  The Atlantic Meridional Overturning Circulation (AMOC) distributes heat and
salt into the Northern Hemisphere via a warm surface current toward the
subpolar North Atlantic, where water sinks and returns southwards as a deep
cold current. There is substantial evidence that the AMOC has slowed down over
the last century. We introduce a conceptual box model for the evolution of
salinity and temperature on the surface of the North Atlantic Ocean, subject to
the influx of meltwater from the Greenland ice sheets. Our model, which extends
a model due to Welander, describes the interaction between a surface box and a
deep-water box of constant temperature and salinity, which may be convective or
non-convective, depending on the density difference. Its two main parameters
$\mu$ and $\eta$ describe the influx of freshwater and the threshold density
between the two boxes, respectively.
  We use bifurcation theory to analyse two cases of the model: instantaneous
switching between convective or non-convective interaction, where the system is
piecewise-smooth (PWS), and the full smooth model with more gradual switching.
For the PWS model we derive analytical expressions for all bifurcations. The
resulting bifurcation diagram in the $(\mu,\eta)$-plane identifies all regions
of possible dynamics, which we show as phase portraits - both at typical
parameter points, as well as at the different transitions between them. We also
present the bifurcation diagram for the case of smooth switching and show how
it arises from that of the PWS case. In this way, we determine exactly where
one finds bistability and self-sustained oscillations of the AMOC in both
versions of the model. In particular, our results show that oscillations
between temperature and salinity on the surface North Atlantic Ocean disappear
completely when the transition between the convective and non-convective
regimes is too slow.
