Emanuel convection

Introduction

The Emanuel convection scheme (Emanuel and Živković-Rothman 1999) is a mass-flux parameterization designed for use in climate models. It emphasizes the accurate representation of convective water fluxes and microphysical processes, which are critical for controlling atmospheric water vapor and its associated feedback on climate.

Unlike many other schemes, it avoids the “entraining plume” model in favor of a buoyancy-sorting hypothesis. This approach assumes that mixing between clouds and their environment is episodic and inhomogeneous, leading to a spectrum of mixtures that ascend or descend to their respective levels of neutral buoyancy.

*Schematic representation of the Emanuel Convection Scheme logic.*

Key Concepts

Subcloud-Layer Quasi-Equilibrium

The scheme assumes that convective mass fluxes adjust to keep the subcloud layer neutrally buoyant when air is displaced upward. The Cloud-Base Mass Flux (CBMF), denoted as $M_b$, is relaxed toward this equilibrium state:

\[\frac{\partial M_b}{\partial t} = \frac{\alpha}{\Delta t} (T_{v,p} - T_v + \Delta T_b)_{LCL} - \frac{D}{\Delta t} M_b\]

where $\alpha$ is a relaxation rate, $D$ is a damping coefficient, and the terms in parentheses represent the buoyancy excess at the Lifted Condensation Level (LCL).

Episodic Mixing and Buoyancy Sorting

The rate of mixing $\delta M$ at any level is proportional to the vertical gradient of buoyancy $B$ of the undilute cloud air:

\[\frac{\delta M}{M_b} = \frac{|\delta B| + \Lambda \delta p}{\sum_{i=1}^N [|\delta B| + \Lambda \delta p]}\]

Air mixed into the cloud forms a spectrum of mixtures. Each mixture’s fate is determined by its own buoyancy: * Positively buoyant mixtures continue to ascend (entrainment). * Negatively buoyant mixtures descend and eventually leave the cloud (detrainment).

Microphysics and Precipitation

The scheme converts cloud water to precipitation using a temperature-dependent threshold $l_c$:

\[l_c = l_0 \left(1 - \frac{T}{T_{crit}}ight)\]

where $l_0$ is the warm-cloud autoconversion threshold and $T_{crit}$ is a critical temperature (typically -55°C) below which all cloud water is converted to precipitation.

Unsaturated Downdrafts

Precipitation falling through unsaturated air evaporates, driving a hydrostatic downdraft. This downdraft transports heat and moisture into the subcloud layer and enhances surface fluxes by increasing the effective surface wind speed.

Implementation Details

The implementation in climt (both the Fortran-wrapped and the pure Python versions) follows these logical steps:

Thermodynamics: Calculate profiles of Enthalpy ($h$), Moist Static Energy ($h_m$), and Virtual Temperature ($T_v$).
Parcel Lifting (TLIFT): Iteratively solve for the temperature and moisture of an undilute parcel lifted from the level of maximum MSE.
Mixing Matrices: Compute the $SIJ$ (mixing fraction) and $MENT$ (entrainment flux) matrices to determine mass exchange at every level.
Downdraft Loop: Integrate downward from the highest level of positive CAPE to find the properties of the precipitating downdraft.
Tendencies: Calculate the net impact on grid-scale Temperature, Moisture, and Momentum.
Conservation: Apply a final adjustment to ensure exact column enthalpy and momentum conservation.

References

Emanuel, K. A., and M. Živković-Rothman. 1999. “Development and Evaluation of a Convection Scheme for Use in Climate Models.” Journal of the Atmospheric Sciences 56 (11): 1766–82. https://doi.org/10.1175/1520-0469(1999)056<1766:DAEOAC>2.0.CO;2.

--- title: "Emanuel convection" bibliography: ../../references.bib --- ## Introduction The Emanuel convection scheme [@emanuel1999] is a mass-flux parameterization designed for use in climate models. It emphasizes the accurate representation of convective water fluxes and microphysical processes, which are critical for controlling atmospheric water vapor and its associated feedback on climate. Unlike many other schemes, it avoids the "entraining plume" model in favor of a **buoyancy-sorting hypothesis**. This approach assumes that mixing between clouds and their environment is episodic and inhomogeneous, leading to a spectrum of mixtures that ascend or descend to their respective levels of neutral buoyancy. <figure class="align-center"> <img src="./emanuel_convection_schematic.png" width="600" alt="Schematic representation of the Emanuel Convection Scheme logic." /> <figcaption aria-hidden="true"><em>Schematic representation of the Emanuel Convection Scheme logic.</em></figcaption> </figure> ## Key Concepts ### Subcloud-Layer Quasi-Equilibrium The scheme assumes that convective mass fluxes adjust to keep the subcloud layer neutrally buoyant when air is displaced upward. The **Cloud-Base Mass Flux (CBMF)**, denoted as $M_b$, is relaxed toward this equilibrium state: $$\frac{\partial M_b}{\partial t} = \frac{\alpha}{\Delta t} (T_{v,p} - T_v + \Delta T_b)_{LCL} - \frac{D}{\Delta t} M_b$$ where $\alpha$ is a relaxation rate, $D$ is a damping coefficient, and the terms in parentheses represent the buoyancy excess at the Lifted Condensation Level (LCL). ### Episodic Mixing and Buoyancy Sorting The rate of mixing $\delta M$ at any level is proportional to the vertical gradient of buoyancy $B$ of the undilute cloud air: $$\frac{\delta M}{M_b} = \frac{|\delta B| + \Lambda \delta p}{\sum_{i=1}^N [|\delta B| + \Lambda \delta p]}$$ Air mixed into the cloud forms a spectrum of mixtures. Each mixture's fate is determined by its own buoyancy: \* **Positively buoyant** mixtures continue to ascend (entrainment). \* **Negatively buoyant** mixtures descend and eventually leave the cloud (detrainment). ### Microphysics and Precipitation The scheme converts cloud water to precipitation using a temperature-dependent threshold $l_c$: $$l_c = l_0 \left(1 - \frac{T}{T_{crit}}ight)$$ where $l_0$ is the warm-cloud autoconversion threshold and $T_{crit}$ is a critical temperature (typically -55°C) below which all cloud water is converted to precipitation. ### Unsaturated Downdrafts Precipitation falling through unsaturated air evaporates, driving a hydrostatic downdraft. This downdraft transports heat and moisture into the subcloud layer and enhances surface fluxes by increasing the effective surface wind speed. ------------------------ Implementation Details ------------------------ The implementation in `climt` (both the Fortran-wrapped and the pure Python versions) follows these logical steps: 1. **Thermodynamics**: Calculate profiles of Enthalpy ($h$), Moist Static Energy ($h_m$), and Virtual Temperature ($T_v$). 2. **Parcel Lifting (TLIFT)**: Iteratively solve for the temperature and moisture of an undilute parcel lifted from the level of maximum MSE. 3. **Mixing Matrices**: Compute the $SIJ$ (mixing fraction) and $MENT$ (entrainment flux) matrices to determine mass exchange at every level. 4. **Downdraft Loop**: Integrate downward from the highest level of positive CAPE to find the properties of the precipitating downdraft. 5. **Tendencies**: Calculate the net impact on grid-scale Temperature, Moisture, and Momentum. 6. **Conservation**: Apply a final adjustment to ensure exact column enthalpy and momentum conservation.