Three-pattern decomposition of global atmospheric circulation: part I—decomposition model and theorems

Abstract

In order to study the interactions between the atmospheric circulations at the middle-high and low latitudes from the global perspective, the authors proposed the mathematical definition of three-pattern 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 three-pattern 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 large-scale interactions between the middle-high latitudes and low latitudes circulations.

Appendix: The proof of Theorem 2

1.1 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

$$\left\{ \begin{array}{l} - \nabla \times \vec{A} = \vec{V}^{{\prime }} , \hfill \\ \nabla \cdot \vec{A} = 0, \hfill \\ \end{array} \right.$$

(6.1)

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

$$\varDelta R = \frac{1}{\sin \theta }\frac{{\partial v^{{\prime }} }}{\partial \lambda } - \frac{1}{\sin \theta }\frac{{\partial (u^{{\prime }} \sin \theta )}}{\partial \theta }.$$

Using the condition of Theorem 2, we have

$$\left\{ {\begin{array}{*{20}l} {\varDelta R = \frac{1}{\sin \theta }\frac{{\partial v^{{\prime }} }}{\partial \lambda } - \frac{1}{\sin \theta }\frac{{\partial (u^{{\prime }} \sin \theta )}}{\partial \theta },\quad (\lambda ,\theta ,\sigma ) \in \varOmega ,} \hfill \\ \\ {\frac{\partial R}{\partial n}|_{\partial \varOmega } = \frac{\partial R}{\partial \sigma }|_{\sigma = 1} = 0,} \hfill \\ \end{array} } \right.$$

(6.2)

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:

$$\varDelta = \frac{1}{{\sin^{2} \theta }}\frac{{\partial^{2} }}{{\partial \lambda^{2} }} + \frac{1}{\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial }{\partial \theta }) + \frac{{\partial^{2} }}{{\partial \sigma^{2} }}.$$

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

$$\left\{ {\begin{array}{*{20}l} {\frac{\partial W}{\partial \sigma } = \frac{\partial R}{\partial \theta } + u^{{\prime }} ,\quad (\lambda ,\theta ,\sigma ) \in \varOmega ,} \hfill \\ \\ {W|_{\partial \varOmega } = W|_{\sigma = 1} = 0,} \hfill \\ \end{array} } \right.$$

(6.3)

$$\left\{ {\begin{array}{*{20}l} {\frac{\partial H}{\partial \sigma } = \frac{1}{\sin \theta }\frac{\partial R}{\partial \lambda } - v^{{\prime }} ,\quad (\lambda ,\theta ,\sigma ) \in \varOmega ,} \hfill \\ \\ {H|_{\partial \varOmega } = H|_{\sigma = 1} = 0.} \hfill \\ \end{array} } \right.$$

(6.4)

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 three-pattern decomposition of global atmospheric circulation uniquely exists.

1.2 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 three-pattern 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

$$\left\{ {\begin{array}{*{20}l} { - \nabla \times \vec{A} = \vec{V}^{\prime },} \hfill \\ {\nabla \cdot \vec{A} = D \ne 0.} \hfill \\ \end{array} } \right.$$

(6.5)

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

$$\left\{ {\begin{array}{*{20}l} { - \nabla \times \vec{A}_{1} = \vec{V}^{{\prime }} ,} \hfill \\ {\nabla \cdot \vec{A}_{1} = 0,} \hfill \\ \end{array} } \right.$$

(6.6)

$$\left\{ {\begin{array}{*{20}l} { - \nabla \times \vec{A}_{2} = 0,} \hfill \\ {\nabla \cdot \vec{A}_{2} = D \ne 0.} \hfill \\ \end{array} } \right.$$

(6.7)

Furthermore, according to the conditions of Theorem 2, we have

$$\left\{ {\begin{array}{*{20}l} {H_{1} = H_{2} = 0,} \hfill \\ {W_{1} = W_{2} = 0,} \hfill \\ {\frac{{\partial R_{1} }}{\partial \sigma } = \frac{{\partial R_{2} }}{\partial \sigma } = 0,} \hfill \\ \end{array} } \right.$$

(6.8)

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

$$\vec{A}_{2} = \nabla \varphi ,$$

i.e.,

$$\vec{A}_{2} = H_{2} \vec{i} + W_{2} \vec{j} + R_{2} \vec{k} = \frac{1}{\sin \theta }\frac{\partial \varphi }{\partial \lambda }\vec{i} + \frac{\partial \varphi }{\partial \theta }\vec{j} + \frac{\partial \varphi }{\partial \sigma }\vec{k}.$$

(6.9)

Thus

$$\nabla \cdot \vec{A}_{2} = \varDelta \varphi = D.$$

(6.10)

Using the boundary conditions of Eqs. (6.8) and (6.9), we have

$$\left\{ {\begin{array}{*{20}l} {\frac{1}{\sin \theta }\frac{\partial \varphi }{\partial \lambda } = H_{2} = 0,} \hfill \\ {\frac{\partial \varphi }{\partial \theta } = W_{2} = 0,} \hfill \\ \end{array} } \right.$$

(6.11)

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 $\tau$

$$\frac{\partial \varphi }{\partial \tau }|_{\partial \varOmega } = \frac{1}{\sin \theta }\frac{\partial \varphi }{\partial \lambda }\tau_{1} + \frac{\partial \varphi }{\partial \theta }\tau_{2} = 0,$$

(6.12)

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

$$\left\{ {\begin{array}{*{20}l} {\varDelta \varphi = D,\quad {\text{in}}\;\varOmega ,} \\ {\frac{\partial \varphi }{\partial \tau }|_{\partial \varOmega } = 0.} \\ \end{array} } \right.$$

(6.13)

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

$$\left\{ {\begin{array}{*{20}l} {\varDelta \varphi = D,\quad {\text{in}}\;\varOmega } \hfill \\ {\varphi |_{\partial \varOmega } = c.} \hfill \\ \end{array} } \right.$$

(6.14)

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.,

$$\left\{ {\begin{array}{*{20}l} {\frac{{\partial W_{2} }}{\partial \sigma } - \frac{{\partial R_{2} }}{\partial \theta } = 0,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial R_{2} }}{\partial \lambda } - \frac{{\partial H_{2} }}{\partial \sigma } = 0,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial (\sin \theta H_{2} )}}{\partial \theta } - \frac{1}{\sin \theta }\frac{{\partial W_{2} }}{\partial \lambda } = 0.} \hfill \\ \end{array} } \right.$$

(6.15)

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

$$\left\{ {\begin{array}{*{20}l} {\frac{{\partial (W_{1} + W_{2} )}}{\partial \sigma } - \frac{{\partial (R_{1} + R_{2} )}}{\partial \theta } = u^{{\prime }} ,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial (R_{1} + R_{2} )}}{\partial \lambda } - \frac{{\partial (H_{1} + H_{2} )}}{\partial \sigma } = v^{{\prime }} ,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial (\sin \theta (H_{1} + H_{2} ))}}{\partial \theta } - \frac{1}{\sin \theta }\frac{{\partial (W_{1} + W_{2} )}}{\partial \lambda } = \dot{\sigma },} \hfill \\ \end{array} } \right.$$

(6.16)

and

$$\left\{ {\begin{array}{*{20}l} {\frac{{\partial W_{1} }}{\partial \sigma } - \frac{{\partial R_{1} }}{\partial \theta } = u^{{\prime }} ,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial R_{1} }}{\partial \lambda } - \frac{{\partial H_{1} }}{\partial \sigma } = v^{{\prime }} ,} \hfill \\ {\frac{1}{\sin \theta }\frac{{\partial (\sin \theta H_{1} )}}{\partial \theta } - \frac{1}{\sin \theta }\frac{{\partial W_{1} }}{\partial \lambda } = \dot{\sigma }.} \hfill \\ \end{array} } \right.$$

(6.17)

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 non-unique 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 three-pattern decomposition of global atmospheric circulation.