Transcript AOSS_401_20070919_L06_Thermo_Energy

AOSS 401, Fall 2007
Lecture 6
September 19, 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 1 graded
• Homework 2 due today
• Homework 3 posted on ctools between
now and Friday
Weather
• NCAR Research Applications Program
– http://www.rap.ucar.edu/weather/
• National Weather Service
– http://www.nws.noaa.gov/dtx/
• Weather Underground
– http://www.wunderground.com/modelmaps/m
aps.asp?model=NAM&domain=US
Outline
1. Review from Monday
•
•
Continuity Equation
Scale Analysis
2. Conservation of Energy
•
•
•
Thermodynamic energy equation and the
first law of thermodynamics
Potential temperature and adiabatic motions
Adiabatic lapse rate and static stability
From last time
Conservation of Mass
• Conservation of mass leads to another
equation; the continuity equation
• Continuity  Continuous
• No holes in a fluid
• Another fundamental property of the
atmosphere
• Need an equation that describes the time
rate of change of mass (density)
Eulerian Form of the
Continuity Equation

   u
t
Dz
In the Eulerian
point of view, our
parcel is a fixed
volume and the
fluid flows
through it.
Dy
Dx
Lagrangian Form of the
Continuity Equation
1 D
   u
 Dt
The change in mass (density) following the motion is
equal to the divergence
Convergence = increase in density (compression)
Divergence = decrease in density (expansion)
Scale Analysis of the
Continuity Equation
• Define a background
pressure field
• “Average” pressure and
density at each level in
the atmosphere
• No variation in x, y, or time
• Hydrostatic balance applies
to the background pressure
and density
p0  p0 ( z )
0  0 ( z)
dp0 ( z )
  0 ( z) g
dz
Scale Analysis of the
Continuity Equation
Total pressure and density = sum of background + perturbations
(perturbations vary in x, y, z, t)
p  p0 ( z )  p' ( x, y, z, t )
  0 ( z )   ' ( x, y, z, t )
Start with the Eulerian form of the continuity equation,
do the scale analysis, and arrive at
   0u  0
Scale Analysis of the
Continuity Equation
• Expand this equation
   0u  0
Remember, ρ0
does not depend
on x or y
 u v w  w d 0

0
 
 
 x y z   0 dz
u v
1 d
 0 w


x y
 0 dz
Scale Analysis of the
Continuity Equation
• The vertical motion on large (synoptic)
scales is closely related to the divergence
of the horizontal wind
u v
1 d
 0 w


x y
 0 dz
1 d
 0 w
  uh  
 0 dz
Scale Analysis of the
Horizontal Momentum Equations
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )
Dt
a
a
 y
  2 ( v )
U·U/L
U·U/a
U·W/
a
ΔP/ρL
Uf
Wf
νU/H2
10-4
10-5
10-8
10-3
10-3
10-6
10-12
Largest Terms
Geostrophic Balance
1 p
 2Ωv sin(  )  fv
 x
1 p
 2Ωu sin(  )   fu
 y
• There is no D( )/Dt term (no acceleration)
• No change in direction of the wind (no rotation)
• No change in speed of the wind along the
direction of the flow (no divergence)
What are the scales of the terms?
For “large-scale” mid-latitude
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )
Dt
a
a
 y
  2 ( v )
U·U/L
U·U/a
U·W/
a
ΔP/ρL
Uf
Wf
νU/H2
10-4
10-5
10-8
10-3
10-3
10-6
10-12
Prediction
(Prognosis)
Ageostrophic
Analysis
(Diagnosis)
Geostrophic
• Remember the definition of geostrophic wind
1 p
vg 
f x
1 p
ug  
f y
• Our prediction equation for large scale midlatitudes
Du
1

Dt

Dv
1

Dt

p
 fv  f v  vg   fvag
x
p
 fu   f u  ug    fuag
y
Ageostrophic Wind and
Vertical Motion
• Remember the scaled continuity equation
1 d
0 w
  uh  
 0 dz
• Vertical motion related to divergence, but
geostrophic wind is nondivergent.
• Divergence of ageostrophic wind leads to
vertical motion on large scales.
Closing Our System of Equations
• We have formed equations to predict changes in
motion (conservation of momentum) and density
(conservation of mass)
• We need one more equation to describe either
the time rate of change of pressure or
temperature (they are linked through the ideal
gas law)
• Conservation of energy is the basic principle
Conservation of Energy: The
thermodynamic equation
• First law of thermodynamics:
• Change in internal energy is equal to the
difference between the heat added to the
system and the work done by the system.
• Internal energy is due to the kinetic energy
of the molecules (temperature)
• Total thermodynamic energy is the internal
energy plus the energy due to the parcel
moving
Thermodynamic Equation For a
Moving Parcel
DT
D
cv
Dt
p
Dt
J
• J represents sources or sinks of energy.
–
–
–
–
radiation
latent heat release (condensation/evaporation, etc)
thermal conductivity
frictional heating.

• cv = 717 J K-1 kg-1, cvT = a measure of internal energy
– specific heat of dry air at constant volume
– amount of energy needed to raise one kg air one degree Kelvin if
the volume stays constant.
Thermodynamic Equation
DT
D
cv
p
J
Dt
Dt
• Involves specific heat at constant volume
1 D
• Remember the material derivative
   u
form of the continuity equation
 Dt
• Following the motion, divergence leads
to a change in volume
• Reformulate the energy equation in terms of specific
heat at constant pressure
Another form of the
Thermodynamic Equation
DT
D
cv
p
J
Dt
Dt
DT
Dp
cp

J
Dt
Dt
• Short derivation
• Take the material derivative of the equation of
state
• Use the chain rule
 and the fact that R=cp-cv
• Substitute in from the thermodynamic energy
equation in Holton
• Leads to a prognostic equation for the material
change in temperature at constant pressure
p  RT
D
DT
( p )  R
Dt
Dt
D
Dp
DT
p

R
Dt
Dt
Dt
D
Dp
DT
DT
p

 cp
 cv
Dt
Dt
Dt
Dt
Substitute in from the thermodynamic
energy equation (Holton, pp. 47-49)
D
Dp
DT
D
p

 cp
p
J
Dt
Dt
Dt
Dt
DT
Dp
cp

J
Dt
Dt
(Ideal gas law)
(Material derivative)
(Chain Rule)
(Use R=cp-cv)
D
 DT

p
 J
 cv
Dt
 Dt

(Cancel terms)
Thermodynamic equation
DT
Dp
cp

J
Dt
Dt
• Prognostic equation that describes the
change in temperature with time
• In combination with the ideal gas law
(equation of state) the set of predictive
equations is complete
Atmospheric Predictive Equations
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )  2 ( v )
Dt
a
a
 y
Dw u 2  v 2
1 p


 g  2Ωu cos( )  2 ( w)
Dt
a
 z
D
    u
Dt
DT
D 1
cv
p
   J
Dt
Dt   
p  RT and  
1

Motions in a Dry (Cloud-Free)
Atmosphere
• For most large-scale motions, the amount of
latent heating in clouds and precipitation is
relatively small
• In absence of sources and sinks of energy in a
parcel, entropy is conserved following the
motion
• Why is this important?
– Large scale vertical motion
– Atmospheric stability (convection)
Motions in a Dry (Cloud-Free)
Atmosphere
• Goal: find a variable that
– Is conserved following the motion if there are no
sources and sinks of energy (J)
– Describes the change in temperature as a parcel
rises or sinks in the atmosphere
• Adiabatic process: “A reversible
thermodynamic process in which no heat is
exchanged with the surroundings”
• Situations in which J=0 referred to as
– Dry adiabatic
– Isentropic
• Why is this useful?
Synoptic Motions
Forced Ascent/Descent
Cooling
Warming
Derivation of Potential Temperature
c p DT Rd Dp J


T Dt
p Dt T
D (ln T )
D (ln p ) J
cp
 Rd
 0
Dt
Dt
T
c p D (ln T )  Rd D (ln p )
(No sources or sinks of energy)
(Adiabatic process)
p
T
c p  D (ln T )  Rd  D (ln p )
T0
p0
 c p (ln T0  ln T )   Rd (ln p0  ln p )
 T0  Rd  p0 
ln   
ln  
 T  cp  p 
p 
T0  T  0 
 p
(Energy Equation
divided by temperature)
Rd
cp

(Integrate between
two levels)
(Use the properties of
the natural logarithm)
(Take exponential of
both sides)
Definition of the
Potential Temperature 
Rd
cp
p0 
  T  
 p 
Note: p0 is defined to be a constant reference level
p0 = 1000 hPa
Interpretation: the potential temperature is the
temperature a parcel has when it is moved from a
(higher or lower )pressure level down to the surface.

 p0 
  T  
 p
Rd
cp
• The temperature at the top of the continental divide is
-10 degrees celsius (about 263 K)
• The pressure is 600 hPa, R=287 J/kg/K, cp=1004 J/kg/K
• Compute
304 K
1. potential temperature at the continental divide
2.
The temperature the air would have if it sinks to the plains
(pressure level of 850 hPa) with no change in potential
temperature
T  290 K  17 oC  64 o F
Dry Adiabatic Lapse Rate
Change in Temperature with Height
• For a dry adiabatic, hydrostatic atmosphere the
potential temperature  does not vary in the
vertical direction:

0
z
• In a dry adiabatic, hydrostatic atmosphere the
temperature T must decrease with height. How
quickly does the temperature decrease?


ln   ln T


 p0 
 
 p
Rd
cp

  ln T  Rd ln  p0 

c p  p 



  Rd  p0 
ln    ln T    ln  
z
z
z  c p  p 
1  1 T Rd 
ln p0  ln p 


 z T z c p z
T  T Rd T 
 ln p 


 z z
c p z

1 p g
 ln p   

z
p z
p
T  T g Rd T


 z z c p p
T  T g


 z z c p
(logarithm of
potential temperature)
(take the vertical derivative)
(Definition of d lnx and
derivative of a constant)
(Multiply through by T)
(Hydrostatic balance)
(Equation of State)
Dry adiabatic lapse rate
The adiabatic change in temperature with height is
T  T g


 z z c p
For dry adiabatic, hydrostatic atmosphere
T g


 d
z c p
d: dry adiabatic lapse rate (approx. 9.8 K/km)
Atmospheric Static Stability
and Potential Temperature
• Static: considering an atmosphere at rest
(no u, v, w)
• Consider what will happen if an air parcel is
forced to rise (or sink)

• Stable: parcel returns to
0
the initial position
z
• Neutral: parcel only rises/sinks

if forcing continues, otherwise
0
z
remains at current level
• Unstable: parcel accelerates

0
away from its current position
z
Static Stability
p  Rd T
p parcel  penv
 parcel Rd Tparcel   env Rd Tenv
 parcelTparcel   envTenv
• Displace an air parcel up or down
• Assume the pressure adjusts instantaneously; the parcel
immediately assumes the pressure of the altitude to which it
is displaced.
• Temperature changes according to the adiabatic lapse rate
Static Stability
• Adiabatic: parcel potential
temperature constant with height
• For instability, the temperature of the
atmosphere has to decrease at
greater than 9.8 K/km
• This is extremely rare…
• Convection (deep and shallow) is
common
• How to reconcile lack of instability
with presence of convection?

0
z

0
z

0
z
Static Stability and Moisture
• The atmosphere is not dry—motion is not dry
adiabatic
• If air reaches saturation (and the conditions are
right for cloud formation), vapor will condense to
liquid or solid and release energy (J≠0)
• Average lapse rate in the troposphere: -6.5 oC/km
• Moist (saturated) adiabatic lapse rate: -5 oC/km
Consider the Upper Atmosphere
Atmosphere in Balance
• Hydrostatic balance (no vertical acceleration)
• Geostrophic balance (no rotation or divergence)
• Adiabatic lapse rate (no clouds or precipitation)
• What we are really interested in is the difference
from balance.
• This balance is like a strong spring, always
pulling back.
• It is easy to know the approximate state.
Difficult to know and predict the actual state.
Next time
• Ricky will be lecturing Friday, Monday, and
Wednesday
• We have essentially completed chapters
1-2 in Holton
• We have derived a set of governing
equations for the atmosphere
• Chapter 3 will introduce simple
applications of these equations
• First exam covers chapters 1-3—three
weeks from today!