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EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
FILTERING OF EQUATIONS FOR ATMOSPHERE (CONT)
LECTURE 6
(Reference: Peixoto & Oort, Chapter 3)
Meridional 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
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2
)/v()v(/ˆ1/v z
zp
afudtd VH
222 //ˆ/u/ huluLpfLuVH
/ˆ10 pa
fu
Horizontal 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
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2
222 //ˆ/u/ huluLpfLuVH
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
Horizontal 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
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2
Geostrophic Balance
“Rossby Number”
Geostrophic Balance Holds when Ro << 1
(zonal) (meridional)
|u||/| 2
fLuRo
1
6410
101010
Lfu||||
222 //ˆ/u/ huluLpfLuVH
/cosˆ1v pa
f
/ˆ1 pa
fu
Horizontal Momentum Balance
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
/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 2cos/2/
ˆ1
ˆ1
pp
afp
afGV
Horizontal Momentum Balance
pkf
ˆˆ
1G V “Geostrophic Wind”
dPGF
CF
V
dp
fVG
1 2cos/2/
ˆ1
ˆ1
pp
afp
afGV
f=2sin
7.27x10-5 s-1
d=600 km
=5.6 m/s
p
ap
ap 1,
cos1
Example:
dp
fVG
1
Let us Revisit e.g. the Meridional 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
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2
)/v()v(/ˆ1/v z
zp
afudtd VH
222 //ˆ/u/ huluLpfLuVH
/ˆ10 pa
fuWhat if the acceleration/non-linear term cannot be neglected?
i.e., Ro 1
Let us Revisit e.g. the Meridional 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
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2
)/v()v(/ˆ1/v z
zp
afudtd VH
222 //ˆ/u/ huluLpfLuVH
/ˆ1v pa
fudtd This applies to flows
with strong curvature
What if the acceleration/non-linear term cannot be neglected?
i.e., Ro 1
Horizontal Momentum Balance:
/ˆ1v pa
fudtd
tcosx Rtsiny R
tsinu Rtcosv R
/cosˆ1vu pa
fdtd
This applies to flows with strong curvature
“Gradient Wind Balance”
Centripetal acceleration
(zonal)
(meridional)
ac=V2/RR
V=R
V=(u2+v2)1/2
tcos/ 2 Rdtdu tsin/v 2 Rdtd
Horizontal Momentum Balance:
tcosx Rtsiny R
tsinu Rtcosv R
This applies to flows with strong curvature
“Gradient Wind Balance”
Centripetal acceleration
(zonal)
(meridional)
ac=V2/RR
V=R
V=(u2+v2)1/2
tcos/ 2 Rdtdu tsin/v 2 Rdtd
/ˆ1tsin2p
afu
RV
/cosˆ1vtcos2
pa
fR
V
Horizontal Momentum Balance:
tcosx Rtsiny R
tsinu Rtcosv R
What about flow near the equator?
“Gradient Wind Balance”
Centripetal acceleration
(zonal)
(meridional)
ac=V2/RR
V=R
V=(u2+v2)1/2
tcos/ 2 Rdtdu tsin/v 2 Rdtd
/ˆ1tsin2p
afu
RV
/cosˆ1vtcos2
pa
fR
V
Horizontal Momentum Balance:
/ˆ1v padt
d
/cosˆ1u padt
d
“Cyclostrophic Balance”
What about flow near the equator?
tcosx Rtsiny R
tsinu Rtcosv R
ac=V2/R
V=R
V=(u2+v2)1/2
tcos/ 2 Rdtdu tsin/v 2 Rdtd
Centripetal acceleration
(zonal)
(meridional)
RNear equator
(e.g. Hurricane),
Coriolis Force is negligible,
and balance is between PGF
and Centripetal acceleration
Horizontal Momentum Balance
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
Generally an excellent approximation for ‘upper level winds’Any evidence of breakdown of Geostrophy?
Horizontal Momentum Balance
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
Relationship Between Temperature and Winds?
Advection
Horizontal Momentum Balance
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
Relationship Between Temperature and Winds?
Advection
dtdp
CpCplatq
Cpradq
zTz
T
zTwTtTdtdT
VH 1)/()(
/)(//
V
Horizontal Momentum Balance
/cosˆ1v pa
f
/ˆ1 pa
fu
Geostrophic Balance
(zonal) (meridional)
What can we say about this term?
dtdp
CpCplatq
Cpradq
zTz
T
zTwTtTdtdT
VH 1)/()(
/)(//
V
