1. © Crown copyright Met Office A Framework for The Analysis of Physics-Dynamics Coupling Strategies Andrew Staniforth, Nigel Wood (Met O Dynamics Research) and Jean Côté (Met Service of Canada)

2. © Crown copyright Met Office Outline  Physics-Dynamics & their coupling  Extending the framework of Caya et al (1998)  Some coupling strategies  Analysis of the coupling strategies  Summary

3. What is dynamics and physics?  Dynamics =  Resolved scale fluid dynamical processes:  Advection/transport, rotation, pressure gradient  Physics =  Non-fluid dynamical processes:  Radiation, microphysics (albeit filtered)  Sub-grid/filter fluid processes:  Turbulence + convection + GWD © Crown copyright Met Office

4. What do we mean by physics-dynamics coupling?  Small  t (how small?) no issue:  All terms handled in the same way (ie most CRMs, LES etc)  Even if not then at converged limit  Large  t (cf. time scale of processes) have to decide how to discretize terms  In principle no different to issues of dynamical terms (split is arbitrary - historical?)  BUT many large scale models have completely separated physics from dynamics Þ inviscid predictor + viscous physics corrector (Note: boundary conditions corrupted) © Crown copyright Met Office

5. Aim of coupling Large scale modelling (  t large):  SISL schemes allow increased  t and hence balancing of spatial and temporal errors  Whilst retaining stability and accuracy (for dynamics at least)  If physics not handled properly then coupling introduces O(  t ) errors & advantage of SISL will be negated  Aim: Couple with O(  t 2 ) accuracy + stability © Crown copyright Met Office

6. Framework for analysing coupling strategies  Numerical analysis of dynamics well established  Some particular physics aspects well understood (eg diffusion) but largely in isolation  Caya, Laprise and Zwack (1998)  simple model of coupling:  Regard as either a simple paradigm or F(t) is amplitude of linear normal mode (Daley 1991)  CLZ98 used this to diagnose problem in their model © Crown copyright Met Office

7. CLZ98’s model  represents:  Damping term (if real and > 0)  Oscillatory term (dynamics) if imaginary G = constant forcing (diabatic forcing in CLZ98) Model useful but:  Neglects advection (& therefore orographic resonance)  Neglects spatio-temporal forcing terms © Crown copyright Met Office

8. Extending CLZ98’s model  Add in advection, and allow more than 1  -type process  In particular, consider 1 dynamics oscillatory process, 1 (damping) physics process:  Solution = sum of free and forced solution: © Crown copyright Met Office

9. Exact Resonant Solution  Resonance occurs when denominator of forced solution vanishes, when:  Solution = sum of free and resonant forced solution: which, as all terms are real, reduces to: © Crown copyright Met Office

10. Application to Coupling Discretizations  Apply semi-Lagrangian advection scheme  Apply semi-implicit scheme to the dynamical terms (e.g. gravity modes)  Consider 4 different coupling schemes for the physics:  Fully Explicit/Implicit  Split-implicit  Symmetrized split-implicit  Apply analysis to each © Crown copyright Met Office

11. Fully Explicit/Implicit  Time-weights: dynamics,  physics, forcing   =0  Explicit physics - simple but stability limited   =1  Implicit physics - stable but expensive © Crown copyright Met Office

12. Split-Implicit Two step predictor corrector approach:  First = Dynamics only predictor (advection + GW)  Second = Physics only corrector © Crown copyright Met Office

13. Symmetrized Split-Implicit Three step predictor-corrector approach:  First = Explicit Physics only predictor  Second = Semi-implicit Dynamics only corrector  Third = Implicit Physics only corrector © Crown copyright Met Office

14. Analysis  Each scheme analysed in terms of its:  Stability  Accuracy  Steady state forced response  Occurrence of spurious resonance © Crown copyright Met Office

15. Stability  Stability can be examined by solving for the free mode by seeking solutions of the form: and requiring the response function to have modulus  1 © Crown copyright Met Office

16. Accuracy  Accuracy of free mode determined by expanding E in powers of  t and comparing with expansion of analytical result: © Crown copyright Met Office

17. Forced Regular Response  Forced response determined by seeking solutions of form:  Accuracy of forced response again determined by comparing with exact analytical result. © Crown copyright Met Office

18. Steady State Response of the Forced Solution  Key aspect of parametrization scheme is its steady state response when  k =0 and  >0  Accuracy of steady-state forced response again determined by comparing with exact analytical result: © Crown copyright Met Office

19. Forced Resonant Solution  Resonance occurs when the denominator of the Forced Response vanishes  For semi-Lagrangian, semi-implicit scheme there can occur spurious resonances in addition to the physical (analytical) one © Crown copyright Met Office

20. Results I  Stability:  Centring or overweighting the Dynamics and Physics ensures the Implicit, Split-Implicit and Symmetrized Split-Implicit schemes are unconditionally stable  Accuracy of response:  All schemes are O(  t) accurate  By centring the Dynamics and Physics the Implicit and Symmetrized Split-Implicit schemes alone, are O(  t 2 ) © Crown copyright Met Office

21. Results II  Steady state response:  Implicit/Explicit give exact response independent of centring  Split-implicit spuriously amplifies/decays steady-state  Symmetrized Split-Implicit exact only if centred  Spurious resonance:  All schemes have same conditions for resonance  Resonance can be avoided by: Applying some diffusion (  >0) or Overweighting the dynamics (at the expense of removing physical resonance) © Crown copyright Met Office

22. Summary  Numerics of Physics-Dynamics coupling key to continued improvement of numerical accuracy of models  Caya et al (1998) extended to include:  Advection (and therefore spurious resonance)  Spatio-temporal forcing  Four (idealised) coupling strategies analysed in terms of:  Stability, Accuracy, Steady-state Forced Response, Spurious Resonance © Crown copyright Met Office

23. Application of this analysis  A simple comparison of four physics-dynamics coupling schemes Andrew Staniforth, Nigel Wood and Jean Côté (2002) Mon. Wea. Rev. 130, 3129-3135  Analysis of the numerics of physics-dynamics coupling Andrew Staniforth, Nigel Wood and Jean Côté (2002) Q. J. Roy. Met. Soc. 128 2779- 2799  Analysis of parallel vs. sequential splitting for time-stepping physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2004) Mon. Wea. Rev. 132, 121-132  Mixed parallel-sequential split schemes for time-stepping multiple physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2005) Mon. Wea. Rev. 133, 989-1002  Some numerical properties of approaches to physics-dynamics coupling for NWP Mark Dubal, Nigel Wood and Andrew Staniforth (2006) Q. J. Roy. Met. Soc. 132, 27-42 (Detailed comparison of Met Office scheme with those of NCAR CCM3, ECMWF and HIRLAM) © Crown copyright Met Office

24. Thank you! Questions?