Weak temperature gradient approximation: Difference between revisions

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{{Short description|Meteorological concept}}

= 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.

{{Orphan|date=January 2025}}

\cite{raymond_modelling_2005}\cite{sobel_modeling_2000}

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 |last1=Raymond |first1=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|url-access=subscription }}</ref><ref name=":1">{{Cite journal |last1=Sobel |first1=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 \citename=":2">{siebesma_clouds_2020{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}\cite}</ref><ref name=":3">{charney_note_1963{Cite journal |last=Charney |first=Jule G. |date=1963-11-01 |title=A Note on Large-Scale Motions in the Tropics |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|doi-access=free }}</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 Explanationexplanation ==

Free tropospheric temperature refers to the [[temperature found]] in the higherupper partlayers of the troposphere where the influence from the surface and the [[boundary layer effects]] is negligible. Although the framework is basedformulated onwith itsthe gradients of free tropospheric temperature, this phenomenon occurs as a result of gradients and fluctuations in [[buoyancy]]. AnyDensity stablyor stratifiedbuoyancy fluidfluctuations whichin undergoesa densitystably orstratified buoyancy fluctuations willfluid lead to the formation of [[gravity waveswave]]s.<ref name=":2" /> \cite{siebesma_clouds_2020}In the tropics, where Coriolis force is negligibly small, these [[gravity waveswave]]s prove to be very effective at smoothing out buoyancy gradients, in a process called gravity-wave adjustment or buoyant equalization. \cite<ref>{bretherton_gravity_1989{Cite journal |last1=Bretherton |first1=Christopher S. |last2=Smolarkiewicz |first2=Piotr K. |date=1989-03-15 |title=Gravity Waves, Compensating Subsidence and Detrainment around Cumulus Clouds |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|doi-access=free }}</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 \citename=":4">{adames_basic_2022{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|url-access=subscription }}</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.\cite{charney_note_1963}<ref name=":3" />

=== Equation Derivationderivation ===

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

Line 20:

</math>

* p: pressure

scale analysis suggests that the difference in pressure at two equal height $h$ 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 heightheights $<math>h$</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>

Line 32 ⟶ 37:

</math>

* <math>\boldsymbol{u}</math> is the horizontal velocity component

Scale analysis now suggests that

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 \cite{charney_note_1963}name=":2" />

== Applications ==

ThisThe assumption of negligible horizontal temperature gradient has ansignificant effectimplications onfor the study of the interactioninteractions between large scale circulation and convection atin the tropics. Although, the WTG does not directly apply to the humidity field, latent heat release from phase changes of phase related to convective activity affects temperature and therefore moisture must also 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 follows,is not only fromsupported by observations, but also becauseby the fact that gravity waves efficiently spreaddisperse the vertical structure of deep convective areas aroundacross the tropics.<ref \cite{siebesma_clouds_2020}name=":2" /> From the conservation of dry static energy, the WTG can be used to derive the WTG balance equation

<math>

Line 55 ⟶ 62:

</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. \cite{siebesma_clouds_2020}\cite{adames_basic_2022}

* <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 \cite{siebesma_clouds_2020}\cite{adames_basic_2022}name=":2" /><ref name=":4" />

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}

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 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.<ref \cite{siebesma_clouds_2020}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. \cite<ref>{sobel_weak_2001{Cite journal |last1=Sobel |first1=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|url-access=subscription }}</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 states or heavily-precipitating states, depending on the stability of the constrained temperature profile.<ref>{{Cite journal |last1=Wong |first1=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 |journal=Geophysical Research Letters |language=en |volume=50 |issue=24 |doi=10.1029/2023GL104350 |issn=0094-8276|doi-access=free }}</ref> The WTG has also been used as a parametrization for large-scale motion in cloud-permitting models.<ref name=":4" />

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 energystability and humidity. ThisDue solvesto the limitationstrong ofcoupling suchbetween modelsvertical ofmotion understandingand precipitation, the distributionWTG approach allows the study of precipitation asdistribution, aalso prescribedin verticalthe motionbulk constrainssetup.<ref precipitation.name=":2" \cite{siebesma_clouds_2020}/><ref \cite{sobel_modeling_2000}name=":1" />

Using the WTG framework, many different processes have been studied and better understood. These include, both synoptic processes such as the Walker Cell<ref>{{Cite journal |last1=Bretherton |first1=Christopher S. |last2=Sobel |first2=Adam H. |date=2002-10-15 |title=A Simple Model of a Convectively Coupled Walker Circulation Using the Weak Temperature Gradient Approximation |url=https://journals.ametsoc.org/view/journals/clim/15/20/1520-0442_2002_015_2907_asmoac_2.0.co_2.xml |journal=Journal of Climate |language=EN |volume=15 |issue=20 |pages=2907–2920 |doi=10.1175/1520-0442(2002)015<2907:ASMOAC>2.0.CO;2 |issn=0894-8755|url-access=subscription }}</ref> and the Madden Julian Oscillation<ref>{{Cite journal |last=Chikira |first=Minoru |date=2014-02-01 |title=Eastward-Propagating Intraseasonal Oscillation Represented by Chikira–Sugiyama Cumulus Parameterization. Part II: Understanding Moisture Variation under Weak Temperature Gradient Balance |url=https://journals.ametsoc.org/view/journals/atsc/71/2/jas-d-13-038.1.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=71 |issue=2 |pages=615–639 |doi=10.1175/JAS-D-13-038.1 |issn=0022-4928}}</ref> and also mesoscale processes such as, the diurnal cycle of convection,<ref>{{Cite journal |last1=Ruppert |first1=James H. |last2=Hohenegger |first2=Cathy |date=2018-06-15 |title=Diurnal Circulation Adjustment and Organized Deep Convection |url=https://journals.ametsoc.org/view/journals/clim/31/12/jcli-d-17-0693.1.xml |journal=Journal of Climate |language=EN |volume=31 |issue=12 |pages=4899–4916 |doi=10.1175/JCLI-D-17-0693.1 |issn=0894-8755|hdl=21.11116/0000-0000-0677-4 |hdl-access=free }}</ref> convective self-aggregation<ref>{{Cite journal |last1=Sessions |first1=Sharon L. |last2=Sugaya |first2=Satomi |last3=Raymond |first3=David J. |last4=Sobel |first4=Adam H. |date=2010-06-27 |title=Multiple equilibria in a cloud-resolving model using the weak temperature gradient approximation |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2009JD013376 |journal=Journal of Geophysical Research: Atmospheres |language=en |volume=115 |issue=D12 |doi=10.1029/2009JD013376 |issn=0148-0227}}</ref> and tropical cyclone formation.<ref>{{Cite journal |last1=Raymond |first1=David J. |last2=Sessions |first2=Sharon L. |date=March 2007 |title=Evolution of convection during tropical cyclogenesis |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2006GL028607 |journal=Geophysical Research Letters |language=en |volume=34 |issue=6 |doi=10.1029/2006GL028607 |issn=0094-8276|url-access=subscription }}</ref>

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}

== References ==

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}

{{Reflist}}

[[Category:Atmospheric temperature]]

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}

[[Category:Tropical meteorology]]