ThermodynamicsM. D. Eastin We need to understand the environment around a moist air parcel in order...

Thermodynamics M. D. Eastin

We need to understand the environment around a moist air parcel in order to determine whether it will rise or sink through the atmosphere

Here we investigate parameters that describe the large-scale environment

Hydrostatics

Thermodynamics M. D. Eastin

Outline:

Review of the Atmospheric Vertical Structure

Hydrostatic Equation

Geopotential Height Application

Hypsometric Equation Applications Layer Thickness Heights of Isobaric Surfaces Reduction of Surface Pressure to Sea Level

Hydrostatics

Thermodynamics M. D. Eastin

Pressure:

• Measures the force per unit area exerted by the weight of all the moist air lying above that height• Decreases with increasing height

Density:

• Mass per unit volume• Decreases with increasing height

Temperature (or Virtual Temperature):

• Related to density and pressure via the Ideal Gas Law for moist air

• Decreases with increasing height

Review of Atmospheric Vertical Structure

vdTRp ρ

z(km)

Tv(K) p(mb)

0

Tropopause12

151013

-60200

Thermodynamics M. D. Eastin

Balance of Forces:

• Consider a vertical column of air

• The mass of air between heights z and z+dz is ρdz and defines a slab of air in the atmosphere

• The downward force acting on this slab is due to the mass of the air above and gravity (g) pulling the mass downward

• The upward force acting on this slab is due to the change in pressure through the slab

Hydrostatic Equation

dzgF

dpF

Thermodynamics M. D. Eastin

Balance of Forces:

• The upward and downward forces must balance (Newton’s laws)

• Simply re-arrange and we arrive at the hydrostatic equation:

Hydrostatic Equation

dzgdp

FF

gdz

dp

Thermodynamics M. D. Eastin

Application:

• Represents a balanced state between the downward directed gravitational force and the upward directed pressure gradient force

• Valid for large horizontal scales (> 1000 km; synoptic) in our atmosphere• Implies no vertical motion occurs on these large scales

The large-scale environment of a moist air parcel is in hydrostatic balance and does not move up or down

Note: Hydrostatic balance is NOT valid for small horizontal scales (i.e. themoist air parcel moving through a thunderstorm)

Hydrostatic Equation

Thermodynamics M. D. Eastin

Definition:

• The geopotential (Φ) at any point in the Earth’s atmosphere is the amount of work that must be done against the gravitational field to raise a mass of 1 kg from sea-level to that height.

• Accounts for the change in gravity (g) with height

Geopotential Height

z

0dzg

Height Gravity z (km) g (m s-2)

0 9.81 1 9.80 10 9.77 100 9.50

dzgd

Thermodynamics M. D. Eastin

Definition:

• The geopotential height (Z) is the actual height normalized by the globally averaged acceleration due to gravity at the Earth’s surface (g0 = 9.81 m s-2), and is defined by:

• Used as the vertical coordinate in most atmospheric applications in which energy plays an important role (i.e. just about everything)• Lucky for us → g ≈ g0 in the troposphere

Geopotential Height

0g

ΦZ

Height Geopotential Height Gravity z (km) Z (km) g (m s-2)

0 0.00 9.81 1 1.00 9.80 10 9.99 9.77 100 98.47 9.50

Thermodynamics M. D. Eastin

Application:

• The geopotential height (Z) is the standard “height” parameter plotted on isobaric charts constructed from daily soundings:

Geopotential Height

500 mb

Geopotential heights (Z)are solid black contours(Ex: Z = 5790 meters)

Air temperatures (T) arered dashed contours(Ex: T = -11ºC)

Winds are shown as barbs

Thermodynamics M. D. Eastin

Derivation:

• If we combine the Hydrostatic Equation with the Ideal Gas Law for moist air and the Geopotential Height, we can derive an equation that defines the thickness of a layer between two pressure levels in the atmosphere

1. Substitute the ideal gas law into the Hydrostatic Equation

Hypsometric Equation

gdz

dpvdTRp ρ

vdTR

gp-

dz

dp

Thermodynamics M. D. Eastin

Derivation:

2. Re-arranging the equation and using the definition of geopotenital height:

3. Integrate this equation between two geopotential heights (Φ1 and Φ2) and the two corresponding pressures (p1 and p2), assuming Tv is constant in the layer

vdTR

gp-

dz

dp

p

dpTRgdzd vd

2

1

2

1 p

dpTRd vd

p

p

Hypsometric Equation

Thermodynamics M. D. Eastin

Derivation:

4. Performing the integration:

5. Dividing both sides by the gravitational acceleration at the surface (g0):

2

1

2

1 p

dpTRd vd

p

p

1

2vd12 p

plnTRΦΦ

1

2

0

vd

0

1

0

2

p

pln

g

TR

g

Φ

g

Φ

Hypsometric Equation

Thermodynamics M. D. Eastin

Derivation:

6. Using the definition of geopotential height:

Defines the geopotential thickness (Z2 – Z1) between any two pressure levels (p1 and p2) in the atmosphere.

1

2

0

vd

0

1

0

2

p

pln

g

TR

g

Φ

g

Φ

1

2

0

vd12 p

pln

g

TRZZ

Hypsometric Equation

HypsometricEquation

Thermodynamics M. D. Eastin

Interpretation:

• The thickness of a layer between two pressure levels is proportional to the mean virtual temperature of that layer.

• If Tv increases, the air between the two pressure levels expands and the layer becomes thicker

• If Tv decreases, the air between the two pressure levels compresses and the layer becomes thinner

1

2

0

vd12 p

pln

g

TRZZ

Hypsometric Equation

Black solid lines are pressure surfaces

Hurricane (warm core) Mid-latitude Low (cold core)

Thermodynamics M. D. Eastin

Interpretation:

Hypsometric Equation

Layer 1:

Layer 2:

p2

p1

p1

p2

Which layer has the warmest mean virtual temperature?

+Z

+Z

Thermodynamics M. D. Eastin

Application: Computing the Thickness of a Layer

A sounding balloon launched last week at Greensboro, NC measured a mean temperature of 10ºC and a mean specific humidity of 6.0 g/kg between the 700 and 500 mb pressure levels. What is the geopotential thickness between these two pressure levels?

T = 10ºC = 283 Kq = 6.0 g/kg = 0.006

p1 = 700 mbp2 = 500 mb

g0 = 9.81 m/s2

Rd = 287 J /kg K

1. Compute the mean Tv → Tv = 284.16 K

2. Compute the layer thickness (Z2 – Z1) → Z2 – Z1 = 2797.2 m

Hypsometric Equation

1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv

Thermodynamics M. D. Eastin

Application: Computing the Height of a Pressure Surface

Last week the surface pressure measured at the Charlotte airport was 1024 mb with a mean temperature and specific humidity of 21ºC and 11 g/kg, respectively, below cloud base. Calculate the geopotential height of the 1000 mb pressure surface.

T = 21ºC = 294 Kq = 11.0 g/kg = 0.011

p1 = 1024 mbp2 = 1000 mb

Z1 = 0 m (at the surface)Z2 = ???

g0 = 9.81 m/s2

Rd = 287 J /kg K

1. Compute the mean Tv → Tv = 295.97 K

2. Compute the height of 1000 mb (Z2) → Z2 = 198.9 m

Hypsometric Equation

1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv

Thermodynamics M. D. Eastin

Application: Reduction of Pressure to Sea Level

• In mountainous regions, the difference in surface pressure from one observing station to the next is largely due to elevation changes

• In weather forecasting, we need to isolate that part of the pressure field that is due to the passage of weather systems (i.e., “Highs” and “Lows”)

• We do this by adjusting all observed surface pressures (psfc) to a common reference level → sea level (where Z = 0 m)

Hypsometric Equation

850 mb

600 mb

700 mb

400 mb

500 mb

Denver

Aspen

Kathmandu

Thermodynamics M. D. Eastin

Application: Reduction of Pressure to Sea Level

Last week the surface pressure measured in Asheville, NC was 934 mb with a surface temperature and specific humidity of 14ºC and 8 g/kg, respectively. If the elevation of Asheville is 650 meters above sea level, compute the surface pressure reduced to sea level.

T = 14ºC = 287 Kq = 8.0 g/kg = 0.008

p1 = ??? (at sea level)p2 = 934 mb (at ground level)

Z1 = 0 m (sea level)Z2 = 650 m (ground elevation)

g0 = 9.81 m/s2

Rd = 287 J /kg K

1. Compute the surface Tv → Tv = 288.40 K2. Solve the hypsometric equation for p1 (at sea level)3. Compute the sea level pressure (p1) → p1 = 1009 mb

Hypsometric Equation

1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv

Thermodynamics M. D. Eastin

Application: Reduction of Pressure to Sea Level

• All pressures plotted on surface weather maps have been “reduced to sea level”

Hypsometric Equation

Thermodynamics M. D. Eastin

In Class ActivityLayer Thickness:

Observations from yesterday’s Charleston, SC sounding:

Pressure (mb) Temperature (ºC) Specific Humidity (g/kg) 850 10.4 9.2 700 1.8 3.5

Compute the thickness of the 850-700 mb layer

Reduction of Pressure to Sea Level:

Observations from the Charlotte Airport: Z = 237 m (elevation above sea level) p = 983 mbT = 10.5ºCq = 15.6 g/kg

Compute the surface pressure reduced to sea level

Write your answers on a sheet of paper and turn in by the end of class…

Thermodynamics M. D. Eastin

Summary:

• Review of the Atmospheric Vertical Structure

• Hydrostatic Equation

• Geopotential Height• Application

• Hypsometric Equation• Applications• Layer Thickness• Heights of Isobaric Surfaces• Reduction of Surface Pressure to Sea Level

Hydrostatics

Thermodynamics M. D. Eastin

References

Houze, R. A. Jr., 1993: Cloud Dynamics, Academic Press, New York, 573 pp.

Markowski, P. M., and Y. Richardson, 2010: Mesoscale Meteorology in Midlatitudes, Wiley Publishing, 397 pp.

Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.

Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp. Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.