Dynamic Meteorology: lecture 2 Sections 1.3-1.5 and Box 1.5 Potential temperature Radiatively determined temperature (boxes 1.1-1.4) Buoyancy (-oscillations) and static instability, Brunt-Vaisala frequency Introduction to project 1 (problem 1.6) (in couples) Convective Available Potential Energy (CAPE) 19/9/2018: tutorial 14/9/2018 (Friday) Problems 1.1, 1.3, 1.4, 1.6 + extra problems in this lecture project 1 in couples Section 1.4 Potential temperature, θ $ θ T p ref ' & ) % p ( κ eq. 1.21 κ R /c p R = c p c v pα = RT Jdt = c v dt + pdα dθ dt = J $ p } Π c p & Π % p ref κ ' ) ( eq. 1.23 Exner-function If J=0 (adiabatic) θ is materially conserved!! 1
Section 1.4 Potential temperature, θ $ θ T p ref ' & ) % p ( κ 350 330 300 Zonal mean and time mean (over the period 1979-2001) stratification of the atmosphere (potential temperature), as deduced from a state of the art dataassimmilation system. Labels are in Kelvin (K): potential increases with height! Intermezzo Boxes 1.1-1.4 (you do not need to study these boxes for the exam): Poten:al temperature determined by radia:on In an ideal atmosphere (well mixed, one greenhouse gas, Solar radiajon absorbed only by earth s surface): Equilibrium temperature, determined by radiajon (see box 1.4) is Absorbed Solar radiajon Problem 1.5 (page 38) T = Q σ a q a p +1 2σ g Stefan-Boltzmann constant 1/ 4 (eq. 1.27a) specific concentraion of the absorber absorpjon cross-secjon Show that this is a sta:cally stable temperature profile for earth-like values of the parameters Q, g, σ a, q a and p 2
Ver:cal mo:on due to buoyancy m,v F p = V p z m hup://en.wikipedia.org/wiki/buoyancy Ver:cal component of equa:on of mo:on: F g = mg dt = V p z mg Ver:cal mo:on due to buoyancy m,v F p = V p z m hup://en.wikipedia.org/wiki/buoyancy Ver:cal component of equa:on of mo:on: F g = mg dt = V p z mg p Hydrosta:c balance: 0 = V 0 0 z mg Perturb this equilibrium state so that: dt = ( V 0 + V' ) ( p0 + p' ) mg z 3
Ver:cal mo:on due to buoyancy F p = V p z m,v m HydrostaJc balance: 0 = V 0 p 0 z mg hup://en.wikipedia.org/wiki/buoyancy F g = mg Ver:cal component of equa:on of mo:on: ( ) p 0 dt = V 0 +V' dt z + p' p mg = V 0 z 0 z V' p 0 z dt = V' p 0 z V p' z = V p z mg V p' z mg Vertical accelerations Previous slide: dt = V' p 0 z V p' z p 0 z = ρ 0 g (hydrostatic balance) Air-parcel accelerates in vertical direction due to gravity and due to the pressure gradient dt = ρ dp' 0gV' V dz Archimedes force ( buoyancy ) Weight of the air displaced Source of a sound wave problem 1.1 Archimedes (287-212BC): A body or element immersed or floating in a fluid at rest experiences an upward thrust which is equal to the weight of the fluid displaced 4
Vertical accelerations Previous slide: dt = ρ dp' 0gV' V dz p' 0 dt ρ 0 gv' (vertical motion due to buoyancy ) T = T 0 + T' Equation of state: V'= Rm T p T 0 Rm T T 0 = Rm ( T T 0 ) p 0 p 0 p 0 p 0 dw dt g T' g θ' T 0 θ 0 Gravity is dynamically important if there are temperature differences Section 1.5 Acceleration due to buoyancy Vertical acceleration: d 2 z dt 2 = g θ' θ 0 At z=z* air parcel has potential temperature, θ = θ * Potential temperature air parcel is conserved! In the environment: θ 0 = θ *+ dθ 0 dz δz 5
Section 1.5 Acceleration due to buoyancy Vertical acceleration: d 2 z dt 2 = g θ' θ 0 At z=z* air parcel has potential temperature, θ = θ * Potential temperature air parcel is conserved! In the environment: θ 0 = θ *+ dθ 0 dz δz Buoyant force is proportional to θ * θ * dθ 0 θ' = 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 Section 1.5 Stability of hydrostatic balance Repeat: d 2 δz dt 2 = g θ 0 dθ 0 dz δz N 2 δz Brunt Väisälä-frequency, N: N 2 g dθ 0 θ 0 dz 6
Section 1.5 Stability of hydrostatic balance d 2 δz dt 2 = g dθ 0 δz N 2δz θ 0 dz g dθ 0 Brunt Väisälä-frequency, N: N 2 θ 0 dz The solution: If If δz = exp(±int ) g dθ 0 <0 Exponential growth θ 0 dz g dθ 0 N2 = >0 oscillation θ 0 dz N2 = instability stability Static instability Formation of cumulus clouds Figure 1.26: (Espy, 1841) (Leslie Bonnema, 2011) 7
Static stability Mountain induced vertical oscillations Brunt Väisälä frequency N 2 g dθ 0 θ 0 dz N is about 0.01-0.02 s -1 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 isothermal lower stratosphere? 8
Project 1: problem 1.6, page 41, lecture notes Parcel model of a buoyancy oscillation in a stratified environment Construct a numerical model (in e.g. Python; see next 2 slides), which calculates the vertical position and vertical velocity of a dry air-parcel, which is initially at rest just above the earth s surface. Initially this air parcel is warmer than its environment. The governing equations are dz dw = w; dt dt = g θ' ; dθ θ 0 dt = 0. The potential temperature of the environment of the air parcel is θ 0 = θ 0 ( 0) + Γz. A more complicated environmental potential temperature is prescribed in problem 1.6 Reference for Python language: https://www.python.org Book about use of Python in Atmospheric Science: http://www.johnny-lin.com/pyintro/ 9
Learn by example Solution problem 0.1 in Python See the Python-script Problem0.1AD.py on Blackboard (Assignments) import modules, numpy and matplotlib define arrays and fill with zero s for-loop Plot result of iteration Box 1.5 Convective Available Potential Energy (CAPE) Governed by: dw dt = g θ' Bg θ 0 B = buoyancy B θ' θ 0 Assuming a stationary state and horizontal homogeneity we can write: dw dt w dw dz = Bg. or wdw = Bgdz. 10
Box 1.5 Convective Available Potential Energy (CAPE) wdw = Bgdz. B θ' θ 0 Integrate this equation from a level z 1 to a level z 2. An air parcel, starting its ascent at a level z 1 with vertical velocity w 1, will have a velocity w 2 at a height z 2 given by w 2 2 = w 1 2 + 2 CAPE, z 2 CAPE g Bdz. z 1 What is CAPE in your model problem of project 1 (problem 1.6)? What is the associated maximum value of w? Does your model reproduce this value? Figure 2 (Box 1.5) 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 fair weather cumulus cloud 1.5 km deep, over a track about 250 m below cloud top 11
Figure 1.28 Updraughts in cumulus clouds Upward velocities of 50 m/s! What value of CAPE is required to get this upward velocity? The vertical motion in (b) a cumulo-nimbus cloud more than 10 km deep, over a track 6 km above the ground 12
Next: Tutorial Wednesday, 19/9/2018 Problems 1.1, 1.3, 1.4, 1.6 + extra problems of lecture 2 project 1 in couples Today (14 September 2018), 13:15-15:00 Python course Room 607 BBG 13
Next lecture: 21/9/2018 (Friday) Dynamic Meteorology: lecture 3 Sections 1.9-1.11, 1.14-1.15 Moist (cumulus) convection Relative humidity, mixing ratio Clausius Clapeyron equation Dew point (lapse rate) Lifted condensation level Equivalent potential temperature 26/9/2018: tutorial Problems 1.7, 1.8 + extra problems 14