^{1}

^{*}

^{1}

To eliminate the irrational supposition that condensed liquid water always falls immediately, specific water m and maximum airborne specific water
*m*
_{m }are introduced into the dynamic framework on non-uniform saturated moist atmosphere (m is the ratio of the airborne liquid water mass to the moist air mass in unit cubage moist air,
*m*
_{m} is its maximum value with
,
,
and
are airborne coefficient, vertical velocity and saturated specific humidity respectively). The balance equation between water vapor and airborne liquid water is derived. From the balance equation, a new formula of precipitate rate is got. The analysis shows that in the air stream with some upward vertical velocity (
), the condensed liquid water can precipitate under the condition with
(q is specific humidity) and
only, otherwise it is detained in the air and becomes airborne liquid water. Not only does precipitating liquid water contain condensed liquid water, but also contains converged and existing airborne liquid water. Following above discussion, improved dynamic equations on non-uniform saturated moist atmosphere are provided.

Due to the important roles of water vapor in the genesis and development of weather systems, many studies have contributed to the dynamical discussion on moist atmosphere, e.g., Betts A. K., (1973) [

To consider the transitional area between the unsaturated and saturated atmosphere, the discontinuity could occur in the latent heat term of the thermodynamic equation when the latent heat is released in the saturated atmosphere whereas latent heat is not released in the unsaturated atmosphere. Traditionally, the discontinuity in the latent heat term of the thermodynamic equation can be summarized by the Dirac delta function. The discontinuity makes the theoretical analysis very difficult for near saturated atmosphere, where the relative humidity is usually large enough to condensate but not saturated (just like rain regions), so these discussions are limit to the condition of saturated atmosphere e.g., Wang L. M., (1980) [

However, the condensation process does not occur so abruptly in realistic atmosphere. According to the observational evidences pointed by Mason, (1971) [

On the basis of EQSNUSMA, a battery of results is got:

Wang X. R. and K. J. Wu, (1995) [

speed and static non-equilibrium parameter

equilibrium wind of moist air is less than that of dry air, because static non-equilibrium parameter is less than 1. It was pointed out that the super-geo-strophic behavior of low level jet streams is caused by non-equilibrium relating to jet strengthening, it is not equilibrium airflow.

The non-uniform saturated moist potential vorticity (NUSMPV), or the generalized moist potential vorticity (GMPV), is thus defined and its tendency equation is derived by Wang X. R., Z. X. Wang and C. N. Shi, (1998) [

Based on the works by Wang X. R., Z. X. Wang and C. N. Shi, (1998) [_{0} is narrow sense NUSMPV.). In this kind of process the non-equilibrium energy is dispersed and lost by gravitational and sound wave. On the contrary, when the

Furthermore, in recent years, some new variables from EQSNUSMA, such as generalized potential temperature, GMPV, generalized convective vorticity vector et al., are applied more to the dynamical analysis of hot and humid weather systems and torrential rain systems, e.g., Gao S. T., Y. S. Zhou, T. Lei et al. (2005) [

Such great progress for dynamic on non-uniform saturated moist atmosphere has been made in theory and application aspects, but, in all studies, almost no one is involved in the discussion on condensed liquid water. Traditionally, it is supposed that condensed liquid water always precipitates immediately. However, it is not true, in realistic atmosphere, the condensed liquid water either falls down in the form of precipitate or is detained in the air in the form of airborne liquid water (fog or cloud). By the analysis of some failure model experiments on dynamic equations on non-uniform saturated moist atmosphere, it is found that the reason of failure is exactly the irrational supposition that condensed liquid water always falls down immediately. So, in the dynamic equations on non-uniform saturated moist atmosphere, it becomes a question how to consider the condensed liquid water in air. By the discussion of airborne liquid water content and the balance equation between water vapor and liquid water, the new idea (not only precipitate contains condensed liquid water also includes airborne liquid water) is firstly raised and an advanced precipitable formula is set up by Wang, X.R., Gao, S.T. (2007) [

To eliminate the irrational supposition that condensed liquid water always falls down immediately, specific water m and maximum airborne specific water

In the condition of

Following Wang X. R., Z. X. Wang and C. N. Shi (1999) [

where

Following Mason, (1971) [

where

where N is the all number of liquid water droplets,

According to the observational evidences pointed by Mason, (1971) [

where

With Equation (5), (6) and (7), Equation (3) becomes

From the physical properties in cumuliform clouds versus height above cloud base (

here a is the coefficient independent of p and T. Because

here

When liquid water precipitates, with Equation (13), a new formula of precipitate rate is got as

From Equation (14), not only does precipitating liquid water contains condensed liquid water, but also contains converged and existing airborne liquid water.

In addition, Equation (14) can be rewritten as

The generalized temperature

in saturated moist air (

In moist adiabatic condition, using thermodynamic equation, state equation and continuity,

Equation (17) can be rewritten as

In real atmosphere,

From Equation (22), (23) and (24), we have

In other words,

be rewritten as

In the condition of

With Equation (30), Equation (2) becomes

From Equation (31), because

In addition, because of down-current, the rising temperature cased by adiabatic expansion restrains condensation process, so

From Equation (32) and (33), we have

Following Wang X. R., Z. X. Wang and C. N. Shi (1999) [

In moist adiabatic condition, from Equations (35)-(41) and the definition of generalized temperature

and the vertical coordinate transform formula based on the non-static equilibrium by Wang X. R., C. E. Chi and Z. X. Wang (1997) [

From Equation (35)-(41) and (43)-(49), the dynamic equations on non-uniform saturated moist atmosphere are improved, because of introducing specific water m and maximum airborne specific water

Because the character of

With the new formula of precipitate rate in p coordinate (49), the new formula of rainfall intensity can be written roughly as

where

When estimating rainfall intensity, it is usually assumed that the local change of meteorological equal to 0, so

If assuming

where

here

where Fc is the confluent function of airborne liquid water, it may be expressed as

On the theoretical plane, using the synchronous analysis data with rainfall provided by RAFS (regional analysis and forecasting system), let the actual rainfall intensity

Here, in the supposition that

The gross-precipitation of this super-heavy rain is 1631.1mm, the maximum rainfall intensity is 189.5 mm/h (in the neighborhood of Banqiao reservoir in August 7, 22:00 Beijing time), in August 7, 20:00 Beijing time (the observation time on schedule), the center rainfall intensity on large scale surface weather chart is 12.5 mm/h, the actual extremes of rainfall intensity is 99.7 mm/h (in the neighborhood of Banqiao reservoir). Although the horizontal distribution graph of

With the new formula of precipitate rate in p coordinate (49), using mean value theorem, the new formula of rainfall intensity can be written roughly as

where

From the analysis report provided by LGSHR1975 (1977) [^{3}×kg^{−1}×K^{−1}, g = 9.8 m×s^{−2}, ^{−3} hPa/s, ^{−3} hPa/s and

Based on 8m/s (the surface wind velocity (SWV) recorded at Suiping weather station, which is the nearest from Banqiao reservoir) and 12m/s (the SWV estimated from the physical phenomena at Banqiao reservoir), with an eye on the small scale orographic influence of Banqiao reservoir (climbing and bell-mouthed effects in the condition of NE wind), the

From

In this study, by introducing specific water and maximum airborne specific water into the dynamic framework on non-uniform saturated moist atmosphere, the irrational supposition, that condensed liquid water always falls immediately, is eliminated. The balance equation between water vapor and airborne liquid water is derived. From the balance equation, a new formula of precipitate rate is derived. Following the above discussion, improved dynamic equations on non-uniform saturated moist atmosphere are provided. Finally, in the supposition that

SWV m/s | ^{−3} hPa/s | ^{−3} hPa/s | ||
---|---|---|---|---|

8 12 | 15 16.4 | 5 5 | 30.7 37.0 | 65.1 81.8 |

It is necessary to point out that the character of

This study was supported by Anhui Provincial Natural Science Foundation under Grant No. 1508085MD64, 1408085MKL60 and China Meteorology Administration Foundation under Grant No. CMAGJ2015M28.