AOSS 401, Fall 2007Lecture 24
November 07, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News November 07, 2007
• Homework 6 (Posted this evening)– Due Next Monday
• Important Dates: – November 16: Next Exam (Review on 14th)– November 21: No Class– December 10: Final Exam
Weather
• National Weather Service– http://www.nws.noaa.gov/– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html
• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/getForecast?
query=ann+arbor
– Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US
Rest of Course
• Wrap up quasi-geostrophic theory (Chapter 6)– Potential vorticity– Vertical velocity– Will NOT do Q vectors
• We will have a lecture on the Eckman layer (Chapter 5)– Boundary layer, mix friction with rotation
• We will have a lecture on Kelvin waves (Chapter 11)– A long wave in the tropics
• There will be a joint lecture with 451 on hurricanes (Chapter 11)
• Computer homework (perhaps lecture) on modeling• Special topics?
Material from Chapter 6
• Quasi-geostrophic theory
• Quasi-geostrophic vorticity– Relation between vorticity and geopotential
• Geopotential prognostic equation
• Quasi-geostrophic potential vorticity
Scaled equations in pressure coordinates (The quasi-geostrophic (QG) equations)
Dg
v g
Dt f0
k v a y
k v g
v g
1
f0
k
ua
x va
y p0
tv g
p
J
p
with R
c p
and stability parameter =RdTo
p
d ln0 dp
momentum equation
continuity equation
thermodynamicequation
geostrophic wind
v v g
v a with
v a
v g O(Ro) 0.1
Dv
Dt
Dg
v g
Dt with
Dg
Dtt ug
x vg
y
v g
1
f0
k
Midlatitude - plane approximation :
f f0 fy
0
y f0 y = f0 2cos(0)
ay
with y = a( - 0) (a = radius of the earth)
constant f0 2sin(0)
Ttot (x,y, p, t)T0(p)T(x,y, p, t) with T0
pTp
Approximations in the quasi-geostrophic (QG) theory
Quasi-geostrophic equations cast in terms of geopotential
and omega.
)1
(1
)()()(
2
00
2
0
0000
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
g
g
V
V
THERMODYNAMIC EQUATION
VORTICITY EQUATION
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
GEOPOTENTIAL TENDENCY EQUATION
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((2
02
00
202
p
f
pf
ff
tp
f
p gg
VV
f0 * Vorticity Advection
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((2
02
00
202
p
f
pf
ff
tp
f
p gg
VV
Thickness Advection
Advection of vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Advection of ζ tries to propagate the wave this way
Advection of f tries to propagate the wave this way
Relationship between upper troposphere and surface
vorticity advection
thickness advection
To think about this
• Read and re-read pages 174-176 in the text.
Idealized vertical cross section
Great web page with current maps:
Real baroclinic disturbances
http://www.meteoblue.ch/More-Maps.79+M5fcef4ad590.0.html
Personalize your maps (create a login):http://my.meteoblue.com
Real baroclinic disturbances:850 hPa temperature and geopot. thickness
warm airadvectioneast of thesurface low,enhances the ridge
cold airadvection,enhances trough
Real baroclinic disturbances:500 hPa rel. vorticity and mean SLP
sea level pressure
Positivevorticity, pos. vorticityadvection,increase incyclonicvorticity
Real baroclinic disturbances:500 hPa geopot. height and mean SLP
Upper level systemslags behind (to the west):system stilldevelops
With the benefit of hindsight and foresight let’s look back.
g
t f0
pv g ( g f )
g
t f0
pv g g vg
QG vorticity equation
Advection of relative vorticity
Advection of planetary vorticity
Stretchingterm
Competing
THINKING ABOUT THESE TERMS
g
t f0
pv g ( g f )
g
t f0
pv g g vg
QG vorticity equation
Advection of relative vorticity
Advection of planetary vorticity
Stretchingterm
Competing
WHAT ABOUT THIS
TERM?
Consider our simple form of potential vorticity
vorticitypotential
0)(
H
fH
f
Dt
Dhorizontal
From scaled equation, with assumption of constant density and temperature.
There was the assumption that the layer of fluid was shallow.
Fluid of changing depth
What if we have something like this, but the fluid is an ideal gas?
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((2
02
00
202
p
f
pf
ff
tp
f
p gg
VV
Still looks a lot like time rate of change of vorticity
Quasi-Geostrophicpotential vorticity (PV) equation
Simplify the last term of the geopotential tendencyequation by applying the chain rule:
v g
p
f02
p
f02
v gp
p
= 0 Why?
stability parameter =RdTo
p
d ln0 dp
Quasi-Geostrophicpotential vorticity (PV) equation
Simplify the last term of the geopotential tendencyequation by applying the chain rule:
v g
p
f02
p
f02
v gp
p
= 0 Why?
stability parameter =RdTo
p
d ln0 dp
THERMAL WIND RELATION
ppf g kv
0
Quasi-Geostrophicpotential vorticity (PV) equation
qtv g q
Dg
Dtq0
Simplify the last term of the geopotential tendencyequation by applying the chain rule:
v g
p
f02
p
f02
v gp
p
q1
f0
2 f p
f0
p
= 0
Leads to the conservation law:
Quasi-geostrophic potential vorticity:
Conserved following the geostrophic motion
Why?
stability parameter =RdTo
p
d ln0 dp
Imagine at the point flow decomposed into two “components”
A “component” that flows around the point.
Vorticity
• Related to shear of the velocity field.∂v/∂x-∂u/∂y
Imagine at the point flow decomposed into two “components”
A “component” that flows into or away from the point.
Divergence
• Related to stretching of the velocity field.∂u/∂x+∂v/∂y
Potential vorticity (PV): Comparison
q1
f0
2 f p
f0
p
Quasi-geostrophic PV:
THESE ARE LIKE STRETCHING IN THE VERTICAL
Barotropic PV:
PV g f
h
s-1
PV ( f )( gp
)Ertel’s PV:
m-1s-1
Units:
K kg-1 m2 s-1
g
t f0
pv g ( g f )
g
t f0
pv g g vg
QG vorticity equation
Advection of relative vorticity
Advection of planetary vorticity
Stretchingterm
Competing
WHAT ABOUT THIS
TERM?
Fluid of changing depthWhat if we have something like this, but the fluid is
an ideal gas?
Conversion of thermodynamic energy to vorticity, kinetic energy. Again the link between the thermal
field and the motion field.
Two important definitions
• barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature (idealized assumption).= (p)
• baroclinic – density depends on pressure and temperature (as in the real world).= (p,T)
Barotropic/baroclinic atmosphere
Barotropic: pp + pp + 2p
pp + pp + 2p
T+2TT+TT
T
T+2TT+T
Baroclinic:
ENERGY IN HERE THAT IS CONVERTED TO MOTION
Barotropic/baroclinic atmosphere
Barotropic: pp + pp + 2p
pp + pp + 2p
T+2TT+TT
T
T+2TT+T
Baroclinic:
DIABATIC HEATING KEEPS BUILDING THIS UP
• NOW WOULD BE A GOOD TIME FOR A SILLY STORY
VERTICAL VELOCITY
Dp
Dtptv h p w
pz
ptv g p
v a p wg
ptv a p wg
with the help of scale analysis (free troposphere)
wg
Vertical motions: The relationship between w and
= 0 hydrostatic equation
≈ 10 hPa/d≈ 1m/s 1Pa/km≈ 1 hPa/d
≈ 100 hPa/d
p v h
p0
p (v g
v a )
p0
ug
xvg
y
p
p (v a )
p0
x
(1
f
y
)y
(1
f
x
)
p
p (v a )
p0
assume f is approximately constant
p p (
v a )
if v h
v g
p0
Links the horizontal and vertical motions. Since geostrophy is such a good balance, the vertical motion is linked to the divergence of the ageostrophic wind (small).
= 0
Link between and the ageostrophic wind
Vertical pressure velocity
( p1)(p2) (p1 p2) u
x v
y
p
For synoptic-scale (large-scale) motions in midlatitudesthe horizontal velocity is nearly in geostrophic balance.Recall: the geostrophic wind is nondivergent (for constant Coriolis parameter), that is
Horizontal divergence is mainly due to small departures from geostrophic balance (ageostrophic wind).Therefore: small errors in evaluating the winds <u> and <v>
lead to large errors in . The kinematic method is inaccurate.
v g
ug
xvg
y0
Think about this ...
• If I have errors in data, noise.
• What happens if you average that data?
• What happens if you take an integral over the data?
• What happens if you take derivatives of the data?
Estimating the vertical velocity: Adiabatic Method
Start from thermodynamic equation in p-coordinates:
Sp 1 Tt uTx vTy
- (Horizontal temperature advection term)
Sp:Stability parameter
Tt uTx vTy Sp
J
c p
Assume that the diabatic heating term J is small (J=0), re-arrange the equation
Estimating the vertical velocity: Adiabatic Method
If T/t = 0 (steady state), J=0 (adiabatic) and Sp > 0 (stable):• then warm air advection: < 0, w ≈ -/g > 0 (ascending air)• then cold air advection: > 0, w ≈ -/g < 0 (descending air)
If local time tendency Tt0 (steady state)
Sp 1 u
Tx vTy
v h T
Sp
Horizontal temperature advection term
Stability parameter
Adiabatic Method
• Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong.
Estimating the vertical velocity: Diabatic Method
Start from thermodynamic equation in p-coordinates:
pp c
JS 1
Diabatic term
Tt uTx vTy Sp
J
c p
If you take an average over space and time, then the advection and time derivatives tend to cancel out.
mean meridional circulation
Conceptual/Heuristic Model
Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002
•Observed characteristic behavior•Theoretical constructs•“Conservation”
•Spatial Average or Scaling•Temporal Average or Scaling
YieldsRelationship between parameters if observations and theory are correct
One more way for vertical velocity
Quasi-geostrophic equations cast in terms of geopotential
and omega.
)1
(1
)()()(
2
00
2
0
0000
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
g
g
V
V
THERMODYNAMIC EQUATION
VORTICITY EQUATION
ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA
Quasi-Geostrophic Omega Equation
1.) Apply the horizontal Laplacian operator to the QG thermodynamic equation
2.) Differentiate the geopotential height tendency equation with respect to p
3.) Combine 1) and 2) and employ the chain rule of differentiation (chapter 6.4.1 in Holton, note factor ‘2’ is missing in Holton Eq. (6.36), typo)
2 f0
2
2
2 p
2 f0
v gp
1
f0
2 f
Advection of absolute vorticityby the thermal wind
Vertical Velocity Summary
• Though small, vertical velocity is in some ways the key to weather and climate. It’s important to waves growing and decaying. It is how far away from “balance” the atmosphere is.
• It is astoundingly difficult to calculate. If you use all of these methods, they should be equal. But using observations, they are NOT!
• In fact, if you are not careful, you will not even to get them to balance in models, because of errors in the numerical approximation.
One more summary of the mid-latitude wave
Idealized (QG) evolution of a baroclinic disturbance(Read and re-read pages 174-176 in the text.)
L
H
+ warm airadvection
- cold airadvection
- neg. vorticityadvection
+ pos. vorticityadvection
500 hPageopotential
p at the surface
p at the surface
Waves• The equations of motion contain many forms
of wave-like solutions, true for the atmosphere and ocean
• Some are of interest depending on the problem: Rossby waves, internal gravity (buoyancy) waves, inertial waves, inertial-gravity waves, topographic waves, shallow water gravity waves
• Some are not of interest to meteorologists, e.g. sound waves
• Waves transport energy, mix the air (especially when breaking)
Waves
• Large-scale mid-latitude waves, are critical for weather forecasting and transport.
• Large-scale waves in the tropics (Kelvin waves, mixed Rossby-gravity waves) are also important, but of very different character.
• This is true for both ocean and atmosphere. • Waves can be unstable. That is they start to
grow, rather than just bounce back and forth.
• And, with that, Chapter 6, of Jim Holton’s book rested comfortably in the mind of the students.
Below
• Basic Background Material
Couple of Links you should know about
• http://www.lib.umich.edu/ejournals/– Library electronic journals
• http://portal.isiknowledge.com/portal.cgi?Init=Yes&SID=4Ajed7dbJbeGB3KcpBh– Web o’ Science
A nice schematic
• http://atschool.eduweb.co.uk/kingworc/departments/geography/nottingham/atmosphere/pages/depressionsalevel.html
Mid-latitude cyclones: Norwegian Cyclone Model
• http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm
Tangential coordinate system
Ω
R
Earth
Place a coordinate system on the surface.
x = east – west (longitude)y = north – south (latitude)
z = local vertical orp = local vertical
Φ
a
R=acos()
Tangential coordinate system
Ω
R
Earth
Relation between latitude, longitude and x and y
dx = acos() dis longitudedy = ad is latitude
dz = drr is distance from center of a “spherical earth”
Φ
a
f=2Ωsin()
=2Ωcos()/a
Equations of motion in pressure coordinates(using Holton’s notation)
written)explicitlynot (often
pressureconstant at sderivative horizontal and time
; )()
re temperatupotential ; velocity horizontal
ln ;
0)(
Dt
Dp
ptDt
D( )
vu
pTS
p
RT
p
c
JST
t
TS
y
Tv
x
Tu
t
T
ppy
v
x
u
fDt
D
pp
p
ppp
p
V
jiV
V
V
VkV
Scale factors for “large-scale” mid-latitude
s 10 /
m 10
m 10
! s cm 1
s m 10
5
4
6
1-
-1
UL
H
L
unitsW
U
1-1-11-
14-0
2
3-
sm10
10
10/
m kg 1
hPa 10
y
f
sf
P
Scaled equations of motion in pressure coordinates
pg
aa
gagg
g
c
R
p
J
pt
py
v
x
u
yfDt
D
f
;
0
1
0
0
V
VkVkV
kV Definition of geostrophic wind
Momentum equation
Continuity equation
ThermodynamicEnergy equation
Weather
• National Weather Service– http://www.nws.noaa.gov/– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html
• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/getForecast?
query=ann+arbor
– Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US