aoss 401, fall 2006 lecture 9 september 26, 2007 richard b. rood (room 2525, srb) [email protected]...
TRANSCRIPT
AOSS 401, Fall 2006Lecture 9
September 26, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Contract with class.– First exam October 10.
• Will make it completely through Chapter 3.
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
Outline
• Thermal Wind
• Equations of motion in pressure coordinates
• Maps
Full equations of motion
1 and
)1
(
)()cos(21v
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
222
22
2
RTp
JDt
Dp
Dt
DTc
Dt
D
wΩugz
p
a
u
Dt
Dw
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
v
u Tangential coordinate system.z, height, as a vertical coordinate.zonal, meridional, vertical
• What are the three major balances we have discussed?– What are the assumptions of those balances?
The geostrophic balance
y
pfu
z
x
pfv
z
1
1
Take a vertical derivative of the equation.
The geostrophic balance
y
T
fT
g
z
u
x
T
fT
g
z
v
z
T
T
u
y
T
fT
g
z
u
z
T
T
v
x
T
fT
g
z
v
Use equation of state to eliminate density.
Thermal wind relationship in height (z) coordinates
moving fluid
Shear? (3)
• Shear is a word used to describe that velocity varies in space.
more slowly moving fluid
wind.zonal ofshear verticalz
u
z
The geostrophic balance
y
T
fT
g
z
u
y
T
fT
g
z
u
What does this equation tell us?
Zonal wind a a level is a function of average meridional temperature BELOW.
Thermal wind relationship in height (z) coordinates
An estimate of the July mean zonal wind
northsummer
southwinter
note the jet streams
Now we return to our march to pressure coordinates.
Pressure altitude
Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart. What does this look like?
Use pressure as a vertical coordinate?
• What do we need.– Pressure gradient force in pressure
coordinates.– Way to express derivatives in pressure
coordinates.– Way to express vertical velocity in pressure
coordinates.
Expressing pressure gradient force
Integrate in altitude
gz
p
z
z
gdzzp
gdzzpp
)(
)()(
Pressure at height z is force (weight) of air above height z.
Concept of geopotential
kF g
gdz
d
Define a variable such that the gradient of is equal to g. This is called a potential function.
We have assumed here that is a function of only z.
Integrating with height
gdz
d
z
z
gdzz
gdzz
gdzd
0
0
)(
0)0(
)0()(
What is geopotential?
• Potential energy that a parcel would have if it was lifted from surface to the height z.
• It is analogous to the height of a pressure surface.– We seek to have an analogue for pressure on
a height surface, which will be height on a pressure surface.
xx
zg
x
p
x
zg
x
p
1
1
Implicit that this is on a constant z surface
Implicit that this is on a constant p surface
Horizontal pressure gradient force in pressure coordinate is the gradient of geopotential
pz
pz
yy
p
xx
p
1
1
Our horizontal momentum equation(rotating coordinate system)
p
pp
pp
fDt
D
fuyDt
D
fxDt
Du
uku
)()v
(
v)()(
Assume no viscosity
Other assumptions in these equations?
2.4) Homework question
• Below the transformation of some scalar Ψ from the vertical coordinate z to the vertical coordinate p, pressure, is expressed symbolically. Write expressions for the horizontal (x and y) derivatives of Ψ and the vertical derivative (z) of Ψ in the pressure coordinate system.
Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t)
What happens if ∂p/∂z = 0?
2.4) Answer
Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t)
z
p
pz
y
p
pyy
x
p
pxx
pz
pz
If ∂p/∂z=0 then the transform cannot be made, because p does not depend on z. It is not a monotonic function.
What do we do with the material derivative?
Tt
T
Dt
DT
z
Tw
y
Tv
x
Tu
t
T
Dt
DT
Dt
Dz
z
T
Dt
Dy
y
T
Dt
Dx
x
T
t
T
Dt
DT
U
By definition: wDt
Dzv
Dt
Dyu
Dt
Dx ,,
Implicit that horizontal and time derivatives at constant z
What do we do with the material derivative?
Dt
Dp
p
T
Dt
Dy
y
T
Dt
Dx
x
T
t
T
Dt
DT
Dt
Dpv
Dt
Dyu
Dt
Dx,,By definition:
Implicit that horizontal and time derivatives at constant p
p
T
y
Tv
x
Tu
t
T
Dt
DT
Continuity equation
0)(
py
v
x
u
pyv
xu
t
Dt
D
p
u
u
Think about this derivation!
Thermodynamic equation
Jp
T
y
Tv
x
Tu
t
Tc
Rcc
JDt
Dp
Dt
DTRc
JDt
Dp
Dt
DTc
p
vp
v
v
)(
)(
Thermodynamic equation
pp
pp
p
c
J
pc
RT
p
T
y
Tv
x
Tu
t
T
c
J
cp
T
y
Tv
x
Tu
t
T
Jp
T
y
Tv
x
Tu
t
Tc
)(
)(
)(
Equation of state
Thermodynamic equation
pp
pp
pp
c
JS
y
Tv
x
Tu
t
T
pc
RT
p
TS
c
J
pc
RT
p
T
y
Tv
x
Tu
t
T
)(
)(
Sp is the static stability parameter.What is static stability?
Equations of motion in pressure coordinates(plus hydrostatic and equation of state)
pp
p
p
c
JS
y
Tv
x
Tu
t
T
py
v
x
u
fDt
D
0)(
uku
Tangential coordinate system.p, pressure, as a vertical coordinate.zonal, meridional, vertical
Full equations of motion
1 and
)1
(
)()cos(21v
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
222
22
2
RTp
JDt
Dp
Dt
DTc
Dt
D
wΩugz
p
a
u
Dt
Dw
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
v
u Tangential coordinate system.z, height, as a vertical coordinate.zonal, meridional, vertical
In the derivation
• Have used the conservation principle.
• Have relied heavily on the hydrostatic assumption.
• Require that the conservation principle holds in all coordinate systems.
• Plus we did some implicit scaling.
Let’s move this to a chart
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
gp
p
fuy
fx
)(
v)( g
Geopotential, Φ, in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
yfug
2
Think a second
• This is the i (east-west, x) component of the geostrophic wind.
• We have estimated the derivatives based on finite differences.– Does this seem like a reverse engineering of the
methods we used to derive the equations?
• There is a consistency– The direction comes out correctly! (towards east)– The strength is proportional to the gradient.
Geopotential, Φ, in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
north
Δy
yfug
Return toGeopotential, Φ, in upper troposphere
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
northDo not assume geostrophic.Are the winds parallel to the height contours?
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
Return toGeopotential, Φ, in upper troposphere
eastwest
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
ΔΦ > 0
south
northDo not assume geostrophic.A qualitative velocity contour. Not the same as geopotential, but usually close.
Next Time
• Natural Coordinates
• Balanced Flows