  We study the problems of controllability and ergodicity of the system of 3D
primitive equations modeling large-scale oceanic and atmospheric motions. The
system is driven by an additive force acting only on a finite number of Fourier
modes in the temperature equation. We first show that the velocity and
temperature components of the equations can be simultaneously approximately
controlled to arbitrary position in the phase space. The proof is based on
Agrachev-Sarychev type geometric control approach.
  Next, we study the controllability of the linearisation of primitive
equations around a non-stationary trajectory of randomly forced system.
Assuming that the probability law of the forcing is decomposable and
observable, we prove almost sure approximate controllability by using the same
Fourier modes as in the nonlinear setting. Finally, combining the
controllability of the linearised system with a criterion from
arXiv:1802.03250, we establish exponential mixing for the nonlinear primitive
equations with a random force.
