2.4 Adiabatic Processes
Adiabatic processes involve no heat exchange with the environment. As air parcels rise and expand (or sink and compress), their temperature changes predictably.
Dry Adiabatic Lapse Rate
$$\Gamma_d = -\frac{dT}{dz} = \frac{g}{c_p} = 9.8 \text{ K/km}$$
Rate of temperature decrease for unsaturated rising air
Derived from the first law of thermodynamics with δQ = 0 and hydrostatic balance. Approximately 10°C per 1000m of ascent.
Derivation
$$\delta Q = 0 \;\Rightarrow\; c_p\,dT = \frac{1}{\rho}\,dp$$
Combining with the hydrostatic equation $dp = -\rho\,g\,dz$:
$$c_p\,dT = -g\,dz \quad\Longrightarrow\quad \Gamma_d = -\frac{dT}{dz} = \frac{g}{c_p} = \frac{9.81}{1005} \approx 9.8 \text{ K/km}$$
Moist Adiabatic Lapse Rate
$$\Gamma_m = \Gamma_d \frac{1 + \frac{L_v r_s}{R_d T}}{1 + \frac{L_v^2 r_s}{c_p R_v T^2}}$$
Varies from ~4-9 K/km depending on temperature and moisture
Warm, Moist Air
Γm ≈ 4-5 K/km
More latent heat release
Cold, Dry Air
Γm ≈ 8-9 K/km
Approaches dry adiabatic
Pseudoadiabatic Energy Balance
$$c_p\,dT + g\,dz + L_v\,dw_s = 0$$
In a saturated ascending parcel, the heat released by condensation ($-L_v\,dw_s > 0$) partially offsets the adiabatic cooling, yielding a smaller lapse rate than $\Gamma_d$. Condensate is assumed to fall out immediately (pseudoadiabatic assumption).
Potential Temperature
$$\theta = T \left(\frac{p_0}{p}\right)^{R_d/c_p} = T \left(\frac{p_0}{p}\right)^{0.286}$$
Temperature a parcel would have if brought adiabatically to 1000 hPa
Key Properties
- • Conserved during dry adiabatic processes
- • Constant on isentropic surfaces
- • ∂θ/∂z > 0 indicates stable stratification
- • Used to compare air masses at different altitudes
Thermodynamic Diagrams
Skew-T Log-P Coordinates
$$x = T - T_0 + k \ln\!\left(\frac{p_0}{p}\right), \qquad y = -\ln\!\left(\frac{p}{p_0}\right)$$
On a Skew-T diagram, isotherms tilt rightward with decreasing pressure (the "skew"), and the vertical axis is $-\ln p$ so that altitude increases upward. The skew factor $k$ is chosen so that dry adiabats and isotherms are nearly perpendicular, making it easier to read temperature and stability.
Tephigram Coordinates
$$x = T, \qquad y = \ln\theta = \ln T - \frac{R_d}{c_p}\ln p + \text{const}$$
On a tephigram the axes are temperature and entropy ($\propto \ln\theta$). Area on the diagram is proportional to energy, so CAPE and CIN can be measured directly as enclosed areas.
Normand's Rule
On any thermodynamic diagram, three lines from the surface observation converge at a single point — the LCL:
Dry adiabat
through surface $T$
Saturation mixing ratio line
through surface $T_d$
Saturated adiabat
through surface $T_w$
$$\theta_{\text{dry}}(T) = \theta_{\text{sat}}(T_w) \quad \text{at} \quad p = p_{\text{LCL}} \quad \text{where} \quad w = w_s(T_d, p)$$
This graphical technique provides $T_w$, $T_d$, and the LCL simultaneously from a single sounding.
Interactive Simulation: Dry & Moist Adiabatic Lapse Rates
PythonPlots temperature vs height for the DALR and MALR at various starting temperatures. Shows how the MALR varies with temperature and pressure, demonstrating that warmer, moister air produces smaller lapse rates.
Click Run to execute the Python code
Code will be executed with Python 3 on the server