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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
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