1

Pressure and mass conservation

Today’s goals • 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: F = ma

“momentum concerned”

• Ideal gas law: p = rRT

expression of mass for air

• First law of thermodynamics

“energy conserved”

Gravity exists g

Earth is round a

There are days and nights

(Earth is rotating about once per day)

Alphabet soup is better in Greek

4

Forces?

• Fg, Gravity

• Fv, Viscosity

• Fp, Pressure gradient

F=ma All have the same units, N

5

Gravity

• Newton’s law of gravity 2r

mGMF e

Me mass of the Earth

m mass of the parcel

r Distance between center of the Earth and air parcel

So acceleration,

Directed exactly toward the

Earth’s center of gravity

20*r

GM

m

Fg e

6

Me

m

dme

r

center of

mass

Integrating all mass

elements of, it can be

shown that the direction

vector r points to the

center of mass

7

Earth is big, atmosphere is small

Radius of earth, a = 6371km

Depth of atmosphere ~30km

So we can use an approximation: r~ a

Thus,

Salby, Fundamentals of Atmospheric Physics, page 11.

Troposphere

Stratosphere

Mesosphere

Ionosphere

Thermosphere

Exosphere

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

13

• Position x,y,x

But there are others!

Horizontal: Longitude, latitude

Vertical: Pressure, potential temperature,….

Velocity: u, v, w

(Change in position with time)

14

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

•Total mass of the atmosphere M = ∫globe dm

T, V, r,

p, q

x

y

z

Volume dV = dx dy dz

Mass dm = r dV

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:

zp

zgpzp dd)(0

r

• Compute sea level pressure:

0

dzgps r

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