Content deleted Content added
Username0088 (talk | contribs) No edit summary |
m Singular |
||
| (24 intermediate revisions by 8 users not shown) | |||
Line 1:
{{Short description|Meteorological concept}}
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>▼
{{Orphan|date=January 2025}}
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. It 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>▼
▲In [[atmospheric science]], the '''
▲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. It is understood to occur as a result of the weak [[Coriolis force]] in the tropics.
In a multitude of theoretical, modelling and observational studies, the WTG has been applied to study [[Synoptic scale meteorology|synoptic]]- and [[Mesoscale meteorology|mesoscale]] phenomena in the tropics.
== Physical
Free tropospheric temperature refers to the [[temperature]] in the upper layers of the troposphere where the influence from the surface and the [[boundary layer]] is negligible. Although the framework is formulated with the gradients of free tropospheric temperature, this
As, buoyancy is closely related to temperature (more specifically the [[virtual temperature]] and the virtual potential temperature) the framework is usually named Weak Temperature Gradient approximation.<ref name=":3" />
=== Equation
This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance
<math>
Line 18 ⟶ 20:
</math>
* p: pressure
* <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>
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>
Line 37 ⟶ 39:
* <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>
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>
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 numbers 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.
== Applications ==
The assumption of negligible horizontal temperature gradient has
<math>
Line 66 ⟶ 68:
where the diabatic heating represents surface fluxes, radiation and latent heat effects. 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 ways 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 used 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.
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
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
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.<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>
== 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. <ref name=":2" /><ref name=":1" />
{{Reflist}}
[[Category:Atmospheric temperature]]
[[Category:Tropical meteorology]]
| |||