Atmospheric Thermodynamics

Atmospheric Composition

78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, and small amounts of other trace gases

Atmospheric Structure

Troposphere (0-10 km), Stratosphere (10-50 km), Mesosphere (50-85 km), and Thermosphere ( 5- 100 km)

Dry air

P = eRdT

Rd: specific gas constant for dry air = 287 J kg-1 k -1

Moist Air

pv = RdTv

Rd: specific gas constant for dry air = 287 J kg-1 k -1

Water vapor

e = evRvT

e = water vapor

Rv = 461.5 J kg -1 K -1 (water vapor specific gas constant)

Work

First Law of Thermodynamics

heat added to the system (dq) → internal energy + kinetic energy → work

du = dq - dw  

Enthalpy

The total heat content of a system

h = p + alpha

Entropy

The ratio change in heat to the absolute temperature

ds = dq/T

Adiabatic Process

Heat is constant; dq = 0

dq = CpdT - αdp

Isobaric Process

Pressure is constant; dp = 0

dq = CpdT - αdp

Isothermal Process

Temperature is constant; dT = 0
dq = CpdT - αdp

Potential Temperature

The temperature an air parcel would have if it were expanded or compressed adiabatically from its existing pressure and temperature to a standard pressure

Potential Temperature Formula

Theta = T (p0/p1)^R/cp
R: gas constant for dry air = 287 J kg -1 k -1

Thermodynamic function: Helmholtz

f = u - Ts

df = du - Tds - sdT

u = internal energy; Ts = entropy

Thermodynamic potential at constant volume

Thermodynamic function: Gibbs

g = u - Ts + pα

dg< - sdT + αdp

u = internal energy; Ts = entropy

Thermodynamic potential at constant pressure

Specific humidity

qv = mv/mv + md

Saturation Mixing Ratio

The maximum amount of water vapor (in grams) that can be present in one kilogram of dry air at any given temperature

w_s = 622e_s/p-e_s

Virtual Temperature

Thee temperature that dry air would need to attain in order to have the same density as the moist air at the same pressure

Tv= T( 1 +0.608qv)

Dew and Frost Point

Isobaric Cooling

Clausius-Clapeyron Equation

es(T) = esTo(mb) x (e^L/Rv) x (1/To - 1/T)

L = 2.5 × 10^6 J K-1 s-1

Rv = 461.5 J kg-1 k-1

Water Droplet Growth

Bigger drops must grow at the expense of the smaller ones

Sounding and Stability of the Layers

Thermodynamic Diagram: Geopotential

d(phi) = gdz

Thermodynamic Diagram: Hydrostatic Balance

dp/dz = (rho)g

the two forces (gravity and PGF) oppose and cancel.

Hypsometric Equation

Δz=Rd​Tv/g ​​​ln (p2/​p1)​​

US Standard Atmosphere

1013.25 mb

Dry Adiabatic Lapse Rate

9.8 deg C/ km

dT/dz = -g/cp = - gamma

Temperature decreases with increasing height

Moist Adiabatic Lapse Rate

The rate at which the temperature of a parcel of saturated air decreases as the parcel is lifted in the atmosphere.

Parcel Method

Mixing between the parcel and its environment is negligible 

The displacement of the parcel does not disturb the local environment

The process is adiabatic (heat is constant)

The pressures of the parcel and environment are identical

Vertical Stability

(dw/dt) = g(Tv - Tvo), where Tv is the virtual temperature

Stability Classification

delta < gamma (stable)

delta = gamma (neutral)

delta > gamma unable

Lifting Condensation Level (LCL)

The point of saturation resulting from an adiabatic expansion

Level of Free Convection (LFC)

That point in which a parcel, having originated from the surface and passed through its LCL, becomes positively buoyant

Convective Available Potential Energy (CAPE)

   g\int_{}^{}\!\frac{\theta^{\prime}\left(z\right)-\theta o\left(z\right)}{\theta o\left(z\right)}\,dz, (J kg^-1)