Two-dimensional turbulence over topography of varying roughness

We numerically study weakly decaying two-dimensional turbulence over topography of varying roughness by comparing long-term flow states from simulations to the minimum enstrophy state (MES) proposed by Bretherton & Haidvogel ( J. Fluid Mech. , vol. 78, 1976, issue 1, pp. 129–154). The presence of isolated vortices in the numerical simulations leads to significant differences from the theory. These vortices are either roaming or topographically locked, with this distinction depending on the topographic roughness, the initial length scale and the initial non-dimensional energy ( $E/E_{\#}$ ), where $E_{\#}$ denotes the critical energy proposed by Siegelman & Young ( Proc. Natl. Acad. Sci. USA , vol. 120, 2023, issue 44, p. e2308018120). For low-energy and weakly rough topography, scatter plots show that the numerical results deviate from the MES due to the presence of vortices. As topographic roughness increases, the number, size and mobility of vortices decrease, leading to closer agreement between the numerical results and the MES. High energy and weakly rough topography lead to roaming vortices on a field of homogenised background potential vorticity. However, we observe topographically trapped eddies on highly rough topographies, even when initial length scales are much smaller than the domain size. The degree to which numerical results deviate from MES scaling is due to the presence of vortices, which, in turn, depend on topographic roughness and initial conditions in complex ways, suggesting a rich range of ways in which turbulence organises the long-term flow.