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

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1 Atmospheric Dynamics: lecture 14 ( Topics Chapter 9: Baroclinic waves and cyclogenesis What is a baroclinic wave? Quasi-geostrophic equations Omega equation Original articles: Baroclinic wave Huib de Swart 1

2 Baroclinic wave Baroclinic wave 2

3 Baroclinic wave Baroclinic wave 3

4 Baroclinic wave Baroclinic wave 4

5 Warm sector Warm sector 5

6 Warm sector Warm sector 6

7 warm seclusion occlusion 7

8 Upward motion 8

9 Atmospheric river warm conveyor belt 9

10 warm conveyor belt warm conveyor belt 10

11 Chapter 9 Quasi-geostrophic theory Quasi-geostrophic approximation Leads to a system of two equations with two unknowns Unknowns: vertical velocity aneopotential. However: neither equation is an explicit equation for the vertical velocity A third equation (the Omega equation ), the solution of which provides the vertical velocity, is derived. This equation gives physical insight into relation frontogenesis, vertical motion and cyclogenesis Section 9.3 Section 1.30: d v = f k ˆ v Φ Φ p = RT p Quasi-geostrophic approximation Primitive equations with pressure as vertical coordinate: (Box 9.1, 12) (Box 9.1, 1) u x + v + ω p = 0 (Box 9.1, 14) t + u x + v S pω = J c p S p α c p p (Box 9.1, 8) 11

12 Section 9.3 Quasi-geostrophic approximation See also GFD d v = f k ˆ v Φ Φ p = RT p u x + v + ω p = 0 (Box 9.1, 12) (Box 9.1, 1) (Box 9.1, 14) t + u x + v S pω = J c p (Box 9.1, 8) Approximations: f f 0 + df dy y f 0 + βy v g = 1 f 0 ˆ k Φ d v d gv g t + v g = t + u g x + v g Section 9.3 Quasigeostrophic equations d v v g = fk ˆ v Φ ( f 0 + βy) ˆ ( v a ) + f 0 ˆ k v g + k v g 12

13 Section 9.3 Quasigeostrophic equations d v vg v g v g >> v a = fk ˆ v Φ ( f 0 + βy) ˆ ( v a ) + f 0 ˆ k v g + = βy ˆk v g + v ( a ) f 0 ˆk va βy ˆk v g f 0 ˆk va k v g Section 9.3 Quasigeostrophic equations d v vg v g v g >> v a = fk ˆ v Φ ( f 0 + βy) ˆ ( v a ) + f 0 ˆ k v g + = βy ˆk v g + v ( a ) f 0 ˆk va βy ˆk v g f 0 ˆk va k v g This is questionable (see fig lecture notes) Quasi-geostrophic approximation is difficult to justify completely from first principals, except under very restricted conditions. The justifications comes from practice: it works (i.e. in hindsight) 13

14 Section 1.32 If β=0 then vg = f 0 ˆk va x-component: u g = 1 f 0 v a Jetstreak d v > 0 v a > 0 d v < 0 v a < 0 Ageostrophic wind perpendicular to acceleration: v a = 1 f 0 ˆk vg Section 9.4 Quasigeostrophic vorticity equation v g = βyk ˆ v g f 0 k ˆ v a u g = f 0 v a + βyv g v g = f 0 u a βyu g 14

15 Section 9.4 Quasigeostrophic vorticity equation v g = βyk ˆ v g f 0 k ˆ v a u g = f 0 v a + βyv g v g u g = 1 Φ f 0 ;v g = 1 Φ f 0 x = f 0 u a βyu g x + v g = 0 Section 9.4 Quasigeostrophic vorticity equation v g = βyk ˆ v g f 0 k ˆ v a u g = f 0 v a + βyv g v g u g = 1 Φ f 0 ;v g = 1 Φ f 0 x = f 0 u a βyu g x + v g = 0 Quasi-geostrophic vorticity: ζ g = v g x = 1 f 0 2 Φ 15

16 Section 9.4 Quasigeostrophic vorticity equation v g = βyk ˆ v g f 0 k ˆ v a u g = f 0 v a + βyv g v g u g = 1 Φ f 0 ;v g = 1 Φ f 0 x = f 0 u a βyu g x + v g = 0 Quasi-geostrophic vorticity: ζ g = v g x = 1 f 0 2 Φ use this: Quasi-geostrophic vorticity eqn: ζ g = 1 f 0 2 Φ ' u = f a 0 x + v * a ), βv g ( + Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: ζ g = 1 f 0 2 Φ ' u = f a 0 x + v * a ), βv g ( + 16

17 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: ζ g = 1 f 0 2 Φ ' u = f a 0 x + v * a ), βv g ( + + continuity equation: u x + v + ω p = 0 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: ζ g = 1 f 0 2 Φ ' u = f a 0 x + v * a ), βv g ( + + continuity equation: becomes: u x + v + ω p = 0 u a x + v a + ω p = 0 because x + v g = 0 17

18 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: ζ g = 1 f 0 2 Φ ' u = f a 0 x + v * a ), βv g ( + + continuity equation: u x + v + ω p = 0 becomes: u a x + v a + ω p = 0 because x + v g = 0 Therefore: 2 Φ 2 ω = f 0 p β Φ x Section 9.4 Quasi-geostrophic thermodynamic quation Eq : Eq : t + u x + v +ω p = αω + J c p c p Φ p = RT p t + u x + v S pω = + J c p S p = α c p p Quasi-geostrophic thermodynamic eq.: Φ p = σω RJ c p p σ R p S p 18

19 Section 9.4 Two equations and two unknowns Closed set of equations Quasi-geostrophic vorticity eq. 2 Φ 2 ω = f 0 p β Φ x Quasi-geostrophic thermodynamic eq. Φ p = σω RJ c p p This set of equations was used by the pioneers of numerical weather prediction Section 9.4 Most important equations Quasi-geostrophic vorticity eq. 2 Φ 2 ω = f 0 p β Φ x 9.21 Quasi-geostrophic thermodynamic eq. Continuity equation Φ p = σω RJ c p p u a x + v a + ω p = Divergence of geostrophic wind = 0 x + v g =

20 An equation for omega Meteorologists are most interested in omega because this variable gives a clear indication of where clouds and precipitation will form. In the following we derive a separate equation for omega, which is called the omega-equation Frontogenesis as a disturbance to thermal balance p = R pf 0 ; v g p = R pf 0 x Derive these two equations now 20

21 Using: Frontogenesis as a disturbance to thermal balance 9.18 u g = f 0 v a + βyv g Derive this now p = R pf 0 ; v g p = R pf 0 x and assuming β=0, and neglecting ageostrophic motion: Using: Frontogenesis as a disturbance to thermal balance u g = f 0 v a + βyv g # & % ( = $ p ' p p = R pf 0 ; and assuming β=0, and neglecting ageostrophic motion: x v g p = R # f 0 p % $ x v g p = R pf 0 x + x & ( ' 21

22 Using: Frontogenesis as a disturbance to thermal balance u g = f 0 v a + βyv g # & % ( = $ p ' p p = R pf 0 ; and assuming β=0, and neglecting ageostrophic motion: x v g p = R # f 0 p % $ x # & % ( = v g $ ' v g p = R pf 0 x + x x * & ( ' (J = 0) * (See also section 1.37, lecture notes) Using: Frontogenesis as a disturbance to thermal balance u g = f 0 v a + βyv g # & % ( = $ p ' p and assuming β=0, and neglecting ageostrophic motion: x v g p = R Subtracting these two equations yields: f 0 p R x + v g = 0 p = 2 p = R pf 0 ; v g p = R pf 0 x # f 0 p % $ x # & % ( = v g $ ' x + v g + x x & ( ' = 2 2Q g2 (J = 0) 22

23 Frontogenesis as a disturbance to thermal balance f 0 p R p = 2 x + v g = 2 2Q g2 y-component of the geostrophic Q-vector Q-vector is vector frontogenesis function see section 1.37 Frontogenesis as a disturbance to thermal balance f 0 p R p = 2 x + v g = 2 2Q g2 y-component of the geostrophic Q-vector` Q-vector is vector frontogenesis function see section 1.37 Disturbance to thermal wind balance 23

24 Frontogenesis as a disturbance to thermal balance f 0 p R p = 2 x + v g = 2 2Q g2 y-component of the geostrophic Q-vector Q-vector is vector frontogenesis function see section 1.37 Disturbance to thermal wind balance Now: let us include the ageostrophic flow Neglecting ageostrophic flow we have (previous slides): f 0 p R p = 2 x + v g = 2 2Q g2 24

25 Neglecting ageostrophic flow we have (previous slides): f 0 p R p = 2 x + v g = 2 2Q g2 Role of ageostrophic flow is to preserve thermal wind balance Neglecting ageostrophic flow we have (previous slides): $ & % f 0 p R p ' $ ) = 2 & ( % x + v g ' ) = 2Q g2 2 $ ' & ) ( % ( Role of ageostrophic flow Is to preserve thermal wind balance 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 25

26 Neglecting ageostrophic flow we have (previous slides): $ & % f 0 p R p ' $ ) = 2 & ( % x + v g ' ) = 2Q g2 2 $ ' & ) ( % ( Role of ageostrophic flow Is to preserve thermal wind balance 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 If there conservation thermal wind balance From previous slide: 2 p 2Q g2 + f 0 R 2 p 2Q g1 + f 0 R v a p pσ R The x-component of thermal wind balance yields: u a p pσ R ω = 0 ω x = 0 26

27 From previous slide: 2Q g2 + f 0 R From this we can derive an equation for the vertical motion: 2 p 2 p 2Q g1 + f 0 R v a p pσ R The x-component of thermal wind balance yields: u a p pσ R ω = 0 ω x = 0 + x From previous slide: 2Q g2 + f 0 R From this we can derive an equation for the vertical motion: 2 p 2 p 2Q g1 + f 0 R v a p pσ R The x-component of thermal wind balance yields: u a p pσ R ω = 0 ω x = 0 + x Omega-equation From the two equations above: Where we have used u a x + v a + ω p = 0 σ 2 ω + f ω p 2 = 2R p Q g and 2 2 x

28 Omega equation:interpretation σ 2 ω + f ω p 2 = 2R p Q g $ Q g1 = x x + v g ' $ & ) ; Q g2 = % x ( x + v g ' & ). % ( Omega equation:interpretation σ 2 ω + f ω p 2 = 2R p Q g $ Q g1 = x x + v g ' $ & ) ; Q g2 = % x ( x + 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 ω w Q g i.e. upward (downward) motion if Q g -vector is convergent(divergent) 28

29 Next lecture: interpretation of the solution of the omega equation 64 N Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue) Fig 1.85 (lower panel) THICK CONTOURS: HEIGHT: m; TEMPERATURE: 0.0 C; CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 C wf bbf 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 29

30 64 N Qg-VECTOR, POTENTIAL TEMPERATURE (cyan) and HEIGHT (blue) Fig 1.62 (lower panel) THICK CONTOURS: HEIGHT: m; TEMPERATURE: 0.0 C; CONTOUR-INTERVAL: HEIGHT: 50.0 m; TEMPERATURE: 5.0 C wf bbf N cf warm sector Q-vector convergence 60 pe model run hPa : Q1 =5*10^-11 K m^-1 s^-1 (min. value:10^-11 K m^-1 s^-1) hrs 64 N VERTICAL VELOCITY (w) (blue: up; red:down) and WIND VECTOR THICK CONTOURS: / / CONTOUR-INTERVAL: w: 1.0 hpa/hr / bbf wf 46 N warm sector cf Q-vector convergence 60 pe model run hPa 10 m/s hrs 30

31 Omega equation: example ω w Q g Upward motion if Q- vector is convergent downward motion Upward motion Analysis of divergence of geostrophic Q-vector at 850 hpa (thick lines, labeled in units of K m -2 s -1 ) and the height of the 850 hpa surface (thin lines labeled in m) on April 4, 2001, 12 UTC. Trough of a Rossby-wave + - Satellite image Upward motion Meteosat satellite image in VISchannel, April 4, 2001, 1200 UTC. downward motion 31

32 Next: Prepare presentation project 2: 6 January 13:15 Each presentation is 15 minutes and consists of presenting the hypothesis, a description of the data used, the cross-correlation matrix, the eigenvectors and eigenvalues, interpretation of the most important principal component(s) and a conclusion. Project 3: Problem 3.2, hand in answer individually before, or during the next lecture on Friday 8 January 2016 Next lecture (8 January): Baroclinic instability, cyclogenesis and frontogenesis Next: Prepare presentation project 2: 6 January 13:15 Each presentation is 15 minutes and consists of presenting the hypothesis, a description of the data used, the cross-correlation matrix, the eigenvectors and eigenvalues, interpretation of the most important principal component(s) and a conclusion. Project 3: Problem 3.2, hand in answer individually before, or during the next lecture on Friday 8 January 2016 Next lecture (8 January): Baroclinic instability, cyclogenesis and frontogenesis Merry Christmas and a happy new year 32