  Context: The first tentative detection of a magnetic field on the Hot Jupiter
type exoplanet $\tau$ Bo\"otis b was recently reported by Turner et al. (2021).
The magnetic field was inferred from observations of circularly-polarized radio
emission obtained with the LOFAR telescopes. The radio emission is possibly a
consequence of the interaction of the surrounding stellar wind with the
planet's magnetic field. Methods: We perform magnetohydrodynamic simulations of
the space environment around $\tau$ Bo\"otis b and its interaction with the
stellar wind using the PLUTO code. We study the magnetospheric energy fluxes
and effects of different magnetic field orientations in order to understand the
physical processes which cause energy fluxes leading to the observed radio
emission given the proposed magnetic field strength in Turner et al. (2021).
Furthermore we study the effect of stellar wind density and pressure on
magnetospheric energy fluxes given the uncertainty of extrasolar stellar wind
predictions. Results: We find in our simulations that the interaction is most
likely super--Alfv\'enic and energy fluxes generated by the stellar
wind--planet interaction are consistent with the observed radio powers.
Magnetospheric Poynting fluxes are of the order of 1-8 $\times10^{18}$ W for
open, semi-open and closed magnetospheres. The Poynting fluxes are
energetically consistent with the radio powers in Turner et al. (2021) for a
magnetospheric Poynting flux-to-radio efficiency $> 10^{-3}$ when the magnetic
fields of the planet and star are aligned. In case of lower efficiency factors
the magnetospheric emission scenario is according to the parameter space
modeled in this study not powerful enough. In case of a magnetic polarity
reversal of the host star towards an anti-aligned field configuration, expected
radio powers in the magnetospheric emission scenario fall below the observable
threshold.
