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UCSD SIO 217A - Midterm Exam

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SIO 217a Atmospheric and Climate Sciences I: Atmospheric Thermodynamics Fall 2011 Midterm Exam (No calculators, notes, books, PDAs.) KEY Curry and Webster, Ch. 1-4 (and Section12.1) Here are some numerical values, some of which may be useful on this exam: Average radius of Earth: 6370 km Mean reflectivity of the Earth: 0.31 Mean molecular weight of dry air: 29 g/mole Mean molecular weight of water vapor: 18 g/mole Gas constant for dry air, Rd: 287 J deg-1 kg-1 Gas constant for water vapor, Rv: 461 J deg-1 kg-1 Specific heat at constant pressure, cp: 1004 J deg-1 kg-1 Specific heat at constant volume, cv: 717 J deg-1 kg-1 Latent heat of vaporization for water at 273K, Llv: 2.5×106 J kg-1 Solar luminosity: 3.92×1026 W Earth-sun distance: 1.50×1011 m Stefan-Boltzmann constant, σ: 5.67×10-8 W m-2 K-4 1. Almost one-third of the Earth’s incoming solar radiation is reflected back to space. a. Name the property of the Earth controls the fraction of incoming light reflected. Albedo is the mean reflectivity of the Earth. b. Calculate the amount of incoming solar radiation at the top of the atmosphere [in W m-2]. Instantaneous at solar noon =(luminosity)/(4pi*ESdistance2) =(3.92×1026)/(4*3.14*(1.50×1011)2)=1390 W m-2; Averaged over Earth surface = (1390 W m-2)/4=342 W m-2. c. What happens to the energy from the incoming radiation that is not reflected? The remaining energy is absorbed by the Earth and the atmosphere and then re-emitted. d. If the amount of light reflected were decreased, how would a simple model with no greenhouse effect predict Earth’s temperature would respond? Give your model and state its assumptions. Assume that: (1) the earth behaves as a blackbody, (2) atmosphere is transparent to non-reflected portion of the solar beam; (3) atmosphere in radiative equilibrium with surface. Then, at equilibrium, the incoming shortwave flux and outgoing longwave flux are equal (i.e. there is no accumulation) so for the normal solar luminosity we can write: FL= σTatm4 (assumption 1; Eqn. 3.20); FS = FL(assumption 2-3; Eqn. 3.20) 0.25*S0(1- αp) = σTatm4 (Eqn. 3.20, Eqn. 12.) As albedo decreases, Tatm will increase. 2. Consider the properties of the standard atmosphere, assuming hydrostatic balance and constant lapse rate of Γ=6.5K km-1. a. Write the equation for the hydrostatic balance and describe where it comes from. b. Derive an expression for the variation of height with pressure z(p), in terms of the surface pressure p0, surface temperature T0 and a constant lapse rate Γ. c. What are typical values for p0 and T0 for modern Earth?d. Name one common application of this relationship. a. dp=-ρgdz: The hydrostatic balance is the equality of upward (pressure gradient) and downward (gravitational) forces in the atmosphere that results in little net vertical motion. b. From the hydrostatic equation for an ideal gas (Eqn. 1.42) € ∂p = −pgRdT∂z and a constant lapse rate € Γ = −dTdz we get € dp = −pgRdTdzdpp= −g(T0− Γz)Rd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ dzdppp0p∫= −gRd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ dz(T0− Γz)0z∫lnpp0=gΓRd⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ln(T0− Γz)T0pp0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ RdΓg⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =T0− ΓzT0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 1−ΓzT0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ z =T0Γ1−pp0⎛ ⎝ ⎜ ⎞ ⎠ ⎟ RdΓg⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ c. p0 =1013 mb; T0= 288K. d. This relationship is used to determine altitude from pressure measurements on aircraft. 3. Define the following terms in 10 words or less; an equation, graph, or sketch may be added if appropriate: a. mechanical equilibrium occurs when two substances have no net exchange of force, i.e. p1=p2. b. adiabatic means there is no exchange of heat into or out of the system; A path in which no heat is lost or gained during the process (Q=0). c. virtual potential temperature is the temperature a parcel would have if there were no water vapor in it and if it were brought adiabatically and reversibly to p0 (usually 1 atm). d. Wien’s law λmax=2890/T; The maximum wavelength emitted is proportional to the inverse temperature. e. exact differential is a function ξ for which dξ has the properties (1) for any closed path € dξ= 0∫, and (2) for ξ(x,y) where x and y are independent, then € dξ=∂ξ∂x⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ydx +∂ξ∂y⎛ ⎝ ⎜ ⎞ ⎠ ⎟ xdy ≡ Mdx + Ndy ⇒∂M∂y⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x=∂N∂x⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y. € θv= T 1+ 0.608qv( )p0p⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Rdcpdf. state variable is a path-independent function ξ for which dξ has the properties (1) for any closed path € dξ= 0∫, and (2) for ξ(x,y) where x and y are independent, then € dξ=∂ξ∂x⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ydx +∂ξ∂y⎛ ⎝ ⎜ ⎞ ⎠ ⎟ xdy ≡ Mdx + Ndy ⇒∂M∂y⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x=∂N∂x⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y. g. Gibbs’ phase rule says that for a system in thermal, chemical, and mechanical equilibrium, then the number of degrees of freedom are given by the Gibbs phase rule: f=χ-ϕ+2 (for χ components in ϕ phases). 4. Skeptics of global warming often criticize records of global temperature based on the lack of comparable standards and other measurement uncertainties. However, only three fundamental results of physics are required to evaluate the response of mean surface temperature to increasing greenhouse gases. Name them. 1) The first law of thermodynamics. 2) The Stefan-Boltzman equation (or Planck’s law of radiation). 3) The absorption of infrared radiation by greenhouse gases. 5. The saturation vapor pressure (of water) at a temperature of 30°C is 42.4 hPa. Consider moist air at 30°C, a pressure of 1000 hPa, and a relative humidity of 25%. Find the values: a. vapor pressure e=H*es = 0.25*42.4 = 10.6 hPa [Eqn. 4.34a]. b. mixing ratio wv=mv/md=(Mv/Md)*(e/(p-e))=0.622*(10.6/(1000-10.6))=0.00666 [Eqn. 4.36]. c. specific humidity qv=mv/(mv+md)=wv/(wv+1)=0.00662 [Eqn. 1.20]. d. virtual temperature Tv=(1+0.608qv)*T=(1+0.608qv)*T=304.2K [Eqn. 1.25]. 6. The saturation vapor pressure (of water) doubles


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UCSD SIO 217A - Midterm Exam

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