Surface effects in QG dynamics

Transcription

1 Surface effects in QG dynamics Ben Harvey Supervisor: Maarten Ambaum 2/11/2010 Introduction

2 Surface Outline QG dynamics Surface QG (temperature) 2-d Euler (vorticity) 2/11/2010 Introduction

3 Outline of talk Introduction: PV basics Quasi-geostrophic theory The surface QG equations Filament instability: Baroclinic instability: 2/11/2010 Introduction

4 Potential vorticity: a Outline (very short) introduction Vorticity is a measure of the local rotation rate of a fluid It s conserved in 2-d incompressible and frictionless flows: (2-d Euler equations) This represents the conservation of angular momentum D 0, Dt 2 2/11/2010 Introduction

5 Potential vorticity: a Outline (very short) introduction Vorticity is a measure of the local rotation rate of a fluid It s conserved in 2-d incompressible and frictionless flows: (2-d Euler equations): This represents the conservation of angular momentum D 0, Dt 2 P f h Two important properties: (i) (ii) P (and q) is conserved by adiabatic and frictionless flows It can be `inverted under suitable balance conditions 2/11/2010 Introduction

6 The simplest such balance Quasi-geostrophic Outline theory Based on the fact that the atmosphere is close to geostrophic and hydrostatic balance (at large scales)...thermal wind balance Or, equivalently, that the Rossby number is small (Ro=U/fL<<1) and the Richardson number is large (Ri=N 2 H 2 /U 2 >>1/Ro) 2/11/2010 Introduction

7 Quasi-geostrophic Outline theory Linearise the PV equation for small h variations (write h = h 0 +h ): f f h' f P 1... h h0 h0 h0 h0 fh' 2 h0 rotation stratification 2/11/2010 Introduction

8 Quasi-geostrophic Outline theory Linearise the PV equation for small h variations (write h = h 0 +h ): f f h' f P 1... h h0 h0 h0 h0 q h P f 0 fg 2 N q 0 q ' z fh' 2 h0 rotation stratification The continuously stratified (Boussinesq, constant N) version takes the form: 2/11/2010 Introduction

9 Quasi-geostrophic Outline theory Linearise the PV equation for small h variations (write h = h 0 +h ): f f h' f fh' P h h0 h0 h0 h0 h0 rotation stratification The continuously stratified (Boussinesq, constant N) version takes the form: Finally, hydrostatic balance gives: So that q h P f q 0 f q0 f q ' g 2 h fg 2 N q /11/2010 Introduction z f N 0 q ' z z (typically N/f=100)

10 But what happens at the surface? Quasi-geostrophic Outline theory 2/11/2010 Introduction

11 But what happens at the surface? Quasi-geostrophic Outline theory? 2/11/2010 Introduction

12 But what happens at the surface? Quasi-geostrophic Outline theory Use that q is conserved The result is a two component system: Interior potential vorticity: Surface temperature: q f q0 f q ' g? 2 h z f N z /11/2010 Introduction

13 The surface Outline QG equations Focus on the surface component by setting q=0 Then, at the surface, Dq Dt 0 With PV inversion given by 2 h f N z q f 0 q g z Key ingredients: A horizontal boundary QG f-plane dynamics Negligible interior PV 2/11/2010 Introduction

14 Key ingredients: The surface Outline QG equations A horizontal boundary QG f-plane dynamics Negligible interior PV Other applications: Tropopause perturbations Juckes (1994) showed that small scale tropopause perturbations can be approximated by surface QG 2/11/2010 Introduction

15 Key ingredients: The surface Outline QG equations A horizontal boundary QG dynamics Negligible interior PV Other applications: Tropopause perturbations Juckes (1994) showed that small scale tropopause perturbations can be approximated by surface QG Upper level ocean eddies Lapeyre and Klein (2006) apply surface QG dynamics to upper level ocean dynamics 2/11/2010 Introduction

16 Dq Dt How to model Outline numerically? 0 2 h f N z q f 0 q g z This is effectively a two-dimensional system Surface QG q n ( xy, ) n ˆ n ˆ ( k) q ( k ) k 2-d Euler n ( xy, ) n ˆ n ˆ ( k) ( k ) 2 k u n v n n y n x u n v n n y n x n q 1 ( xy, ) n 1 ( xy, ) 2/11/2010 Introduction

17 Surface Outline QG dynamics A turbulence simulation: (animation) 2/11/2010 Introduction

18 Surface Outline QG dynamics A turbulence simulation: (animation) 2/11/2010 Introduction

19 Surface Outline QG dynamics Surface QG (temperature) 2-d Euler (vorticity) Both have: large coherent vortices plus complicated small scale structure But: SQG vortices are less tightly bound and small scale structure is more messy 2/11/2010 Introduction

20 Outline of talk Introduction: PV basics Quasi-geostrophic theory The surface QG equations Filament instability: Motivation The effects of straining Baroclinic instability: 2/11/2010 Introduction

21 Motivation: Filament Outline instability Both 2-d Euler vorticity filaments and surface QG temperature filaments are unstable in isolation (barotropic instability, the Rayleigh problem,...): However...the presence of external flows often act to stabilise vorticity filaments (Dritschel et al 1991) e.g., straining: The filament remains coherent if the strain rate is large enough: s 0.25 The question: What happens for surface QG dynamics? 0 2/11/2010 Part I

22 Calculated by Juckes (1995). Basic state: Filament Outline instability 2/11/2010 Part I

23 Calculated by Juckes (1995). Basic state: Filament Outline instability Perturbation: where 2/11/2010 Part I

24 Calculated by Juckes (1995). Basic state: Filament Outline instability Perturbation: where Growth rates proportional to q0 L 2/11/2010 Part I

25 A numerical simulation: Filament Outline instability 2/11/2010 Part I

26 A numerical simulation: Filament Outline instability time 2/11/2010 Part I

27 Now add a straining flow: (s is the strain rate) Filament Outline instability The filament will be stretched and squashed exponentially: Which normally would act to stabilise any irregularities along the filament. 2/11/2010 Part I

28 Outline time 2/11/2010 Part I

29 Filament Outline instability The equation for linear perturbations takes the same form: Except that now the wavenumber and filament width are functions of time: L L e st 0 2 e st 0 2/11/2010 Part I

30 Filament Outline instability The equation for linear perturbations takes the same form: Except that now the wavenumber and filament width are functions of time: L L e st 0 2 e st 0 There are three competing effects: 1. Kinematic decay: 2. Growth rate increase: A e 2st q / L 0 Stable until L L q s c 0 / 3. Wave-number decrease 2/11/2010 Part I

31 Filament Outline instability The equation for linear perturbations takes the same form: Except that now the wavenumber and filament width are functions of time: L L e st 0 2 e st 0 There are three competing effects: 1. Kinematic decay: 2. Growth rate increase: A e 2st q / L 0 3. Wave-number decrease 2/11/2010 Part I

32 Log(A) Outline There are three competing effects: 1. Kinematic decay: 2. Growth rate increase: A e 2st q / L 0 3. Wave-number decrease 2/11/2010 Part I

33 Log(A) Outline There are three competing effects: 1. Kinematic decay: 2. Growth rate increase: A e 2st q / L 0 3. Wave-number decrease 2/11/2010 Part I

34 Log(A) Outline There are three competing effects: 1. Kinematic decay: 2. Growth rate increase: A e 2st q / L 0 3. Wave-number decrease 2/11/2010 Part I

35 The conceptual picture: Filament Outline instability In both models filaments are formed by the presence of straining, then... In 2-d Euler dynamics: the straining keeps the filaments stable. Instability only occurs if the straining stops or the filament moves away from it. In surface QG: the straining keeps the filaments stable, but only for a short time. Once they reach a critical width perturbation growth dominates. 2/11/2010 Part I

36 Outline of talk Introduction: PV basics Quasi-geostrophic theory The surface QG equations Filament instability: Motivation The effects of straining Baroclinic instability: Uniform PV models A new model 2/11/2010 Introduction

37 Baroclinic instability Motivation: 2/11/2010 Part II

38 Baroclinic instability Motivation: 2/11/2010 Part II

39 Baroclinic instability Motivation: 2/11/2010 Part II

40 Baroclinic instability: Uniform PV models Example Eady (1949) Juckes (1998) 2/11/2010 Part II

41 Growth rates Baroclinic instability: Uniform PV models Example Eady (1949) Juckes (1998) wavenumber wavenumber 2/11/2010 Part II

42 Baroclinic instability: Uniform PV models Yet another model: Interesting features: No longer restricted to tropopause-surface symmetry Nonlinear evolution may be more realistic due to circular geometry - is the direction of wavebreaking simply related to the basic state? 2/11/2010 Part II

43 height Baroclinic instability: Uniform PV models Linear dynamics: First consider a single temperature patch on the tropopause radius 2/11/2010 Part II

44 Baroclinic instability: Uniform PV models Linear dynamics: Next add an opposing patch at the surface 2/11/2010 Part II

45 Baroclinic instability: Uniform PV models Linear dynamics: A non-symmetric example 2/11/2010 Part II

46 Nonlinear simulations: Baroclinic instability: Uniform PV models 2/11/2010 Part II

47 Nonlinear simulations: Baroclinic instability: Uniform PV models 2/11/2010 Part II

48 Conclusions Outline Small scale stratospheric intrusions can be modelled as filaments in the surface QG equations Filaments in the surface QG model behave very differently to those of the more familiar 2-d Euler system, because......straining does not stabilise them In fact, straining just constrains when the instability occurs. Uniform PV quasi-geostrophic models provide analytically tractable examples of baroclinic instability We ve formulated a new circular model as an extension to the `polar-front model of Juckes (1998) The linear theory gives realistic growth rates for atmospheric parameter values, and the nonlinear evolutions may be insightful for understanding wavebreaking 2/11/2010