Step 1: Understanding the Concept:
The Dry Adiabatic Lapse Rate (DALR) refers to the rate at which the temperature of a dry unsaturated parcel of air decreases as it rises through the atmosphere under adiabatic conditions.
Adiabatic means no heat is exchanged between the air parcel and its surroundings.
Step 2: Key Formula or Approach:
The DALR ($\Gamma_d$) is derived from the first law of thermodynamics and the hydrostatic equation:
\[ \Gamma_d = \frac{g}{c_p} \]
Where:
\( g \) is the acceleration due to gravity ($\approx 9.81$ m/s$^{2}$).
\( c_p \) is the specific heat of dry air at constant pressure ($\approx 1004$ J/kg$\cdot$K).
Step 3: Detailed Explanation:
By substituting the values:
\[ \Gamma_d \approx \frac{9.81}{1004} \approx 0.0098 \text{ K/m} \]
Converting this to kilometers:
\[ 0.0098 \times 1000 = 9.8 \text{ K/km or } 9.8^{\circ}\text{C/km} \]
This constant represents the cooling of rising dry air solely due to expansion as atmospheric pressure decreases with height.
Step 4: Final Answer:
The standard value for DALR is 9.8$^{\circ}$C/km.