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

doi: 10.1007/s00376-016-5292-y

  • The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms——linear and nonlinear terms, and diabatic heating——on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.
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    Koch S. E., P. B. Dorian, 1988: A mesoscale gravity wave event observed during CCOPE. Part III: Wave environment and probable source mechanisms. Mon. Wea. Rev., 116, 2570- 2592.10.1175/1520-0493(1988)1162.0.CO; This paper presents the results of a very detailed investigation into the effects of preexisting gravity waves upon convective systems, as well as the feedback effects of convection of varying intensity upon the waves. The analysis is based on the synthesis of synoptic surface and barograph data with high-resolution surface mesonetwork, radar, and satellite data collected during a gravity wave event described by Koch and Golus in Part I of this series of papers. Use is also made of the synoptic barograph data and satellite imagery to trace the waves beyond the mesonetwork and thus determine their apparent source region just upstream of the mesonetwork. It is shown that two of the gravity waves modulated convection within a weak squall line as they propagated across the line. The other six waves remained closely linked with convective systems that they appeared to trigger. However, it is shown that the waves were not excited by convection. Furthermore, the waves retained their signatures in the surface mesonetwork fields in the presence of rainshowers. Two episodes of strongest gravity wave activity are identified, each of which consisted of a packet of four wave troughs and ridges displaying wavelengths of 150 km. A Mesoscale Convective Complex (MCC) forms rapidly from very strong or severe thunderstorms apparently triggered by the individual members of the second wave packet. It is suggested that the large size and long duration of this complex were due in part to the periodic renewal and organization provided by this wave packet. Strong convection appears to substantially affect the gravity waves locally by augmenting the wave amplitude, reducing its wavelength, distorting the wave shape, altering the wave phase velocity, and greatly weakening the in-phase covariance between the perturbation wind and pressure ( p u *) fields. These convective effects upon the gravity waves are explained in terms of hydrostatic and nonhydrostatic pressure forces and gust front processes associated with thunderstorms. Despite the implication from these findings of the loss or obscuration of the original wave signal, the gravity wave signal remained intact just outside of the active storm cores and the entire wave-storm system exhibited outstanding spatial coherence over hundreds of kilometers. The observations are also compared to the predictions from wave-CISK theory. Although qualitative agreement is found, quantitative comparisons give rather unimpressive agreement, due in large measure to simplifications inherent to the theory.
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    Lac C., J.-P. Lafore, and J.-L. Redelsperger, 2002: Role of gravity waves in triggering deep convection during TOGA COARE. J. Atmos. Sci., 59, 1293- 1316.10.1175/1520-0469(2002)0592.0.CO; The role of gravity waves in the initiation of convection over oceanic regions during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) experiment is investigated. First, an autocorrelation method is applied to infrared temperature observations of convective events from satellite images. It reveals that new deep convective cells often occur a few hours after a previous intense event at a typical distance of a few hundred kilometers. Such fast moving modes (faster than 15 m s611) are interpreted as the trace of gravity waves excited by previous convection and contributing to trigger further convection. Second, the specific case of 11–12 December 1992, during which an active squall line is generated after the collapse of a previous mesoscale convective system (MCS) nearby, is analyzed with a nonhydrostatic model. The triggering of the second MCS is well reproduced explicitly, owing to the use of the two-way interactive grid nesting. The convective source ...
    Lindzen R. S., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31, 156- 179.10.1175/1520-0469(1974)0312.0.CO; CISK (Conditional Instability of the Second Kind) is examined for internal waves where low-level convergence is due to the inviscid wave fields rather than to Ekman pumping. It is found that CISK-induced waves must give rise to mean cumulus activity (since there are no negative clouds), and it is suggested that this mean activity plays an important role in the finite-amplitude equilibration of the system. The most unstable CISK waves will be associated with very short vertical wavelengths [O(3 km)] in order to maximize (in some crude sense) subcloud convergence. Thus, the vertical scale is largely determined by the height of cloud base. The vertical scale, in turn, determines the dispersive relations between horizontal and temporal scales. It is found that there exists a wave-CISK mode which is independent of longitude, and hence independent of the mean zonal flow. Because of this independence, the period of this oscillation should form a prominent line in tropical spectra. This period turns out to be about 4.8 days which is indeed a prominent feature of tropical spectra. It is shown, due to longitudinal inhomogeneities in the tropics (such as land-sea), that the above oscillation must be accompanied by traveling disturbances whose period with respect to the ground will also be 4.8 days and whose longitudinal scales will typically be from 1000–3000 km depending on the mean zonal flow. It is further shown that the existence of the above oscillatory system has two additional implications: The above system is, itself, unstable with respect to gravity waves with horizontal scales on the order of 100–200 km. Such waves may be associated with cloud clusters. The above system leads to maximum low-level convergence (and hence, a tendency toward mean cumulus activity) in regions centered about ±6°–7° latitude, thus providing a possible explanation for the position of the ITCZ.
    Liu L., L. K. Ran, and X. G. Sun, 2015: Analysis of the structure and propagation of a simulated squall line on 14 June 2009. Adv. Atmos. Sci.,32(4), 1049-1062, doi: 10.1007/s00376-014-4100-9.10.1007/ squall line on 14 June 2009 in the provinces of Jiangsu and Anhui was well simulated using the Advanced Regional Prediction System (ARPS) model. Based on high resolution spatial and temporal data, a detailed analysis of the structural features and propagation mechanisms of the squall line was conducted. The dynamic and thermodynamic structural characteristics and their causes were analyzed in detail. Unbalanced flows were found to play a key role in initiating gravity waves during the squall line’s development. The spread and development of the gravity waves were sustained by convection in the wave-CISK process. The squall line’s propagation and development mainly relied on the combined effect of gravity waves at the midlevel and cold outflow along the gust front. New cells were continuously forced by the cold pool outflow and were enhanced and lifted by the intense upward motion. At a particular phase, the new cells merged with the updraft of the gravity waves, leading to an intense updraft that strengthened the squall line.
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    O'Sullivan, D., T. J. Dunkerton, 1995: Generation of inertia-gravity waves in a simulated life cycle of baroclinic instability. J. Atmos. Sci., 52, 3695- 3716.10.1175/1520-0469(1995)0522.0.CO; excitation and propagation of inertia-gravity waves (IGWs) generated by an unstable baroclinic wave was examined with a high-resolution 3D nonlinear numerical model. IGWs arose spontaneously as the tropospheric jetstream was distorted by baroclinic instability and strong parcel accelerations took place, primarily in the jetstream exit region of the upper troposphere. Subsequent propagation of IGWs occurred in regions of strong windspeed-in the tropospheric and stratospheric jets, and in a cutoff low formed during the baroclinic lifecycle. IGWs on the flanks of these jets were rotated inward by differential advection and subsequently absorbed by the model's hyperdiffusion. Although absorption of IGWs at the sidewalls of the jet is an artifact of the model, IGW propagation was for the most pan confined to regions with an intrinsic period shorter than the local inertial period. Only a few IGWs were able to penetrate the middle stratosphere, due to weak winds or an unfavorable alignment of wavevector with respect to the mean flow.IGWs are important both as a synoptic signal in the jetstream, which may influence subsequent tropospheric developments, and as a source of isentropic or cross-isentropic mixing in the lower stratosphere. The authors' results demonstrated for the first time numerically a significant isentropic displacement of potential vorticity isopleths due to IGWs above the tropopause. Since conditions for IGW propagation are favorable within a jet, a region of strong isentropic potential vorticity gradient, it is likely that inertia-gravity waves affect the permeability of the lower stratospheric vortex and may in some instances lead to stratosphere-troposphere exchange.
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    Sato K., 1994: A statistical study of the structure, saturation and sources of inertio-gravity waves in the lower stratosphere observed with the MU radar. J. Atmos. Terr. Phys., 56, 755- 774.10.1016/0021-9169(94), in order to examine the generation mechanisms of the IGWs, an analysis is made from two points of view: geostrophic adjustment of the westerly wind jet and the topographic effect. The meridional propagation direction of IGWs is examined in a section of latitude and altitude relative to the jet axis using ECMWF operational data. Most of the IGWs observed in the 12–18 km height region (above the ground) in winter propagate meridionally toward the jet axis, indicating that geostrophic adjustment at least just as the jet axis is not the main generation mechanism of the IGWs. On the other hand, the characteristics of intensive IGWs propagating westward relative to the background wind in the 18–22 km height region in winter are in good accord with mountain waves excited in strong westerly winds near the surface.
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    Smith R. B., 1979: The influence of mountains on the atmosphere. Advances in Geophysics, 21, 87- chapter reviews the meteorological phenomena that are associated with topography. The study of airflow past mountains is complicated by the wide range of scales that must be considered. The ratios of the mountain width to each of the natural length scales are important in determining the physical regime of the flow. This idea is emphasized in the chapter by treating the effects of boundary layers and buoyancy. The theory of two-dimensional mountain waves with the help of its governing equations is presented and the observations of mountain waves are presented. The chapter also examines the influence of the boundary layer on mountain flows and slope winds and mountain and valley winds. It considers the perturbation to the wind flow caused by a mountain of intermediate scale where the rotation of the Earth cannot be neglected. For this the flow near mesoscale and synoptic-scale mountains, quasi-geostrophic flow over a mountain, the effect of inertia on the flow over mesoscale mountains, and theories of lee cyclogenesis are discussed. Finally the chapter describes planetary-scale mountain waves; a vertically integrated model of topographically forced planetary waves; the vertical structure of planetary waves; models of stationary planetary waves allowing meridional propagation and lateral; and variation in the background wind.
    Snyder C., R. Plougonven, and D. J. Muraki, 2009: Mechanisms for spontaneous gravity wave generation within a dipole vortex. J. Atmos. Sci., 66, 3464-
    Song I.-S., H.-Y. Chun, and T. P. Lane, 2003: Generation mechanisms of convectively forced internal gravity waves and their propagation to the stratosphere. J. Atmos. Sci., 60, 1960- 1980.10.1175/1520-0469(2003)0602.0.CO; Characteristics of gravity waves induced by mesoscale convective storms and the gravity wave sources are investigated using a two-dimensional cloud-resolving numerical model. In a nonlinear moist (control) simulation, the convective system reaches a quasi-steady state after 4 h in which convective cells are periodically regenerated from a gust front updraft. In the convective storms, there are two types of wave forcing: nonlinear forcing in the form of the divergences of momentum and heat flux, and diabatic forcing. The magnitude of the nonlinear source is 2 to 3 times larger than the diabatic source, especially in the upper troposphere. Three quasi-linear dry simulations forced by the wave sources obtained from the control (CTL) simulation are performed to investigate characteristics of gravity waves induced by the various wave source mechanisms. In the three dry simulations, the magnitudes of the perturbations produced in the stratosphere are comparable, yet much larger than those in the CTL si...
    Tulich S. N., B. E. Mapes, 2008: Multiscale convective wave disturbances in the tropics: Insights from a two-dimensional cloud-resolving model. J. Atmos. Sci., 65, 140- 155.10.1175/ convective wave disturbances with structures broadly resembling observed tropical waves are found to emerge spontaneously in a nonrotating, two-dimensional cloud model forced by uniform cooling. To articulate the dynamics of these waves, model outputs are objectively analyzed in a discrete truncated space consisting of three cloud types (shallow convective, deep convective, and stratiform) and three dynamical vertical wavelength bands. Model experiments confirm that diabatic processes in deep convective and stratiform regions are essential to the formation of multiscale convective wave patterns. Specifically, upper-level heating (together with low-level cooling) serves to preferentially excite discrete horizontally propagating wave packets with roughly a full-wavelength structure in troposphere and “dry” phase speeds in the range 16–18 m s. These wave packets enhance the triggering of new deep convective cloud systems, via low-level destabilization. The new convection in turn causes additional heating over cooling, through delayed development of high-based deep convective cells with persistent stratiform anvils. This delayed forcing leads to an intensification and then widening of the low-level cold phases of wave packets as they move through convecting regions. Additional widening occurs when slower-moving (658 m s)“gust front” wave packets excited by cooling just above the boundary layer trigger additional deep convection in the vicinity of earlier convection. Shallow convection, meanwhile, provides positive forcing that reduces convective wave speeds and destroys relatively small-amplitude-sized waves. Experiments with prescribed modal wind damping establish the critical role of short vertical wavelengths in setting the equivalent depth of the waves. However, damping of deep vertical wavelengths prevents the clustering of mesoscale convective wave disturbances into larger-scale envelopes, so these circulations are important as well.
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    Wang S. G., F. Q. Zhang, 2007: Sensitivity of mesoscale gravity waves to the baroclinicity of jet-front systems. Mon. Wea. Rev., 135, 670- 688.10.1175/ study investigates the sensitivity of mesoscale gravity waves to the baroclinicity of the background jet-front systems by simulating different life cycles of baroclinic waves with a high-resolution mesoscale model. Four simulations are made starting from two-dimensional baroclinic jets having different static stability and wind shear in order to obtain baroclinic waves with significantly different growth rates. In all experiments, vertically propagating mesoscale gravity waves are simulated in the exit region of upper-tropospheric jet streaks. A two-dimensional spectral analysis demonstrates that these gravity waves have multiple components with different wave characteristics. The short-scale wave components that are preserved by a high-pass filter with a cutoff wavelength of 200 km have horizontal wavelengths of 8509“161 km and intrinsic frequencies of 309“11 times the Coriolis parameter. The medium-scale waves that are preserved by a bandpass filter (with 200- and 600-km cutoff wavelengths) have horizontal wavelengths of 25009“350 km and intrinsic frequencies less than 3 times the Coriolis parameter. The intrinsic frequencies of these gravity waves tend to increase with the growth rate of the baroclinic waves; gravity waves with similar frequency are found in the experiments with similar average baroclinic wave growth rate but with significantly different initial tropospheric static stability and tropopause geometry. The residuals of the nonlinear balance equation are used to assess the flow imbalance. In all experiments, the developing background baroclinic waves evolve from an initially balanced state to the strongly unbalanced state especially near the exit region of upper-level jet fronts before mature mesoscale gravity waves are generated. It is found that the growth rate of flow imbalance also correlates well to the growth rate of baroclinic waves and thus correlates to the frequency of gravity waves.
    Wang S. G., F. Q. Zhang, and C. Snyder, 2009: Generation and propagation of inertia-gravity waves from vortex dipoles and jets. J. Atmos. Sci., 66, 1294-
    Wang S. G., F. Q. Zhang, 2010: Source of gravity waves within a vortex-dipole jet revealed by a linear model. J. Atmos. Sci., 67, 1438-
    Wang S. G., F. Q. Zhang, and C. C. Epifanio, 2010: Forced gravity wave response near the jet exit region in a linear model. Quart. J. Roy. Meteor. Soc., 136, 1773- 1787.10.1002/ This study investigates the propagation of gravity waves in the region of significant horizontal and vertical shear associated with a localized atmospheric jet using a linear model. Gravity waves are produced in the linear model by imposing prescribed divergence/convergence forcing of various scales near the core of an idealized local jet. The spatial structures of these forced gravity waves are nearly steady after a few inertial periods, despite the amplitudes slowly increasing with time. Linear model simulated wave response to prescribed forcing shows limited dependence on the scales of the forcing. It is found that the wave structure (e.g. horizontal/vertical wavelengths, phases and locations) away from the forcing are largely constrained by the environmental wind shear through the wave capture mechanism. Consequently, simulated gravity wave activities have the tendency to be focused on the vicinity where the line of constant shear aspect ratio approximates to the characteristic large-scale environmental aspect ratio ( f/N ). Ray tracing analysis is further used to demonstrate that wave capturing is the consequence of different influences of the horizontal and vertical shears upon longer and shorter waves. Copyright 2010 Royal Meteorological Society
    Williams P. D., T. W. N. Haine, and P. L. Read, 2005: On the generation mechanisms of short-scale unbalanced modes in rotating two-layer flows with vertical shear. J. Fluid Mech., 528, 1- 22.10.1017/ report on the results of a laboratory investigation using a rotating two-layer annulus experiment, which exhibits both large-scale vortical modes and short-scale divergent modes. A sophisticated visualization method allows us to observe the flow at very high spatial and temporal resolution. The balanced long-wavelength modes appear only when the Froude number is supercritical (i.e. F > F upi^2/2), and are therefore consistent with generation by a baroclinic instability. The unbalanced short-wavelength modes appear locally in every single baroclinically unstable flow, providing perhaps the first direct experimental evidence that all evolving vortical flows will tend to emit freely propagating inertia-gravity waves. The short-wavelength modes also appear in certain baroclinically stable flows.
    Wu D. L., F. Q. Zhang, 2004: A study of mesoscale gravity waves over the North Atlantic with satellite observations and a mesoscale model. J. Geophys. Res., 109, D22104.10.1029/ microwave data are used to study gravity wave properties and variabilities over the northeastern United States and the North Atlantic in the December-January periods. The gravity waves in this region, found in many winters, can reach the stratopause with growing amplitude. The Advanced Microwave Sounding Unit-A (AMSU-A) observations show that the wave occurrences are correlated well with the intensity and location of the tropospheric baroclinic jet front systems. To further investigate the cause(s) and properties of the North Atlantic gravity waves, we focus on a series of wave events during 19-21 January 2003 and compare AMSU-A observations to simulations from a mesoscale model (MM5). The simulated gravity waves compare qualitatively well with the satellite observations in terms of wave structures, timing, and overall morphology. Excitation mechanisms of these large-amplitude waves in the troposphere are complex and subject to further investigations.
    Xue, M., Coauthors, 2001: The Advanced Regional Prediction System (ARPS) multi-scale nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and applications. Meteor. Atmos. Phys., 76, 143- 165.
    Zhang F. Q., 2004: Generation of mesoscale gravity waves in upper-tropospheric jet-front systems. J. Atmos. Sci., 61, 440- 457.10.1175/1520-0469(2004)061<0440:GOMGWI>2.0.CO; nested mesoscale numerical simulations with horizontal resolution up to 3.3 km are performed to study the generation of mesoscale gravity waves during the life cycle of idealized baroclinic jet–front systems. Long-lived vertically propagating mesoscale gravity waves with horizontal wavelengths 65100–200 km are simulated originating from the exit region of the upper-tropospheric jet streak, in a manner consistent with past observational studies. The residual of the nonlinear balance equation is found to be a useful index in diagnosing flow imbalance and predicting wave generation. The imbalance diagnosis and model simulations suggest that balance adjustment, as a generalization of geostrophic adjustment, is likely responsible for generating these mesoscale gravity waves. It is hypothesized that, through balance adjustment, the continuous generation of flow imbalance from the developing baroclinic wave will lead to the continuous radiation of gravity waves.
    Zhang F. Q., S. E. Koch, C. A. Davis, and M. L. Kaplan, 2000: A survey of unbalanced flow diagnostics and their application. Adv. Atmos. Sci.,17, 165-183, doi: 10.1007/s00376-000-0001-1.10.1007/
    Zhang F. Q., C. A. Davis, M. L. Kaplan, and S. E. Koch, 2001: Wavelet analysis and the governing dynamics of a large amplitude mesoscale gravity wave event along the east coast of the United States. Quart. J. Roy. Meteor. Soc., 127, 2209- 2245.10.1002/ Available
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Manuscript received: 04 February 2016
Manuscript revised: 20 July 2016
Manuscript accepted: 08 August 2016
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Diagnosis of the Forcing of Inertial-gravity Waves in a Severe Convection System

  • 1. Institute of Atmospheric Physics, Chinese Academy Sciences, Beijing 100029, China
  • 2. Meteorological Bureau of Jilin Province, Changchun 130062, China

Abstract: The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms——linear and nonlinear terms, and diabatic heating——on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.

1. Introduction
  • 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.

2. The non-hydrostatic wave equation
  • 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 cp 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.

3. Forcing of IGWs in a severe 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.

4. Discussion
  • 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.

5. Conclusion
  • 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).




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