The turbulent eddy fluxes in three-layer retrograde quasi-geostrophic (QG) flows are investigated theoretically and numerically, with a focus on the joint impacts from an interior potential vorticity (PV) gradient and a bottom slope with isobaths aligned with the thermal shear. While a bottom slope overall suppresses geostrophic turbulence, this suppression is selective in the baroclinic modes. In a surface-intensified thermal shear, a bottom slope depletes (replenishes) turbulent energy in the first (second) baroclinic mode, alleviating its suppression effect on the second baroclinic mode. By contrast, in a bottom-intensified thermal shear, a bottom slope shunts turbulent energy from both baroclinic modes into the barotropic mode; yet by occupying a broader spectral range, the second baroclinic mode remains less suppressed. The resulting eddy buoyancy fluxes across a sloping bottom become more intensified towards the surface (bottom) in a surface-intensified (bottom-intensified) thermal shear than over a flat bottom. As the bottom steepens, up-gradient eddy buoyancy fluxes emerge on the near-bottom isopycnal interface in a surface-intensified thermal shear, consistent with the tendency of geostrophic turbulence to homogenise the bottom-layer PV imprinted by strong topography. This tendency is ultimately overwhelmed by strong bottom drag, inducing topographically insensitive down-gradient eddy buoyancy fluxes. Our theory and model solutions provide guidance to adapt an eddy closure of weakly-dissipated two-layer QG turbulence. The adapted closure theory adequately quantifies the strength and vertical structure of eddy fluxes in our three-layer retrograde QG flow simulations. This work offers insights into parameterising depth-dependent mesoscale eddy fluxes across sloping seafloor in climate models.

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