Dynamic Meteorology: lecture 12

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1 30/11/2018 (Friday) Dynamic Meteorology: lecture 12 Sections 9.5, 9.6 Omega equa)on Large scale ver)cal mo)on in a quasi-geostrophic atmosphere Two level model of the atmosphere 5/12/2017 (Wednesday!!!): lecture 13 (sections ) Two-level model (con)nued) Baroclinic instability of planetary waves in the two-level model 7/12/2017 (Friday!!!): tutorial 11 Problems 9.4, 9.6, 9.7 (a.j.vandelden@uu.nl) (

2 Section 9.5 Quasi-geostrophic omega-equa)on Meteorologists are most interested in vertical motion ( omega =ω), because this variable indicates where clouds and precipitation will form, i.e. in regions where ω=dp/<0. In the following we derive a separate quasi-geostrophic equation for omega, called omega-equation. We will follow the didactic derivation due to Hoskins, Draghici and Davies (1978) (see list of references in chapter 9).

3 Section 9.5 The atmosphere is usually in a state close to thermal balance Thermal wind balance: (in pressure coordinates) p = R ; v g p = R Derive these two equations

4 Section 9.5 Frontogenesis is associated with changing temperature gradients: it disturbs thermal balance Thermal wind balance: (in pressure coordinates) p = R v g p = R Changing temperature gradient must be associated with changing thermal wind Let us evaluate the consequences of this process in a quasi-geostrophic atmosphere

5 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance and p = R v g p = R t + u + v S pω = + J c p (lecture 11) Assume β=0, neglect ageostrophic motion and assume J=0 then t + u g + v g = 0 also becomes t + u g t + u g + v g + v g v g = f 0 v a + βyv g v g = 0 (1.247a)

6 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance t + u g p = R Assume β=0, neglect ageostrophic motion and assume J=0 + v g = 0 v g p = R t + u g + v g v g = 0 # % $ p & ( = v g ' = p v g p Are these equa)ons consistent with thermal wind balance?

7 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R t + u g + v g = 0 t + u g + v g v g = 0 # % $ # % $ p & ( = v g ' & ( = ' p v g p = R # f 0 p % $ + & ( '

8 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R t + u g + v g = 0 t + u g + v g v g = 0 # % $ p & ( = v g ' = p v g p = R f 0 p v g + + v g = 0

9 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R Subtract: f 0 p R = v g p f 0 p R = v g p + = 2 v g +

10 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R Assume β=0, neglect ageostrophic motion and assume J=0 Previous slide f 0 p R p = 2 v g +

11 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R Assume β=0, neglect ageostrophic motion and assume J=0 or f 0 p R p = 2 + v g = 2 2Q g2 y-component of the geostrophic Q-vector` Q-vector is vector frontogenesis function, see section 1.40

12 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R Assume β=0, neglect ageostrophic motion and assume J=0 or f 0 p R p = 2 + v g = 2 2Q g2 y-component of the geostrophic Q-vector` Q-vector is vector frontogenesis function, see section 1.40 Maintenance of thermal wind balance requires that f 0 p R p = 0

13 Section 9.5 Dynamical consequences of frontogenesis in a quasi-geostrophic atmosphere Thermal wind balance p = R v g p = R Assume β=0, neglect ageostrophic motion and assume J=0 or f 0 p R p = 2 + v g = 2 2Q g2 y-component of the geostrophic Q-vector` Q-vector is vector frontogenesis function, see section 1.40 Maintenance of thermal wind balance requires that f 0 p R p Thermal wind balance cannot be sustained = 0 if Q g2 0

14 The role of ageostrophic flow is to preserve thermal wind balance Neglecting ageostrophic flow we have (previous slides): f 0 p R p = 2Q g2 Repeating the derivation of the previous slides including ageostrophic flow yields $ & % f 0 p R p ' ) = 2Q g2 + f 0 ( R 2 p v a p pσ R ω (β=0 and J=0)

15 The role of ageostrophic flow is to preserve thermal wind balance Neglecting ageostrophic flow we have (previous slides): f 0 p R p = 2Q g2 Repeating the derivation of the previous slides including ageostrophic flow yields $ & % f 0 p R p ' ) = 2Q g2 + f 0 ( R 2 p v a p pσ R ω =0 Conservation of thermal wind balance

16 The role of ageostrophic flow is to preserve thermal wind balance From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 Likewise, the x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω = 0

17 The role of ageostrophic flow is to preserve thermal wind balance From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 Likewise, the x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω = 0 From this we can derive an equation for the vertical motion: +

18 The role of ageostrophic flow is to preserve thermal wind balance From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 Likewise, the x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω = 0 From this we can derive an equation for the vertical motion: + From the two equations above: σ 2 ω + f ω p 2 = 2R p ( omega equa)on )! Q! g where we have used and u a + v a + ω p =

19 Section 9.5 Omega equa)on: interpreta)on σ 2 ω + f ω p 2 = 2R p! Q! g $ Q g1 = & % + v g ' $ ) ; Q g2 = & ( % + v g ' ). ( Since both T, u g and v g can all be expressed as a function of Φ, we can can calculate the vertical motion from the distribution of Φ only!!!

20 Section 9.5 Omega equa)on: interpreta)on σ 2 ω + f ω p 2 = 2R p! Q! g $ Q g1 = & % + v g ' $ ) ; Q g2 = & ( % + v g ' ). ( Since both T, u g and v g can all be expressed as a function of Φ, we can can calculate the vertical motion from the distribution of Φ only!!! A simplified version of the omega equation is ω w!! Q g i.e. upward (downward) motion if Q g -vector is convergent (divergent)

21 64 N Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue) THICK CONTOURS: HEIGHT: m; TEMPERATURE: 0.0 C; CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 C wf bbf Fig (lower panel) N cf warm sector pe model run hPa : Q1 =5*10^-11 K m^-1 s^-1 (min. value:10^-11 K m^-1 s^-1) hrs

22 64 N Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue) THICK CONTOURS: HEIGHT: m; TEMPERATURE: 0.0 C; CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 C wf bbf Fig (lower panel) N cf warm sector 0 15 Q-vector convergence pe model run hPa : Q1 =5*10^-11 K m^-1 s^-1 (min. value:10^-11 K m^-1 s^-1) hrs

23 VERTICAL VELOCITY (w) (blue: up; red:down) and WIND VECTOR 64 N THICK CONTOURS: / / CONTOUR-INTERVAL: w: 1.0 hpa/hr / Upward (downward) motion if Q g -vector is convergent (divergent) bbf wf 46 N warm sector cf Q-vector convergence 60 pe model run hPa 10 m/s hrs

24 Conceptual model of life-cycle of a middle-latitude cyclone Shapiro and Keyser, 1990 Pressure and fronts: about hours temperature:

25 Conceptual model of life-cycle of a middle-latitude cyclone Shapiro and Keyser, 1990 Pressure and fronts: about hours Manifestation of the instability wave-like perturbations superposed on a westerly current in thermal wind balance temperature:

26 Cyclogenesis and the life-cycle of a midlatitude cyclone An example: March March About 24 hours

27 Analysis of the stability of a zonal current in thermal wind balance to wave-like perturbations Procedure 1. Model formulation 2. Model equations 3. Steady state 4. Linearisation around steady state 5. Investigate dynamics of wave-like perturbations

28 Section 9.4 Quasi-geostrophic equations (adiaba)c atmosphere) Quasi-geostrophic vorticity eq. 2 Φ 2 ω = f 0 p β Φ Quasi-geostrophic thermodynamic eq. Φ p = σω Parameters represen)ng sta)c stability: σ R p S p S p α c p p

29 Section 9.6 Two level model Unknowns are: Δp Φ 1, Φ 3 and ω 2 (ψ=φ/f 0 ) Δp

30 Section 9.6 Closed set of equations Δp Δp (ψ 1 =Φ 1 /f 0 ) (ψ 3 =Φ 3 /f 0 ) Unknowns are: Φ 1, Φ 3 and ω 2 (ψ=φ/f 0 ) Apply vorticity equation to levels 1 and 3 Apply thermodynamic equation to level 2 Φ p = σω 2 Φ 2 ω = f 0 p β Φ

31 Section 9.6 Numerical approximations Apply thermodynamic equation to level 2: (ψ 1 =Φ 1 /f 0 ) (ψ 3 =Φ 3 /f 0 ) ' ) ) ) ( Φ p *, = σω,, + 2

32 Section 9.6 Numerical approximations Apply thermodynamic equation to level 2: (ψ 1 =Φ 1 /f 0 ) (ψ 3 =Φ 3 /f 0 ) ' Φ ) p ) ) ( *, = σω,, + 2 t Φ p 2 + ( v! ) g 2! Φ p 2 = σω 2

33 Section 9.6 Numerical approximations Apply thermodynamic equation to level 2: (ψ 1 =Φ 1 /f 0 ) (ψ 3 =Φ 3 /f 0 ) t ' Φ * ) p, ) = σω, ), ( + Φ p 2 + ( v! ) g 2! 2 Φ p 2 = σω 2 Finite difference approximation: $ Φ' & ) % p ( 2 Φ 1 Φ 3 = Φ 1 Φ 3 p 1 p 3 Δp

34 Next lecture: 5/12/2018 (Wednesday!!!), 13:15-15:00 Dynamic Meteorology: lecture 13 Sections 9.6, 9.7 Two-level model (con)nued) Baroclinic instability of planetary waves in the two-level model 7/12/2017 (Friday!!!): tutorial 11 Problems 9.4, 9.6, 9.7

Baroclinic wave. Atmospheric Dynamics: lecture 14 18/12/15. Topics. Chapter 9: Baroclinic waves and cyclogenesis. What is a baroclinic wave?

Baroclinic wave. Atmospheric Dynamics: lecture 14 18/12/15. Topics. Chapter 9: Baroclinic waves and cyclogenesis. What is a baroclinic wave? Atmospheric Dynamics: lecture 14 (http://www.staff.science.uu.nl/~delde102/) Topics Chapter 9: Baroclinic waves and cyclogenesis What is a baroclinic wave? Quasi-geostrophic equations Omega equation Original