Next 4 weeks:Atmospheric temperature profiles

Stability[1-week on Fronts from Hugh Pumphrey]

ThunderstormsAir Pollution

David Stevenson

Crew Building Room 314

dstevens@staffmail.ed.ac.uk

L13 Physics of Dry Air

• Vertical pressure gradient through the atmosphere: – The Hydrostatic Equation (revision)

• Measuring temperature profiles: radiosondes

• 1st Law of Thermodynamics: conservation of energy

• ‘Air parcels’

• The ‘Dry Adiabatic Lapse Rate’

• At what height do clouds form?• Will air rise or sink?

Atmospheric stability

=> Rain showers/thunderstorms caused by moist air rising and cooling

Inversions – e.g. fog Pollution dispersion –

e.g. Buncefield fire

Why are vertical profiles important?

Vertical pressure gradient

• We know pressure (p) decreases as you go up through the atmosphere.

• Q: So why doesn’t air flow from high p at surface to low p at altitude? (i.e. why hasn’t Earth lost its atmosphere to space?)

• A: Gravity attracts it towards the centre of the Earth.

• The balance of gravity and vertical pressure gradient is ‘hydrostatic balance’

Downward force on airin shaded slab, due topressure of air above

Upward force on air inshaded slab due to pressure of air below

Net upward force on slab, due to the pressure gradient= -p

must balancedownwardforce due to weight of slab= gz

Hydrostatic Equation (see Lecture 8)

zgp

gz

p In limit as

z → 0,

First law of thermodynamicsConservation of energy for a parcel of air:

dw = Work doneby air parcel

dp

dwdudq du = Change in internal energy

of air parcel

)( dTcp

dq = heat added to air parcel from its

surroundings

dpdTcdq p

Cp is the specific heat capacity at constant pressureα is the specific volume (1/density)

Pumping up/letting down a tyre

• Pump up tyre: air compresses, work is done on the air in the tyre (dw is –ve; the air in the tyre doesn’t do work). If we assume dq=0, then dT is +ve, the air heats up (valve gets hot).

• Let down tyre: air expands, the air in the tyre does work on its surroundings (dw is +ve). If we assume dq=0, then dT is –ve, the air cools down (valve gets cold).

dpdTcdq p

Radiosondes

Temperature (and humidity, pressure) sensor, attached to weather balloon, with radio transmitter to send data back to earth.

Radiosonde

• Rises to between 20-30 km • then balloon bursts and radiosonde returns to surface by parachute

• pressure, Temperature and moisture are all measured by sensors and the signals are transmitted to a base station by radio

How do we measure temperature and moisture in the atmosphere?

Ikarus project: http://www.youtube.com/watch?v=MCBBRRp9DOQ&NR=1

Wind speed and directionare not directly measuredbut inferred(radar echo,onboard radio receiveror GPS-based systems).

All measurements in the profile are attributed to the nominal hour of the ascent. This is the hour at which the sonde reaches 100 mb. It takes approximately an hour for the balloon to rise to this level and thus the sondes are released one hour before the synoptic hours.

Automatic balloon releases

Radiosonde data

http://weather.uwyo.edu/upperair/sounding.html

Radiosonde data is reported up to four times per day at the synoptic hours of 00, 06, 12 and 18 GMT. The number of ascents varies widely between countries and stations.

You can get worldwide radiosonde data from:

Temperature of an ascending air parcel

• Start with the 1st Law of thermodynamics:

• Assume the ascent is adiabatic, i.e. dq=0• Use the hydrostatic equation:• Gives:

gdzdp

dpdTcdq p

)(0 gdzdTcp

gdzdTcp 0pc

g

dz

dT or:

remember:

1

Dry adiabatic lapse rateAcceleration due to gravity, g = 9.81 m s-2

Specific heat capacity dry air (at constant P), Cp = 1004 J K-1 kg-1

So:

Check units: remember a Joule, J, can be expressed in fundamental SI units (e.g., kinetic energy = ½ m v2):

1 J = 1 kg (m s-1)2 = 1 kg m2 s-2

So units of Cp, J K-1 kg-1 = kg m2 s-2 K-1 kg-1 = m2 s-2 K-1 So the temperature gradient has units:

11 8.90098.01004

81.9

KkmKmc

g

dz

dT

p

1122

2

Kmm

K

Ksm

ms

c

g

dz

dT

p

Z

p

Concept: an ‘air parcel’

It’s a usefulconcept to imagine what will happen as a mass (‘parcel’) of air moves up or down in the atmosphere.

Assumptions for a parcel of air

- No exchange of mass with environment- No exchange of heat with surrounding- Adjusts to pressure of environment

(And moves slowly enough to neglect energy of movement of air parcel)

Summary

• Air temperature generally decreases with increasing altitude (e.g. radiosonde data)

• Using some physics (hydrostatic equation, 1st Law of thermodynamics), we can derive a theoretical expression for the temperature gradient of an adiabatically ascending dry air parcel: -9.8 K/km

• This is quite often a good approximation of the real atmosphere

• Main complications involve moisture condensing and releasing latent heat – next lecture.

Current Weather

Hugh Pumphrey’s web-pages:

https://www.geos.ed.ac.uk/homes/hcp/currentmet.html

Potential Temperature (θ)

• The potential temperature of an air parcel is its temperature when compressed (or expanded) adiabatically to surface pressure (p0) (defined as a standard pressure of 1000 hPa).

• Again, start from the 1st Law of Thermodynamics, and make dq=0:

0 dpdTcp

Ideal Gas Law (see Lecture 8)

RTp p

RT

1so:

0 dpdTcp

0 dpp

RTdTcp

substitute in α:

Divide by RT:

0p

dp

T

dT

R

cp

Integrate both sides, from the starting (p,T) tothe surface (p0,T0), noting cp/R is a constant:

p

p

T

T

p

p

dp

T

dT

R

c

00

Remember integral of 1/x is natural log of x:

1

212 lnlnln

1 ln 2

1

2

1x

xxxdx

xx

x

x

x

x

0

lnlnp

pT

R

cp

Remember: abba ln)(ln

0p

pT R

cp

pc

R

p

pT

0

Hence: or:

Rearrange to give potential temperature, θ:

pc

R

p

pT

0

Integrating: