|
= 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 [[Troposphere|tropospheric]] temperature in the [[tropics]] has negligible horizontal (and temporal) gradients compared to its vertical gradient.<ref name=":0">{{Cite journal |last=Raymond |first=David J. |last2=Zeng |first2=Xiping |date=2005-04-01 |title=Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation |url=http://doi.wiley.com/10.1256/qj.03.97 |journal=Quarterly Journal of the Royal Meteorological Society |language=en |volume=131 |issue=608 |pages=1301–1320 |doi=10.1256/qj.03.97}}</ref><ref name=":1">{{Cite journal |last=Sobel |first=Adam H. |last2=Bretherton |first2=Christopher S. |date=2000-12-15 |title=Modeling Tropical Precipitation in a Single Column |url=https://journals.ametsoc.org/view/journals/clim/13/24/1520-0442_2000_013_4378_mtpias_2.0.co_2.xml |journal=Journal of Climate |language=EN |volume=13 |issue=24 |pages=4378–4392 |doi=10.1175/1520-0442(2000)013<4378:MTPIAS>2.0.CO;2 |issn=0894-8755}}</ref>
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 itIt is understood to occur as a result of the weak [[Coriolis force]] in the tropics. <ref name=":2">{{Cite book |url=https://www.cambridge.org/core/books/clouds-and-climate/7B47159F7B050B71625111E40795D182 |title=Clouds and Climate: Climate Science's Greatest Challenge |date=2020 |publisher=Cambridge University Press |isbn=978-1-107-06107-1 |editor-last=Siebesma |editor-first=A. Pier |___location=Cambridge |editor-last2=Bony |editor-first2=Sandrine |editor-last3=Jakob |editor-first3=Christian |editor-last4=Stevens |editor-first4=Bjorn}}</ref><ref name=":3">{{Cite journal |last=Charney |first=Jule G. |date=1963-11-01 |title=A Note on Large-Scale Motions in the Tropics |url=https://journals.ametsoc.org/view/journals/atsc/20/6/1520-0469_1963_020_0607_anolsm_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=20 |issue=6 |pages=607–609 |doi=10.1175/1520-0469(1963)020<0607:ANOLSM>2.0.CO;2 |issn=0022-4928}}</ref>
ThroughIn a multitude of theoretical studies, modelling and observationsobservational studies, the WTG has been often applied to study [[Synoptic scale meteorology|synoptic]]- and [[Mesoscale meteorology|mesoscale]] phenomena in the tropics.
== Physical Explanation ==
Free tropospheric temperature refers to the temperature found in the higherupper partlayers of the troposphere where the influence from the surface and [[boundary layer]] effects is negligible. Although the framework is basedformulated onwith itsthe gradients of free tropospheric temperature, this occurs as a result of gradients and fluctuations in [[buoyancy]]. Density or buoyancy fluctuations in a stably stratified fluid lead to the formation of gravity waves.<ref name=":2" /> In the tropics, where Coriolis force is negligibly small, these [[Gravity wave|gravity waves]] prove to be very effective at smoothing out buoyancy gradients, in a process called gravity-wave adjustment or buoyant equalization.<ref>{{Cite journal |last=Bretherton |first=Christopher S. |last2=Smolarkiewicz |first2=Piotr K. |date=1989-03-15 |title=Gravity Waves, Compensating Subsidence and Detrainment around Cumulus Clouds |url=https://journals.ametsoc.org/view/journals/atsc/46/6/1520-0469_1989_046_0740_gwcsad_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=46 |issue=6 |pages=740–759 |doi=10.1175/1520-0469(1989)046<0740:GWCSAD>2.0.CO;2 |issn=0022-4928}}</ref> This effectively redistributes temperature between convective,regions of precipitating regionsconvection and dryerclear-sky regionsregion. 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. <ref name=":4">{{Cite journal |last=Adames |first=Ángel F. |date=2022-08-01 |title=The Basic Equations under Weak Temperature Gradient Balance: Formulation, Scaling, and Types of Convectively Coupled Motions |url=https://journals.ametsoc.org/view/journals/atsc/79/8/JAS-D-21-0215.1.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=79 |issue=8 |pages=2087–2108 |doi=10.1175/JAS-D-21-0215.1 |issn=0022-4928}}</ref>
BuoyancyAs, buoyancy is closely related to temperature, (more specifically the [[virtual temperature]] and the virtual potential temperature,) leadingthe toframework theis nameusually named Weak Temperature Gradient approximation.<ref name=":3" />
=== Equation Derivation ===
This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance
<math>
</math>
* p: pressure
scale analysis suggests that the difference in pressure at two equal heights <math>h</math> is ▼* <math>\rho</math>: density
* g: gravitational acceleration
* z: height above surface
▲[[Scale analysis (mathematics)|scale analysis ]] suggests that the difference (<math>\delta</math>) in pressure at two equal heights <math>h</math> is
<math>
\delta p \sim g h \delta \rho
</math> <ref name=":2" />
These pressure differences can also be analyzed using the [[Navier–Stokes equations|Navier-Stokes]] momentum equation in the tropics with the [[Coriolis parameter]] <math>f \sim 0</math>
<math>
\frac{d\boldsymbol{u}}{dt}=-\frac{1}{\rho}\delta p
</math>
* <math>\boldsymbol{u}</math> is the horizontal velocity component
Scale analysis now suggests that
<math>
\frac{\delta \rho}{\rho}\sim \frac{\delta p}{p}\sim \frac{\delta \theta}{\theta}\sim \mathcal{F}_r
</math> <ref name=":3" />
where <math>\mathcal{F}_r=\frac{U^2}{g h}</math> is the [[Froude number]], defined as the ratio of vertical inertial force to the gravitational force; <math>U</math> is a horizontal velocity scale. Whereas the same approach for extra-tropical regions would yield
<math>
\frac{\delta \rho}{\rho}\sim \frac{\delta\theta}{\theta}\sim \frac{\mathcal{F}_r}{R_o}
</math> <ref name=":3" />
where <math>R_o=\frac{U}{f L}</math> is the [[Rossby number]] with L a characteristic horizontal length scale. This shows that for small Rossby numbernumbers in the extra-tropics, density (and with it temperature) perturbations are much larger than in the tropical regions. <ref name=":3" />
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. <ref name=":32" />
== Applications ==
ThisThe 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 phase changes related to convective activity must be considered.<ref name=":2" /> The WTG approximation allows for models and studies to fix the free tropospheric temperature profile, usually using the reversible moist adiabat. The use of the moist adiabat is supported not only by observations but also by the fact that gravity waves efficiently disperse the vertical structure of deep convective areas across the tropics.<ref name=":2" /> From the conservation of dry static energy, the WTG can be used to derive the WTG balance equation
<math>
</math>
* <math>\eta_d</math>: dry static energy
where Q is the diabatic heating from surface fluxes and latent heat effects, and <math>\omega</math> 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.<ref name=":2" /><ref name=":4" /> ▼* <math>\omega</math>: vertical pressure velocity
* Q: diabatic heating
▲where Q is the diabatic heating fromrepresents surface fluxes , radiation and latent heat effects , and <math>\omega</math> 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.<ref name=":2" /><ref name=":4" />
There are two wayways to interpret this conclusion. The first, classical interpretation is that the large scale circulation creates conditions for atmospheric convection to occur.<ref name=":2" /> The alternate, more important interpretation is that the surface fluxes and latent heat effects are 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.<ref name=":2" /> Using this framework, a coupling between large scale vertical motion and diabatic heating in the tropics is achieved.
=== Models ===
The weak temperature gradient approximation is often useused 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. <ref>{{Cite journal |last=Sobel |first=Adam H. |last2=Nilsson |first2=Johan |last3=Polvani |first3=Lorenzo M. |date=2001-12-01 |title=The Weak Temperature Gradient Approximation and Balanced Tropical Moisture Waves |url=https://journals.ametsoc.org/view/journals/atsc/58/23/1520-0469_2001_058_3650_twtgaa_2.0.co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=58 |issue=23 |pages=3650–3665 |doi=10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2 |issn=0022-4928}}</ref>
Many studies implemented the WTG constraint in radiative-convective equilibrium (RCE) models, by fixing the mean virtual temperature profile.<ref name=":0" /> Often this creates opposing results with either dry, non-precipitating results or heavily-precipitating states, depending on the stability of the constrained temperature profile.<ref>{{Cite journal |last=Wong |first=N. Z. |last2=Kuang |first2=Z. |date=2023-12-28 |title=The Effect of Different Implementations of the Weak Temperature Gradient Approximation in Cloud Resolving Models |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2023GL104350 |journal=Geophysical Research Letters |language=en |volume=50 |issue=24 |doi=10.1029/2023GL104350 |issn=0094-8276}}</ref>
| 