© 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)

© 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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,  Analysis of the numerics of physics-dynamics coupling Andrew Staniforth, Nigel Wood and Jean Côté (2002) Q. J. Roy. Met. Soc  Analysis of parallel vs. sequential splitting for time-stepping physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2004) Mon. Wea. Rev. 132,  Mixed parallel-sequential split schemes for time-stepping multiple physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2005) Mon. Wea. Rev. 133,  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, (Detailed comparison of Met Office scheme with those of NCAR CCM3, ECMWF and HIRLAM) © Crown copyright Met Office

Thank you! Questions?