AOSS 401, Fall 2006Lecture 10
September 28, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Homework 2 returned today
• Homework 3 due today (questions?)
• Homework 4 posted Monday
• Exam 1 October 10—covers chapters 1-3 in Holton
Weather
• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/
• National Weather Service– http://www.nws.noaa.gov/dtx/
Correction…
• I made a mistake in my last set of lectures (September 19th)
• Geostrophic wind is only non-divergent if pressure is the vertical coordinate…
• Corrected lecture 6 posted to ctools by Monday.
Today:Material from Chapter 3
• Natural coordinates
• Balanced flow
Another Coordinate System?
• We want to simplify the equations of motion• For horizontal motions on many scales, the
atmosphere is in balance– Mass (p, Φ) fields in balance with wind (u)– It is easy to observe the pressure or geopotential
height, much more difficult to observe the wind
• Balance provides a way to infer the wind from the observed (p, Φ)
• Wind is useful for prediction (remember the advection homework and in-class problems?)
The horizontal momentum equation
p
pp
pp
fDt
D
fuydt
d
fxdt
du
uku
v
v
Assume no viscosity and no vertical wind
Geostrophic balance
High Pressure
Low Pressure
Flow initiated by pressure gradient
Flow turned by Coriolis force
Geostrophic & observed wind 300 mb
Describe previous figure. What do we see?
• At upper levels (where friction is negligible) the observed wind is parallel to geopotential height contours.
• (On a constant pressure surface)
• Wind is faster when height contours are close together.
• Wind is slower when height contours are farther apart.
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
δΦ = Φ0 – (Φ0+2ΔΦ)
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
yy
2
The horizontal momentum equation
p
pp
pp
fDt
D
fuydt
d
fxdt
du
uku
v
v
Assume no viscosity
Geostrophic approximation
g
p
p
fuy
fx
gv
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
yfug
2
Geopotential (Φ) in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
yfug
Geopotential (Φ) in upper troposphere
• Think about the observed wind– Flow is parallel to geopotential height lines– There is curvature in the flow
Geostrophic & observed wind 300 hPa
Geopotential (Φ) in upper troposphere
• Think about the observed (upper level) wind– Flow is parallel to geopotential height lines– There is curvature in the flow
• Geostrophic balance describes flow parallel to geopotential height lines
• Geostrophic balance does not account for curvature
• How to best describe balanced flow with curvature?
Another Coordinate System?
• We want to simplify the equations of motion• For horizontal motions on many scales, the
atmosphere is in balance– Mass (p, Φ) fields in balance with wind (u)– It is easy to observe the pressure or geopotential
height, much more difficult to observe the wind
• Balance provides a way to infer the wind from the observed (p, Φ)
• Need to describe balance between pressure gradient, coriolis, and curvature
“Natural” Coordinate System
• Follow the flow
• From hydrodynamics—assumes no local changes – No local change in geopotential height– No local change in wind speed or direction
• Assume– Horizontal flow only (no vertical component)– No friction
Return to Geopotential (Φ) in upper troposphere
eastwest
Φ0+3ΔΦ
Φ0
ΔΦ > 0
south
northDefine one component of the horizontal wind as
tangent to the direction of the wind. t
t t
t
Return to Geopotential (Φ) in upper troposphere
eastwest
Φ0+3ΔΦ
Φ0
ΔΦ > 0
south
north
t t
t
Define the other component of the horizontal wind
as normal to the direction of the wind. n
nn n
• Regardless of position (i,j)– t always points in the direction of flow– n always points perpendicular to the direction of
the flow toward the left
• Remember the “right hand rule” for vectors? Take k x t to get n
• Assume– Pressure as a vertical coordinate– Flow parallel to contours of geopotential height
“Natural” Coordinate System
• Advantage: We can look at a height (on a pressure surface) and pressure (on a height surface) and estimate the wind.– It is difficult to directly measure winds– We estimate winds from pressure (or
hydrostatically equivalent height), a thermodynamic variable.
– Natural coordinates are useful for diagnostics and interpretation.
“Natural” Coordinate System
“Natural” Coordinate System
• For diagnostics and interpretation of flows, we need an equation…
Return to Geopotential (Φ) in upper troposphere
eastwest
ΔΦ > 0
south
north
t t
t
Geostrophic assumption. Do you notice that those n vectors point towards something out in the distance?
nn n
HIGH
Low
Return to Geopotential (Φ) in upper troposphere
eastwestsouth
north
HIGH t t
tnn nLow
Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.
Time to look at themathematics
First simplification: the velocity
• Always positive
• Always points in the positive t direction
V
tV
V
VDefine velocity as:
Definition of magnitude:
One direction: no (u,v)
Goal: Quantify Acceleration
Dt
DV
Dt
DV
Dt
DDt
VD
Dt
D
tt
V
tV
)(
acceleration is:
Change in speed
Change in Direction
(Chain Rule)
How to get as a function of V, R ?Dt
Dt
Remember our circle geometry…
this is not rotation of the Earth!It is an element of curvature in the flow.
Δφ
R=radius of curvature t
t
Δtt+Δt
Δs
Δs=RΔφ
Remember our circle geometry…
this is not rotation of the Earth!It is an element of curvature in the flow.
Δφ
R=radius of curvature t
t
Δtt+Δt
Δs
Δs=RΔφ
n
n
tt
t
R
s
Remember our circle geometry…
If Δs is very small, Δt is parallel to n.So, Δt points in the direction of n
Δφ
R=radius of curvature t
t
Δtt+Δt
Δs
Δs=RΔφ
n
n
tt
t
R
s
nt
ntt
nt
tt
t
t
R
V
Dt
D
VDt
DsDt
Ds
RDt
Ds
Ds
D
Dt
D
RDs
D
s
t
R
s
RVDt
D
st
lim
), of (function ?
0,
Remember, we want anexpression for
From circle geometrywe have:
Rearrange and take the limit
Use the chain rule
Remember the definitionof velocity
Goal: Quantify Acceleration
ntV
nt
tt
V
tV
R
V
Dt
DV
Dt
D
R
V
Dt
DDt
DV
Dt
DV
Dt
DDt
VD
Dt
D
2
)(
acceleration defined as:
(Chain Rule)
We just derived:
So the total acceleration is
Acceleration in Natural Coordinates
ntV
R
V
Dt
DV
Dt
D 2
Along-flowspeed change
?
Acceleration in Natural Coordinates
222
2
1
RRRR
V
RDt
DRV
RsDt
DsV
R
V
Dt
DV
Dt
D
nn
ntVThe total
acceleration is
Definition ofwind speed
Circle geometry
Plug in for Δs
Centrifugal force
angular velocity
angle of rotation
Acceleration in Natural Coordinates
CentrifugalAcceleration
ntV
R
V
Dt
DV
Dt
D 2
Along-flowspeed change
We have seen that Coriolis force is normal to the velocity.
nVk fVf - force coriolis
Pressure gradient (by definition)
nsp nt
The horizontal momentum equation
nsfV
R
V
Dt
DV
fDt
Dp
ntnnt
uku
2
onssubstitutiour all make we
The horizontal momentum equation(in natural coordinates)
nfV
R
V
sDt
DV
nsfV
R
V
Dt
DV
2
2
formcomponent in and
ntnnt
Along-flowdirection (t)
Across-flowdirection (n)
• Simplification?
• Which coordinate system is easier to interpret?
fuyDt
D
fxDt
Du
pp
pp
v
v
nfV
R
V
sDt
DV
2
• We are only looking at flow parallel to geopotential height contours
00
nfV
R
V
2
• Simplification?
• Which coordinate system is easier to interpret?
• We are only looking at flow parallel to geopotential height contours
fuyDt
D
fxDt
Du
pp
pp
v
v
nfV
R
V
2
Curved flow (Centrifugal Force)
Coriolis Pressure Gradient
One Diagnostic Equation
Uses of Natural Coordinates
• Geostrophic balance– Definition: coriolis and pressure gradient in
exact balance.– Parallel to contours straight line R is
infinite
nfV
R
V
2
0
Geostrophic balance in natural coordinates
nfV
Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight.
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
northn
fVg
Δn
Therefore
• If the contours are curved then the real wind is not geostrophic.
How does curvature affect the wind?(cyclonic flow/low pressure)
nfV
R
V
2
R
t
n
Δn
Φ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low
1. Mathematical Perspective
ag
gag
agg
fVR
V
nfVfV
R
V
nVVf
R
V
nfV
R
V
2
2
2
2
Equation of motion
Split Coriolis intogeostrophic and
ageostrophic parts
Use definition of geostrophic wind
1. Mathematical Perspective
g
agg
ag
ag
VV
VVV
V
fVR
V
0
0 0
2Total centrifugal force balances ageostrophic
part of coriolis
Total wind is sum of its parts
Real wind speed is slower than geostrophic
for cyclonic flow!
Look at sign of terms (R > 0)
2. Physical Perspective
V
PGF
COR
Geostrophic balance Add curvature (centrifugal force)
V
PGF
COR
CEN
Pressure gradient force is the same in each case. With curvature less coriolis force is needed to balance the pressure gradient.
gfV fV>
Geostrophic & observed wind 300 hPa
Geostrophic & observed wind 300 hPa
Observed:95 knots
Geostrophic:140 knots
How does curvature affect the wind?(anticyclonic flow/high pressure)
nfV
R
V
2
R
t
n
Δn
Φ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low
1. Mathematical Perspective
g
agg
ag
ag
VV
VVV
V
fVR
V
0
0 0
2
Total wind is sum of its parts
Real wind speed is faster than geostrophic
for anticyclonic flow!
Look at sign of terms (R > 0)
Total centrifugal force balances ageostrophic
part of coriolis
2. Physical Perspective
V
PGF
COR
Geostrophic balance Add curvature
V
PGF
COR
CEN
Pressure gradient force is the same in each case. With curvature more coriolis force is needed to balance the pressure gradient.
gfV fV<
Geostrophic & observed wind 300 hPa
Geostrophic & observed wind 300 hPa
Observed:30 knots
Geostrophic:25 knots
What did we just do?
• Found a way to describe balances between pressure gradient, coriolis, and curvature
• We assumed friction was unimportant and only looked at flow at a particular level
• We assumed flow was on pressure surfaces• We saw that the simplified system can be used
to describe real flows in the atmosphere• Can we describe other flow patterns? (Different
scales? Different regions of the Earth?)
Next Time:Finish balanced flows
• Cyclostrophic flow (tornados, water spouts, dust devils)
• Gradient wind (general description of curved flow anywhere on the globe—if friction is not important…)