi-nantly governing the plasma plume expansion in the moderate magnetic field are aptly describedin the ideal MHD regime. The studies thus show that the laser generated plasma plume can beutilized to carry out interesting experiments on MHD phenomena in a simple laboratory set up. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] F e b . INTRODUCTION One often encounters the interaction of flowing plasma with an external magnetic in vari-ous astrophysical and laboratory scenarios. Supernova explosions in the interstellar medium,ejection of solar coronal mass in the planetary magnetosphere, artificial plasma release inmagnetosphere and propagation of laser-generated plasma plume in an external magneticfield are just a few examples to cite [1–3]. Besides the fundamental aspect, the fact that amultitude of applications relies on the interaction of plasma with the magnetic field makesthis an important topic of research. For instance, in pulsed laser deposition, the magneticfield is used to guide the plasma towards the coating site. Moreover, laser fusion schemesemploy external magnetic field to optimize the conditions necessary for ignition and con-finement. The physical understanding of the processes underlying the expansion of plasmaplume in the presence of external magnetic field is, therefore, important. Several researchershave carried out experimental and theoretical investigations over the past many years in thefield of laser-produced plasma plume expansion in the presence of external magnetic field[4, 5] and shown that the dynamics as well as shape, size, and structure of the plasma plumeevolving in the external magnetic field depend significantly on plasma parameters and theexternal magnetic field. While many simulation studies exists for plasma plume expansionin external magnetic field parallel to the direction of propagation of plasma plume, [6–10]there are only a few 3D simulation studies pertaining to plasma plume propagation in trans-verse magnetic field [11]. In this manuscript we present results of 3-D simulation based ona ideal Magnetohydrodynamics model for the evolution of a laser produced plasma plumein moderate transverse external field.The paper has been organized as follows. We provide a brief description of the simulationset up and geometry along with the boundary conditions that have been employed in sectionII. In section III the simulation results are presented along with the comparison with a recentexperimental observations. Section IV contains the summary and conclusion.2
I. SIMULATION SETUP AND BOUNDARY CONDITIONS
We aim to model the evolution of the laser produced plasma plume against an externalbackground magnetic field which is aligned perpendicular to plasma plume flow direction. Inorder to relate the simulation closely to the experimental scenario, we perform the simulationin two stages. The first stage comprises production of plasma by laser-target interaction andthe second stage comprises evolution of plasma in the external magnetic field. During theproduction phase the laser energy heats the target, causing the melting and the vaporizationand possible ionization of its material. The initial plasma formation is modeled using 1Dlagrangian VLL (Virtual laser laboratory) code [12]. VLL code, models interaction of laserpulse with the target material and account for laser light absorption, plume expansion, elec-tron thermal conductivity, two-temperature effects, phase transitions, material spallation.The Lagrangian plasma profiles of density, pressure and velocity are then used to constructinflow profiles for the problem. This inflow profiles are used as an input to the 3D idealMHD code PLUTO [13]. The PLUTO code is a modular Godunov-type parallel code thatnumerically solves the following set of MHD equations: ∂∂t ρρ (cid:42) vE (cid:42) B + (cid:42) ∇ · ρ (cid:42) vρ (cid:42) v (cid:42) v − (cid:42) B (cid:42) B + Ip t ( E + Ip t ) (cid:42) v − (cid:42) B (cid:16) (cid:42) v · (cid:42) B (cid:17) (cid:42) v (cid:42) B − (cid:42) B (cid:42) v T = (1)Here ρ is the mass density, ρ (cid:42) v is the momentum density, (cid:42) v is the fluid velocity, p is thethermal pressure, (cid:42) B is magnetic field and E t is the total energy density. E t = ρe + 12 ρv + B IG. 1: Geometry and boundary condition for the computational domain.
The schematic in Fig. 1 shows the computational domain and the boundary conditionsimposed at the boundaries. In order to compare the simulation results with the experimen-tal observations we have chosen the extent of the computational domain in both x and y direction to be 1 . . . .
13 T has been applied along the ˆ y direction. Except for theshaded region, reflective boundary condition is imposed on the xy plane at z = 0. Inflowboundary condition is imposed in the shaded region till the inflow persists (here it is forabout 10 ns). Further, for the VLL simulation, we chose the target material to be Aluminumand a Gaussian profile for the incident laser pulse. The intensity, wavelength and the pulsewidth of the laser pulse were 2 . × W / cm , 1 . µ m and 10 ns respectively. This cor-responded to a fluence of 19 . / cm . The maximum value of the mass density, pressureand velocity of the inflowing plasma plume as obtained from the VLL simulation are about1 . × − g / cm , 1 . × Ba, and 2 × cm / s respectively. These parameters are sim-ilar to the experimental parameters considered in [14]. Once the inflow stops, reflectiveboundary condition is imposed on the shaded/inflow region.4 II. SIMULATION RESULTS
Many researchers have experimentally investigated the dynamical and geometrical fea-tures of the laser-generated plasma evolving in the external transverse magnetic field [15, 16].These experiments have demonstrated the formation and collapse of the diamagnetic cavityon the plane perpendicular to the applied magnetic field. In present work we have per-formed ideal MHD simulations to understand the role of magnetic field in the evolution ofthe plasma plume. We have chosen the simulation parameters such that the experimentalfindings reported in the reference [14] could be compared with the simulation results. Wehave ignored the effect of viscosity, thermal conductivity, and Hall term and employed pureideal MHD equations to simulate plasma plume evolution [17–19] and would show later thateven with the absence of these effects, the experimental observations of cavity formationand its collapse is well reproduced.In Fig.2 we have shown a series 3D surface plots of plasma density of the plume atvarious times over a course of its evolution. It can be observed that the evolution followstwo distinct phases. The plasma bulges out in the transverse direction in the first phaseforming a magnetic cavity and in the second phase the cavity collapses.
FIG. 2: Isosurface plots of the plasma plume density at different times. The figure displays fourdifferent normalized densities depicted in the color bar.
This is also corroborated by the plot of magnetic field lines shown in Fig.3 which showsthe initial bulging of the magnetic field lines in the transverse direction. However, at a5ater stage the lines appear to collapse back. This can be understood by realizing thatinitially (0 − ns ) the plasma expands unabated under the effect of large thermal pressurewhich pushes the magnetic field lines from the high pressure region. The flow velocity ofthe plasma plume front is of the order of 1 × cm / s. As time passes the plasma plumeexpands in all directions and a cavity starts appearing gradually in the plume. The cavityis observed to attain the maximum size at about 300 ns and after this it begins to collapse.The change in the shape of the plasma plume observed in the Fig.2 is related to the changein the topology of the applied magnetic field and the thermal pressure of the plasma. Fig.3shows the topology of the external magnetic field lines at various times We observe that as FIG. 3: The magnetic field lines at four different planes at different times. the plasma plume expands (and propagates at the velocity exceeding the Alfven velocity( V A ∼ × cm / s)) across the background magnetic field, it sweeps up the magnetic layersin accordance to the the frozen-in condition in the ideal MHD framework.The magnetic fieldlines advect with the plasma flow which results in bending and accumulation of field lines atthe expanding plume front. The electromagnetic force density, J × B , acting on the plasmaplume due the change in the topology of the magnetic field could be understood in termsof lateral magnetic pressure ∇ ⊥ (cid:16) B π (cid:17) and the magnetic tension B π acting along the lines ofmagnetic field.The in-homogeneity in the magnetic field leads to the force density pushing the field linesfrom the regions of high density to regions of low density in the direction perpendicular6o the field. The bending and stretching of the magnetic field line leads to the magnetictension. The magnetic field lines have a tendency to shorten and hence this force due to themagnetic tension points in a direction opposite to the direction of the convex curvature ofthe bent field lines. Once the plasma pressure drops as the result of expansion the magneticfield lines snap back. The interplay between the magnetic field gradient, magnetic tensionand the plasma pressure gradient govern the evolution of the plasma plume. The role ofthe magnetic tension and the magnetic pressure in governing the plasma plume structure isclearly evident from the Fig.3 and Fig.2. The segregation and stretching of the magneticfield lines gives rise to (cid:126)J × (cid:126)B force which opposes the expansion of plasma and there isincrease in the magnetic field energy. FIG. 4: Kinetic and magnetic field energy as function of time.
In Fig.4 we show the plot of plasma kinetic energy and magnetic field energy as functionof time. The total energy which is the sum of these energies has also been plotted. It canbe observed that both kinetic and magnetic energy (and thereby total energy) increases upto 80 ns as the plume enters the simulation box. Thereafter, till 500 ns the total energyremains constant. At 500 ns the plasma plume hits the other boundary and starts movingout of the simulation box which shows up as the drop in total energy. In the interveningperiod of 80 ns to 500 ns for which the total energy is a constant, there is an exchange ofenergy between the kinetic and magnetic parts. The kinetic energy associated with plasmadecreases and the magnetic energy in the box increases. This happens as the plasma pushes7he magnetic field in the transverse direction, thereby stretching the magnetic field linesas can be seen from Fig.3. Thus, as plasma expands, its thermal pressure continuouslydecreases and the effect of (cid:126)J × (cid:126)B force dominates. As seen in the Fig.2, the plasma from thecore of the plume accumulates at the plume boundary thereby forming a cavity surroundedby a shell of shocked plasma. A distinct cavity begins to appear at around 120 ns andit reaches its maximum size at around 300 ns . Further, it is observed that as the plasmaplume translates across the magnetic field, it expands without much deceleration in thedirection of the applied magnetic field while the expansion of plasma experiences notabledeceleration in the direction perpendicular to both magnetic field and the flow direction .This observation is well corroborated by the Fig.5 which shows the pseudo-color plots ofdensity in the cut-plane parallel and perpendicular to the direction of the applied magneticfield. FIG. 5: The subplot on the left and right shows the cross-sectional view of the plasma plumedensity in the planes parallel and perpendicular to the magnetic field respectively. The planespass through the center of the plume.
Further, the plasma plume is cooled initially due to free expansion and as the cavitybegins to form the plasma shell is heated by the adiabatic compression at the field-plasmainterface. This is depicted in Fig.6 where the cut-plane plot of the plasma temperatureshows a bubble structure.The second phase of the plasma plume evolution commences once the cavity attains itsmaximum size. In this phase the plasma plume gradually begins to collapse under the8
IG. 6: Cross-sectional view of the plasma plume temperature (in ev)in the planes passingthrough center of plume and parallel and perpendicular to the externally applied magnetic fieldrespectively. action of J × B force. Since the plasma expands freely across the plane containing themagnetic field, the plasma plume assumes the shape of a flat pancake as depicted by thelast subplot in the Fig.2. We now compare the results of the MHD simulation with theexperimental observations. In this experiment a laser fluence of 19 . / cm was used forablation and the resulting plasma plume was allowed to expand in external magnetic field of0 .
13 T oriented in direction perpendicular to direction of propagation of the plasma plume.Further details of the experiment can be found in the reference [14]. In order to comparethe diamagnetic cavity formed in the experiment and the simulation, we plot the imagesof the plasma plume obtained from the experiment and plume density obtained simulationtogether. It is important to note that the intensity of the ICCD images of plasma plume iscommensurate with the plume density. In fact, the intensity recorded by ICCD is sum totalof the intensities coming from the planes constituting the plasma plume.The subplots of the top row of Fig.7 shows the images of the expanding plasma plumeobtained with an ICCD camera oriented in direction parallel to the applied magnetic field.In other words, these experimental images correspond to plasma plume density in planeperpendicular to the magnetic field. The middle row shows the pseudo color plots of theplasma density in the cut plane perpendicular to the magnetic field and passing throughthe center of the plume from simulation. The bottom row the pseudo color plot is obtained9
IG. 7: Slice of plasma plume taken in a plane (x-z ) perpendicular to the direction of magneticfield (ˆ y ). The top row shows the images of plume at different times as obtained in theexperiment. The center row shows the plume density in the plane passing through the center.Thebottom row shows the plume density obtained after adding all the layers of the plumeperpendicular to the magnetic field. after adding up the density of all the planes perpendicular to the magnetic field of the 3Dplume image. This should approximately correspond to the images captured by the ICCD.It can be seen that various aspects of the experimentally observed features are well capturedin our simulations. The plume location as well as the dimensions of the diamagnetic cavityare accurately reproduced in simulations. It is observed that once the cavity has attainedmaximum size it begins to collapse into a narrow jet like structure.In order to compare the plume size obtained in the experiment with the plume sizeobtained in the simulation, we have plotted the plume size in the direction of expansionand plume size in direction perpendicular to both magnetic field and direction of the plumepropagation as function of time for both simulation and experiment (Fig.8). The data fromthe experiment and simulation match quiet well.Fig.9 shows the pseudo color plots of plasma plume density in the plane parallel to theexternal magnetic field obtained from simulation and experiment. We see that the size ofthe plume is large as compared to the images presented in the Fig.7. Further, at later times,the decelerating plasma plume is engulfed by the stretched magnetic field line and at sametime there is mass density gradient at the surface of the plasma plume. This condition isideal for magnetic Rayleigh Taylor instability to occur [11]. We believe that the striation10 IG. 8: Comparison of plume size as obtained from the experiment and that obtained from thesimulation. like structures that appear in both simulation and experimental observations could be duethe magnetic Rayleigh Taylor instability.
FIG. 9: Slice of plasma plume taken in a plane parallel to the direction of magnetic field(y-zplane). The top row shows the images of plume at different times as obtained in the experiment.The center row shows the plume density in the plane passing through the center as obtained insimulation.The bottom row shows the plume density obtained after adding all the layers of theplume parallel to magnetic field as obtained in simulation. V. SUMMARY AND CONCLUSION
We have presented a 3D ideal MHD simulation of laser generated plasma plume expan-sion in external transverse magnetic field and compared the results of the simulation withthe experimental results. The simulation studies shows that the transverse magnetic fieldsignificantly governs the expansion of plasma plume and the observations made by thisstudy have been found to compare reasonably well with the experimental observations. Forinstance, diamagnetic cavity formation and collapse of the plasma plume into a jet like struc-ture are the important experimental observations that are all observed in the ideal MHDsimulation. Besides this, the plume size as function of time matches with that obtained inthe experiment. The simulation results conclusively shows that all the major experimentalobservations could be attributed to the interplay between the −→ J × −→ B force and the forcedue thermal pressure. Furthermore, our simulation studies which are based on ideal MHDclearly show that for the plasma plume expansion in the low magnetic field, the plasmabased aspects like electrical resistivity, thermal diffusion, and Hall effect do not play anycrucial role in the evolution of the plume. 12 cknowledgments We acknowledge that the results reported in this article are the outcome of the simulationsperformed on the Antya cluster at Institute for Plasma Research. We thank Dr. BhargavViadya for useful discussions and suggestions regarding the PLUTO code. [1] Gerhard Haerendel. Experiments with plasmas artificially injected into near-earth space.
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