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Just to be clear, I am not asking you to solve this problem. This is a homework problem, and all I am asking for is if there was a mistake in the question. The basic setup of the question is that a parcel of air is displaced upwards to a certain height, then dropped. The question asks to prove that the height of the parcel is given by $(h-h_0)\cos(Nt)$. The question states that the Brunt–Väisälä frequency is given by $N=\sqrt \frac{g(\Gamma-\Gamma_A)}{T}$, where:

  • $g$ is acceleration due to gravity, $\Gamma$ is the lapse rate of a parcel of air
  • $\Gamma$ is lapse rate of the air parcel
  • $\Gamma_A$ is the lapse rate of the surrounding atmosphere
  • $T$ is the temperature of the surrounding atmosphere from which it is dropped
  • $h_0$ is the initial height of the air parcel (before it was lifted up)

I'm asking to verify that the formula they gave for the Brunt–Väisälä frequency frequency is correct — after all, if it were, would it not be 0 in this case? My thoughts were that the lapse rate of the air parcel and the surrounding atmosphere would be the same (because it is the same air).

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The lapse rate measures temperature variation with altitude but it does itself also vary with altitude and also with moisture content. If you displace some air with a certain moisture content vertically by some "magical" means, perhaps for the purposes of constructing a question (which isn't always easy), it's lapse rate will be that from the original altitude, and it's moisture content is unchanged.

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  • $\begingroup$ The question also explicitly states that dry adiabatic lapse rate $\Gamma$ and it has ambient air lapse rate $\Gamma$: I quote, "if the dry adiabatic lapse rate is Γ and the ambient air lapse rate is Γ...". $\endgroup$
    – Andi Gu
    Dec 27, 2016 at 1:47
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    $\begingroup$ It must be the second differential, then $d^2T/dZ^2$ is not zero. That's my best guess. Look at these T vs height plots, they are curved slightly. en.wikipedia.org/wiki/Lapse_rate#/media/File:Emagram.GIF Anyway the question doesn't really ask you to explain why $\Gamma_A$ and $\Gamma$ are different, but it does say they are. (given that it has defined two separate variables and if they were equal N would be zero.) $\endgroup$
    – JMLCarter
    Dec 27, 2016 at 1:53
  • $\begingroup$ Unless it's a trick question, in which case start your answer by stating the assumption that they are not equal otherwise N would be zero. (aka answer both cases) $\endgroup$
    – JMLCarter
    Dec 27, 2016 at 1:58
  • $\begingroup$ Be sure to add what you just told me (above) $\endgroup$
    – JMLCarter
    Dec 27, 2016 at 2:07
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Why do you think actual lapse rate of atmosphere must always be equal to dry adiabatic lapse rate? Dry adiabatic lapse rate is an idealized lapse rate that would be followed by air parcels that undergo isentropic expansion as they ascend in atmosphere. Actual processes may not conform to these idealizations. You are right in saying that if $\Gamma=\Gamma_A$, then $N=0$. This means that air parcel would not oscillate, because it would still be in equilibrium when displaced from its original position, while oscillation requires some restoring force towards origin. Putting $N=0$ in the proposed solution says just as much. So there is nothing wrong with the given expression for Brunt-Vaisala frequency.

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