Direct numerical simulations are performed to study the dynamics of an inhomogeneous stratified shear flow that models an atmospheric jet centred at the tropopause across which the density stratification is non-uniform. Small to moderate background stratifications are selected, and simulations are conducted for a range of Reynolds and Froude numbers. A spectral domain decomposition method that is particularly suitable for simulations of non-uniformly stratified shear flows is developed to simulate the desired turbulent jet, and quasi-equilibrium flow fields are obtained by long-time integration of governing equations. The structures of the mean flow and turbulence fields are calculated, which are interpreted using relevant length scales (Ozmidov, buoyancy, shear, Ellison) and Richardson number profiles. The ratios of the Ellison to buoyancy scales are much smaller than unity at the jet core and approach unity at the edges, confirming that mechanical turbulence prevails in the jet core, while nonlinear waves and stratification effects are dominating at the jet edges. The jet core is found to support sustained mechanical (active) turbulence, outside which lay a region of patchy turbulence and nonlinear gravity wave activity characterized by spatially decaying velocity fluctuations and strong temperature fluctuations. Detailed energy budgets show how energy is partitioned within the flow, including the transport of energy from the jet to its immediate vicinity by nonlinear gravity waves.