The Milbrandt-Yau (2005) PSD

The Milbrandt-Yau (2005) PSD used in the GEM model uses a modified gamma distribution of the form:

\[N(D_{max}) = N_0 D_{max}^\nu \exp(- \lambda D_{max} ^ \mu)\]

In this two-moment scheme, the number density \(\rho_n\) and mass density \(\rho_m\) are used as predictive moments. They can be related to the \(N_0\) and \(\lambda\) parameters of the PSD using

\[ \begin{align}\begin{aligned}\rho_n = \frac{N_0}{\mu} \lambda^{\frac{\nu}{\mu}} \Gamma (\frac{\nu + 1}{\mu})\\\rho_m = a\frac{N_0}{\mu} \lambda^{\frac{\nu + b}{\mu}} \Gamma ( \frac{\nu + 1 + b}{\mu} ),\end{aligned}\end{align} \]

where \(a, b\) are the parameters of the mass-size relationship

\[M(D) = a \cdot D_{max}^b.\]

class parts.scattering.psd.my05.MY05(nu, mu, a, b, hydrometeor_type=None, number_density=None, mass_density=None)

Bases: parts.scattering.psd.arts.arts_psd.ArtsPSD

The MY05 class describes the size distributions of particles in an atmosphere using the number density \(\rho_n\) and mass density \(\rho_m\) as predictive moments.

The \(\nu\) and \(\mu\) parameters take on fixed values depending on the hydrometeor type.

evaluate(x)

Compute a numeric representation of the PSD data.

Parameters:

x(numpy.ndarray): Array containing the values of the size parameter at which to evaluate the PSD.

Returns:

PSDData object containing the numeric PSD data corresponding to this PSD.

classmethod from_psd_data(psd, nu, mu, a, b)

Create a MY05 PSD from given psd data.

Parameters:

psd(PSDData or other PSD): PSD data from which to create the MY05 representation.

nu(float): \(\nu\) parameter of MY05 PSD

mu(float): \(\mu\) parameter of MY05 PSD

a(float): \(a\) coefficient of the mass-size relationship

b(float): \(b\) coefficient of the mass-size relationship

get_mass_density()

Returns:

The numpy.ndarray containing the mass density data of the PSD.

get_moment(p, reference_size_parameter=None)

Analytically computes the \(p\) th moment :maht:`M(p)` of the PSD using

\[M(p) = \frac{N_0}{\mu} \lambda ^{-\frac{\nu + p + 1}{\mu}} \Gamma ( \frac{\nu + p + 1}{\mu})\]

Parameters:

p(float): Which moment of the distribution to compute.

Returns:

numpy.ndarray containing the \(p\) th moment for all volume elements described by the PSD.

reference_size_parameter(:class: SizeParameter): Size parameter with respect to which the moment should be computed.

moment_names

The names of the predictive moments of the PSD.

pnd_call_agenda

The ARTS WSM implementing the MY05 PSD.