  We interpret penumbral filaments as due to convection in field-free, radially
aligned gaps just below the visible surface of the penumbra, intruding into a
nearly potential field above. This solves the classical discrepancy between the
large heat flux and the low vertical velocities observed in the penumbra. The
presence of the gaps causes strong small-scale fluctuations in inclination,
azimuth angle and field strength, but without strong forces acting on the gas.
The field is nearly horizontal in a region around the cusp-shaped top of the
gap, thereby providing an environment for Evershed flows. We identify this
region with the recently discovered dark penumbral cores. Its darkness has the
same cause as the dark lanes in umbral light-bridges, reproduced in numerical
simulations by Nordlund and Stein (2005). We predict that the large vertical
and horizontal gradients of the magnetic field inclination and azimuth in the
potential field model will produce the net circular polarization seen in
observations. The model also explains the significant elevation of bright
filaments above their surroundings. It predicts that dark areas in the penumbra
are of two different kinds: dark filament cores containing the most inclined
(horizontal) fields, and regions between bright filaments, containing the least
inclined field lines.
