ThermodynamicsM. 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

ThermodynamicsM. 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

ThermodynamicsM. 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 z (km) T v (K) p (mb) 0 Tropopause

ThermodynamicsM. 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

ThermodynamicsM. 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

ThermodynamicsM. 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. the moist air parcel moving through a thunderstorm) Hydrostatic Equation

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

ThermodynamicsM. 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 (g 0 = 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 ≈ g 0 in the troposphere Geopotential Height HeightGeopotential Height Gravity z (km) Z (km) g (m s -2 )

ThermodynamicsM. 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) are red dashed contours (Ex: T = -11ºC) Winds are shown as barbs

ThermodynamicsM. 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

ThermodynamicsM. D. Eastin Derivation: 2. Re-arranging the equation and using the definition of geopotenital height: 3. Integrate this equation between two geopotential heights (Φ 1 and Φ 2 ) and the two corresponding pressures (p 1 and p 2 ), assuming T v is constant in the layer Hypsometric Equation

ThermodynamicsM. D. Eastin Derivation: 4. Performing the integration: 5. Dividing both sides by the gravitational acceleration at the surface (g 0 ): Hypsometric Equation

ThermodynamicsM. D. Eastin Derivation: 6. Using the definition of geopotential height:  Defines the geopotential thickness (Z 2 – Z 1 ) between any two pressure levels (p 1 and p 2 ) in the atmosphere. Hypsometric Equation Hypsometric Equation

ThermodynamicsM. D. Eastin Interpretation: The thickness of a layer between two pressure levels is proportional to the mean virtual temperature of that layer. If T v increases, the air between the two pressure levels expands and the layer becomes thicker If T v decreases, the air between the two pressure levels compresses and the layer becomes thinner Hypsometric Equation Black solid lines are pressure surfaces Hurricane (warm core)Mid-latitude Low (cold core)

ThermodynamicsM. D. Eastin Interpretation: Hypsometric Equation Layer 1: Layer 2: p2p2 p1p1 p1p1 p2p2 Which layer has the warmest mean virtual temperature? +Z

ThermodynamicsM. 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 K q = 6.0 g/kg = p 1 = 700 mb p 2 = 500 mb g 0 = 9.81 m/s 2 R d = 287 J /kg K 1. Compute the mean T v → T v = K 2. Compute the layer thickness (Z 2 – Z 1 ) → Z 2 – Z 1 = m Hypsometric Equation

ThermodynamicsM. 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 K q = 11.0 g/kg = p 1 = 1024 mb p 2 = 1000 mb Z 1 = 0 m (at the surface) Z 2 = ??? g 0 = 9.81 m/s 2 R d = 287 J /kg K 1. Compute the mean T v → T v = K 2. Compute the height of 1000 mb (Z 2 ) → Z 2 = m Hypsometric Equation

ThermodynamicsM. 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 (p sfc ) 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

ThermodynamicsM. 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 K q = 8.0 g/kg = p 1 = ??? (at sea level) p 2 = 934 mb (at ground level) Z 1 = 0 m (sea level) Z 2 = 650 m (ground elevation) g 0 = 9.81 m/s 2 R d = 287 J /kg K 1. Compute the surface T v → T v = K 2. Solve the hypsometric equation for p 1 (at sea level) 3. Compute the sea level pressure (p 1 ) → p 1 = 1009 mb Hypsometric Equation

ThermodynamicsM. D. Eastin Application: Reduction of Pressure to Sea Level All pressures plotted on surface weather maps have been “reduced to sea level” Hypsometric Equation

ThermodynamicsM. D. Eastin In Class Activity Layer Thickness: Observations from yesterday’s Charleston, SC sounding: Pressure (mb)Temperature (ºC) Specific Humidity (g/kg) Compute the thickness of the mb layer Reduction of Pressure to Sea Level: Observations from the Charlotte Airport: Z = 237 m (elevation above sea level) p = 983 mb T = 10.5ºC q = 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…

ThermodynamicsM. 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

ThermodynamicsM. 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.