The basic equations in isobaric coordinates (Holton, 1979) are \begin{eqnarray} \label{eq1} \frac{du}{dt}&=&-\frac{\partial\phi}{\partial x}+fv+F_x ,(1)\\ \label{eq2} \frac{dv}{dt}&=&-\frac{\partial\phi}{\partial y}-fu+F_y , (2)\end{eqnarray} and \begin{equation} \label{eq3} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial\omega}{\partial p}=0 , (3)\end{equation} where u, v, and ω are the zonal wind, meridional wind and vertical velocity, respectively; p is the pressure; φ is the geopotential; f is the Coriolis parameter; and Fx and Fy are the frictional forces along the zonal and meridional directions, respectively. The operator is $$ \frac{d}{dt}=\frac{\partial}{\partial t}+V_h \cdot \bigtriangledown_h+\omega\frac{\partial}{\partial p} , $$ where Vh=ui+vj and $$\bigtriangledown_h=\frac{\partial}{\partial x}i+\frac{\partial}{\partial y}j$$ with i and j as the unit vectors pointing east and north, respectively.
The mean state of a given variable, e.g. A, is determined by the time average
\[ \bar{A}=\frac{1}{T_2-T_1}\int^{T_2}_{T_1}Adt , \]
where T1 and T2 represent the period for the time average, and the overbar indicates the time average of the variable. The perturbation of a variable from the mean state is indicated by a prime, A', and given by \(A'=A-\bar{A}\). Ignoring friction and taking a time average of Eqs. (1) and (2) leads to the following expressions: \begin{eqnarray} \label{eq4} \frac{\partial\bar{u}}{\partial t}&=&-\overline{u\frac{\partial u}{\partial x}}-\overline{v\frac{\partial u}{\partial y}}- \overline{\omega\frac{\partial u}{\partial p}}-\frac{\partial\bar{\phi}}{\partial x}+f\bar{v} ;(4)\\ \label{eq5} \frac{\partial\bar{v}}{\partial t}&=&-\overline{u\frac{\partial v}{\partial x}}-\overline{v\frac{\partial v}{\partial y}}- \overline{\omega\frac{\partial v}{\partial p}}-\frac{\partial\bar{\phi}}{\partial y}-f\bar{u} . (5)\end{eqnarray} It should be noted that the left-hand side of Eqs. (5) and (6), which contain a time mean operator (i.e., the overbar), do not equal to zero, because the time derivative is calculated first and then the time mean is calculated (if the reverse calculation order is true, it is zero). In fact, by moving the time mean overbar into the time derivative operator, the time average is generalized to a running mean (derivation not shown). A running mean operator is a low-pass filter, which decomposes the real meteorological field into a mean part (background circulation) and an eddy part (smaller-scale systems). If we use N as the time window for calculating the running mean, signals with periods greater than N are retained (mean part), while the eddy part only contains signals with periods less than or equal to N. Thus, the left-hand side of Eqs. (5) and (6) denotes the local variation of the mean flow (Fu et al., 2016).
According to the Helmholtz theorem, u=uψ+uχ and v=vψ+vχ, where the subscript ψ stands for the wind that is related to the stream function, i.e., the rotational wind; likewise, the subscript χ stands for the wind determined by the potential function, i.e., the divergent wind. Suppose the mean state of the rotational wind satisfies the quasi-geostrophic balance (Hoskins et al., 1978, 1985; Holton, 1979), it can be derived that \begin{equation} \label{eq6} \frac{\partial\bar{u}_\psi}{\partial t}=-\bar{u}_\psi\frac{\partial\bar{u}_\psi}{\partial x}- \bar{v}_\psi\frac{\partial\bar{u}_\psi}{\partial y}-\frac{\partial\bar{\phi}}{\partial x}+f\bar{v} (6)\end{equation} and \begin{equation} \label{eq7} \frac{\partial\bar{v}_\psi}{\partial t}=-\bar{u}_\psi\frac{\partial\bar{v}_\psi}{\partial x}-\bar{v}_\psi\frac{\partial\bar{v}_\psi}{\partial y}- \frac{\partial\bar{\phi}}{\partial y}-f\bar{u}. (7)\end{equation} By subtracting Eqs. (7) and (8) from Eqs. (5) and (6), respectively, it follows that \begin{equation} \label{eq8} \frac{\partial\bar{u}_\chi}{\partial t}=-\overline{u\frac{\partial u}{\partial x}}- \overline{v\frac{\partial u}{\partial y}}-\overline{\omega\frac{\partial u}{\partial p}}+ \left(\bar {u}_\psi\frac{\partial\bar{u}_\psi}{\partial x}+\bar{v}_\psi\frac{\partial\bar{u}_\psi}{\partial y}\right), (8)\end{equation} and \begin{equation} \label{eq9} \frac{\partial\bar{v}_\chi}{\partial t}=-\overline{u\frac{\partial v}{\partial x}}- \overline{v\frac{\partial v}{\partial y}}-\overline{\omega\frac{\partial v}{\partial p}}+ \left(\bar{u}_\psi \frac{\partial\bar{v}_\psi}{\partial x}+\bar{v}_\psi\frac{\partial\bar{v}_\psi}{\partial y}\right).(9) \end{equation} Using the Helmholtz theorem and decomposing the wind field into the mean and perturbed components leads to: \(u=\bar{u}+u'\), \(v=\bar{v}+v'\), \(\omega=\bar\omega+\omega'\), \(u_\chi=\bar{u}_\chi+u'_\chi\), and \(u_\psi=\bar{u}_\psi+u'_\psi\). The advection terms on the right-hand side of Eq. (9) can be rewritten as \begin{eqnarray} \label{eq10} &\!\!\!\!&-\overline{u\frac{\partial u}{\partial x}}-\overline{v\frac{\partial u}{\partial y}}-\overline{\omega\frac{\partial u}{\partial p}}\nonumber\\ &\!\!\!\!&=-\left(\bar{u}\frac{\partial\bar{u}_\chi}{\partial x}+\bar{v}\frac{\partial\bar{u}_\chi}{\partial y}+\bar{\omega}\frac{\partial\bar{u}_\chi}{\partial p}\right)- \left(\bar{u}\frac{\partial\bar{u}_\psi}{\partial x}+\bar{v}\frac{\partial\bar{u}_\psi}{\partial y}+\bar{\omega}\frac{\partial\bar{u}_\psi}{\partial p}\right)-\nonumber\\ &\!\!\!\!&\quad\left(\overline{u'\frac{\partial u'_\chi}{\partial x}}+\overline{v'\frac{\partial u'_\chi}{\partial y}}+\overline{\omega'\frac{\partial u'_\chi}{\partial p}}\right)- \left(\overline{u'\frac{\partial u'_\psi}{\partial x}}+\overline{v'\frac{\partial u'_\psi}{\partial y}}+\overline{\omega'\frac{\partial u'_\psi}{\partial p}}\right) .\nonumber\\ (10)\end{eqnarray} Taking the time average of Eq. (3) results in \begin{equation} \label{eq11} \frac{\partial\bar{\omega}}{\partial p}=-\left(\frac{\partial\bar{u}}{\partial x}+\frac{\partial\bar{v}}{\partial y}\right). (11)\end{equation} Based on Eq. (11), the second term of the right-hand side of Eq. (10) can be merged with the last term of Eq. (8):
\begin{eqnarray} \label{eq12} &\!\!\!\!&-\left(\bar{u}\frac{\partial\bar{u}_\psi}{\partial x}+\bar{v}\frac{\partial\bar{u}_\psi}{\partial y}+ \bar{\omega}\frac{\partial\bar{u}_\psi}{\partial p}\right)+\left(\bar{u}_\psi\frac{\partial\bar{u}_\psi} {\partial x}+\bar{v}_\psi\frac{\partial\bar{u}_\psi}{\partial y}\right)\nonumber\\ &\!\!\!\!&=-\frac{\partial\bar{u}_\chi\bar{u}_\psi}{\partial x}-\frac{\partial\bar{v}_\chi\bar{u}_\psi}{\partial y}- \frac{\partial\bar{\omega}\bar{u}_\psi}{\partial p} .(12) \end{eqnarray} Subtracting Eq. (11) from Eq. (3) yields \begin{equation} \label{eq13} \frac{\partial\omega'}{\partial p}=-\left(\frac{\partial u'}{\partial x}+\frac{\partial v'}{\partial y}\right) , (13)\end{equation} and from substituting Eq. (13) into the third term of Eq. (10), it follows that \begin{equation} \label{eq14} \overline {u'\frac{\partial u'_\chi}{\partial x}}+\overline{v'\frac{\partial u'_\chi}{\partial y}}+ \overline{\omega'\frac{\partial u'_\chi}{\partial p}}=\frac{\partial\overline{u'u'_\chi}}{\partial x}+ \frac{\partial\overline{v'u'_\chi}}{\partial y}+\frac{\partial\overline{\omega'u'_\chi}}{\partial p} . (14)\end{equation} Similarly, the last term of Eq. (10) can be rewritten as \begin{equation} \label{eq15} \overline{u'\frac{\partial u'_\psi}{\partial x}}+\overline{v'\frac{\partial u'_\psi}{\partial y}}+ \overline{\omega'\frac{\partial u'_\psi}{\partial p}}=\frac{\partial\overline{u'u'_\psi}}{\partial x}+ \frac{\partial\overline{v'u'_\psi}}{\partial y}+\frac{\partial\overline{\omega'u'_\psi}}{\partial p} , (15)\end{equation} and substituting Eqs. (10), (12), (14) and (15) into Eq. (8) results in \begin{eqnarray} \label{eq16} \frac{\partial\bar{u}_\chi}{\partial t}&\!=\!&-\left(\bar{u}\frac{\partial\bar{u}_\chi}{\partial x}+ \bar{v}\frac{\partial\bar{u}_\chi}{\partial y}+\bar{\omega}\frac{\partial\bar{u}_\chi}{\partial p}\right)\!-\! \left(\frac{\partial\bar{u}_\chi\bar{u}_\psi}{\partial x}\!+\!\frac{\partial\bar{v}_\chi\bar{u}_\psi}{\partial y}\!+\! \frac{\partial\bar{\omega}\bar{u}_\psi}{\partial p}\right)-\nonumber\\ &\!\!&\left(\frac{\partial\overline{u'u'_\chi}}{\partial x}\!+\!\frac{\partial\overline{v'u'_\chi}}{\partial y}\!+\! \frac{\partial\overline{\omega'u'_\chi}}{\partial p}\right)\!-\! \left(\frac{\partial\overline{u'u'_\psi}}{\partial x}\!+\!\frac{\partial\overline{v'u'_\psi}}{\partial y}\!+\! \frac{\partial\overline{\omega'u'_\psi}}{\partial p}\right) .\nonumber\\(16) \end{eqnarray} Similarly, using Eqs. (9), (11) and (13), Eq. (17) is obtained: \begin{eqnarray} \label{eq17} \frac{\partial\bar{v}_\chi}{\partial t}&\!=\!&-\left(\bar{u}\frac{\partial\bar{v}_\chi}{\partial x}\!+\! \bar{v}\frac{\partial\bar{v}_\chi}{\partial y}\!+\!\bar{\omega}\frac{\partial\bar{v}_\chi}{\partial p}\right)- \left(\frac{\partial\bar{u}_\chi\bar{v}_\psi}{\partial x}\!+\!\frac{\partial\bar{v}_\chi\bar{v}_\psi}{\partial y}\!+\! \frac{\partial\bar{\omega}\bar{v}_\psi}{\partial p}\right)-\nonumber\\ &\!\!&\left(\frac{\partial\overline{u'v'_\chi}}{\partial x}\!+\!\frac{\partial\overline{v'v'_\chi}}{\partial y}\!+\! \frac{\partial\overline{\omega'v'_\chi}}{\partial p}\right)\!-\!\left(\frac{\partial\overline{u'v'_\psi}}{\partial x}\!+\! \frac{\partial\overline{v'v'_\psi}}{\partial y}\!+\!\frac{\partial\overline{\omega'v'_\psi}}{\partial p}\right) .\nonumber\\(17) \end{eqnarray} If the differences (Duχ and Dvχ) between the left-hand side and the right-hand side of Eqs. (17) and (18) are caused by the friction, subgrid processes, and calculation uncertainties, then Eqs. (18) and (19) can be obtained:
$$\frac{\partial\bar{u}_\chi}{\partial t}=-\left(\underbrace{\bar{u}\frac{\partial\bar{u}_\chi}{\partial x}+ \bar{v}\frac{\partial\bar{u}_\chi}{\partial y}}\limits_{\scriptsize\hbox{MHAU}}+\mathop{\bar{\omega}\frac{\partial\bar{u}_\chi}{\partial p}}\limits_{\scriptsize\hbox{MVAU}}\right)-
\left(\underbrace{\frac{\partial\bar{u}_\chi\bar{u}_\psi}{\partial x}+\frac{\partial\bar{v}_\chi\bar{u}_\psi}{\partial y}}\limits_{\scriptsize\hbox{MIHU}}+ \mathop{\frac{\partial\bar{\omega}\bar{u}_\psi}{\partial p}}\limits_{\scriptsize\hbox{MIVU}}\right) \\
-\left(\underbrace{\frac{\partial\overline{u'u'_\chi}}{\partial x}+\frac{\partial\overline{v'u'_\chi}}{\partial y}}\limits_{\scriptsize\hbox{EPHU}}+ \mathop{\frac{\partial\overline{\omega'u'_\chi}}{\partial p}}\limits_{\scriptsize\hbox{EPVU}}\right)-
\left(\underbrace{\frac{\partial\overline{u'u'_\psi}}{\partial x}+ \frac{\partial\overline{v'u'_\psi}}{\partial y}}\limits_{\scriptsize\hbox{ESHU}}+\mathop{\frac{\partial\overline{\omega'u_\psi'}}{\partial p}}\limits_{\scriptsize\hbox{ESVU}}\right)+\mathop{D_{u\chi}}\limits_{\scriptsize\hbox{RES}};(18)$$
$$ \frac{\partial\bar{v}_\chi}{\partial t}=-\left(\underbrace{\bar{u}\frac{\partial\bar{v}_\chi}{\partial x}+ \bar{v}\frac{\partial\bar{v}_\chi}{\partial y}}\limits_{\scriptsize\hbox{MHAV}}+\mathop{\bar{\omega}\frac{\partial\bar{v}_\chi}{\partial p}}\limits_{\scriptsize\hbox{MVAV}}\right)-
\left(\underbrace{\frac{\partial\bar{u}_\chi\bar{v}_\psi}{\partial x}+\frac{\partial\bar{v}_\chi\bar{v}_\psi}{\partial y}}\limits_{\scriptsize\hbox{MIHV}}+ \mathop{\frac{\partial\bar{\omega}\bar{v}_\psi}{\partial p}}\limits_{\scriptsize\hbox{MIVV}}\right) \\
-\left(\underbrace{\frac{\partial\overline{u'v'_\chi}}{\partial x}+\frac{\partial\overline{v'v'_\chi}}{\partial y}}\limits_{\scriptsize\hbox{EPHV}}+ \mathop{\frac{\partial\overline{\omega'v'_\chi}}{\partial p}}\limits_{\scriptsize\hbox{EPVV}}\right)-
\left(\underbrace{\frac{\partial\overline{u'v'_\psi}}{\partial x}+ \frac{\partial\overline{v'v'_\psi}}{\partial y}}\limits_{\scriptsize\hbox{ESHV}}+\mathop{\frac{\partial\overline{\omega'v'_\psi}}{\partial p}}\limits_{\scriptsize\hbox{ESVV}}\right)+\mathop{D_{v\chi}}\limits_{\scriptsize\hbox{RES}}.(19)$$
As Eq. (18) shows, the term labeled MHAU represents the horizontal transport of \(\bar{u}_\chi\) by the mean flow, and MVAU denotes the vertical transport of \(\bar{u}_\chi\) by the mean flow. MIHU stands for the horizontal transport of \(\bar{u}_\psi\) by the mean divergent wind; this term can also be regarded as the interaction between the mean divergent wind and mean rotational wind. MIVU represents the vertical transport of \(\bar{u}_\psi\) by the mean vertical motion. EPHU and EPVU denote the entire effect (during the time-average calculation period) of the transport of u'χ by the horizontal and vertical eddy flows, respectively. ESHU and ESVU represent the entire effect of the transport of u'ψ by the horizontal and vertical eddy flows, respectively. RES is a residual term including the effects of friction, subgrid processes, and calculation uncertainties. The sum of all right-hand side terms (except for RES) of Eq. (19) is defined as term total (TOT). All the terms of Eq. (20) are similarly explained. As shown by Eqs. (18) and (19), instead of using the potential function χ, which satisfies the relationship Vχ=∇χ (i.e., the divergent wind is the gradient of potential function) (Holton, 1979; Krishnamurti and Ramanathan, 1982; Ding and Liu, 1985; Hoskins et al., 1985), this study directly focuses on the divergent wind, because (2) compared with the potential function, the divergent wind is a more direct way to show the divergent flow, and (3) the divergent wind has a unique solution, whereas there is no unique solution for the potential function (Cao and Xu, 2011; Xu et al., 2011).
It should be noted that, because the quasi-geostrophic balance is used to decompose the two equations (assuming the mean rotational flow satisfies the quasi-geostrophic balance), there should be a lower limit of the time window for determining the mean flow. According to previous studies (Hoskins et al., 1978, 1985; Holton, 1979), a five-day period can be used as the lower limit of the running mean. In Eqs. (18) and (19), some terms are similar to the Reynolds stress (Holton, 1979), i.e., in the form of the time mean of perturbation variables (e.g., EPHU, EPVU). These terms represent the accumulated effects of the smaller-scale weather systems on the mean flow during the time-mean period, which can be regarded as the feedback effects of the eddy flows on the background circulation. Terms with \(\bar\omega\) denote the effects associated with the mean flows' convection activities, while terms with ω' represent the effects from the eddy flows' convective activities.