P3 Liquid Relaxation Timescale

This document describes the P3 routine calc_liq_relaxation_timescale. Despite the routine name, the outputs epsr and epsc are relaxation-rate coefficients used in the liquid adjustment calculation. This document also describes the physical-property tests that verify the implementation.

Purpose

The routine computes two liquid-phase relaxation-rate coefficients used by P3:

epsr, the rain relaxation-rate coefficient
epsc, the cloud-liquid relaxation-rate coefficient

The rain expression combines an analytic gamma-distribution term with a ventilation-enhanced lookup-table term. The cloud-liquid expression is a direct closed-form product.

Physical Formulation

For active rain lanes, defined by

\[ q_{r,\mathrm{incld}} \ge Q_{\mathrm{small}} \quad\text{and}\quad \mathrm{context}=\mathrm{true}, \]

define

\[ T(\mu_r,\lambda_r) = \mathrm{apply\_table}(\mathrm{revap\_table\_vals}, \mathrm{lookup}(\mu_r,\lambda_r)), \]

\[ A = \frac{\Gamma(\mu_r+2)}{\lambda_r}, \qquad B = \left(\frac{\rho}{\mu}\right)^{1/2} Sc^{1/3} T(\mu_r,\lambda_r). \]

Then the rain relaxation-rate coefficient is

\[ \epsilon_r = 2\pi C_r \rho D_v \left[ f_{1r} A + f_{2r} B \right]. \]

For active cloud-liquid lanes, defined by

\[ q_{c,\mathrm{incld}} \ge Q_{\mathrm{small}} \quad\text{and}\quad \mathrm{context}=\mathrm{true}, \]

the cloud-liquid relaxation-rate coefficient is

\[ \epsilon_c = 2\pi \rho D_v C_c. \]

For cloud-small lanes inside context,

\[ q_{c,\mathrm{incld}} < Q_{\mathrm{small}}, \]

the implementation sets

\[ \epsilon_c = 0. \]

For rain-small lanes, and for lanes outside context, the implementation returns epsr = 0 because it initializes the entire epsr pack to zero before overwriting active rain lanes.

Variable Definitions

\(\rho\): air density
\(D_v\): vapor diffusivity, dv
\(\mu\): dynamic viscosity used in the ventilation term
\(Sc\): Schmidt number, sc
\(\mu_r\): rain size-distribution shape parameter
\(\lambda_r\): rain size-distribution slope parameter, lamr
\(C_r\): rain prefactor, cdistr
\(C_c\): cloud-liquid prefactor, cdist
\(f_{1r}\): analytic rain coefficient
\(f_{2r}\): ventilation rain coefficient
\(Q_{\mathrm{small}}\): hydrometeor activity threshold used separately for qr_incld and qc_incld
\(T(\mu_r,\lambda_r)\): rain evaporation lookup-table value

Implementation Details

Source: components/eamxx/src/physics/p3/impl/p3_calc_liq_relaxation_timescale_impl.hpp
The threshold comparison is inclusive:
qr_incld >= QSMALL
qc_incld >= QSMALL
epsr is initialized globally before the context-filtered rain overwrite:

epsr = 0;

This means out-of-context lanes are guaranteed to return epsr = 0.

epsc is only modified through .set(... && context, ...) operations. Therefore, when context == false, the routine does not guarantee that epsc is overwritten.

Lookup-Table Role

The lookup-table contribution enters only the rain expression through \(T(\mu_r,\lambda_r)\). This term depends on both mu_r and lamr, so tests that vary those inputs in the full rain expression can mix algebraic scaling with interpolation effects.

To avoid false positives, the property tests isolate exact algebraic identities by holding mu_r and lamr fixed whenever the lookup-table value must remain unchanged.

Mathematical Properties

For active rain lanes, the implementation can be decomposed into an analytic gamma-distribution term and a ventilation-enhanced table term:

\[ \epsilon_r = 2\pi C_r \rho D_v \left[ f_{1r}\frac{\Gamma(\mu_r+2)}{\lambda_r} + f_{2r}\left(\frac{\rho}{\mu}\right)^{1/2} Sc^{1/3} T(\mu_r,\lambda_r) \right]. \]

This decomposition implies exact additivity between the analytic and ventilation terms. It also implies exact linear scaling with \(D_v\) and \(C_r\). When the ventilation term is disabled, the expression is linear in \(\rho\), \(f_{1r}\), and \(\lambda_r^{-1}\). When the analytic term is disabled and the lookup-table term is held fixed, the expression scales as \(\rho^{3/2}\), \(\mu^{-1/2}\), and \(Sc^{1/3}\).

For active cloud-liquid lanes,

\[ \epsilon_c = 2\pi \rho D_v C_c, \]

so the cloud-liquid coefficient is exactly linear in \(\rho\), \(D_v\), and \(C_c\).

The mixed rain expression also yields a robust density bound when both terms are active and non-negative:

\[ 2\epsilon_r(\rho) \le \epsilon_r(2\rho) \le 2\sqrt{2}\,\epsilon_r(\rho). \]

Connection to Bulk Microphysics

The cloud-liquid expression has the form of a diffusion-limited vapor exchange coefficient. It is linear in vapor diffusivity, air density, and the cloud geometric distribution factor. The rain expression adds a ventilation-enhanced term. The factor \((\rho/\mu)^{1/2}Sc^{1/3}\) represents the expected Reynolds/Schmidt-number dependence of ventilation corrections in bulk microphysics.

This structure implies that increasing diffusivity, air density, or the distribution prefactors increases the relaxation-rate coefficients. Increasing viscosity damps the ventilation contribution, while increasing Schmidt number enhances it. The tests encode these expectations as exact scaling identities when the lookup-table term is held fixed.

Property Tests

The unit tests live in components/eamxx/src/physics/p3/tests/p3_calc_liq_relaxation_timescale_unit_tests.cpp.

Test Organization

The Catch2 test case p3_calc_liq_relaxation_timescale has two top-level sections:

properties
bfb

Inside properties, the nested sections cover:

context and threshold masking
cloud-liquid identity and exact scaling
rain additivity between the analytic and ventilation terms
full rain prefactor scaling with dv and cdistr
analytic-only rain identities
ventilation-only rain power-law identities
mixed microphysical scaling with rho, mu, and sc
mixed-term density bound
separation of cloud-liquid and rain-only factors
zero-coefficient and coefficient-locality behavior
non-negativity and finite-output checks

Tolerance Philosophy

Test Category	Tolerance Type	Value	Rationale
Exact identities and scaling	Identity	`100 * epsilon`	Tight algebraic checks with floating-point slack
Bounds and non-negativity	Inequality	`100 * epsilon`-scaled	Stable comparisons for one-sided checks
Expected-zero branches	Absolute	`1e-30`	Numerical zero floor for thresholded and zero-coefficient cases

Here, epsilon is machine epsilon for EAMxx Real (std::numeric_limits<Real>::epsilon()).

Context-Mask Coverage

The suite explicitly documents the asymmetric output behavior:

epsr is zero for out-of-context lanes because the routine zeroes the full pack before applying the rain mask.
epsc is not guaranteed to be written outside context, so the tests initialize it with a sentinel value and verify that the sentinel is preserved.

Table-Sensitive Identities

The suite avoids asserting lookup-sensitive identities that would vary with the interpolated table value. In particular:

all non-coefficient inputs are held fixed for the rain-additivity check
mu_r and lamr are held fixed for ventilation-only scaling checks
the exact lamr^{-1} scaling is tested only in the analytic-only regime

BFB Coverage

The BFB regression path is preserved to ensure the property tests do not change production behavior. It uses representative randomized inputs spanning active and inactive threshold states and compares the serialized outputs against the established baseline format.