Transcript lecture_6_2010_buoyancy

Stability &
Convection
Buoyancy,
Lapse Rate
Lecture 6. EPS 5: 09 Feb. 2010
1. Review the concept of the barometric law (hydrostatic balance: each layer of
atmosphere must support the weight of the overlying column mass of atmosphere).
Discuss the distribution of pressure with altitude in the atmosphere, or depth in the
ocean.
2. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.
3. Archimedes principle: the buoyancy force on an object is equal to the weight of the
fluid displaced by the object. Role of gravity.
4. The buoyancy of warm air.
5. A brief look at global weather patterns—sea surface temperature and buoyancy.
6. Introducing the properties of water.
Relationship between density, pressure and
altitude
Z
Z
P
2
2
1
P
1
By how much is P1 > P2? The weight of the slab of fluid between
Z1 and Z2 is given by the density,  , multiplied by volume of the
slab) and g
weight of slab =  ×(area × height) ×g.
Set the area of the column to 1 m2, the weight is  g × (Z2 -Z1):
If the atmosphere is not being accelerated, there must be a
difference in pressure (P2 - P1) across the slab that
exactly balances the force of gravity (weight of the
slab).
ocean
atmosphere
1 bar = 105 N/m2
Buoyancy
Buoyancy is the tendency for less dense fluids to be forced upwards by more dense
fluids under the influence of gravity. Buoyancy arises when the pressure forces on an
object are not perfectly balanced. Buoyancy is extremely significant as a driving force
for motions in the atmosphere and oceans, and hence we will examine the concept very
carefully here.
The mass density of air  is given by mn, where m is the mean mass of an air molecule
(4.81×10-26 kg molecule-1 for dry air), and n is the number density of air (n =2.69 ×
1025 molecules m-3 at T=0o C, or 273.15 K). Therefore the density of dry air at 0 C is  =
1.29 kg m-3. If we raise the temperature to 10° C (283.15 K), the density is about 4%
less, or 1.24 kg m-3. This seemingly small difference in density would cause air to move
in the atmosphere, i.e. to cause winds.
P1x
Buoyancy force: Forces on a
solid body immersed in a tank of
water. The solid is assumed
less dense than water and to
have area A (e.g. 1m2 ) on all
sides. P1 is the fluid pressure at
level 1, and P1x is the
downward pressure exerted by
the weight of overlying
atmosphere, plus fluid between
the top of the tank and level 2,
plus the object. The buoyancy
force is P1 – P1x (up ↑) per
unit area of the submerged
block.
Net Force (Net pressure
forces – Gravity)
The buoyancy force and Archimedes principle.
1. Force on the top of the block: P2 × A =  water D2 A g
(A = area of top)
weight of the water in the volume above the block
2. Upward force on the bottom of the block = P1 × A =  water D1 A g
3. Downward force on the bottom of the block = weight of the water in the
volume above block + weight of block =  water D2 A g +  block (D1 - D2) A g
Unbalanced, Upward force on the block ( [2] – [3] ):
Fb =  water D1 A g – [  water D2 A +  block (D1 - D2) A ] g
=  water g Vblock –  block g Vblock = ( water –  block) (D1 – D2) A g
weight of block
BUOYANCY FORCE = weight of the water (fluid) displaced by the block
Volume of the block = (D1 – D2) A
Archimedes principle: the buoyancy force on an
object is equal to the weight of the fluid displaced by
the object
•object immersed in a fluid
•weight of fluid displaced
•for the fluid itself, there will be a net upward force (buoyancy force exceeds
object weight) on parcels less dense than the surrounding fluid, a net
downward force on a parcel that is more dense.
•buoyancy can accelerate parcels in the vertical direction (unbalanced force).
•the derivation of the barometric law assumed that every air parcel
experienced completely balanced forces, thus didn't accelerate. Buoyancy
exactly balanced the weight of the parcel (“neutrally buoyant”) – this is
approximately true even if the acceleration due to unbalanced forces is quite
noticeable, because the total forces on an air parcel are really huge
(100,000 N/m2), and thus only small imbalances are needed to produce
significant accelerations.
Write down your answer. We will do this experiment in the next class, and
invite a volunteer to explain the result using Archimedes principle.
A closer look at the U-tube experiment…
compute the density of the paint thinner :
 w h1 =  w h3 +  p h 2
 w (h1 – h3) =  p h2
h1
U
h2
h3
buoyancy force:  whi g –  phi g = ( w –  p )hi g
Looks a lot like Archimedes' principle
2 hi = h1 + h2 + h3
Lecture 6. EPS 5
1. Review the concept of the barometric law (hydrostatic balance: each layer of
atmosphere must support the weight of the overlying column mass of atmosphere).
Discuss the distribution of pressure with altitude in the atmosphere, or depth in the
ocean.
2. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.
3. Archimedes principle: the buoyancy force on an object is equal to the weight of the
fluid displaced by the object. Role of gravity.
4. The buoyancy of warm air.
Buoyancy and air temperature.
Consider two air parcels at the same pressure, but different temperatures.
P =  1 (k/m) T1 =  2 (k/m) T2
Then
 1/ 2 = T2/T1 ; if T1 > T2,  1 <  2 . Warmer air, lower density!
Cold, relatively dense air has
higher density than adjacent warm
air, the warm air is buoyant (the cold
air is "negatively buoyant"). The
"warm air rises" (is buoyant!) .
Lecture 6. EPS 5
1. Review the concept of the barometric law (hydrostatic balance: each layer of
atmosphere must support the weight of the overlying column mass of atmosphere).
Discuss the distribution of pressure with altitude in the atmosphere, or depth in the
ocean.
2. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.
3. Archimedes principle: the buoyancy force on an object is equal to the weight of the
fluid displaced by the object. Role of gravity.
4. The buoyancy of warm air.
5. Introduce properties of water vapor.
Lecture 6. EPS 5: 09 Feb. 2010
Review the concept of the barometric law (hydrostatic balance: each layer of
atmosphere must support the weight of the overlying column mass of atmosphere).
Discuss the distribution of pressure with altitude in the atmosphere, or depth in the
ocean.
1. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.
2. Archimedes principle: the buoyancy force on an object is equal to the weight of the
fluid displaced by the object. Role of gravity.
3. The buoyancy of warm air.
4. A brief look at global weather patterns—sea surface temperature and buoyancy.
5. Introducing the properties of water.
Global Sea Surface Temperatures February 2002
Global Sea Surface Temperature Anomalies, December 2001
10 Feb 2002
GOES ir image
10 Feb 2002
GOES ir image
/www.cira.colostate.edu/Special/CurrWx/g8full40.asp
Feb 2003
Global Sea Surface Temperature Anomalies, Dec. 2001, Jan 2003
12-2001
La Niña
01-2003
El Niño
Lecture 6. EPS 5: 09 Feb. 2010
Review the concept of the barometric law (hydrostatic balance: each layer of
atmosphere must support the weight of the overlying column mass of atmosphere).
Discuss the distribution of pressure with altitude in the atmosphere, or depth in the
ocean.
1. Introduce buoyancy. Pressure force "upwards" on an object immersed in a fluid.
2. Archimedes principle: the buoyancy force on an object is equal to the weight of the
fluid displaced by the object. Role of gravity.
3. The buoyancy of warm air.
4. A brief look at global weather patterns—sea surface temperature and buoyancy.
5. Introducing the properties of water.