  The traditional Hamiltonian structure of the equations governing conservative
Rayleigh-B\'enard convection (RBC) is singular, i.e. it's Poisson bracket
possesses nontrivial Casimir functionals. We show that a special form of one of
these Casimirs can be used to extend the bilinear Poisson bracket to a
trilinear generalised Nambu bracket. It is further shown that the equations
governing dissipative RBC can be written as the superposition of the
conservative Nambu bracket with a dissipative symmetric bracket. This leads to
a Nambu-metriplectic system, which completes the geometrical picture of RBC.

Fluid Dynamics;Atmospheric and Oceanic Physics