Need help understanding rain rate formulas from drop sizes

Question

I have come across different formulas by which the rain rate [mm/s] or [mm/h] can be computed from the drop size distribution. I'm somehow unable to convert the different formulas into each other. It is maybe a stupid question with some unit error, but I cannot find it.

A well-known formula seems: \begin{equation}R_1 = \frac{\pi}{6} \int_0^{\infty} v(D) D^3 N(D) dD \end{equation} This should be the instantaneous rain rate in [mm/s]?? Unfortunately most publications just quote the formula without units...

I'm supposing that $D$ is in [mm], $N(D)$ in [mm$^{-1}$ m$^{-3}$], and $v(D)$ in [m s$^{-1}$].

$v(D)$ is usually approximated as $3.778\cdot D^{0.67}$.

Another formula I have come across is \begin{equation} R_2 = 6\pi \cdot 10^{-4} \int_0^{\infty} v(D) D^3 N(D) dD, \end{equation} which is said to be [mm/h], then. Also for $D$ in [mm], $N(D)$ in [mm$^{-1}$ m$^{-3}$], and $v(D)$ in [m s$^{-1}$].

Now apparently $R_1 \cdot 0.0036 = R_2$. But why?

If $R_1$ is in [mm/s], then we should multiply by 3600 to get to [mm/h]. Can someone spot where I am wrong? Is it right that $R_1$ is in [mm/s]? (given the units of D, N(D) as above)

Answer 1

The first equation does not match up with the units. But first things first... What you're calculating when you calculate rain rate is an accumulation rate, which is volume flux through a plane (mass flux divided by density) divided by the area of that plane. For this reason, it doesn't matter whether it is computed over a square meter or a square kilometer -- since volume flux is proportional to the area, it cancels out

Also it doesn't really matter whether $N(D)$ has units of $mm^{-1} m^{-3}$ or $cm^{-1} m^{-3}$, since that first unit cancels out in integration.

If the units are as you say they are in the first example, then the expression should read $$R_1 = 10^{-6}\frac{\pi}{6} \int_0^\infty v(D)D^3N(D)dD$$ to get a result in $mm/s$.

Multiplying this by 3600 to get a result in $mm/hr$ yields the the second expression.