  Two-dimensional simulations of hot Jupiter upper atmospheres including the
planet's magnetic field are presented. The goal is to explore magnetic effects
on the layer of the atmosphere that is ionized and heated by stellar EUV
radiation, and the imprint of these effects on the Ly$\alpha$ transmission
spectrum. The simulations are axisymmetric, isothermal, and include both
rotation and azimuth-averaged stellar tides. Mass density is converted to
atomic hydrogen density through the assumption of ionization equilibrium. The
three-zone structure -- polar dead zone, mid-latitude wind zone, and equatorial
dead zone -- found in previous analytic calculations is confirmed. For a
magnetic field comparable to that of Jupiter, the equatorial dead zone, which
is confined by the magnetic field and corotates with the planet, contributes at
least half of the transit signal. For even stronger fields, the gas escaping in
the mid-latitude wind zone is found to have a smaller contribution to the
transit depth than the equatorial dead zone. Transmission spectra computed from
the simulations are compared to HST STIS and ACS data for HD 209458b and HD
189733b, and the range of model parameters consistent with the data is found.
The central result of this paper is that the transit depth increases strongly
with magnetic field strength when the hydrogen ionization layer is magnetically
dominated, for dipole magnetic field $B_0 > 10\ {\rm G}$. Hence transit depth
is sensitive to magnetic field strength, in addition to standard quantities
such as the ratio of thermal to gravitational binding energies.
