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On the Variation of Divergent Flow: An Eddy-flux Form Equation Based on the Quasi-geostrophic Balance and Its Application


doi: 10.1007/s00376-016-6212-x

  • Based on basic equations in isobaric coordinates and the quasi-geostrophic balance, an eddy-flux form budget equation of the divergent wind has been derived. This newly derived budget equation has evident physical significance. It can show the intensity of a weather system, the variation of its flow pattern, and the feedback effects from smaller-scale systems (eddy flows). The usefulness of this new budget equation is examined by calculating budgets for the strong divergent-wind centers associated with the South Asian high, and the strong divergence centers over the Tibetan Plateau, during summer (June-August) 2010. The results indicate that the South Asian high significantly interacts with eddy flows. Compared with effects from the mean flow (background circulation), the eddy flows' feedback influences are of greater importance in determining the flow pattern of the South Asian high. Although the positive divergence centers over the Tibetan Plateau intensify through different mechanisms, certain similarities are also obvious. First, the effects from mean flow are dominant in the rapid intensification process of the positive divergence center. Second, an intense offsetting mechanism exists between the effects associated with the eddy flows' horizontal component and the effects related to the eddy flows' convection activities, which weakens the total effects of the eddy flows significantly. Finally, compared with the effects associated with the convection activities of the mean flow, the accumulated effects of the eddy flows' convection activities may be more favorable for the enhancement of the positive-divergence centers.
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Manuscript received: 14 August 2016
Manuscript revised: 26 November 2016
Manuscript accepted: 26 November 2016
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On the Variation of Divergent Flow: An Eddy-flux Form Equation Based on the Quasi-geostrophic Balance and Its Application

  • 1. International Center for Climate and Environment Sciences, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
  • 2. Laboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
  • 3. Institute of Plateau Meteorology, China Meteorological Administration, Chengdu 610072, China

Abstract: Based on basic equations in isobaric coordinates and the quasi-geostrophic balance, an eddy-flux form budget equation of the divergent wind has been derived. This newly derived budget equation has evident physical significance. It can show the intensity of a weather system, the variation of its flow pattern, and the feedback effects from smaller-scale systems (eddy flows). The usefulness of this new budget equation is examined by calculating budgets for the strong divergent-wind centers associated with the South Asian high, and the strong divergence centers over the Tibetan Plateau, during summer (June-August) 2010. The results indicate that the South Asian high significantly interacts with eddy flows. Compared with effects from the mean flow (background circulation), the eddy flows' feedback influences are of greater importance in determining the flow pattern of the South Asian high. Although the positive divergence centers over the Tibetan Plateau intensify through different mechanisms, certain similarities are also obvious. First, the effects from mean flow are dominant in the rapid intensification process of the positive divergence center. Second, an intense offsetting mechanism exists between the effects associated with the eddy flows' horizontal component and the effects related to the eddy flows' convection activities, which weakens the total effects of the eddy flows significantly. Finally, compared with the effects associated with the convection activities of the mean flow, the accumulated effects of the eddy flows' convection activities may be more favorable for the enhancement of the positive-divergence centers.

1. Introduction
  • According to the Helmholtz theorem, the horizontal wind field can be decomposed into two separate parts: the divergent wind, which is determined only by divergence; and the rotational wind, which is solely related to vorticity (Hawkins and Rosenthal, 1965; Bijlsma et al., 1986; Lynch, 1988). This decomposition has been proven to be very useful in meteorology (Lynch, 1988; Xu, 2005; Cao and Xu, 2011; Fu et al., 2013). The divergent wind and its associated velocity potential, as well as the rotational wind and its associated streamfunction, are used widely and effectively in weather and climate map analyses (Hendon, 1986; Trenberth and Chen, 1988; Grist et al., 2009), dynamical diagnoses (Buechler and Fuelberg, 1986; Fu et al., 2011), and data assimilation studies (Daley, 1991; Parrish and Derber, 1992; Daley and Barker, 2001; Xu, 2005; Xu et al., 2006, 2007).

    In general, divergence is smaller than vorticity (Graham, 1953; Landers, 1956); nevertheless, the divergent wind associated with divergence still plays an important role in the atmospheric general circulation (Chen and Wiin-Nielsen, 1976; Chen et al., 1978; Chen, 1980; Chen and Yen, 1991), particularly in severe convective activities (Wiin-Nielsen and Drake, 1966; Ding and Liu, 1985; Buechler and Fuelberg, 1986; Fu et al., 2011). As reported by (Chen and Wiin-Nielsen, 1976), divergent wind is crucial to the energy balance of the atmosphere, because it acts as a type of catalyst in converting the available potential energy into kinetic energy. This point was also confirmed by (Krishnamurti and Ramanathan, 1982) while analyzing the Arabian Sea monsoon. Furthermore, (Buechler and Fuelberg, 1986) found that the divergent-wind's kinetic energy is strikingly enhanced during intense convection processes, and this increase in the divergent wind may account for the major variation of these processes. It should be noted that, thus far, divergent wind has mainly been discussed in the form of energy (Wiin-Nielsen and Drake, 1966; Chen and Wiin-Nielsen, 1976; Chen et al., 1978; Chen, 1980; Krishnamurti and Ramanathan, 1982; Ding and Liu, 1985; Buechler and Fuelberg, 1986). Although energy budget equations are effective, the energy characteristics mainly describe the intensity of a weather system, while the exact flow pattern (e.g., divergence or convergence, etc.) of the weather system cannot be determined from these energy features. For instance, within a fixed region, the flow pattern can be changed by the advection and/or a non-power force (power remains constant at zero, and thus work cannot be done), such as the Coriolis force, while the area-averaged kinetic energy remains constant. Under this condition, a weather system may change greatly, while the energy status changes only slightly. Thus, a direct budget of the divergent wind that can reflect the variation of the flow pattern is indeed necessary. In addition, the traditional divergent-wind kinetic energy budget equation cannot show scale interactions among weather systems of various scales. Therefore, the primary purpose of this study was to design a set of budget equations that can describe the variation of the divergent flow and its associated scale interactions. Moreover, as an application example, the newly derived divergent-wind budget equation is used to investigate the strong divergent-wind center associated with the South Asian high (Reiter and Gao, 1982), and the strong divergence centers over the Tibetan Plateau (Ye, 1981), during summer 2010, to understand the scale interactions that sustained these weather systems.

    The remainder of this paper is organized as follows: the derivation and physical significance of the budget equation are shown in section 2; application of the divergent-wind budget equation is presented in section 3; and a summary and discussion is given in section 4.

2. Derivation of the budget equations and their physical significance
  • 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 (D and D) 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:

    $$\frac{\partial\bar{u}_\chi}{\partial t}=-\left(\underbrace{\bar{u}\frac{\partial\bar{u}_\chi}{\partial x}+ \bar{v}\frac{\partial\bar{u}_\chi}{\partial y}}\limits_{\scriptsize\hbox{MHAU}}+\mathop{\bar{\omega}\frac{\partial\bar{u}_\chi}{\partial p}}\limits_{\scriptsize\hbox{MVAU}}\right)- \left(\underbrace{\frac{\partial\bar{u}_\chi\bar{u}_\psi}{\partial x}+\frac{\partial\bar{v}_\chi\bar{u}_\psi}{\partial y}}\limits_{\scriptsize\hbox{MIHU}}+ \mathop{\frac{\partial\bar{\omega}\bar{u}_\psi}{\partial p}}\limits_{\scriptsize\hbox{MIVU}}\right) \\ -\left(\underbrace{\frac{\partial\overline{u'u'_\chi}}{\partial x}+\frac{\partial\overline{v'u'_\chi}}{\partial y}}\limits_{\scriptsize\hbox{EPHU}}+ \mathop{\frac{\partial\overline{\omega'u'_\chi}}{\partial p}}\limits_{\scriptsize\hbox{EPVU}}\right)- \left(\underbrace{\frac{\partial\overline{u'u'_\psi}}{\partial x}+ \frac{\partial\overline{v'u'_\psi}}{\partial y}}\limits_{\scriptsize\hbox{ESHU}}+\mathop{\frac{\partial\overline{\omega'u_\psi'}}{\partial p}}\limits_{\scriptsize\hbox{ESVU}}\right)+\mathop{D_{u\chi}}\limits_{\scriptsize\hbox{RES}};(18)$$ $$ \frac{\partial\bar{v}_\chi}{\partial t}=-\left(\underbrace{\bar{u}\frac{\partial\bar{v}_\chi}{\partial x}+ \bar{v}\frac{\partial\bar{v}_\chi}{\partial y}}\limits_{\scriptsize\hbox{MHAV}}+\mathop{\bar{\omega}\frac{\partial\bar{v}_\chi}{\partial p}}\limits_{\scriptsize\hbox{MVAV}}\right)- \left(\underbrace{\frac{\partial\bar{u}_\chi\bar{v}_\psi}{\partial x}+\frac{\partial\bar{v}_\chi\bar{v}_\psi}{\partial y}}\limits_{\scriptsize\hbox{MIHV}}+ \mathop{\frac{\partial\bar{\omega}\bar{v}_\psi}{\partial p}}\limits_{\scriptsize\hbox{MIVV}}\right) \\ -\left(\underbrace{\frac{\partial\overline{u'v'_\chi}}{\partial x}+\frac{\partial\overline{v'v'_\chi}}{\partial y}}\limits_{\scriptsize\hbox{EPHV}}+ \mathop{\frac{\partial\overline{\omega'v'_\chi}}{\partial p}}\limits_{\scriptsize\hbox{EPVV}}\right)- \left(\underbrace{\frac{\partial\overline{u'v'_\psi}}{\partial x}+ \frac{\partial\overline{v'v'_\psi}}{\partial y}}\limits_{\scriptsize\hbox{ESHV}}+\mathop{\frac{\partial\overline{\omega'v'_\psi}}{\partial p}}\limits_{\scriptsize\hbox{ESVV}}\right)+\mathop{D_{v\chi}}\limits_{\scriptsize\hbox{RES}}.(19)$$

    Figure 1.  (a) Summer-averaged geopotential height (black solid line; units: gpm), temperature (red dashed line; units: °C) and divergence (shaded; units: 10-6 s-1) at 200 hPa. (b) Summer-averaged vertical motions (shaded; units: cm s-1), uχ (black lines; units: m s-1), and vχ (red lines, units: m s-1) at 200 hPa. The thick blue solid line outlines the terrain height of 3000 m, and the purple dashed rectangles mark the typical areas for uχ and vχ respectively.

    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.

    Figure 2.  Horizontal-averaged budget terms of (a) uχ and (b) vχ within typical areas A and B, shown in Fig. 1, respectively (units: 10-6 m s-2).

3. Application of the divergent-wind budget equation
  • The South Asian high (also known as the Tibetan High) is a large-scale semi-permanent high that appears around the Tibetan Plateau in the upper troposphere (Fig. 1a) during boreal summer (Ye, 1981; Tao and Ding, 1981; Reiter and Gao, 1982; Qu and Huang, 2012). The South Asian high is characterized by significant upper-level divergence (Fig. 1). The divergence may contribute to convective activities in the middle and lower troposphere (Ye, 1981), resulting in many disastrous weather systems in East Asia (Tao and Ding, 1981; Qu and Huang, 2012). In this study, the strong divergent-wind centers and the intense divergence centers associated with the South Asian high are investigated using the newly derived divergent-wind budget equation. The six-hourly Climate Forecast System Reanalysis dataset, with a resolution of 0.5°× 0.5°, from the National Centers for Environmental Prediction (NCEP) (Saha et al., 2010), is used in the calculations.

  • As shown by Eqs. (18) and (19), the rotational wind and divergent wind should be determined before calculating the budget equations. (Sangster, 1960) and (Bijlsma et al., 1986) solved this problem for a global area with periodic or no boundary conditions. However, for a limited domain, these methods are inaccurate and expensive to calculate. (Chen and Kuo, 1992) developed two iterative methods for computing rotational wind and divergent wind in a limited domain. This method is accurate and convergent, but only appropriate for rectangular domains with no data hole. (Xu et al., 2011) adopted classical integral formulae (i.e., Green's function) for calculating rotational wind and divergent wind within limited domains of arbitrary shape. (Cao and Xu, 2011) furthered this method and designed several numerical schemes to decrease the discretization errors. Their main idea was to separate the total solution into: (1) the internal part, which is determined by the distribution of vorticity only, and can be expressed by the divergence inside the domain; and (2) the external part, which is determined by the boundary conditions. In this study, the rotational wind and divergent wind are calculated by using the method reported by (Xu et al., 2011) and (Cao and Xu, 2011), which has been proven to be accurate and effective in solving the problem in a limited domain of arbitrary shape. During the study period, the difference between the divergence calculated by the original wind and divergent wind is approximately two orders of magnitude smaller than the original divergence. This guarantees the calculation accuracy for the budget equations in this study.

    As Fig. 1a illustrates, during summer 2010, the South Asian high is quasi-stable, with its central area mainly located within the region (20°-35°N, 40°-105°E). The South Asian high features a wide warm area, with a strong warm core of -46°C appearing around the western boundary of the Tibetan Plateau. Divergence associated with the South Asian high is remarkable and is mainly located to the south of 23.5°N, around the Tibetan Plateau, and over the East China Sea. Within these three upper-level divergence regions, convection activities are obvious (Fig. 1b).

    The summer-averaged divergent wind is shown in Fig. 1b, from which it is obvious that \(\bar{u}_\chi\) is negative to the west of 92°E, whereas it is positive to the east of 92°E. This configuration forms a basic flow pattern of divergence. A strong negative center of \(\bar{u}_\chi\) that persists around the western boundary of the Tibetan Plateau is marked as typical area A (Fig. 1b). Equation (19) is used to understand the maintenance of the strong negative \(\bar{u}_\chi\) center within area A. To the north of 13°N, \(\bar{v}_\chi\) is mainly positive, whereas to the south of 13°N, it is mainly negative (Fig. 1b). A strong positive center of \(\bar{v}_\chi\) appears over the northern Tibetan Plateau, which is marked as typical area B. Over the Tibetan Plateau, \(\bar{v}_\chi\) mainly decreases from north to south, which also contributes to a basic flow pattern of divergence. Equation (20) is used to investigate the maintenance of the strong positive \(\bar{v}_\chi\) center within area B.

    Figure 3.  Dominant terms for the maintenance of uχ (shaded; units: 10-6 m s-2), and the summer-averaged uχ (black lines; units: m s-1), where the thick blue solid line outlines the terrain height of 3000 m and the red dashed rectangle marks typical area A.

    Figure 2a illustrates the area-A-averaged budget of \(\bar{u}_\chi\), from which it can be seen that the RES term (which is mainly due to friction-related effects, subgrid processes, and calculation uncertainties) accounts for 30% of TOT, implying that the budget equation balances at an 70% level. Therefore, the budget equation has captured the main features during the evolution of divergent flow. As Fig. 2a shows, during the whole summer, the total effect of all the budget terms of Eq. (19), except for RES (i.e., TOT), remains negative. This means that the conditions are favorable for the maintenance of the negative \(\bar{u}_\chi\) center. Except for the ESVU and MHAU terms, all other terms favor the maintenance of the negative \(\bar{u}_\chi\) center within area A (Fig. 2a). The ESHU term is the most favorable factor for the negative center's persistence, and the MIVU term is the second most important. The horizontal distribution of the ESHU term is shown in Fig. 3a, from which it can be seen that negative ESHU dominates area A. This means area A mainly exports positive u'ψ momentum or imports negative u'ψ momentum, favoring the maintenance of negative \(\bar{u}_\chi\). Figure 3b illustrates the MIVU term, showing that negative MIVU dominates the western area A, whereas positive MIVU dominates the eastern area A; this configuration may favor the westward movement of the negative center. From Figs. 1b and 3b, it can be seen that the distribution of the MIVU term is closely related to the distribution of the mean vertical motions. As mentioned above, compared with the effects of the mean flow, the eddy flows' feedback effects on the mean flow may be of greater importance in maintaining the negative \(\bar{u}_\chi\) center.

    Figure 4.  Dominant terms for the maintenance of vχ (shaded; units: m s-2), and the summer-averaged vχ (black lines; units: m s-1), where the thick blue solid line outlines the terrain height of 3000 m and the red dashed rectangle marks typical area B.

    Figure 5.  (a) Summer-averaged divergence at 200 hPa around the Tibetan Plateau (shaded; units: 10-6 s-1), and the three typical areas (I-III) over the plateau (blue dashed rectangles), where the thick black solid line outlines the terrain height of 3000 m. (b-d) Area-averaged divergence (black line; units: 10-6 s-1) and its five-day moving average (red line; units: 10-6 s-1) (left ordinate), as well as the five-day moving average of the area-averaged vertical velocity (blue line; units: cm s-1) (right ordinate), where the green dashed lines show the typical stage for analysis.

    Figure 2b illustrates the area-B-averaged budget of \(\bar{v}_\chi\). As it shows, the RES term accounts for 28% of TOT; or in other words, this equation balances at an 72% level. This confirms that the budget equation captures the main characteristics of the divergent flow's evolution. From Fig. 2b, it can be seen that TOT is negative, meaning that the positive \(\bar{v}_\chi\) center within area B weakens gradually during the summer. The ESHV term is the dominant factor accounting for the positive \(\bar{v}_\chi\) center's weakening, whereas the ESVV and MIVV terms are the main factors favoring the positive center's maintenance. Figure 4 shows the distribution of the ESVV and MIVV terms, from which it can be seen that positive ESVV dominates area B. This means that the vertical eddy transport of the v'ψ momentum converges within area B, which contributes to maintaining the positive \(\bar{v}_\chi\) center. As can be seen in Fig. 2b, the area-B-averaged ESHV reaches a negative extremum, acting to weaken the positive center. The total effect of the ESHV and ESVV terms is negative, implying that the total three-dimensional eddy transport of v'ψ diverges (the vertical eddy flow transports positive v'ψ momentum into area B, and then the horizontal eddy flow transports it out of area B). From Fig. 4b, it can be seen that the positive MIVV dominates area B, which favors the positive \(\bar{v}_\chi\) center's maintenance. As Fig. 1b illustrates, mean ascending motions also dominate area B, but the distribution of the mean vertical motions differs from the distribution of the MIVV term significantly (Fig. 4b). This implies that the distribution of \(\bar{v}_\psi\) features significant unevenness within area B (not shown).

  • During summer 2010, the divergence and ascending motions over the Tibetan Plateau remained strong (Fig. 1), with horizontal distributions all characterized by significant unevenness. There are three typical areas of divergence over the Tibetan Plateau, shown as areas I-III in Fig. 5a, of which areas II and III feature ascending motions (Fig. 1b); whereas, ascending motions only appear in the northern and middle parts of area I.

    Figures 5b-d illustrate the area-averaged divergence and vertical motion within areas I-III. To clarify the main mechanisms accounting for the production of positive divergence within these three typical areas, typical stages that include the entire successive variation process from the minimum divergence to maximum divergence are selected, as indicated by the green dashed lines in Figs. 5b-d.

    To examine the main mechanisms accounting for the enhancement of the divergence within the three typical areas, the calculation Eq. (18)/∂ x+∂Eq. (19)/∂ y is conducted, and the resultant terms are as follows: \(\rm MHD=\partial\rm MHAU/\partial x+\partial \rm MHAV/\partial y\) and \(\rm MVD=\partial\rm MVAU/\partial x+\partial\rm MVAV/\partial y\) denotes the divergence variation due to the mean horizontal and vertical transport, respectively; \(\rm IHD=\partial\rm MIHU/\partial x+\partial\rm MIHV/\partial y\) stands for the divergence production/extinction due to the horizontal interaction between the mean divergent wind and mean rotational wind; \(\rm IVD=\partial\rm MIVU/\partial x+\partial\rm MIVV/\partial y\) represents the divergence variation due to the mean vertical transport of the mean rotational wind momentum; \(\rm PHD=\partial\rm EPHU/\partial x+\partial\rm EPHV/\partial y\) stands for the divergence production/extinction due to the horizontal eddy transport of the perturbation divergent wind; \(\rm PVD=\partial\rm EPVU/\partial x+\partial\rm EPVV/\partial y\) denotes the divergence variation due to the vertical eddy transport of the perturbation divergent wind; \(\rm SHD=\partial \rm ESHU/\partial x+\partial\rm ESHV/\partial y\) and \(\rm SVD=\partial\rm ESVU/\partial x+ \partial \rm ESVV/\partial y\) represent the divergence production/extinction due to the horizontal and vertical eddy transport of perturbation rotational wind, respectively. The total effect of all the budget terms is: TOT = MHD + MVD + IHD + IVD + PHD + PVD + SHD + SVD (not including friction-related effects).

    Figure 6.  Horizontal-averaged budget terms of the divergence (units: 10-10 s-2) within areas I-III.

    Figure 7.  Dominant terms for the variation of the divergence within areas I-III (units: 10-10 s-2), where the black solid is the time-averaged geopotential height at 200 hPa, the red dashed rectangles show the three typical areas, and the thick grey solid line outlines the terrain height of 3000 m.

    Figure 8.  Time-averaged horizontal wind field (barbs; units: m s-1), vertical motions (shaded; units: cm s-1), and geopotential height at 200 hPa (blue solid line; units: gpm) during the three typical stages shown in Figs. 5b-d. The red dashed rectangles show the three typical areas, and the thick grey solid line outlines the terrain height of 3000 m.

    Figure 9.  Time-averaged horizontal eddy kinetic energy (blue solid line; units: m2 s-2), vertical eddy kinetic energy (shaded; units: 10-2 m2 s-2), and geopotential height at 200 hPa (black solid line; units: gpm) during the three typical periods shown in Figs. 5b-d. The red dashed rectangles show the three typical areas, and the thick grey solid line outlines the terrain height of 3000 m.

    Figure 10.  Horizontal-averaged budget terms of the divergence (units: 10-10 s-2) within areas I-III.

    As Fig. 6 shows, all three areas feature divergence intensification (TOT > 0), and the divergence within area I enhances most rapidly. This result is consistent with the variation of divergence within these three areas (Figs. 5b-d). The dominant factors accounting for the variation of divergence within these three areas are very different: for area I, the MHD term is the most favorable factor for the divergence's enhancement, whereas the PHD term is the most detrimental factor; for area II, the IHD term is the most favorable factor, whereas the IVD term is the most detrimental factor; and for area III, the SVD and SHD terms are the most favorable and most detrimental factors, respectively.

    Figure 7 illustrates the horizontal distribution of the dominant terms for the variation of the divergence within areas I-III. From Fig. 8a, it can be seen that area I is dominated by the strong southwesterly; correspondingly, the MHD term is in a wave-like pattern, orientated from the southwest to the northeast (Fig. 7a), and the positive MHD, related to divergence of the mean transport, dominates area I. From Fig. 7a, it can be seen that negative PHD dominates area I, and acts to decrease the positive divergence. This term is determined by the horizontal eddy transport of the perturbation divergent wind because, as Fig. 9a shows, the horizontal eddy kinetic energy is strong within area I, which may provide favorable conditions for the horizontal eddy transport.

    Figure 7b shows the dominant terms for the divergence within area II. As it illustrates, positive IHD dominates area II, favoring the enhancement of the divergence. The IVD term is closely related to the mean convective activities, and acts to slow down the positive divergence's enhancement, with its strong negative centers mainly located in the western part of area II. As Fig. 8b shows, ascending motions are mainly located in the southern part of area II, whereas the northern part of area II features descending motions. The significant difference between the IVD term and the mean vertical motions is due to the unevenness of the mean rotational flow within area II (not shown).

    The SVD term dominates the divergence's enhancement within area III (Fig. 6). As Fig. 7c shows, strong positive SVD centers mainly appear around the northern boundary of this area; also, the vertical eddy kinetic energy, which can represent the intensity of the eddy flows' convective activities, is strong there (Fig. 9c). Strong negative SHD centers mainly appear around the northern boundary of area III, superposing on the positive SVD centers (Fig. 7c). From Fig. 9c, it can be seen that the horizontal eddy kinetic energy, which can reflect the horizontal eddy flows' intensity, decreases from north to south within area III.

    As Figs. 6 and 7b-c show, an offset mechanism may exist between the horizontal and vertical components of the terms determined by the same mechanism (i.e., MHD and MVD, IHD and IVD, PHD and PVD, and SHD and SVD). To investigate the total effects related to the three-dimensional mean transport, MHD + MVD is calculated. Similarly, IHD + IVD is calculated to examine the effect of the three-dimensional interaction between the mean rotational wind and mean divergent wind. PHD + PVD is calculated to reflect the effect of the three-dimensional eddy transport associated with the perturbation divergent-wind momentum; and SHD + SVD is calculated to investigate the effect of the three-dimensional eddy transport associated with perturbation rotational-wind momentum.

    As Fig. 10a shows, for the positive divergence within areas I-II, MHD + MVD is the most favorable factor; whereas for area III, IHD + IVD is the most favorable factor. Moreover, for all three areas, similar to the mean flows' effects, which favor the enhancement of the positive divergence, the effects of the eddy flows are also important to the variation of the divergence.

    The effects associated with the mean flows' convection activities and eddy flows' convection activities are shown in Fig. 10b. Within the three areas, the effects associated with the mean flows' convective activities and the effects associated with the eddy flows' convective activities are generally equally important. The effects associated with the mean flows' convection activities are generally detrimental for the enhancement of the positive divergence. From Fig. 10b, it can be seen that PVD and SVD offset each other; on the whole, for area I, the effects associated with the eddy flows' convection activities are detrimental to the positive divergence's enhancement; whereas for areas II and III, the eddy flows' convection activities are favorable for the enhancement of the divergence. Figure 9 shows that the intensity of the eddy flows' convection activities are closely related to their effects on the divergence: the more intense the eddy flows' convection activities, the more favorable they are to the enhancement of positive divergence.

4. Summary and discussion
  • In this study, based on basic equations in isobaric coordinates and the quasi-geostrophic balance, an eddy-flux form of the divergent-wind budget equation with evident physical significance has been derived. Compared with the traditional energy budget equations associated with the divergent wind, aside from the intensity of the weather system, the new divergent-wind budget equation can also reveal the variation of the flow pattern, which is closely related to the evolution of weather systems. In addition, this new budget equation includes both the influence from the mean flow and the effect from the eddy flow, which can reflect the scale interactions.

    As the divergent-wind budget equation is based on the quasi-geostrophic balance, i.e., the mean rotational flow should satisfy the quasi-geostrophic balance, it is better to use a time window larger than five days (Holton, 1979) to calculate the time mean in Eqs. (18) and (19); or, in other words, this equation can guarantee its accuracy for systems with a period larger than five days. To solve different kinds of scientific questions, the divergent-wind budget equation can be reformed accordingly. For instance, the divergent-wind budget equation can be reformed into the budget equation of the mean divergent-wind kinetic energy by taking Eq. (18) \(\times\bar{u}_\chi\) + Eq. (19) \(\times\bar{v}_\chi\), or into the mean divergence budget equation by applying Eq. (19)/∂ x+∂ Eq. (20)/∂ y.

    The strong divergent-wind centers associated with the South Asian high during summer (June-August) 2010 are investigated using the divergent-wind budget equation. The results indicate that the accumulated effects of the eddy flows' feedback on the mean flow during the whole summer are a dominant factor in the evolution of the divergent-wind centers, whereas the effects associated with the mean flow are relatively smaller. Therefore, the South Asian high interacts with the eddy flows significantly: on the one hand, the South Asian high provides a background environment for the evolution of eddy flows; on the other hand, the eddy flows' feedback effects are of great importance in determining the flow pattern of the South Asian high.

    To examine the main mechanisms accounting for the intensification of the three typical divergence centers over the Tibetan Plateau, a divergence-related transform of Eqs. (19) and (20) is conducted. Within the three typical divergent areas, the dominant mechanisms for the enhancement of divergence centers are different. Generally, the mean flow's effects dominate the rapid intensification process of the positive divergence. This differs from the results of the divergent-wind center's budget shown above, mainly due to (2) an intense offset mechanism in the divergence budget that weakens the total effects of the eddy flows is maintained significantly; and (3) the period used for the divergence budget is much less than the divergent wind budget (generally, the accumulation of the eddy flows' effects may be more important as the accumulation period becomes longer). Moreover, compared with the effects associated with the mean flows' convection activities, the accumulated effects of the eddy flows' convection activities may be more favorable for the positive divergence's intensification over the Tibetan Plateau.

    In the future, we intend to use this newly derived eddy-flux form divergent-wind budget equation to investigate the mechanisms accounting for the evolution of other weather systems that are also closely related to divergent wind. Moreover, we hope to apply new balance equations suitable for describing sub-synoptic/mesoscale weather systems to the divergent wind budget equation. Subsequently, the divergent-wind budget equation could be used to investigate the evolution of sub-synoptic/mesoscale weather systems.

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