Does one body of water freeze over first if they have different depths?
May 6, 2017 7:37 AM   Subscribe

I saw this question recently on a Bennett mechanics exam and it has me stumped. If two bodies of water (lake or pond let's say) have equal surface area, but different depths, will one freeze over before the other? I know that the shallower pond will freeze completely before the deeper one, but the question is only asking about the surface freezing over.

My thought is this: assuming they are subject to the same environment (air temperature, ground temperature) that they would both freeze over at the same time/rate due to the equal surface area.

I've done lots of Googling and so far haven't found anything that specifically addresses this question.
posted by robadobdob to Science & Nature (11 answers total) 3 users marked this as a favorite
Water holds a lot of heat, and if you have more water without more surface area it will take longer to cool down and start freezing. (Water will circulate and keep the surface liquid until the whole lake is fairly close to 0º all the way through)

If you somehow start both bodies of water already at like 0º, then yeah, probably.
posted by aubilenon at 7:42 AM on May 6 [5 favorites]

You can compare Lakes Ontario and Erie. Lake Erie has a surface area of about 10,000 square miles. Its average depth is 62 feet. Lake Ontario has a surface area of about 7500 square miles but its average depth is 283 feet. The winter climate around Erie is slightly warmer than Ontario. But despite the warmer Erie climate, it is not uncommon during winter for the surface of Erie to freeze over completely while at the same time Ontario remains ice free.
posted by Seymour Zamboni at 7:52 AM on May 6 [6 favorites]

Yeah, deeper lake with same surface area will freeze after the shallower one.

To be careful, you could set up some differential equations and make a bunch of assumptions (lakes are right circular cylinders, no wind, all the water is perfectly still, etc.).

And then you'd see this result fall out of basic heat equation in three dimensions with those boundary conditions. But that's kind of a pain, and only worth it if you really want to understand the gritty details of the math and physics. The basic reason is it will talke more time to cool more water, because the water helps keep the other water warm. This is called heat capacity.
posted by SaltySalticid at 7:55 AM on May 6 [1 favorite]

Water has minimum volume at 4°C. So if you introduce the two bodies of water into a negative C environment when they are above 4°C then yes, the deeper one will have to churn for longer before reaching a uniform 4°C and being able to begin cooling down more. But if they are both at or below 4°C to begin with, I would have thought they should freeze over simultaneously (all other factors such as wind, fish, and so on disregarded) because they would simply stratify. Was there any more detail in the original question?
posted by labberdasher at 8:26 AM on May 6 [3 favorites]

Yeah, I don't know what a Bennett mechanics exam is and how much it considers the real world versus reasoning from first principles, but limnologically, this question is actually pretty complicated and certainly depends on, as labberdasher notes, the mixing/stratification of the lake, which is primarily determined by climate, although extreme variation in depth (very deep lakes may never mix, very shallow lakes may always be well mixed) or biological activity (causing buildup of CO2 in deep water and sudden, unpredictable turnover) can also play a role. Other factors that can affect the motion of the surface of the lake also affect icing over, including wind and biological activity. And of course in addition to air temperature, the amount of direct energy inputs from sunlight matters, so climate factors like day length and cloud cover, obviously, but also the light extinction coefficient--how far down the light penetrates--which is largely a function of local geology, human development, and biological activity. Finally, differences in water chemistry, especially salinity, can change the specific heat capacity of the water.

If all of those things were exactly the same between your two lakes, rather than just the surface area as seems to have been in your question, then all that would matter would be the total heat stored in the water, in which case the shallower lake would freeze more quickly.

I just gave my final exam in Limnology, unfortunately, or I would have totally put this on there.
posted by hydropsyche at 8:41 AM on May 6 [7 favorites]

(I thought a "mechanics" exam would be looking for basic thermodynamics and highly idealized physics, not anything to do with complicated real limnology. The question can be answered based simple on concepts (or relatively simple equations) with the given information if we assume this is spherical cow stuff, but otherwise it cannot be answered with the given info, for the reasons hydropsyche describes.)
posted by SaltySalticid at 8:55 AM on May 6

Just to pile on, it seems to me that since the earth at deeper depths pretty much stays the same temp (much above freezing), if the two bodies were exactly alike except for depth, the deeper one would have more water exposed to the surrounding earth.
posted by rudd135 at 9:11 AM on May 6

Assuming they are chemically similar and depth is the only variable of importance, the shallower lake freezes over first. If saltiness varies considerably, this might change the answer. High salt content changes the freezing point of water, which is why we salt roads in winter.
posted by Michele in California at 9:59 AM on May 6

Just to pile on, it seems to me that since the earth at deeper depths pretty much stays the same temp (much above freezing), if the two bodies were exactly alike except for depth, the deeper one would have more water exposed to the surrounding earth.

I think that would also be a factor, but ground temperature varies slightly with latitude. So they would have to be at the exact same latitude, too.
posted by hydropsyche at 10:01 AM on May 6

There are a ton of things that contribute to icing but once you get below freeing its mostly how much does the surface water move. So in practice, the fetch, prevailing winds, topography, inflows and outflows, any vertical or horizontal currents etc would be much more important than depth. Having said that we have a series of small ornamental ponds next to each other in the garden and in the winter they all start to form ice at the edges at the same time but the smallest one does indeed start to ice over in the middle first. Which is the sign to drain them all.
posted by fshgrl at 1:54 PM on May 6

I think the deep lake's surface would freeze faster too, but I believe the intent of the question was to point up that the two would freeze at much more nearly the same time than you would guess if you judged water by the behavior of other freezing liquids.

The difference is due to the fact that labberdasher mentions, that water reaches maximum density well above its freezing point (at 4º C) rather than densifying all the way down to freezing the way the vast, vast majority of other liquids do.

As soon as the air drops below 4º C, the water the air cools no longer sinks, it sits right there at the surface because it's suddenly less dense than the water immediately below it.

As the surface layer continues to cool with falling air temp, it sits there right at the surface in deep and shallow lake alike until it starts to freeze, and then it continues to sit there as it freezes.

In almost all other liquids, the frozen part would sink to the bottom, and you wouldn't see anything frozen on the surface until almost the entire body froze, and that would take much longer in a deeper body of liquid.
posted by jamjam at 4:04 PM on May 6 [2 favorites]

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