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) Problem 1.2 (p.16); problem 1.6 (p. 44); problem 1.7 (p.45) (a.j.vandelden@uu.nl) (http://www.phys.uu.nl/~nvdelden/dynmeteorology.htm) Topics for BOX (about 1000 words) Moist adiabat and moist convective adjustment How to calculate the temperature of the lifting condensation level Convective inhibition Conservation of potential vorticity (Ertel s theorem) Difference between confluence and convergence Storm tracks Cold front, warm front, occluded front, backbent front Polar stratospheric vortex Breaking planetary waves Sudden stratospheric warming North Atlantic Oscillation and Arctic Oscillation Seasonal cycle of the Hadley circulation The Inter-Tropical Convergence Zone (ITCZ) Brewer-Dobson circulation ENSO and Walker circulation Difference between a tornado and a tropical cyclone 1
Examples of a BOX Data for case study 2
20092011-13 Tropical cyclone 21092011-03 21092011-00 COLD FRONT 3
warm sector below cloud band 4
cold sector *see geophysical fluid dynamics or Holton and sections 1.7 and 1.8 of lecture notes The equations* momentum dv dt = α p gk ˆ 2Ω v + Fr α 1 ρ Pressure gradient (1.5) Gravity (1.4) Coriolis (1.6&1.7) Friction (1.3) mass energy state dρ dt = ρ v Jdt = c v dt + pdα pα = RT eq. 1.7b eqs. 1.4a,b,c Unknowns are: v,ρ,t, p 5
advection Material derivative of a scalar Scalar is a function of x, y, z and t d dt = t + v = t + u x + v y + w z eq. 1.6 Material derivative Local derivative Advection Non-linear!!! Section 1.7 cloud advection & stationary gravity waves d dt (...) = 0 t (...) = 0 6
cloud advection & stationary gravity waves d dt (...) = 0 t (...) = 0 Material derivative of a vector dv dt du dt uvtanφ + uw a a ˆ i + dv dt + u2 tanφ + vw ˆ j + dw a a dt u2 +v 2 k ˆ a Additional terms due to curved coordinate system!! These terms are frequently neglected in theoretical analysis eq. 1.5 (see geophysical fluid dynamics) Section 1.7 7
Section 1.7 Coriolis effect Ω v = ( wωcosφ vωsinφ)ˆ i + ( uωsinφ)ˆ j - ( uωcosφ)ˆ k From scale 2Ω analysis :* v ( 2Ωvsinφ)ˆ i + ( 2Ωusinφ)ˆ j fvˆ i + fuˆ j 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. 8
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 -1 kg -1 ), while n is the molecular number density (in numbers per m 3 ). 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), q ρ v ρ Section 1.9 Clausius Clapeyron Clausius-Clapeyron equation for the water vapour pressure, p e, which is in equilibrium with the liquid phase: p e T = p e L v R v T 2 e s =p e L v and R v are, respectively, the socalled latent heat of evaporation (2.5 10 6 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 L v is constant (=2.5 10 6 J K -1 ) 9
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 10
Zonal mean heating J<0 J>0 http://www.ecmwf.int/research/era/era-40_atlas/ Section 1.13 Potential temperature, θ θ T p ref p κ eq. 1.54 κ R /c p pα = RT Jdt = c v dt + pdα dθ dt = J p } Π c p Π p ref κ eq. 1.55 Exner-function If J=0 (adiabatic) θ is materially conserved!! 11
Equations in terms of potential temperature and Exner-function dθ dt = J Π dv dt = θ Π gk ˆ 2Ω v + Fr dπ dt = RΠ v + RJ c v θ c v eq. 1.55 eq. 1.56 eq. 1.58 problem 1.6 Three differential equations with three unknowns! Archimedes principle & buoyancy *see Holton, chapter 2 Section 1.4 An element immersed in a fluid at rest experiences an upward thrust which is equal to the weight of the fluid displaced. If ρ 0 is the density of the fluid and V 1 is the volume of the object, the upward thrust is therefore equal to gρ 0 V 1. The net upward force, F (the socalled buoyancy force), on the object is equal to (gρ 0 V 1 -gρ 1 V 1 ), where ρ 1 is the density of the object. With the equation of state and some additional approximations we can derive that F mg T 0 T 1 T 0 Gravity is dynamically important if there are temperature differences problem 1.2 (15 minutes to do this problem) 12
Acceleration under buoyancy Force on air parcel: Air parcel has temperature θ =θ * F = m d2 z dt 2 mgt 1 T 0 T 0 mg θ 1 θ 0 θ 0 Potential temperature environment of air parcel: θ 0 = θ * + dθ 0 dz δz Acceleration under buoyancy Force on air parcel: F = m d2 z dt 2 mgt 1 T 0 T 0 mg θ 1 θ 0 θ 0 Air parcel has temperature θ =θ * Potential temperature environment of air parcel: θ 0 = θ * + dθ 0 dz δz Buoyant force is proportional to θ θ 1 θ 1 θ * dθ 0 0 = dz δz dθ 0 = dz δz θ 0 θ 0 θ 0 Therefore d 2 δz dt 2 = g θ 0 dθ 0 dz δz 13
Stability of hydrostatic balance d 2 δz dt 2 = g θ 0 dθ 0 dz δz N 2 δz N 2 g dθ 0 θ 0 dz The solution: δz = exp ( ±int) If N 2 = g dθ 0 θ 0 dz < 0 Exponential growth instability If N 2 = g dθ 0 θ 0 dz > 0 oscillation stability Brunt Väisälä-frequency, N Brunt Väisälä frequency N 2 g dθ 0 θ 0 dz Extra problem: Demonstrate that the Brunt-Väisälä frequency is constant in an isothermal atmosphere. What is the typical time-period of a buoyancy oscillation in the atmosphere? 14