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Snowflake Size |
From an empirical study of the distribution of raindrop sizes J.S. Marshall and W.M. Palmer concluded that such distributions could be adequately represented as negative exponential functions. They gave emprical formulas for the parameter of the distributions as a function of the rainfall intensity. Often when Marshall and Palmer's results are presented the parameter is considered to be determined by the rainfall intensity. The causality however runs the other direction. The intensity of rainfall is a function of the negative exponential parameter and reasonable estimates of the relationship can be derived a priori that approximate the empirical relationship found by Marshall and Palmer.
Marshall and Palmer worked with a frequency distribution n(r) whereas for presentation of the material here it is more convenient to work with the probability distribution f(r). For the negative exponential distribution
The negative exponential distribution has some very convenient properties; i.e., the average or expected values of the powers of r are easily determined:
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
Thus,
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 L^{2} so
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 r^{2}. Thus for this flow regime the average velocity is a function of the average r^{2}. Thus
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:
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 r^{2}. For this flow regime the relationship between R and λ would be
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
The formula found by Gunn and Marshall for the parameter of the negative exponential λ and the intensity of snowfall R was
Snowflakes are planar and the volume is a function of r^{2}, where r would now stand for a diameter measure of the snowflake. Since the expected value of r^{2} is 2!/λ^{2} the relation between R and λ would be
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
Taking into account this λ dependence of v, the relationship of R would be:
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
The negative exponential provides a better fit to empirical distributions if it is modified to take into account a minimum particle size; i.e.,
Then the formula for the moments that applies is
The values of E{r^{n}} can be found by noting that r = r_{0} + (r-r_{0}) so
where the ^{n}C_{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:
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