Computational challenges in Numerical Weather Prediction

Transcription

1 Computational challenges in Numerical Weather Prediction Mike Cullen Oxford 15 September 2008

2 Contents This presentation covers the following areas Historical background Current challenges Why does it work? Linear and nonlinear methods Validation of methods

3 Historical background

4 Numerical prediction W.Bjerknes and L.F.Richardson recognised in the early 1900s that the equations of motion and thermodynamics gave a prognostic set of equations which were sufficient to advance a given atmospheric state in time. Richardson did an actual experiment, though it did not give realistic results.

5 Numerical prediction II Bjerknes, Rossby, Charney and others explained the observed behaviour of extra-tropical weather systems in terms of simpler sets of equations. Such systems were used for NWP in the 1950s, but were not very successful. All prediction since the 1960s has been done using (almost) the complete laws of physics. The predictions have improved as more computing power became available-and also more observations.

6 Improved forecast skill

7 Current challenges

8 Smaller-scale forecasts Now we have much more powerful computers, forecasts of small-scale severe weather events is feasible. Not clear how predictable such systems are. Ensemble forecasts are used to give an idea of the possible outcomes. Competition for computing resources between ensemble size and model resolution-both are inadequate at present.

9 Floods in July 2007

10 UK computer forecast from midday, 19 July

11 42-hour ensemble forecasts of rain for 20 July 2007

12 Comments All ensemble members predicted heavy rain over Southern England. The basic weather system was highly predictable. There was significant spread in where the heavy rain would fall. At shorter lead times, these uncertainties were reduced and high probabilities were concentrated on the SW Midlands. Not clear that a single deterministic forecast of extreme conditions will ever be useful.

13 Why does it work?

14 Basic issue The solution of the governing equations is far to complicated to compute explicitly. A grid length of 0.001m would be required, requiring computers 28 orders of magnitude more powerful than those available now. In practice the equations are averaged implicitly within the discretisation. Success only possible either if the equations are linear, or there are nonlinear solutions which accurately describe large scales independently

15 Time series of wind speed

16 Typical satellite picture

17 Corresponding weather map

18 Large-scale flows Need to identify scale separation. Large scales computed explicitly Small scales estimated statistically by sub-grid modelling This will only work if there is a physical scale separation in the system. The earth s rotation defines a separation timescale, but it is rather coarse.

19 Mathematical procedure Define asymptotic regime of interest by assuming Lagrangian timescale greater than that associated with the Earth s rotation. Define the Rossby number Ro as the ratio of these time scales. Ro >0.1 means fluid 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).

20 Example of real trajectories Met Office global model back trajectories for 11 January 2005, 4 day period, marked every 12 hr.

21 Mathematical theory Mathematicians have proved that an inviscid system of equations (the semi-geostrophic system) derived using this scale separation can be solved for large times. The equations consist of a single scalar conservation law, together with an elliptic problem to derive the pressure, winds and temperatures. The proof makes use of the property that Lagrangian transport cannot create new values of the transported quantity, together with the polar factorisation theorem of Brenier et al. to show that the transported quantities can be rearranged to form the gradient of a convex function. The convexity property ensures that a sequence of approximate solutions converges to a limit.

22 Mathematical theory The solutions contain stable large-scale disturbances corresponding to weather systems. The solutions are an accurate approximation to the full governing equations for small Ro. Thus the full equations will inherit the same large-scale behaviour. The averaging scale implied is much coarser than that attainable in operational forecasting. It is therefore desirable to ensure that these solutions are sufficiently accurately reproduced in operational models which use a finer averaging scale. This is non-trivial because the solutions have singularities, and are Lagrangian rather than Eulerian in nature.

23 Linear and nonlinear methods

24 Linear and nonlinear methods Numerical methods for NWP have traditionally been developed with the linearised equations in mind. Thus even-order finite-difference schemes with centred time differencing were favoured. The spectral method, which gives an exact spatial representation of a truncated linear system was almost universally used for global NWP for many years.

25 Methods II The importance of the large-scale solutions was recognised by resolving only these in time. Fast wave solutions were treated implicitly, to allow longer timesteps. The ratio of spatial horizontal averaging scale toi vertical averaging scale corresponded to a speed of 50ms -1. The typical aspect ratio of the large-scale solutions (about 0.01) was used to choose the ratio of vertical to horizontal resolution.

26 Nonlinear regimes As resolution increases, more nonlinearity is resolved. The nonlinear theory of large-scale solutions suggests that quasi-monotone advection should be used for all variables. This is also applicable to some, but not all, small-scale regimes. Enforcement of asymptotic behaviour suggests use of decentred time integration for all waves not resolved well in time. Explicit use of artificial viscosity should be avoided-as relevant limit solutions are inviscid.

27 Validation of methods

28 Validation demonstration Compare solutions of full governing equations with those of semi-geostrophic system. To allow accurate numerical solution, work with variables depending on (x,z) only. Three-dimensional effects come in through forcing terms. Problem initialised with unstable linear mode, when solution becomes nonlinear the growth saturates. SG solution is discontinuous in physical space, but not in a Lagrangian sense (a weather front). Full solution should converge to SG at rate Ro 2.

29 Demonstration II Choose scaling of problem so that SG solution is independent of Ro. Illustrate numerical solution of compressible Euler equations for Ro=0.125, 0.062, Illustrate convergence of solutions to that for Ro=0.031 as Ro decreases with and without quasi-monotone advection. Illustrate convergence to geostrophic balance (which defines SG solution) as Ro decreases.

30 v field for various Ro Ro=0.125 Ro=0.062 Ro=0.031

31 Convergence of v field at days 3,6,7,9 Non-monotone Monotone

32 Convergence of geostrophic departure, days 3,6,7,9 Non-monotone Monotone

33 Comments Numerical technique has little effect while solution is smooth. Quasi-monotone advection improves convergence to limit solution, once discontinuous. Theoretical second order convergence only achieved for smooth solutions.

34 Questions and answers