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. 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- and mesoscale phenomena in the tropics.
Physical explanation 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 phenomenon 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. 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. This effectively redistributes temperature between regions of precipitating convection and clear-sky region. 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. 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.
Equation derivation This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance
∂ p ∂ z = − ρ g {\displaystyle {\frac {\partial p}{\partial z}}=-\rho g}
p: pressure
ρ {\displaystyle \rho } : density g: gravitational acceleration z: height above surface scale analysis suggests that the difference ( δ {\displaystyle \delta } ) in pressure at two equal heights h {\displaystyle h} is
δ p ∼ g h δ ρ {\displaystyle \delta p\sim gh\delta \rho }
These pressure differences can also be analyzed using the Navier-Stokes momentum equation in the tropics with the Coriolis parameter f ∼ 0 {\displaystyle f\sim 0}
d u d t = − 1 ρ δ p {\displaystyle {\frac {d{\boldsymbol {u}}}{dt}}=-{\frac {1}{\rho }}\delta p}
u {\displaystyle {\boldsymbol {u}}} is the horizontal velocity component Scale analysis now suggests that
δ ρ ρ ∼ δ p p ∼ δ θ θ ∼ F r {\displaystyle {\frac {\delta \rho }{\rho }}\sim {\frac {\delta p}{p}}\sim {\frac {\delta \theta }{\theta }}\sim {\mathcal {F}}_{r}}
where F r = U 2 g h {\displaystyle {\mathcal {F}}_{r}={\frac {U^{2}}{gh}}} is the Froude number, defined as the ratio of vertical inertial force to the gravitational force; U {\displaystyle U} is a horizontal velocity scale. Whereas the same approach for extra-tropical regions would yield
δ ρ ρ ∼ δ θ θ ∼ F r R o {\displaystyle {\frac {\delta \rho }{\rho }}\sim {\frac {\delta \theta }{\theta }}\sim {\frac {{\mathcal {F}}_{r}}{R_{o}}}}
where R o = U f L {\displaystyle R_{o}={\frac {U}{fL}}} 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.
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