Buoyancy force

Buoyancy, 

Lapse Rate 

Stability & 

Convection

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.

Z 2 

Z 1 

P 2 

P 1 

Relationship between density, pressure and 

altitude 

By how much is P1 > P2? The weight of the slab of fluid between 

Z1 and Z2 is given by the density, ρ, multiplied by volume of the 

slab) and g 

weight of slab = ρ×(area × height) ×g. 

Set the area of the column to 1 m 2 , the weight is ρ 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 = 10 5 N/m 2

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 ρ 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 × 10 25 

molecules m -3 at T=0 o C, or 273.15 K). Therefore the density of dry air at 0 C is ρ = 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. 1m 2 ) 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 = ρ water D 2 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 = ρ water D 1 A g 

3. Downward force on the bottom of the block = weight of the water in the 

volume above block + weight of block = ρ water D 2 A g + ρ block (D 1 - D 2 ) A g 

Unbalanced, Upward force on the block ( [2] – [3] ): 

F b = ρ water D 1 A g – [ ρ water D 2 A + ρ block (D 1 - D 2 ) A ] g 

= ρ water g V block – ρ block g V block = (ρ water – ρ block ) (D 1 – D 2 ) A g 

weight of block 

BUOYANCY FORCE = weight of the water (fluid) displaced by the block 

Volume of the block = (D 1 – D 2 ) 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/m 2 ), 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 : 

h 1 

U 

h 2 

h 3 

ρ w h 1 = ρ w h 3 + ρ p h 2 

ρ w (h 1 – h 3 ) = ρ p h 2 

buoyancy force: ρ w h i g – ρ p h i g = (ρ w – ρ p )h i g 

Looks a lot like Archimedes' principle 

2 h i = h 1 + h 2 + h 3

Lecture 6. EPS 5 




ocean. 




4. The buoyancy of warm air.

Buoyancy and air temperature. 

Consider two air parcels at the same pressure, but different temperatures. 

P = ρ 1 (k/m) T 1 = ρ 2 (k/m) T 2 

Then ρ 1 /ρ 2 = T 2 /T 1 ; if T 1 > T 2 , ρ 1 < ρ 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 




ocean. 





5. Introduce properties of water vapor.


Review the concept of the barometric law (hydrostatic balance: each layer of 



ocean. 







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

http://www.cira.colostate.edu/Special/CurrWx/g8full40.asp 

10 Feb 2003 

GOES ir image

12-2001 

La Niña 

01-2003 

El Niño 

Global Sea Surface Temperature Anomalies, Dec. 2001, Jan 2003


Review the concept of the barometric law (hydrostatic balance: each layer of 



ocean.

Buoyancy force

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