  In this paper, Hamiltonian and energy preserving reduced-order models are developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with the state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with skew-symmetric finite-differences, that preserve the Hamiltonian structure. The resulting skew-gradient system is integrated in time with the energy preserving average vector field (AVF) method. The reduced-order model (ROM) is constructed in the same way as the full order model (FOM), preserving the reduced skew-symmetric structure and integrating in time with the AVF method. Relying on structure-preserving discretizations in space and time and applying proper orthogonal decomposition (POD) with the Galerkin projection, an energy preserving reduced order model (ROM) is constructed. The nonlinearities in the ROM are computed by applying the discrete empirical interpolation (DEIM) method to reduce the computational cost. The computation of the reduced-order solutions is accelerated further by the use of tensor techniques. The overall procedure yields a clear separation of the offline and online computational cost of the reduced solutions. The accuracy and computational efficiency of the ROMs are demonstrated for a numerical test problem. Preservation of the energy (Hamiltonian), and other conserved quantities, i.e. mass, buoyancy, and total vorticity show that the reduced-order solutions ensure the long-term stability of the solutions while exhibiting several orders of magnitude computational speedup over the FOM. 
Energy preserving reduced-order modelling of thermal shallow water   equation