# Marshall-Palmer Drop Size Distribution and Rainfall Intensity ## 1. Marshall-Palmer DSD Formula The **Marshall-Palmer** drop size distribution is given by: \[ N(D) = N_ \exp(-\Lambda D) \] where: - \( N(D) \): Number of drops per unit volume per unit size interval (usually \(\text{m}^{-3}\,\text{mm}^{-1}\)) - \( N_ \): Intercept parameter (\(800\, \text{m}^{-3} \text{mm}^{-1}\)) - \( D \): Drop diameter (mm) - \( \Lambda \): Slope parameter, related to rainfall intensity \( R \) ## 2. Relationship Between \( \Lambda \) and \( R \) \[ \Lambda = 4.1\, R^{-.21} \] where \( R \) is in mm/hr. ## 3. Step-by-Step Calculations ### Step 1: Calculate \( \Lambda \) for Various \( R \) Let's use a range of rainfall intensities \( R = 1, 5, 10, 20, 50 \) mm/hr. | \( R \) (mm/hr) | \( \Lambda = 4.1 R^{-.21} \) | |-----------------|------------------------------| | 1 | 4.10 | | 5 | 2.81 | | 10 | 2.39 | | 20 | 2.03 | | 50 | 1.61 | ### Step 2: Compute \( N(D) \) for \( D = .5,\,1,\,2 \) mm Use \( N_ = 800 \). \[ N(D) = 800 \cdot \exp(-\Lambda D) \] #### Example Calculation for \( R = 5\, \text{mm/hr} \), \( \Lambda = 2.81 \): - \( D = .5 \): \( N(.5) = 800 \exp(-2.81 \cdot .5) \approx 800 \exp(-1.405) \approx 800 \times .245 \approx 196 \) - \( D = 1 \): \( N(1) = 800 \exp(-2.81 \cdot 1) \approx 800 \exp(-2.81) \approx 800 \times .060 \approx 480 \) - \( D = 2 \): \( N(2) = 800 \exp(-2.81 \cdot 2) \approx 800 \exp(-5.62) \approx 800 \times .004 \approx 32 \) #### Calculated Table | \( R \) | \( \Lambda \) | \( N(.5) \) | \( N(1) \) | \( N(2) \) | |---------|---------------|--------------|------------|------------| | 1 | 4.10 | 1203 | 131 | 2.16 | | 5 | 2.81 | 196 | 480 | 32 | | 10 | 2.39 | 245 | 763 | 73 | | 20 | 2.03 | 298 | 1157 | 180 | | 50 | 1.61 | 378 | 1867 | 514 | ## 4. Graphical Representation ```plaintext Number of Drops N(D) vs Rainfall Intensity R (logarithmic y-axis) N(D) ^ | o D=.5mm | o | o | o | o | o | o | o | o | o +-------------------------------> R (mm/hr) (You would plot three curves, one each for D=.5, 1, 2 mm, showing N(D) increasing with R) ``` *To plot, use a logarithmic scale for \( N(D) \) due to the wide range.* ## 5. Discussion - **Trend**: As rainfall intensity \( R \) increases, \( \Lambda \) decreases, so \( N(D) \) increases for all drop sizes. - More intense rain means more drops of all sizes. - **Drop Size Effect**: - For small drops (\( D = .5 \) mm), \( N(D) \) is always highest. - For larger drops (\( D = 2 \) mm), \( N(D) \) is much smaller, but the increase with \( R \) is more pronounced. - **Physical Meaning**: - Light rain: Mostly small drops, very few large drops. - Heavy rain: Both small and large drops are more numerous. ## 6. Final Answer **The number of drops \( N(D) \) at any fixed drop size increases with rainfall intensity \( R \). The increase is more dramatic for larger drops. This is because the parameter \( \Lambda \) decreases with increasing \( R \), making the exponential drop-off less steep. Graphically, for each \( D \), the curve of \( N(D) \) vs \( R \) is upward-sloping, and the curves for larger \( D \) are lower but rise faster with \( R \).** --- **Sample Plot**  *Alt text: Line plot showing N(D) versus rainfall intensity R for three drop sizes according to Marshall-Palmer distribution; N(D) increases with R, with steeper increases for larger D.* --- **Key Result:** - \( N(D) \) increases with \( R \) for all \( D \), but more rapidly for larger drops. - The Marshall-Palmer distribution explains why heavier rain includes more large drops.