Transcript AOSS_401_20071003_L13_Thermal_Vertical.ppt

AOSS 401, Fall 2007
Lecture 13
October 5, 2007
Richard B. Rood (Room 2525, SRB)
[email protected]
734-647-3530
Derek Posselt (Room 2517D, SRB)
[email protected]
734-936-0502
Class News
• Homework.
• Homework due Monday
• Answers to last homework are posted on ctools
• We will go over the review questions on Friday
– Think about them
• Exam next Wednesday
• Today we will talk about thermal wind and vertical velocity some
more
• Friday and Monday we will look at the material in different ways and
more thoroughly
• Also have your questions
• Mid-term evaluation
– “students will be notified soon thereafter that they can fill out the
midterm evaluations between October 8 and October 14”
Some information about the exam
General Exam Instructions
•
•
•
Logistics
– Write name on all pages.
– Honor system signature required - see first page after intro material.
– Turn in the exam sheets that are handed out.
– No books, no calculators, no computers.
• Personal notes allowed (~ 2 pages)
– Mathematical formulas and identities will be provided.
General instructions.
– You may use the provided equation numbers to refer to equations.
– Draw pictures.
– Show your work. A good start is worth points!
– Work from fundamentals.
– You have seen these ideas and techniques in your homework problems.
• Don’t make the questions more complicated than they are.
– Read all questions first.
There are N problems.
– Point distribution. Problem#(Points)
• P1(n1), P2(n2), ...
• Total points for exam = 30
Variables and constants
•
•
•
•
p is pressure
ρ is density
T is temperature
α is specific volume
• u is velocity vector (ui, vj,
wk)
• g is gravity force (assume
constant)
f  2sin(  )  Coriolis parameter,
at latitude   45  ,  10 -4 s 1
• R is the gas constant for
dry air
• cv is specific heat at
constant volume
• cp is specific heat at
constant pressure = (cv +
R)
• a is radius of Earth
• Ω is angular velocity of
Earth’s rotation
• n is kinematic viscosity
coefficient
Units
kg m
Units of Force, Newton  N  2
s
N kg m 1
Units of Pressure, Pascal  Pa  2  2
m
s m2
100 Pa  1 hPa
Equations of motion
(z, height, as vertical coordinate)
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( ) n 2 (u ) (Eq. 1)
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  ) n 2 ( v)
Dt
a
a
 y
Dw u 2  v 2
1 p


 g  2Ωu cos( ) n 2 ( w)
Dt
a
 z
D
    u
Dt
c p DT R Dp J
DT
D
cv
p
 J or


Dt
Dt
T Dt P Dt T
1
p  RT and  

(Eq. 2)
(Eq. 3)
(Eq. 4)
(Eq. 5)
(Eq. 6)
tangential coordinate system on Earth’s surface
(x, y, z) = (+ east, + north, + local vertical)
Equations of motion
(p, pressure, as a vertical coordinate)
DV
 fk  V  
Dt
u v


(  )p 
 V 
0
x y
p
p
T
T
T
T
J
u
v
 S p 
 V   T  S p 
t
x
y
t
cp
(Eq. 7)
(Eq. 8)
(Eq. 9)

RT
   
(Eq. 10)
p
p
V  ui  vj  horizontal velocity ;   potential temperatu re
D( ) 

 ) p  (V  ) p  
Dt
t
p
;
Dp
 ln 
; S p  T
Dt
p
tangential coordinate system on Earth’s surface
(x, y, p) = (+ east, + north, pressure is vertical coordinate)
Mathematical Expressions
(for arbitrary x)
f x 2  2 f
f ( x  x)  f ( x)  x 
 ...
2
x
2! x
(1  x) 1  1  x  x 2  x 3  ...
x3 x5
sin x  x 

 ...
3! 5!
x2 x4
cos x  1 

 ...
2! 4!
1
x
d ln x  dx, ln x  ln y  ln
x
y
Scale factors for “large-scale” mid-latitude
U  10 m s
-1
P  10 hPa
W  1 cm s units!
  1 kg m
L  10 m
 /   10  2
-1
6
H  10 m
-3
4
L / U  10 s
5
f 0  10-4 s 1
f

 10-11 m -1s -1
y
Honor code
NAME: ___________________________
signature
"I have neither given nor received unauthorized
aid on this examination, nor have I concealed
any violations of the Honor Code."
Types of questions
• Usually 6-7 questions
• First two will be about definitions, what things
are.
– This equation represents what physical principle?
– This term means?
• Second two will be very similar to homework
problems, but a little different
– Pay attention to the definition of the problem
• Last two will require putting together the
concepts and tools in different ways.
Any questions about the test
Material from Chapter 3
• Thermal wind
• Vertical Velocity
• Review
Equations of motion in pressure coordinates
(plus hydrostatic and equation of state)
Du
 fk  u   p 
Dt
u v

(  )p 
0
x y
p
T
T
T
J
u
v
 S p 
t
x
y
cp
Linking thermal field with wind field.
• The Thermal Wind
Geostrophic wind
1 
1 
ug  
, vg 
f y
f x
Hydrostatic Balance

RT

p
p
Geostrophic wind
Take derivative wrt p.
u g
1   v g 1

,

p
f y p
p
f
u g 1  RT
v g
1

,

p
f y p
p
f
 
x p
 RT
x p
Links horizontal temperature gradient
with vertical wind gradient.
Thermal wind
u g
v g
1  RT
1  RT

,

p
f y p
p
f x p
u g R T
v g
R T
p

, p

p
f y
p
f x
U g
R
  k   pT
 ln p
f
p is an independent variable, a coordinate.
Hence, x and y derivatives are taken with p
constant.
Relation between zonal mean
temperature and wind is strong
• This is a good diagnostic – an excellent
check of consistency of temperature and
winds observations.
• We see the presence of jet streams in the
east-west direction, which are persistent
on seasonal time scales.
• Is this true in the tropics?
Thermal wind
U g
R
  k   pT
 ln p
f
R
dU g   k   pTd ln p
f
U@p
p
R
dU g    k   pTd ln p

f p0
U @ p0
Thermal wind
U@p
p
R
dU g    k   pTd ln p

f p0
U @ p0
assume that at any (x, y) T in a layer is
represente d by and average T
R
U g ( p )  U g ( p0 )   k   p T
f
p
 d ln p
p0
p0
R
U g ( p )  U g ( p0 )  k   p T ln
f
p
Thermal wind
p0
R
U g ( p )  U g ( p0 )  k   p T ln
f
p
p0
R T 
 ln
uT  
f y  p
p
p0
R T 
vT 
 ln
f x  p
p
Thermal wind
p0
R
U g ( p )  U g ( p0 )  k   p T ln
f
p
1 
uT  
(   0 )
f y
?
1 
vT 
(   0 )
f x
From Previous Lectures
Thickness
Note link of thermodynamic
variables, and similarity to scale
heights calculated in idealized
atmospheres.
( z )
Z
g0
R
Z 2  Z1 
g0

p1
p2
Td ln p
Z2-Z1 = ZT ≡ Thickness - is proportional to
temperature is often used in weather forecasting to
determine, for instance, the rain-snow transition.
(We will return to this.)
Similarity of the equations
R
Z 2  Z1 
T
g0

p1
p2
d ln p
R
U g ( p)  U g ( p0 )   k   p T
f
p
 d ln p
p0
There is clearly a relationship between thermal
wind and thickness.
Schematic of thermal wind.
Thickness of
layers related to
temperature.
Causing a tilt of
the pressure
surfaces.
from Brad Muller
Another excursion into the atmosphere.
X
X
850 hPa surface
X
300 hPa surface
from Brad Muller
Another excursion into the atmosphere.
X
X
X
850 hPa surface
300 hPa surface
from Brad Muller
Another excursion into the atmosphere.
850 hPa surface
300 hPa surface
from Brad Muller
Another excursion into the atmosphere.
850 hPa surface
300 hPa surface
from Brad Muller
Weather
• National Weather Service
– http://www.nws.noaa.gov/
– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/findweather/getForecast?query=ann+arbor
– Model forecasts:
http://www.wunderground.com/modelmaps/maps.asp
?model=NAM&domain=US
A summary of ideas.
• In general, these large-scale, middle latitude
dynamical features tilt westward with height.
• The way the wind changes direction with altitude
is related to the advection of temperature,
warming or cooling in the atmosphere below a
level.
– This is related to the growth and decay of these
disturbances.
– Lifting and sinking of geopotential surfaces.
Some simple links to vertical velocity
Equations of motion in pressure coordinates
(plus hydrostatic and equation of state)
Du
 fk  u   p 
Dt
u v

(  )p 
0
x y
p
T
T
T
J
u
v
 S p 
t
x
y
cp
Let’s think about growing and
decaying disturbances.
Mass continuity equation.
u v

(  )p 
0
x y
p
u v
d  (  ) p dp
x y
Let’s think about growing and
decaying disturbances.
u v
d  (  ) p dp
x y
 @ p 0
 d  
 @ p sfc
p 0

p sfc
(
u v
 ) p dp
x y
p 0
u v
 ( p  0)   ( psfc )    (  ) p dp
x y
p sfc
p sfc
p sfc
u v
 ( psfc )    (  ) p dp     p  u dp
x y
p 0
p 0
Formally links vertical wind and divergence.
Let’s think about growing and
decaying disturbances.
From homework problem 4.5 (scaled)
psfc
p sfc
p sfc
u v
   (  ) p dp     p  u dp
t
x y
p 0
p 0
Convergence (divergence) of mass into (from)
column above the surface will increase
(decrease) surface pressure.
Another link to vertical velocity
Du
 fk  u   p 
Dt
u v

(  )p 
0
x y
p
T
T
T
J
u
v
 S p 
t
x
y
cp
Another link to vertical velocity
T
T
T
J
u
v
 S p 
t
x
y
cp
assume small diabatic heating
T
T
T
u
v
 S p
t
x
y
1

Sp
 T
T
T 


u
v
x
y 
 t
Estimating vertical velocity
• First way: Mass continuity, the divergence
of the horizontal wind
– Errors in estimates of wind often dominate the
calculation
• Errors generally amplified when you take
derivatives
• Second way: Temperature field, advection
of temperature
– Generally easier to measure temperature and
to get an estimate of temperature advection
Some take away messages
Some things that we learned (1)
• Organizing structure provided by rotation.
• Rotation is less important in the tropics, which is
clearly observable in the atmosphere.
• There is a theoretical limit on pressure gradients
associated with high pressure systems.
– Highs tend to be smeared out; they tend to have
moderate wind speeds.
• There is not such a limit for low pressure
systems.
– Lows can be very intense; The highest wind speeds
are associated with lows.
Some things that we learned (2)
• There is the possibility of “anomalous”
circulations.
– Possibility of cyclonic highs
– Possibility of anti-cyclonic lows
• We can estimate frictional dissipation
based on the angle between lines of
constant pressure, or height, and the
observed wind.
Some things that learned (3)
• Dynamical features can isolate air and
allow the evolution of extraordinary
chemical processes.
Dynamics is scale dependent
• Planetary waves: 107 meters, 10,000 km
– Have we seen one of these in our lectures?
• Synoptic waves: Our large-scale, middlelatitude, 106 meters, 1000 km
– What’s a synoptic wave? What does synoptic mean?
•
•
•
•
•
Hurricanes: 105 meters, 100 km
Fronts: 104 meters, 10 km
Cumulonimbus clouds: 103 meters, 1 km
Tornadoes: 102 meters, 0.1 km
Dust devils: 1 - 10 meters
Review Problems
• The following three problems are good
review problems for the test.
• We will solve them in class on 10/5
4.1 Review Problem
The material derivative is written below in two different coordinate
systems. One uses pressure as a vertical coordinate; the other
uses height.
DT T
T
T
T

u
v

Dt
t
x
y
p
DT T
T
T
T

u
v
w
Dt
t
x
y
z
What are the dependent and independent variables? When
taking partial derivatives in one variable, what other parameters
are held constant? Write down the definition of w and ω in terms
of differentials of the independent variables. What are the units
of ω?
4.2 Review Problem
Problem 3.3 from Text (Holton, 4th Edition)
A tornado rotates with constant angular velocity ω. Show that the
surface pressure at the center of the tornado is given by
  2 r02
p  p0 exp(
)
2 RT
where p0 is the surface pressure at the distance r0 from the
center and T is the temperature (assumed constant). If the
temperature is 15o C and pressure and wind speed at 100 m from
the center are 1000 hPa and 100 m s-1, respectively, what is the
central pressure?
4.3) Review Problem
The horizontal momentum equations in
pressure coordinates can be written as:
k is a constant which represents
frictional dissipation.
du

) p  ( ) p  fv  ku
dt
x
dv

( ) p  ( ) p  fu  kv
dt
y
Du
 fk  u   p   ku
Dt
(
Assume that the lines of constant geopotential are oriented in the
east-west direction and that the flow is balanced. Derive equations
that express the velocity components, u and v, in terms of pressure
gradient, k and f. Draw a picture of this flow field, showing,
quantitatively, the relation of the velocity field to the lines of
geopotential.
Questions
How do these natural coordinates
relate to the tangential coordinates?
Ω
• They are still tangential, but
the unit vectors do not point
west to east and south to
north.
• The coordinate system turns
a
with the wind.
Φ = latitude • And if it turns with the wind,
what do we expect to happen
to the forces?
Earth
Looking down from above
Looking down from above
Looking down from above
Looking down from above
Looking down from above
Balanced flows in natural coordinates
(balanced, here, means steady)

fV  
n
V2


R
n
V2

 fV  
R
n
geostrophi c
cyclostrop hic
gradient