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

2. 2 Outline What do we mean by physics? Extending the framework of Caya et al (1998) Some coupling strategies Analysis of the coupling strategies Summary

3. 3 What do we mean by physics? Molecular effects Radiation Microphysics (conversion terms, latent heating) Underpinning fluid dynamical equations: Physics = Boundary Conditions+RHS:

4. 4 Filtered equation set Added sub-filter physics (neglect molecular effects): Modified Boundary Conditions Turbulence Convection Gravity-wave stress

5. 5 Physics Physics split into two parts: –True physics (radiation, microphysics) (albeit filtered) –Sub-filter physics = function of filter scale (NOT a function of grid scale though often assumed to be! Convergence issues Mason and Callen ‘86) »CRMs, LES etc = turbulence »NWP/GCMs = turbulence + convection + GWD »Mesoscale = turbulence + ?  Boundary conditions also!

6. 6 What do we mean by physics-dynamics coupling? For explicit models with small  t no issue - all terms (physics/dynamics) handled in the same way (ie most CRMs, LES etc) and even if not then at converged limit Real issue only when  t large compared to time scale of processes, then have to decide how to discretize terms - but in principle no different to issues of dynamical terms (split is arbitrary - historical?) BUT many large scale models have completely separated physics from dynamics  inviscid predictor + viscid physics corrector (boundary conditions corrupted)

7. 7 Holy grail of coupling Large scale modelling (  t large): SISL schemes allow increased  t and therefore a balancing of the spatial and temporal errors whilst retaining stability and accuracy (for dynamics at least) If physics is not handled properly then the coupling will introduce O(  t ) errors and the advantage of SISL will be negated Aim: Couple with O(  t 2 ) accuracy + stability

8. 8 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)

9. 9 CLZ98’s model  represents either a damping term (if real and > 0) or oscillatory term (dynamics) if imaginary G = const. forcing (diabatic forcing in CLZ98) Model useful (CLZ98 diagnosed problem in their model) but: –neglects advection (and therefore cannot analyse eg SL orographic resonance) –neglects spatio-temporal forcing terms

10. 10 Extending CLZ98’s model These effects can be easily included whilst retaining the analytical tractability of CLZ98: Since Eq is linear can Fourier decompose  where is the k th Fourier mode of Assumed only one frequency of oscillation

11. 11 Exact Regular Solution Consider only 1 dynamics oscillatory process, 1 (damping) physics process: Solution = sum of free and forced solution:

12. 12 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:

13. 13 Application to Coupling Discretizations Assume a semi-Lagrangian advection scheme Apply a 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

14. 14 Fully Explicit/Implicit Time-weights: dynamics,  physics, forcing  =0  Explicit physics - simple but stability limited  =1  Implicit physics - stable but expensive

15. 15 Split-Implicit Two step predictor corrector approach: First = Dynamics only predictor (advection + GW) Second = Physics only corrector

16. 16 Symmetrized Split-Implicit Three step predictor-corrector approach: First = Explicit Physics only predictor Second = Semi-implicit Dynamics only corrector Third = Implicit Physics only corrector

17. 17 Analysis Each of the 4 schemes has been analysed in terms of its: –Stability –Accuracy –Steady state forced response –Occurrence of spurious resonance

18. 18 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

19. 19 Accuracy Accuracy of the free mode is determined by expanding E in powers of  t and comparing with the expansion of the analytical result:

20. 20 Forced Regular Response As for the free mode, the forced response can be determined by seeking solutions of the form: Accuracy of the forced response can again be determined by comparison with the exact analytical result.

21. 21 Steady State Response of the Forced Solution A key aspect of a parametrization scheme (and often the only fully understood aspect) is its steady state response when  k =0 and  >0. Accuracy of the steady-state forced response can again be determined by comparison with the exact analytical result:

22. 22 Forced Resonant Solution Resonance occurs when the denominator of the Forced Response vanishes For a semi-Lagrangian, semi-implicit scheme there can occur spurious resonances in addition to the physical (analytical) one

23. 23 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 )

24. 24 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)

25. 25 Summary Numerics of Physics-Dynamics coupling key to continued improvement of numerical accuracy of models Caya et al (1998) has been 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