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The ice would sink, and, particularly in the higher latitudes, the sinking ice would allow more ice to form in winter. Eventually, the northern and southern oceans would be solid from the surface to the bottom, as would northern and southern lakes, and slower moving rivers.

It depends on how the ice formed:

# Land ice. If you had a glacier sliding off into the ocean (like on Greenland), then the ice would be fresh water (no salt). Fresh water is lighter than the salt water in the ocean, so the glacial ice may continue to float. # Sea ice. If you are just talking about normal lakes and oceans freezing because of cold air, then convection currents in the water would prevent any ice from forming at the top. Which would mean you would not have an 'sinking ice'. No ice would form at all until the entire water body cooled down to 0°C. That would mean that none of the oceans would freeze at all. They would remain open water even during cold, long winters. Small lakes would freeze solid (and take much longer to thaw in the summer), but larger lakes like the Great Lakes on the US/Canada border would likely remain completely ice free. There would be some ice near the shores of both the large lakes and oceans (where it got shallow enough to freeze solid). Most the aquatic life in the smaller lakes would die. Large rivers would remain ice free, but smaller rivers would freeze solid. If the water source of the smaller rivers continued producing (like an under-ground spring), then you would end up with an icicle effect where the water keeps freezing over top other layers of ice and the river would become this enormous mound of ice resembling the Hawaiian volcanoes. It is possible that more ice could form in natural bodies of water but claims of freezing from bottom to top are wildly speculative and unjustified by analysis in my opinion. The following considerations from the disciplines of thermodynamics and heat transfer are the primary determining factors.

Sinking ice brings about two effects at the surface. (1) An increased temperature difference between the water and the atmosphere due to warmer water replacing the sinking ice. This effect would cause greater heat transfer from the water to the atmosphere. (2) A reduction of thermal conductivity across the layer formerly occupied by the ice since the thermal conductivity of water is lower than that of ice by almost a factor of four near freezing temperature. This effect would cause the heat flow from the water to the atmosphere to decrease. If the former effect is predominant then there will be an increased heat flow from the surface to the atmosphere. If it is also assumed that any change in the heat flow from the earth into the body of water as a result of the ice falling to the bottom is small compared to the heat transfer at the surface then in accordance with the First law of thermodynamics, there would be increased cooling of the body of water under the condition of ice sinking over the condition of reality, floating ice. Thus under these assumptions, more freezing of natural bodies would occur if ice did not float.

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