  This paper experimentally investigates the convection in a fast rotating
Tangent Cylinder (TC), for Ekman numbers down to $E=3.36\times10^{-6}$, in a
configuration relevant to the liquid core of the Earth. In the apparatus, the
TC results from the Proudman-Taylor constraint incurred by rotating a
hemispherical fluid vessel heated in its centre by a protruding heating element
of cylindrical shape. The resulting convection that develops above the heater,
i.e within the TC, is shown to set in for critical Rayleigh numbers and
wavenumbers respectively scaling as $Ra_c\sim E^{4/3}$ and $a_c\sim E^{1/3}$
with the Ekman number $E$. Though exhibiting the same exponents as for plane
rotating convection, these laws are indicative of much larger convective plumes
at onset. The structure and dynamics of these plumes are in fact closer to
those found in solid rotating cylinders heated from below, suggesting that the
confinement within the TC induced by the Taylor-Proudman constraint influences
convection in a similar way as solid walls would do. There is further
similarity in that the critical modes in the TC all exhibit a slow retrograde
precession at onset. In supercritical regimes, the precession evolves into a
thermal wind with a complex structure featuring retrograde rotation at high
latitude and either prograde or retrograde rotation at low latitudes (close to
the heater), depending on the criticality and the Ekman number. Nevertheless
the intensity of the thermal wind measured by the Rossby number scales as
$Ro\sim 0.85(Ra_q^*)^{0.41}$ with the Rayleigh number based on the heat flux
$Ra_q^*$. This scaling suggests that the convection in the TC is driven by
quasi-geostrophic dynamics, a finding supported by the scaling for the
rotation-normalised Nusselt number $Nu^{*} \sim (Ra_{q}^{*})^{5/9}$.
