  Recent numerical work in helioseismology has shown that a periodically
varying subsurface magnetic field leads to a fanning of the $f$-mode, which
emerges from the density jump at the surface. In an attempt to model a more
realistic situation, we now modulate this periodic variation with an envelope,
giving thus more emphasis on localized bipolar magnetic structures in the
middle of the domain. Some notable findings are: (i) compared to the purely
hydrodynamic case, the strength of the $f$-mode is significantly larger at high
horizontal wavenumbers $k$, but the fanning is weaker for the localized
subsurface magnetic field concentrations investigated here than the periodic
ones studied earlier; (ii) when the strength of the magnetic field is enhanced
at a fixed depth below the surface, the fanning of the $f$-mode in the
$k\omega$ diagram increases proportionally in such a way that the normalized
$f$-mode strengths remain nearly the same in different such cases; (iii) the
unstable Bloch modes reported previously in case of harmonically varying
magnetic fields are now completely absent when more realistic localized
magnetic field concentrations are imposed beneath the surface, thus suggesting
that the Bloch modes are unlikely to be supported during most phases of the
solar cycle; (iv) the $f$-mode strength appears to depend also on the depth of
magnetic field concentrations such that it shows a relative decrement when the
maximum of the magnetic field is moved to a deeper layer. We argue that
detections of $f$-mode perturbations such as those being explored here could be
effective tracers of solar magnetic fields below the photosphere before these
are directly detectable as visible manifestations in terms of active regions or
sunspots.
