Abstract
In order to study the interactions between the atmospheric circulations at the middlehigh and low latitudes from the global perspective, the authors proposed the mathematical definition of threepattern circulations, i.e., horizontal, meridional and zonal circulations with which the actual atmospheric circulation is expanded. This novel decomposition method is proved to accurately describe the actual atmospheric circulation dynamics. The authors used the NCEP/NCAR reanalysis data to calculate the climate characteristics of those threepattern circulations, and found that the decomposition model agreed with the observed results. Further dynamical analysis indicates that the decomposition model is more accurate to capture the major features of global three dimensional atmospheric motions, compared to the traditional definitions of Rossby wave, Hadley circulation and Walker circulation. The decomposition model for the first time realized the decomposition of global atmospheric circulation using three orthogonal circulations within the horizontal, meridional and zonal planes, offering new opportunities to study the largescale interactions between the middlehigh latitudes and low latitudes circulations.
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01 March 2017
An erratum to this article has been published.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (40805034 and 41475068), the Special Scientific Research Project for Public Interest (GYHY201206009) and the Fundamental Research Funds for the Central Universities of China (lzujbky201213).
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An erratum to this article is available at https://doi.org/10.1007/s0038201735548.
Appendix: The proof of Theorem 2
Appendix: The proof of Theorem 2
Sufficient conditions
If Eq. (3.5) holds, i.e., \(\nabla \cdot \vec{A} = 0\), the stream function vector \(\vec{A} = H\vec{i} + W\vec{j} + R\vec{k}\) satisfies
from Definition 1. We need to prove that Eq. (6.1) will lead to the conclusion that a zero velocity field can only be decomposed into three zero velocity fields, under the boundary conditions of \(H = W = 0,\frac{\partial R}{\partial \sigma } = 0\) if \(\sigma = 1\). In order to prove this, we apply vertical vorticity operation onto both sides of the first equation in Eq. (6.1) using the definition of 3D vorticity vector in Eq. (2.5), and fitting the second equation in Eq. (6.1) into it. We then obtain
Using the condition of Theorem 2, we have
where \(\varOmega = S^{2} \times [0,1]\), \(S^{2} = \{ ( (\lambda ,\theta )0 \le \lambda \le 2\pi ,0 \le \theta \le \pi \}\) is the surface of unit sphere, \(\partial \varOmega\) is the boundary of \(\varOmega\), i.e., the surface of a unit sphere, and \(n\) is the unit outer normal vector of \(\partial \varOmega\). The operator \(\varDelta\) is the 3D Laplacian in the spherical \(\sigma\)coordinate system:
We know from the existing mathematical theorems (Taylor 1999) that the solution of Eq. (6.2) is existed and unique up to a constant. When \(\vec{V}^{{\prime }} \equiv 0\) (\(u^{{\prime }} ,v^{{\prime }}\) are zeros), the stream function \(R\) is a constant, and we must have \(\vec{V}_{R}^{{\prime }} \equiv 0\). Using Eq. (6.1) and the conditions in Theorem 2, we have
Thus, \(W = \int_{1}^{\sigma } {\left( {\frac{\partial R}{\partial \theta } + u^{{\prime }} } \right)d\sigma }\) and \(H = \int_{1}^{\sigma } {\left( {\frac{1}{\sin \theta }\frac{\partial R}{\partial \lambda }  v^{{\prime }} } \right)d\sigma }\) uniquely exist. When \(\vec{V}^{{\prime }} \equiv 0\) (\(u^{{\prime }} ,v^{{\prime }}\) are zeros), \(R\) is a constant, and thus the stream function \(H,W\) are zeros, leading to \(\vec{V}_{H}^{{\prime }} \equiv \vec{V}_{W}^{{\prime }} \equiv 0\). This proves that Eq. (3.5) guarantees the threepattern decomposition of global atmospheric circulation uniquely exists.
Necessary condition
We will approach to the proof by contradiction. According to Definition 1 and the conditions of Theorem 1 and Theorem 2, we can assume that the threepattern decomposition of global atmospheric circulation uniquely exists, i.e., for given \(\vec{V}^{{\prime }} = \vec{i}u^{{\prime }} + \vec{j}v^{{\prime }} + \vec{k}\dot{\sigma }\) and \(\nabla \cdot \vec{V}^{{\prime }} = 0\) there exists a unique stream function vector \(\vec{A} = H\vec{i} + W\vec{j} + R\vec{k}\) satisfying

1.
\( \nabla \times \vec{A} = \vec{V}^{{\prime }}\), i.e., \(\vec{V}_{H}^{{\prime }} + \vec{V}_{W}^{{\prime }} + \vec{V}_{R}^{{\prime }} = \vec{V}^{{\prime }}\);

2.
\(\vec{V}^{{\prime }} \equiv 0\) will lead to \(\vec{V}_{H}^{{\prime }} \equiv \vec{V}_{W}^{{\prime }} \equiv \vec{V}_{R}^{{\prime }} \equiv 0\);
we must have \(\nabla \cdot \vec{A} = 0\). This because, otherwise, if we assume that \(\nabla \cdot \vec{A} = D \ne 0\), then \(\vec{A}\) must satisfy
According to theories of fluid dynamics, \(\vec{A}\) can be uniquely decomposed into \(\vec{A} = \vec{A}_{1} + \vec{A}_{2}\), where \(\vec{A}_{1} = H_{1} \vec{i} + W_{1} \vec{j} + R_{1} \vec{k}\) and \(\vec{A}_{2} = H_{2} \vec{i} + W_{2} \vec{j} + R_{2} \vec{k}\) are satisfying
Furthermore, according to the conditions of Theorem 2, we have
when \(\sigma = 1\). Applying the above discussions about the sufficient conditions to Eq. (6.6), we know that \(\vec{A}_{1}\) uniquely exists up to a constant when \(\vec{V}^{{\prime }} \equiv 0\) (\(u^{{\prime }} ,v^{{\prime }}\) are zeros), and we then must have \(\vec{{V}_{H}}{_{1}^{{\prime }}} \equiv \vec{{V}_{W}}{_{1}^{{\prime }}} \equiv \vec{{V}_{R}}{_{1}^{{\prime }}} \equiv 0\). Since \(\nabla \times \vec{A}_{2} = 0\) in Eq. (6.7), there must exists a scalar function \(\varphi (\lambda ,\theta ,\sigma )\) such that
i.e.,
Thus
Using the boundary conditions of Eqs. (6.8) and (6.9), we have
when \(\sigma = 1\). If we denote \(\tau = \tau_{1} \vec{i} + \tau_{2} \vec{j}\) the horizontal tangent of any point on \(\partial \varOmega\), we obtain the direction derivative of the function \(\varphi\) along \(\tau\)
from the definition of direction derivative and using Eq. (6.11). From Eqs. (6.10) and (6.12) we know that the scalar function \(\varphi (\lambda ,\theta ,\sigma )\) satisfies
Note that \(\frac{\partial \varphi }{\partial \tau }_{\partial \varOmega } = 0\) is equivalent to \(\varphi _{\partial \varOmega } = c\), where \(c\) is a constant. In other words, the zero direction derivative of \(\varphi\) along any horizontal tangent \(\tau\) on \(\partial \varOmega\) will lead to the fact that \(\varphi\) is a constant on the sphere surface \(\partial \varOmega\). Thus, Eq. (6.13) is equivalent to
Apparently, the solution of Eq. (6.14) is uniquely existed and the gradient \(\nabla \varphi = \vec{A}_{2}\) for \(\varphi\) satisfying Eq. (6.14) is not a vector constant (otherwise we must have \(\nabla \cdot \vec{A}_{2} = \varDelta \varphi = D \equiv 0\)). We then have\(\nabla \times \vec{A}_{2} = \nabla \times \nabla \varphi = 0,\) i.e.,
Using Eq. (6.6) we have \( \nabla \times \vec{A} =  \nabla \times (\vec{A}_{1} + \vec{A}_{2} ) =  \nabla \times \vec{A}_{1} = \vec{V}^{{\prime }} ,\) which can be expressed as
and
From Eqs. (6.15) to (6.17) we know that when \(\vec{V}^{{\prime }} \equiv 0,\) a zero velocity field will be decomposed into three nonzero velocity fields, a contradiction to the definition of appropriate circulation decomposition. The hypothesis of \(\nabla \cdot \vec{A} = D \ne 0\) does not hold, we must have \(\nabla \cdot \vec{A} = 0\). The proof is completed.
In the above proof, from Eqs. (6.15) to (6.17) we note that the vector \(\vec{A}_{2}\) has no contribution to the original velocity field \(\vec{V}\), but it contributes to the decomposed velocity fields \(\vec{V}_{H} ,\vec{V}_{W} ,\vec{V}_{R}\). The existence of \(\vec{A}_{2}\) not only gives nonunique decomposition, but also generates inconsistent decomposition results, i.e., zero velocity is decomposed into two nonzero velocity with equal magnitude and opposite direction. The condition of \(\nabla \cdot \vec{A} = 0\) in the theorems ensures that \(\vec{A}_{2} \equiv 0\), and thus leads to the existence and uniqueness of our threepattern decomposition of global atmospheric circulation.
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Hu, S., Chou, J. & Cheng, J. Threepattern decomposition of global atmospheric circulation: part I—decomposition model and theorems. Clim Dyn 50, 2355–2368 (2018). https://doi.org/10.1007/s0038201528184
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DOI: https://doi.org/10.1007/s0038201528184
Keywords
 Horizontal circulation
 Meridional circulation
 Zonal circulation
 Threepattern decomposition of global atmospheric circulation