Diagnosis of the Forcing of Inertial-gravity Waves in a Severe Convection System

Gravity waves exist widely and play an important role in the momentum budget of the atmosphere (Fritts and Alexander, 2003). Both observational (Plougonven et al., 2003) and numerical investigations (Zhang, 2004) have revealed that the atmospheric imbalance plays an import role in exciting inertial-gravity waves (IGWs). Many diagnostic measures have been proposed to discuss the generation of IGWs; for example, the cross-stream Lagrangian Rossby number (Koch and Dorian, 1988), the Lagrangian Rossby number (O'Sullivan and Dunkerton, 1995), the residual of the nonlinear balance equation (Zhang et al., 2001), and the Richardson Number. It has been suggested that jets and fronts produce low-frequency IGWs (e.g., Uccellini and Koch, 1987; Sato, 1994; Plougonven and Teitelbaum, 2003; Wu and Zhang, 2004; Wang and Zhang, 2007, 2009). Besides the atmospheric imbalance, convection is thought to be another source of gravity waves (Pandya et al., 2000; Kim et al., 2003). In turn, gravity waves are conducive to the production of secondary circulations surrounding clouds in the troposphere (Schmidt and Cotton, 1990), and influence the transfer of momentum between mesoscale waves and large-scale flow. The transfer has been incorporated into GCMs by using gravity-wave drag parameterization schemes (Charron and Manzini, 2002).

The contributions of IGWs to the organization of tropical convection have been widely discussed (e.g., Oouchi, 1999; Peng et al., 2001; Lac et al., 2002; Tulich and Mapes, 2008). It has been found that gravity waves could release unstable energy that triggers new convections in front of a storm when they propagate at speeds faster than the storm. The mechanism of wave conditional instability was adopted to explain the propagation and maintenance of mesoscale convective bands in the midlatitudes (Lindzen, 1974; Raymond, 1987; Koch et al., 2001; Zhang et al., 2001).

The source of gravity waves is an important subject attracting much attention (Ford, 1994a, 1994b; Reeder and Griffiths, 1996; Plougonven and Zeitlin, 2002; Williams et al., 2005). (Ford, 1994a) rewrote a linear wave equation with forcing terms on the rhs linked to vertical motions. Using scaling arguments, (Plougonven and Zhang, 2007) obtained a wave equation with forcing terms resulting from the primary flow, which can be used as an indicator of the excitation of IGWs. (Wang and Zhang, 2010) developed a linear numerical model to explore the source of the gravity waves within a vortex dipole. (Song et al., 2003) investigated the generation mechanisms for two-dimensional IGWs in the stratosphere. They found that both the nonlinear and diabatic forcing are important to the generation of waves. Compared with the diabatic sources, the nonlinear sources are inefficient in generating linear gravity waves that propagate vertically into the stratosphere due to the vertical propagation condition, such as basic-state wind and its vertical shear.

In order to obtain a wave equation, it is necessary to separate a flow into two components. Temporal-scale separation (Gill, 1982; Ford et al., 2000, 2002; Reznik et al., 2001; Saujani and Shepherd, 2002) and spatial-scale separation (Davis and Emanuel, 1991; Plougonven and Zhang, 2007; Snyder et al., 2009; Wang et al., 2010) are often employed to decompose a flow into large-scale balanced flow and IGWs. This separation partially leads to the weak interactions between the two types of motions. (Wu and Zhang, 2004) examined the forcing of gravity waves imposed in nonlinear balance flow through PV inversion. The nonlinear balance had higher-order accuracy than quasigeostrophy and was applicable to both synoptic- and mesoscale systems (Gent and McWilliams, 1982; Zhang et al., 2000).

Built in hydrostatic dynamics, most previous wave equations have rarely been applied to mesoscale convection. Therefore, in the present study, our motivation was to provide a non-hydrostatic wave equation with the forcing of IGWs. Based on the wave equation, we then identified what was responsible for the excitation of IGWs in a real case of convection simulated by ARPS. Following this introduction, the non-hydrostatic wave equation is presented in section 2. The high-resolution simulation of the convection case is addressed in section 3. The forcing of IGWs in the convection case is explored in section 4. The generation mechanism of IGWs is discussed in section 5. Conclusions are given in section 6.

A diabatic and inviscid flow is considered. The non-hydrostatic equations under Boussinesq approximation on an f plane in Cartesian coordinates are used (Xue et al., 2001): \begin{eqnarray} &&\frac{\partial u}{\partial t}+V\cdot{\pmb{\nabla}}u-fv=-\frac{\partial}{\partial x}\left(\frac{{p}'}{\bar{\rho}}\right) ,(1)\\ \label{eq1} &&\frac{\partial v}{\partial t}+V\cdot{\pmb{\nabla}}v+fu=-\frac{\partial}{\partial y}\left(\frac{{p}'}{\bar{\rho}}\right) ,(2)\\ \label{eq2} &&\frac{\partial w}{\partial t}+V\cdot{\pmb{\nabla}}w=-\frac{1}{\bar{\rho}}\frac{\partial{p}'}{\partial z}+\frac{{\theta}'}{\bar{\theta}}g ,(3)\\ &&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0 ,(4)\\ &&\frac{\partial\theta}{\partial t}+V\cdot{\pmb{\nabla}}\theta=Q , (5)\end{eqnarray} where v=(u,v,w) is the three-dimensional velocity vector; f is the Coriolis constant; p is the pressure; ρ is the density; g is gravitational acceleration; θ is the potential temperature; Q is the diabatic heating; \(\pmb\nabla=\partial/\partial xi+\partial/\partial yj+\partial/\partial zk\) is the three-dimensional gradient operator; and i, j and k are the unit vectors in the x,y and z directions, respectively. An overbar denotes an average field and a prime denotes a perturbation. The average density is subjected to the hydrostatic balance and static equation; namely, \begin{eqnarray} \bar{\rho}&=&-\dfrac{1}{g}\dfrac{\partial\bar{p}}{\partial z} ,(6)\\ \overline{T}&=&\dfrac{\bar{p}}{\bar{\rho}R} , (7)\end{eqnarray} where R is the gas constant. The average potential temperature is given by \begin{equation} \label{eq3} \bar{\theta}=\overline{T}\left(\frac{p_{\rm s}}{\bar{p}}\right)^{\frac{R}{c_p}} , (8)\end{equation} where p _s is the reference surface pressure and c_p is the specific heat at constant pressure. Therefore, as long as the average pressure is known, the averages of density and potential temperature are derived from Eqs. (6)-(8).

The above equations are employed to identify the form of the IGW equation with forcing terms. The scale separation of flow is an important problem. For arbitrary average fields, a wave equation for IGWs is given by Eq. (A23) (the detailed derivation can be seen in the Appendix). (Plougonven and Zhang, 2007) used the scaling argument to identify the form that the wave equation and its forcing terms should take. (Wang et al., 2010) adopted the balanced flow from PV inversion to evaluate the forcing term. The derivation in the Appendix does not involve approximation, and so Eq. (A23) is applicable for various average fields. For simplification, the temporal and horizontal domain average is adopted to separate the flow in this study. The average fields are steady and horizontally homogeneous, and just vary with height. Thus, Eq. (A23) in the Appendix is reduced to \begin{equation} \label{eq4} -{\pmb{\nabla}}^2\frac{\partial^2{w}'}{\partial t^2}-N^2{\pmb{\nabla}}_h^2{w}'-f^2\frac{\partial^2{w}'}{\partial z^2}={\rm FG1}+{\rm FG2}+{\rm FGS} , (9)\end{equation} where w' is the vertical velocity perturbation; N² is the stratification stability parameter; and \(\pmb\nabla^2\) and \(\pmb\nabla_h^2\) denote the three-dimensional and two-dimensional Laplacian operators, respectively. The three forcing terms on the right-hand side are given by \begin{eqnarray} \label{eq5} {\rm FG1}&=&-g{\pmb{\nabla}}_h^2{\rm FT1}-{\pmb{\nabla}}_h^2\frac{\partial}{\partial t}{\rm FW1}+ \frac{\partial^2}{\partial z\partial t}{\rm FD1}+f\frac{\partial}{\partial z}{\rm FV1} ,(10)\\ \label{eq6} {\rm FG2}&=&-g{\pmb{\nabla}}_h^2{\rm FT2}-{\pmb{\nabla}}_h^2\frac{\partial}{\partial t}{\rm FW2}+ \frac{\partial^2}{\partial z\partial t}{\rm FD2}+f\frac{\partial}{\partial z}{\rm FV2} ,\qquad (11)\\ {\rm FGS}&=&-g{\pmb{\nabla}}_h^2{\rm FTS} , (12)\end{eqnarray} where \begin{eqnarray} \label{eq7} {\rm FD1}&=&\bar{V}_h\cdot{\pmb{\nabla}}\left(\frac{\partial{w}'}{\partial z}\right)- \left(\frac{\partial{w}'}{\partial x}\frac{\partial\bar{u}}{\partial z}+\frac{\partial{w}'}{\partial y}\frac{\partial\bar{v}}{\partial z}\right) ,(13)\\ \label{eq8} {\rm FD2}&=&V'\cdot{\pmb{\nabla}}\left(\frac{\partial{w}'}{\partial z}\right)+ 2\left(\frac{\partial{u}'}{\partial x}\frac{\partial{v}'}{\partial y}-\frac{\partial{v}'}{\partial x}\frac{\partial{u}'}{\partial y}\right)-\nonumber\\ &&\left(\frac{\partial{w}'}{\partial x}\frac{\partial{u}'}{\partial z}+\frac{\partial{w}'}{\partial y}\frac{\partial{v}'}{\partial z}\right)- \left(\frac{\partial{w}'}{\partial z}\right)^2 , (14)\end{eqnarray} \begin{eqnarray} \label{eq9} {\rm FV1}&=&-\bar{V}_h\cdot{\pmb{\nabla}}{\zeta}'-\frac{\partial{w}'}{\partial x}\frac{\partial\bar{v}}{\partial z}+ \frac{\partial{w}'}{\partial y}\frac{\partial\bar{u}}{\partial z} ,(15)\\ \label{eq10} {\rm FV2}&=&-{V}'\cdot{\pmb{\nabla}}{\zeta}'+{\zeta}'\frac{\partial{w}'}{\partial z}-\frac{\partial{w}'}{\partial x}\frac{\partial{v}'}{\partial z}+ \frac{\partial{w}'}{\partial y}\frac{\partial{u}'}{\partial z} ,(16)\\ \label{eq11} {\rm FW1}&=&-\bar{V}_h\cdot{\pmb{\nabla}}{w}'-\frac{{p}'}{\bar{\rho}^2}\frac{\partial\bar{\rho}}{\partial z} ,(17)\\ \label{eq12} {\rm FW2}&=&-{V}'\cdot{\pmb{\nabla}}{w}' ,(18)\\ \label{eq13} {\rm FT1}&=&-\bar{V}_h\cdot{\pmb{\nabla}}\frac{{\theta}'}{\bar{\theta}} ,(19)\\ \label{eq14} {\rm FT2}&=&-\frac{{\theta}'}{\bar{\theta}}{w}'\frac{\partial}{\partial z}(\ln\bar{\theta}) -{V}'\cdot{\pmb{\nabla}}\left(\frac{{\theta}'}{\bar{\theta}}\right) ,(20)\\ \label{eq15} {\rm FTS}&=&\frac{Q}{\bar{\theta}} .(21) \end{eqnarray} The wave equation, Eq. (5), is similar in form to Eq. (10) in (Plougonven and Zhang, 2007). A linear wave operator associated with the amplitude of the IGW is retained on the lhs, and the forcing due to linear and nonlinear perturbations and diabatic heating is on the rhs. For an adiabatic flow with stationary background winds, the equation yields a dispersion relation of standard IGWs after ignoring the quadratic terms in perturbation fields. The rhs forcing is grouped into three categories: (1) The linear forcing (FG1), which is associated with FD1, FV1, FW1 and FT1, representing the linear forcing terms in perturbation fields in the equations of divergence, vertical vorticity, vertical velocity and potential temperature, respectively. (2) The nonlinear forcing (FG2), which is associated with FD2, FV2, FW2 and FT2, representing the quadratic forcing terms in perturbation fields in the equations of divergence, vertical vorticity, vertical velocity and potential temperature, respectively. (3) The diabatic forcing (FGS), which is associated with the diabatic heat that is generally composed of latent heating and radiation heating in a numerical model. The wave equation, Eq. (6), indicates linear and nonlinear forcing and diabatic forcing are responsible for sources of gravity waves imposed on a steady, horizontal uniform flow. (Song et al., 2003) obtained a two-dimensional wave equation that is different from the wave equation, Eq. (6). Due to the anelastic approximation and symmetry in the meridional direction, the wave equation in (Song et al., 2003), Eq. (11), involved the nonlinear forcing and diabatic forcing except the linear forcing.

The linear forcing FG1 involves the Laplacian of potential-temperature linear forcing, the Laplacian of local change in vertical-velocity linear forcing, the vertical gradient of local change in divergence linear forcing, and the vertical gradient of vorticity linear forcing. The nonlinear forcing FG2 is composed of the Laplacian of potential-temperature nonlinear forcing, the Laplacian of local change in vertical-velocity nonlinear forcing, the vertical gradient of local change in divergence nonlinear forcing, and the vertical gradient of vorticity nonlinear forcing. The Laplacian of diabatic heating constitutes the diabatic forcing FGS.

There are two differences between Eq. (5) and Eq. (10) of (Plougonven and Zhang, 2007). The first is that the wave equation, Eq. (5), is built in a non-hydrostatic dynamic framework and suitable to mesoscale convection systems, like squall lines. The second is that for the steady and horizontal homogeneous basic state, the basic-state forcing only composed of average fields disappears in the wave equation, Eq. (5). The source of IGWs is dominated by the linear and nonlinear forcing and diabatic forcing. For the basic state given by a running average, the wave equation, Eq. (A23), in the Appendix, involves the forcing term FGO that is only composed of averaged fields. It presents the large-scale forcing, composed of temporal and vertical derivatives of the residual of the nonlinear balance equation, the vertical gradient of the residual of the vorticity equation, and the Laplacian of the residual of the potential temperature equation.

The three kinds of forcing on the rhs of Eq. (5) can be applied to determine the source of gravity waves in a real atmosphere. Since the wave equation is built in the framework of non-hydrostatic and ageostrophic dynamics, it is applicable to high-impact mesoscale weather. Therefore, as reported in the following section, the three kinds of forcing were adopted to examine the source of IGWs in a convection case.

The method of analyzing the source of IGWs, based on Eq. (5), was applied to a severe convection event that occurred on 9 June 2009. The convection case, presenting the features of a squall line, was simulated with a horizontal resolution of 1 km, and discussed by (Liu et al., 2015). It was found (Fig. 1) that the observed reflectivity presented a band shape, and moved southward. The simulated reflectivity showed a slightly broader and stronger band, and spread southward. The simulated reflectivity center and movement were consistent with the observations. To a degree, the simulation reproduced the evolution of the severe convection case. As reported in the following section, the simulation data used by (Liu et al., 2015) were employed to examine the source of IGWs in the convection case.

Figure 1.  The observed radar composite reflectivity (units: dB$Z$) at (a) 0830 UTC, (c) 1030 UTC and (e) 1300 UTC, and the simulation at (b) 0830 UTC, (d) 1030UTC and (f) 1300 UTC 14 June 2009. The black solid line indicates the location of the cross section referred to below. The figure is reproduced from Liu et al. (2015).

Figure 2.  Cross sections of (a) vertical velocity (color-shaded; units: m s$^-1$), (b) diabatic heating from cloud microphysical processes (color-shaded; units: 10$^-5$ K s$^-1$) and stream line, (c) divergence perturbation (color-shaded; units: 10$^-5$ s$^-1$) and (d) vorticity perturbation (color-shaded; units: 10$^-4$ s$^-1$) along 119$^\circ$E at 1030 UTC 14 June 2009, where the contour line denotes hydrometeors (units: 10$^-4$ g g$^-1$), and the green line denotes the precipitation rate (units: 10$^-3$ mm h$^-1$).

Figure 3.  As in Fig. 2, except at 1230 UTC 14 June 2009.

Figure 4.  Temporal variation of (a) vertical velocity (units: m s$^-1$), (b) divergence perturbation (units: 10$^-5$ s$^-1$), (c) vorticity perturbation (units: 10$^-4$ s$^-1$) and (d) liquid hydrometeors (units: 10$^-4$ g g$^-1$) at the altitude of 1.75 km along 119$^\circ$E during 0800-1300 UTC 14 June 2009.

Figure 5.  Temporal-vertical cross sections of (a) total forcing, (b) diabatic forcing, (c) linear forcing and (d) nonlinear forcing (units: 10$^-12$ m$^-1$ s$^-3$) at (32.25$^\circ$N, 119$^\circ$E).

In the cross section (119^°E) along the intensive reflectivity (Fig. 2) in the mature stage of the convection event, the vertical velocity fluctuated in the convection region (32^°-32.5^°N). The flow ascended in the heating area and descended in the cooling area. Both presented an undulating pattern. The divergence and vorticity perturbations fluctuated sharply. In the decaying stage (Fig. 3), the vertical velocity, diabatic heating, divergence and vorticity perturbations possessed fluctuating characteristics in the convection region (31^°-31.7^°N). Their intensity was weaker than that in the mature stage. In the temporal-meridional cross sections, the vertical velocity, vorticity and divergence perturbations, and liquid hydrometeors, were characterized by fluctuation and propagated southwards (Fig. 4). (Wang et al., 2010) suggested that the fluctuation in the convection region is associated with IGWs. Next, we focused on the forcing of IGWs.

The three forcing terms on the rhs of Eq. (5) were calculated to analyze the source of IGWs. The total forcing was prominent at 1030 UTC (mature stage) (Fig. 5) and 1230 UTC (decaying stage) (Fig. 6) 14 June 2009. In the mature stage, the diabatic forcing served as the dominant forcing of waves in the middle and lower troposphere. The linear forcing took place in the middle and upper levels. The nonlinear forcing was intense throughout the whole troposphere. The comparison showed the upper-level total forcing was determined by the linear forcing. The middle-level and lower-level total forcing was dominated by the diabatic forcing. On the other hand, the nonlinear forcing was offset by the linear forcing in the upper level, and by diabatic forcing in the middle and lower levels. In the decaying stage (Fig. 6), the total forcing had a complicated structure. The total forcing derived mainly from the diabatic forcing. The linear forcing exerted an impact in the middle and upper levels. The nonlinear forcing became strong near the ground, but was offset by the diabatic forcing.

Figure 6.  As in Fig.5, except at (31.25$^\circ$N, 119$^\circ$E).

Figure 7.  Vertical cross sections of (a) total forcing, (b) diabatic forcing, (c) linear forcing and (d) nonlinear forcing (color-shaded; units: 10$^-12$ m$^-1$ s$^-3$) along 119$^\circ$E at 1230 UTC 14 June 2009, where the contour line denotes hydrometeors (units: 10$^-4$ g g$^-1$) and the green line denotes the precipitation rate (units: 10$^-3$ mm h$^-1$).

The vertical-meridional cross section of wave forcing at 1230 UTC 14 June 2009 is presented in Fig. 7. The total forcing was confined to the fully developed convection area (31.1^°-31.5^°N) throughout the whole troposphere. The diabatic forcing appeared within the inner area of the developed convection, with intense magnitude in the middle and lower levels. The high values of linear forcing were located in middle and upper levels. The upper-level and lower-level nonlinear forcings were dominant. It is inferred that the upper-level total forcing mainly resulted from the linear and nonlinear forcing, and the diabatic and nonlinear forcings were the chief contributors to the lower-level total forcing.

The forcing in three specific levels was examined (Fig. 8). Near the ground (Fig. 8a), the forcing of waves derived mainly from the nonlinear forcing (31^°-31.2^°N) and diabatic forcing (31.35^°-31.45^°N). This might have been caused by momentum and thermal flux due to turbulence and latent heating release, due to water vapor phase change in the boundary layer. In the middle level (Fig. 8b), the diabatic forcing played important roles in the forcing of waves. The reason was that the microphysical processes releasing heat, such as the condensation of water vapor and the melting of snow and hail, took place in the middle level. The linear forcing in the belt of 31.5^°-31.25^°N made a contribution in the middle level. In the upper level (Fig. 8c), the linear forcing was highly influential.

Figure 8.  The total forcing (red line), diabatic forcing (black line), linear forcing (green line) and nonlinear forcing (blue line) (units: 10$^-12$ m$^-1$ s$^-3$) at the altitudes of (a) 0.75 km, (b) 5.75 km and (c)12.75 km, along 119$^\circ$E at 1230 UTC 14 June 2009.

The 3-moment bulk microphysics parameterization used in the simulation involves 28 kinds of microphysical processes that release or absorb heat (Xue et al., 2001). The latent forcing from six microphysical processes with large magnitude is shown in Fig. 9. The chief latent forcing resulted from moisture condensation to cloud water in the lower and middle levels in the strong convection region (Fig. 9d). This was because the moisture condensation released the greatest amount of latent heating. The latent forcing due to the rest of the microphysical processes was secondary. For example, hail melting (Fig. 9c) and hail collection of rain (Fig. 9f) produced latent forcing in the lower level. The latent forcing of snow sublimation (Fig. 9b) and snow deposition (Fig. 9e) appeared in the upper level, and that of cloud evaporation (Fig. 9a) lay in the middle level. These distribution patterns were associated with the location where the microphysical processes took place. The above-mentioned diabatic forcing reflected the direct influence of latent heating due to water phase change on the source of IGWs. On the other hand, numerical experiments revealed that latent heating also changed dynamic and thermodynamic fields, and implemented an indirect excitation of IGWs (not included here).

For the linear forcing (FG1) (Fig. 10), the Laplacian of potential-temperature linear forcing, presenting a thermodynamic linear forcing, was the dominant component. It almost appeared throughout the whole troposphere in the strong convection region. The latent heating produced intense perturbation of potential temperature in the convection region, leading to thermodynamic discontinuity and baroclinicity with a large horizontal gradient. The perturbation of potential temperature appeared in the discontinuity region, and resulted in the intense linear forcing of waves. Other components of linear forcing were smaller than the thermodynamic linear forcing component. For the nonlinear forcing quadratic terms in the perturbation fields (Fig. 11), the primary component came from the Laplacian of potential-temperature nonlinear forcing in the convection region. This indicates that the vertical component of thermal flux and the divergence of thermal flux were important for the excitation of IGWs in the upper and lower levels. We concluded that, although the linear, nonlinear and diabatic forcings played roles in exciting gravity waves, they also exerted an influence on the different positions of the convection.

Figure 9.  Vertical cross sections of diabatic forcing components due to the (a) cloud evaporation rate, (b) snow sublimation rate, (c) hail melting rate, (d) cloud condensation rate, (e) snow deposition rate and (f) rain-to-hail collection rate (color-shaded; units: 10$^-12$ m$^-1$ s$^-3$), along 119$^\circ$E at 1230 UTC 14 June 2009, where the contour line denotes hydrometeors (units: 10$^-4$ g g$^-1$) and the green line denotes the precipitation rate (units: 10$^-3$ mm h$^-1$).

Figure 10.  Vertical cross sections of linear forcing components along 119$^\circ$E at 1230 UTC 14 June 2009, where the contour line denotes hydrometeors (units: 10$^-4$ g g$^-1$) and the green line denotes the precipitation rate (units: 10$^-3$ mm h$^-1$): (a) $-g\pmb\nabla_h^2\rm FT1$, (b) $(\partial^2/\partial z\partial t)\rm FD1$, (c) $-\pmb\nabla_h^2(\partial/\partial t)\rm FW1$ and (d) $f(\partial/\partial z)\rm FV1$ (color-shaded; units: 10$^-12$ m$^-1$ s$^-3$).

Figure 11.  Vertical cross sections of nonlinear forcing components along 119$^\circ$E at 1230 UTC 14 June 2009, where the contour line denotes hydrometeors (units: 10$^-4$ g g$^-1$) and the green line denotes the precipitation rate (units: 10$^-3$ mm h$^-1$): (a) $-g\pmb\nabla_h^2\rm FT2$, (b) $-\pmb\nabla_h^2(\partial/\partial t)\rm FW2$, (c) $(\partial^2/\partial z\partial t)\rm FD2$ and (d) $f(\partial/\partial z)\rm FV2$ (color-shaded; units: 10$^-12$ m$^-1$ s$^-3$).

Figure 12.  Temporal variations of (a) total forcing, (b) diabatic heating forcing, (c) linear forcing and (d) nonlinear forcing (units: 10$^-12$ m$^-1$ s$^-3$), along 119$^\circ$E at the altitude of 1.75 km.

Figure 13.  As in Fig.12, except at the altitude of 12.75 km.

From the perspective of temporal variation (Fig. 12), it was found that in the lower level (altitude of 1.75 km), the total forcing of gravity waves followed the convection to move southwards during 0930-1300 UTC 14 June 2009. The diabatic forcing had a magnitude larger than the other forcings, and possessed a pattern similar to the total forcing. So, the source of gravity waves came mostly from the diabatic forcing. This is because, in the lower and middle levels, there was much latent heating due to moisture condensation. In the upper level (Fig. 13), the intense total forcing took place during the period of 0930-1100 UTC, in which the convection was located in the mature stage. The primary forcing of IGWs turned to the linear and nonlinear forcing. Relatively, the diabatic forcing played an unimportant role because of the lack of latent heating due to moisture condensation in the upper level. In the decaying stage (1200-1250 UTC), because of weakening microphysical processes, the heating forcing became negligible. The chief total forcing resulted from the linear forcing. This was because, generally, the upper-level flow was imbalanced near a jet stream. To reach a new equilibrium state, the geostrophic adjustment process took place to produce gravity waves and spread energy far away. On the other hand, the average flow was strong in the upper level, such as the jet stream at 200 hPa. This resulted in the intense advection of potential temperature perturbation. Consequently, the Laplacian of potential-temperature linear forcing became dominant.

In short, in the middle and lower levels, the diabatic forcing due to latent heating from moisture condensation appeared to be the primary source of gravity waves. In the upper level, the primary source was composed of linear and nonlinear forcing, while the heating forcing contributed little due to the lack of latent heating. In the linear and nonlinear forcing of waves, the Laplacian of potential-temperature linear and nonlinear forcing was the chief component. The inference was that the thermodynamic forcing dominated the source of gravity waves in the convection region.

The above analysis showed that the diabatic forcing, linear and nonlinear forcing associated with potential temperature perturbation were the primary sources of IGWs in the convection region of the case examined here. This could be explained through the vertical motion equation, Eq. (3), and the thermodynamic equation, Eq. (A8). It can be seen from Eq. (3) that, for gravity waves, the individual change in vertical velocity is driven by the vertical component of the pressure gradient force and buoyancy. As a chief restoring force, the buoyancy in Eq. (29) is dominated by the diabatic heating and potential-temperature linear and nonlinear forcing. Therefore, the vertical velocity is indirectly influenced by these thermodynamic elements. These were the chief sources of gravity waves in the present case.

Song et al. (2013) compared the relative importance of nonlinear forcing and diabatic forcing through three quasilinear dry numerical experiments. The results revealed that the magnitude of the nonlinear forcing was two to three times larger than that of the diabatic forcing. The nonlinear sources were inefficient in generating linear gravity waves that could propagate vertically into the stratosphere, due to the condition of the basic-state wind and its vertical shear. In the present study, the source of waves came from the linear and nonlinear forcing and diabatic forcing, in a real convection case. The nonlinear forcing was intense in the upper and lower levels, and a little smaller in magnitude than the linear forcing and diabatic forcing, except near the ground. The nonlinear forcing was out of phase with the linear forcing and diabatic forcing, and was canceled out by the latter. The influence of linear forcing reached the stratosphere, and the nonlinear forcing was confined to the troposphere. This indicated that the nonlinear forcing might not generate waves propagating into the stratosphere. To a degree, this is consistent with Song et al. (2013). However, a quantitative investigation should be conducted through numerical experiments. This is planned in future work.

Topography is an important stationary source of gravity waves, and plays a significant role in the excitation and maintenance of gravity waves (Koch et al., 2001). (Smith, 1979) presented a theory of two-dimensional mountain waves, and observations of mountain waves. He also discussed the effect of inertia on the flow over mesoscale mountains, and theories of lee cyclogenesis. Since there was no large-scale topography in the convection region investigated in this study, the influence of topography was excluded from the analysis of the source of gravity waves.

Considering the importance of IGWs in mesoscale convection systems, a non-hydrostatic wave equation under Boussinesq approximation in Cartesian coordinates was derived to investigate the source of IGWs. The approach similar to (Ford, 1994a) and (Plougonven and Zhang, 2007) was adopted in the derivation. The temporal and horizontal domain average was used to separate the flow. The wave equation is characterized by a linear wave operator on the lhs, and forcing involving linear and nonlinear terms and diabatic heating on the rhs. The forcing is suitable for diagnosing the source of gravity waves.

The wave equation was applied to a real case of convection that occurred in East China on 14 June 2009. The case was simulated using ARPS with a horizontal resolution of 1 km. It was found that the fields of vertical velocity, divergence and vorticity perturbations, and liquid water, fluctuated in the severe convection region, presenting a wave feature. The diabatic, linear and nonlinear forcing in the wave equation presented large magnitude at different altitudes in the severe convection region. The strongest forcing of waves came from the latent forcing due to latent heating of water vapor condensation in the middle and lower levels. The linear forcing was located in the middle and upper troposphere in the severe convection region, mostly because the average flow was strong in upper level, such as the jet stream at 200 hPa. The nonlinear forcing appeared throughout the whole troposphere. In particular, near the surface, intense nonlinear forcing was associated with turbulence in the boundary layer.

In the severe convection region, the latent heating due to condensation of vapor to cloud water was the most important among the microphysical processes. So, the diabatic forcing from condensational latent heating played a role in the generation of gravity waves. Among the components of linear and nonlinear forcing, the Laplacian of potential-temperature linear and nonlinear forcing were the chief forcing sources of gravity waves. This can be explained as follows: For gravity waves, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles is dominated by the buoyancy. On the other hand, the diabatic heating, potential-temperature linear and nonlinear forcing determine the tendency of buoyancy. Subsequently, they indirectly influence the evolution of gravity waves.

Derivation of the Non-hydrostatic Wave Equation

The approach proposed by (Ford, 1994a) was employed to derive the non-hydrostatic wave equation. The divergence equation can be obtained by taking [(∂/∂ x)(1)+(∂/∂ y)(2)] \begin{eqnarray} -\dfrac{\partial^2w}{\partial t\partial z}-V\cdot{\pmb{\nabla}}\left(\dfrac{\partial w}{\partial z}\right) &=&\zeta f-{\pmb{\nabla}}_h^2\left(\dfrac{{p}'}{\bar{\rho}}\right)-\left(\frac{\partial w}{\partial z}\right)^2+\nonumber\\ &&2\left(\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial v}{\partial x}\frac{\partial u}{\partial y}\right) \!-\!\left(\frac{\partial w}{\partial x}\frac{\partial u}{\partial z}+\frac{\partial w}{\partial y}\frac{\partial v}{\partial z}\right) ,(A1)\nonumber\\ \end{eqnarray} where (∂ u/∂ x)+(∂ v/∂ y)=-(∂ w/∂ y) is used. Taking [(∂/∂ x)(2)-(∂/∂ y)(1)] yields the vorticity equation \begin{eqnarray} \frac{\partial\zeta}{\partial t}+V\cdot{\pmb{\nabla}}\zeta=({\zeta +f})\frac{\partial w}{\partial z}-\frac{\partial w}{\partial x}\frac{\partial v}{\partial z}+ \frac{\partial w}{\partial y}\frac{\partial u}{\partial z} , (A2)\end{eqnarray} where ζ=(∂ v/∂ x)(∂ u/∂ y) is the vertical component of relative vorticity.

Assume the flow is separated into two portions; namely, \begin{equation} \left( \begin{array}{c} u \\ v \\ w \\ \zeta \\ \theta \end{array} \right)=\left( \begin{array}{c} \bar{u}+{u}' \\ \bar{v}+{v}' \\ {w}' \\ \bar{\zeta}+{\zeta}' \\ \bar{\theta}+{\theta}' \end{array} \right) (A3)\end{equation} where the overbar denotes the average field and the prime denotes the perturbation. Note that the average field of vertical velocity is supposed to be zero (Plougonven and Zhang, 2007). By using Eq. (A3), one can rewrite Eqs. (A1), (A2), (3), (4) and (5) as follows: \begin{eqnarray} -\frac{\partial}{\partial t}\left(\frac{\partial{w}'}{\partial z}\right)-{\zeta}'f+{\pmb{\nabla}}_h^2\left(\frac{{p}'}{\bar{\rho}}\right)&=&{\rm FD0}+{\rm FD1}+{\rm FD2} ,(A4)\\ \frac{\partial{\zeta}'}{\partial t}-f\frac{\partial{w}'}{\partial z}&=&{\rm FV0}+{\rm FV1}+{\rm FV2} ,(A5)\\ \frac{\partial{w}'}{\partial t}+\frac{\partial}{\partial z}\left(\frac{{p}'}{\bar{\rho}}\right)-\frac{{\theta}'}{\bar{\theta}}g&=&{\rm FW0}+{\rm FW1}+{\rm FW2} ,(A6)\qquad\\ \frac{\partial u'}{\partial x}+\frac{\partial v'}{\partial y}+\frac{\partial w'}{\partial z}&=&0 ,(A7)\\ \frac{\partial}{\partial t}\left(\frac{{\theta}'}{\bar{\theta}}\right)+\frac{N^2}{g}{w}'&=&{\rm FT0}+{\rm FT1}+{\rm FT2} ,(A8) \end{eqnarray} where the different groups of rhs terms are \begin{eqnarray} {\rm FD0}&=&\bar{\zeta}f+2\left(\frac{\partial\bar{u}}{\partial x}\frac{\partial\bar{v}}{\partial y}- \frac{\partial\bar{v}}{\partial x}\frac{\partial\bar{u}}{\partial y}\right) ,(A9)\\ {\rm FD1}&=&\bar{V}_h\cdot{\pmb{\nabla}}\left(\frac{\partial{w}'}{\partial z}\right)+2\Bigg(\frac{\partial\bar{u}}{\partial x}\frac{\partial{v}'}{\partial y}- \frac{\partial{v}'}{\partial x}\frac{\partial\bar{u}}{\partial y}+\nonumber\\ &&\frac{\partial{u}'}{\partial x}\frac{\partial\bar{v}}{\partial y}- \frac{\partial\bar{v}}{\partial x}\frac{\partial{u}'}{\partial y}\Bigg)-\left(\frac{\partial{w}'}{\partial x}\frac{\partial\bar{u}}{\partial z}+ \frac{\partial{w}'}{\partial y}\frac{\partial\bar{v}}{\partial z}\right) ,(A10)\\ {\rm FD2}&=&V'\cdot{\pmb{\nabla}}\left(\frac{\partial{w}'}{\partial z}\right)+2\left(\frac{\partial{u}'}{\partial x}\frac{\partial{v}'}{\partial y}- \frac{\partial{v}'}{\partial x}\frac{\partial{u}'}{\partial y}\right)-\nonumber\\ &&\left(\frac{\partial{w}'}{\partial x}\frac{\partial{u}'}{\partial z}+ \frac{\partial{w}'}{\partial y}\frac{\partial{v}'}{\partial z}\right)-\left(\frac{\partial{w}'}{\partial z}\right)^2 ,(A11)\\ {\rm FV0}&=&-\dfrac{\partial\bar{\zeta}}{\partial t}-\bar{V}_h\cdot{\pmb{\nabla}}\bar{\zeta} ,(A12)\\ {\rm FV1}&=&-\bar{V}_h\cdot{\pmb{\nabla}}{\zeta}'-{V}'\cdot{\pmb{\nabla}}\bar{\zeta}+\bar{\zeta}\left(\frac{\partial{w}'}{\partial z}\right)- \frac{\partial{w}'}{\partial x}\frac{\partial\bar{v}}{\partial z}+\frac{\partial{w}'}{\partial y}\frac{\partial\bar{u}}{\partial z} ,\nonumber (A13)\\ \\ {\rm FV2}&=&-{V}'\cdot{\pmb{\nabla}}{\zeta}'+{\zeta}'\frac{\partial{w}'}{\partial z}-\frac{\partial{w}'}{\partial x}\frac{\partial{v}'}{\partial z}+ \frac{\partial{w}'}{\partial y}\frac{\partial{u}'}{\partial z} ,(A14)\\ {\rm FW1}&=&-\bar{V}_h\cdot{\pmb{\nabla}}{w}'-\frac{{p}'}{\bar{\rho}^2}\frac{\partial\bar{\rho}}{\partial z} ,(A15)\\ {\rm FW2}&=&-{V}'\cdot{\pmb{\nabla}}{w}' ,(A16)\\ {\rm FT0}&=&-\frac{\partial\ln\bar{\theta}}{\partial t}-\bar{V}_h\cdot{\pmb{\nabla}}\ln\bar{\theta} ,(A17)\\ {\rm FT1}&=&-\left(\frac{\partial\ln\bar{\theta}}{\partial t}+\bar{V}_h\cdot{\pmb{\nabla}}\ln\bar{\theta}\right)\frac{{\theta}'}{\bar{\theta}}- {V}'_h\cdot{\pmb{\nabla}}\ln\bar{\theta}-\bar{V}_h\cdot{\pmb{\nabla}}\frac{{\theta}'}{\bar{\theta}} ,\nonumber (A18)\\ \\ {\rm FT2}&=&-\frac{{\theta}'}{\bar{\theta}}{V}'\cdot{\pmb{\nabla}}(\ln\bar{\theta})-{V}'\cdot{\pmb{\nabla}}\frac{{\theta}'}{\bar{\theta}} ,(A19)\\ {\rm FTS}&=&\frac{Q}{\bar{\theta}} .(A20) \end{eqnarray} The lhs terms in Eqs. (A4)-(A8) are linear in perturbation fields, which can build a wave operator. The rhs terms represent dynamic and thermodynamic forcing on the gravity waves. They are grouped into three categories: the first is composed by the averaged fields, denoted by F*0; the second is linear in perturbation fields, denoted by F*1; and the third is quadratic in perturbation fields, denoted by F*2. The wildcards "*" present D,V,W,T, respectively.

By taking (∂/∂ z)[(∂/∂ t)( A4)+f·( A5)], one has \begin{eqnarray} &&-\frac{\partial^4{w}'}{\partial z^2\partial t^2}+{\pmb{\nabla}}_h^2\left[\frac{\partial^2}{\partial z\partial t}\left(\frac{{p}'}{\bar{\rho}}\right)\right]- f^2\frac{\partial^2{w}'}{\partial z^2}\nonumber\\ &&=\frac{\partial^2}{\partial z\partial t}{\rm FD0}+f\frac{\partial}{\partial z}{\rm FV0} +\frac{\partial^2}{\partial z\partial t}{\rm FD1}+\nonumber\\ &&f\frac{\partial}{\partial z}{\rm FV1}+\frac{\partial^2}{\partial z\partial t}{\rm FD2}+f\frac{\partial}{\partial z}{\rm FV2} . (A21)\end{eqnarray} After taking \([(\rm A22)-(\pmb\nabla_h^2(\partial/\partial t)(\rm A6)+g\cdot\pmb\nabla_h^2(\rm A8))]\), the wave equation is given by \begin{eqnarray} -{\pmb{\nabla}}^2\frac{\partial^2{w}'}{\partial t^2}-N^2{\pmb{\nabla}}_h^2{w}'-f^2\frac{\partial^2{w}'}{\partial z^2}={\rm FG0}+{\rm FG1}+{\rm FG2}+{\rm FGS} ,\nonumber (A22)\\ \end{eqnarray} where \begin{eqnarray} {\rm FG0}&=&-g{\pmb{\nabla}}_h^2{\rm FT0}+\frac{\partial^2}{\partial z\partial t}{\rm FD0}+f\frac{\partial}{\partial z}{\rm FV0} ,(A23)\\ {\rm FG1}&=&-g{\pmb{\nabla}}_h^2{\rm FT1}-{\pmb{\nabla}}_h^2\dfrac{\partial}{\partial t}{\rm FW1}+ \dfrac{\partial^2}{\partial z\partial t}{\rm FD1}+f\frac{\partial}{\partial z}{\rm FV1} ,(A24) \nonumber\\ \\ {\rm FG2}&=&-g{\pmb{\nabla}}_h^2{\rm FT2}-{\pmb{\nabla}}_h^2\frac{\partial}{\partial t}{\rm FW2}+ \frac{\partial^2}{\partial z\partial t}{\rm FD2}+f\frac{\partial}{\partial z}{\rm FV2} ,(A25)\nonumber\\ \\ {\rm FGS}&=&-g{\pmb{\nabla}}_h^2{\rm FTS} .(A26) \end{eqnarray} The lhs terms in Eq. (A23) are linear in the perturbation field of vertical velocity, interpreted as the gravity wave operator. The rhs terms, involving the average state, linear and nonlinear terms and diabatic heating, represent the forcing of gravity waves. The wave equation, Eq. (A23), is analogous to those of previous studies (Ford, 1994a; Griffiths and Reeder, 1996; Plougonven and Zhang, 2007). Note that the average is not specific, Eq. (A23) is applicable for various separations of flow. On the other hand, Eq. (A23) is built in three-dimensional coordinates with Boussinesq approximation. When anelastic approximation is adopted, two-dimensional coordinates are needed to derive a wave equation (Song et al., 2003).