Upload
denis-cuthbert-scott
View
220
Download
0
Tags:
Embed Size (px)
Citation preview
AOSS 401, Fall 2007Lecture 21
October 31, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News October 31, 2007
• Homework 5 (Due Friday)– Posted to web– Computing assignment posted to ctools under
the Homework section of Resources
• Next Test November 16
• Thanks for the comments on the Midterms evaluations
A Diversion
• Santa Ana Winds
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
Material from Chapter 6
• Quasi-geostrophic theory
• Quasi-geostrophic vorticity– Relation between vorticity and geopotential
Quasi-Geostrophic
• Derive a system of equations that is close to geostrophic and hydrostatic balance, but includes the effects of ageostrophic wind
• Comes from scale analysis of equations of motion in pressure coordinates
• Scale analysis = make assumptions(where do these assumptions break down?)
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
From the scale analysis we introduced the non-dimensional Rossby number
114
15
00
0
1010
10
Number Rossby
s
s
Lf
U
f
RoLf
U
A measure of planetary vorticity compared to relative vorticity.A measure of the importance of rotation.
Scale analysis of equations in pressure coordinates
• Start:– horizontal flow is approximately geostrophic– vertical velocity much smaller than horizontal
velocity
We will scale the material derivative
yv
xu
tDt
( )D
Dt
( )D
tDt
D( )
Dt
Dp
ptDt
D( )
ggg
gpgp
pp
)()
; )()
V
V
Ignore // small
This is for use in the advection of temperature and momentum.
ω comes from div(ageostrophic wind)
Variation of Coriolis parameter
• L, length scale, is small compared to the radius of the Earth
• In the calculation of geostrophic wind, assume f is constant; f = f0
• We cannot assume f is constant in the Coriolis terms…
...0 ydy
dfff
Variation of Coriolis parameter
0
0
00
0
at 0
cos2)
...
0
y
ady
df
yfydy
dfff
ydy
dfff
Variation of Coriolis parameter
1sin
cos
0
0
0
00
Roa
L
f
L
yfydy
dfff
Scale of first two terms.
Continuity equation becomes
0
0)(
py
v
x
u
ppy
v
x
u
aa
p
V
Thermodynamic equation
pTS
c
JST
t
TS
y
Tv
x
Tu
t
T
p
ppp
ln
V
geostrophic wind can be used here.
static stability, Sp, is large; ω cannot be ignored
Thermodynamic equation(use the fact that atmosphere is near hydrostatic balance)
p
T
dp
dT
tpyxTpTtpyxT
pTS
c
JST
t
TS
y
Tv
x
Tu
t
T
tot
p
ppgpgg
0
0 ),,,()(),,,(
ln
V
split temperature into basic state plus deviation
Thermodynamic equation(and with the hydrostatic equation)
pg
pg
pg
c
R
p
J
pt
c
J
R
p
ptR
p
dp
d
p
RT
p
RT
p
c
J
R
pT
t
T
;
ln ; 00
V
V
V
note the inverse relation of heating with pressure
The Momentum Equation
horizontal flow is approximately geostrophic
• L, length scale, is small compared to the radius of the Earth
• In the calculation of geostrophic wind, assume f is constant; f = f0
windicageostroph
1
0
a
ag
g f
V
VVV
kV
horizontal flow is approximately geostrophic
110
Rog
a
ag
V
V
VV
Forcing terms in momentum equation
gag
ag
ag
fyf
yf
ff
fDt
D
VkVVk
VVk
VVkVk
VkV
00
0
)()(
)()(
)(
approx of coriolis parameter Use definition of geostrophic wind
in the pressure gradient force
def’n of the full wind
Forcing terms in momentum equation
ga
gag
ag
ag
yf
fyf
yf
ff
VkVk
VkVVk
VVk
VVkVk
0
00
0
)()(
)()(
)(
Approximate horizontal momentum equation
gagg yf
Dt
DVkVk
V 0
This equation states that the time rate of change of the geostrophic wind is related to1. the coriolis force due to the ageostrophic
wind and 2. the part of the coriolis force due to the
variability of the coriolis force with latitude and the geostrophic wind.
Both of these terms are smaller than the geostrophic wind itself.
A Point
• All of the terms in the equation for the CHANGE in the geostrophic wind, which is really a measure of the difference from geostrophic balance, are order Ro (Rossby number).
– Again, reflects the importance of rotation to the dynamics of the atmosphere and ocean
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
What is the point?
• Set of equations that describes synoptic-scale motions and includes the effects of ageostrophic wind (vertical motion)
Scale Analysis = Make Assumptions
Quasi-geostrophic system is good for:
• Synoptic scales
• Middle latitudes
• Situations in which Va is important
• Flows in approximate geostrophic and hydrostatic balance
• Mid-latitude cyclones
Scale Analysis = Make Assumptions
Quasi-geostrophic system is not good for:
• Very small or very large scales
• Flows with large vertical velocities
• Situations in which Va ≈ Vg• Flows not in approximate geostrophic and
hydrostatic balance
• Thunderstorms/convection, boundary layer, tropics, etc…
What will we do next?
• Derive a vorticity equation for these scaled equations.– Actually provides a “suitable” prognostic
equation because need to include div(ageostrophic wind) in the prognostics.
– Remember the importance of divergence in vorticity equations.
Derive a vorticity equation
• Going to spend some time with this.
Vorticity
fy
u
x
v
a
Uk
vorticityofcomponent Vertical
relative vorticityvelocity in (x,y) plane
shear of velocity suggests rotation
Vorticity
fy
u
x
v
a
Uk
vorticityofcomponent Vertical
view this as the definition of relative vorticity
If we want an equation for the conservation of vorticity, then
• We want an equation that represents the time rate of change of vorticity in terms of sources and sinks of vorticity.
Conservation (continuity) principle
• dM/dt = Production – Loss
Newton’s Law of Motion
mdt
d/F
v
Which is the vector form of the momentum equation.(Conservation of momentum)
Where F is the sum of forces acting on a parcel, m mass, v velocity
What are the forces?
• Total Force is the sum of all of these forces– Pressure gradient force– Gravitational force– Viscous force– Apparent forces
• Derived these forces from first principles
If we want an equation for the conservation of vorticity, then
• We could approach it the same way as momentum, define the sources and sinks of vorticity from first principles.– But that is hard to do. What are the first
principle sources of vorticity?
• Or we could use the conservation of momentum, and the definition of vorticity to derive the equation.
Newton’s Law of Motion(components)
y
u
x
v
mFdt
dv
mFdt
du
y
x
vorticityof definition and
/
/
Combine definition and conservation principle
y
u
x
v
mFdt
dv
mFdt
du
y
x
vorticityof definition and
/
/
Operate on momentum equation
)/()(
)/()(
mFxdt
dv
x
mFydt
du
y
y
x
Subtract
)/()/()()( mFy
mFxdt
du
ydt
dv
x xy
so a time rate of change of vorticity will come from here.
Subtract
)/()/()()( mFy
mFxdt
du
ydt
dv
x xy
so details will depend on d( )/dt.For an Eulerian fluid d( )/dt = D( )Dt, material derivative.
For a Lagrangian description could write immediately.
Expand derivative
)/()/(
)()(
)/()/()()(
)/()/()()(
mFy
mFx
uyt
u
yv
xt
v
x
mFy
mFx
ut
u
yv
t
v
x
mFy
mFxDt
Du
yDt
Dv
x
xy
xy
xy
vv
vv
Expand derivative
)/()/(
)/()/(
)()(
mFy
mFx
uy
uyt
u
y
vx
vxt
v
x
mFy
mFx
uyt
u
yv
xt
v
x
xy
xy
vv
vv
vv
Expand derivative
)/()/(
mFy
mFx
uy
uyt
u
y
vx
vxt
v
x
xy
vv
vv
Dζ/Dt comes from here.
Expand derivative
)/()/(
mFy
mFx
uy
uyt
u
y
vx
vxt
v
x
xy
vv
vv
Other things comes from here.
Collect terms
)/()/( mFy
mFx
uy
vxDt
Dxy
vv
Let’s return to our quasi-geostrophic formulation
Scaled horizontal momentum in pressure coordinates
0
0
0
0
0
gagg
gagg
gagg
yuufDt
vD
yvvfDt
uD
yfDt
D
VkVkV
Use definition of vorticity vorticity equation
gaagg
gagg
gagg
vy
v
x
uf
Dt
D
yuufDt
vD
x
yvvfDt
uD
y
)(
0)(
0)(
0
0
0
An equation for geopotential tendency
gg
gaag
ggg
gaagg
vfp
fDt
D
vfy
v
x
uf
Dt
D
fy
u
x
v
vy
v
x
uf
Dt
D
02
02
02
02
2
0
0
)(
1
)(
Barotropic fluid
gg
gg
vfDt
D
vfp
fDt
D
02
02
02
barotropic
Perturbation equation
xxu
t
vvuuu
vfDt
D
g
ggggg
gg
2
02
)(
equation of formon perturbati
;
Wave like solutions
0))((
)Re(
)(
22
)(0
2
klkuk
e
xxu
t
g
tlykxi
g
Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.
Stationary wave
g
g
tlykxi
ulk
klkuk
e
22
22
)(0
0))((
0
)Re(
Wind must be positive, from the west, for a wave.
Study Questions
Study question 1Thermodynamic equation
pg
pg
pg
c
R
p
J
pt
c
J
R
p
ptR
p
p
RT
p
c
J
R
pT
t
T
; V
V
V
Why can we pull this p outside of the derivative operators?
Study Question 2Forcing terms in momentum equation
ga
gag
ag
ag
yf
fyf
yf
ff
VkVk
VkVVk
VVk
VVkVk
0
00
0
)()(
)()(
)(
Do the derivation of this approximation.
Study Question 3An estimate of the July mean zonal wind
northsummer
southwinter
Compare this figure to the similar figure for January. What is similar and different between the tropospheric jets? (magnitude, position). Without referring to the plots of temperature, what can you say about the temperature structure?
Study question 4
y
fv
Dt
Df
f
thatShow parameter. Coriolis theis
Study question 5
What is the scale of the horizontal divergence of the wind to the total vorticity in middle latitudes. Would you expect the same in the tropics?
What are the units of potential vorticity?
Study question 6
Use definition of vorticity vorticity equation
gaagg
gagg
gagg
vy
v
x
uf
Dt
D
yuufDt
vD
x
yvvfDt
uD
y
)(
0)(
0)(
0
0
0
Carry out this derivation.
Scaled equations of motion in pressure coordinates
pg
aa
g
gagg
c
R
p
J
pt
py
v
x
u
f
yfDt
D
;
0
1
0
0
V
kV
VkVkV
Set up and start the derivation for the time rate of change of divergence of the horizontal divergence. Show how to extract D(div(uhorizontal)/Dt= ….