Transcript AOSS 321, Fall 2006 Earth Systems Dynamics 10/9/2006

AOSS 321, Winter 2009
Earth Systems Dynamics
Lecture 12
2/17/2009
Christiane Jablonowski
[email protected]
734-763-6238
Eric Hetland
[email protected]
734-615-3177
Today’s class
•
•
•
Transformation of the vertical coordinate: from
height coordinates to pressure coordinates
Geostrophic wind equations on constant
pressure surfaces
Variation of the geostrophic wind with height:
Thermal wind
Generalized vertical coordinate
Candidates for vertical coordinate: any vertical
coordinate ‘s’ that is a single-valued monotonic
function of height with ∂s /∂z ≠ 0.
Constant surface s0
z
Δx
Δz
p3
x
p1
p2
Pressure values
Pressure profile:
Pressure as a vertical coordinate?
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.
Some basics of the atmosphere:
Temperature profile
Temperature as a
vertical coordinate?
What about potential
temperature?
Troposphere: depth ~ 1.0 x 104 m
Generalized vertical coordinate
For the vertical coordinate ‘s’, the generalized form
of the coordinate transformation becomes:
p3  p1 p3  p2 z p2  p1


x
z x
x
p  p z  p 
       
x s z x s x z
Using the identity
p s p

z z s
we can also derive the

p  s z  p p 
alternate form:        
x s z x s s x z

Example: ‘s’ could be the potential temperature
s= (isentropes) or pressure s=p (isobars), check it!
Pressure gradient in p coordinates
p  p  p z 
       
x z x s z x s
z
p0
p0+Δp

Δz
Δx
x
Pressure gradient
force
Here: use s = p
This derivation leads to:
p  p0  ( p0  p)  z 
   
  
x z 
x x p
z
p  p z 
  
 
x z z x p
p 
z 
   g 
x z
x p
(hydrostatic
equation used)
z 
(gz) 

1 p 
    g   



 
x p
 x p
x p
 x z
Pressure gradient forces
in z and p coodinates
1 p
z

 g
 x
x
1 p
z


 g  
 x
x
x
Implicit 
that this is
on a constant z
surface
Implicit that this is
on a constant p
surface
Horizontal pressure gradient forces
 
1 p 
     
 x z x p
 
1 p 
     
 y z y p
in pressure coordinates:
the pressure gradient
forces are determined
by the gradients of the
geopotential 
The subscripts ‘z’ and ‘p’ indicate which variables
are held constant in the two representations.
Our approximated horizontal momentum
equations (in p coordinates)
No viscosity, no metric terms, no cos-Coriolis terms
du
 
      fv
dt p
x p
 
dv 
      fu
dt p
y p
Dv h 

  f k  v h   p 
 Dt p
Subscript h: horizontal

Subscript p: constant p surfaces!
Sometimes subscript is omitted,
 tells you that this is on p surfaces
Material derivative in p coordinates
By definition:
Dx
Dy
Dp
 u,
 v,

Dt
Dt
Dt
Total derivative on constant pressure surfaces:
() 
D()  () 
() 
()
      u   v   
Dt p t p
x p
p
y p
different !
DT T
T
T
T
e.g. DT/Dt:

u
v

Dt t
x
y
p
(subscript p omitted)
Geostrophic Wind
(in p coordinates)
Component form:
1  
ug    
f y p
1  
v g   
f x p
Vector form:
1
v g  k   p
f
Here we implicitly assume that the partial
derivatives in x and 
y direction are computed on

constant p surfaces.
Advantage of the p coordinates: Simplicity of the equation!
Density no longer appears explicitly.
Example: 300hPa geopotential
height map
A map that
shows you the
geopotential height
(here in deka m)
on an isobaric level
immediately
lets you compute
the direction and
magnitude of the
geostrophic wind
ug and vg!
Exercise:
compute the
geostrophic wind
ug and vg at the
position x in m/s
(assume f=const).
Linking the thermal field and wind field:
The Thermal Wind Relationship
• The vertical shear of the geostrophic wind is
related to the so-called thermal wind.
• The thermal wind is not really a wind, but a
vector difference between the geostrophic wind
vg at an upper level and at a lower level.
• The vertical shear of vg is directly related to the
horizontal temperature gradient.
• The thermal wind vector points such that cold air
is to the left, and warm air is to the right, parallel
to the isotherms. (Northern Hemisphere).

Geostrophic wind
in pressure coordinates
1  
ug     ,
f y p
1  
v g   
f x p
Hydrostatic Balance
in pressure coordinates

RT

p
p
Vertical wind shear
Take derivative with respect to p:
u g
(Hydrostatic
equation used)
1   vg 1

,

p
f y p
p f
u g 1  RT vg
1

,

p
f y p
p
f
 
x p
 RT
x p
Links horizontal temperature gradient
with vertical wind gradient.
Vertical wind shear

ug 1   Rd T 
 

p
f y p p
v g
ug Rd T 

 
p pf y p
v g
Vector notation:

1   Rd T 
  

p
f x p p
Rd T 
   
p
pf x p
v g
Rd
  k   pT
ln p
f
These are equations that describe the vertical shear
of the geostrophic
wind.
Derivation of the thermal wind:
Integrate
v g
Rd
  k   pT
ln p
f
Rd
d v g   k   p T d ln p
f
v ( p2 )
Rd
 d vg   f
v ( p1 )
p2
 k 
p
T d ln p
p1
with p2 < p1
Index ‘1’ indicates the lower level, ‘2’ the upper level.
Thermal wind
v ( p2 )
Rd
 dv g   f
v ( p1 )
p2
 k 
p
Td ln p
p1
assume that at any (x,
y) T in a layer is
represented by a layer-mean average T
Rd
v g ( p2 )  v g ( p1 )   k   p T
f
p2
 d ln p
p1
 p1 
Rd
v g ( p2 )  v g ( p1 ) 
k   p T ln 
f
p2 
with p2 < p1
Thermal wind vT
(in pressure coordinates)
 p1 
Rd
vT  v g ( p2 )  v g ( p1 ) 
k   p T ln  vectorf
p2  form
Rd  T   p1 
uT   
 ln 
f  y p p2 
componentRd  T   p1 
vT 

 ln 
f  x p p2 
form
The thermal wind vT is the vector difference between
geostrophic winds at two pressure levels with p2 < p1.
T : mean temperature in the layer bounded by p2 & p1
Example: Thermal wind vT between
500 hPa and 1000 hPa
vg (p2 = 500 hPa)
vT thermal wind
vg (p1 = 1000 hPa)
The thermal wind vT is the vector difference between
geostrophic winds at an upper level and lower level.
An excursion into the atmosphere:
Geopotential height and wind
X
850 hPa surface
from Brad Muller
X
X
300 hPa surface
An excursion into the atmosphere:
Geopotential height and wind
X
X
850 hPa surface
from Brad Muller
X
300 hPa surface
Thermal wind
850 hPa surface
300 hPa surface
Thermal wind vT
vg

Geostrophic wind
at 850 hPa
from Brad Muller
Geostrophic
wind at
300 hPa
Another excursion into the atmosphere.
850 hPa surface
300 hPa surface
Geostrophic
wind at
300 hPa
Thermal
wind vT
Geostrophic wind
at 850 hPa
from Brad Muller
Annual mean zonal mean temperature T
(hPa)
P
r
e
s
s
u
r
e
100
Kelvin
260
230
220
200
210
245
1000
260
North Pole
300
Equator
250
South Pole
y
Exercise:
What do you know about the zonal wind
when looking at the y-p temperature plot?
In midlatitudes:
Assume the zonal wind at the surface is westerly (u > 0 m/s).
Look at the zonal-mean temperature map and estimate the
sign of the gradient ∂T/∂y from the equator to the poles at
three different pressure levels (marked by the red lines) in
both hemispheres (midlatitudes).
Sketch a rough latitude-pressure profile of the zonal-mean
zonal wind u in midlatitudes using the thermal wind
relationship.

Thermal wind component uT
(in pressure coordinates)
Rd  T   p1 
uT  u2  u1   
 ln 
f  y p p2 
Index 1: lower pressure level p1
Index 2: upper pressure level, p2 < p1
f = 2 sin()
Coriolis parameter
Annual mean zonal mean zonal wind u
(hPa)
100
P
r
e
s
s
u
r
e
1000
North Pole
5
15
20
25
10
15
2
m/s
5
Equator
South Pole
y
The mean zonal wind follows the thermal wind relationship.
Thermal wind
Good online resource (see examples below):
http://www.aos.wisc.edu/~aalopez/aos101/wk12.html
The thermal wind describes the vertical geostrophic
wind shear
Veering winds with height
(clockwise rotation with height)
Backing winds with height
(counterclockwise rotation
with height)
Thermal wind and temperature advection
Direction of the thermal wind determines the thermal
structure of the atmosphere.
The thermal wind always points parallel to lines of
constant thickness with lower thicknesses to the left,
therefore, the thermal wind always has the colder air
to the left (in Northern Hemisphere).
Veering winds:
warm air advection
Backing winds:
cold air advection
Thermal wind and temperature advection
Direction of the thermal wind determines the thermal
structure of the atmosphere.
In NH, the thermal wind always points parallel to lines
of constant thickness (parallel to isotherms) with lower
thicknesses to the left. Therefore, the thermal wind
always has the colder air to the left.
Veering winds (clockwise
rotation with height) :
warm air advection
Backing winds (counterclockwise
rotation with height):
cold air advection
Thermal wind: Big idea
• The thermal wind describes the wind shear
between the geostrophic winds at two different
levels in the atmosphere (upper level - lower level).
• The way the wind changes direction and speed
with height (wind shear) is related to the advection
of temperature, warming or cooling, in the
atmosphere.
• The thermal wind vector points such that cold air is
to the left, and warm air is to the right, parallel to
the isotherms (Northern Hemisphere).

Alternative form of the thermal wind
So far:
p1 
Rd
vT  v g ( p2 )  v g (p1) 
k   p T ln 
f
p2 
Alternative form:
1
vT  v g ( p2 )  v g ( p1 )  k   p ( 2  1 )
f
1 
uT  
( 2  1 )
f y
Why is this correct?
1 
vT 
( 2  1 )
f x
Index ‘1’ indicates the lower level, ‘2’ the upper level.

Recall: Hypsometric equation
Thickness
(z)
Z
g
Rd
Z 2  Z1 
g

p1
p2
Td ln p
This equation links
thermodynamics and
dynamics
 p1 
 2  1  Rd T ln 
p2 
Z2 - Z1 = ZT ≡ Thickness - is proportional to layer
mean temperature
Index ‘1’ indicates the lower level, ‘2’ the upper level.

Similarity of the equations
Rd
ZT  Z 2  Z1 
T
g
p1
 d ln p
p2
Rd
vT  v g ( p2 )  v g ( p1 ) 
k  p T
f
p1
 d ln p
p2
There is clearly a relationship between thermal
wind vT and thickness ZT.
Index ‘1’ indicates the lower level, ‘2’ the upper level.
Thickness and thermal wind
Pressure
surfaces
(isobars)
z
geostrophic wind (here: increase
with height)
Thickness of
layers related to
temperature.
Causing a tilt of
the pressure
y surfaces.
x
There is vertical shear in the geostrophic wind field!
The shear is expressed by the thermal wind relationship.
Drawing from Brad Muller