[Paper Review] Analytic Solutions of Transverse Magneto-hydrodynamics under Bjorken Expansion

Shi Pu, Di-Lun Yang|arXiv (Cornell University)|Nov 15, 2016

Fluid Dynamics and Turbulent Flows4 citations

TL;DR

This paper presents analytic solutions for transverse magneto-hydrodynamics under Bjorken expansion, showing that time- and space-dependent magnetic fields induce transverse fluid flow whose direction depends on the magnetic field's decay exponent. The key result is that for $ n < -1 $, flow is inward toward the center, while for $ n > -1 $, flow is outward, with vanishing flow at $ n = -1 $, explained by magnetic flux conservation.

ABSTRACT

We review the recent developments of analytic solutions in transverse magneto-hydrodynamics under Bjorken expansion. It is found that the time dependence of magnetic fields can either increase or reduce the energy density depending on the decay exponent of magnetic fields. Moreover, perturbative solutions under weak magnetic fields with spatial inhomogeneity results in transverse flow, where the directions of flow also depend on the decay exponent of magnetic fields in time.

Motivation & Objective

• To develop analytic solutions for relativistic magneto-hydrodynamics in the presence of spacetime-dependent magnetic fields under Bjorken expansion.
• To investigate how time-dependent and spatially inhomogeneous magnetic fields affect energy density and fluid velocity in quark-gluon plasma.
• To clarify the physical origin of transverse flow induced by magnetic fields, particularly the role of magnetic flux conservation.
• To provide a simplified, analytically tractable framework to gain physical insight before numerical simulations in heavy-ion collisions.

Proposed method

• Formulate the energy-momentum tensor for a conformal fluid coupled to a transverse magnetic field $ B_y( au) $, with $ p = rac{1}{3} ho $, in Milne coordinates.
• Apply the conservation law $ abla_ u T^{ ueta} = 0 $, projecting along longitudinal and transverse directions to derive evolution equations for energy density and fluid velocity.
• Solve the energy density equation analytically for arbitrary time-dependent $ B_y( au) $, yielding $ ho( au) ightarrow au^{-4/3} imes ext{integral of } B_y^2 + au B_y rac{dB_y}{d au} $.
• For spatial inhomogeneity, expand the magnetic field in Fourier modes $ B_y( au,x) = ilde{B}_k( au) e^{ikx} $, leading to a second-order ODE for each mode $ b_k( au) $.
• Use asymptotic boundary conditions at late times (vanishing $ B $) to fix integration constants, enabling full analytic solution of $ ho_1( au,x) $ and $ u_x( au,x) $.

Experimental results

Research questions

• RQ1 How does a time-dependent transverse magnetic field affect the energy density evolution in Bjorken-expanding quark-gluon plasma?
• RQ2 What is the origin of transverse fluid flow induced by spatially inhomogeneous magnetic fields in relativistic hydrodynamics?
• RQ3 Why does the direction of transverse flow reverse depending on the decay exponent $ n $ of the magnetic field $ B_y ightarrow au^n $?
• RQ4 How does magnetic flux conservation explain the observed reversal of flow direction at $ n = -1 $?
• RQ5 Can analytic solutions with spatial inhomogeneity provide a physical mechanism for magnetic-field-induced collective flow in heavy-ion collisions?

Key findings

• The energy density $ ho( au) $ is modified by the time evolution of $ B_y^2 $ and its derivative, with the solution given explicitly as $ ho( au) = au^{-4/3} imes ext{integral of } (B_y^2 + au B_y rac{dB_y}{d au}) $.
• For spatially inhomogeneous magnetic fields, transverse flow $ u_x $ is induced, with its direction determined by the decay exponent $ n $ of $ B_y( au) ightarrow au^n $.
• When $ n < -1 $, the transverse flow is directed inward toward $ x = 0 $, consistent with magnetic flux conservation as the field decays faster than expansion.
• When $ n > -1 $, the flow is directed outward, as the expanding medium experiences increasing flux due to slow field decay.
• At $ n = -1 $, the flow vanishes, indicating a balance between magnetic flux decrease and medium expansion.
• For $ n = -1 $, the solution simplifies to $ u_x = 0 $, $ ho_1 = -rac{3}{2 au} ilde{B}_y^2(x) $, confirming the absence of transverse flow.