Stationary relativistic jets
- Serguei S Komissarov^{1, 2}Email author,
- Oliver Porth^{1, 3} and
- Maxim Lyutikov^{2}
https://doi.org/10.1186/s40668-015-0013-y
© Komissarov et al. 2015
Received: 18 May 2015
Accepted: 23 October 2015
Published: 5 November 2015
Abstract
In this paper we describe a simple numerical approach which allows to study the structure of steady-state axisymmetric relativistic jets using one-dimensional time-dependent simulations. It is based on the fact that for narrow jets with \(v_{z}\approx c\) the steady-state equations of relativistic magnetohydrodynamics can be accurately approximated by the one-dimensional time-dependent equations after the substitution \(z=ct\). Since only the time-dependent codes are now publicly available this is a valuable and efficient alternative to the development of a high-specialised code for the time-independent equations. The approach is also much cheaper and more robust compared to the relaxation method. We tested this technique against numerical and analytical solutions found in literature as well as solutions we obtained using the relaxation method and found it sufficiently accurate. In the process, we discovered the reason for the failure of the self-similar analytical model of the jet reconfinement in relatively flat atmospheres and elucidated the nature of radial oscillations of steady-state jets.
Keywords
1 Introduction
Highly collimated flows of plasma from compact objects of stellar mass, like young stars, neutron stars and black holes, as well as supermassive black holes residing in the centers of active galaxies is a wide-spread phenomenon which has been and will remain the focal point of many research programs, both observational and theoretical. Some features of these cosmic jets, like moving knots, are best described using time-dependent fluid models. However, most of these jets have sufficiently regular global structure, which is indicative of steady production and propagation and promotes development of stationary models. Such models are also easier to analyze, and they are very helpful in our attempts to figure out the key factors of the jet physics.
The simplest approach to steady-state flows is to completely ignore the variation of flow parameters across the jet. This allows to reduce the complicated system of non-linear partial differential equations (PDEs) describing the jet dynamics to a set of ordinary differential equations (ODEs) which can be integrated more easily (e.g. Blandford and Rees 1974; Komissarov 1994). A similar reduction in the dimensionality is achieved in self-similar models, where unknown functions depend only on a combination of independent variables known as a self-similar variable. This also allows to reduce the original PDEs to a set of ODEs (e.g. Blandford and Payne 1982; Vlahakis and Tsinganos 1998). While providing important test cases and useful insights, this approach is not sufficiently robust - boundary and other conditions that select such exceptional solutions are not always present in nature.
Obviously, such shocked flows cannot be described by one-dimensional (1D) and self-similar models, which we mentioned earlier, and more complex, at least two-dimensional (2D), models have to be applied instead. The system of steady-state equations of compressible fluid dynamics, not to mention magnetohydrodynamics, is already very complicated and generally requires numerical treatment. One of the ways of finding its solutions involves integration of the original time-dependent equations in anticipation that if the boundary conditions are time-independent then the time-dependent numerical solution will naturally evolve towards a steady-state (e.g. Ustyugova et al. 1999; Komissarov et al. 2009; Tchekhovskoy et al. 2008). One clear advantage of this approach is that it allows to use standard codes for time-dependent fluid dynamics. Such codes are now well advanced and widely available. However, this type of the relaxation approach is characterized by slow convergence and hence rather expensive.
In order to speed up the convergence, one can use other relaxation methods, which are developed specifically for integrating steady-state equations (e.g. May and Jameson 2011). They often involve a relaxation variable which is called ‘pseudo time’. However, this time evolution is not realistic but designed to drive solutions towards a steady-state in the fastest way possible. The only disadvantage of this approach is that it involves development of a specialised computer code dedicated to solving only steady-state problems. The authors are not aware of such codes for relativistic hydro- and magnetohydrodynamics.
For supersonic flows, the system of steady-state equations turns out to be hyperbolic, with one of spatial coordinates playing the role of time (Glaz and Wardlaw 1985). (In the case of magnetic jets, the speed of sound is replaced with the fast magneto-sonic speed and we classify flows as sub-, tran-, or super-sonic based on its value compared to the flow speed.) In this case, one can find steady-state solutions utilising numerical methods which were designed specifically for hyperbolic systems, like the method of characteristics or ‘marching’ schemes. These methods have been used in the past in applications to relativistic jets (e.g. Daly and Marscher 1988; Wilson and Falle 1985; Wilson 1987; Bowman 1994; Bowman et al. 1996) but publicly available codes do not exist yet. Their development is as time-consuming as that of time-dependent codes whereas the range of applications is much more limited. This explains their current unavailability. Moreover, when flow becomes subsonic, even very locally, this approach fails.
In this paper, we propose a new approach, which allows to find approximate numerical steady-state jet solutions rather cheaply and using widely available computer codes. To be more precise, we focus on highly relativistic narrow axisymmetric jets and show that in this regime the 2D steady-state equations of Special Relativistic MHD (SRMHD) are well approximated by 1D time-dependent equations of SRMHD. Like in the standard marching schemes, the spatial coordinate along the jet plays the role of time. This allows us to find steady-state structure of axisymmetric jets by carrying out basic 1D SRMHD simulations, which can be done with very high resolution even on a very basic personal computer. In such simulations, no special effort is needed to preserve the magnetic field divergence-free and the computational errors associated with multi-dimensionality are eliminated. As the result, more extreme conditions can be tackled. Here we focus only on relativistic jets, because of our interest to AGN and GRB jets, but we see no reason why this approach cannot be applied to non-relativistic hypersonic jets as well. Our approach is closely related to the so-called ‘frozen pulse’ approximation, which also utilizes the similarity between the steady-state and time-dependent equations describing ultra-relativistic flows (Piran et al. 1993; Vlahakis and Königl 2003; Sapountzis and Vlahakis 2013). In this approximation, the steady-state equations are used to analyze the dynamics of time-dependent flows. The similarity between 1D time-dependent models and 2D steady-state jet solutions has been noted before, in particular in Matsumoto et al. (2012).
In order to study the potential of this new approach we have carried out a number of test simulations and compared the results obtained in this way with both analytical models and numerical solutions obtained with more traditional methods. The results are very encouraging and allow us to conclude that this method is viable and can be used in a wide range of astrophysical applications.
2 Approximation
Given that in relativistic fluid dynamics small differences between the magnitudes of energy and momentum may result in huge variations of Lorentz factor and even lead to inconsistency, one could feel uneasy about the approximations we make. However, the final result is exactly the system of 1D time-dependent SRMHD and this means that self-consistency is not compromised. For example, the flow speed will not exceed the speed of light because of the errors of our approximation.
Our approach is similar to ‘marching’ - we compute solution for a downstream jet cross-section using only the previously found solutions for upstream cross-sections. Strictly speaking, this requires the flow to be super-sonic for unmagnetized jets and super-fast-magnetosonic for magnetized ones (Wilson and Falle 1985; Dubal and Pantano 1993). However, in our derivations we never had to utilize this condition. This suggests that it is not required when we wish to find only approximate solutions. For example, one may argue that the fact that information can propagate upstream does not necessarily imply that this always has a strong effect on the flow - the upstream-propagating waves could be rather weak. If so, we may still apply our method to jets where the supersonic condition is not fully satisfied, but we always need to check that the conditions (11)-(15) of our approximation hold for obtained solutions.
3 Numerical implementation
The analysis of Section 2 shows that as long as they are applied to narrow jets with high Lorentz factor, the axisymmetric steady-state equations of SRMHD are very close to 1D time-dependent equations of SRMHD in cylindrical geometry. This suggests that it may be possible to use time-dependent simulations with 1D SRMHD codes to study the 2D structure of steady-state jet solutions. However in order to be able to do this, we also need to find a way of accommodating the 2D boundary conditions of steady-state problems in such simulations.
For 2D supersonic flows we need to fix all flow parameters at the jet inlet and impose some conditions at the jet boundary, consistent with it being a stationary contact wave. No boundary conditions are needed for the outlet boundary - its flow parameters are part of the solution. In the corresponding 1D problem, the 2D boundary conditions at the inlet boundary simply become the initial conditions of the 1D Cauchy problem. The final 1D solution corresponds to the slice of the 2D solution at the outlet boundary. As to the contact discontinuity at the 2D jet boundary, the situation is not that trivial.
Suppose that the total pressure at this boundary is a function of z, \(p=p_{\mathrm {b}}(z)\). When we replace z with t this becomes \(p=p_{b}(t)\). Thus we need somehow to impose time-dependent boundary conditions. In the simulations presented below, the following approach was utilised: (1) we extend the computational domain so that it includes the external gas, (2) we track the point separating the jet from the external gas and (3) we reset the external gas parameters according to the prescribed functions of time every computational time step.
After the reset, the 1D jet boundary is no longer a contact but a more general discontinuity. In particular, the jet plasma will generally have radial velocity component. If it is positive, but in the external gas it is set to zero, then a shock wave will launched into the jet when this discontinuity is resolved. If it is negative, then this will be a rarefaction wave. On the one hand, this reflects how the information about changing environment is communicated to the interior of a steady-state jet. On the other hand, in 1D simulations the strength of the emitted wave depends on the external density - higher density, and hence lower temperature, will result in stronger waves moving into the jet. This is obviously not so for 2D steady-state jets, which react only to the external pressure. Thus additional measures need to be undertaken. First, in order to negate the effect of the radial velocity jump at the jet boundary, the radial velocity of the external gas is reset not to zero but to its value at the last jet cell. Second, in order reduce the role of the external gas inertia, it helps to set its density to a low value, so that its sound speed becomes relativistic. Although we have not tried this, one could set the polytropic index of the external gas to \(\Gamma=2\), which would make the sound speed of ultra-relativistically hot gas equal to the speed of light.
4 Examples
4.1 Bowman’s jet
Bowman’s solution is shown in the top part of Figure 1. As the external pressure decreases rapidly, the jet quickly becomes under-expanded and enters the phase of almost free expansion. When it enters the outer region of constant pressure it becomes over-expanded and a reconfinement shock is pushed towards its axis, where it gets reflected. Gas passed though these two shocks becomes hot and its pressure rises. As a result, the jet becomes somewhat under-expanded again and begins to expand for the second time. Then it becomes over-expanded again and another shock is pushed into the jet and so on.
In the bottom part of this figure, we show the results of our 1D simulations for this jet using exactly the same visualization technique as in the original paper. The agreement between the two solutions is quite remarkable. A very good match for the maximal radial extension and the oscillation-length of the jet is obtained. The successive reconfinement shocks are somewhat sharper than in B94, most likely due to the application of a shock-capturing scheme. We checked our approach against other numerical models of B94 as well. In all models, the results for profile of jet radius and Mach number are in good agreement. Noticeable but still minor differences arise only for the colder models, most likely due to the different equation of state used in our simulations.
4.2 Self-similar models of jet reconfinement
The problem of reconfinement of initially free-expanding steady-state jets is quite important and a number of authors have tried to find simple analytic of semi-analytic solutions. Falle (1991) and Komissarov and Falle (1997) used the Kompaneets approximation, which assumes that the gas pressure immediately downstream of the reconfinement shock is equal to the external pressure at the same distance, to derive a simple ODE for the shock radius. Assuming particular flow profiles in the shocked layer, one can also determine the location of the jet boundary (e.g. Bromberg and Levinson 2007). The Kompaneets approximation is accurate only for very narrow jets. To improve on it, one also has to take into account the variation of the gas pressure across the shocked layer (Nalewajko and Sikora 2009). In our second test, we compare our results with the semi-analytical model by Kohler et al. (2012), thereafter KBB12, who assumed self-similarity of the flow in this layer. This assumption is more suitable for the case where the reconfinement shock never reaches the jet axis, because otherwise the distance where this occurs sets a characteristic length scale.
The plots in Figure 3 also reveal a thin layer of decreased entropy stretching along the jet boundary. As in this layer the entropy is lower than anywhere in the initial solution, this is definitely a numerical artifact. We have checked that it becomes less pronounced with increased numerical resolution. Moreover, this layer forms well inside the jet and thus its origin is not related to the resetting procedure but is a property of our time-dependent code.
4.3 Magnetized jets. 1D versus 2D solutions
The steady-state structure of magnetized jets is more complex, mainly due to the non-trivial contribution of the magnetic tension to the force balance. A number of authors have tackled this problem analytically using various approximations (e.g. Zakamska et al. 2008; Lyubarsky 2009; Lyubarsky 2010; Kohler and Begelman 2012). However, none of these studies deliver a model suitable for detailed testing of our numerical approach. Dubal and Pantano (1993) studied the steady-state structure of relativistic jets with azimuthal magnetic field using the method of characteristics. This would be a good test case but the setup of their simulations is ambiguous. We have tried several variants of the setup but each time failed to reproduce the results. The mechanisms of magnetic collimation and acceleration of relativistic jets were studied numerically by Komissarov et al. (2007), Komissarov et al. (2009) and Tchekhovskoy et al. (2010) using a ‘rigid wall’ outer boundary. While this allows for a well-controlled experiment, Komissarov et al. (2009) have shown that the connection between the shape of the boundary and the external pressure gradient is not straightforward, with significant degeneracy. For this reason, we concluded that in the magnetic case the best way of testing the performance of our 1D approach would be via new 2D axisymmetric time-dependent simulations using the relativistic AMRVAC code (Keppens et al. 2012; Porth et al. 2014).
We considered two models, A and B. In the models A, the magnetic field is purely azimuthal and the other parameters are \(r_{j}=1\), \(r_{m}=0.37\), \(b_{m}=1\), \(\rho_{0}=1\), \(z_{0}=1\), \(\beta_{m}=0.34\), \(\Gamma_{0}=10\). The local magnetization parameter \(\sigma=b^{2}/w\) does not exceed \(\sigma_{\max}=0.7\) in this model and thus the jet is only moderately magnetized. The jet core is relativistically hot, with the gas pressure reaching \(p_{\max}=\rho\) at the axis, which opens the possibility of efficient hydrodynamic acceleration once the jet is allowed to expand. In the simulations we used the adiabatic equation of state \(w=\rho+(\gamma/\gamma-1)p\) with \(\gamma=4/3\).
Model B turned out too stiff for our 2D code, but presented no problems in 1D simulations. For this reason we compare here the 1D and 2D results for model A only. In these simulations we used the atmosphere with \(\kappa=1\). The computational domain is \(20 r_{j}\) in the radial direction and \(800 r_{j}\) in the axial direction.
The initial solution in our 2D simulations was constructed via interpolation of the converged 1D solution onto the 2D cylindrical grid. Since we did not include gravity to balance the pressure gradient in the external atmosphere, in order to preserve the atmosphere in its initial state the atmospheric parameters were reset to their initial values every time step, just like this was done in the 1D case. In order to test the convergence of 2D solutions, we made three runs with doubled resolution, \(N_{r}=400, 800, \mbox{and }1{,}600\) cells in the radial direction. The number of cell in the axial direction was always twice the number of cells in the radial one.
Typically, the 2D solutions exhibited some evolution at first but then quickly settled into a stationary state. For example in the case of \(N_{r}=400\), the timestep-to-timestep relative variation of the conserved flow variables dropped below \(6\times10^{-6}\) at \(t=1{,}000\) and remained approximately constant thereafter. Furthermore, the relative \(L_{1}\) error of density between times \(t=1{,}000\) and \(t=3{,}000\) was \(2.8\times10^{-4}\), indicating that a stationary state had been reached. The 2D solutions converge with the grid-convergence index \(\eta>1.25\) over the entire simulated time.
4.4 Magnetized jets in power-law atmospheres
In order find λ we note that for cold jets \(a_{m}^{2} \propto(p/\rho) \propto z^{-\kappa(1-1/\gamma)}\) and hence Eq. (44) yields Eq. (43) independently of the value of γ. For hot jets, \(a_{m} \simeq\mbox{const}\) and Eq. (44) still leads to Eq. (43) if we use \(\gamma=4/3\). Thus, the law (43) for the wavelength of oscillations is very robust.
The deviation from the force-balance corresponding to the secular jet expansion is due to the finite propagation speed of the waves - as they move across the jet they are also advected downstream by the supersonic flow. As the result, the jet interior reacts to the changes in the external pressure with a delay. It keeps expanding when the internal pressure is already too low and keeps contracting when it is already too high. As κ increases, the wavelength of the oscillation increases as well. This is expected as the more rapid overall expansion of the jet in an atmosphere with larger κ means that it takes longer for a magneto-sonic wave to traverse the jet, not only as the result of the larger jet radius but also as the result of its higher Mach number (and hence smaller Mach angle).
Overall, this is very similar to the well-known evolution of under-expanded supersonic jets studied in laboratories. Normally, their compressive transverse waves steepen into shocks. In our model A with \(\kappa=1\) we also detect shocks, but they become progressively weaker, suggesting that they may disappear further out along the jet. For \(\kappa=0.5\), shocks do not form at all. The exact reason for this in not yet clear.
5 Conclusions
In this paper we presented a novel numerical approach, which can be used to determine the structure of steady-state relativistic jets. It is based on the similarity between the two-dimensional steady-state equations and the one-dimensional time-dependent equations of SRMHD with the cylindrical symmetry in problems involving narrow highly-relativistic (\(v_{z}\approx c\)) flows. Such similarity has already been utilised in the so-called ‘frozen pulse’ approximation where dynamics of time-dependent relativistic flows is analyzed using the steady-state equations (Piran et al. 1993; Vlahakis and Königl 2003; Sapountzis and Vlahakis 2013). Here we do the opposite and construct approximate steady-state solutions via numerical integration of the time-dependent equations. The main advantage of this approach is utilitarian. First, it allows us to use computer codes for relativistic MHD (or hydrodynamics in the case of unmagnetized flows), which are now widely available, in place of highly-specialised codes for integrating steady-state equations, which are not openly available at the moment. Moreover, the reduced dimensionality means that the computational facilities can be very modest - a basic laptop will suffice. In contrast, the relaxation method based on integration of two-dimensional time-dependent equations can be computationally quite expensive.
We compared numerical solutions obtained with this approach with analytical models and numerical solutions obtained with other techniques. The considered problems involved a variety of flows both magnetized and unmagnetized, with different equations of state and external conditions. The results show that the method is sufficiently accurate and robust.
Although we focused only on relativistic flows, we see no reason why this approach cannot be applied to non-relativistic hypersonic flows. For such flows, the axial velocity of bulk motion plays the role of the speed of light in the substitution \(z=ct\) used in our derivations.
As a byproduct of our test simulations, we obtained two results of astrophysical interest. We demonstrated that the failure of the self-similar model of the jet reconfinement in power-law atmospheres with the index \(\kappa<8/3\) (Kohler et al. 2012) is rooted in the assumption of isentropy of the shocked layer, which is made in this model. In reality, the reconfinement shock becomes stronger with the distance along the jet, resulting in a strong spatial variation of the entropy. We also found that the radial oscillations of steady-state jets, discovered in the analytical models of Poynting-dominated jets (Lyubarsky 2009) is a generic part of the jet adjustment to the space-variable external pressure and not specific to the high-magnetization regime only. The oscillations are standing waves induced by the variation.
The steady-state solutions are useful for elucidating some key factors in flow dynamics and may closely describe some of the observed phenomena in astrophysical jets. However, they are often subject to various instabilities which may dramatically modify the flow properties. Most instability studies, both analytical and numerical, deal with very simple problems where the steady-state solution is readily available. In more realistic setup, the issue of finding the steady-state solution, which can then be subjected to perturbations, becomes more involved and this is where our method can be applied in the instability studies.
Declarations
Acknowledgements
SSK and OP were supported by STFC under the standard grant ST/I001816/1. SSK and ML were supported by NASA under the grant NNX12AF92G. OP thanks Purdue University for hospitality during his visits in 2014. We thank the anonymous reviewers for constructive comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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