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If you have a well insulated vertically aligned pipe open at the top and bottom to ambient air, and then continuously cool the air at the top entry cross-sectional diameter to create a free-falling cold air parcel and flow through the tube, does the air actually compress itself as it gets denser lower in the tube, and if so, is that due as well to any effect of gravity over its transit through the tube? (given a high enough tube length)

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