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The objective of this work is to understand how the characteristics of relativistic MHD turbulence may differ from those of nonrelativistic MHD turbulence. We accomplish this by studying the ideal invariants in the relativistic case and comparing them to what we know of nonrelativistic turbulence. Although much work has been done to understand the dynamics of nonrelativistic systems (mostly for ideal incompressible fluids), there is minimal literature explicitly describing the dynamics of relativistic MHD turbulence using numerical simulations. Many researchers simply assume that relativistic turbulence has the same invariants and obeys the same dynamics as non-relativistic systems. Our results show that this assumption may be incorrect.

Many studies in numerical relativity and high-energy astrophysics depend on the dynamics of relativistic plasmas. These include phenomena such as primordial turbulence, neutron stars, active galactic nuclei, and accretion disks near black holes [

In the following report we will first discuss what is currently known about the dynamics of nonrelativistic MHD systems. We introduce the standard incompressible and compressible nonrelativistic MHD evolution equations as well as the ideal invariants for those systems. In Section 4, we will introduce the relativistic MHD equations and the relativistic equivalents of the nonrelativistic ideal MHD invariants. We then describe our numerical experiment and present our results for a relativistic MHD code. We conclude by discussing the similarities and differences between the different systems.

Work by Shebalin [

By varying the mean magnetic field (

For an incompressible fluid u(k, t) is the Fourier coefficient of turbulent velocity and b(k, t) is the Fourier coefficient of the turbulent magnetic field. The energy, cross helicity and magnetic helicity can be expressed in terms of these as:

A cubic computational domain with N grid points in each direction is assumed. The statistical mechanics of the system is defined by the Gaussian canonical probability density function (PDF):

Case | Mean field | Angular velocity | Invariants |
---|---|---|---|

I | 0 | 0 | |

II | 0 | ||

III | 0 | ||

IV | |||

V | E |

where Z is the partition function and

averages using the PDF while

it is non-ergodic. Here

be found as a function of

Phase portraits resulting from computer simulations of Shebalin’s five cases show that coherent structures formed in many systems where the magnetofluid was experiencing turbulence [

Compressible MHD systems have not been studied as much as incompressible systems so here we will focus primarily on their invariants. We will assume that both incompressible and compressible systems share the same statistical mechanics and dynamics whenever the same invariants apply. In a nonrelativistic compressible MHD system; Energy and the Incompressible form of Cross Helicity are always conserved for a nondissipative system [

Given that our relativistic system is by default a compressible system, we naively expect to see that the same ideal invariants will apply for the relativistic system as the nonrelativistic compressible system. The equations for ideal compressible MHD are similar to those of the incompressible system with the exception of the first equation.

Case | Mean field | Invariants |
---|---|---|

I | 0 | |

II |

The fluid and electromagnetic components of the relativistic MHD equations are developed from several well-known equations [

Here, P is the fluid pressure,

terms of the energy density using the

and

Here,

Notice that unlike the nonrelativistic system, we use the stress-energy tensor within the evolution equations so that 4-momentum conservation is built into the system. This results in a set of equations that look very different from that of the nonrelativistic system. According to work by Yoshida [

Here the canonical 4-momentum density

For this numerical experiment, we calculate Energy Density (E), Relativistic Helicity Density (

Here the magnetic field is related to the vector potential by the equation,

In order to study the invariants of the relativistic MHD equations, we used a code called FixedCosmo which was originally written by one of us [

Because the objective of this study is to test the ideal relativistic MHD system, we complete a series of runs in a “high-energy” regime. The parameters used approximate that of the early universe around the electroweak scale. This is done so we can apply the results to any relativistic MHD system.

Each data run utilized Fourier spectral differencing on a grid with 64 × 64 × 64 internal data points. We ran these simulations for about 7500 iterations or over 10^{−9} s of physical time. The electron oscillation time for the “high energy” regime is about

Truncation errors were found by doubling the resolution and measuring the change in the observed total errors. By assuming that the Euler Method, used to calculate

If the truncation errors are within an order of magnitude of the normalized errors, we can conclude that the system is invariant. We also found it impossible to completely eliminate the mean magnetic field and mean angular momentum in all cases. A mean magnetic field (on the order of 1% of the maximum field) remained in every case. Also, each case seemed to have a small angular velocity, also less than 1% of the fluid velocities within the simulation. The authors feel that these residual quantities where not enough to significantly disrupt the system and could be safely ignored.

Variables | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|

Max velocity (c) | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |

0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0.35 | 0.35 | 0.35 | |

Init temperature (K) | 2.8e15 | 2.8e15 | 2.8e15 | 2.8e15 | 2.8e15 |

Init density (kg/m^{3}) | 9.7e29 | 9.7e29 | 9.7e29 | 9.7e29 | 9.7e29 |

Max magnetic field (G) | 1.0e13 | 1.0e13 | 1.0e13 | 1.0e13 | 1.0e13 |

0 | 2.0e13 | 0 | 0 | 2.0e13 | |

0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 2.0e13 | 0 |

Mean magnitude of errors | E | |||
---|---|---|---|---|

Case 1 | 1.8e−9 | 2.8e−14 | 9.6e−5 | 1.0e−2 |

Case 2 | 1.8e−9 | 2.8e−14 | 9.7e−5 | 2.8e−2 |

Case 3 | 8.1e−10 | 0.0 | 6.6e−5 | 7.1e−3 |

Case 4 | 7.7e−10 | 0.0 | 6.0e−5 | 1.9e−2 |

Case 5 | 7.8e−10 | 0.0 | 6.6e−5 | 1.8e−2 |

Truncation errors | 2.1e−9 | 2.2e−14 | 2.9e−5 | 2.0e−3 |

The results in

Our results show that in the high-energy Relativistic MHD regime only Energy and Relativistic Helicity are clearly conserved. We are not able to conclusively prove Cross Helicity conservation. Magnetic Helicity conservation is questionable in this system. This is not an unexpected result but it does raise several interesting questions which lie beyond the scope of this article. Does the potential lack of Cross and Magnetic Helicity Conservation effect the dynamics of the relativistic system when it comes to phenomena such as inverse Energy Cascade or the Kolmogorov Energy Spectrum? How do magnetic dynamos in relativistic MHD systems function? Are there any other overlooked dynamics in relativistic MHD systems? These are all questions which we hope to address in future numerical studies.

The authors declare that there is no conflict of interests regarding the publication of this article.

The authors would like to acknowledge the support of the University of Houston Center for Advanced Computing and Data Systems for access to the high performance computing resources used for the completion of this project. The authors would also like to thank John Shebalin for several useful conversations and helpful suggestions.

DavidGarrison,PhuNguyen, (2016) Invariants in Relativistic MHD Turbulence. Journal of Modern Physics,07,281-289. doi: 10.4236/jmp.2016.73028