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Atmospheric Dynamics: Lecture 2 the Equations*

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Atmospheric Dynamics: Lecture 2 the Equations* 9/20/12 Atmospheric Dynamics: lecture 2 Topics Some aspects of advection and the Coriolis-effect (1.7) Composition of the atmosphere (figure 1.6) Equation of state (1.8&1.9) Water vapour in the atmosphere (1.10) Potential temperature and Exner function (1.13) Buoyancy (1.4) Vertical accelerations and instability > convection (1.15) Latent heat release and conditional instability (1.16) Equivalent potential temperature (1.16) Problem 1.2 (sec. 1.4); problem 1.6 (sec. 1.13) and 3 extra problems ([email protected]) (http://www.phys.uu.nl/~nvdelden/dynmeteorology.htm) *see geophysical fluid dynamics or “Holton” and sections 1.7 and 1.8 of lecture notes The equations* 1 dv ˆ α ≡ momentum = −α∇ p − gk − 2Ω × v + Fr ρ dt Pressure gradient (1.5) Gravity (1.4) Coriolis (1.6&1.7) Friction (1.3) € dρ € mass = −ρ∇ ⋅ v eqs. 1.4a,b,c dt energy Jdt = cvdT + pdα € pα = RT state Unknowns are: v ,ρ,T, p € eq. 1.7b € € 1 9/20/12 advection Material derivative of a scalar Scalar is a function of x, y, z and t d ∂ ∂ ∂ ∂ ∂ = + v ⋅ ∇ = + u + v + w eq. 1.6 dt t t x y z ∂ ∂ ∂ ∂ ∂ Non-linear!!! Material Local “Advection” derivative derivative € Section 1.7 cloud advection & stationary gravity waves ∂ d (...) = 0 (...) = 0 ∂t dt € € 2 9/20/12 cloud advection & stationary gravity waves ∂ d (...) = 0 (...) = 0 ∂t dt € € Material derivative of a vector dv ⎛ du uvtanφ uw⎞ ⎛ dv u2tanφ vw⎞ ⎛ dw u2 +v 2 ⎞ ≡ ⎜ − + ⎟ ˆi + ⎜ + + ⎟ ˆj +⎜ − ⎟k ˆ dt ⎝ dt a a ⎠ dt a a dt a ⎝ ⎠ ⎝ ⎠ Additional terms due to curved coordinate system!! eq. 1.5 These terms are frequently neglected in theoretical analysis € (see geophysical fluid dynamics) Section 1.7 3 9/20/12 Section 1.7 Coriolis effect ˆ ˆ ˆ Ω × v = (wΩcosφ − vΩsinφ)i + (uΩsinφ)j - (uΩcosφ)k From “scale analysis”:* € ˆ ˆ ˆ ˆ 2 Ω × v ≈ −(2Ωvsinφ)i + (2Ωusinφ) j ≡ − fvi + fuj f is the Coriolis Parameter € *w<<v and w<<u , see Holton, chapter 2 or geophysical fluid dynamics Figure 1.6 Composition of the atmosphere Annual and global average concentration of various constituents in the atmosphere of Earth, as function of height above the Earth’s surface. The concentration is expressed as a fraction of the total molecule number density. This fraction is proportional to the mixing ratio. F11 and F12 denote the chlorinated fluorocarbons Freon-11 and Freon-12. Note that the concentration of carbon dioxide is constant up to a height of 100 km, while the concentrtaion of water vapour decreases by several orders of magnitude in the lowest 20 km. 4 9/20/12 Section 1.8 Equation of state p = nkT p = ρRT Here k is Boltzman’s constant (=1.381 10-23 J K-1). If the air is dry, R is the specific gas constant for dry air (=287 J K-1kg-1), while n is the molecular number density (in numbers per m3). If air is a mixture of dry air and water vapour, R is the €specific gas "constant"€ for this mixture Section 1.8 Equation of state p = nkT p = ρRT Here k is Boltzman’s constant (=1.381 10-23 J K-1). If the air is dry, R is the specific gas constant for dry air (=287 J K-1kg-1), while n is the molecular number density (in numbers per m3). If air is a mixture of dry air and water vapour, R is the €specific gas "constant"€ for this mixture The water vapour concentration in the atmosphere is expressed in terms of either the fraction of the total number of molecules, or as the fraction of the mass density of air (specific humidity), ρ q ≡ v ρ Or as fraction of the mass density of dry air (mixing ratio), ρ r ≡ v ρ € d € 5 9/20/12 Section 1.9 Clausius Clapeyron Clausius-Clapeyron equation for the water vapour pressure, pe, which is in equilibrium with the liquid phase: ∂pe peLv = 2 ∂T RvT L and R are, respectively, the so- e =p v v s e called latent heat of evaporation/ condensation (2.5×106 J kg-1) and €the gas constant for water vapour (461.5 J K-1 kg-1). Equilibrium water vapour pressure as a function of temperature, according to the Clausius Clapeyron equation assuming Lv is constant (=2.5×106 J K-1) Section 1.9 Clausius Clapeyron and relative humidity Clausius-Clapeyron equation for the water vapour pressure, pe, which is in equilibrium with the liquid phase: ∂pe peLv = 2 ∂T RvT Relative humidity is defined as the es=pe ratio e € RH ≡ es € Actual vapour pressure 6 9/20/12 Section 1.9 Clausius Clapeyron and latent heat Air cools when moving upward and becomes saturated at some point. If it continues rising it releases its latent heat at a rate of 2.5×106 J per kg of condensed water vapour! es=pe Extra problem: How much latent heat (in J) is released into the atmosphere per year if annual average, global average precipitation is 1000 kg m-2? Is this number big? Section 1.10 Clausius Clapeyron and precipitable water In an atmosphere with constant relative humidity the density profile of water vapour is (using Clausius Clapeyron eq.): ⎛ z ⎞ ρv (z) = ρv (0)exp⎜ − ⎟ ⎝ Hv ⎠ Hv is the scale height of water vapour (≈2 km). Integration of this equation gives € ∞ PW = ∫ ρv (z)dz = ρv (0)Hv 0 PW is the precipitable water Monthly mean precipitable water as a function of€ the monthly mean mass density of water vapour at the Earth’s surface for the 12-year period running from 1997 until 2008. 7 9/20/12 Section 1.11 Water cycle Precipitable water http://www.ecmwf.int/research/era/ERA-40_Atlas/ Hadley circulation The average meridional circulation in the tropics, called the Hadley circulation, is thought to be driven by latent heat release in large convective clouds in the ITCZ. The subsidence in the subtropics leads to warming of the air and a concomitant reduction of the relative humidity ITCZ 8 9/20/12 Zonal mean heating J<0 J>0 http://www.ecmwf.int/research/era/ERA-40_Atlas/ Section 1.13 Potential temperature, θ κ ⎛ pref ⎞ θ ≡ T⎜ ⎟ eq. 1.54 κ ≡ R /cp ⎝ p ⎠ ⎛ ⎞κ pα = RT dθ J € p € } = Π ≡ cp⎜ ⎟ dt Π ⎝ pref ⎠ Jdt = cvdT + pdα € eq. 1.55 Exner-function € € € € If J=0 (adiabatic) θ is materially conserved!! 9 9/20/12 κ Section 1.15 ⎛ pref ⎞ θ ≡ T⎜ ⎟ ⎝ p ⎠ Potential temperature, θ Nineteenth century depiction of isentropes in the atmosphere encircling the Earth. PN indicates North Pole; PS indicates South Pole. The 300 K isentrope usually grazes the Earth’s surface in the tropics. A relatively isolated canopy of potentially cold air exists €over the poles, especially in winter. 350 300 Twentieth century (ERA-40) depiction of the zonal mean and time mean (over the period 1979-2001) stratification of the atmosphere (potentail temperature) as deduced from a state of the art data-assimmilation system. Labels are in K. Equations in terms of potential temperature and Exner-function dθ J = eq. 1.55 dt Π dv = −θ∇ Π − gkˆ − 2Ω × v + Fr eq. 1.56 dt € dΠ RΠ RJ = − ∇ ⋅ v + eq. 1.58 dt cv cvθ € problem 1.6 Three€ differential equations with three unknowns! 10 9/20/12 Types of clouds and precipitation 1.static instability There are two types of clouds: 1.convective clouds: showery precipitation with thunder if clouds are deep. 2.layered clouds: steady precipitation, for example drizzle 2.slow adjustment to balance Formation of cumulus clouds Figure 1.27: (Espy, 1841) 11 9/20/12 Relative humidity in convective layer Simulation with cloud model (labels %): FIGURE 2.31. Model simulation of the relative humidity in the atmospheric boundary layer at midday in June in The Netherlands. The hatched regions correspond to clouds (regions where the relative humidity is 100%). The degree of cloud cover in this case is about 20%. Relative humidity is reasonably constant at the surface, but not in the cloud layer!! Van Delden & Oerlemans, 1982 Figure 1.28 Updraughts in cumulus clouds The relatively sharp downdraughts at the edge of the cumulus cloud are a typical feature of cumulus clouds. What effect is responsible for these downdraughts? The vertical motion in (a) a fair weather cumulus cloud 1.5 km deep, over a track about 250 m below cloud top 12 9/20/12 Figure 1.28 Updraughts in cumulus clouds The vertical motion in (a) a fair weather cumulus cloud 1.5 km deep, over a track about 250 m below cloud top and, (b) a supercell thunderstorm over a track about 6000 m above the ground. Section 1.4 Archimedes principle & buoyancy An element immersed in a fluid at rest experiences an upward thrust which is equal to the weight of the fluid displaced. If ρ0 is the density of the fluid and V1 is the volume of the object, the upward thrust is therefore equal to gρ0V1 . The net upward force, F (the so- called buoyancy force), on the object is equal to (gρ0V1-gρ1V1), where ρ1 is the density of the object. With the equation of state and some additional approximations we can derive that T −T F ≈ mg 0 1 T0 Gravity is dynamically important if there are temperature differences problem 1.2 (15 minutes to do this problem) *section 1.4 € 13 9/20/12 Section 1.15 Acceleration under buoyancy 2 Force on air parcel: d z T1 −T0 θ1 −θ0 F = m 2 ≈ mg ≈ mg dt T0 θ0 Air parcel has temperature θ =θ * dθ Potential temperature€ environment of air parcel: θ = θ *+ 0 δz 0 dz € € Section 1.15 Acceleration under buoyancy 2 Force on air parcel: d z T1 −T0 θ1 −θ0 F = m 2 ≈ mg ≈ mg dt T0 θ0 Air parcel has temperature θ =θ * dθ Potential temperature€ environment of air parcel: θ = θ *+ 0 δz 0 dz € dθ dθ θ −θ *− 0 δz − 0 δz Buoyant force is proportional to θ −θ 1 1 0 = dz = dz θ0 € θ0 θ0 2 Therefore€ d δz g dθ0 2 = − δz dt θ0 dz € 14 9/20/12 Section 1.15 Stability of hydrostatic balance 2 d δz g dθ0 2 2 = − δz ≡ −N δz dt θ0 dz 2 g dθ0 Brunt Väisälä-frequency, N: N ≡ N is about 0.01-0.02 s-1 θ dz € 0 The solution: € δz = exp(±iNt) 2 g dθ If N = 0 < 0 Exponential growth instability θ0 dz € 2 g dθ If N = 0 > 0 oscillation stability θ0 dz € € Section 1.15 Brunt Väisälä frequency g dθ N 2 ≡ 0 θ0 dz Extra problem€ : Demonstrate that the Brunt-Väisälä frequency is constant in an isothermal atmosphere that is hydrostatically stable.
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