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UCSD SIO 217A - Lecture

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1!Chapter 3 Homework!“At zenith angle Z”!“Isotropic”!“Radiance”!Ch. 3 Homework!• Correction!• Extra Credit! +2 pts for Latex Answer Key (submit by email by Friday 12:30pm)!Lecture Ch. 4a!• Equilibrium!• Phase changes!• Enthalpy changes from phase changes!– Latent heat!– Clapeyron equation!– Clausius-Clapeyron equation!Curry and Webster, Ch. 4 (pp. 96-115; skip 4.5 (except 4.5.1), 4.6)!For Wednesday: Homework Problem Ch.4 Prob. 4, 5!2!Atmospheric “Components”!Phase Diagrams!• Pressure-temperature diagrams!• Degrees of freedom!• Pressure-volume diagrams!dG=0!€ ∂p∂T⎛ ⎝ ⎜ ⎞ ⎠ ⎟ gdpdT≈ΔpΔT=p2− p1T2− T1Consider:!Tangent vs. Average!Phase Equilibrium!• Thermal equilibrium!• Mechanical equilibrium!• Chemical equilibrium!Reminders!• Virtual Temperature: The temperature air would have at the given pressure and density if there were no water vapor in it !• Potential Temperature: The temperature a parcel would have if it were brought adiabatically and reversibly to p0 (usually 1 atm) !• Virtual Potential Temperature: The temperature a parcel would have if there were no water vapor in it (only condensed water) and if it were brought adiabatically and reversibly to p0 (usually 1 atm)!€ θv= T 1+ 0.608qv( )p0p⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Rdcpd€ Tv= T 1+ 0.608qv( )€ θ= Tp0p⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Rdcpd3!Water Vapor Metrics!Chemical Equilibrium!• Two phases in equilibrium!– Constant T, P!• Phase changes!– Constant T, P!• (What was G?)!€ ΔGT ,P= 0€ ΔGT ,P= 0Gibbs (Free) Energy!Enthalpy Change!• Enthalpy for phase transition!• Define latent heat!Curry and Webster, Ch. 4!Quiz !• What is mechanical equilibrium? Give an equation.!• What 3 equilibria are required for the Gibbs phase rule?!• Give the equation for the Gibbs phase rule. !• What is the change in free energy for a phase change at constant T, P?!• What type of pressure change is described by the Clausius-Clapeyron equation? i.e. what changes as a function of what under what conditions?!Answer briefly and clearly, with appropriate equations or diagrams. !Clapeyron Equation!• Enthalpy change for any phase transition!Exact! (tangent) (Not exact but usually good) Exact! (average)4!Clausius-Clapeyron "Equation!• Latent heat of vaporization!Phase Change Relationships!• Clapeyron equation!– All phase changes!– Non-ideal equations of state!• Clausius-Clapeyron equation!– Liquid-vapor equilibrium only: vL << vV!– Ideal gas law for vapor: vV = RT/p!Ch. 4: Problem 4, 5!Answers:!Water Vapor Metrics!Mixing ratio!Specific humidity!Relative humidity!Water vapor by mass!Water vapor by partial pressure!Water saturation!Virtual temperature!Virtual potential temperature!€ qv=mvmd+ mv€ wv=mvmd=ρvρd€ qv= 0.622ep − 1 − 0.622( )e⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € ws= 0.622esp − es⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € H ≈wvws€ H =ees€ θv= T 1 + 0.608qv( )p0p⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Rdcpd€ Tv= T 1 + 0.608qv( )Water Saturation Pressures !es doubles with every 10C!!T(C)!eS (hPa)!10!12.3!20!23.4!30!42.4!40!73.8!(this is one consequence of Clausius-Clapeyron’s equation)!Review Problems!• Hydrostatic equilibrium!– Special cases!– Pressure altitude dependence!• More Midterm Review problems!– Terminology review!Curry and Webster, Ch. 4 (pp. 96-115; skip 4.5 (except 4.5.1), 4.6)!Wednesday, Oct. 20: Homework, Review (bring questions), and Read Ch. 5!Wednesday, Oct. 27: Midterm!5!Chapter 1, Problem 11!Special Cases of Hydrostatic Equilibrium!• 1. rho=constant (homogeneous)!– H=8 km =RT/g=scale height eq. 1.39!• 2. constant lapse rate (e.g. if hydrostatic, homogeneous, and ideal gas)!– -dT/dz=constant=-g/R=-34/deg/km!• 3. isothermal T=constant (and ideal gas)!– p=p_0*exp(-z/H)!Special Cases of Hydrostatic Equilibrium!• Hydrostatic: Force balance on gravity and upward pressure!Homogeneous Atmosphere!Homogeneous Constant density Constant lapse rate Isothermal Atmosphere!Hydrostatic Equilibrium Example!Consider a planet with an atmosphere in hydrostatic equilibrium. Assume that theatmosphere is an ideal gas. Also assume that the temperature is a maximum at thesurface of the planet, and, as height increases, the temperature in the atmospheredecreases linearly (in other words, temperature decreases with height at a constant rate).Derive a formula for atmospheric density as a function of height in this atmosphere.6!Hydrostatic Equilibrium Example!Consider a planet with an atmosphere in hydrostatic equilibrium. Assume that theatmosphere is an ideal gas. Also assume that the temperature is a maximum at thesurface of the planet, and, as height increases, the temperature in the atmospheredecreases linearly (in other words, temperature decreases with height at a constant rate).Derive a formula for atmospheric density as a function of height in this atmosphere.From the hydrostatic equation for an ideal gas (Eqn. 1.42)€ ∂p = −pgRdT∂zand a constant lapse rate € Γ = −dTdz we get€ dp = −pgRdTdz−dTdzΓ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = −pgΓRddTTdpp=gΓRd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ dTTdpp=gΓRd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ dTT∫∫lnpp0=gΓRd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ lnTT0p = p0TT0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ gRdΓ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ which is Eqn 1.48. Then dividing both sides by RT and noting that for an ideal gas€ ρ=pRT, we get€ pRT=ρ=p0RTTT0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ gRdΓ=p0R T0− Γz( )T0− ΓzT0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ gRdΓClausius Clapeyron Example!The saturation vapor pressure at a temperature of 30°C is 42.4 hPa. The gas constant fordry air is 287 J K-1 kg-1. The gas constant for water vapor is 461 J K-1 kg-1.In addition to the constants given above, here is one more: the saturation vapor pressureat a temperature of 40°C is 73.8 hPa. Assuming that the latent heat of vaporization isconstant, use this information to calculate the numerical value for this latent heat.Clausius Clapeyron Example!The saturation vapor pressure at a temperature of 30°C is 42.4 hPa. The gas constant fordry air is 287 J K-1


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UCSD SIO 217A - Lecture

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