The negative exponential distribution has some very convenient properties; i.e., the average or expected values of the powers of r are easily determined:

E{rⁿ} = ∫^∞₀rⁿf(r)dr = n!/λⁿ

This property makes computation of various averages very easy.

The rainfall intensity (mm/hr) is equal to the product of average droplet volume and the intensity of droplet downfall I (number of droplets per unit area per hour). The average or expected value of droplet volume for a negative exponential distribution is simply

E{(4/3)πr³} = (4/3)π(3!)/λ³

Thus,

R = 8πI/λ³
and hence if I
is independent of λ
λ = CR^-1/3
where C = (8πI)^1/3

This is reasonably close to the empirical formula of λ = CR^-0.21 reported by Marshall and Palmer.

But is the droplet downfall intensity I really independent of λ? Let L be the average distance between raindroplets, both vertically and horizontally. The number of droplets hitting a plane per unit time is the ratio of the average vertical velocity and the average distance between them, v/L. Each droplet would be associated with an area of L² so

I = (v/L)/L² = v/L³

The terminal velocity of a droplet for small Reynold's number is a function of its cross section area of the droplet and hence depends upon r². Thus for this flow regime the average velocity is a function of the average r². Thus

E{v} = kE{r²} = k2!/λ²

where k depends up the density of water, the gravitational acceleration and the dynamic viscosity of air [(2/9)gρ_L/μ].

When the dependence of I on λ is taken into account the relationship between rainfall intensity R and λ is:

R = 16πk/(L³λ⁵)
and hence if L
is independent of λ
λ = CR^-1/5

This is almost precisely the formula given by Marshall and Palmer.

For high Reynolds number the flow regime results in the terminal velocity depending on the square root of r rather than r². For this flow regime the relationship between R and λ would be

λ = C₃R^-2/7.

For an intermeditate range of Reynolds numbers the velocity depends upon the first power of r and hence the relationship between R and λ would be

λ = C₄R^-1/3.

The formula found by Gunn and Marshall for the parameter of the negative exponential λ and the intensity of snowfall R was

λ = CR^-0.48.

Snowflakes are planar and the volume is a function of r², where r would now stand for a diameter measure of the snowflake. Since the expected value of r² is 2!/λ² the relation between R and λ would be

R = CI/λ²
and hence if I
were independent of λ
λ = C'λ^-1/2

For snowflakes Nakaya and Terada found the fall velocity v to be indepent of size. Based upon their result the about relationship should apply. Others assert that the dependence of velocity on size to be of the form

v = cr^0.3

Taking into account this λ dependence of v, the relationship of R would be:

R = C/(λ²λ^0.3)
= Cλ^-2.3
and hence
λ = c'R^-0.435

This is nearly the same as the relationship found by Gunn and Marshall and it is notable that the exponent value for snowflakes relative to that for raindrops is in keeping with the two dimensional nature of snowflakes compared to the three dimensional nature of rain drops.

Graupel, in contrast to planar snowflakes, is three dimensional in its nature. The fall velocity of graupel particles was found by Nakaya and Terada to depend up size raised to the 0.6 power. Therefore the intensity R is inversely proportional to λ raised to the 3.6 power and hence for hailstorms

R = c/λ^3.6
and thus
λ = c'R^-0.278

The negative exponential provides a better fit to empirical distributions if it is modified to take into account a minimum particle size; i.e.,

f(r) = λexp[-λ(r-r₀)] for r ≥ r₀
and f(r) = 0 otherwise.

Then the formula for the moments that applies is

E{(r-r₀)ⁿ} = n!/λⁿ

The values of E{rⁿ} can be found by noting that r = r₀ + (r-r₀) so

E{rⁿ} = Σ(ⁿC_i)r₀^n-i(i!)/λⁱ

where the ⁿC_i's are the coefficents for the binomial expansion. Thus if R is a function of a power of r the solution for λ as a function of R is not simple but the solution does exist as the root of a polynomial in (1/λ).

Sources:

• R.R. Rodgers and M.K. Yau, A Short Course in Cloud Physics, Butterworth-Heinemann, Woburn, MA, 1989.
• J.S. Marshall and W.M. Palmer, "The distribution of raindrops with size," Journal of Meteorology, Vol. 5 (1948), pp. 165-166.
• K.L.S. Gunn and J.S. Marshall, "The distribution with size of aggregate snowflakes," Journal of Meteorology, Vol.15 (1958), pp. 452-461.
• N.H. Fletcher, The Physics of Rainclouds, Cambridge University Press, Cambridge, U.K., 1962.