  We investigate the null controllability property of the parabolic equation
associated with the Grushin operator defined by the canonical almost-Riemannian
structure on the 2-dimensional sphere $\mathbb S^2$. This is the natural
generalization of the Grushin operator $\mathcal G = \partial_x^2 +
x^2\partial_y^2$ on $\mathbb R^2$ to this curved setting, and presents a
degeneracy at the equator of $\mathbb S^2$. Namely, its expression in
latitude-longitude spherical coordinates $(x,y)\in(-\pi/2,\pi/2)\times
[0,2\pi)$ is
  $$\mathcal{L} = \partial_{x}^{2}-\tan
x\partial_{x}+\tan^2x\partial_{y}^{2}.$$
  We prove that the null controllability is verified in large time when the
control acts as a source term distributed on a subset $\omega = \{
(x_1,x_2,x_3)\in \mathbb S^2\mid \alpha<|x_3|<\beta \}$ for some
$0\le\alpha<\beta\le 1$. More precisely, we show the existence of a positive
time $T^{*}>0$ such that the system is null controllable from $\omega$ in any
time $T>T^{*}$, and that the minimal time of control from $\omega$ satisfies
$T_{\min}\ge\log(1/\sqrt{1-\alpha^2})$. Here, the lower bound corresponds to
the Agmon distance of $\omega$ from the equator.
  These results are obtained by applying the same ideas as in (Beauchard et
al., 2014), consisting in proving a suitable Carleman estimate, and then taking
advantage of a Poincar\'e-Hardy type inequality due to (Chisholm and Everitt,
1970) to show the positive null-controllability result. On the other hand, the
negative statement is proved by exploiting an appropriate family of spherical
harmonics, which concentrate at the equator, to falsify the uniform
observability inequality.
