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


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


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


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


dzgF

dpF


Balance of Forces:

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

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


dzgdp

FF

gdz

dp


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)



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


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


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


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


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



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



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


HypsometricEquation


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


Black solid lines are pressure surfaces

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


Interpretation:


Layer 1:

Layer 2:

p2

p1

p1

p2

Which layer has the warmest mean virtual temperature?

+Z

+Z


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


1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv


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


1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv


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)


850 mb

600 mb

700 mb

400 mb

500 mb

Denver

Aspen

Kathmandu



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


1

2

0

vd12 p

pln

g

TRZZ

T0.61q)(1Tv



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



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…


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


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.

thermodynamicsm. d. eastin we need to understand the environment around a moist air parcel in order...

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