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Motivated by the recently found realization of the $1+1$ dimensional Bjorkenflow in ideal and nonideal relativistic magnetohydrodynamics (MHD), we useappropriate symmetry arguments, and determine the evolution of magnetic fieldsarising from the $3+1$ dimensional self-similar and Gubser flows in aninfinitely conductive relativistic fluid (ideal MHD). In the case of the $3+1$dimensional self-similar flow, we arrive at a family of solutions, that arerelated through a differential equation arising from the corresponding Eulerequation. To find the magnetic field evolution from the Gubser flow, we solvethe MHD equations of a stationary fluid in a conformally flat $dS^{3}\timesE^{1}$ spacetime. The results are then Weyl transformed back into the Minkowskispacetime. In this case, the temporal evolution of the resulting magnetic fieldis shown to exhibit a transition between an early time $1/t$ decay to a$1/t^{3}$ decay at a late time. Here, $t$ is the time coordinate. Transverseand longitudinal components of the magnetic fields arising from these flows arealso found. The latter turns out to be sensitive to the transverse size of thefluid. In contrast to the result arising from the Gubser flow, the radialdomain of validity of the magnetic field arising from the self-similar flow ishighly restricted. A comparison of the results suggests that the (conformal)Gubser MHD may give a more appropriate qualitative picture of the magneticfield decay in the plasma of quarks and gluons created in heavy ion collisions.

Evolution of magnetic fields from the 3+1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics