The formation of heavy precipitation typically requires transport and convergence of large amounts of water vapour in addition to microphysical processes inside clouds. Because the set of moist dynamic equations describe both dynamic and thermodynamic aspects of water vapour, certain physical variables derived from a moist dynamic system, such as a moist potential vorticity and other derived variables, can be used to diagnose and potentially predict the occurrence of heavy precipitation.
One of the fundamental aspects in moist dynamics is to determine the thermal property of a parcel of air with significant water content. In order to do this, scientists have proposed various temperature parameters describing moist air. One such parameter, equivalent potential temperature (θe), is conventionally used when dealing with a moist atmosphere (e.g. Holton, 1992). Ninomiya (1984), for instance, suggested that for studying Meiyu fronts θe should be used instead of potential temperature. Tripoli and Cotton (1981) used an ice–liquid water potential temperature to describe thermodynamic processes in deep convective cloud systems. Pointin (1984) further developed this idea by introducing the use of ice–liquid water potential temperature to handle some non-reversible processes, such as freezing of supercooled water, evaporation in unsaturated air, and precipitation. Hauf and Höller (1987) pointed out that a generalized entropy temperature could unify all of these different kinds of potential temperature.
The introduction of these various forms of thermodynamic potential temperature allows for definitions of various dynamic variables (e.g. potential vorticity) to be used for different physical situations. Schubert et al(2001) derived a generalized potential vorticity in a moist atmosphere, using a virtual potential temperature. They proved that the solenoidal term was eliminated when virtual potential temperature was a function of total density and pressure only. They also noted several problems with the potential vorticity derived from equivalent potential temperature. Gray (1999) discussed a mass-forcing problem in a mesoscale convective system which used mass forcing in the potential vorticity equation to diagnose potential vorticity anomalies. Rivas Soriano and García Diez (1997) derived a form of generalized potential vorticity based on Hauf and Holler's (1987) entropy temperature. They argued that the solenoidal forcing on potential vorticity generated by spatial gradients of ice concentration is of the same order of magnitude as the classic solenoid term.
Because of their conservative property in a dry or moist adiabatic process, potential vorticity and equivalent potential vorticity have been widely used in analyses of atmospheric stability (Bennetts and Hoskins, 1979; Bennetts and Sharp, 1982; Robinson, 1989); diagnostic studies of tropical cyclones (Schubert and Hack, 1983; Thorpe, 1985; Montgomery and Farrell, 1993; Wu and Emanuel, 1995) and extratropical cyclones (Hoskins and Berrisford, 1988; Davis and Emanuel, 1991; Cao and Cho, 1995), investigations of fronts (Shapiro, 1974, 1976; Montgomery and Farrell, 1991; Fulton and Schubert, 1991); analyses of rain bands (Chan and Cho, 1989; Xu, 1992), and mesoscale convective systems (Raymond and Jiang, 1990; Lu et al, 1997; Gao et al, 2005). However, the use of potential vorticity derived from potential temperature assumes an absolutely dry atmosphere, while the use of traditional moist potential vorticity derived from equivalent potential temperature assumes a completely saturated atmosphere everywhere. In many of the weather systems discussed above, the atmosphere can neither be absolutely dry nor completely saturated everywhere. A realistic atmosphere may thus present a non-uniformly saturated situation. Furthermore, observations show that condensation may take place at relative humidities as low as 75% (Mason, 1971), depending on local pressure and temperature inside a cloud. As a result, many air parcels may never reach saturation because water is condensed out before the air parcel becomes saturated.
In order to depict a non-uniformly saturated atmosphere, Gao et al(2004) (hereafter GWZ) and Gao et al(2005) introduced a generalized potential temperature which captures absolutely dry and completely saturated atmospheres as limiting cases but is also valid if the atmosphere does not reach saturation. Using this new thermodynamic variable, a generalized moist potential vorticity (GMPV) can be defined. GMPV combines moist thermodynamic and dynamic processes together, providing a potential new tool to examine mesoscale convective problems. In particular, Gao et al(2005) applied GMPV to the detection and prediction of severely humid weather. Yang et al(2007) used GMPV to derive an ageostrophic moist Q-vector for the diagnosis of heavy precipitation. Since potential vorticity has been widely used in diagnosing storm systems, it is natural to inquire whether GMPV can be a good dynamic tracer for heavy precipitation systems.
In this study, we employ the generalized moist potential vorticity dynamic and thermodynamic system to examine a Meiyu rainfall event in China. In particular, we diagnose how mass loss due to heavy rainfall can generate an anomaly of GMPV. In light of this non-conservative property of GMPV, we investigate its possible use as both a diagnostic as well as a predictive tool to track heavy rainfall systems. With this particular purpose in mind, we compare diagnostic results from both GMPV and moist potential vorticity (MPV). In section 2, we re-examine the concept of equivalent potential temperature and generalized potential temperature. A GMPV equation is then derived and compared with an MPV equation based on θe. In section 3, we present the synoptic situation for a heavy precipitation event that occurred in the Huai River basin in eastern China. We then conduct a mesoscale modelling study of this heavy rainfall event (section 4). In section 5, we apply both GMPV and MPV to analyse forecasts of this heavy precipitation event. Two more real cases are further examined in section 6, and we summarize and conclude our findings in section 7.
2. Moist potential vorticity and generalized moist potential vorticity in a non-uniformly saturated atmosphere
2.1. Equivalent potential temperature and generalized potential temperature
In considering moisture in the atmosphere, one simple assumption is that at any one time the air is saturated everywhere. In this case, condensational heating contributes to the thermodynamic equation such that (Holton, 1992),
where , R is the gas constant, cp is specific heat at constant pressure, L is the latent heat coefficient, T is atmospheric temperature, p and p0 are pressure and a reference pressure respectively, qs is saturated specific humidity, and Q is diabatic heating. In the following, we extend the analysis presented in Gao et al(2004). One may rewrite (1) in the form
The first term on the right-hand side (rhs) of the above equation can be combined with the left-hand side term (lhs), while the second rhs term represents the effect of changing temperature on the latent heat, and thus can be considered as part of the diabatic heating. Therefore, the above expression can be expressed as
From this result, one can define the equivalent potential temperature as
Equivalent potential temperature widens the usefulness of potential temperature as a diagnostic tool by including atmospheric moisture. However, it assumes that the atmosphere is saturated everywhere. Also note that with this definition of equivalent potential temperature, the thermodynamics equation now can be written in the form
where denotes the total diabatic heating.
GWZ proposed a generalized equivalent potential temperature, which can be used to describe a non-uniformly saturated atmosphere. This generalized equivalent potential temperature is of the form
where q is specific humidity and (q/qs)k is termed the condensation probability function, and where k can take different values for different moist atmospheric situations (GWZ). One can see that in an absolutely dry atmosphere, q = 0, and θ* reduces to θ; while in a completely saturated atmosphere, q = qs, and θ* reduces to θe. In a more realistic atmosphere, 0 < q < qs, thus θ* will be located somewhere between θ and θe.
Substituting (6) into the thermodynamic equation, one obtains a generalized thermodynamic equation similar to (5):
where . Equation (7) thus expresses a conservation principle for θ* instead of θe. Notice that the diabatic heating term here is defined differently than that in (5).
2.2. Mass continuity equation with precipitation sink
When precipitation occurs, a large amount of water mass is removed from the moist atmosphere associated with a storm, thus causing mass loss. In this case, the mass continuity equation should include a mass sink term. To derive a generalized mass continuity equation, let us denote ρ as total air density, which is a sum of
where ρd is density for dry air, ρv is density for water vapour, and ρr is density for rain. Each of these components satisfies their own mass continuity equation, in the forms
where Sv is the conversion rate from water vapour to rain, v is three-dimensional velocity, and Vt is terminal velocity for rain drops. Upon adding these three equations together, we obtain
Qr = − ∇•(ρrVt) thus defines a mass sink in the total mass conservation equation due to precipitation.
2.3. Moist potential vorticity and generalized moist potential vorticity
A vector vorticity equation can be expressed as
where α is specific volume, is the absolute vorticity vector, and all external forces have been neglected. One can obtain a moist potential vorticity equation for a saturated atmosphere by combining (5), (12) and (13), yielding
where the moist potential vorticity (MPV) is defined as . Substituting the definition for equivalent potential temperature (4) and the expression for specific volume , (14) becomes
This result seems to suggest that the generation of MPV depends on the resultant effect of the inner product of the solenoidal forcing and gradient of saturated specific humidity (term 1), the differential of diabatic heating and mass forcing on MPV (term 2), and gradient of diabatic heating on an integrated MPV (term 3). In particular, the first term is dependent upon the angle between the gradient of saturated specific humidity and the solenoidal term comprising the pressure and temperature gradients. However, since qs≃qs(p, T) approximately, this angle will be near 90 degrees and hence the generation of MPV by solenoidal forcing will approach zero. This implies that there is no direct MPV production due to an inclusion of saturated moist dynamics, which, as we will see in the latter sections, puts a limitation on the use of MPV as a diagnostic variable for precipitation.
Now let us examine what we obtain using a generalized potential temperature instead of equivalent potential temperature. Taking the inner product of (13) with the gradient of (7) and using (12), we arrive at
where now represents a generalized moist potential vorticity (GMPV) corresponding to the use of generalized potential temperature. Substituting the generalized potential temperature and specific volume expressions into (16) gives
Comparing (17) with (15), one can see that a similar set of physical forcings is responsible for the generation of GMPV as for the generation of MPV. However, the production of GMPV due to solenoidal forcing (term 1) depends on an inner product of the solenoidal term with the gradient of specific humidity, rather than the gradient of saturated specific humidity as in the expression for MPV in (15). In a transition between raining and non-raining zones, the moisture gradient is understandably large. Therefore, the production of GMPV can also be large. In particular, (17) indicates that interaction between atmospheric baroclinicity and the moisture gradient can lead to GMPV production through solenoidal forcing in such a way that large GMPV will result when a large moisture gradient lines up with large baroclinicity. In this sense, GMPV reflects the effects of both atmospheric dynamics and moist thermodynamics. Since heavy rainfall in the midlatitudes typically occurs in strong baroclinic regions where large amounts of water vapour converge, there is strong potential for GMPV to be successfully used to diagnose as well as predict heavy rainfall events.
3. Synopsis of the 4 July heavy precipitation event in the Huai River region
During a period of activity along the East Asian Meiyu in June and July 2003, China's Yangtze and Huai river region experienced two major precipitation episodes. The first, on and off from 21 to 28 June, was characterized by heavy precipitation mainly in the Yangtze River basin. During the second episode, between 29 June and 11 July, the rain band moved over the Huai River basin. In this study, we concentrate on an especially intense event during this second episode in the Huai River basin. The Huai River (within the grey shaded region on Figure 1 denoting China) is indicated on the figure by the highlighted curve east of the letters H.R. This heavy precipitation event lasted from 0000 UTC 4 July to 1200 UTC 5 July.
Figure 1 displays plots of the geopotential height and horizontal wind vectors at the 500 hPa pressure level at 12-hour time intervals between 0000 UTC 4 July and 0000 UTC 5 July 2003. The data are from the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR) reanalysis dataset (Kalnay et al, 1996) with a horizontal grid spacing of 1°× 1° latitude and longitude. During this heavy rainfall event, there was a low-pressure centre located in northeast Asia, from which a trough extended initially in the southwest to northeast direction, and over time gradually turned toward a southeast to northwest orientation. The movement of this trough led to a southward intrusion of cold air into a large region of eastern China. Despite the change of orientation of the trough line, the large-scale atmospheric circulation displayed a quite persistent pattern. The subtropical high was located in the western Pacific off the coast of southeast China. On its northwest flank, the wind was mostly from the southwest, bringing warm moist air and providing abundant water vapour for the Meiyu rain band. The convergence of the cold dry air from the north and the warm moist flow from the south marked a typical Meiyu frontogenetic condition, and was conducive to heavy precipitation in this convergence zone. The Huai River basin was located along the boundary of these two air masses, and was thus naturally subject to the resulting frontal rain band.
4. Mesoscale model simulation of the 4 July heavy precipitation event
In this section, we describe a mesoscale numerical model simulation of the 4 July heavy precipitation event. For these simulations the Weather Research and Forecasting (WRF) model, jointly developed by the National Oceanic and Atmospheric Administration (NOAA) and NCAR, is used. The WRF model is fully compressible and non-hydrostatic. The Arakawa C grid is used for the horizontal grid points and a terrain-following mass coordinate is used in the vertical. Either a third-order or fourth-order Runge–Kutta iteration scheme is used for time integration. Model initialization used the 1°× 1° NCEP/NCAR reanalysis dataset; the lateral boundary conditions also used these 6-hour interval data. The simulation was initiated at 0000 UTC 4 July 2003 and ended at 1200 UTC 5 July 2003. One outer domain and one nested domain were used (see Figure 2), with the centre for both located at (32.5°N, 115.0°E). The grid spacing for the outer domain is 30 km, with a domain size of 90 × 90 horizontal grid points, while that for the nested domain is 10 km, with a domain size 145 × 136 horizontal grid points. In the vertical, 28 levels were used. Table I lists various physical parametrization schemes selected for this study.
Table I. Physical parametrization schemes used in the WRF model simulations for both coarse-grid and fine-grid domains.
Physical paramet. scheme
Fine-grid (nested) domain
Planetary Boundary Layer
Yonsei Univ. scheme
Yonsei Univ. scheme
Cumulus paramet. scheme
In Figure 3(a), model-simulated 24-hour accumulated precipitation amounts in the nested domain are shown. The simulated rainfall is verified against observed rainfall for this particular event, using rain-gauge data from 179 surface stations in eastern China (provided by the Chinese Meteorological Administration). These rainfall data are then interpolated onto the model grids using the GrADS graphic package (available at http://www.iges.org/grads/). Comparing the simulated precipitation fields with the analysis of surface rain-gauge observations for the same period (Figure 3(b)), it is clear that the model captured the location and orientation of the rain band reasonably well. However, the model underpredicted rain area and rain intensity in most locations. Since the focus of this study is on the use of dynamic variables as diagnostic tools, the simulated data is adequate for this purpose. On Figure 4, hourly precipitation from the WRF forecasts for 0900, 1000, 1100 and 1200 UTC 4 July 2003 is plotted. The rain band did maintain itself and intensify during the period.
5. Diagnostic studies of heavy precipitation using MPV and GMPV
With the set of simulated model data, we now diagnose how MPV and GMPV respond to the dynamic and thermodynamic forcing due to precipitation across the Meiyu front. Figure 5 displays the horizontal distribution of MPV at the 900 hPa level in the nested domain for the same simulation times as Figure 4. One can see that although a band of negative MPV seems to correspond to the rain band, the signals spread almost everywhere in the domain. This result is consistent with the theoretical analysis conducted in section 2, in the sense that moist dynamics does not provide forcing for MPV due to the vanishing of the solenoidal term in (15).
Figure 6 is the same plot as Figure 5, except for GMPV. Contrary to the previous result for MPV, GMPV did show strong signals aligned with the rain band. The fact that both large positive and negative values of GMPV occur indicates that the solenoidal forcing changes sign across the rain band. From (17), it is seen that the moist gradient would indeed provide such a sign change for GMPV, when moisture increases toward the centre of the rain band, but decreases on both sides of the centre, while for MPV no such mechanism existed.
To further verify these results, vertical cross-sections of MPV and GMPV are shown along a given longitude (120°E or 119°E) in Figures 7 and 8, respectively. The equivalent potential temperature does show a broad maximum in the rain-band centre above the 900 hPa level, while MPV displays banded signals across the entire domain below this level. This feature is consistent with the thermodynamic field, in which the equivalent temperature (the contours) exhibits almost uniformly large values due to saturated condition below 900 hPa. The vertical MPV pattern matches the horizontal distribution in Figure 5, i.e. there is no physical signal indicated by MPV that relates to the precipitation field. In Figure 8, on the other hand, the generalized potential temperature displays a tighter vertical band than that for equivalent potential temperature. The corresponding GMPV field also exhibits sharp-banded structures collocated with the rain band. The transition of positive to negative GMPV across the rain band is consistent with the sign change of the moisture gradient.
6. Additional case-studies
To confirm that GMPV possesses a better diagnosis for heavy rainfall than MPV, results from two additional similar cases are described in this section. These two heavy rainfall events both occurred in the Yangtze–Huai river basin during, respectively, 8–10 July 2007 (hereafter referred to as case 2) and 28–30 June 2009 (case 3). Figure 9(a) and (b) show the 24 h accumulated rainfall and distributions for these two events, both of which exhibit the typical summer rainfall pattern of the East Asian Meiyu front.
To simulate these two cases, we again used the WRF model set-up exactly as described in the first case. Figure 10(a) and (b) show model-simulated hourly rainfall for case 2 on 0000 and 0100 UTC 9 July 2007. The diagnosed GMPV fields for these times (Figure 10(c) and (d)) are compared with corresponding MPV fields (Figure 10(e) and (f)). Both fields seem to capture the banded structure of rainfall. However, the GMPV shows a much stronger signal than does MPV. The diagnosed MPV fields also present scattered false rains on the south side of the Meiyu front. This can be clearly seen on the vertical cross-sections transverse to the front (Figure 11). Figure 11(a) and (b) show isotherms of generalized potential temperature (contours) and GMPV (colour shades) at the two diagnostic times, while Figure 11(c) and (d) show the isotherms of equivalent potential temperature (contours) and MPV (colour shades).
Numerical experiments are also conducted for case 3. Hourly simulated rainfall is shown in Figure 12(a) and (b) at 2300 and 0000 UTC 29–30 June 2009, respectively. The diagnosed GMPV (Figure 12(c) and (d)) and MPV (Figure 12(e) and (f)) both show broader features in this case. However, GMPV still captures a relatively clear banded structure following the location and orientation of the observed rainfall. As with the two previous cases, vertical cross-sections at these two times for generalized potential temperature and GMPV (Figure 13(a) and (b)) show better frontal structure than equivalent potential temperature and MPV (Figure 13(c) and (d)).
This study examined the linkage between a precipitation field and combined dynamic and thermodynamic fields. To incorporate a non-uniformly saturated atmosphere, a generalized potential temperature was utilized, which improved the depiction of a moist atmosphere by allowing for variations of water content. In contrast, the traditional equivalent potential temperature assumes saturation everywhere for a moist atmosphere. With this newly defined thermodynamic variable, a generalized moist potential vorticity (GMPV) can be derived.
The forced GMPV captures gradients of both temperature and humidity. Therefore it is ideal for tracking, diagnosing and predicting combined dynamic and thermodynamic processes in moist atmospheres. To illustrate its use, GMPV was applied to the diagnostic study of three heavy precipitation events. The results indicate that GMPV presents a significant signal in the regions of heavy precipitation. In contrast, the traditional MPV based on equivalent potential temperature could not capture moist dynamic processes necessary to track precipitation fields.
A final point should be made relating to the usefulness of GMPV. Because of the nature of potential vorticity, which combines both dynamic and thermodynamic effects, GMPV can successfully couple regions of large vorticity due to storm development to the moisture gradients produced by convergence of water vapour in the storm region. This implies that GMPV is not only useful as a diagnostic tool (as shown in this study), but may also serve as a predictive variable for heavy precipitation. Given predicted wind or vorticity fields, coupled with predicted temperature and moisture fields, one can construct a future GMPV from which heavy rainfall can be inferred before it occurs.
This research was supported by the National Natural Sciences Foundation and State Key Laboratory Open Project of China under grants No. 40775031 and No. 2008LASW-A01, respectively. C. Lu was also supported by the Wang Kuangcheng fellowship for this work, and he is grateful to the Chinese Academy of Sciences (CAS) for the support of a short-term visit to the Institute of Atmospheric Physics (IAP), CAS.