Download Pressure and mass conservation

Pressure and mass conservation
1
Today’s goals
•
•
•
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Some definitions:
Gravity. What is a shallow atomsphere
Coordinates.
The concept of a “parcel”
(or finite element in the fluid)
• Vertical variations in pressure
• Vertical variation in temperature
(Seemingly obvious, but fascinating research topic)
Some initial things to remember
• Newton’s second law:
“momentum concerned”
• Ideal gas law:
expression of mass for air
• First law of thermodynamics
“energy conserved”
F = ma
Gravity exists
Earth is round
There are days and nights
(Earth is rotating about once per day)
g
a
Alphabet soup is better in Greek
p = rRT

Forces?
• Fg, Gravity
• Fv, Viscosity
• Fp, Pressure gradient
F=ma
All have the same units, N
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Gravity
• Newton’s law of gravity
Me
m
r
GM e m
F 
r2
mass of the Earth
mass of the parcel
Distance between center of the Earth and air parcel
So acceleration,
Directed exactly toward the
Earth’s center of gravity
GM e
F
g0 *    2
m
r
5
r
m
dme
Me
center of
mass
Integrating all mass
elements of, it can be
shown that the direction
vector r points to the
center of mass
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Earth is big, atmosphere is small
Radius of earth, a = 6371km
Depth of atmosphere ~30km
So we can use an approximation: r~
a
Thus,
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Exosphere
Ionosphere
Thermosphere
Mesosphere
Stratosphere
Troposphere
Salby, Fundamentals of Atmospheric Physics, page 11.
Composition of the atmosphere
• Air is a mixture of various gases
• Particularly, nitrogen, oxygen, water vapor,
carbon dioxide
• We can assume air behaves like an ideal gas
Definition:
Lapse rate
Pressure is mass
• Pressure defined as force (imparted on
parcel walls by molecules) per units area
p=F/A
• For horizontal surface the force is weight
F = mg
• So, mass:
m = F/g = pA/g
Define some coordinates
• Position x,y,x
But there are others!
Horizontal: Longitude, latitude
Vertical: Pressure, potential temperature,….
Velocity: u, v, w
(Change in position with time)
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The atmosphere is a fluid continuum
• Can be described atmosphere by a set field variables
that are continuous functions (in space and time)
e.g. temperature, pressure, wind, mass, water content
• It is convenient to consider these properties applying
to a “small” parcel of air
Volume dV = dx dy dz
T, V, r,
p, q
x
z
Mass dm = r dV
y
•Total mass of the atmosphere M = ∫globe dm
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Special look at mass….err pressure
• What is pressure?
NCEP Reanalysis: http://www.cdc.noaa.gov/
Pressure is mass
• Pressure defined as force (imparted on
parcel walls by molecules) per units area
p=F/A
• For horizontal surface the force is weight
F = mg
• So, mass:
m = F/g = pA/g
• What about change in forces on a parcel?
Pressure and mass
• Compute pressure at some height by integrating:
p
z
p( z )   dp   g  rdz

0
• Compute sea level pressure:
0
ps   g  rdz

If the surface pressure is 1000hPa, what is the pressure with
half the atmospheric mass above?
Half! (500hPa)
Relationship
between
temperature,
pressure and
density
The
hydrostatic
equation
Exercises
Practice question:
• Holton 1.13, 1.16, 1.17, 1.18
Reading to next week
• Holton2.6
• Holton 2.7
• Start on RY CH1