1. Page 1© Crown copyright 2007 Why is the atmosphere so predictable? M.J.P.Cullen 22 November 2007

2. Page 2© Crown copyright 2007 Introduction

3. Page 3© Crown copyright 2007 Large scale flow Aim is to predict the atmosphere as depicted on a weather chart as seen in the media, not to predict all the detail. We see ‘weather systems’, with fronts and jet streams. Evolution is unsteady and non-periodic. Qualitatively, the weather map looks much the same from day to day (at a given season). Weather systems are qualitatively different in the tropics.

4. Page 4© Crown copyright 2007 Typical satellite picture

5. Page 5© Crown copyright 2007 Corresponding weather map

6. Page 6© Crown copyright 2007 Governing equations On all relevant scales, the atmosphere is governed by the compressible Navier-Stokes equations, the laws of thermodynamics, phase changes and source terms. The solutions of these equations are very complicated, reflecting the complex nature of observed flows. The accurate solution of these equations would require computers 10 30 times faster than now available.

7. Page 7© Crown copyright 2007 Weather forecasting The fact that numerical weather prediction is possible, and quite successful, implies that the evolution is largely controlled by large-scale dynamics. Thus the ‘butterfly’ effect must be limited. Identify this large-scale control by choosing system of equations which only describes large scales. Prove that this system can be solved independently-so that no knowledge of small scales is required. Estimate the difference between this solution and that of the real problem.

8. Page 8© Crown copyright 2007 Large-scale balances

9. Page 9© Crown copyright 2007 Large-scale flows-hydrostatic balance Because the atmosphere is thin, flows on large horizontal scales have a small aspect ratio. The rest state of the atmosphere is described by a balance of forces between gravity and a radial basic state pressure gradient The balance of forces determining atmospheric motion involves the perturbation to this balance due to horizontal density variations: where ρ´,p´ are the density and pressure perturbations.

10. Page 10© Crown copyright 2007 Large-scale flows-geostrophic balance Large-scale flow is also dominated by the Earth’s rotation. The key balance of forces is expressed as where Ω is the Earth’s angular velocity vector and p is the pressure. The flow is also within the ‘thin’ atmospheric shell and so is nearly horizontal. These requirements are contradictory for northward flow at the equator.

11. Page 11© Crown copyright 2007 Illustration Shows the direction of the ‘geostrophic’ wind, and its projection in a horizontal direction. Ω

12. Page 12© Crown copyright 2007 Lagrangian and Eulerian dynamics Descriptive theory of large-scale meteorology has usually been in the Lagrangian frame. Thus we speak of ‘air masses’ with long-lived characteristics (temperature and moisture). Illustration follows of Lagrangian and Eulerian phenomena (forced by hills) coexisting in the atmosphere. Consider stability of geostrophic and hydrostatic states to Lagrangian displacements of the fluid-ignoring consequential pressure perturbations.

13. Page 13© Crown copyright 2007 Lagrangian and Eulerian dynamics Animation removed

14. Page 14© Crown copyright 2007 Static stability A state of rest in hydrostatic balance is stable if the density decreases with height. In the compressible atmosphere, this is expressed by the ‘potential temperature’ increasing with height. In terms of potential temperature perturbations, hydrostatic balance becomes where Π´ is a function of p´ and θ´ is the potential temperature perturbation.

15. Page 15© Crown copyright 2007 Static stability II Stability requires θ ´ to increase with r, so that ∂Π ´/ ∂r increases with r, so that Π´ is a convex function of r. If this condition is satisfied, a displaced parcel with fixed θ´ will feel a restoring pressure force, assuming the pressure is not changed. If a large-scale flow is statically unstable, it will self-destruct, ceasing to be large-scale. When moisture is included, spontaneous large- scale violations of static stability can be generated, leading to thunderstorms.

16. Page 16© Crown copyright 2007 An unstable state

17. Page 17© Crown copyright 2007 Inertial stability The stability of a geostrophic state has to be considered in an inertial frame. The defining equation in the plane normal to the rotation axis is then The condition that a displaced parcel with fixed momentum will feel a restoring pressure force is that P is a convex function of (x,y). X=x+v/2Ω, Y=y-u/2Ω have to be increasing functions of (x,y) respectively.

18. Page 18© Crown copyright 2007 Inertial stability II For straight flow, condition is that the absolute geostrophic vorticity multiplied by 2Ω is positive (different condition for axisymmetric flow). This is a condition on the pressure field. Allows depressions to be small-scale and intense, but not anticyclones. Pressure gradients have to decrease towards the equator.

19. Page 19© Crown copyright 2007 Example of stable state.

20. Page 20© Crown copyright 2007 Scale separation The natural time-scale associated with large- scale flow is the frequency about which the fluid would oscillate about a stable equilibrium. A scale separation can be achieved by assuming that the Lagrangian time-scale of the flow is greater than that associated with any such oscillation. This requires it to be greater than (2Ω) -1 and also N -1, where N is the buoyancy frequency associated with the static stability condition. The condition involving Ω is usually more stringent.

21. Page 21© Crown copyright 2007 Mathematical procedure Define asymptotic regime of interest by assuming Lagrangian timescale greater than that associated with the Earth’s rotation. This requires the rate of change of wind direction following a fluid trajectory to be Ro /2Ω where Ro<<1. Ro is the Rossby number. This corresponds to a 12 hour period at the poles and 24 hour period at 30° latitude. Ro =0.1 means trajectory changes direction by more than 45° in 24 hours at 60° latitude. Illustrate with actual example in active spell of weather, most trajectories curve less than this (allow for Mercator projection).

22. Page 22© Crown copyright 2007 Example of ‘real’ trajectories Met Office global model back trajectories for 11 January 2005, 4 day period, marked every 12 hr.

23. Page 23© Crown copyright 2007 Solution procedure

24. Page 24© Crown copyright 2007 Construction of equations First identify a stable state in geostrophic and hydrostatic balance as an energy minimising state with respect to Lagrangian displacements conserving mass, momentum and potential temperature. Prove that such a state exists for a given specification of momentum and potential temperature. Formulate and solve the evolution equations for such states.

25. Page 25© Crown copyright 2007 Geostrophic and hydrostatic balance Write equations for compressible atmosphere in Cartesian coordinates with uniform rotation Ω:  Hydrostatic balance  Geostrophic balance

26. Page 26© Crown copyright 2007 Energy integral The energy conserved by the compressible Euler equations in this geometry is

27. Page 27© Crown copyright 2007 New variables Define (as in definition of inertial stability) The energy becomes The conditions for geostrophic and hydrostatic balance become

28. Page 28© Crown copyright 2007 Energy principle Follow Cullen and Feldman (2006). Define Lagrangian map F(t,x) as position at time t of particle initially at x and Lagrangian variable Z by Define virtual displacement by Can show energy is stationary with respect to these displacements if geostrophic and hydrostatic conditions satisfied

29. Page 29© Crown copyright 2007 Energy minimisation Can also show that if the energy is minimised, the static and inertial stability conditions are satisfied. Existence of a minimiser can be proved by mass transportation methods: Cullen and Maroofi (2003) following Brenier (1991) and Cullen and Gangbo (2001).

30. Page 30© Crown copyright 2007 Evolution equations The evolution of the minimum energy states can be described by the semi-geostrophic equations. In physical space these are

31. Page 31© Crown copyright 2007 Relation to full equations Standard a priori estimates are that the difference between SG and the compressible Euler equations is of order Ro(aspect ratio)². Since Ro is defined in a Lagrangian sense, discontinuities in physical space are permitted, such as weather fronts. Steeply sloping fronts, which are unstable to 3d disturbances, are not well described. Shallow sloping fronts are well-described.

32. Page 32© Crown copyright 2007 Shallow water example

33. Page 33© Crown copyright 2007 Example using shallow water equations Applicability of SG model can be demonstrated using shallow water equations. h is depth of water with mean value H.

34. Page 34© Crown copyright 2007 Experimental strategy In shallow water flow there are two small parameters available. Large rotation, 2Ω, leads to geostrophic balance. Large mean depth, H, leads to non- divergence. Define Ro=U/fL, Fr=U/ √gH. L is a length scale. Consider fixed Ro. Allow Fr to vary by changing mean depth. Shallow water solutions converge to 2d Euler at rate Fr². Shallow water solutions converge to SG at rate Ro(Ro/Fr)² for Ro<Fr.

35. Page 35© Crown copyright 2007 Demonstration Data with typical max wind speed 15ms -1. Mean depth chosen to give gravity wave speed 65ms -1 to 360ms -1. (Observed 500hpa evolution best matched using speed 140ms -1.) Gives Ro~0.1, Fr~0.05-0.3. Shallow water version of Met Office UM. SG and 2d Euler codes as similar as possible.

36. Page 36© Crown copyright 2007 Data for test

37. Page 37© Crown copyright 2007 Differences in depth UM-SG differences UM-2d Euler differences

38. Page 38© Crown copyright 2007 Differences in winds UM-SG differences UM-2d Euler differences

39. Page 39© Crown copyright 2007 Comments SG results for geostrophic variables show linear convergence in Ro/Fr, Ro/Fr not small enough to give quadratic convergence. Corresponds to condition on aspect ratio in 3d flow. Ro/Fr=(H/L)(2Ω/N). 2d Euler shows expected convergence in winds, depth asymptotes to non-zero difference. SG differences from UM much smaller than 2d Euler differences from UM.

40. Page 40© Crown copyright 2007 Evolution of SG equations

41. Page 41© Crown copyright 2007 Lagrangian form of SG equations Defining Z as before, the Lagrangian form of the semi-geostrophic equations is

42. Page 42© Crown copyright 2007 Transport equation These equations can be written as an evolution equation for the mass density σ in Σ: This is a transport equation. The velocity U is BV because of the ‘convexity’ properties. It is therefore well-posed (Ambrosio (2004)). The trajectory in Σ can then be mapped back to physical space giving weak existence of the Lagrangian form of the equations: Cullen and Feldman (2006).

43. Page 43© Crown copyright 2007 Consequences This argument proves existence of weak solutions to a Lagrangian form of the equations. The solutions are thus insensitive to changes on a set of measure zero. Thus there is no ‘butterfly’ effect, though chaotic evolution is possible if a finite volume of fluid is perturbed. The trajectories can be traced backwards in time, important for atmospheric composition, transport and pollution issues.

44. Page 44© Crown copyright 2007 Evolutionary properties of the equations Semi-geostrophic dynamics strongly constrained by ‘convexity’ property (e.g. inertial stability). Shows by greatly reduced growth of PV gradients in solutions of shallow water model. Thus in real system: for Ro<<Fr we get stable long-lived disturbances (like semi-geostrophic dynamics) For Ro>>Fr we get layered two-dimensional vortex dynamics. Vertical scale collapse and upscale horizontal cascade bring Ro/Fr back to O(1)

45. Page 45© Crown copyright 2007 Growth of PV gradients for Ro>>Fr

46. Page 46© Crown copyright 2007 Growth of PV gradients for Ro<<Fr

47. Page 47© Crown copyright 2007 Spherical geometry

48. Page 48© Crown copyright 2007 Solution in spherical geometry The clash between the direction of the geostrophic wind and the vertical reduces the symmetry of the problem. The factor 2Ω becomes f=2Ωsinφ where φ is the latitude. The problem cannot be formulated as a mass transport problem and transport equation without further approximations. Illustrate (formal) method of solution without further approximations. Use Cartesian geometry with variable f for ease of presentation.

49. Page 49© Crown copyright 2007 Lagrangian variations Define Minimise energy with respect to Lagrangian displacement satisfying

50. Page 50© Crown copyright 2007 Energy principle Can show E is stationary with respect to these displacements if For E to be minimised requires at least θ increasing with z and

51. Page 51© Crown copyright 2007 Formal proof Can show formally that E can be minimised by using an explicit descent algorithm. The resulting p satisfies a ‘convexity’ property. Can also show formally that the equations can be solved by timestepping

52. Page 52© Crown copyright 2007 Summary Demonstrated that large-scale equations governing atmospheric motions can be solved independently. Solutions are weak Lagrangian solutions, allowing discontinuities between air masses. Solutions are insensitive to very small-scale perturbations-no butterflies. Fluid trajectories can be traced back in time, important for many applications.

53. Page 53© Crown copyright 2007 Questions and answers