Abstract
The Earth’s atmosphere is generally a stochastic dynamical system in which physical processes take place within a wide range of temporal and spatial scales. Dynamical systems associated with small-scale perturbations often have a characteristically low predictability due to our lacked understanding of various physical processes and their feedback mechanisms. In contrast, the atmospheric motion at the meso- to synoptic-scales tends to possess a slow manifold with more predictable behaviours. It is this slow manifold component, often referred to as the balanced flow to be distinguished from the fast smaller-scale motion, that is more of our interest because it plays a critical role in evolution of common weather systems such as midlatitude baroclinic disturbances, mesoscale convective systems (MCSs), or tropical cyclones (TCs). Any weather system that differs too far from a balanced state will undergo a brief period of rapid adjustment, i.e., the so-called adjustment processes. From the balanced perspective, the evolution of mesoscale and larger-scale systems can be viewed as a series of continuous balanced adjustments, often referred to as the quasi-balanced dynamics. Herein, the quasi-balanced dynamic is simply that the mean flows are in a near-balanced state, but the (weak) superimposed perturbation flows are not, which are often related to the development of secondary circulation beyond the framework of balanced appropriations.
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Notes
- 1.
One can see this directly by doing a simple coordinate transformation from z to p-coordinates as follows. Define the coordinate transformation as: x p  = x z ; y p  = y z ; p = p (x z , y z , z); t p  = t z , and then the Jacobian determinant will be J = │p∂p/∂z│. Under this coordinate transformation, the relative vorticity defined as ζ i ≡ ε ijk ∂u j /∂x k , where ε ijk is Levi-Civia symbol, will transform to \( {\zeta}_i^p=\left|J\right|\partial {x}_j^p/\partial {x}_i{\zeta}_i^z \). Apparently, |J| ≠ 1, and thus the horizontal component of the relative vorticity in z-coordinates is no longer preserved as it is transformed to p-coordinates. Therefore, the correct definition of PV in curvilinear coordinate needs to take into account properly the pseudo tensor properties of the coordinate transformation.
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Kieu, C.Q., Zhang, DL. (2016). Balanced Dynamics in Tropical Cyclones. In: Mohanty, U.C., Gopalakrishnan, S.G. (eds) Advanced Numerical Modeling and Data Assimilation Techniques for Tropical Cyclone Prediction. Springer, Dordrecht. https://doi.org/10.5822/978-94-024-0896-6_24
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