Atmospheric Dynamics: lecture 11

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Atmospheric Dynamics: lecture 11 (http://www.staff.science.uu.

1 Atmospheric Dynamics: lecture 11 ( Chapter 9 Baroclinic waves and cyclogenesis What is a baroclinic wave? Quasi-geostrophic equations Omega equation Original articles:

2 Huib de Swart Baroclinic wave

3 Baroclinic wave

4 Baroclinic wave

5 Baroclinic wave

6 Baroclinic wave

7 Baroclinic wave

8 Baroclinic wave

9 Warm sector

10 Warm sector

11 Warm sector

12 Warm sector

13 warm seclusion occlusion

16 Upward motion

17 Atmospheric river

18 warm conveyor belt

19 warm conveyor belt

20 warm conveyor belt

21 Quasi-geostrophic theory Next we introduce the quasi-geostrophic approximation This leads to a system of two equations with two unknowns: vertical velocity and geopotential Omega (vert. vel.) equation gives insight into relation frontogenesis, vertical motion and cyclogenesis

22 Section 9.3 Section 1.30: Quasi-geostrophic approximation Primitive equations with pressure as vertical coordinate: dv = f k ˆ v Φ Φ p = RT p (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)

23 Section 9.3 Quasi-geostrophic approximation See also GFD dv = 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 dv d gv g d g t + v g = t + u g x + v g

24 Section 9.3 Quasigeostrophic equations dv d gv g = fk ˆ v Φ ( f 0 + βy) ˆ ( v a ) + f 0 ˆ k v g + k v g

25 Section 9.3 Quasigeostrophic equations dv d gv g = fk ˆ v Φ ( f 0 + βy) ˆ d g v g = βyk ˆ v g + v g >> v a ( v ) a f 0 ˆ ( v a ) + f 0 ˆ k v g + k v a βyk ˆ v g f 0 k ˆ v a k v g

26 Section 9.3 Quasigeostrophic equations dv d gv g = fk ˆ v Φ ( f 0 + βy) ˆ d g v g = βyk ˆ v g + v g >> v a ( v ) a f 0 ˆ ( v a ) + f 0 ˆ k v g + k v a βyk ˆ v g f 0 k ˆ v a 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)

27 Section 1.35 Jetstreak If β=0 then k ˆ d v v a = 1 f d v > 0 d v < 0 v a < 0 du = fv a v a > 0

28 Section 9.4 Quasigeostrophic vorticity d g v g equation = βy ˆ k v g f 0 ˆ k v a d g u g = f 0 v a + βyv g d g v g = f 0 u a βyu g

29 Section 9.4 Quasigeostrophic vorticity d g v g equation = βy ˆ k v g f 0 ˆ k v a d g u g = f 0 v a + βyv g d 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

30 Section 9.4 Quasigeostrophic vorticity d g v g equation = βy ˆ k v g f 0 ˆ k v a d g u g = f 0 v a + βyv g d 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

31 Section 9.4 Quasigeostrophic vorticity d g v g equation = βy ˆ k v g f 0 ˆ k v a d g u g = f 0 v a + βyv g d 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 Quasi-geostrophic vorticity eqn: d g ζ g = 1 f 0 d g 2 Φ u = f a 0 x + v a βv g

32 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: d g ζ g = 1 f 0 d g 2 Φ u = f a 0 x + v a βv g

33 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: d g ζ g = 1 f 0 d g 2 Φ u = f a 0 x + v a βv g + continuity equation: u x + v + ω p = 0

34 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: d g ζ g = 1 f 0 d g 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

35 Section 9.4 Geopotential and omega as unknowns Quasi-geostrophic vorticity eqn: d g ζ g = 1 f 0 d g 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: d g 2 Φ 2 ω = f 0 p β Φ x

36 Section 9.4 Quasi-geostrophic thermodynamic quation t + u x + v S pω = J c p Φ p = RT p From eq. 1.7c Derive an expression for S p

37 Section 9.4 Quasi-geostrophic thermodynamic quation t + u x + v S pω = J c p Φ p = RT p From eq. 1.7c Derive an expression for S p S p = α c p p

38 Section 9.4 Quasi-geostrophic thermodynamic quation t + u x + v S pω = J c p Φ p = RT p Quasi-geostrophic thermodynamic eq.: d g Φ p = σω RJ c p p σ R p S p

39 Section 9.4 Two equations and two unknowns Closed set of equations Quasi-geostrophic vorticity eq. d g 2 Φ 2 ω = f 0 p β Φ x Quasi-geostrophic thermodynamic eq. d g Φ p = σω RJ c p p This set of equations was used by the pioneers of numerical weather prediction

40 Section 9.4 Most important equations Quasi-geostrophic vorticity eq. d g 2 Φ 2 ω = f 0 p β Φ x Quasi-geostrophic thermodynamic eq. d g Φ p = σω RJ c p p Continuity equation u a x + v a + ω p = 0 Divergence of geostrophic wind = 0 x + v g = 0

41 Homework Problem 9.3 Geopotential field and Vertical motion (taken from Holton, 2004) Given the following expression for the geopotential field: Φ = Φ 0 p ( ) + l 1 sin l x ct ( ) + cf 0 y[ cos( πp / p 0 ) +1] [ ( )] where Φ 0 is a function of p alone, c is a constant speed, l a zonal wave number, and p 0 =1000hPa. (a) Use the quasi-geostrophic vorticity equation to obtain the horizontal divergence field consistent with this Φ-field. (Assume df/dy=0) (b) Assuming that ω(p 0 )=0, obtain an expression for ω(x, y, p, t) by integrating the continuity equation with respect to pressure. (c) Sketch the geopotential fields at 750 hpa and at 250 hpa. Indicate regions of positive and negative vertical velocity at 500 hpa.

42 Section 9.5 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

43 Section 9.5 Frontogenesis as a disturbance to thermal balance p = R pf 0 ; v g p = R pf 0 x Derive these two equations now

44 Section 9.5 Frontogenesis as a disturbance to thermal balance Using: d g 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:

45 Section 9.5 Frontogenesis as a Using: disturbance to thermal balance d g u g d g p = f 0 v a + βyv g = 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

46 Section 9.5 Frontogenesis as a Using: disturbance to thermal balance d g u g d g p = f 0 v a + βyv g = p p = R pf 0 ; and assuming β=0, and neglecting ageostrophic motion: x v g p d g = R f 0 p x = v g v g p = R pf 0 x + x x * (J = 0) * (See also section 1.33, lecture notes)

47 Section 9.5 Frontogenesis as a Using: disturbance to thermal balance d g u g d g p = f 0 v a + βyv g = p p = R pf 0 ; and assuming β=0, and neglecting ageostrophic motion: x v g p d g = R f 0 p x = v g? * v g p = R pf 0 x + x x * (J = 0) (See also section 1.33, lecture notes)

48 Section 9.5 Frontogenesis as a Using: disturbance to thermal balance d g u g d g p = f 0 v a + βyv g = p and assuming β=0, and neglecting ageostrophic motion: x v g p d g = R f 0 p x Subtracting these two equations yields: d g f 0 p R x + v g = 0 p = 2 p = R pf 0 ; v g p = R pf 0 x = v g x + v g + x x * * (See also section 8.1, lecture notes) = 2Q g2 2 d g (J = 0)

49 Section 9.5 Frontogenesis as a disturbance to thermal balance d g f 0 p R p = 2 x + v g = 2Q g2 2 d g y-component of the geostrophic Q-vector Q-vector is vector frontogenesis function see section 1.33

50 Section 9.5 Frontogenesis as a disturbance to thermal balance d g f 0 p R p = 2 x + v g = 2Q g2 2 d g Disturbance to thermal wind balance y-component of the geostrophic Q-vector Q-vector is vector frontogenesis function see section 1.33

51 Section 9.5 Frontogenesis as a disturbance to thermal balance d g f 0 p R p = 2 x + v g = 2Q g2 2 d g Disturbance to thermal wind balance y-component of the geostrophic Q-vector Q-vector is vector frontogenesis function see sections 1.33 & 8.1 Now: let us include the ageostrophic flow

52 Section 9.5 Neglecting ageostrophic flow we have (previous slides): d g f 0 p R p = 2 x + v g = 2Q g2 2 d g

53 Section 9.5 Neglecting ageostrophic flow we have (previous slides): d g f 0 p R p = 2 x + v g = 2Q g2 2 d g Role of ageostrophic flow is to preserve thermal wind balance

54 Section 9.5 Neglecting ageostrophic flow we have (previous slides): d g f 0 p R p = 2 x + v g = 2Q g2 2 d g Role of ageostrophic flow Is to preserve thermal wind balance Repeating the derivation of the previous slides including ageostrophic flow yields d g f 0 p R p = 2Q g2 + f 0 R 2 p v a p pσ R ω

55 Section 9.5 Neglecting ageostrophic flow we have (previous slides): d g f 0 p R p = 2 x + v g = 2Q g2 2 d g Role of ageostrophic flow Is to preserve thermal wind balance Repeating the derivation of the previous slides including ageostrophic flow yields d g 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

56 Section 9.5 From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 The x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω x = 0

57 Section 9.5 From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 The x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω x = 0 From this we can derive an equation for the vertical motion: + x

58 Section 9.5 From previous slide: 2Q g2 + f 2 0 p v a R p pσ R ω = 0 The x-component of thermal wind balance yields: 2Q g1 + f 2 0 p u a R p pσ R ω x = 0 From this we can derive an equation for the vertical motion: + x Omega-equation From the two equations above: σ 2 ω + f ω p 2 = 2R p Q g Where we have used u a x + v a + ω p = 0 and 2 2 x

59 Section 9.5 Omega equation:interpretation σ 2 ω + f ω p 2 = 2R p Q g Q g1 = x x + v g x ; Q g2 = x + v g.

60 Section 9.5 Omega equation:interpretation σ 2 ω + f ω p 2 = 2R p Q g Q g1 = x x + v g x ; Q g2 = 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)

61 Next week: interpretation of the solution of the omega equation

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