Temperature and Thermodynamics, Part II

• Topics to be Covered – Profiles of Temperature in the Boundary Layer – Potential temperature – Adiabatic Lapse Rate – Thermal Stratification

1/8/17 ESPM 129 Biometeorology Why are We Interested in Thermal Stratification of the Atmosphere?

• Compare Temperatures measured at Different Heights and Altitudes • Creates or Suppresses Turbulence and Diffusion • Adiabatic Lifting can Cause Cooling – It can cause water to change phases – Condensation Promotes the Formation of Clouds and Rain • Adiabatic Compression can Cause Heating – Downslope Santa Ana winds, Hot and Dry • Microclimate Modification – Wind machines Break the Nocturnal Inversion Layer and Prevent Frost

ESPM 129 Biometeorology Changes in Air Pressure and Density with Height, Ideal Conditions

3000

3000 2500

2500 2000

2000 1500

1500 Height, m Height,

1000 m Height, 1000 dP=−ρ g dz mPea ()− a ρ = 500 a 500 RT

0 0 65 70 75 80 85 90 95 100 105 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Pressure, kPa Air Density, kg m-3

Air Parcel Lifted Expands and Becomes Less Dense because Surrounding Pressure is Less, Consequently Its Temperature Drops as the Parcel Expends

ESPM 129 Biometeorology Temperature Profiles in the Boundary Layer:

Adiabatic Lapse Rate

3000

2500

−1 2000 Γ=−9.8Kkm

1500 Height, m Height,

1000

500

0 265 270 275 280 285 290 295 300 305

Tair, K

ESPM 129 Biometeorology Potential Temperature, θ

The Temperature a Parcel of Air would have if it were Expanded or Compressed, Adiabatically, from its Existing Temperature and Pressure to a Standard, near sea level (P=101.3 kPa):

Wallace and Hobbs, 1977

P R/(m c ) P R/C θ = T( 0 ) ⋅ p θ = T( 0 ) p P P

T, Temperature (K) R, Universal Gas Constant, 8.314 P, Pressure, kPa

P0, reference pressure, eg 101.3 kPa Cp, Heat Capacity of air at constant pressure cp, specific heat capacity of air at constant pressure

ESPM 129 Biometeorology Temperature Profiles in the Boundary Layer:

Adiabatic Lapse Rate and Potential Temperature

70 3000

75 2500

80 2000

85 1500 Height, m Height, 90 1000 Pressure, kPa

95 500

100 0 265 270 275 280 285 290 295 300 305 296 298 300 302 304 T , K air Potential Temperature, K

ESPM 129 Biometeorology Profiles of Air and Potential Temperature

June 1, 1999, 1800 UT Oak Ridge, TN

700

Virtual Potential Temperature 750 Dry Bulb Temperature

800

Stable 850

Pressure,mb neutral 900

950 unstable

1000 280 290 300 310 320 Temperature, K

ESPM 129 Biometeorology Fundamentals of Thermal Stratification

stable thermal unstable stratification near neutral thermal stability stratifiation Tparcel < Tair Tparcel > Tair m m m Height, ∂T ∂T Height, Height, =−Γ = −Γ ∂T z ∂z =−Γ ∂ ∂z m m

∂θ Height, ∂θ

=0 =0 m ∂θ Height, ∂z ∂z =0 Height, ∂z

ESPM 129 Biometeorology Thermal stratification causes the atmosphere to be either buoyant or stable

∂θ ∂T • The atmosphere is neutrally stratified if: v = 0 v = −Γ ∂z ∂z

• The atmosphere is unstably stratified if: ∂θ v ∂Tv < 0 < −Γ ∂z ∂z

• The atmosphere is stably stratified if: ∂θ v ∂T > 0 v > −Γ ∂z ∂z

ESPM 129 Biometeorology Upcoming Features

• Describe Adiabatic Processes and Potential Temperature in terms of variables we measure, P and T • Start with 1st Law of Thermodynamics • Normalize 1st Law by Mass, yielding ‘Specific’ Equation • Substitute Terms with Gas Laws, converting Equation from a function of heat capacity for volume, Cv, to that for pressure, Cp • Rearrange Equation so T is on one side and P is on the other • Integrate both sides, yielding equations Ln(T) and Ln(P) • Take anti-log and solve for potential temperature

ESPM 129 Biometeorology Adiabatic Process

An air parcel changes its physical state (either its pressure, volume or temperature) without heat being added or withdrawn from the surrounding environment

. dQ = 0

ESPM 129 Biometeorology Understanding Adiabatic Compression and Expansion, And its Impact on Temperature Profiles With the First Law of Thermodynamics

dQ = CV dT + PdV = 0

CV dT = −PdV Change in Change in Work internal Energy

No Heat is Exchanged, But Temperature Can Change

Change in Internal Energy Equals Change in Work Done on the Parcel

ESPM 129 Biometeorology Understanding Adiabatic Processes in the Atmosphere by considering Parcels of Air per unit Mass (m), Specific quantities

Q q = Change in Volume per unit Mass is m Equivalent to the change of the inverse in density, ρ W w = m U dV 1 u = = d( ) = dα m m ρ C c = v v m C c = p p m

ESPM 129 Biometeorology The Specific Form of the 1st Law of Thermodynamics

1 dq = c dT + Pd( ) = c dT + Pdα v ρ v

ESPM 129 Biometeorology Algebraic Tricks to Express dq as f(T, P) and solve for Pdα

ρRT RT P = Pα = m m

1 1 d(Pα) = αdP + Pdα = d(RT) = RdT m m

R Pdα = −adP + dT m

ESPM 129 Biometeorology Algebraic Tricks to Express dq as f(T, P), things we can measure

R Pdα = −adP + dT m

dq= cv dT− Pdα

R dq = (c + )dT −αdP v m

ESPM 129 Biometeorology Change in Specific Heat, dq, is a function of the

Specific Heat Capacity, cp, times a change in Temperature, minus A change in pressure, dP….terms we can MEASURE

R dq=+() c dT −α dP v m R cc=+ pvm

dq= cp dT−α dP

ESPM 129 Biometeorology Define specific heat capacity at constant P

Isobaric case, dP equals 0

R dq = (c + )dT −αdP v m

dq R | = c + = c dT p v m p

ESPM 129 Biometeorology Adiabatic Process, dq=0

dq==0 cp dT −α dP

dP c dT==α dP p ρ

ESPM 129 Biometeorology dP mP cdT==α dP ρ = p ρ RT

RTdP P: Pressure cp dT = V: volume mP n: number of moles R: Universal Gas Constant m: mass per mole T: absolute air temperature dT R dP ρ: air density c = p TmP

ESPM 129 Biometeorology dT R dP R dP == TmcPCP⋅ pp

T dT R P dP = ∫ T m c ∫ P θ ⋅ p P0

T R P ln = ln θ m⋅cp P0

ESPM 129 Biometeorology T R P ln = ln θ m⋅cp P0

TRP exp(ln )= exp( ln ) θ mc⋅ p P0

R

TPmc⋅ p = θ P0

ESPM 129 Biometeorology Potential Temperature

P R/(m c ) θ = T( 0 ) ⋅ p P

P R/C θ = T( 0 ) p P

ESPM 129 Biometeorology Adiabatic lapse rate, How Temperature Changes with Height

0 =ρ cp dT− dP

ρ cp dT= dP

dP Hydrostatic Equation = −ρg dz

Change in Pressure with height is a function of the density of Air times the acceleration due to gravity, g

ESPM 129 Biometeorology Dry Adiabatic Lapse Rate

dT dP ρρcg==− p dz dz

dT g |adiabatic =−=Γ dz cp

The dry adiabatic lapse rate, 9.8 K km-1

ESPM 129 Biometeorology Moist Adiabatic

water vapor condenses and latent heat of condensation is released.

dT g | =−=Γ dz moistde moist c + λ s p dT

λ is the latent heat of vaporization des/dT is the slope of the saturation vapor pressure-temperature curve.

ESPM 129 Biometeorology Key Points

• Define Thermal Stratification – Neutral, Stable and Unstable Conditions • Define Potential Temperature • Define Adiabatic Processes – Dry and Moist Adiabatic Lapse Rate • Define First Law of Thermodynamics

ESPM 129 Biometeorology Summary

• Potential temperature is the temperature of a parcel of air that is moved Adiabatically from a level in the atmosphere to a reference Pressure. – It is defined for conditions when changes heat energy are zero (dQ=0). • Thermal stratification causes the ∂θ atmosphere to be either buoyant or v = 0 stable ∂z ∂θ – The atmosphere is neutrally stratified if: v < 0 – The atmosphere is unstably stratified if: ∂z ∂θ – The atmosphere is stably stratified if: v > 0 ∂z

ESPM 129 Biometeorology