P3 Autoconversion

This document details the implementation and verification of the cloud water to rain autoconversion process in the P3 microphysics scheme.

Physical Formulation

The autoconversion parameterization follows the formulation of Khairoutdinov and Kogan (2000). It represents the collision and coalescence of cloud droplets to form rain drops.

Autoconversion Rate

The time tendency of rain mass mixing ratio due to autoconversion, $\left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}}$, is calculated as a power law function of the cloud water mixing ratio ($q_c$) and the cloud droplet number concentration ($N_c$).

\[ \left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}} = A \, q_c^{\alpha} \, (N_{c, \text{vol}})^{-\beta} \]

Where:

$A$ is the autoconversion prefactor ( autoconversion_prefactor ).
$\alpha$ is the cloud water exponent ( autoconversion_qc_exponent ).
$\beta$ is the cloud number exponent ( autoconversion_nc_exponent ).
Note: $\beta$ is stored as a positive value (1.79 in default configuration) following KK2000 convention. In the implementation this appears as pow(nc_incld * sp(1.e-6) * rho, -autoconversion_nc_exponent), which is the code equivalent of the analytic form $\mathrm{pow}(N_{c, \text{vol}}, -\beta)$.
$N_{c, \text{vol}}$ is the cloud droplet number concentration in units of $\text{cm}^{-3}$.

In the implementation, $N_{c, \text{vol}}$ is derived from the in-cloud number mixing ratio $N_c$ ($\#/\text{kg}$) and air density $\rho$ ($\text{kg}/\text{m}^3$), where $N_c$ corresponds to the variable nc_incld:

\[ N_{c,\text{vol}} = N_c \cdot \rho \cdot 10^{-6} \;\equiv\; \texttt{nc\_incld} \cdot \texttt{sp(1.e-6)} \cdot \rho \]

\equiv \texttt{nc_incld} \cdot \texttt{sp(1.e-6)} \cdot \rho

Number Tendencies

Consistent with the 2-moment approach, the parameterization calculates the tendencies for both mass and number.

1. Cloud-to-Rain Number Transfer

The rate at which cloud droplet number converts to rain droplet number is determined by preserving the specific number concentration:

\[ \left|\frac{\partial N_c}{\partial t}\right|_{\text{auto}} = \left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}} \frac{N_c}{q_c} \]

Implementation: The variable nc2nr_autoconv_tend stores the positive magnitude of this transfer rate. The actual tendency of cloud droplet number is the negative of this value:

\[ \frac{\partial N_c}{\partial t} = -\texttt{nc2nr\_autoconv\_tend} \]

2. Rain Droplet Number Source

The source of rain droplet number is determined by the characteristic mass of newly formed rain embryos:

\[ \left(\frac{\partial N_r}{\partial t}\right)_{\text{auto}} = \frac{\left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}}}{m_{\text{drop}}(r_{\text{auto}})} \]

where $m_{\text{drop}}(r) = \frac{4}{3}\pi\rho_w r^3$ and $r_{\text{auto}}$ is the characteristic autoconversion radius (default: 25 $\mu\text{m}$).

Implementation: The variable ncautr stores this rain number source. It is computed as:

ncautr = qc2qr_autoconv_tend * CONS3

where CONS3 is computed locally as:

\[ \text{CONS3} = \frac{1}{\frac{4\pi}{3}\rho_w r_{\text{auto}}^3} = \frac{1}{m_{\text{drop}}(r_{\text{auto}})} \]

Important: Note that ncautr (rain number source) is not equal to nc2nr_autoconv_tend (cloud number transfer magnitude). The relationship is: $$ \texttt{ncautr} = \texttt{nc2nr_autoconv_tend} \times \frac{q_c}{N_c} \times \texttt{CONS3} $$

Implementation Details

Reference: This logic is implemented in cloud_water_autoconversion() within p3_autoconversion_impl.hpp.
Thresholds: The process is active only when $q_c > 10^{-8} \, \text{kg}/\text{kg}$.
Consistency Enforcement: The implementation ensures that if either the mass or number tendency is zero, both are set to zero, preventing numerical artifacts in extreme regimes:

nc2nr_autoconv_tend.set(qc2qr_autoconv_tend == 0 && context, 0);
qc2qr_autoconv_tend.set(nc2nr_autoconv_tend == 0 && context, 0);

Subgrid Variance: The code includes a placeholder for subgrid variance scaling (Jensen's inequality effect), but it is currently disabled:

// sgs_var_coef = subgrid_variance_scaling(inv_qc_relvar, sp(2.47) );
sgs_var_coef = 1;

Note: When enabled, the scaling factor uses the same exponent ($\alpha = 2.47$) as the $q_c$ dependency, consistent with Jensen's inequality for power-law functions:

$$ \langle q_c^{\alpha} \rangle > \langle q_c \rangle^{\alpha} $$

when variance exists.

Property Tests

The unit testing suite (p3_autoconversion_unit_tests.cpp) employs a "Physics Property Test" strategy. We validate that the implementation adheres to fundamental physical principles across a wide parameter space.

Tolerance Philosophy

The test suite uses two categories of tolerances based on what is being validated:

1. Identity Tolerances (Precision-Dependent)

These validate that the implementation follows its documented mathematical formulas exactly. Since floating-point precision differs between single and double precision builds, these tolerances are conditionally compiled:

Double precision: 1e-14 (~100× machine epsilon)
Single precision: 1e-7 (~1000× machine epsilon)

Examples:

Verifying nc2nr_autoconv_tend = qc2qr_autoconv_tend × (Nc/qc)
Conservation laws with exact arithmetic

Rationale: These are code correctness checks. Failure indicates implementation bugs, not physics issues. The tolerance accounts for ~10 floating-point operations while remaining strict enough to catch formula errors.

2. Physics Tolerances (Precision-Independent)

These validate physical approximations and configuration parameters. They do not depend on floating-point precision:

Standard: 1% (0.01)

Examples:

Embryo radius consistency (configuration parameter)
KK2000 parameterization accuracy
Monotonicity detection thresholds

Rationale: Physics uncertainties exceed numerical precision. A 1% parameterization error is 1% regardless of whether calculations use float or double.

3. Regime Thresholds (Physical Relevance)

These define when values are physically negligible rather than mathematically zero:

Small droplet regime floor: 1e-15 $\text{kg}/\text{kg}/\text{s}$ (physically irrelevant rate for $r < 1 \, \mu\text{m}$)
Absolute floor: 1e-30 (proxy for numerical zero)

Rationale: Power-law formulas produce non-zero values even in physically irrelevant regimes. The small droplet threshold distinguishes when mean droplet sizes are too small for efficient collision-coalescence ($r < 1 \, \mu\text{m}$, representing incipient cloud or sub-threshold conditions) from regimes where autoconversion is physically active.

Test Strategy

We perform a dense sampling of the phase space:

Cloud Water ($q_c$): Logarithmic sweep from $5 \times 10^{-9}$ (sub-threshold regime) to $10^{-2} \, \text{kg}/\text{kg}$ (heavy precipitation).
Below $10^{-8}$: autoconversion inactive (incipient cloud)
Above $10^{-8}$: active cloud-to-rain transition
Cloud Number ($N_c$): Logarithmic sweep from $10^{6}$ to $10^{9} \, \#/\text{m}^3$.

Parameter Validation

Before running property checks, the test suite verifies that the runtime configuration parameters match the expected Khairoutdinov and Kogan (2000) constants (e.g., $A=1350$, $\alpha=2.47$, $r_{\text{auto}}=25 \, \mu\text{m}$).

1. Threshold Behavior

For $q_c < 10^{-8} \, \text{kg}/\text{kg}$, the test verifies that the autoconversion rate is strictly zero.

2. Monotonicity & Sensitivity

We verify the sign of the partial derivatives with respect to the input state variables:

Colloidal Stability (Inverse $N_c$ Dependency): Increasing droplet number concentration while holding water content fixed must strictly decrease the rate. The test enforces this with no relative tolerance, as the physics dictates a strict inverse relationship.

$$ \frac{\partial}{\partial N_c} \left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}} < 0 $$

Water Content Sensitivity (Positive $q_c$ Dependency): Increasing water content while holding number concentration fixed should increase the rate.

$$ \frac{\partial}{\partial q_c} \left(\frac{\partial q_r}{\partial t}\right)_{\text{auto}} > 0 $$

3. Consistency Constraints

We check that the derived number tendencies match their physical definitions.

Specific Loss Conservation (Mathematical Identity): The code checks that the number loss rate satisfies the conservation relationship: $(dNc/dt)_{\text{auto}} = (dqr/dt)_{\text{auto}} \cdot (N_c / q_c)$.

Tolerance: Precision-dependent (1e-14 for double, 1e-7 for float). This is an exact mathematical identity in the implementation, so failures indicate code bugs rather than physics issues.

Rain Embryo Mass (Configuration Consistency): The implicit mass of the newly formed rain drops is recovered from the ratio of mass tendency to number tendency:

$$ m_{\text{effective}} = \frac{\left(\partial q_r / \partial t\right)}{\left(\partial N_r / \partial t\right)} $$

The test verifies that this mass corresponds to the characteristic radius $r_{\text{auto}}$ (25 $\mu\text{m}$) within 1% tolerance (physics-based, precision-independent). This confirms the autoconversion_radius configuration parameter is correctly preserved through the calculation.

4. Physical Limits

Small Droplet Regime Limit (Physical Relevance Check): When the mean droplet radius is very small ($r < 1 \, \mu\text{m}$, characteristic of incipient cloud or sub-threshold conditions), autoconversion must be physically negligible due to low collision efficiency. The test uses a threshold of $10^{-15} \, \text{kg}/\text{kg}/\text{s}$ to distinguish "physically irrelevant" from "mathematically zero." This accounts for the power-law formula producing non-zero values in regimes where collision-coalescence has no physical significance. This is a regime threshold, not a precision-dependent tolerance.

5. Variance Scaling (Disabled)

The test suite checks for subgrid variance scaling.

Principle: Due to the nonlinearity of the autoconversion rate, subgrid variability in $q_c$ should enhance the grid-mean rate.
Status: Since the feature is currently disabled in the implementation (sgs_var_coef = 1), the test detects this state and reports it as "DISABLED" without failing. If the rates differ significantly but do not show enhancement, it flags a failure.

Tolerance Summary

Test Category	Tolerance Type	Value (Double/Single)	Rationale
Specific Loss Conservation	Identity	`1e-14` / `1e-7`	Mathematical identity (2 FLOPs)
Rain Embryo Mass	Physics	`1%` (both)	Configuration parameter accuracy
Monotonicity (Nc)	Strict	No tolerance	Physics requires strict inequality
Monotonicity (qc)	Physics	`0.1%`	Physical sensitivity detection
Small Droplet Regime	Physical	`1e-15` $\text{kg}/\text{kg}/\text{s}$	Relevance threshold ($r < 1 \, \mu\text{m}$)
Variance Detection	Feature	`0.1%`	Distinguish on/off state