2013 년도 1 학기

Chapter 4.2-3

종관기상학

4.2 The equation of continuity

4.2.1. The equation of continuity in height coordinates

dSdVdVt

t

t

Dt

D

Dt

D

vvvnvv

vvv

vv

v

v

ˆ)(

)(

1

If incompressible,

0,0 vDt

D → 3-dimensionally nondivergent

→water

대기는 compressible, but shallow → 연직적으로 변화 적음

→→

→→

→ → → →

→ → → → → →

→→ →

→ →

4.2.2. The equation of continuity in pressure coordinates

0

0

p

g

pyx

Dt

Dzyx

Dt

D

p

v ; diagnostic

Pressure coordinate 에서의 3 차원 바람장은 , compressible 대기에서도

nondivergent

→ column 이 수평면적에 decrease 되면

연직적으로 팽창하면서 보상하게 된다 .

→ total mean conserve

→ material volume 의 incompressible 처럼

행동한다

p

pp

pp 2

[ 그림 4.33]

→→

0

0

0

0)(

)()(

0

Dt

Dppp

t

Dt

Dpp

Dt

Dppp

t

pDtD

p

Dt

Dpp

t

p

y

vp

x

u

pyx

Dt

D

yxp

Dt

Dpx

Dt

yDpy

Dt

xD

pyx

Dt

D

v

vv

v

Local time rate of change of the inverse of

static stability

Horizontal flux out from a volume of the inverse

of static stability

Vertical flux out from a volume of the inverse of static stability

4.2.3. Isentropic 좌표계에서의 연속방정식

→ →

→ → → →

→ →

→ inverse static stability – tendency equation

좌표계에서의

※ Static Stability

; parcel method

no mix with surrounding air

주위가 hydrostatic equilibrium

parcel : adiabatic

p

↓P

Cp

g

dz

dT

dpdTCdq

ad

d

p

0

z

pg

z

pg

dt

dw

A

1

0

1Parcel

environment

Ad

AA

dA

A

AA

A

Tgzdt

dw

zzTzT

zzTzT

zTzT

TTTgdt

dw

RTp

gdt

dw

/

)0()(

)0()(

)0()0(

/)(

)(

/)(

→ buoyancy force

Initial z,

d

d

d

A

d

z

T

; dry adiabatic lapse rate

; environment lapse rate

; stable

; neutral

; unstable

그런데 ,

ApA

A

A

A

A

A

RT

pg

pC

R

Tz

p

pz

T

Tz

zp

p

T

T

p

pT

111

...

, log 0 미분에서

dAA

d

A TTT

11

z

TAd

z

z

z

z

1

0

0

0 ; statically stable equilibrium

; unstable

; neutral equilibrium

; static stability quantity

)( d Warm

airCold air

)( d Warm

air

Cold air

0

0

0

p

p

p

; statically stable equilibrium

z

T

gp

T

Cp

T

C

R AA

p

A

p 1

,

pT

p

T

p

T

Tz

p

pz

T

Tz

zgzT

z

TgzTgz

dt

dw

AAA

AA

A

AA

Ad

...11

///

)( dg

02

2

2

2

zz

g

dt

zd

zz

g

td

zd

dt

dw

2

1

z

gN

; Brunt-väisälä frequency

0

sss

s

ppt

p

p

p

v

4.2.4. The kinematic boundary condition

1V

nV ˆ1

2V

nV ˆ2

; 질량 보존 원리밀도가 지속적으로 변하는 boundary 를 질러이 경계조건 , 역학적 원리를 포함하지 않고motion field 에서 결정

→ Kinematic boundary condition

4.2.5. The dynamic boundary condition→ finite P.G.F ; synoptic, sub-synoptic

Side 1

Side 2p1=p2 ; dynamic boundary condition

4.3 The Thermodynamic equation

4.3.1. Dry thermodynamics ; expansion

no heat exchange ; adiabatic ; compression

heating rate

Dt

DTC

dt

dQ

dpRdTpd

RdTdppd

RTp

RTp

Q

pdTCQ

p

v

0 0

0

p

p

0lim,

t

t

11 deg1004: kgJRCC vp

)( 22 smkg

pp

p

p

pp

Air parcelexpands ; increase

decreaseT

Air parcelrises

Air parcelsinks

Air parcelcontracks ; decrease

increaseT

dt

dQ

Cp

T

CT

dt

dQ

CCp

TT

t

T

p

TT

t

TC

dt

dQ

ppp

ppp

pp

1

1

v

v

v

quasihorizontal temperature advection

pC

; vertical displacement heat work done 과 관련된 adiabatic temperature change

p

T

; vertical temperature advection

dt

dQ

C p

1; diabatic heating

pT

p

T

CRT

RT

p

p

pC

Cp

R

p

p

p

p

pCp

T

C

p

CR

p

p

CRCR

pp

p

pp

ln

0

00

; Static stability parameter

)( RTp 그런데 ,

10

00

,

ppp

p

pp

T

p

pT

→ →

→ →

→ →

T=

,ln

ppp

RT

Static stability ≈ zero, neutral

lapse rate, ~

dry adiabatic lapse rate

p

1

dt

dQ

Cp

T

CT

t

T

ppp

1

v

p

T

dt

dQ

CR

PT

t

T

pp

1

v

R

PT

p

T

R

P

p

T

C p

,pCp

T

dpC

g

z

T

dz

T

;0

→ →

→ →

diabatic heating 이 중요한 경우 , 열역학 제 1 법칙

TDt

DT

T

C

Dt

DC

Dt

Ds

dt

dQ

Tp

p

ln1

dt

dp

lndCT

dQp

dpp

R

T

dTCd

C

p

p

C

R

T

dTd

p

pT

pp

p

1

0

dpp

R

T

dTC

T

dQ

dpdTCdQ

p

p

lnpCs s : specific entropy, (unit mass)

dt

dQ

TCpt

dt

dQ

TCDt

Ddt

dQ

TDt

DCDt

DT

R

pT

t

T

pp

p

p

p

v

v

1

ln

Potential temperature 에 관한 열역학 제 1 법칙

Density 가 p, T 둘 다의 함수 일 때 , baroclinic

verticaladvection

Adiabaticheating

TpRT

p,

quasi-horizontaltemps advection

cut across

solenoid

→ →

→ →

barotropic,

; thermal wind Ⅹ, no quasi horizontal temperature gradient

Tropic, subtropic, midlatitude 여름 ~ barotropic 근사

equivalent barotropic :

)(~ pTpR

pT

p

0 Tpv

Baroclinic

No geostrophic temperature advection

→ isotherm~height contour 에 나란 [ 그림 4.38]

→ geostrophic wind direction 이 같음 .

or (180°, opposite 모든 level 에서 barotropic vorticity eq. 만족

0 Tk pp

kf

T gpg

1,0 vv

→ →→ → ^

→ →

→→^

4.3.2. Moist thermodynamics

Moist, Mv = mass of water vapor Md = mass of dry air Ml = mass of liquid water Mi = mass of ice

Mixing ratio,

Specific humidity

Specific entropy,

ilv

iillvvd

ilvd

iillvvdd

vd

v

d

ii

d

ll

d

vv

rrr

srsrsrs

MMMM

sMsMsMsMs

MM

Mq

M

Mr

M

Mr

M

Mr

1

333

333

33

lnln

lnln

,

lnln

lnlnln

dp

pd

p

ppp

sepRTCconst

constepRTCs

eT

constpRTCs

pRdTdCp

dpR

T

dTCdCds

T

dQ

; Triple point

triple point 에서의 단위질량당 entropy

열역학적평형 ; reversible,

triple point; water vapor

liquid water

ice

333

lnln dpd sep

epR

T

TCs

dT

dIC

pddITdsdQ

p

dpR

T

dTCds

T

dQ

mbkPaeKT

v

p

)11.6(611.0,16.273 33

단위질량당 내부에너지

그런데 ,

; 물을 증발시키기 위해 , 열이 intermolecular bond 를 깰 수 있을 정도로 흡수되어야 한다 .

물의 단위 질량을 증발하는데 필요한 열 ,

- A vapor 의 internal energy liquid internal energy

; vaporigation 의 잠열 ①

ilv ,

)(

51.461,

)(

11

RTp

KkgJRTRe

L

eIILssT

vvv

v

vlvvlv

)10501.2( 16 kgJ

dIdTC

dT

dIC

T

T

T

T v

v

33

②

for vapor, ; 수증기의 정적비열

For liquid, ③

; liquid 의 비열

①, ②, ③ → A

33

33

TTCII

IITTC

vvvv

vvvv

)1410( 11 KkgJCvv

)( 33

33

TTCII

dIdTC

lll

T

T

T

T l

lC

333 TCTTCTRTCIILSST vvlvvvlvvlv

)1870( 11 KkgJTC pv