q vector analysis of torrential rainfall from meiyu front ... · imum while the myfc showed its...

68 ACTA METEOROLOGICA SINICA VOL.23

Q Vector Analysis of Torrential Rainfall from Meiyu Front Cyclone:

A Case Study∗

YUE Caijun1,2†(��

)

1 Shanghai Typhoon Institute of China Meteorological Administration, Shanghai 200030

2 Laboratory of Typhoon Forecast Technique/China Meteorological Administration, Shanghai 200030

(Received October 29, 2008)

ABSTRACT

Following similar derivation of quasi-geostrophic Q vector (QG), a new Q vector (QN) is constructed inthis study. Their difference is that the geostrophic wind in quasi-geostrophic Q vector is replaced by the windin QN vector. The diagnostic analysis of QN vector is compared with that of QG vector in the case studyof a typical Meiyu front cyclone (MYFC) occurred over Changjiang-Huaihe regions during 5–6 July 1991.The results show that the QN vector has more diagnostic advantages than QG vector does. Convergence ofQN vector at 700 hPa is found to be a good indicator to mimic the horizontal distribution of precipitation.QN vector is further partitioned into four components: QN

alst (along-stream stretching), QN

curv (curvature),QN

shdv(shear advection), and QN

crst(cross-stream stretching) in a natural coordinate system with isohypse

(PG partitioning). The application of QN PG partitioning in the MYFC torrential rain indicates that PGpartitioning of QN can identify dominant physical processes. The horizontal distribution of 2∇ · QN

alstis

similar to that of 2∇ · QN and mainly accounts for 2∇ · QN during the entire period of Meiyu. The effectsof QN

curv on rainfall enhancement fade from the mature stage to decay stage. QN

shdv enhances precipitationsignificantly as the MYFC develops, and the effect weakens rapidly when the MYFC decays during itseastward propagation. QN

crstshows little impacts on rainfall during the onset and mature phases whereas it

displays significant role during the decay phase. QN

alstand QN

curv, QN

shdvand QN

crstshow cancellation only

during the decay period.

Key words: Q vector analysis, QN vector, partitioning of QN vector, Meiyu front cyclone, torrential

rainfall

1. Introduction

The diagnosis of Q vector is one of most efficient

methods for studying vertical motion associated with

development of severe storms since it contains informa-

tion of dynamic and thermodynamic processes (e.g.,

Hoskins et al., 1978). The quasi-geostrophic Q vector

(QG) was introduced to study severe storms and re-

lated weathers in China during the late 1980s (e.g.,

Bai, 1988). The semi-geostrophic Q vector, modi-

fied Q vector, moist Q vector, and improved moist Q

vector were further proposed to study torrential rain-

fall processes associated with Meiyu fronts, Southwest

China lows, and landfall of typhoons (e.g., Li and Li,

1997; Zhang, 1998, 1999; Yao and Yu, 2000, 2001; Yao

et al., 2004; Yue et al., 2003b), and their differences

were compared (Yue, 1999; Yue et al., 2005). Q vec-

tor can be partitioned in a natural coordinate with

isentrope (PT partitioning) to study weather process

physical mechanisms (e.g., Keyser et al., 1988, 1992;

Kurz, 1992; Barnes and Colman, 1993; Schar and

Wernli, 1993; Martin, 1999; Morgan, 1999). However,

this partition is mainly applied to quasi-geostrophic

Q vector that has limitations. Recently, Yue et al.

(2003a) applied PT partitioning to moist Q vector.

Meanwhile, Jusem and Atlas (1998) and Donnadille

et al. (2001) proposed another partitioning method,

which is conducted in a natural coordinate with iso-

hypse (PG partitioning). PT partitioning method is

mainly applied to quantitative diagnosis of multiscale

impacts on vertical motion, whereas PG partitioning

method is mainly used to impacts of flow structures

on vertical motion. The study of Jusem and

Atlas (1998) and Donnadille et al. (2001) is based

∗Supported by National Natural Science Foundation of China under Grant Nos. 40875025, 40405009, and 40205008, andShanghai Natural Science Foundation of China under Grant No. 08ZR1422900.

†Corresponding author: [email protected].

NO.1 YUE Caijun 69

on quasi-geostrophic Q vector. In this study, Q vec-

tor modified by Zhang (1999) is further modified (QN

vector) so that it can be calculated with a real atmo-

spheric wind field. The new QN vector can be applied

to diagnose both large scale and mesoscale weathers

while it has the similar to formation QG vector has.

Since torrential rainfall is usually associated with the

Meiyu front (e.g., Liao and Tan, 2005; Chen and Gao,

2006; Zhao et al., 2007), torrential rainfall occurred

from 2000 BT 5 to 2000 BT 6 July 1991 (Tao and

Huang, 1994; Shou et al., 2001) will be diagnosed with

the new QN vector in this study. This rainfall event

was analyzed with PT partitioning method of moist

Q vector (Yue et al., 2003a), and it was found differ-

ent synoptic scales have different impacts on rainfall

development. The main subject of this study is to

quantitatively analyze flow structures and their roles

in producing vertical motion during rainfall develop-

ment. The new QN vector is derived in next section.

The rainfall event and data are briefly described in

Section 3. The partitioning of new QN vector is ap-

plied to this rainfall event in Section 4. Conclusions

and a discussion will be given in Section 5.

2. Derivation of QN vector

Following Zhang (1999), a modified Q vector

is obtained from stationary, adiabatic, and friction-

less primitive equations on f -plane, whose zonal and

meridional components can be, respectively, expressed

by

Q∗x =

1

2[f(

∂v

∂p

∂u

∂x− ∂u

∂p

∂v

∂x) − h

∂V

∂x· ∇θ], (1)

Q∗y =

1

2[f(

∂v

∂p

∂u

∂y− ∂u

∂p

∂v

∂y) − h

∂V

∂y· ∇θ], (2)

where h =R

p(

p

1000)R/cp , θ = T (

1000

P)R/cp , V =

ui+ vj, u and v are zonal and meridional components

of wind, respectively. Equations (1) and (2) can also

be written as

Q∗x =

1

2[f(

∂v

∂p

∂u

∂x−

∂u

∂p

∂v

∂x) − (

∂u

∂x

∂α

∂x+

∂v

∂x

∂α

∂y)], (3)

Q∗y =

1

2[f(

∂v

∂p

∂u

∂y− ∂u

∂p

∂v

∂y) − (

∂u

∂y

∂α

∂x+

∂v

∂y

∂α

∂y)], (4)

where α =1

ρ=

RT

p. Following Dutton (1976),

∂u

∂p≈ ∂ug

∂p,∂v

∂p≈ ∂vg

∂p. Thus, Eqs. (3) and (4) be-

come,

QN

x =1

2[∂(fvg)

∂p

∂u

∂x− ∂(fug)

∂p

∂v

∂x

−(∂u

∂x

∂α

∂x+

∂v

∂x

∂α

∂y)], (5)

QN

y =1

2[∂(fvg)

∂p

∂u

∂y− ∂(fug)

∂p

∂v

∂y

−(∂u

∂y

∂α

∂x+

∂v

∂y

∂α

∂y)]. (6)

Substituting geostrophic relation that fvg =∂ϕ

∂x

and fug = −∂ϕ

∂y, and relation that

∂ϕ

∂p= −α into

Eqs. (5) and (6) leads to

QN = (QN

x , QN

y ) = −i(∂u

∂x

∂α

∂x+

∂v

∂x

∂α

∂y) (7)

−j(∂u

∂y

∂α

∂x+

∂v

∂y

∂α

∂y).

Equation (7) is the expression of QN vector, which

is similar to Eq. (2.5) in Jusem and Atlas (1998). The

difference between them is that real wind is used in

this study whereas quasi-geostrophic wind is used in

Jusem and Atlas (1998). If real wind is replaced by

quasi-geostrophic wind, QN vector will be degraded to

quasi-geostrophic Q vector (e.g., Hoskins et al., 1978).

The ω equation with the forcing of divergence of

QN vector can be written as

∇2(σω) + f2∂2ω

∂p2= −2∇ · QN. (8)

With the assumption of wavelike solution, the left

side is proportional to ω. Thus, ∇ · QN ∝ ω. ω < 0

when ∇ · QN < 0, whereas ω > 0 when ∇ · QN > 0.

3. Rainfall event and data

The torrential rainfall event occurred from 2000

BT 5 to 2000 BT 6 July 1991 was a typical heavy rain-

fall event associated with Changjiang-Huaihe Meiyu

front. At 2000 BT 5 July, there was a narrow low

pressure zone at 700 hPa (Fig. 1a). Surface rainfall

just started with scattered patterns of four centers:

A (30◦N, 110.5◦E), B (30.1◦N, 112.2◦E), C (32.0◦N,


Fig.1. Geopotential height (contour; gpm) and wind fields (arrow; m s−1) at 700 hPa at (a) 2000 BT 5 July, (b) 0800

BT 6 July, and (c) 2000 BT 6 July 1991.

112.1◦E), and D (31.5◦N, 113.9◦E). From 2000 BT 5 to

0800 BT 6 July, the low moved eastward while inten-

sifying. The surface rainfall was enhanced (Fig. 2a).

At 0800 BT 6 July, a Meiyu front cyclone (MYFC)

was formed with a center of (31.5◦N, 113◦E) at 700

hPa (Fig. 1b). Meanwhile, surface rainfall increased

significantly, and the raining area showed a band dis-

tribution with an east-west orientation with its center

located at E (32.5◦N, 116.2◦E) (Fig. 2b). From 0800

BT to 2000 BT 6 July, surface rainfall reached its max-

imum while the MYFC showed its maximum intensity.

At 2000 BT 6 July, the MYFC continued to move east-

ward and into the ocean (Figs. 1c and 2c).

Thus, 2000 BT 5, 0800 BT 6, and 2000 BT 6 July

1991 represent onset, mature, and decay stages of this

event, respectively. The data horizontal resolution is

NO.1 YUE Caijun 71

Fig.2. One-hour cumulative precipitation amount (mm) from observations at (a) 2000 BT 5 July, (b) 0800 BT 6 July,

and (c) 2000 BT 6 July 1991. A, B, C, D, and so on denote maximum rainfall centers.

30 km×30 km, and 15 vertical layers are 1000, 950,

900, 850, 750, 700, 650, 600, 550, 500, 400, 300, 200,

and 100 hPa. The domain for analysis is 29.25◦–

34.80◦N, 109.72◦–120.33◦E.

4. PG partitioning of QN vector and its appli-

cations

Following Jusem and Atlas (1998), Eq. (7) can

be expressed in the natural coordinate with isohypses

as

QN = −t(∂S∗

∂s

∂α

∂s+ KsS

∗ ∂α

∂n)

−n(∂S∗

∂n

∂α

∂s+ KnS∗ ∂α

∂n), (9)

where axis s is parallel to the isohypse line and its unit

vector is t that denotes the wind direction; axis n is

perpendicular to the isohypse line and its unit vector


is n; (t, n, k) complies with the right handed law, k is

a unit vector in vertical coordinate; Ks is curvature of

isohypse line, and Ks > 0 for cyclonic circulations and

Ks < 0 for anticyclonic circulations in the Northern

Hemisphere; Kn is curvature of the line that is perpen-

dicular to the isohypse line, and Kn > 0 for diffluence

and Kn < 0 for confluence; S∗ =√

u2 + v2. Equation

(9) can be partitioned into four components:

QN

alst = −t∂S∗

∂s

∂α

∂s, (10)

QN

curv = −tS∗Ks∂α

∂n, (11)

QN

shdv = −n∂S∗

∂n

∂α

∂s, (12)

QN

crst = −nS∗Kn∂α

∂n. (13)

Here, Eq. (10) is an along-stream stretching

term that denotes enhancement/weakening of temper-

ature gradient along the air flow resulted from con-

tract/extension of horizontal area between isohypse

lines. Equation (11) is a curvature term that denotes

vertical motion caused by flux curvature. Equation

(12) is a shear advection term that denotes temper-

ature advection induced by the shear of horizontal

winds. Equation (13) is a cross-stream stretching term

that denotes enhancement/weakening of temperature

gradient crossing the air flow resulted from diffluence

or confluence. The detailed physical meaning can be

found in Jusem and Atlas (1998) and Donnadille et al.

(2001). Following Jusem and Atlas (1998), Eqs. (10)–

(13) are transferred into Cartesian coordinates, which

are shown in Appendix. Calculations of QN vector at

850, 700, and 500 hPa show that the divergence and

convergence of QN vector at 700 hPa match the sur-

face rainfall better than those at 850 and 500 hPa.

Thus, QN vector at 700 hPa will be analyzed next.

Figures 3a and 3b show that the rainfall centers A,

B, and C are located in convergence zones of 2∇·QN

alst

and 2∇ · QN

curv at 2000 BT 5 July 1991. The rainfall

centers B and C are located in divergence zones of

QN

shdv(Fig. 3c). While the divergence field of QN

crst

does not have any important impacts on surface rain-

fall (Fig. 3d). See summary in Table 1.

Table 1. Summary of collocations of centers of convergence QN vector at 700 hPa and 1-h cumulative rainfall

amount at 2000 BT 5 July 19911 h 700 hPa

2∇ · QN

alst2∇ · QN

curv 2∇ · QN

shdv2∇ · QN

crst

A (30.0◦N, 110.5◦E) −0.6 −0.2 / /

B (30.1◦N, 112.5◦E) −0.6 −0.2 −0.2 /

C (32.1◦N, 112.1◦E) −0.4 −0.4 −0.2 /

D (31.5◦N, 113.9◦E) / / / /

Convergence zones of 2∇·QN

alst and 2∇·QN

curv have

the similar horizontal distributions to rainfall amount,

and they have an important contribution to 2∇ · QN

(Fig. 3e). The rainfall center A is located in con-

vergence zones of 2∇ · QN

alst and 2∇ · QN

curv. This

suggests that the rainfall center A is caused by as-

cending motion forced by QN

alst and QN

curv. The rain-

fall centers B and C also are located in convergence

zones of 2∇·QN

alst, 2∇·QN

curv, and 2∇·QN

shdv, indicat-

ing the roles of QN

alst, QN

curv, and QN

shdv. Thus, QN

alst

and QN

curv are mainly responsible for the rainfall on-

set while QN

shdv plays a secondary role. QN

crst does not

have impacts on precipitation. The rainfall center D

does not show any relation to 2∇ · QN. This suggests

that the rainfall may be caused by convective conden-

sational heating.

At 0800 BT 6 July 1991, rainfall areas are gen-

erally located in divergence zones of QN

alst (Fig. 4a).

Non-uniform distribution of 2∇·QN

alst basically reflects

that of surface rainfall. A main rain-band between 32◦

and 33◦N are located in weak convergence zones of

QN

curv (Fig. 4b). 2∇ · QN

curv has a similar distribution

to rainfall but it is weaker than 2∇ · QN

alst. Mean-

while, raining area south of the main rain-band is not

affected by 2∇ ·QN

curv. Figure 4c shows that the rain-

fall center E is collocated with convergence center of

2∇·QN

shdv but the east-west oriented narrow rain-band

can be seen in distribution of convergence of 2∇·QN

shdv.

Over the entire rainfall area, 2∇ ·QN

crst (Fig. 4d) does

not show any significant convergence and divergence,

NO.1 YUE Caijun 73

Fig.3. Distribution of divergence (10−15hPa−1 s−3) of QN vector at 700 hPa at 2000 BT 5 July 1991. Solid (dashed)

contours represent divergence (convergence). A, B, C, and D denote the centers of 1-h cumulative rainfall. (a), (b), (c),

(d), and (e) denote 2∇ · QN

alst, 2∇ · QN

curv, 2∇ · QN

shdv, 2∇ · QN

crst, and 2∇ · QN, respectively.


Fig.4. As in Fig. 3, but for 0800 BT 6 July 1991. E, F, G, and H denote the centers of 1-h cumulative rainfall.

Table 2. As in Table 1, but for 0800 BT 6 July 1991

1 h 700 hPa

2∇ · QN

alst2∇ · QN

curv 2∇ · QN

shdv2∇ · QN

crst

E (32.5◦N, 116.2◦E) −0.4 −0.2 −0.6 /

F (32.0◦N, 114.2◦E) / / / /

G (30.5◦N, 114.0◦E) −0.6 / −1.0 /

H (31.0◦N, 115.5◦E) −1.2 / −0.4 /

NO.1 YUE Caijun 75

Fig.5. As in Fig. 3, but for 2000 BT 6 July 1991.


and 2∇ · QN

crst also shows no effects on precipitation

(Table 2).

Generally, convergence of QN

alst has similar hori-

zontal patterns to entire raining area as well as main

rain-bands. Centers of convergence of QN

shdv and rain-

fall are collocated but the horizontal distribution of

convergence of QN

shdv does not show band structure as

the main raining area has. The horizontal patterns

of 2∇ · QN

alst and 2∇ · QN

shdv are very similar to that

of 2∇ · QN (Fig.4e), and the formers contribute more

to the latter than the other terms do. Convergence

of QN

curv is not collocated with the main rain-band.

This indicates that the precipitation is mainly forced

by QN

alst and QN

shdv. QN

crst does not affect rainfall de-

velopment.

At 2000 BT 6 July 1991, convergence fields of

QN

alst and QN

crst have similar horizontal distribution to

that of QN, and their centers are collocated (Fig. 5).

This indicates that 2∇·QN

alst and 2∇·QN

crst have a ma-

jor contribution to 2∇ · QN. The divergence zone of

2∇·QN

curv (Fig. 5b) is corresponded to the convergence

zone of 2∇ · QN. This suggests that QN

curv does not

enhance precipitation. Instead, it induces downward

motion to suppress the surface rainfall. In the conver-

gence zone of 2∇ · QN (Fig. 5e), 2∇ · QN

shdv shows an

alternated distribution of divergence and convergence

with small magnitudes (Fig. 5c), which implies small

impacts on rainfall development. During this period,

2∇ · QN

alst and 2∇ · QN

curv, 2∇ · QN

shdv and 2∇ · QN

crst

are out of phase and they cancel each other out. This

only occurs at 2000 BT 6 July 1991.

In the life cycle of MYFC, QN

alst favors develop-

ment of MYFC and associated surface rainfall; QN

curv

enhances precipitation associated with the onset of

MYFC, the enhancement fades during the MYFC ma-

ture period, and it suppresses surface rainfall during

the MYFC decay period; QN

shdv enhances surface rain-

fall, the onset and mature periods, and decrease during

decay period; QN

crst has no impact on surface rainfall

during the onset and mature periods, whereas it has

significant impacts on enhancement of surface rainfall

during the decay period. The analysis reveals that

forcing exists while precipitation occurs during the

entire period of Meiyu. Different terms play differ-

ent roles in producing surface rainfall during different

stages of Meiyu.

5. Conclusions and discussion

In this study, Q∗ vector is modified into QN, par-

titioned in the natural coordinates with isohypse, and

applied to study a typical event of development of

MYFC and associated surface rainfall. The results

include:

(1) Observed wind is directly used in calculations

of QN vector, which shows capacity to study weather

events, in particular, with ageostrophic circulations.

(2) Convergence field of QN vector is calculated

at 700 hPa, which indicates the horizontal distribution

of surface rainfall better than at other pressure levels.

(3) PG partitioning of QN vector can be con-

ducted for discussions of physical processes responsi-

ble for development of weather systems and associated

precipitation.

(4) During the entire period of development of

MYFC, convergence of QN

alst shows similar horizon-

tal distribution to that of QN since it has major con-

tribution to convergence of QN. Thus, it is a major

forcing factor for development of surface rainfall. Ef-

fects of QN

curv on precipitation change from enhance-

ment during onset and mature periods to suppression

during the decay period. QN

shdv has positive impact

on enhancement of precipitation during the onset and

mature phases whereas this positive impact fades as

MYFC moves eastward. QN

crst does not show any im-

pacts during the onset and mature periods. It has pos-

itive impacts on enhancement of surface rainfall during

the decay phase, which is as important as QN

alst is.

In this study, Q∗ vector is modified into QN

vector, which has a similar formulation as quasi-

geostrophic Q vector. QN vector can be calcu-

lated with observed wind without quasi-geostrophic

assumption. Furthermore, the terms after PG parti-

tioning of QN vector have physical meanings, which

is easily used for process study. When real wind is

replaced with geostrophic wind, the QN vector is de-

NO.1 YUE Caijun 77

graded into a quasi-geostrophic Q vector. The physi-

cal meanings of terms after PG partitioning of quasi-

geostrophic Q vector can be found in Jusem and At-

las (1998). Jusem and Atlas (1998) pointed out that

the PG of quasi-geostrophic Q vector can be used in

broad-sense Q vector under alternative balance (AB)

conditions (Davies-Jones, 1991). This supports our

study. The quasi-geostrophic Q vector, broad-sense

Q vector under AB conditions, and QN vector are de-

rived under adiabatic conditions. This is the reason

why we do not conduct PG partitioning directly for

moist Q vector (Zhang, 1998; Yao and Yu, 2000, 2001;

Yao et al., 2004) and modified moist Q vector (Yue et

al., 2003b). We cannot understand physical meanings

for the terms after PG partitioning of moist Q vec-

tor including adiabatic processes. This study should

be considered as extension and continuation of Jusem

and Atlas (1998).

The analysis of PG partitioning of QN vector re-

veals different effects of forcing during different stages

of MYFC. During the onset period, QN

alst, QN

curv, and

QN

shdv have impacts on enhancement of precipitation.

In particular, QN

alst and QN

curv lead to the increase

of surface rainfall. During the mature period, QN

alst

and QN

shdv serve as the major forcing for precipita-

tion. During the decay period, QN

alst and QN

crst play

the major role in producing precipitation. This in-

dicates that the major physical processes responsible

for rainfall development can be identified through the

analysis of PG partitioning of QN vector. During the

life span of MYFC, precipitation lasts due to persis-

tent forcing. However, the forcing is different in dif-

ferent stages of MYFC. Even during the same stage of

Meiyu, forcing could be different over different areas.

The different forcing can be identified by conducting

PG partitioning of QN vector. Finally, it should be

notice that both QN

alst and QN

curv are along isohypse

lines, and both QN

alst and QN

curv cross isohypse lines so

that 2∇·QN

alst and 2∇·QN

curv, 2∇·QN

shdv and 2∇·QN

crst

may cancel each other out. This is a case study. The

generality of QN vector PG partitioning needs further

studies.

Acknowledgement. The author thanks anony-

mous reviewers for their valuable comments and edi-

tors for their editorial efforts.

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APPENDIX

Derivation of components of partitioned QN vector in

pressure coordinate:

Basic relations are defined as

t = ci + sj, (A1)

n = −si + cj, (A2)

c =u

S∗= cosβ, (A3)

s =v

S∗= sinβ. (A4)

(A1) and (A2) lead to that t · n=0.

Relations between t, n, i, j and c, s, β are shown

as follows.

S∗ = cu + sv (A5)

S∗ =√

u2 + v2, (A6)

c2 + s2 = 1, (A7)

∂

∂s= c

∂

∂x+ s

∂

∂y, (A8)

∂

∂n= −s

∂

∂x+ s

∂

∂y, (A9)

NO.1 YUE Caijun 79

(A8) yields

∂S∗

∂s= c

∂S∗

∂x+ s

∂S∗

∂y(A10)

since

∂S∗

∂x= c

∂u

∂x+ s

∂v

∂x, (A11)

and

∂S∗

∂y= c

∂u

∂y+ s

∂v

∂y. (A12)

Substituting (A11) and (A12) into (A10) leads to

∂S∗

∂s= c2 ∂u

∂x+ cs(

∂v

∂x+

∂u

∂y) + s2 ∂v

∂y. (A13)

(A9) yields

∂S∗

∂n= −s

∂S∗

∂x+ c

∂S∗

∂y. (A14)

Substituting (A11) and (A12) into (A14) yields

∂S∗

∂n= c2 ∂u

∂y+ cs(

∂v

∂y− ∂u

∂x) − s2 ∂v

∂x, (A15)

because

Ks =∂β

∂s= c

∂β

∂x+ s

∂β

∂y=

∂s

∂x− ∂c

∂y, (A16)

∂s

∂x=

c

S∗(c

∂v

∂x− s

∂u

∂x), (A17)

and

∂c

∂y=

s

S∗(s

∂u

∂y− c

∂v

∂y). (A18)


Ks =1

S∗[c2

∂v

∂x+ cs(

∂v

∂y−

∂u

∂x) − s2

∂u

∂y]. (A19)

Thus,

S∗Ks = c2∂v

∂x+ cs(

∂v

∂y− ∂u

∂x) − s2

∂u

∂y, (A20)

because

Kn =∂β

∂n= −s

∂β

∂x+ c

∂β

∂y=

∂c

∂x+

∂s

∂y, (A21)

∂c

∂x=

s

S∗(s

∂u

∂x− c

∂v

∂x), (A22)

and

∂s

∂y=

c

S∗(c

∂v

∂y− s

∂u

∂y). (A23)


Kn =1

S∗[c2

∂v

∂y− cs(

∂v

∂x+

∂u

∂y) + s2

∂u

∂x]. (A24)

Thus,

S∗Kn = c2∂v

∂y− cs(

∂v

∂x+

∂u

∂y) + s2

∂u

∂x. (A25)

Finally, (A8) and (A9) lead to

∂α

∂s= c

∂α

∂x+ s

∂α

∂y, (A26)

∂α

∂n= −s

∂α

∂x+ c

∂α

∂y. (A27)

Based on above relations, we can obtain expres-

sions of Eps.(10)–(13) in pressure coordinate.

Substituting (A1), (A13), and (A26) into Eq.(10)

leads to

QN

alst = −t∂S∗

∂s

∂α

∂x= −(ci + sj)[c2

∂u

∂x+ cs(

∂v

∂x+

∂u

∂y) + s2

∂v

∂y](c

∂α

∂x+ s

∂α

∂y). (A28)

Since QN

alst = QN

alstxi + QN

alstyj,

QN

alstx = −[c2∂u

∂x+ cs(

∂v

∂x+

∂u

∂y) + s2

∂v

∂y](c

∂α

∂x+ s

∂α

∂y)c,

(A29)

QN

alsty = −[c2∂u

∂x+ cs(

∂v

∂x+

∂u

∂y) + s2

∂v

∂y](c

∂α

∂x+ s

∂α

∂y)s.

(A30)

Substituting (A3), (A4), (A6), and relation α =

RT/p into (A29) and (A30) yields

QN

alstx = −[u2 ∂u

∂x+ uv(

∂v

∂x+

∂u

∂y) + v2 ∂v

∂y]

(u∂T

∂x+ v

∂T

∂y) · uR

P (u2 + v2)2, (A31)

QN

alsty = −[u2∂u

∂x+ uv(

∂v

∂x+

∂u

∂y) + v2

∂v

∂y]

(u∂T

∂x+ v

∂T

∂y) ·

vR

P (u2 + v2)2. (A32)


leads to

QN

curv = −tS∗Ks∂α

∂n= −(ci + sj)[c2

∂v

∂x+

cs(∂v

∂y− ∂u

∂x) − s2

∂u

∂y](−s

∂α

∂x+ c

∂α

∂y).

(A33)


Since QN

curv = QN

curvxi + QN

curvyj,

QN

curvx = −[c2∂v

∂x+ cs(

∂v

∂y− ∂u

∂x) − s2

∂u

∂y]

(−s∂α

∂x+ c

∂α

∂y)c, (A34)

QN

curvy = −[c2∂v

∂x+ cs(

∂v

∂y− ∂u

∂x) − s2

∂u

∂y]

(−s∂α

∂x+ c

∂α

∂y)s. (A35)



QN

curvx = −[u2 ∂v

∂x+ uv(

∂v

∂y− ∂u

∂x) − v2 ∂u

∂y]

(−v∂T

∂x+ u

∂T

∂y) · uR

p(u2 + v2)2, (A36)

QN

curvy = −[u2∂v

∂x+ uv(

∂v

∂y−

∂u

∂x) − v2

∂u

∂y]

(−v∂T

∂x+ u

∂T

∂y) ·

vR

p(u2 + v2)2. (A37)


leads to

QN

shdv = −n∂S∗

∂n

∂α

∂s= −(−si + cj)[c2

∂u

∂y+

cs(∂v

∂y−

∂u

∂x) − s2

∂v

∂x](c

∂α

∂x+ s

∂α

∂y).

(A38)

Since QN

shdv = QN

shdvxi + QN

shdvyj,

QN

shdvx = [c2∂u

∂y+ cs(

∂v

∂y− ∂u

∂x) − s2

∂v

∂x]

(c∂α

∂x+ s

∂α

∂y)s, (A39)

QN

shdvy = −[c2∂u

∂y+ cs(

∂v

∂y−

∂u

∂x) − s2

∂v

∂x]

(c∂α

∂x+ s

∂α

∂y)c. (A40)

Substituting (A3), (A4), (A6) and relation α =


QN

shdvx = [u2∂u

∂y+ uv(

∂v

∂y− ∂u

∂x) − v2

∂v

∂x]

(u∂T

∂x+ v

∂T

∂y) · vR

p(u2 + v2)2, (A41)

QN

shdvy = −[u2∂u

∂y+ uv(

∂v

∂y− ∂u

∂x) − v2

∂v

∂x]

(u∂T

∂x+ v

∂T

∂y) · uR

p(u2 + v2)2. (A42)


leads to

QN

crst = −nS∗Kn∂α

∂n= −(−si + cj)

[c2∂v

∂y− cs(

∂v

∂x+

∂u

∂y) + s2

∂u

∂x]

(−s∂α

∂x+ c

∂α

∂y). (A43)

Since QN

crst = QN

crstxi + QN

crstyj,

QN

crstx = [c2∂v

∂y− cs(

∂v

∂x+

∂u

∂y) + s2

∂u

∂x]

(−s∂α

∂x+ c

∂α

∂y)s, (A44)

QN

crsty = −[c2∂v

∂y− cs(

∂v

∂x+

∂u

∂y) + s2

∂u

∂x]

(−s∂α

∂x+ c

∂α

∂y)c. (A45)



QN

crstx = [u2∂v

∂y− uv(

∂v

∂x+

∂u

∂y) + v2

∂u

∂x]

(−v∂T

∂x+ u

∂T

∂y) ·

vR

p(u2 + v2)2, (A46)

QN

crsty = −[u2∂v

∂y− uv(

∂v

∂x+

∂u

∂y) + v2

∂u

∂x]

(−v∂T

∂x+ u

∂T

∂y) · uR

p(u2 + v2)2. (A47)

QN

alst, QN

curv, QN

shdv, andQN

crst in pressure coordi-

nate can be calculated using (A31) and (A32), (A36)

and (A37), (A41) and (A42), and (A46) and (A47),

and their convergence can be calculated by

∇ · QN

alst =∂QN

alstx

∂x+

∂QN

alsty

∂y, (A48)

∇ · QN

curv =∂QN

curvx

∂x+

∂QN

curvy

∂y, (A49)

∇ · QN

shdv =∂QN

shdvx

∂x+

∂QN

shdvy

∂y, (A50)

∇ · QN

crst =∂QN

crstx

∂x+

∂QN

crsty

∂y. (A51)

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