Weak temperature gradient approximation

Weak Temperature Gradient Approximation (WTG)

In atmospheric science, the Weak Temperature Gradient approximation (WTG) is a theoretical framework used to simplify the equations governing tropical atmospheric dynamics and circulation. The WTG approximation assumes that free tropospheric temperature in the tropics has negligible horizontal (and temporal) gradients compared to its vertical gradient.

\cite{raymond_modelling_2005}\cite{sobel_modeling_2000}

The assumption of horizontal homogeneity of temperature follows from observations of free tropospheric temperature in the tropical regions as well as early work on the simplified equations governing tropical circulation, and it is understood to occur as a result of the weak Coriolis force in the tropics. \cite{siebesma_clouds_2020}\cite{charney_note_1963}

Through a multitude of theoretical studies, modelling and observations, the WTG has been often applied to study synoptic- and mesoscale phenomena in the tropics.

Physical Explanation

Free tropospheric temperature refers to the temperature found in the higher part of the troposphere where the influence from boundary layer effects is negligible. Although the framework is based on its gradients, this occurs as a result of gradients and fluctuations in buoyancy. Any stably stratified fluid which undergoes density or buoyancy fluctuations will lead to the formation of gravity waves. \cite{siebesma_clouds_2020}In the tropics, where Coriolis force is negligibly small, these gravity waves prove to be very effective at smoothing out buoyancy gradients, in a process called gravity-wave adjustment or buoyant equalization. \cite{bretherton_gravity_1989} This effectively redistributes temperature between convective, precipitating regions and dryer regions. Due to the speed with which the gravity-wave adjustment occurs, the WTG not only considers negligible horizontal buoyancy gradients but also negligibly small temporal gradients. \cite{adames_basic_2022}

Buoyancy is closely related to temperature, more specifically virtual temperature and virtual potential temperature, leading to the name Weak Temperature Gradient.\cite{charney_note_1963}

Equation Derivation

This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance

${\frac {\partial p}{\partial z}}=-\rho g$

scale analysis suggests that the difference in pressure at two equal height $h$ is

$\delta p\sim gh\delta \rho$

These pressure differences can also be analyzed using the momentum equation in the tropics with the Coriolis parameter $f\sim 0$

${\frac {d{\boldsymbol {u}}}{dt}}=-{\frac {1}{\rho }}\delta p$

Scale analysis now suggests that

${\frac {\delta \rho }{\rho }}\sim {\frac {\delta p}{p}}\sim {\frac {\delta \theta }{\theta }}\sim {\mathcal {F}}_{r}$

where ${\mathcal {F}}_{r}={\frac {U^{2}}{gh}}$ is the Froude number, defined as the ratio of vertical inertial force to the gravitational force; U is a horizontal velocity scale. Whereas the same approach for extra-tropical regions would yield

${\frac {\delta \rho }{\rho }}\sim {\frac {\delta \theta }{\theta }}\sim {\frac {{\mathcal {F}}_{r}}{R_{o}}}$

where $R_{o}={\frac {U}{fL}}$ is the Rossby number with L a characteristic horizontal length scale. This shows that for small Rossby number in the extra-tropics, density (and with it temperature) perturbations are much larger than in the tropical regions.

The pressure gradients mentioned above can be understood to be smoothed out by pressure gradient forces which in the tropics, unlike the mid-latitudes, are not balanced by Coriolis force and thus efficiently remove horizontal gradients. \cite{charney_note_1963}

Applications

This assumption of negligible horizontal temperature gradient has an effect on the study of the interaction between large scale circulation and convection at the tropics. Although, the WTG does not apply to the humidity field, latent heat release from changes of phase related to convective activity must be considered. The WTG approximation allows for models and studies to fix free tropospheric temperature, usually using the reversible moist adiabat. The use of the moist adiabat follows, not only from observations, but also because gravity waves efficiently spread the vertical structure of deep convective areas around the tropics. \cite{siebesma_clouds_2020} From the conservation of dry static energy, the WTG can be used to derive the WTG balance equation

$\omega {\frac {\partial \eta _{d}}{\partial p}}\sim Q$

where Q is the diabatic heating from surface fluxes and latent heat effects, and $\omega$ is the pressure velocity. This suggests that variations in a diabatic atmosphere allow for a formulation of equations for which temperature variations must follow a balance between vertical motions and diabatic heating. \cite{siebesma_clouds_2020}\cite{adames_basic_2022}

There are two way to interpret this conclusion. The first, classical interpretation is that the large scale circulation creates conditions for atmospheric convection to occur. The alternate interpretation is that the surface fluxes and latent heat effects are the processes which control the large scale circulation. In this case, a heat source would cause a temperature anomaly which, in the WTG, would get smoothed out by gravity waves. Due to energetic constraints, this would lead to a large-scale vertical motion to cool the column. \cite{siebesma_clouds_2020}

Models

The weak temperature gradient approximation is often use in models with limited domains as a way to couple large-scale vertical motion and small scale diabatic heating. Generally, this has been done by neglecting horizontal free-tropospheric temperature variations (to first order), while explicitly retaining fluid dynamical aspects and diabatic processes. \cite{sobel_weak_2001}

Many studies implemented the WTG constraint in radiative-convective equilibrium (RCE) models, by fixing the mean virtual temperature profile. \cite{raymond_modelling_2005} Often this creates opposing results with either dry, non-precipitating results or heavily-precipitating states, depending on the stability of the constrained temperature profile. \cite{wong_effect_2023}

Bulk, single column models, can also be developed with the WTG. Although these models usually treat temperature prognostically while constraining the large-scale vertical motion, using the WTG approximation, large scale vertical motion becomes a diagnostic variable, dependent on static energy and humidity. This solves the limitation of such models of understanding the distribution of precipitation as a prescribed vertical motion constrains precipitation. \cite{siebesma_clouds_2020} \cite{sobel_modeling_2000}

Using the WTG framework, many different processes have been studied and better understood. These include, the Walker Cell, the diurnal cycle of convection, self-aggregation, tropical cyclone formation, the Madden Julian Oscillation... The WTG has also been used as a parametrization in for large-scale motion in cloud-permitting models. \cite{adames_basic_2022}