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Tropical cyclone (TC) refers to strong weather scale storms with warm core structure and low pressure center generated on the tropical ocean (Chen et al.[1]; Chen[2]). Located on the west coast of the Northwest Pacific Ocean, China has become one of the countries with the most landffalling typhoons and the most affected by typhoons due to the monsoon climate (Huang and Chen[3]; Ren and Yang[4]; Chen[5]; Xu et al.[6]; Zhang et al.[7]; Wu et al.[8]). On the one hand, landfalling typhoons brought sufficient water vapor, which alleviated the drought to a certain extent and ensured bumper harvest. On the other hand, extreme weather such as strong winds, rainstorms, storm surges, and even tornadoes or hail caused by landfalling typhoons and meteorological disasters such as floods and mudslides are extremely destructive, posing a serious threat to the safety of people's lives and property (Yue[9]; Duan et al.[10-11]; Lei et al.[12]; Chen and Xu[13]).
Typhoon rainstorm is the focus of landfalling typhoon research. The landfalling typhoon rainstorm is affected by many factors (Frunk[14]). Terrain, water vapor transport, and the initiation and development of mesoscale strong convection system have an important impact on the intensity and distribution of rainstorm (Zhou et al.[15]; Wu and Li[16]; Zhang et al.[17]; Chen and Zhang[18]; Dong et al.[19-20]; Xu et al.[21]; Ren et al.[22]). In recent years, it has been found that wave activity is also closely related to the occurrence and development of mesoscale systems, especially typhoon precipitation systems (Shen et al.[23-24]; Fei and Lu[25]; Kou and Lu[26]; Zhong et al.[27]; Qiu et al.[28]).
Wave-flow interaction is an important research field in atmospheric science. Many important weather phenomena can be explained by the theory of wave-flow interaction, such as quasi-biennial oscillation of east-west wind in equatorial stratosphere (Lindzen and Holton[29]), explosive warming in the stratosphere (Matsuno[30]), and acceleration of upper jet (Shepherd[31]; Pfeffer[32]).
Wave action (also known as wave action density) is an important physical parameter in the theory of wave-flow interaction, which is usually defined as a (conserved) disturbance parameter. It is the second or higher power term of disturbance amplitude in the small amplitude approximation, and satisfies the wave action equation in the following flux form:
$$ \frac{\partial A}{\partial t}+\nabla \cdot {\bf{F}}=S, $$ where A is the wave action density and F is the wave action flux. S represents the wave action source and sink term composed of forced factors such as non-adiabatic heating and turbulent dissipation. In the past few decades, people have carried out a great deal of research on the wave-flow interaction, and achieved fruitful research results, expanding the wave-flow interaction theory from two-dimensional to three-dimensional, from quasi-geostrophic approximation to non-geostrophic equilibrium, from planetary scale system to meso-small scale system, from linear theory to nonlinear theory, and from dry air to considering water vapor effect. The research enriched and developed the existing wave-flow interaction theory (Dickinson[33]; Dunkerton et al.[34]; Edmon et al.[35]; Eliassen and Palm[36]; Haynes[37]; Plumb[38-39]; Duan et al.[40]; Ding[41]; Gao et al.[42-44]; Huang[45]; Ran et al.[46-47]; Tao[48]).
Wave-flow interaction theory is not only widely used in the study of large-scale planetary wave propagation, but also can be used in the dynamic analysis of disturbances closely related to rainstorm process. Gao and Ran adopted the potential vortex theory, considered the water vapor effect, and established the water vapor wave action equation and thermal wave action equation suitable for mesoscale disturbance system that caused heavy rainfall[49]. Ran et al. studied the dynamic process of rain-belt formation of landfalling typhoon Wilpa by using the disturbance thermal shear advection parameter (essentially a wave action density)[50]. Chu et al. used the disturbance thermal shear advection parameters to diagnose, analyze, and forecast the wave action density of potential shear and deformation in the rainstorm process of landfalling typhoon Morakot[51].
Based on previous studies, we set out to diagnose and analyze the wave activity characteristics of mesoscale disturbance in the storm belt of landfalling platform, and discuss the influence of wave-flow interaction on the development and evolution of the disturbance. Wave-flow interaction essentially includes two aspects: the forcing effect of basic airflow on disturbance (or fluctuation) and the feedback effect of fluctuation on basic airflow. However, in the past, these two aspects were often studied independently. To solve this problem, in this paper, we deduce the tendency equation of disturbance thermal advection parameter (i.e., wave action equation, which can represent the forcing effect of basic state on disturbance) and its corresponding basic state thermal advection parameter equation (which can describe the feedback effect of disturbance on basic state) in local rectangular coordinate system. This is how the two aspects of waveflow interaction are connected. Based on this, the influence of momentum and heat exchange between basic state and disturbance on the development and evolution of wave action density are further analyzed.
The rest of this paper is organized as follows. The theoretical derivations of wave action density in local rectangular coordinate system are presented in Section 2. Section 3 introduces the ARPS data and presents an overview of typhoon Morakot. Section 4 analyzes the application of wave action density in the rainstorm process of landfalling typhoon Morakot. Section 5 verifies the relationship between wave action density and observed precipitation from a longer time series. Conclusions are summarized in the last section.
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The actual atmosphere is often neither absolutely dry nor saturated everywhere, but is in a kind of nonuniform saturated state: saturated in some parts and unsaturated in other places. Therefore, people use generalized potential temperature (Gao et al.[52]) to describe the thermodynamic state of this non-uniform saturated wet atmosphere, i.e.,
$${\theta ^*} = \theta \exp \left[ {\frac{{{L_v}{q_{{\rm{vs}}}}}}{{{c_p}{T_c}}}{{\left( {\frac{{{q_v}}}{{{q_{{\rm{vs}}}}}}} \right)}^k}} \right],$$ (1) where qvs is the specific humidity of saturated water vapor, Tc is the temperature at the rising condensation height and k is the empirical constant. Using thermodynamic equation, the tendency equation of generalized potential temperature can be written as:
$$\frac{{{\rm{d}}{\theta ^*}}}{{{\rm{d}}t}} = {S_{{\theta ^*}}}, $$ (2) Assuming that a physical parameter can be divided into two parts: the basic state (expressed by subscript "0") and the perturbed state (expressed by subscript "e"), we can get the basic state equations and the corresponding linear perturbed equations by linearization:
$$\frac{{\partial {u_0}}}{{\partial t}} + {{\bf{v}}_0} \cdot \nabla {u_0} - f{v_{a0}} = - {{\bf{v}}_e} \cdot \nabla {u_e}, $$ (3) $$\frac{{\partial {v_0}}}{{\partial t}} + {{\bf{v}}_0} \cdot \nabla {v_0} + f{u_{a0}} = - {{\bf{v}}_e} \cdot \nabla {v_e}, $$ (4) $$\frac{{\partial \theta _0^*}}{{\partial t}} + {{\bf{v}}_0} \cdot \nabla \theta _0^* = - {{\bf{v}}_e} \cdot \nabla \theta _e^*, $$ (5) $$\frac{{\partial {u_e}}}{{\partial t}} = - {{\bf{v}}_e} \cdot \nabla {u_0} - {{\bf{v}}_0} \cdot \nabla {u_e} + f{v_{ae}}, $$ (6) $$\frac{{\partial {v_e}}}{{\partial t}} = - {{\bf{v}}_e} \cdot \nabla {v_0} - {{\bf{v}}_0} \cdot \nabla {v_e} - f{u_{ae}}, $$ (7) $$\frac{{\partial \theta _e^*}}{{\partial t}} = - {{\bf{v}}_e} \cdot \nabla \theta _0^* - {{\bf{v}}_0} \cdot \nabla \theta _e^* + {S_{{\theta ^*}}}, $$ (8) In equations (3)-(8), ve= (ue, ve, we) is disturbance vector, and v0=(u0, v0, w0) is basic state velocity vector.
The purpose of this paper is to study the development and evolution of disturbance in the rain-belt of landfalling typhoon. Therefore, firstly, we use disturbance thermal shear advection parameters to describe disturbance, which is defined as (Zhou et al.[53])
$$M = - \left( {\frac{{\partial {u_e}}}{{\partial z}}\frac{{\partial \theta _e^*}}{{\partial x}} + \frac{{\partial {v_e}}}{{\partial z}}\frac{{\partial {\theta _e}^*}}{{\partial y}}} \right) + \left( {\frac{{\partial {u_e}}}{{\partial x}} + \frac{{\partial {v_e}}}{{\partial y}}} \right)\frac{{\partial {\theta _e}^*}}{{\partial z}}$$ (9) Because M is a second-order disturbance parameter, it is essentially a disturbance energy, which represents the comprehensive characteristics of dynamic disturbance and thermal disturbance. The corresponding basic thermal shear advection parameters can be written as
$${M_0} = - \left( {\frac{{\partial {u_0}}}{{\partial z}}\frac{{\partial {\theta _0}^*}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial z}}\frac{{\partial {\theta _0}^*}}{{\partial y}}} \right) + \left( {\frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right)\frac{{\partial {\theta _0}^*}}{{\partial z}}$$ (10) From the above two equations, it can be seen that M (or M0), the disturbance (or basic state) of the vertical component of the convective vorticity vector, is organically related to the vertical gradient of the horizontal divergence disturbance (or basic state) and the generalized potential temperature disturbance (or basic state), which represents the coupling effect of the shear of the horizontal wind field disturbance (or basic state) and the generalized potential temperature disturbance (or basic state) gradient, and this physical parameter combines the dynamic field disturbance (or basic state) and the generalized potential temperature disturbance (or basic state) gradient.
Equations (9) and (10) are rewritten into flux form:
$$M = - \left[ {\frac{\partial }{{\partial x}}\left( {{\theta _e}^*\frac{{\partial {u_e}}}{{\partial z}}} \right) + \frac{\partial }{{\partial y}}\left( {{\theta _e}^*\frac{{\partial {v_e}}}{{\partial z}}} \right)} \right] + \frac{\partial }{{\partial z}}\left[ {{\theta _e}^*\left( {\frac{{\partial {u_e}}}{{\partial x}} + \frac{{\partial {v_e}}}{{\partial y}}} \right)} \right], $$ (11) $${M_0} = - \left[ {\frac{\partial }{{\partial x}}\left( {{\theta _0}^*\frac{{\partial {u_0}}}{{\partial z}}} \right) + \frac{\partial }{{\partial y}}\left( {{\theta _0}^*\frac{{\partial {v_0}}}{{\partial z}}} \right)} \right] + \frac{\partial }{{\partial z}}\left[ {{\theta _0}^*\left( {\frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right)} \right].$$ (12) Taking the time partial derivative at both ends of (11) and (12), we can get:
$$\frac{{\partial M}}{{\partial t}} = - \frac{\partial }{{\partial x}}\left[ {\left( {\frac{{\partial {\theta _e}^*}}{{\partial t}}\frac{{\partial {u_e}}}{{\partial z}} + {\theta _e}^*\frac{{{\partial ^2}{u_e}}}{{\partial z\partial t}}} \right)} \right] - \frac{\partial }{{\partial y}}\left[ {\left( {\frac{{\partial {\theta _e}^*}}{{\partial t}}\frac{{\partial {v_e}}}{{\partial z}} + {\theta _e}^*\frac{{{\partial ^2}{v_e}}}{{\partial z\partial t}}} \right)} \right] + \frac{\partial }{{\partial z}}\left\{ {\left[ {\frac{{\partial {\theta _e}^*}}{{\partial t}}\left( {\frac{{\partial {u_e}}}{{\partial x}} + \frac{{\partial {v_e}}}{{\partial y}}} \right) + {\theta _e}^*\left( {\frac{{{\partial ^2}{u_e}}}{{\partial x\partial t}} + \frac{{{\partial ^2}{v_e}}}{{\partial y\partial t}}} \right)} \right]} \right\}, $$ (13) $$\frac{{\partial {M_0}}}{{\partial t}} = - \frac{\partial }{{\partial x}}\left[ {\left( {\frac{{\partial {\theta _0}^*}}{{\partial t}}\frac{{\partial {u_0}}}{{\partial z}} + {\theta _0}^*\frac{{{\partial ^2}{u_0}}}{{\partial z\partial t}}} \right)} \right] - \frac{\partial }{{\partial y}}\left[ {\left( {\frac{{\partial {\theta _0}^*}}{{\partial t}}\frac{{\partial {v_0}}}{{\partial z}} + {\theta _0}^*\frac{{{\partial ^2}{v_0}}}{{\partial z\partial t}}} \right)} \right] + \frac{\partial }{{\partial z}}\left\{ {\left[ {\frac{{\partial {\theta _0}^*}}{{\partial t}}\left( {\frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right) + {\theta _0}^*\left( {\frac{{{\partial ^2}{u_0}}}{{\partial x\partial t}} + \frac{{{\partial ^2}{v_0}}}{{\partial y\partial t}}} \right)} \right]} \right\}.$$ (14) By eliminating the local variation terms on the right side of equations (13) and (14) by using the basic state and perturbation equations (3) - (8), the tendency equation of M and M0 in local rectangular coordinate system can be obtained, i.e.,
$$\frac{{\partial M}}{{\partial t}} = \nabla \cdot {{\bf{F}}_1} + \nabla \cdot {{\bf{F}}_2} + \nabla \cdot {{\bf{F}}_e} + \nabla \cdot \left[ {{S_{\theta _e^*}}\nabla \times \left( {{\bf{k}} \times {{\bf{v}}_{he}}} \right)} \right], $$ (15) $$\frac{{\partial {M_0}}}{{\partial t}} = \nabla \cdot {{\bf{F}}_{01}} + \nabla \cdot {{\bf{F}}_{02}} + \nabla \cdot {{\bf{F}}_{03}} - \nabla \cdot {{\bf{F}}_e}, $$ (16) In equations (15) and (16),
$${{\bf{F}}_e} = {{\bf{F}}_{eD}} + {{\bf{F}}_{eT}}, $$ (17) $${{\bf{F}}_{eD}} = \left( {\begin{array}{*{20}{l}} {{w_e}\frac{{\partial {u_e}}}{{\partial z}}\frac{{\partial {\theta _0}^*}}{{\partial z}}}\\ {{w_e}\frac{{\partial {v_e}}}{{\partial z}}\frac{{\partial {\theta _0}^*}}{{\partial z}}}\\ { - {u_e}\frac{{\partial {u_e}}}{{\partial x}}\frac{{\partial {\theta _0}^*}}{{\partial x}} - {v_e}\frac{{\partial {v_e}}}{{\partial y}}\frac{{\partial {\theta _0}^*}}{{\partial y}}} \end{array}} \right), $$ (18) $${{\bf{F}}_{eT}} = \left( {\begin{array}{*{20}{l}} { - {w_e}\frac{{\partial {\theta _e}^*}}{{\partial z}}\frac{{\partial {u_0}}}{{\partial z}}}\\ { - {w_e}\frac{{\partial {\theta _e}^*}}{{\partial z}}\frac{{\partial {v_0}}}{{\partial z}}}\\ {{u_e}\frac{{\partial {\theta _e}^*}}{{\partial x}}\frac{{\partial {u_0}}}{{\partial x}} + {v_e}\frac{{\partial {\theta _e}^*}}{{\partial y}}\frac{{\partial {v_0}}}{{\partial y}}} \end{array}} \right), $$ (19) In equation (15), the left side is the local change term of M, the first three terms on the right side are the second-order flux divergence terms, and the fourth term on the right side is the source and sink term of M. As the M and flux divergence terms of equation (15) are both second-order disturbances and satisfy the trend equation of flux form, they are typical wave action density, and equation (15) is a typical wave action equation. In addition, because M represents a disturbance energy, the wave action equation (15) can describe the development and evolution of this disturbance energy and the forcing effect of the fundamental state on the disturbance.
In the first term at the right side of equation (15), F1 represents the comprehensive effect of the spatial gradient of non-geostrophic wind disturbance and generalized potential temperature disturbance. This term can be also written as ∇⋅F1 = (∇ × fvae)⋅∇θe*, so this term is essentially a geostrophic wind potential vortex disturbance caused by geostrophic wind disturbance. The second term at the right end of equation (15) F2 represents the coupling effect of momentum and heat advection of first-order disturbance and momentum and heat shear of first-order disturbance, so ∇⋅F2 describes the divergence of the coupling effect of momentum and heat advection of first-order disturbance and disturbance shear. The third term at the right end of equation (15) Fe consists of two parts: FeD and FeT, in which, FeD represents the coupling effect of momentum advection transport of second-order disturbance and generalized potential temperature basic state gradient; FeT represents the coupling effect of second-order disturbance advection transport of heat and basic airflow shear, so ∇⋅Fe describes the divergence of the coupling effect of second-order disturbance advection transport and basic state gradient. The fourth term at the right end of equation (15) represents the source-sink term of M composed of the generalized potential temperature source-sink term and the horizontal wind field disturbance, which reflects the coupling effect of the curl after the horizontal wind field disturbance rotates by 90 degrees and the gradient of the generalized potential temperature source-sink term.
In equation (16), the left side is the local change term of M0, the first two terms on the right side are the flux divergence terms of the basic state, and the second two terms on the right side are the flux divergence terms of the second-order disturbance, which represent the feedback effect of the disturbance on the basic state. The first term on the right side of equation (16) represents the influence of the fundamental state of geostrophic potential vorticity caused by the fundamental state of geostrophic wind on the local change of M0, because this term is also written as ∇⋅F01 = (∇ × fva0)⋅∇θ0*. The second term on the right side of equation (16) represents the divergence on the coupling effect of the basic airflow on the momentum and heat advection of the basic state and the spatial gradient of the basic state. The third and fourth terms on the right side of equation (16) together can represent the divergence of the coupling effect of momentum advection and heat advection of second-order disturbance with the fundamental state space gradient.
This term is an exchange term between waves and flows, and represents the momentum and heat transfer between the basic state and disturbance, so this term connects the two aspects of wave-flow interaction. The right side of the wave equation (15) and the basic state equation (16) both contain ∇⋅Fe at the same time, but they have opposite signs, indicating that this term is an exchange term between waves and flows, and represents the momentum and heat transfer between the basic state and disturbance. We apply the previous theory to diagnose and evaluate the rainy process generated by landfalling typhoon Morakot in 2009 in order to examine the characteristics of fluctuation in the landfalling typhoon rain-belt.
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Typhoon Morakot, which made landfall in 2009, originated in the northwest Pacific Ocean early on August 4th, strengthened into a typhoon on August 5th, and landed around 23: 45 on August 7th in Hualian, Taiwan Province, China. It landed around 16:20 on the 9th at Xiapu, Fujian Province, weakened into a strong tropical storm on the evening of the 9th, weakened into a tropical storm in the early hours of the 10th, and vanished about 02: 00 on the 12th (shown in Fig. 1a). Morakot wreaked havoc on numerous Chinese provinces and cities over the course of nine days, with Taiwan Province suffered from most of the destruction.
Figure 1. The track of Typhoon Morakot (a), accumulated precipitation of Morakot in five provinces east of China during August 6-11 (b), and accumulated precipitation of Morakot in Taiwan Province from during August 6-9 (c) (units: mm).
Since Morakot wandered at the coastal areas of East China Sea for a very long time, together with the plenty water vapor carried by the southwest monsoon that moved northwards, there was a torrential rain in Taiwan, east and northern part of Fujian and Zhejiang, southern part of Jiangsu, southeast of Anhui, and Shanghai, as well as heavy torrential rain in Taiwan, northeast of Fujian, and southeast of Zhejiang (shown in Fig. 1b and 1c). In Fujian, Zhejiang and Taiwan, the range of the amount exceeding 200 mm covered over 125 thousand km2. In particular, the amount of precipitation at Tuorong of Fujian, Taishun of Zhejiang and Wencheng are 734.4mm, 1250mm (750mm for 24h) and 881mm respectively during Morakot affecting time. At Ali mountain of Jiayi in Taiwan, the accumulated precipitation is 3059.5mm from 501-1100 (1623.5mm for 24h) local time. The accumulated precipitation exceeded the history record of typhoon-induced precipitation in Zhejiang, Fujian and Taiwan.
We apply the wave-flow interaction theory of disturbance thermal shear advection parameters to diagnose and assess the precipitation process induced by the landfalling typhoon Morakot from 00 UTC on August 7th to 00 UTC on August 11th, 2009, based on the ARPS model's nudging assimilation data. The horizontal resolution is 27 km, the number of grid points is 177×177×43, the vertical stratification is 26 layers, the regional center is (120 °E, 25 °N), and the output data interval is 1 h. The following are the procedures for creating nudging assimilation data in ARPS mode: First, the ARPS mode's ADAS module is utilized to read in objective analysis data generated by NCEP / NCAR global final analysis data as well as traditional ground sounding observation data. The empirical formula is as follows:
$${q_{{v_{ - {{nudging }}}}}} = {\mathop{\rm Max}\nolimits} \left[ {0, {q_v} + \gamma \Delta t\left( {{q_{{v_s}}} - {q_v}} \right)\left( {0.1 + \frac{{Ra{\rm{ in }}}}{{Ra{\rm{ in}}{{\rm{ }}_{\max }}}}\alpha \beta } \right)} \right]$$ (20) ARPS main model is used to adjust the specific humidity of water vapor for 900s by nudging assimilation. In equation (20), qv and qvs are the specific humidity of water vapor and saturated specific humidity before nudging adjustment for each time step, Rain and Rainmax are the 6 hours observed accumulated precipitation and the maximum precipitation in the model area, γ, α and β are the coefficient of nudging adjustment for each time step respectively, and qv_nudging is the specific humidity of water vapor after nudging adjustment. The physical significance of equation (20) lies in the nudging assimilation adjustment of the specific humidity of water vapor over the observed precipitation area to the saturated specific humidity of momentum field, heat field and the mixing ratio content of transport condensation, thus making the dynamic field and thermal field of ARPS model tend to be coordinated.
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At 12 UTC on August 8th, 2009, the abnormal values of the disturbance thermal shear advection parameter M were mainly located in the height range of 2-7 km to the right over the observed rain area (as shown in Fig. 2), corresponding to the dense area of disturbance generalized potential temperature isoline there. The disturbance thermal shear advection parameter in the lower troposphere over the rain area were relatively weak. Further analysis shows that the abnormal value area of disturbance thermal shear advection parameter in the middle and lower troposphere is mainly caused by the coupling term (M1 = $ - \frac{{\partial {u_e}}}{{\partial z}}\frac{{\partial {\theta _e}^*}}{{\partial x}}$) of vertical shear of zonal airflow disturbance and zonal gradient of generalized potential temperature disturbance and the coupling term (M =$ \left( {\frac{{\partial {u_e}}}{{\partial x}} + \frac{{\partial {v_e}}}{{\partial y}}} \right)\frac{{\partial {\theta _e}^*}}{{\partial z}}$) of horizontal divergence disturbance and vertical gradient of generalized potential temperature disturbance, and M1 is the main one. The abnormal value area of the troposphere bottom over the rain area is mainly caused by the coupling term (M2 = -$ \frac{{\partial {u_e}}}{{\partial z}}\frac{{\partial {\theta _e}^*}}{{\partial y}}$) of the vertical gradient of meridional wind disturbance and the meridional gradient of generalized potential temperature disturbance, which is still relatively small. The zonal vertical distribution of thermal shear advection parameter M has similar characteristics, and the coupling term M3 of horizontal divergence disturbance and generalized potential temperature disturbance vertical gradient mainly appears in the middle and lower troposphere over the observed rain area. The absolute value of thermal shear advection parameter |M| can represent the intensity of disturbance, and its vertical distribution can represent the vertical structure of disturbance intensity. The positive high value area |M| is located above the observed 6-hour accumulated heavy rainfall area on the ground, and the positive high value center is located in the middle and lower troposphere, which indicates that there is a significant wave disturbance in the middle and lower troposphere above the rain area, which is closely related to the occurrence and development of rainstorm.
Figure 2. The meridional cross sections of |M|(a), M(b), Mx(c), My(d) and Mz(e) (units: 10-5kg m-1s-1) along 119°E at 12 UTC 8 on August 8th, 2009, where the blue thin bar denotes the observation of 6h-accumulated surface rainfall (units: mm).
In order to study the overall characteristics of disturbance over the rain area in the troposphere, we analyze the vertical distribution characteristics of the absolute value of disturbance thermal shear advection parameter 〈|M|〉 of vertical integration. As shown in Fig. 3, the abnormal value area of 〈|M|〉 is mainly located within the typhoon influence area, and the high value center of 〈|M|〉 is basically corresponding to the heavy precipitation center. The observed accumulated surface precipitation zone in the next 6 hours is located within the abnormal value area of 〈|M|〉, and there is always a strong abnormal value area over Taiwan Island, China, but there is no obvious observed precipitation, which is possibly due to the lack of precipitation observation. No matter before, during or after the typhoon landing, the vertical distribution of 〈|M|〉 has a certain correspondence with the observed rain area, and the high value area of 〈|M|〉 basically covers large precipitation centers; it is worth noting that some local precipitation areas are also accompanied by weak abnormal value areas of 〈|M|〉. The above analysis shows that the vertical gradient, horizontal divergence disturbance, and spatial gradient of generalized potential temperature disturbance in the middle and lower troposphere over the rain area are obvious. The thermal shear advection parameter of disturbance can effectively reveal the structural characteristics of this dynamic field and thermal position disturbance over the rain area, so it is closely related to surface precipitation.
Figure 3. The vertical distribution of integrated thermodynamic shear advection parameter (units: 10-8kg m-1s-1) at 00 UTC (a), 06 UTC (b), 12 UTC (c), and 18 UTC (d) on August 8th, 2009, where the red line denotes the observation of 6h-accumulated surface rainfall (units: mm).
In order to further analyze the main factors affecting the development and evolution of disturbance thermal shear advection parameter, we use the nudging assimilation adjustment data of ARPS model to calculate the ∇⋅F1, ∇⋅F2 and ∇⋅Fe on the right side of wave action equation (15), so as to study the main possible reasons affecting the development and evolution of disturbance characterized by thermal shear advection. As shown in Fig. 4, corresponding to the abnormal value area of M, the divergence of wave action flux ∇⋅F=∇⋅F1+∇⋅F2+∇⋅Fe is mainly located in the middle and lower troposphere slightly to the right over the observed rain area, with positive and negative value areas distributed alternately. Because of the uncertainty of symbols, whether the divergence (∇⋅F > 0) or convergence (∇⋅F < 0) of wave action flux promotes or inhibits the development of disturbance depends on the symbols of M. The vertical distribution structure of ∇⋅F1 is similar to that of ∇⋅F wave action flux divergence, but overall its intensity is slightly weaker than the latter, which indicates that ∇⋅F2 is the main component of wave action flux divergence or the main characteristic of wave action flux divergence. The abnormal value area of ∇⋅F2 is mainly located in the middle and lower troposphere slightly to the right above the observed rain area, which corresponds to the high value area of M, but the sign of both is opposite, which indicates that the contribution of ∇⋅F2 is mainly to weaken the thermal shear advection parameter of disturbance and restrain the development of disturbance. The abnormal value area of ∇⋅Fe is weaker than that of ∇⋅F1 and ∇⋅F2, and mainly appears in the bottom troposphere above the rain area. Similarly, the influence of this term on the development and evolution of disturbance depends on the sign of disturbance thermal shear advection parameter in the bottom troposphere.
Figure 4. The meridional cross sections of ∇⋅F (a, 10-9K m-1s-2), ∇⋅F1 (b, 10-9K m-1s-2), ∇⋅F2 (c, 10-9K m-1s-2), ∇⋅Fe (d, 10-10K m-1s-2), ∇⋅Fed (e, 10-10K m-1s-2) and ∇⋅Fef (f, 10-10K m-1s-2) along 119oE at 12 UTC on August 8th, 2009, where the blue thin bars denote the observed 6h-accumulated surface rainfall (units: mm).
In this paper, two components of the exchange term ∇⋅Fe are further calculated: the divergence term of vertical advection transport of disturbance momentum (∇⋅FeD) and the divergence term of vertical advection transport of disturbance heat (∇⋅FeT). As can be seen from Figs. 4e and 4f, the vertical structure of ∇⋅FeT is basically consistent with the vertical distribution of ∇⋅Fe, and the values of both are also equivalent. Relatively, ∇⋅FeD is much weaker, which indicates that the wave-flow interaction in the convective bottom layer over the rain area is mainly characterized by vertical advection of disturbance heat, and there is disturbance heat exchange between the basic state and the disturbance dynamic state, or that the wave-flow interaction is mainly realized by vertical advection of disturbance heat.
From the time evolution trend, as shown in Fig. 5, as the typhoon landed and moved northward, the abnormal value area of wave action flux divergence 〈|∇⋅F|〉 gradually approached the mainland and moved northward, with the meridional range of about 4 latitudes, and the observed rain area was always within the abnormal value area of 〈|∇⋅F|〉, which indicated that the disturbance thermal shear advection parameter or disturbance caused by wave action flux divergence over the rain area constantly developed and changed. By comparison, it can be seen that among the three components of wave flux divergence, although their abnormal value areas all cover the rain area, their intensities are different. Among them, 〈|∇⋅F1|〉 is the strongest, 〈|∇⋅F2|〉 is the second strongest and 〈|∇⋅Fe|〉 is relatively weak, which indicates that the geostrophic wind potential vortex disturbance is the main forcing term that affects the development and evolution of thermal shear advection, while the coupling term of the first-order disturbance advection and the disturbance gradient of dynamic field and thermal field is the secondary forcing term, and the basic state and disturbance are also the secondary forcing term.
Figure 5. The temporal variations of vertically integrated ∇⋅F(a), ∇⋅F1(b), ∇⋅F2(c) and ∇⋅F3(d) (units: 10-8kg m-1s-1) along 119°E from 00 UTC on August 7th to 00 UTC on August 11th, 2009, where the shaded area denotes the observed 6h-accumulated surface rainfall (units: mm).
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In order to verify the relationship between wave action density and observed precipitation from a longer time series, this paper uses the NCEP/GFS analysis field data (horizontal resolution of 0.5°, vertical layer of 26 layers, and time interval of 6 hours) to diagnose and analyze the relationship between wave action density and observed precipitation in Taiyuan and Fuzhou areas of China during the summer months of 1 June - 1 October 2009.
As shown in Fig. 6, there were about 10 intense precipitation processes with 6-hour cumulative precipitation greater than 5 mm observed in Taiyuan area in the summer of 2009, mainly in July and August; the intensity of these processes reached the maximum value during most of the intense precipitation periods and weakened significantly during the weak or no precipitation periods. The correlation coefficient between Taiyuan and precipitation is 0.84487 and the root mean square RMS of precipitation is 0.0821, which is smaller than the standard deviation of precipitation Sd (0.1327), indicating that the correlation between Taiyuan and precipitation passes the significance test and the wave action density is closely related to the observed precipitation. The intensity of precipitation in the Fuzhou area was significantly greater than that in the Taiyuan area in the summer of 2009, with strong precipitation processes corresponding to the maximum values and weak precipitation processes with less intensity, fluctuating with the amount of precipitation. The correlation analysis shows that the correlation coefficient between Fuzhou area and precipitation is 0.78226, and the RMS of the two is 0.11117, which is smaller than the Sd of precipitation (0.12046), indicating that Fuzhou area is closely related to the observed precipitation, and the correlation between the two passes the significance test. The above analysis shows that the wave action density shows a strong signal during the strong precipitation period, implying intense wave activity, while during the non-precipitation period or weak precipitation period, the value of wave action density is relatively small and shows a weak signal, implying insignificant wave activity. Therefore, the above statistical analysis shows that the wave action density proposed in this paper has some applicability and can be used not only for the diagnostic analysis of typhoon and rainstorm processes, but also for the analysis of wave action density in the summer of 2009. It can be used not only for the diagnostic analysis of storm processes, but also for the study of other precipitation processes. This is mainly because the anomalies of wave action density in the lower troposphere of the precipitation area are largely dependent on the vertical distribution of the generalized potential temperature, and the tilted and dense generalized potential temperature contours in the wall of the typhoon eye with obvious horizontal gradients and vertical shear of the wind field can describe these comprehensive characteristics and establish a link with the ground precipitation. On the other hand, the tilted dense area of this generalized temperature contour also exists in other precipitation processes, so the wave action density has some correlation with this precipitation process.
Figure 6. The temporal variations of (the dashed line, 10-4 kgm-3 Ks-1) and the observed 6-h accumulated rainfall (the solid line, mm) averaged on Taiyuan (a, 36°-39°N, 110°-113°E) and Fuzhou (b, 25°-28°N, 118°-121°E) during 00 UTC on June 1st, 2009-00 UTC on October 1st, 2009.
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This theory is suitable for describing the development and evolution of mesoscale systems. The wave action equation contains the same term as the basic state equation, but the divergence terms of the second-order disturbance momentum and heat advection are opposite. The signs of the two terms are opposite, indicating that they represent the disturbance momentum between the disturbance and the initial state.
The rainstorm process of landfalling typhoon Morakot from 00UTC on August 7th to 00 UTC on August 11th, 2009 was diagnosed and analyzed by wave-flow interaction theory. Wave action density can accurately and comprehensively characterize the vertical structure of typical dynamic field and thermal field disturbance of heavy precipitation system. The analysis of wave flux divergence shows that the geostrophic potential vortex disturbance is the main forcing term that affects the development and evolution of the disturbance thermal shear advection. Moreover, the statistical analysis reveals a certain correspondence between the thermodynamic shear advection parameter and the observed 6-hour accumulated surface rainfall in the summer of 2009, suggesting that they are significantly related.
However, the wave action density is a macroscopic dynamic parameter, which cannot fully reflect many complex microphysical processes contained in precipitation and is limited by the time resolution of NCEP/NCAR real-time analysis data. This also suggests that rainstorm is a very complicated weather phenomenon, and the strong wave-flow interaction over the precipitation area is only a necessary condition for the occurrence of heavy precipitation. In order to accurately forecast the occurrence and development of heavy rainfall, it is necessary to consider various restrictive factors that affect the rainstorm process.
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