How does toroidal magnetization govern thermal energy retention in relativistic active galactic nucleus jet cocoons?

Abstract Relativistic jets launched from supermassive black holes at the centres of Active Galactic Nuclei (AGNs) inject vast thermal power into their environments by inflating an overpressured cocoon of shocked plasma. The fraction of this power that thermalizes as internal energy within the cocoon – expressed as the internal energy conversion efficiency per unit cocoon volume, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="bold-italic">η</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">internal</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant="bold">E</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">int</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">cocoon</m:mi> </m:mrow> </m:msubsup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="bold">E</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">injected</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="bold">V</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">cocoon</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:math> ${\boldsymbol{\eta }}_{\mathbf{internal}}=\frac{{\mathbf{E}}_{\mathbf{int}}^{\mathbf{cocoon}}}{{\mathbf{E}}^{\mathbf{injected}}{\times}{\mathbf{V}}_{\mathbf{cocoon}}}$ , provides a physically meaningful measure of the thermalization efficiency by accounting for the continuous expansion of the cocoon during jet propagation. We present three-dimensional special relativistic magnetohydrodynamic jet from Active Galactic Nucleus with Lorentz factor Γ = 10 , Mach number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="bold-script">M</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="bold">6</m:mi> </m:math> $\boldsymbol{\mathcal{M}}=\mathbf{6}$ , and toroidal magnetization σ ϕ ∈ {0.25, 0.50, 0.75} at fixed σ ϕ + σ z = 1 . Three-dimensional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="bold-script">Q</m:mi> </m:math> $\boldsymbol{\mathcal{Q}}$ iso-surfaces demonstrate that increasing σ ϕ systematically enhances jet collimation while suppressing lateral vortex penetration into the cocoon. As a direct consequence, η internal decreases monotonically from 3.15 % at σ ϕ = 0.25–2.70 % at σ ϕ = 0.75 , indicating a systematic reduction in the internal energy conversion efficiency per unit cocoon volume with increasing toroidal magnetization. This behaviour arises because stronger toroidal confinement simultaneously reduces the cocoon’s thermal energy reservoir and amplifies the jet inflow (nozzle) energy flux. Within the investigated parameter regime, the present results demonstrate that increasing toroidal magnetization systematically influences cocoon thermalization, indicating that the toroidal magnetic field plays an important role in regulating the internal energy conversion efficiency per unit cocoon volume rather than establishing a universal regulatory mechanism for all Active Galactic Nucleus jets.