How cold would the Earth have to be for the oceans to freeze entirely?
I imagine it is difficult for the ocean to freeze solid due to the low freezing point of salt water and the insulating effect of the formation of sea ice over the cool but still liquid ocean water.
Certainly for the ocean to freeze entirely, I think first the entire ocean would have to cool to near freezing even in the tropics. I also think that even supposing the Earth's atmosphere were at a uniform temperature below freezing (say -20 C worldwide), the ocean may still not freeze since it receives approx. 0.1 W/m2 in heat flux from the Earth's interior.
Assuming a simple model with an atmosphere of uniform global temperature, and assuming that the layer of sea ice accomodates all of the temperature difference between the surface/air and the still unfrozen water which is I assumed to be isothermal at near freezing, with simplifying assumptions about the properties of ice as an insulator and heat flux from the mantle,
I related the heat flux from the cold surface/air layer to the heat flux from the mantle, assuming the isothermal ocean had reached a state of constant temperature
h*(T - to ) = Qg
where h is a thermal property of the ice (k/d where d is the ice thickness)
T is the constant water temperature of near 275 K
t0 is the external temperature
And Qg is the heat flux from the mantle
From this, I calculated that in this very simple model the required temperature of the air to create a layer of sea ice of thickness d can be found with this expression:
t0 = T - (Qg*d)/k
Using a thickness of 4000 m, which would be a layer of sea ice that fills most of the ocean, I guess that the surface temperature of the Earth would have to be around 120 K for the oceans to freeze solid.
I don't think that I am right- obviously much is neglected in this model and some of my assumptions may not be warranted. Clearly, the Earth would have to be much colder than any time in its history for the oceans to freeze at great depth, but my question is would it have to be even colder than I have guessed, or could the oceans freeze completely or almost completely at warmer global temperatures than 120 K? 912