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

1

P 2

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 m2, 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).

Relationship between density, pressure and altitude

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 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 × 1025 molecules m-3 at T=0o 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.

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.

P1x

Net Force (Net pressure forces – Gravity)

The buoyancy force and Archimedes principle. 1. Force on the top of the block: P2 × A = ρ 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 = ρ 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 = ρ water D2 A g + ρ block (D1 - D2) A g

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

Fb = ρ water D1 A g – [ ρ water D2 A + ρ block (D1 - D2) A ] g

= ρ water g Vblock – ρ block g Vblock = (ρ water – ρ 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 :

U h1

h2

h3

ρw h1 = ρw h3 + ρp h2

ρw (h1 – h3) = ρp h2

buoyancy force: ρwhi g – ρphi g = (ρw – ρp )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.

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!) .

Buoyancy and air temperature.

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

P = ρ1 (k/m) T1 = ρ2 (k/m) T2

Then ρ1/ρ2 = T2/T1 ; if T1 > T2, ρ1 < ρ2 . Warmer air, lower density!

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

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

10 Feb 2003

GOES ir image

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.