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EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
FILTERING OF EQUATIONS OF MOTION FOR ATMOSPHERE
(CONT)
LECTURE 7
(Reference: Peixoto & Oort, Chapter 3,7)
Geostrophic Balance
/cosˆ1v pa
f
/cosˆ1v
Gp
af
/ˆ1 pa
fu
/ˆ1 paf
uG
pkf
ˆˆ
1G V
p
ap
ap 1,
cos1
“Geostrophic Wind”
dPGF
CF
V
dp
fVG
1
Recall from previous lecture…
Exercise:
0GV
under Boussinesq approximation and assumption f[constant]
show
Defines a ‘streamfunction’
0GV
Defines a ‘streamfunction’
Winds don’t parallel the streamfunction!
under Boussinesq approximation and assumption f[constant]
CONVERGENCE AND DIVERGENCE
Northern or Southern Hemisphere?
Winds don’t parallel the streamfunction!
CONVERGENCE AND DIVERGENCE
Northern or Southern Hemisphere?
0)/v()v(/ˆ1
zz
pa
fu VH
0)/()(/cosˆ1v
zuz
upa
f VH
Quasigeostrophic
CONVERGENCE AND DIVERGENCE
Northern or Southern Hemisphere?
0)/v()v(/ˆ1
zz
pa
fu VH
0)/()(/cosˆ1v
zuz
upa
f VH
Near the surface, friction leads to horizontal convergence
CONVERGENCE AND DIVERGENCE
0)/v()v(/ˆ1
zz
pa
fu VH
0)/()(/cosˆ1v
zuz
upa
f VH
Near the surface, friction leads to horizontal convergence
Quasigeostrophic
CONVERGENCE AND DIVERGENCE
0)/v()v(/ˆ1
zz
pa
fu VH
0)/()(/cosˆ1v
zuz
upa
f VH
Near the surface, friction leads to horizontal convergence
Relationship between horizontal convergence and vertical motion
Quasigeostrophic
dzdw
dtd V
1
22 //ˆ/u'/ hwlwHpgfLwuVHs
Vertical Momentum Balance:
Length scale: L106m, l102m
Depth scale: H104m, h 102m
Horizontal velocity scale: u,v 10 ms-1
Vertical velocity scale: w 10-2 ms-1
Horizontal pressure scale: p 10 mb = 1000 Pa
Time Scale: L/u 105s or H/w 106s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Air: 1 kg m-3
Horizontal Eddy Viscosity: H 10-1 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
10-7 ms-2 10-3 ms-2 10 ms-2 10 ms-2 10-7 ms-2 10-7 ms-2
)/()w(/ˆ1
ˆ'/ zw
zzpgufdtdw VH
zpg /ˆ1
ˆ0
Vertical Momentum Balance:
Length scale: L106m, l102m
Depth scale: H104m, h 102m
Horizontal velocity scale: u,v 10 ms-1
Vertical velocity scale: w 10-2 ms-1
Horizontal pressure scale: p 10 mb = 1000 Pa
Time Scale: L/u 105s or H/w 106s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Air: 1 kg m-3
Horizontal Eddy Viscosity: H 10-1 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
)/()w(/ˆ1
ˆ'/ zw
zzpgufdtdw VH
zpg /ˆ1
ˆ0
gzp /Hydrostatic Balance
Vertical Momentum Balance:ATMOSPHERIC
PRESSUREgzp / TRp
d
TRpgzpd
//
dzTRgppd
]/[/
)(___
)/1()/()/ln(00zzTRgpp
d
shzzpp /)(exp00 Hypsometric Equation
shzz /)(0
km3.8 “Scale height”
What’s the solution?
___)/1(/ TgRh
dswhere
gzp /Hydrostatic Balance
and
Combining these,
rearranging,
Vertical Momentum Balance:
shzzpp /)0
(exp0 Hypsometric Equation
rearrangingTdRp/
TdRshzzp //)(exp
00
shzz /)(exp00
TdRp /
00where
gzp / TRpd
and ATMOSPHERICPRESSURE
Vertical Momentum Balance:
shzzpp /)0
(exp0 Hypsometric Equation
rearrangingTdRp/
TdRshzzp //)(exp
00
shzz /)(exp00
TdRp /
00where
gzp / TRpd
and
Vertical Momentum Balance (revisited):
Length scale: L106m, l102m
Depth scale: H104m, h 102m
Horizontal velocity scale: u,v 10 ms-1
Vertical velocity scale: w 10-2 ms-1
Horizontal pressure scale: p 10 mb = 1000 Pa
Time Scale: L/u 105s or H/w 106s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Air: 1 kg m-3
Horizontal Eddy Viscosity: H 10-1 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
10-7 ms-2 10-3 ms-2 10 ms-2 10 ms-2 10-7 ms-2 10-7 ms-2
)/()w(/ˆ1
ˆ'/ zw
zzpgufdtdw VH
22 //ˆ/u'/ hwlwHpgfLwuVHs
zpg /ˆ1
ˆ0
?
Vertical Momentum Balance (revisited):
Length scale: L106m, l102m
Depth scale: H104m, h 102m
Horizontal velocity scale: u,v 10 ms-1
Vertical velocity scale: w 10-2 ms-1
Horizontal pressure scale: p 10 mb = 1000 Pa
Time Scale: L/u 105s or H/w 106s
Radius of Earth: a=6.37x 106m
Coriolis parameter: f,f' 10-4 s-1
Density of Air: 1 kg m-3
Horizontal Eddy Viscosity: H 10-1 m2s-1
Vertical Eddy Viscosity: V 10-1 m2s-1
10-7 ms-2 10-3 ms-2 10 ms-2 10 ms-2 10-7 ms-2 10-7 ms-2
)/()w(/ˆ1
ˆ'/ zw
zzpgufdtdw VH
22 //ˆ/u'/ hwlwHpgfLwuVHs
zpgdtdw /
ˆ1
ˆ ?
Vertical Momentum Balance (revisited):
zpgdtdw /
ˆ1
ˆ
Consider a parcel displaced displaced from hydrostatic equilibrium:
'www ' pp0dtwd
zpg /ˆ1
ˆ0
zpgdtdw /
ˆ1
ˆ'
ˆ
)(' gdtdw
zpg /ˆ1
ˆ0
(1)
(2)
(2)-(1)
g
dtdw )('
Buoyancy Force
Vertical Momentum Balance (revisited):
zpgdtdw /
ˆ1
ˆ
Consider a parcel displaced displaced from hydrostatic equilibrium:
g
dtdw )('
Buoyancy Force
TRpd
TRpd
/
TdTd //
TTT
)(
Vertical Momentum Balance (revisited):
zpgdtdw /
ˆ1
ˆ
Consider a parcel displaced displaced from hydrostatic equilibrium:
Buoyancy Force
TRpd
TRpd
/
TdTd //
TTT
)(
TgTT
dtdw )('
Vertical Momentum Balance (revisited):
Now, consider the Thermodynamics:
dtdp
CpCplatq
Cpradq
zTz
TdtdTVH 1)/()(/
We consider parcel motion with no diffusion of heat and no fluxes of heat across the parcel boundary (Q=0):
“Adiabatic”
dtpd
pCR
TQpC
dtTd d ln/1/ln
For an ideal gas we can rewrite this:
dtdp
CpCpQ
1
0ln/ln dtpd
pCR
dtTd d )/(/00ppTT
CpRd
[or “isentropic” (since ds/dt=Q/T)]
Vertical Momentum Balance (revisited):
Now, consider the Thermodynamics:
dtdp
CpCplatq
Cpradq
zTz
TdtdTVH 1)/()(/
dtpd
pCR
TQpC
dtTd d ln/1/ln
For an ideal gas we can rewrite this:
dtdp
CpCpQ
1
Potential Temperature )( /0ppTDefine
What is useful about this quantity?
dtpddtTddtd ln/ln/ln
CpRd
TQCp
/1
TQpCdt
d /11
is conserved for adiabatic motion
Vertical Momentum Balance (revisited):
Now, consider the Thermodynamics:
11
2
kgJK1005
ms81.9/
CpgdzdT
addDry Adiabatic lapse rate
dtdp
CpCpQdtdT
1/ for adiabatic motiondtdp
Cp1
dpCp
dT 1
dzdp
CpdzdT
1/
gdzdp But
Tgzdtdwd
/)(/'
dCpgdzdT /
TgTT
dtdw )(' Recall
m/K01.0
AdTgzz /)(
0 zz )(dT
g Stability Properties?
AssumezTT
0
zTTd
0
Vertical Momentum Balance (revisited):
Now, consider the Thermodynamics:
Tgzdtdwd
/)(/'
TgTT
dtdw )(' Recall
AdTgzz /)(
0 zz )(dT
g Stability Properties?
AssumezTT
0
zTTd
0
)( /0ppT Cp
Rd
Exercise:
dA
z
T
Show
Thus:
0z
0z
0z
stable
neutral
unstable
Vertical Momentum Balance (revisited):
0z
0z
0z
stable
neutral
unstabledA
z
T
Vertical Momentum Balance (revisited):
0z
0z
0z
stable
neutral
unstabledA
z
T
sE
E
A
z
T
