An integral balance is developed for steady fluid flows relating dissipation in volumes bounded by isosurfaces of a tracer (quasi-conserved quantity) and solid boundaries to the covariance of the tracer value and surface fluxes across the boundaries. The balance is used to estimate upper bounds for vertical eddy diffusion coefficients for temperature and salinity in various volumes of the ocean. The vertical temperature diffusivity is calculated to be small, O(0.1 × 10−4 m2 s−1), except for the warmest and coldest volumes of the ocean. The vertical salinity diffusivity for the volume that makes up most of the deep ocean is estimated to be O(1 × 10−4 m2 s−1). Sources of error in these calculations are discussed, and the sensitivity to errors in the surface flux data is evaluated. The dissipation integral is also applied to demonstrate some related results concerning extrema and homogenization. The Prandtl–Batchelor theorem is a special case of one of these results. As a consequence of these results, if turbulent transfer is downgradient and there are no internal sources or sinks, a necessary (but not sufficient) condition for a climatological tracer distribution to be in a steady state is the absence of internal extrema. The climatological salinity distribution does not appear to violate this condition.