Determining the depth of atmospheric winds in the outer planets of the Solar System is a key topic in planetary science. We provide constraints on these depths in Uranus and Neptune via the total induced Ohmic dissipation, due to the interaction of the zonal flows and the planetary magnetic fields. An upper bound can be placed on the induced dissipation via energy and entropy flux throughout the interior. The induced Ohmic dissipation is directly linked to the electrical conductivity profile of the materials involved in the flow. We present a method for calculating electrical conductivity profiles of ionically conducting hydrogen-helium-water mixtures under planetary conditions, using results from ab initio simulations. We apply this prescription on several ice giant interior structure models available in the literature, where all the heavy elements are represented by water. According to the energy (entropy) flux budget, the maximum penetration depth for Uranus lies above $0.93R_{mathrm{scriptscriptstyle{U}}}$ ($0.90R_{mathrm{scriptscriptstyle{U}}}$) and for Neptune above $0.95R_{mathrm{scriptscriptstyle{N}}}$ ($0.92R_{mathrm{scriptscriptstyle{N}}}$). These results for the penetration depths are upper bounds, and are consistent with previous estimates based on the contribution of the zonal winds to the gravity field. As expected, interior structure models with higher water abundance in the outer regions have also a higher electrical conductivity and therefore reach the Ohmic limit at shallower regions. Thus, our study shows that the likelihood of deep-seated winds on Uranus and Neptune drops significantly with the presence of water in the outer layers.
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