We know that the lower atmosphere has high pressure and as we go up, the pressure decreases, if it's so then why doesn't all gases fly up into the upper atmosphere from the lower following Bernoulli's theorem? I do expect that gravitational effect on gases isn't that worth notable. Do correct me for my mistake if it exists!

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Why don't you try to compare the gravitational and pressure forces for a layer of atmosphere close to Earth. You get the gravitation force from mass of such a layer(assume a thickness, say 1 m) and get force due to pressure difference by pressure difference*Area of layer. By comparing the two answers you'll get your answer. :) – Cheeku Mar 4 '13 at 12:02
That the gravitational effect is unnotable is your mistake. It is exactly gravity which counteracts the gradient in pressure. – Fabian Mar 4 '13 at 12:03
your thinking is right but the gas in the lower surface do not heat up so much that they expand and their density fall too much and they start rising up. – Dimensionless Mar 4 '13 at 12:40
Akash, were you actually trying to convince me abt bernoulli's theorem wid ur answer? – idiosincrasia23 Mar 4 '13 at 14:06

Exactly the same is true of the atmosphere. The pressure at ground level is 101,325 Pa because each square metre of the ground has about 10,329 kg of air above it (10329 kg times the acceleration due to gravity 9.81 m/sec$^2$ = 101325 Pa). If you could magically remove the 100 km or so of atmosphere that's above some patch of air at ground level that air would indeed immediately expand upwards.