Transcript Hypsometric Equation - UNC Charlotte Pages

Hydrostatics
We need to understand the environment around a moist air parcel in order to
determine whether it will rise or sink through the atmosphere
Here we investigate parameters that describe the large-scale environment
Thermodynamics
M. D. Eastin
Hydrostatics
Outline:
 Review of the Atmospheric Vertical Structure
 Hydrostatic Equation
 Geopotential Height
 Application
 Hypsometric Equation
 Applications
 Layer Thickness
 Heights of Isobaric Surfaces
 Reduction of Surface Pressure to Sea Level
Thermodynamics
M. D. Eastin
Review of Atmospheric Vertical Structure
Pressure:
• Measures the force per unit area exerted
by the weight of all the moist air lying
above that height
• Decreases with increasing height
z (km)
Tropopause
12
Density:
• Mass per unit volume
• Decreases with increasing height
Temperature (or Virtual Temperature):
• Related to density and pressure via the
Ideal Gas Law for moist air
p  ρRd Tv
0
-60
200
15
1013
Tv (K)
p (mb)
• Decreases with increasing height
Thermodynamics
M. D. Eastin
Hydrostatic Equation
Balance of Forces:
• Consider a vertical column of air
• The mass of air between heights z and
z+dz is ρdz and defines a slab of air
in the atmosphere
• The downward force acting on this slab
is due to the mass of the air above and
gravity (g) pulling the mass downward
F  g  dz
• The upward force acting on this slab is
due to the change in pressure through
the slab
F  dp
Thermodynamics
M. D. Eastin
Hydrostatic Equation
Balance of Forces:
• The upward and downward forces must
balance (Newton’s laws)
F  F
 dp  g  dz
• Simply re-arrange and we arrive at the
hydrostatic equation:
dp
  g
dz
Thermodynamics
M. D. Eastin
Hydrostatic Equation
Application:
• Represents a balanced state between the downward directed
gravitational force and the upward directed pressure gradient force
• Valid for large horizontal scales (> 1000 km; synoptic) in our atmosphere
• Implies no vertical motion occurs on these large scales
The large-scale environment of a moist air parcel
is in hydrostatic balance and does not move up or down
Note: Hydrostatic balance is NOT valid for small horizontal scales (i.e. the
moist air parcel moving through a thunderstorm)
Thermodynamics
M. D. Eastin
Geopotential Height
Definition:
• The geopotential (Φ) at any point in the Earth’s atmosphere is the amount of
work that must be done against the gravitational field to raise a mass of 1 kg
from sea-level to that height.
d  g dz
z
   g dz
0
• Accounts for the change in gravity (g) with height
Thermodynamics
Height
z (km)
Gravity
g (m s-2)
0
1
10
100
9.81
9.80
9.77
9.50
M. D. Eastin
Geopotential Height
Definition:
• The geopotential height (Z) is the actual height normalized by the globally
averaged acceleration due to gravity at the Earth’s surface (g0 = 9.81 m s-2),
and is defined by:
Φ
Z
g0
• Used as the vertical coordinate in most atmospheric applications in which
energy plays an important role (i.e. just about everything)
• Lucky for us → g ≈ g0 in the troposphere
Thermodynamics
Height
z (km)
Geopotential Height
Z (km)
Gravity
g (m s-2)
0
1
10
100
0.00
1.00
9.99
98.47
9.81
9.80
9.77
9.50
M. D. Eastin
Geopotential Height
Application:
• The geopotential height (Z) is the standard “height” parameter plotted on
isobaric charts constructed from daily soundings:
500 mb
Geopotential heights (Z)
are solid black contours
(Ex: Z = 5790 meters)
Air temperatures (T) are
red dashed contours
(Ex: T = -11ºC)
Winds are shown as barbs
Thermodynamics
M. D. Eastin
Hypsometric Equation
Derivation:
• If we combine the Hydrostatic Equation with the Ideal Gas Law for moist air
and the Geopotential Height, we can derive an equation that defines the
thickness of a layer between two pressure levels in the atmosphere
1. Substitute the ideal gas law into the Hydrostatic Equation
dp
  g
dz
p  ρRd Tv
dp - p g

dz Rd Tv
Thermodynamics
M. D. Eastin
Hypsometric Equation
Derivation:
2. Re-arranging the equation and using the definition of geopotenital height:
dp - p g

dz Rd Tv
dp
d  gdz   Rd Tv
p
3. Integrate this equation between two geopotential heights (Φ1 and Φ2) and
the two corresponding pressures (p1 and p2), assuming Tv is constant in
the layer

2
1
Thermodynamics
d   Rd Tv 
p2
p1
dp
p
M. D. Eastin
Hypsometric Equation
Derivation:
4. Performing the integration:

2
1
d   Rd Tv 
p2
p1
dp
p
 p2 

Φ2  Φ1   Rd Tv ln
 p1 
5. Dividing both sides by the gravitational acceleration at the surface (g0):
Rd Tv  p2 
Φ2 Φ1



ln
g0 g0
g0
 p1 
Thermodynamics
M. D. Eastin
Hypsometric Equation
Derivation:
6. Using the definition of geopotential height:
Rd Tv  p2 
Φ2 Φ1



ln
g0 g0
g0
 p1 
Rd Tv  p2 

Z 2  Z1  
ln
g0
 p1 
Hypsometric
Equation
 Defines the geopotential thickness (Z2 – Z1) between any two pressure levels
(p1 and p2) in the atmosphere.
Thermodynamics
M. D. Eastin
Hypsometric Equation
Interpretation:
• The thickness of a layer between two pressure levels is proportional to the
mean virtual temperature of that layer.
• If Tv increases, the air between the
two pressure levels expands and
the layer becomes thicker
• If Tv decreases, the air between the
two pressure levels compresses
and the layer becomes thinner
Rd Tv  p2 

Z 2  Z1  
ln
g0
 p1 
Black solid lines are
pressure surfaces
Hurricane (warm core)
Thermodynamics
Mid-latitude Low (cold core)
M. D. Eastin
Hypsometric Equation
Interpretation:
p2
+Z
Layer 1:
p1
p2
Layer 2:
+Z
p1
Which layer has the warmest mean virtual temperature?
Thermodynamics
M. D. Eastin
Hypsometric Equation
Application: Computing the Thickness of a Layer
A sounding balloon launched last week at Greensboro, NC measured a mean
temperature of 10ºC and a mean specific humidity of 6.0 g/kg between the 700
and 500 mb pressure levels. What is the geopotential thickness between these
two pressure levels?
Tv  (1  0.61q) T
T = 10ºC = 283 K
q = 6.0 g/kg = 0.006
Rd Tv  p2 

Z 2  Z1  
ln
g0
 p1 
p1 = 700 mb
p2 = 500 mb
g0 = 9.81 m/s2
Rd = 287 J /kg K
1. Compute the mean Tv
→
Tv = 284.16 K
2. Compute the layer thickness (Z2 – Z1)
Thermodynamics
→
Z2 – Z1 = 2797.2 m
M. D. Eastin
Hypsometric Equation
Application: Computing the Height of a Pressure Surface
Last week the surface pressure measured at the Charlotte airport was 1024 mb
with a mean temperature and specific humidity of 21ºC and 11 g/kg, respectively,
below cloud base. Calculate the geopotential height of the 1000 mb pressure
surface.
T = 21ºC = 294 K
q = 11.0 g/kg = 0.011
Tv  (1  0.61q) T
p1 = 1024 mb
p2 = 1000 mb
Rd Tv  p2 

Z 2  Z1  
ln
g0
 p1 
Z1 = 0 m (at the surface)
Z2 = ???
g0 = 9.81 m/s2
Rd = 287 J /kg K
1. Compute the mean Tv
→
Tv = 295.97 K
2. Compute the height of 1000 mb (Z2)
Thermodynamics
→
Z2 = 198.9 m
M. D. Eastin
Hypsometric Equation
Application: Reduction of Pressure to Sea Level
• In mountainous regions, the difference in surface pressure from one observing
station to the next is largely due to elevation changes
400 mb
500 mb
Kathmandu
600 mb
700 mb
850 mb
Aspen
Denver
• In weather forecasting, we need to isolate that part of the pressure field that
is due to the passage of weather systems (i.e., “Highs” and “Lows”)
• We do this by adjusting all observed surface pressures (psfc) to a common
reference level → sea level (where Z = 0 m)
Thermodynamics
M. D. Eastin
Hypsometric Equation
Application: Reduction of Pressure to Sea Level
Last week the surface pressure measured in Asheville, NC was 934 mb with a
surface temperature and specific humidity of 14ºC and 8 g/kg, respectively. If the
elevation of Asheville is 650 meters above sea level, compute the surface pressure
reduced to sea level.
T = 14ºC = 287 K
q = 8.0 g/kg = 0.008
p1 = ??? (at sea level)
p2 = 934 mb (at ground level)
Z1 = 0 m (sea level)
Z2 = 650 m (ground elevation)
Tv  (1  0.61q) T
Rd Tv  p2 

Z 2  Z1  
ln
g0
 p1 
g0 = 9.81 m/s2
Rd = 287 J /kg K
1. Compute the surface Tv → Tv = 288.40 K
2. Solve the hypsometric equation for p1 (at sea level)
3. Compute the sea level pressure (p1) → p1 = 1009 mb
Thermodynamics
M. D. Eastin
Hypsometric Equation
Application: Reduction of Pressure to Sea Level
• All pressures plotted on surface weather maps have been “reduced to sea level”
Thermodynamics
M. D. Eastin
In Class Activity
Layer Thickness:
Observations from yesterday’s Charleston, SC sounding:
Pressure (mb)
850
700
Temperature (ºC)
10.4
1.8
Specific Humidity (g/kg)
9.2
3.5
Compute the thickness of the 850-700 mb layer
Reduction of Pressure to Sea Level:
Observations from the Charlotte Airport:
Z = 237 m (elevation above sea level)
p = 983 mb
T = 10.5ºC
q = 15.6 g/kg
Compute the surface pressure reduced to sea level
Write your answers on a sheet of paper and turn in by the end of class…
Thermodynamics
M. D. Eastin
Hydrostatics
Summary:
• Review of the Atmospheric Vertical Structure
• Hydrostatic Equation
• Geopotential Height
• Application
• Hypsometric Equation
• Applications
• Layer Thickness
• Heights of Isobaric Surfaces
• Reduction of Surface Pressure to Sea Level
Thermodynamics
M. D. Eastin
References
Houze, R. A. Jr., 1993: Cloud Dynamics, Academic Press, New York, 573 pp.
Markowski, P. M., and Y. Richardson, 2010: Mesoscale Meteorology in Midlatitudes, Wiley Publishing, 397 pp.
Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp.
Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.
Thermodynamics
M. D. Eastin