  The limit of decreasing laser frequency can not be considered independently
from nondipole effects due to increase in the laser-induced continuum electron
speed in this limit. Therefore, in this work, tunneling ionization in the
adiabatic limit is considered for an effective field that includes effects
beyond the electric-dipole term to first order in $1/c$, with $c$ the speed of
light. The nondipole term describes the interaction resulting from the electric
dipole-induced velocity of the electron and the magnetic field component of the
laser. The impact of this term on the ionization rate, tunnel exit point,
momentum at the tunnel exit and electron dynamics is discussed. In the
appropriate limit, the results of a nondipole strong-field approximation
approach and those of the strict adiabatic limit, where time and field strength
are parameters, are consistent. The nondipole strong-field approximation
approach is used to identify nonadiabatic modifications of the initial
conditions. The results open up an avenue to include nondipole effects in the
initial tunneling ionization step in semiclassical models of strong-field and
attosecond physics.
