7 – Group 임지유, 김도형, 방주희

Consider a layer of the atmosphere in which ( Γ<Γ d) Unsaturated air T A < T B Ideal equation, P= ρ RT P A =P B, R = constant ρ ∝ 1/T ρ A > ρ B

T C > T D ρ C < ρ D Γ < Γ d Unsaturated air Positive static stability C D

T A > T B, ρ A < ρ B T C ρ D Γ > Γ d Unsaturated air Negative static instability Consider a layer of the atmosphere in which ( Γ>Γ d) C D

(ρ=p/RT, ρ’= P/ RT’) An unsaturated parcel of air has density ρ′ and temperature T ′, and the density and temperature of the ambient air are ρ and T. Derive an expression for the upward acceleration of the air parcel in term of T, T ′, g Net upward force F = ρ g - ρ′ g F = ( ρ - ρ′ )g

Derive an expression that describes the subsequent vertical displacement of the air parcel as a function of time in terms of T, the lapse rate of the ambient air ( Γ ), and the dry adiabatic lapse rate( Γ d). The temperature of the air parcel is T′= T 0 -( Γ d )z′ The ambient temperature is T′-T = -( Γ d – Γ )z′ T = T 0 -( Γ )z′ The decrement of air parcel temperature is

T ′ -T = -( Γ d – Γ )z ′ T′-T + N = 0 N : the Brunt- Vaisala frequency

This equation is the second term differential equation z'= A cos Nt + B sin Nt t = 0, namely that z' = z'(0) & dz'/dt = 0 at t = 0, it follows that z'(t) = z'(0) cos Nt It means the parcel executes a buoyancy oscillation about its equilibrium level z with amplitude equal to its initial displacement z'(0), and frequency N.

The Brunt-Vaisala frequency is thus a measure of the static stability. The higher the frequency, the greater the ambient stability.

A layer of unsaturated air flows over mountainous terrain in which the ridges are 10Km apart in the direction of the flow. The lapse rate is 5ºC/Km and the temperature is 20 ºC. For what value of the wind speed U will the period of the orographic forcing match the period of a buoyancy oscillation Solution: For the period τ of the orographic forciong to match the period of the buoyancy oscillation it is required that, τ L U N 2π2π [L : the spacing between the ridges ]

τ L U N 2π2π L U N 2π2π L U 2π2π In SI units, U 2π2π 293K 9.8m/s 2 ((9.8/m – 5.0/m)) m = 20m/s U

Show that if the potential temperature ɵ increasing altitude the atmosphere is stable with respect to the displacement of unsaturated air parcels. Letting and Dividing through by yield

Under certain conditions, the normal vertical temperature gradient is inverted such that the air is colder near the surface of the Earth. An inversion is also produced whenever radiation from the surface of the earth is less than the amount of radiation received from the sun, which commonly occurs at night, or during the winter when the angle of the sun is very low in the sky.

A parcel of unsaturated air displaced Upward from O will arrive at A with a temperature greater than that of its environment. Therefore, it will be less dense than the ambient air. if left to itself, will continue to rise. If the parcel is displaced downward it will be cooler than the ambient air, and it will continue to if left to itself.

Height Temperature ΓsΓs ΓdΓd Γ Γ < Γ s Stable Height Temperature ΓsΓs Γ Γ = Γ s Neutral Height Temperature ΓsΓs Γ Γ > Γ s Unstable Air containing the maximum amount of water vapor possible at a given temperature the partial pressure of the water vapor is equal to the vapor pressure of water at the same temperature.