Investigation of the Mei-yu Front Using a New Deformation Frontogenesis Function

YANG ShuaiGAO Shoutingand Chungu LU

1Laboratory of Cloud-Precipitation Physics and Severe Storms,Institute of Atmospheric Physics, Chinese Academy of Sciences,Beijing100029

2State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences,Beijing100081

3U.S.National Science Foundation,Arlington VA22230,USA

Investigation of the Mei-yu Front Using a New Deformation Frontogenesis Function

YANG Shuai?1,GAO Shouting1,2,and Chungu LU3

1Laboratory of Cloud-Precipitation Physics and Severe Storms,Institute of Atmospheric Physics, Chinese Academy of Sciences,Beijing100029

2State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences,Beijing100081

3U.S.National Science Foundation,Arlington VA22230,USA

A new frontogenesis function is developed and analyzed on the basis of a local change rate of the absolute horizontal gradient of the resultant deformation.Different from the traditional frontogenesis function,the newly de fi ned deformation frontogenesis is derived from the viewpoint of dynamics rather than thermodynamics.Thus,it is more intuitive for the study of frontogenesis because the compaction of isolines of both temperature and moisture can be directly induced by the change of a fl ow fi eld.This new frontogenesis function is particularly useful for studying the mei-yu front in China because mei-yu rainbands typically consist of a much stronger moisture gradient than temperature gradient,and involve large deformation fl ow.An analysis of real mei-yu frontal rainfall events indicates that the deformation frontogenesis function works remarkably well,producing a clearer mei-yu front than the traditional frontogenesis function based on a measure of the potential temperature gradient.More importantly,the deformation frontogenesis shows close correlation with the subsequent (6 h later)precipitation pattern and covers the rainband well,bearing signi fi cance for the prognosis or even prediction of future precipitation.

deformation,frontogenesis,precipitation

1. Introduction

Frontogenesis has been studied widely for many years due to its importance in weather analyses and forecasting (Margules,1906;Bergeron,1928;Newton,1954;Reed, 1955;Reed and Danielsen,1958;Hoskins and Bretherton,1972;Fang and Wu,1998,2001;Schultz and Steenburgh,1999;Wu and Fang,2001;Schultz and Sanders,2002; Schultz and Trapp,2003;Wu et al.,2004;Li et al.,2013). From a kinematic point of view,Petterssen(1936,1956) de fi ned the scalar frontogenesis function as the Lagrangian change of the magnitude of horizontal potential temperature gradientd|?θ|/dt.Miller(1948)discussed the change of front intensity by analyzing the scalar quantity.Keyser et al. (1988)extended Petterssen’s frontogenesis function to vector formd(?θ)/dt,which is divided into the frontogenetical (or scalar)componentFn=(1/2)|?θ|(D-Ecos2δ)and rotational componentFs=(1/2)|?θ|(ζ+Esin2δ)withD,ζandEdenotingdivergence,vorticityandresultantdeforma-tion,respectively,andδrepresenting the angle between the dilatation-axis orientation and the isentrope in the horizontal plane[0°＜δ＜90°;Keyser et al.(1988),Fig.1].Note that theFnformula is given with the addition of the minus sign on Petterssen’s scalar frontogenesis function.

Inspection of the de fi nitions of scalar and vector frontogenesis functions reveals the signi fi cance of deformation in frontogenesis.Divergence and vorticity contribute toFnandFsrespectively,while deformation works in both frontogenesis components.Furthermore,Dandζmay be positive or negative,and therefore the roles ofDinFqnandζinFsare determined by their signs.However,E(=is always positive regardless of the sign of stretching deformationEstand shearing deformationEsh.Thus,the role ofEis completely determined by the angleδ.Asδ＞45°,deformation leads toFsfrontogenesis by rotating isentropes toward the dilatation axis,which makesδtend to be less than 45°.Certainly,it is well known that deformation is frontogenetical inFnprovided thatδ＜45°.In other words,in classical frontogenesis theory,deformation can either make the isolines of potentialtemperature tend to compactdue to fl ow con fl uence, or it leads to frontogenesis by causing the angle between iso-lines of potential temperature and the dilation axis to rotate towards a favorable con fi guration for frontogenesis.Therefore,deformation is an important fl ow pattern for frontogenesis.

The importance of deformation in frontogenesis has been discussed in many studies.Sawyer(1956)and Eliassen (1962)derived the frontal secondary circulation equation by adopting a geostrophic approximation with the assumption of no change of potential temperature along a front.They illuminated the roles of terms associated with geostrophic deformation and shear deformation in frontal secondary circulation.Both Davies-Jones(1982,1985)and Doswell(1984) studied the role of deformation in frontogenesis.Ninomiya (1984,2000)showed that the deformation in the subtropical zone is also a primary contributor to mei-yu frontogenesis. Schultz et al.(1998)addressed the role of large-scale diffl uent and con fl uent fl ow in low-level cyclone/front structure and evolution through a combined observational and idealized modeling approach.Gao et al.(2008)demonstrated the con fl uence effect of the resultant deformation(termed“total deformation”in their study)on moisture transport by both an idealized non-divergent and irrotational deformation fl ow pattern and case studies of heavy precipitation events associated with deformation-dominant fl ow.

The above-mentioned studies and theories emphasize the role of deformation in frontogenesis.However,previous studies have been based on the classical thermodynamic frontogenesis function,which relies on the Lagrangian change of potential temperature(θ)or equivalent potential temperature (θe)gradients(e.g.,Ninomiya,1984)as the criterion for frontogenesis/frontolysis(Petterssen,1936,1956;Davies-Jones, 1982;Doswell,1984;Davies-Jones,1985;Keyser et al., 1988).In such a de fi ned frontogenesis function,a different tracer,θorθe,must be adopted for either a temperature front or a moisture front.On the other hand,the con fl uence of air fl ow induced by deformation can drive the isolines of both temperature and moisture close together,dynamically,implying that deformation may have a key role in frontogenesis (Keyser et al.,1986,1988).Furthermore,for certain types of precipitation systems,such as those associated with the meiyu front in eastern China,the large-scale fl ow pattern often exhibits a dominant deformation pattern at low levels(Yang et al.,2007;Ran et al.,2009;Yang et al.,2009;Gao et al., 2010;Yang et al.,2014),while the divergence and vorticity are sometimes of smaller magnitude(Gao et al.,2008).In addition,a deformation and con fl uence shear line frequently presents in a mei-yu front and its associated frontal precipitation in eastern China.For these types of systems,it is important to understand the role of deformation,in the triggering and maintenance of the precipitation,and to gain insight into the temporal evolution of the deformation frontogenesis process and its correlation with precipitation.

In the present study,a deformation frontogenesis function is developed and applied to real meteorological cases. Different from the traditional frontogenesis equation,which evaluates a local change rate of the absolute horizontal gradient of potential(or equivalent potential)temperature,we derive a frontogenesis function by using a local change rate of the absolute horizontal gradient of resultant deformation. This new function is more intuitive and essential in dynamics compared to the traditional function,since it is the wind fi eld that drives the compactness of isolines of temperature and moisture.In section 2,the dynamic connections and interactions among resultant deformation,the horizontal gradient ofθe,and the vertical motionωare elucidated.In sections 3 and 4,the expressions of deformation frontogenesis function are derived.The results from diagnoses and analyses of the deformation frontogenesis process in real mei-yu frontal episodes are reported in section 5.Conclusions are given in section 6.

2. Deformation-induced geostrophic circulation

Deformation is important in frontogenesis through shearing and stretching effects.It is conceivable that fl ow shearingand stretching can cause an increase in the horizontal temperature gradient,and therefore baroclinicity through advection, which leads to frontogenesis.Such effects will act upon any physical variable,regardless of whether it is temperature or atmospheric moisture.However,the question is:Can such horizontal variability in fl ow fi elds result in circulations in the vertical direction(an essential dynamics feature typifying baroclinicity)?To elucidate the dynamic connections and interactions among deformation,the horizontal gradient ofθe, and vertical motionω,we conduct the following analysis using quasigeostrophic dynamics.

In the formulation of the quasigeostrophicωequation, two“primary forcing functions”—differential vorticity advection and temperature advection—are considered(e.g., Bluestein,1992).These forcing functions can be written as

whereandζgare quasigeostrophic velocity vector and vertical relative vorticity,fandf0are Coriolis parameter and its constant approximation,σis static stability parameter,Tis absolute temperature,Ris ideal gas constant,and other notations are standard meteorological notation,and the problem is analyzed in apvertical coordinate.

The differential vorticity-advection term,i.e.,the fi rst term in Eq.(1),can be separated into the following two parts:

The fi rst term in Eq.(2)represents advection of geostrophic absolute vorticity by thermal wind.The second term represents advection of thermal vorticity by geostrophic wind.The temperature-advection term[the second term in Eq.(1)]can be expanded similarly:

Using the thermal-wind relation

and enduring some algebraic manipulations,we fi nd that the temperature-advection term[Eq.(3)]can be broken up into the following three parts:

where the relation?ug/?x+?vg/?y=0 has been used,and the geostrophic stretching and shearing deformation are respectively in the form

One can see that the fi rst two terms in Eq.(5)are now being converted to similar terms in Eq.(2),which means that the parts of thermal advection are physically equivalent to the corresponding parts of vorticity advection.The equability of temperature and vorticity advection is a well-known concept in synoptic meteorology.However,the most important result from the above derivation comes from the third term in Eq.(5),which presents an additional forcing mechanism for vertical motion.This forcing measures the net effect of the permutation of shearing/stretching and thermal shearing/stretching deformation.

Combining the vorticity-advection and temperatureadvection forcing functions using Eqs.(2)and(5),one can obtain the following frictionless and adiabatic form of the quasigeostrophicω-equation:

Thisequationimpliesacirculationintheverticalplanewithin the geostrophic dynamics framework.The vertical differential advection of vorticity,and the vertical differential stretching and shearing deformation in the horizontal wind can all contribute a vertical motion.

Because a frontal circulation can be directly related to a con fi guration of the deformation fi eld,it is dynamically more sensible to adopt deformation as a key variable to examine frontogenesis/frontollysis.As mentioned in the previous section,it does not matter what dynamical tracer,be itθorθe,is involved in a particular frontal circulation,as long as a fl ow feature that transports these tracers is captured.The deformation-induced difference in vertical motion inside and outside the frontal zone makes a difference in the pumping up of low-level air,which will cause low-level convergence and thus increase the horizontal gradient ofθe.Therefore,the connections and interactions amongE,θeandωdevelop.

3. Derivation of the deformation frontogenesis function

From the above analyses,one can see that fl ow deformation may serve as a precursor for dynamic frontogenesis and a driving factor for any dynamical tracer to be concentrated upon.Therefore,it is reasonable to de fi ne a frontogenesis function using a deformation fi eld.Such a de fi ned frontogenesis function will be independent of the tracer type,and thus can be used in situations where a temperature/moisture front occurs in the absence of a moisture/temperature front.However,since a frontal circulation may be forced by both shear-ingandstretchingdeformation,weneedtode fi neamoregeneral fi eld to include the effects of both.

Based on Petterssen’s velocity decomposition(1956),the 2D velocity-differential tensor can be written as

whereui=(u,v)and?xj=(?x,?y).Thevelocity-differentialsD=(?u/?x+?v/?y),ζ=(?v/?x-?u/?y),Est=(?u/?x-?v/?y),andEsh=(?v/?x+?u/?y)are,respectively, the divergence,vorticity,stretching deformation,and shearing deformation.

For the fi rst matrix on the right-hand side of Eq.(7),a resultant deformation is de fi ned to generalize the resultant effect of both stretching and shearing deformations.In order to do so,we take determinant for the fi rst matrix and the resultant deformation can thus be de fi ned as

The resultant deformation has been used by meteorologists for many decades(e.g.,Petterssen,1956;Keyser et al.,1986; Norbury,2002).When the(x,y)coordinate system is rotated to a new coordinate system(x′,y′),it can be proven that the resultant deformation preserves symmetry under a coordinate rotation.That is,the resultant deformation is independent of the coordinate rotation.

Note that the operator?/?t(but notd/dt)is utilized in Eq.(9)in the present study,which is based on two points. First,Wu et al.(2004)discussed the intensity change of frontal systems by utilizing both local and Lagrangian frontogenesis functions,and then further studied the problem of geostrophic adjustment during frontogenesis.Obviously,local frontogenesis has already been used in the past.Second, the resultant deformationEacted by operator?/?therein is just a driver(from the dynamical viewpoint)but not a tracer (in the thermodynamic sense).

Beginning from horizontal motion equations,Gao et al. (2008)derived a prognostic equation of the resultant deformation:

It can be seen from Eq.(10)that the local time rate of change of resultant deformation is related to the advection of deformation(both horizontally and vertically:A2andA3),is produced by the interaction of deformation and divergence (A4),is affected by theβeffect(A5),and is related to various second-orderderivativesofgeopotentialactingonthestretching and shearing deformations(A6andA7),the tilting effects (A8andA9),and friction and/or turbulence mixture(A10andA11).Furthermore,by the effects ofA8andA9,the in fl uence of vertical motion onEis present.

On the basis of the above resultant deformation equation,the deformation frontogenesis function is derived in detail:

where|?E|/0 is assumed.

Let us write Eq.(10)in symbolic form,

whereA2,A3,···,A11are the terms in Eq.(10).Thus,Eq. (11)becomes

where

In Eqs.(12–14),is a three-dimensional velocity vector,?his horizontal divergence,andω=dp/dtis the vertical velocity in thep-coordinate system.

According to Eq.(13),F1represents the contribution of the advection of the absolute value of the horizontal total deformation gradient to local frontogenesis.The sign of this term is determined by projection of the 3D velocity vector on the gradient of|?E|.As the angle betweenand?3|?E|is less thanπ/2,the projection is positive,F1＜0,and advection leads to frontolysis.Otherwise,F1＞0 leads to frontogenesis.In addition,the deformation frontogenesis equation is also associated with horizontal deformation(F2),horizontal divergence(F3),theβeffect(F4),pressure gradient force (F5),thetiltingeffect(F6),andfrictionand/orturbulencemixture(F7).

4. Alternative derivation of the deformation frontogenesis function

In section 3,the deformation frontogenesis function[Eq. (13)]was derived based on the resultant deformation equation[Eq.(10)]of Gao et al.(2008).However,both equations include too many terms,which increase their complexity.At least in form,the advantage of the above deformation frontogenesis function[Eq.(13)]is not too evident compared with the traditional frontogenesis function.Therefore,simpli fi cation,or another derivation,is necessary to cut down the complexity of the deformation frontogenesis function.An alternative derivation of the deformation frontogenesis function is performed as follows:

The horizontal motion equation and its component forms can be written as

From Eq.(8),we have

Combining with Eq.(15),?Est/?tand?Esh/?tin Eq.(16) become

Note that,similarly,we have the relations

Thus,Eq.(16)becomes

From Eq.(17),Eq.(11)can be rewritten as

Equation(18)can be further simpli fi ed to

with=?E/|?E|,tgα=Est/Eand ctgα=Esh/E.

For convenience of calculation and analysis,Eq.(19)is expanded and noted as the sum operator of several forcing factorsXm(m=1,2,3,4):

where

In Eq.(20b),Xmdenotes the co-action of two kinds of deformation effects.One is the deformation of the wind fi eld itself (tgα=Est/E,ctgα=Esh/E),and the other is the stretching deformation(Jst,m)and shearing deformation(Jsh,m)of four forcing terms in the horizontal wind local change tendency equation[Eq.(15)].Takingm=1(or 2,3,4)as an example, the second type of deformation effect(Jst,mandJsh,m)represents the advection(or pressure gradient force,Coriolis force, friction and/or turbulence mixture)forcing deformation.For brevity,Xm(m=1,2,3,4)is termed advection forcing(X1), pressure gradient forcing(X2),Coriolis forcing(X3),and friction forcing(X4)of the deformation frontogenesis function.

This form of the deformation frontogenesis function[Eq. (20)],recorded as a sum operator,looks simple and offers an easy explanation relative to the expression[Eq.(13)]derived in section 3.Equation(20)includes four terms with a clear physical sense for each term.At least in form,it is not more complicated than the traditional frontogenesis function.Furthermore,it is not harder for it to provide a clear physical explanation for each forcing term in Eq.(20).Also,based on its advantage from the dynamic viewpoint(e.g.,the confl uence of air fl ow induced by deformation can dynamically drive the isolines of any one tracer,be itθorθe,closer together),the deformation frontogenesis function will exert its maximal advantage if its diagnostic analysis for frontogenesis and associated precipitation via a case study is good.

The difference and relationship between Eqs.(13)and (20)should be discussed to develop contextual linkages, since the latter is an alternative form of the former.By rigorous derivation,we have the relationsX1=F1+F2+F3+F6,X2=F5,X3=F4,andX4=F7.After careful derivation and repeated validation,it is shown that the above relations are rigorous and without any conditions of assumptions and simpli fi cations.All these analyses indicate that,in addition to being concise in form,Eq.(20)can re fl ect all of the information contained in Eq.(13).Therefore,the calculation in the case study in the next section based on Eq.(20)can be considered reasonable.

Note that Eq.(13)is retained in the paper for two reasons:

(1)Because the deformation frontogenesis functionFis de fi ned as the local change rate of|?E|,it is natural to deriveFbased on theEequation.It is logical to derive theEequation fi rst,and thenFbased on theEequation.Furthermore, fromEequation to its extended application(F),a set of complete and systematic derivation about deformation theory is built up,just as have done in the fi rst author’s Ph.D dissertation(Yang,2007).Because too many terms are included in Eq.(13),Eq.(20)is derived as an alternative form of the deformation frontogenesis function.

(2)The origin of Eq.(13)is important as a“true”value to validate Eq.(20),both in theory and in calculation.By testing,Eq.(20)can re fl ect all of the information contained in Eq.(13).Therefore,their calculation results in a real case analysis are also consistent.

Certainly,Eq.(20)can be further simpli fi ed under certain conditions.For example,if pure stretching deformation is assumed(which can be achieved by rotating the coordinate system),Esh=0 andE=Est,which lead to tgα=1 and ctgα=0.Therefore,Eq.(20b)becomes

which leads to the frontogenesis function,Eqs.(20a)and (19),to become

At this moment,the shear deformations of both the wind fi eld itself and the forcing terms of wind change disappear.Only the effects of their stretching deformations work.In the next section,a case study is reported in which we validate the application of the deformation frontogenesis function.

5. Case study

A heavy rainfall event associated with a mei-yu front occurred over the middle and lower reaches of the Yangtze River in China from 0000 UTC 10 to 0000 UTC 12 July 2010 (Fig.1a).The rainband was oriented in the west-southwest to east-northeast direction.The circulation background was as follows.A line of strong con fl uence shear and a typical“saddle” fl ow pattern existed in the lower troposphere(Figs. 2a–c).This con fl uence shear zone was a result of the largescale fl ow pattern that consisted of a subtropical high located inthenorthwesternPaci fi c,alowcenteredat(33.7°N,118°E) and a southwest vortex at(28.7°N,106.5°E),as well as another high at(34.5°N,109°E)(Fig.2a).The“saddle” fl ow pattern stretched vertically to the 600 hPa level(not shown).

Along this shear region,there was a zone of high equivalent potential temperatureθethat clearly marked the mei-yu frontal zone stretching in the northeast–southwest direction between 25°N and 36°N(Fig.3a).In particular,Ninomiya (1984)argued that,different from polar frontogenesis where a gradient of potential temperature is conventionally used,a measure of frontogenesis in mei-yu rainbands should adopt a gradient of equivalent potential temperature.Therefore, the strong gradient of equivalent potential temperature is still used to denote the frontal zone,but the traditional frontogenesis function is replaced by the deformation frontogenesis function.

The fronts(with compact isolines ofθe)formed before the onset of precipitation and maintained during the course of precipitation.Unlike the classical cold/warm front,this case showed a strong gradient in humidity but not in temperature over the middle and lower reaches of the Yangtze River in China(Figs.3b and c).

To obtain a set of fi ne-scale data to perform the diagnosis of the frontogenesis function,the Weather Research and Forecasting(WRF)model is used to produce a mesoscale numerical simulation.Yu et al.(2012)simulated the rainfall event using the WRF model and obtained good results according to its validation.Therefore,the same con fi guration,schemes,initial and boundary fi elds are adopted in our simulation.The details of the con fi guration of the simulation can be found in Yu et al.(2012).The resolutions of the outer and inner domains of the simulation are 36 km and 12 km.The inner-domain model outputs are used to analyze the frontogenesis features for the case.Figure 1b shows the simulated 48-h precipitation between 0000 UTC 10 and 0000 UTC 12 July 2010.The orientations of rainbands between the simulation and observation are similar(Figs.1a and b). The observed precipitation has a maximal center of 270 mm (48 h)-1,while the simulated precipitation has four centers of considerable strength[＞300 mm(48 h)-1].The typical deformation stream fi eld is also reproduced(Figs.2c and d).The simulated geopotential height and temperature fi elds werevalidatedinFig.2ofYuetal.(2012).Themodeldataset is used to diagnose the frontogenesis function.

From the distribution of humidity(Fig.3c),equivalent potential temperature(Fig.3a),and the resultant deformation (Fig.4a),we can see that the intense humidity gradient is closely correlated with a large value of the resultant deformation.The pattern,orientation and coverage of the large-valueregions between the resultant deformation and humidity consistently correspond with each other(Figs.4a and 3c).Furthermore,the strength of the resultant deformation is nearly two times larger than vorticity/divergence in this case,reaching up to 1.0×10-4s-1(Fig.4).On the basis of the above analyses,deformation was large in magnitude compared to vorticity and divergence,and showed a strong presence in the large humidity gradient regions.Thus,this was likely the driving factor for frontogenesis in the frontal rainfall case. With this analysis,the deformation frontogenesis process by taking accounting of the gradient of total deformation is justi fi ed and is examined below.

As a comparison,a traditional frontogenesis function using equivalent potential temperature was also calculated and analyzed.Following Ninomiya(1984,2000)and Sun and Du (1996),this frontogenesis function can be written as

where FG1,FG2,FG3,and FG4are de fi ned,respectively,as

whereθeis equivalent potential temperature;andu,v,andωare the zonal,meridional and vertical velocity components, respectively.FG1,FG2,FG3,and FG4are diabatic heating, convergence,deformation,and the slantwise terms,respectively.

From Fig.5,the deformation frontogenesis zones[with the friction forcing termX4in Eq.(20)neglected]correspond fronts(compact equivalent potential temperature zone)better than the traditional frontogenesis zones[the calculation of the traditional frontogenesis function neglects the diabatic heating,i.e.,FG1in Eq.(23)].The pattern,orientation and coverage of the large-value regions between the deformation frontogenesis andθegradients are consistent with each other (Figs.5a,c and e),while the scattered distributions of traditional frontogenesis do not completely cover the compact regions ofθeisolines(Figs.5b,d and f).More importantly,the pattern of deformation frontogenesis and the rainband closely resemble each other(Figs.5a,c and e,and Figs.6a–c).For instance,the deformation frontogenesis regions and the interlinked double rainbands stretch northeastwards to the west of 115°E in Figs.5a and 6a,while the traditional frontogenesis function cannot re fl ect the pattern of precipitation(Figs. 5b and d).The“Y-shaped”patterns of precipitation(red line in Fig.6b)and the deformation frontogenesis function(red line in Fig.5c),and the maximal precipitation center(red rectangular box in Fig.6c)and the intensity center of the deformation frontogenesis function(red rectangular box in Fig. 5e)located at(30°N,110°E),do not appear in the distribution charts of the traditional frontogenesis function(Figs.5d andf),evenwhenthecontourintervalofboththedeformation frontogenesis function(by reducing to 0.5)and the traditional frontogenesis function(by reducing to 0.1)are changed.This shows that the coverage of both frontogenesis functions is spread out and wider than those of the rainbands.However, the traditional function does not perform better.Note that no deformation frontogenesis signals in Fig.5a cover the dashed-line part of the southern branch of double rainbands (east of 117°E and south of 29°N),which may be explained as follows.The calculation of the deformation frontogenesis function is based on the deformation fi eld.The dashed-line part in Fig.6a is outside the deformation-dominant con fl uence shear region(Figs.2a and c),and therefore is not located in the large-value band of the deformation fi eld(Fig. 4a).Thus,the deformation frontogenesis function based on the deformation fi eld does not work at this location.It is possible that this represents one of the limitations of this kind of frontogenesis function,exerting its great advantage in the deformation-dominant fl ow pattern.Furthermore,the deformation frontogenesis strengthens[(3.5–6)×10-14m-1s-2] with the increase of rainfall amount(80–130 mm)from 10 to 11 July(Figs.5a and e),while the traditional frontogenesis function[(3–1.5)×10-9K s-1m-1](Figs.5b and f)cannot re fl ect the variation in the rainfall amount(Figs.6a and c).

Note that the area of the deformation frontogenesis function is a little larger than the rainband,which can be further explained by the following four aspects:

(1)The area of the deformation frontogenesis function is larger than the rainband,which is partly related to the choice of contour interval.If the contour interval of precipitation (Fig.6)is changed(e.g.,reduced to 1 mm),the rainbands become wider.

(2)The deformation frontogenesis function is based on the resultant deformation fi eld,which after all is a pure dynamical variable.Precipitation events are very complicated processes related to clouds and microphysics,possibly partly explaining why their coverage does not completely match.

(3)Furthermore,the more important and useful aspect for precipitation is the prediction ability rather than the diagnosis.Therefore,the frontogenesis functions(Figs.5a,c and e)are compared with the precipitation 6 h later(Fig.6)to test their prediction ability.For example,the frontogenesis functions at 0000 UTC 10 July(Figs.5a and b)are compared with precipitation at 0600 UTC 10 July(Fig.6a),and the frontogenesis functions at 0600 UTC(Figs.5c and d)are compared with 6 h precipitation at 1200 UTC(Fig.6b),and so on.This may also partly explain why their coverage does not completely match.

(4)The correlation between the deformation frontogenesis function with precipitation should be compared with that ofthetraditionalfrontogenesisfunctionwithprecipitationbecause these two frontogenesis functions are equivalent—both have a role in frontogenesis and induce the development of weather systems and precipitation.By comparison,the deformation function without considering friction(which is diffi cult to accurately calculate)performs better than the traditional frontogenesis function when neglecting diabatic heating because it is too dif fi cult to accurately calculate.

But why can the new deformation frontogenesis function re fl ect the frontogenesis process better than the traditional function?Because from the above analyses the resultant deformation can drive the contraction ofθeisolines,the problem of frontogenesis is considered from the dynamic rather than the thermodynamic aspect in this paper.From the connections and interactions amongE,θe(Fig.7a)andω(Fig. 7b),the mechanism is further explained as follows.According to traditional understanding,the con fl uence associated with the deformation and the transportation of warm and moistairbysucha fl owmakesthemassandmoistureconcentrated distinctly towards the con fl uence region of the saddle fi eld.Therefore,theθeisolines become dense,indicating a buildup of strongθegradient along the con fl uence zone,i.e. frontogenesis(Fig.7a).In the 3D case,vertical motion along the con fl uence zone will cause the upward transportation of warm and moist air brought in by the horizontal deformation fl ow.Furthermore,the difference in vertical motion inside and outside the frontal zone(Fig.7b)induced by the deformation fl ow makes a difference in the pumping up of lowlevel air,which will cause low-level convergence and thus increase the horizontal gradient ofθe.Thus,frontogenesis presents and it sets up a favorable condition for precipitation, whichpossibly offersan explanation.Furthermore,thedeformation frontogenesis function avoids the in fl uence of inaccuracy brought about by the calculation of the diabatic heating term in the traditional frontogenesis formulation.

Figure 8 shows the variations of the terms in the deformation frontogenesis function[in Eq.(20)].The evolution tendency of the terms(Fig.8)shows that,although the Coriolis forcing term(X3)is negative and nearly equal to zero,the advection forcing term(X1)and the pressure gradient forcing term(X2)in the frontal zone work together to sustain the strong gradient ofθe,which manifests the major factors affecting deformation frontogenesis.The results of this case show that the focus of the mass fi eld towards the con fl uence zone,caused by the advection forcing effect of deformation frontogenesis,has some positive feedback on the pressure gradient between both sides of the con fl uence zone.Therefore,the pressure gradient forcing is related to the change of local deformation frontogenesis.Certainly,to begin with,it is the con fl uence associated with the advection forcing that provides a favorable condition for moisture convection.The fact that the advection forcing term is the largest in magnitude in this case further emphasizes the role of the wind fi eld, which also veri fi es the appropriateness of directly de fi ning a frontogenesis function from the dynamical viewpoint,since it is the wind fi eld that drives the compactness of the isolines of any one tracer to lead to frontogenesis.

6. Conclusion

In this study,we investigated a mei-yu front using a new deformation frontogenesis function.The new frontogenesis function is de fi ned based on the local change rate of the absolute gradient in the resultant deformation.Different from the traditional frontogenesis function,the new deformation frontogenesis function places emphasis on the fl ow itself rather than on the fl ow-driving tracers,such as potential temperature or equivalent potential temperature.Such a definition of the frontogenesis function is independent of the tracer type,and thus can be used in situations where a temperature/moisture front occurs in the absence of a moisture/temperature front.Therefore,it is more intuitive for the study of mei-yu frontogenesis since the isolines of both temperature and moisture are derived by deformation in a deformation-dominant fl ow pattern.

Within the framework of quasigeostrophic theory,anωequation was derived,with shearing and stretching deformation presented as forcing functions.This result suggested an importance of deformation in frontal circulation.

A deformation-frontogenesis function was derived and applied to a mei-yu front precipitation case.In comparison with traditional frontogenesis dynamics,which evaluate the local change rate of the absolute gradient of equivalent potential temperature,the deformation-frontogenesis equation presented a better depiction of fronts in deformation-dominant fl ow situations.More importantly,the deformation frontogenesis regions covered the rainband well and showed close correlation with the precipitation pattern,which is signi fi cant in terms of the prognosis of precipitation.

To obtain more general conclusions and to explore the relationship between deformation frontogenesis and precipitation in more detail,we plan in future work to examine more cases using data with higher temporal and spatial resolutions.

Acknowledgements.The authors were supported by the National 973 Fundamental Research Program of the Ministry of Science and Technology of China(Grant No.2013CB430105),the Key Research Program of the Chinese Academy of Sciences(Grant No. KZZD-EW-05-01),the Special Scienti fi c Research Fund of the Meteorological Public Welfare of the Ministry of Sciences and Technology,China(Grant No.GYHY201406003),the National Natural Science Foundation of China(Grant Nos.41375054,41375052 and 40805001),and the Opening Foundation of the State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences(Grant Nos.2012LASW-B02 and 2013LASW-A06).

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(Received 1 July 2014;revised 7 September 2014;accepted 15 September 2014)

?Corresponding author:YANG Shuai

Email:ys ys@126.com