THERMAL PHYSICS COMPUTER LAB #3 : Stability of Dry Air and Brunt-Vaisala Oscillations

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1 THERMAL PHYSICS COMPUTER LAB #3 : Sabiliy of Dry Air and Brun-Vaisala Oscillaions Consider a parcel of dry air of volume V, emperaure T and densiy ρ. I displace he same volume V of surrounding air of emperaure T' and densiy ρ'. The parce is pushed upwards by he buoyancy force and downwards by is weigh. We hen wrie Newon's second law: ma = F = F buoyancy - F weigh. Now m = ρv and F weigh = ρvg. According o Archimedes' principle he buoyancy force is he weigh of he displaced fluid: F buoyancy =ρ'vg. By subsiuing hese formulas ino Newon's second law we ge: a = g(ρ'/ρ - 1. By using he ideal gas equaion of sae, ρ = pm/rt, ρ' = pm/rt', we find: ρ'/ρ = T/T'. The equaliy of pressures inside and ouside he parcel is required by mechanical equilibrium. Hence he acceleraion of he air parcel is: a = g(t/t'-1. BUOYANCY FORCE DRY AIR PARCEL WEIGHT 1

2 Exercise 1: Consider a parcel of cold air T = 0 C surrounded by warmer air T = 10C. Compue he acceleraion of he parcel. a( T, T' := T T' T := 73 T' := 83 a( T, T' = 0.347m s Wha is he significance of he minus sign? Exercise : Repea Exercise 1 wih he following changes: parcel of warm air T = 5 C surrounded by cold air T' = 5 C. T := T' := a( T, T' = 0.706m s We now sudy he moion of he parcel by using Newon's second law and hermodynamic argumens for emperaure profiles. The emperaure profile for he parcel is T = Tnaugh - Γ(z - znaugh where Γ = Mg/c p. Γ is called he dry adiabaic lapse rae. This formula is derived by assuming he parcel o be an ideal gas ha undergoes an adiabaic process, i.e. no hea exchange beween he parcel and he environmen. The molar mass for air is 9g/mole and since air is diaomic c p = 7R/. M := g := 9.8 R := 8.3 c p := R M g Γ := Γ = This is he dry adiabaic lapse rae c p in K/m. Every 100m he parcel's emperaure is lowered by abou 1 C. Tnaugh := 73 znaugh := 100 The emperaure profile of he parcel: T( z := Tnaugh Γ ( z znaugh MIRON KAUFMAN, 009

3 We denoe γ he ambien lapse rae. Peixoo and Oor, pg 141, find ha he mean γ is 0.6K/100m, i.e. in average for a 1km climb he emperaure drops by 6 C. γ := The emperaure profile of he ambian air: T' ( z := Tnaugh γ ( z znaugh We solve numerically: a = d z/d = g(t(z/t'(z-1 by using he Odesolve funcion which is differenial equaion solver. To solve an iniial value or boundary value problem, we follow he following seps. 1. Wrie he keyword Given as mah, no ex.. Ener he differenial equaion. Remember: *use he Boolean = for wriing equaions; *use he d d and d d operaors for specifying derivaives; *use z(, no z, for he unknown funcion; *sae he iniial or boundary values. Use he prime noaion, i.e. z'(, for derivaive values. * Close he solve block wih an assignmen of he form: funcion name := Odesolve(independen variable, inerval endpoin, # of poins. A = 0 he parcel is a z = znaugh and has a velociy v = vnaugh. vnaugh := 1 Given d z( d = T( z( T' ( z( z( 0 = znaugh z' ( 0 = vnaugh MIRON KAUFMAN, 009 ( z := Odesolve, 000,

4 The velociy: v( := d d z( The acceleraion: a( := T( z( T' ( z( We graph he posiion z versus ime and noe ha he parcel undergoes so calle Brun-Vaisala oscillaions. The period of he oscillaions is: period := π ( Γ γ Tnaugh period = seconds. This is abou 9 minues. Challenge problem: Prove he Brun-Vaisala period formula. z( period period znaugh period period v(

5 period period a( * If Γ > γ, as in our simulaion, he parcel undergoes oscillaory moion. The dr air is sable agains he verical moion of a parcel. Smaller he difference Γ - γ, longer he period is. * If Γ < γ he Brun-Vaisala period is an imaginay number, meaning ha he moion is no oscillaory. The dry amosphere is unsable wih respec o verical moion of a parcel. The parcel does no reurn o is iniial locaion. Assignmen: Run he simulaion for γ = 0.03C/m. Do you sill observe oscillaions? For an inversion layer he ambien lapse rae γ is negaive, as he warmer laye is above he colder layer. Since Γ > 0 > γ he inversion air is sable agains he verical moion of a parcel. Compue he ambian lapse rae γ if he period is 3 minues. From he Brun-Vaisala formula we compue γ: period := π ( Γ γ Tnaugh period := 180 Γ 4 π Tnaugh period = 0.04 so γ = -4 C/km. g References: C. F. Bohren, B. A. Albrech: "Amospheric Thermodynamics" pg R. R. Rogers, M. K. Yau: "A Shor Course in Cloud Physics" pg 9-3 J. P. Peixoo, A. H. Oor: "Physics of Climae" pg D. Brun, Quaerly Journal of he Royal Meeorological Sociey, 53, pg 30-3, 197 5