Atmospheric systems incorporating thermal dynamics must be stable with
respect to both energy and entropy. While energy conservation can be enforced
via the preservation of the skew-symmetric structure of the Hamiltonian form of
the equations of motion, entropy conservation is typically derived as an
additional invariant of the Hamiltonian system, and satisfied via the exact
preservation of the chain rule. This is particularly challenging since the
function spaces used to represent the thermodynamic variables in compatible
finite element discretisations are typically discontinuous at element
boundaries. In the present work we negate this problem by constructing our
equations of motion via weighted averages of skew-symmetric formulations using
both flux form and material form advection of thermodynamic variables, which
allow for the necessary cancellations required to conserve entropy without the
chain rule. We show that such formulations allow for stable simulations of both
the thermal shallow water and 3D compressible Euler equations on the sphere
using mixed compatible finite elements without entropy damping