Hydrostatic Equilibrium
Chapt 3, page 28
z+dz
p-dp

p
: pressure gradient force
z
g : gravity
p+dp
z-dz
dp
in equilibriu m  
g
dz
(one of the best approximations in meteorology)
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
1
Remember pressure is force per unit area so pressure per
unit height is weight (mg) per unit volume (Nm-2 m-1 = kg
m s-2 m-3). In this class we will usually ignore horizontal
variations in thermodynamic variables (T* is virtual temp)
and write
p dp

  g
z dz
but   p / RT *
dp
g
or

dz
*
p
RT
* dp
RT
  gdz   d
p
Copyright © 2010 R. R. Dickerson
& Z.Q. Li
2
dp
So RT
p
Increment of geopotential energy.
Thus the thermodynamic coordinates -RlnP, T
yield areas proportional to energy (emagram)
Consider the case for g constant (good assumption).
Integrate the hydrostatic equation from p = po to p and
zo to z.
 g

p( z )  po exp 
( z  zo ) 
*
 RT

Copyright © 2010 R. R. Dickerson
& Z.Q. Li
3
To find the mean virtual temperature take the weighted average:
dp
T

p
p
*
T  p
o dp
p p
po
*
T*
p
-Rlnp
po
T*
Thus thermodynamic diagrams can be used to determine the
geopotential thickness between pressure levels.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
4
Geopotential
dp
gdz   RT
p
*
Define an increment of geopotential d  gdz (energy/ma ss)
g has a slight variation with latitude and altitude which can
usually be ignored; in AOSC652 you will integrate for g = f(z).
So we can define an increment of geopotential thickness, or
height.
dY  d/go; go = 9.8 m/s2
RT * dp
dY  
go p
RT *
Y2  Y1  
ln( p2 p1 )
go
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
5
Moisture effects on Geopotential
RT *
Y  
ln( p2 p1 )
go
RT

(1  0.6 w ) ln( p2 p1 )
go
If moisture were not included :
RT
Yo  
ln( p2 p1 )
go
(Y )  Y  Yo  Y (0.6 w )
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
6
Pressure Variation with z for
“Special” Atmospheres
1. Constant Density
  o
(Homogenous Atmosphere)
dP   gdz   o gdz
0
H
 dP    g  dz
o
po
0
Po
 o RT RTo
H


o g
o g
g
H → “Scale Height” = height of const. T (288K), homogenous
atmosphere ~ 8 km.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
7
2. Constant Lapse Rate Atmosphere
Define Lapse Rate
T
 
z
P
dP = -r gdz = gdz
RT
dP
g
gdz
=dz = P
RT
R(To - g z)
Students should integrate to find P as a
function of z : P = P(z)
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
8
3. Isothermal Atmosphere :  = 0
dP
g
dz

dz  
P
RTo
H
P( z )  Po e
z / H
The isothermal approximation is good to about 30% for the
troposphere. Problem for students: What fraction of the
mass of atmosphere lies between 800 and 700 hPa?
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
9
Stability Criteria
Air parcel
properties
Environment properties
P, T, , , w
Assume
P’, T’, ’, ’, w’
1. Parcel and environment are in instantaneous
dynamic equilibrium: P = P’
2. Atmosphere is in hydrostatic equilibrium.
3. Parcel and environment do not mix.
4. No compensating motion by environment as
an air parcel moves.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
10
Consider a dry adiabatic displacement by an air parcel:
d ¢q = 0 = c p dT - a dP
dT
dP
a
or c p
=a
=- g
dz
dz
a¢
RT *
RT ¢*
with a =
and a ¢ =
P
P
dT
g æ T* ö
Therefore
=- ç *÷
dz
cp è T ¢ ø
T*
But
~ 1 for all conditions
*
T¢
dT
g
so
= - = 9.8 K km ~ 1 K 100m
dz
cp
æ dT ö
Define Dry Adiabatic Lapse Rate G d º - ç ÷ = g / c p
è dz øj
Copyright © 2013 R. R. Dickerson
11
& Z.Q. Li
For R&Y HW 4.6
Convective Available Potential Energy: CAPE
The maximum energy available to an ascending parcel, in parcel theory.
On a thermodynamic diagram this is called positive area, and can be seen as
the region between the lifted parcel process curve and the environmental
sounding, from the parcel's level of free convection to its level of neutral
buoyancy. Defined as
where αe is the environmental specific volume profile, αp is the specific
volume of a parcel moving upward moist-adiabatically from the level of free
convection, pf is the pressure at the level of free convection, and pn is the
pressure at the level of neutral buoyancy. The value depends on whether the
moist-adiabatic process is considered reversible or irreversible
(conventionally irreversible) and whether the latent heat of freezing is
considered (conventionally not). AMS Glossary of Meteorology.
12
Consider a Saturated Adiabatic displacement (pseudo adiabatic):
d q   Lv dwo  c p dT  dP
or
dwo
dT
dP
 Lv
 cp

dz
dz
dz
Note that wo= f(T,P)
We want to derive dT/dz for this process using the hydrostatic
equation and assuming /’ ~ 1.
so
dwo
dT
 Lv
 cp
g
dz
dz
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
13
but ws @
e es (T )
P
So by differentiating wrt z and dividing by ws :
1 dws 1 des dT 1 dP
=
ws dz es dT dz P dz
1 dP 1 dP¢
r ¢g -g
g
but
=
==
»P dz P¢ dz
P¢ RT ¢
RT
The above can be applied to the Clausius Clapeyron Equation:
dws Lv ws dT gws
=
+
2
dz
RvT dz RT
æ Lv ws ö
ç1+
÷
dT
g è
RT ø
=
º Gs
dz
c p æ e L2v ws ö
çç1+
÷
2 ÷
c pT R ø
è
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
14
Saturated Adiabatic Lapse Rate
The table below lists values of Gs in oK/km
temperature
pressure
1000 mb
700 mb
500 mb
-30 C
9.2
9.0
8.7
0C
6.5
5.8
5.1
20 C
4.3
3.7
3.3
Note that G < G
Copyright © 2013 R. R. Dickerson
d
& Z.Q. Lis
15
Buoyant Force on an Air Parcel
The environment is in hydrostatic equilibrium (no acceleration)
d 2z
P
so
 0  g  
2
dt
z
In general, an air parcel WILL be subject to an acceleration due
to density differences with the environment, so CAPE is the
integral of  (from the wet adiabat) wrt p; for the parcel:
P
z   g  
z
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
16
P P

since
z
z
(   )
z  g
then

RT *

but
P
(T *  T * )
so acceleration z  g
T *
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
17
Stability Criteria
zo
We are interested in small displacements of a parcel from its
original location.
If with a small
displacement we
find that
z  0
z  0
z  0
The environment is
said to be
Unstable
Neutral
Stable
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
18
For convenience, consider the parcel location zo to be zero. The
temperature of the environment (with prime) may be written as:
*
2
*


d
T
d
T
*
*
2
1

T  T0 
z 2
z  ...
2
dz
dz
If the displacement z is sufficiently small,
*

d
T
T *  T0* 
z
dz
or T *  T0*  z
dT *
 
dz
the " environmen tal lapse rate"
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
19
For the parcel we may write:
*
dT
T T 
z
dz
*
*
or T  T0  Gp z
*
*
0
Gp = parcel lapse rate.
remember
(T *  T * )
z  g
T *
z 
g (  Gp ) z
T0*  z
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
20
z 
g (  Gp ) z
T  z
*
0
Thus if…
g > Gp
z > 0 ® unstable case
g = Gp
z=0®
neutral case
g < Gp
z<0®
stable case
For dry displacements, use Gp= Gd
For saturated displacements, use Gp= Gs
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
21
Since Gs< Gd, we must also consider conditions between
the two stability criteria for dry and saturated.
  Gs
  Gs
Gs    Gd
  Gd
  Gd
Absolutely stable
Saturated neutral
Conditiona lly unstable
Dry neutral
Absolutely unstable
G = parcel lapse rate
 = environment lapse rate
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
22
z
DIURNAL CYCLE OF SURFACE
HEATING/COOLING:
Subsidence
inversion
MIDDAY
1 km
Mixing
depth
NIGHT
0
MORNING
T
NIGHT
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
MORNING AFTERNOON
23
*** Emagram ***
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
24
Convective Instability
dry air
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
25
Creterion for Convective Stability
¶q w
> 0, Convectively Stable
¶z
¶q w
< 0, Convectively Unstable
¶z
¶q w
= 0, Convectively Neutral
¶z
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
26
Criterion for Convective Instability
 w
 0, Stable
z
 w
 0, Neutral
z
 w
 0, unstable
z
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
27
DIURNAL CYCLE OF SURFACE
HEATING/COOLING:
z
Subsidence
inversion
MIDDAY
1 km
Mixing
depth
NIGHT
0
MORNING
T
NIGHT
MORNING AFTERNOON
ATMOSPHERIC LAPSE RATE AND STABILITY
“Lapse rate” = -dT/dz
Consider an air parcel at z lifted to z+dz and released.
It cools upon lifting (expansion). Assuming lifting to be
adiabatic, the cooling follows the adiabatic lapse rate G :
z
stable
G = 9.8 K km-1
g
G  dT / dz 
 9.8 K km-1
Cp
z
unstable
inversion
unstable
What happens following release depends on the
local lapse rate –dTATM/dz:
ATM
• -dTATM/dz > G e upward buoyancy amplifies
(observed) initial perturbation: atmosphere is unstable
• -dTATM/dz = G e zero buoyancy does not alter
perturbation: atmosphere is neutral
• -dTATM/dz < G e downward buoyancy relaxes
T
initial perturbation: atmosphere is stable
• dTATM/dz > 0 (“inversion”): very stable
The stability of the atmosphere
against
mixing is solely determined
Copyright
© 2013vertical
R. R. Dickerson
29
by its lapse rate
& Z.Q. Li
EFFECT OF STABILITY ON VERTICAL STRUCTURE
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
30
WHAT DETERMINES THE LAPSE RATE OF THE
ATMOSPHERE?
•
•
An atmosphere left to evolve adiabatically from an initial state would
eventually tend to neutral conditions (-dT/dz = G ) at equilibrium
Solar heating of surface disrupts that equilibrium and produces an
unstable atmosphere:
z
z
ATM
G
z
final
G
ATM
T
Initial equilibrium
state: - dT/dz = G
G
initial
T
Solar heating of
surface: unstable
atmosphere
T
buoyant motions relax
unstable atmosphere to
–dT/dz = G
• Fast vertical mixing in an unstable atmosphere maintains the lapse rate to G.
Observation of -dT/dz = G Copyright
is sure©indicator
of an unstable atmosphere.31
2013 R. R. Dickerson
& Z.Q. Li
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
32
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
33
Plume looping, Baltimore ~2pm.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
34
Plume Lofting, Beijing in Winter ~7am.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
35