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A non-vanishing entropy production rate is a hallmark of non-equilibriumstationary states and is therefore at the heart of non-equilibriumthermodynamics. It is a manifestation of a steady circulation $J_{\rm inv}$along the level sets of the invariant density $\rho_{\rm inv}$, and is thusgenerically used to quantify how far a steady system is driven out ofequilibrium. While it is well known that there exists a continuum of distinctsteady states with the same invariant measure, the question how the geometryand topology of the invariant current a priori affect dissipation remainedelusive. For confined irreversible diffusions we identify two minimaldescriptors, the $\rho_{\rm inv}$-weighted vorticity and the variation of$J_{\rm inv}$ along level sets of $\rho_{\rm inv}$, and prove that thesejointly bound from above the steady-state entropy production rate. In regionswhere $\rho_{\rm inv}$ is close to Gaussian the bound is dominated solely bythe vorticity of the drift field and in the low-noise (Freidlin-Wentzel) limitby any non-potential contribution to the drift, rendering $J_{\rm inv}$virtually constant along the level sets of $\rho_{\rm inv}$.

Vorticity and level-set variations of invariant current bound steady-state dissipation