We present results from two-dimensional numerical simulations of a sup
ersonic turbulent flow with parameters characteristic of the interstel
lar medium at the 1 kpc scale in the plane of the Galactic disk, incor
porating shear, thresholded and discrete star formation (SF), self-gra
vity, rotation, and magnetic fields. A test of the model in the linear
regime supports the results of the linear theory presented by Elmegre
en in 1991. At low shear, a weak azimuthal magnetic field stabilizes t
he medium by opposing collapse of radial perturbations, while a strong
field is destabilizing by preventing Coriolis spin-up of azimuthal pe
rturbations (magnetic braking). At high shear, azimuthal perturbations
are sheared into the radial direction before they have time to collap
se, and the magnetic field becomes stabilizing again. In the fully non
linear turbulent regime, while some results of the linear theory persi
st, new effects also emerge. The production of turbulent density fluct
uations appears to be affected by the magnetic field as in the linear
regime: moderate field strengths cause a decrease in the time-integrat
ed star formation rate, while larger values cause an increase. A resul
t not predicted by the linear theory is that, for very large field str
engths, a decrease in the integrated SFR obtains again, indicating a '
'rigidization'' of the medium due to the magnetic field. Other exclusi
vely nonlinear effects are the following: (1) Even though there is no
dynamo in two dimensions, the simulations are able to maintain or incr
ease their net magnetic energy in the presence of a seed uniform azimu
thal component. (2) A well-defined power-law magnetic spectrum and an
inverse magnetic cascade are observed in the simulations, indicating f
ull MHD turbulence. Thus, magnetic field energy is generated in region
s of SF and cascades up to the largest scales. (3) The field has a sli
ght but noticeable tendency to be aligned with density features. This
appears to be as much a consequence of the gas pushing on the magnetic
field as due to constraints on gas motions because of the presence of
the magnetic field. (4) A ''pressure-cooker'' effect is observed in w
hich the magnetic field prevents H II regions from expanding freely, a
s in the recent results of Slavin and Cox. (5) The orientation of the
large-scale azimuthal field appears to follow that of the large-scale
Galactic shear. (6) A tendency to exhibit less filamentary structures
at stronger values of the uniform component of the magnetic field is p
resent in several magnetic runs. Possible mechanisms that may lead to
this result are discussed. (7) For fiducial values of the parameters,
the flow in general appears to be in rough equipartition between magne
tic and kinetic energy. There is no clear domination of either the mag
netic or the inertial forces. (8) A median value of the magnetic field
strength within clouds is similar to 12 mu G, while for the interclou
d medium a value of similar to 3 mu G is found. Maximum contrasts of u
p to a factor of similar to 10 are observed.