Mountains and Weather (Part II)

Recall the vertical momentum balance for an air parcel in the   atmosphere. (The vertical acceleration of an air parcel is due to the   pressure gradient force in competition with gravity.)

Dw/Dt = PGF_z +   F_g

Dw/Dt = -1/ρ(∂p/∂z) –g

w is vertical velocity, t   is time, ρ is density, z is height, and g is the acceleration due to gravity.

If the pressure gradient force balances gravity, then we have  hydrostatic balance.

∂p/∂z = -ρg

Consider an environment where hydrostatic balance holds. Let’s  say we perturb an air parcel resting comfortably at some height H moving east with speed µ. The motion of the parcel in the  vertical will be determined by the vertical momentum balance.

Dw_parcel/Dt   = -1/ρ_parcel(∂p_parcel/∂z) – g

Now we make an assumption. We will assume the motion of the parcel is quasi-static, i.e. the pressure of the parcel always matches that of the environment.

Dw_parcel/Dt  = -1/ρ_parcel(∂p_environment/∂z) –g

We had made a previous assumption that the environment is in   hydrostatic balance.

Dw_parcel/Dt   = -1/ρ_parcel(-ρ_environmentg) – g

Dw_parcel/Dt  =  (ρ_environment –ρ_parcel)g/ρ_parcel

Instead of describing vertical acceleration in terms of density, we prefer potential temperature θ.   (Consider potential temperature as a measure of energy.)

Dw_parcel/Dt  =   (θ_parcel– θ_environment)g/θ_environment

For example, if the parcel has more energy than its   surroundings, then the vertical acceleration will be positive. Therefore, the parcel will rise.

Dw_parcel/Dt =   (θ_parcel/θ_environment – 1)g

The energy of the parcel is always conserved. The energy of the   surroundings, however, changes with height. If ∂θ/∂z is the change in energy of the environment with respect to height, then the energy of the surroundings can be written in terms of θ_parcel and the change in height of the parcel δz. In other words, θ_environment = θ_parcel   + ∂θ_environment/∂z(δz).

Dw_parcel/Dt =   {θ_parcel/[θ_parcel + ∂θ_environment/∂z(δz)] –   1}g

Using the Taylor series expansion of 1/(1 – x), we can approximate θ_parcel/[θ_parcel + ∂θ_environment/∂z(δz)].

Dw_parcel/Dt   =-gδz/θ_parcel(∂θ_environment/∂z)

This is a homogenous, second-order linear differential equation  in δz with constant coefficients.   Recall δz is a function which  represents the height of the parcel as a function of time.

D²(δz)/Dt² +   g/θ_parcel(∂θ_environment/∂z)(δz) = 0

Since we have wave motion, it comes as no surprise that the solution to this differential equation should be a combination of sine and cosine. It   can be shown that the period of oscillation of this solution is the   following.

T = 2π[g/θ_parcel(∂θ_environment/∂z)]^-1/2

Why would any of this information be important? Well, if the initial disturbance to the air parcel came from a mountain, then the parcel would begin to oscillate vertically after it passes over it at speed μ. If the air parcel has water vapor in it, there is a chance that at some height during the parcel’s journey clouds will form. If so, the resulting pattern looks like this:

Given the distance between the clouds and mean horizontal speed μ of the air parcels, we can work backwards in our analysis to recover the equation of motion for the air parcel in the vertical and get a better understanding of the atmosphere’s stability.