Relativistic magnetohydrodynamics in one dimension

📝 Abstract

We derive a number of solution for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes. First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum. Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov potential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation.

💡 Analysis

We derive a number of solution for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes. First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum. Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov potential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation.

📄 Content

Relativistic magnetohydrodynamics in one dimension Maxim Lyutikov and Samuel Hadden Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036 Abstract We derive a number of solution for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes. First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum. Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov po- tential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation. 1 arXiv:1112.0249v1 [astro-ph.HE] 1 Dec 2011 I. INTRODUCTION Expansion of plasma into vacuum is a basic problem in fluid mechanics that has a wide range of applications from heavy nuclei collisions to astrophysics. In nuclear physics, Belenkij and Landau [1] used the hydrodynamical approach to study multiparticle pro- duction in heavy ion collisions. A head-on collision of two highly relativistic nuclei cre- ates a relativistically-compressed hot layer of quarkgluon plasma that expands quasi-one- dimensionally [2]. On a very different scale, a wide variety of astrophysical objects like jets from Active Galactic Nuclei (AGN) [3], Gamma Ray Burst (GRB) [4] and pulsar winds and magnetospheres of a special type of neutron stars - magnetars [5]- may contain relativistic strongly magnetized plasma, in which the energy density of the magnetic field dominates over the matter energy density (including kinetic and thermal energies), B2 ≥ρc2, P. During explosions the strongly magnetized plasma created by the central source suddenly expands into a surrounding low density medium. In case of magnetar flares, the initial dissipation event creates relativistically hot, strongly magnetized fireball, somewhat analogous to Solar coronal mass ejections [6]; the fireball accelerates to relativistic velocities [7]. In long GRBs, when a hot magnetically dominated jet reaches the surface of the star it breaks into low density medium [4, 8]. Similar dynamics may occur in non-stationary outflows in AGNs as well [9]. In all the above mentioned cases, it is expected that at some distances from the source, the magnetic field is dominated by the toroidal component, while motion is preferentially radial. Thus, velocity is nearly perpendicular to magnetic field - this type of motion is sometimes called transverse magnetohydrodynamics. Equations of transverse MHD reduce to fluid equations, but with a complicated equation of state [10]. Qualitatively, such plasma behaves like a mixture of fluids with different adiabatic indices, Γ = 2 for the magnetic field, and some Γ for the kinetic pressure, usually taken to be Γ = 4/3. Thus, many results of fluid dynamics, which assume a single adiabatic index, become invalid. Numerical investigation of these phenomena requires the use of relativistic MHD codes that can handle high magnetization and high kinetic pressures exceeding the rest mass density. Exact, explicit non-linear solutions of fluid equations, and especially relativistic MHD equations, are rare. Yet they are important for benchmark estimates of the overall dynamical behavior in numerical simulations of relativistic flows and strongly magnetized 2 outflows in particular. In §II we find analytical expressions for self-similar expansion into vacuum of a hot magnetized plasma, considering both the Newtonian and relativistic cases with arbitrary ratios of kinetic and magnetic pressures to rest mass density. In addition, at later times, when the whole initial state of plasma is affected by the expansion, the expansion dynamics becomes non-self-similar. A classical related problem is then the expansion of a slab of finite length. Arbitrary isentropic one-dimensional motion of a fluid is fully integrable. Mathematically this is achieved by exchanging the independent variables {t, x} and dependent variables {ρ, v}. As a result a system of two non-linear equations (of mass conservation and Euler’s law plus assumed isentropic equation of state) is reduced to a single linear equation for the (Legendre-transformed) Bernoulli (Khalatnikov) potential [e.g. 11]. This is always possible if the coefficients in these equations do not depend explicitly on time and coordinate. In the case of isentr

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