Novel three-pattern decomposition of global atmospheric circulation: generalization of traditional two-dimensional decomposition

Abstract

This study investigates the differences and connections between the three-pattern decomposition of global atmospheric circulation, the representation of the horizontal vortex circulation in the middle–high latitudes and the local partitioning of the overturning circulation in the tropics. It concludes that the latter two methods are based on the traditional two-dimensional (2D) decomposition of the vortex and divergent circulations in the fluid dynamics and that the three-pattern decomposition model is not a simple superposition of the two traditional methods but a new three-dimensional (3D) decomposition of global atmospheric circulation. The three-pattern decomposition model can decompose the vertical vorticity of atmosphere into three parts: one part is caused by the horizontal circulation, whereas the other two parts are induced by divergent motions, which correspond to the zonal and meridional circulations. The diagnostic results from the decomposed vertical vorticities accord well with the classic theory: the atmospheric motion at 500 hPa is quasi-horizontal and nondivergent and can represent the vertical mean state of the entire atmosphere. The analysis of the climate characteristics shows that the vertical vorticities of the zonal and meridional circulations are the main cause of the differences between the three-pattern circulations and traditional circulations. The decomposition of the vertical vorticity by the three-pattern decomposition model offers new opportunities to quantitatively study the interaction mechanisms of the Rossby, Hadley and Walker circulations using the vorticity equation.

Appendix

1.1 Appendix A The 2D decomposition of the vortex and divergent circulation

In general, the velocity of the fluid motions contains both the vortex part and divergent part. Thus, the 2D horizontal velocities $\vec{V}(\lambda ,\varphi )=u(\lambda ,\varphi )\vec{i}+v(\lambda ,\varphi )\vec{j}$ can be quantitatively decomposed into two parts:

$$\vec{V}(\lambda ,\varphi )={{\vec{V}}_{div}}(\lambda ,\varphi )+{{\vec{V}}_{rot}}(\lambda ,\varphi ),$$

(46)

where ${{\vec{V}}_{div}}={{u}_{div}}\vec{i}+{{v}_{div}}\vec{j}$ is the divergent (irrotational) part and ${{\vec{V}}_{rot}}={{u}_{rot}}\vec{i}+{{v}_{rot}}\vec{j}$ is the vortex (nondivergent) part. In this section, we will detailly introduce the calculation process of ${{\vec{V}}_{div}}$ and ${{\vec{V}}_{rot}}$.

Since the vortex part ${{\vec{V}}_{rot}}$ is nondivergent, we have the following divergence formula using Eq. (46):

$${{\nabla }_{2}}\cdot \vec{V}={{\nabla }_{2}}\cdot {{\vec{V}}_{div}}=D,$$

(47)

where ${{\nabla }_{2}}=\frac{1}{a\cos \varphi }\frac{\partial }{\partial \lambda }\vec{i}+\frac{1}{a}\frac{\partial }{\partial \varphi }\vec{j}$ is the 2D gradient operator on a horizontal plane, $D$ is the divergence field of the velocity field $\vec{V}$ and $a$ is the earth radius. Because the gradient of arbitrary scalar function is irrotational, there exists a potential function $\chi$ that satisfy

$${{\vec{V}}_{div}}=\nabla \chi .$$

(48)

Furthermore, by inserting Eq. (48) into Eq. (47), we obtain

$${{\Delta }_{2}}\chi =D,$$

(49)

where ${{\Delta }_{2}}=\frac{1}{{{a}^{2}}{{\cos }^{2}}\varphi }\frac{{{\partial }^{2}}}{\partial {{\lambda }^{2}}}+\frac{1}{{{a}^{2}}\cos \varphi }\frac{\partial }{\partial \varphi }(\cos \varphi \frac{\partial }{\partial \varphi })$ is the 2D Laplacian. The potential function $\chi$ can be obtained by solving Poisson Eq. (49) and the divergent part ${{\vec{V}}_{div}}$ is then obtained using Eq. (48).

Similar to the velocity potential function $\chi$, the streamline and its intuitive physical picture are important for depicting the nondivergent vortex motions. For vortex part ${{\vec{V}}_{rot}}$, we have ${{\vec{V}}_{rot}}\times d\vec{S}=0$ according to the definition of streamline, i.e.,

$${{v}_{rot}}a\cos \varphi d\lambda -{{u}_{rot}}ad\varphi =0,$$

(50)

where $d\vec{S}=a\cos \varphi d\lambda \vec{i}+ad\varphi \vec{j}$ represents the streamline element on the isobaric surface. Equation (50) is the streamline equation of the velocity field ${{\vec{V}}_{rot}}$. The following nondivergent condition is then established:

$${{\nabla }_{2}}\cdot {{\vec{V}}_{rot}}=\frac{1}{a\cos \varphi }\frac{\partial {{u}_{rot}}}{\partial \lambda }+\frac{1}{a\cos \varphi }\frac{\partial {{v}_{rot}}\cos \varphi }{\partial \varphi }=0,$$

(51)

and Eq. (51) is the necessary and sufficient condition that Eq. (50) can be expressed as the total derivative of a stream function $\psi (\lambda ,\varphi ,t)$, i.e.,

$$d\psi ={{v}_{rot}}a\cos \varphi d\lambda -{{u}_{rot}}ad\varphi =0.$$

(52)

We then have

$${{u}_{rot}}=-\frac{1}{a}\frac{\partial \psi }{\partial \varphi },\ {{v}_{rot}}=\frac{1}{a\cos \varphi }\frac{\partial \psi }{\partial \lambda },$$

(53)

where $\psi$ is the stream function of the velocity ${{\vec{V}}_{rot}}$.

Since the divergent part ${{\vec{V}}_{div}}$ is irrotational, we have the following vorticity formula using Eq. (46):

$${{\nabla }_{2}}\times \vec{V}={{\nabla }_{2}}\times {{\vec{V}}_{rot}}=\zeta \vec{k},$$

(54)

where $\zeta =\frac{1}{a\cos \varphi }\frac{\partial v}{\partial \lambda }-\frac{1}{a\cos \varphi }\frac{\partial u\cos \varphi }{\partial \varphi }$ represents the vertical vorticity field of velocity field $\vec{V}$. According to Eq. (54), $\zeta$can be rewritten as the following:

$$\zeta =\frac{1}{a\cos \varphi }\frac{\partial {{v}_{rot}}}{\partial \lambda }-\frac{1}{a\cos \varphi }\frac{\partial {{u}_{rot}}\cos \varphi }{\partial \varphi }.$$

(55)

Fitting Eq. (53) into Eq. (55), we then have

$${{\Delta }_{2}}\psi =\zeta .$$

(56)

Similar to the calculation process of ${{\vec{V}}_{div}}$, we can get the stream function $\psi$ by solving Poisson Eq. (56) and obtain the vortex part ${{\vec{V}}_{rot}}$ using Eq. (53).

1.2 Appendix B The traditional mass stream functions

In order to describe the Hadley circulation using traditional mass stream function, we usually zonally average the continuity Eq. (4) of the vertical circulations in the pressure coordinate (Hartmann 1994; Trenberth et al. 2000) as follows:

$$\frac{1}{2\pi }\int_{0}^{2\pi }{\left( \frac{1}{a\cos \varphi }\frac{\partial {{u}_{div}}}{\partial \lambda }+\frac{1}{a\cos \varphi }\frac{\partial {{v}_{div}}\cos \varphi }{\partial \varphi }+\frac{\partial \omega }{\partial p} \right)}d\lambda =0.$$

(57)

If we use $[{{v}_{div}}]=\frac{1}{2\pi }\int_{0}^{2\pi }{{{v}_{div}}d\lambda }$ and $[\omega ]=\frac{1}{2\pi }\int_{0}^{2\pi }{\omega d\lambda }$ to represent the global zonal mean of ${{v}_{div}}$ and $\omega$, respectively, Eq. (57) can then be expressed as the following according to the $2\pi$ periodicity of ${{u}_{div}}$ about the longitude $\lambda$:

$$\frac{1}{a\cos \varphi }\frac{\partial [{{v}_{div}}]\cos \varphi }{\partial \varphi }+\frac{\partial [\omega ]}{\partial p}=0.$$

(58)

In order to use the mass flow to describe the Hadley circulation, we should rewrite Eq. (58) as follows:

$$\frac{2\pi a}{g}\frac{\partial [{{v}_{div}}]\cos \varphi }{\partial \varphi }+\frac{2\pi {{a}^{2}}\cos \varphi }{g}\frac{\partial [\omega ]}{\partial p}=0.$$

(59)

Similar to the process of Eq. (53), for the Hadley circulation, there exists a 2D mass stream function ${{\psi }_{H}}$ such that the velocities of the Hadley circulation satisfy the following formulas using Eq. (59),

$$[{{v}_{div}}]=\frac{g}{2\pi a\cos \varphi }\frac{\partial {{\psi }_{H}}}{\partial p},\ [\omega ]=-\frac{g}{2\pi {{a}^{2}}\cos \varphi }\frac{\partial {{\psi }_{H}}}{\partial \varphi }.$$

(60)

Thus, the mass stream function ${{\psi }_{H}}$ can be expressed as follows:

$${{\psi }_{H}}=\frac{2\pi a\cos \varphi }{g}\int_{0}^{p}{[{{v}_{div}}]dp},$$

(61)

or

$${{\psi }_{H}}=-\frac{2\pi {{a}^{2}}}{g}\int_{-\frac{\pi }{2}}^{\varphi }{\cos \varphi [\omega ]d\varphi }.$$

(62)

Because the vertical velocity $\omega$ is not the observing variable, we usually use Eq. (61) to calculate ${{\psi }_{H}}$.

Similarly, in order to describe the Walker circulation, we often meridionally average the continuity Eq. (4) after multiplying both sides with $\cos \varphi$, i.e.,

$$\frac{1}{\pi }\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\left( \frac{1}{a}\frac{\partial {{u}_{div}}}{\partial \lambda }+\frac{1}{a}\frac{\partial {{v}_{div}}\cos \varphi }{\partial \varphi }+\frac{\partial \omega \cos \varphi }{\partial p} \right)}d\varphi =0.$$

(63)

We use $\left\langle{{u}_{div}} \right\rangle=\frac{1}{\pi }\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{{{u}_{div}}d\varphi }$ and $\left\langle\omega \cos \varphi \right\rangle=\frac{1}{\pi }\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\cos \varphi \omega d\varphi }$ to represent the global meridional mean of ${{u}_{div}}$ and $\omega \cos \varphi$, respectively. We then obtain

$$\frac{1}{a}\frac{\partial \left\langle{{u}_{div}} \right\rangle}{\partial \lambda }+\frac{\partial \left\langle\omega \cos \varphi \right\rangle}{\partial p}=0,$$

(64)

where ‘< >’ represents global meridional mean. We rewrite Eq. (64) as the following:

$$\frac{\pi a}{g}\frac{\partial \left\langle{{u}_{div}} \right\rangle}{\partial \lambda }+\frac{\pi {{a}^{2}}}{g}\frac{\partial \left\langle\omega \cos \varphi \right\rangle}{\partial p}=0.$$

(65)

Thus, for the Walker circulation, Eq. (65) determines a 2D mass stream function ${{\psi }_{W}}$ such that the velocities of the Walker circulation satisfy

$$\left\langle{{u}_{div}}\right\rangle=\frac{g}{\pi a}\frac{\partial {{\psi }_{W}}}{\partial p},\ \left\langle\omega \cos \varphi \right\rangle=-\frac{g}{\pi {{a}^{2}}}\frac{\partial {{\psi }_{W}}}{\partial \lambda },$$

(66)

The stream function ${{\psi }_{W}}$ can then be expressed as

$$\psi _{W} = \frac{{\pi a}}{g}\int_{0}^{p} {\left\langle {u_{{div}} } \right\rangle dp} ,$$

(67)

or

$$\psi _{W} = - \frac{{\pi a^{2} }}{g}\int_{0}^{\lambda } {\left\langle {\omega \cos \varphi } \right\rangle d\varphi } .$$

(68)

Similarly, we use Eq. (67) to calculate ${{\psi }_{W}}$ since the vertical velocity $\omega$ is not the observing variable. However, it is well-known that the Walker circulation is restricted in the tropics, we usually use the meridional mean $\left\langle{{u}_{div}} \right\rangle_{{{5}^{\circ }}S}^{{{5}^{\circ }}N}$ in the low latitudes instead of the global meridional mean $\left\langle{{u}_{div}} \right\rangle$ in Eq. (67) to calculate mass stream fuction ${{\psi }_{W}}$.