Merge pull request #438 from CliMA/ro/p3_aspect_docs

P3 aspect ratio derivation

docs/src/P3Scheme.md

@@ -181,6 +181,8 @@ V(D) = \phi^{\kappa} \sum_{i=1}^{j} \; a_i D^{b_i} e^{-c_i \; D}
 where ``\phi = (16 \rho_{ice}^2 A(D)^3) / (9 \pi m(D)^2)`` is the aspect ratio,
 and ``\kappa``, ``a_i``, ``b_i`` and ``c_i`` are the free parameters.
 
+The aspect ratio of a spheroid is defined ``\phi = \frac{a}{c}``, where ``a`` is the equatorial radius and ``c`` is the distance from the pole to the center. In terms of ``a`` and ``c``, a spheroid's volume can be represented as ``V = \frac{4}{3} \pi a^2 c``, and its cross-sectional area can be assumed ``A(a, c) = \pi a c``. We use ``m(D)`` and ``A(D)`` from P3, so by substituting ``m(a, c) = \rho_{ice} * V(a, c)``, ``A(a, c)`` for ``m(D)``, ``A(D)`` into the formulation of aspect ratio above, we demonstrate agreement with the definition ``\phi = \frac{a}{c}``.
+
 Note that ``\phi = 1`` corresponds to spherical particles
   (small spherical ice (``D < D_{th}``) and graupel (``D_{gr} < D < D_{cr}``)).
 The plot provided below helps to visualize the transitions between spherical and nonspherical regimes.