  We investigate the interplay of magnetic fluctuations and Cooper pairing in
twisted bilayer graphene from a purely microscopic model within a large-scale
tight-binding approach resolving the \AA ngstr\"om scale. For local onsite
repulsive interactions and using the random-phase approximation for spin
fluctuations, we derive a microscopic effective pairing interaction that we use
for self-consistent solutions of the Bogoliubov-de-Gennes equations of
superconductivity. We study the predominant pairing types as function of
interaction strength, temperature and band filling. For large regions of this
parameter space, we find chiral $d$-wave pairing regimes, spontaneously
breaking time-reversal symmetry, separated by magnetic instabilities at integer
band fillings. Interestingly, the $d$-wave pairing is strongly concentrated in
the AA regions of the moir\'e unit cell and exhibits phase windings of integer
multiples of $2\pi$ around these superconducting islands, i.e. pinned vortices.
The spontaneous circulating current creates a distinctive magnetic field
pattern. This signature of the chiral pairing should be measurable by
state-of-the-art experimental techniques.
