1. Markus Gross PDC14 Setting the scene

2. http://pc-mgross.cicese.mx/mediawiki/index.php/PDC14 ftp://ftp.cicese.mx/pub/divOC/ocefisica/PDC14

3. 1.1997: “Believable Scales and Parameterizations in a Spectral Transform Model”[1]. Lander and Hoskins. 1. coarse ‘‘physics’’ grid 2.Grid scale of the dynamics is not “believable”. 2.1998: “Consequences of Using the Splitting Method for Implementing Physical Forcings in a Semi-Implicit Semi-Lagrangian Model” [2]. Caya, Laprice and Zwack. 1.Effect of “splitting” 2.parameterization subsequently to time stepping the dynamics, in combination with long time steps (15 min) 3.the splitting error can become unacceptably large. 3.1999: “Convergence of atmospheric simulations with increasing horizontal resolution and fixed forcing scales” [3] Williamson 1.grid and scale of the physical parameterizations fixed 2.horizontal resolution of the dynamical core is increased. 3.aid the convergence of tropical Hadley circulation 4.does not converge if the physics grid is not held constant. 5.parameterization at the coarser grid do not include their own forcings from the finer scale, which appears to be contradictory to the purpose of parameterizations in first place. Some History

4. 1.1999: “Running GCM physics and dynamics on different grids: algorithm and tests” [4] Molod 1. The vertical grid is refined for parameterization. 2.benefits fields which are computed directly in the physical parametrization, 3.and in the vertical structure of rh and mass stream function. 2.1999: “The numerical coupling of the physical parameterizations to the "dynamical" equations in a forecast model” [5]. Nils Wedi. 1.SLAVEPP 2.Parameterizations split into two groups 3.predictors are used 4.demonstrates second order accuracy, 5.increase in stability, 6.reduction of the time step dependence and numerical noise, 7.improved mass conservation, 8.more accurate forecasts with respect to RMS and anomaly correlations and 9.improved tropical cyclone tracks, when compared with a simpler fractional stepping (or sequential or time split scheme).

5. 1.2002: “Analysis of the numerics of physics-dynamics coupling” [6] and “A Simple Comparison of Four Physics Dynamics Coupling Schemes” [7]. Staniforth, Wood and Côté. 1.explicit, implicit, split-implicit and symmetrized split-implicit coupling. 2.extends [2] in complexity of the sample problems as well as the coupling mechanisms (by adding the implicit and symmetrized split-implicit option) 3.stability of the explicit coupling is very restrictive for fast damping processes (vertical diffusion in the boundary layer at high resolution) 4.could be addressed by implicit coupling, 5.leads to “a highly nonlinear and computationally difficult and expensive problem to solve. “ 6.split implicit coupling addresses this but reduces the accuracy. 7.symmetrized split-implicit coupling addresses this, albeit at cost. 2.2002: “Time-Split versus Process-Split Coupling of Parameterizations and Dynamical Core” [8]. Williamson 1.Simulations based on time-split (sequential) and process-split (parallel) couplings compared to NCAR CCM3 2.overall differences were small, 3.probably due to the already very small time step. 4.Nevertheless statistically relevant differences were found.

6. 1.2003: “On the use of a predictor corrector scheme to couple the dynamics with the physical parametrization in the ECMWF model” [9]. Cullen and Salmond. 1.predictor–corrector scheme 2.advantages of a fully-implicit scheme 3.Use of more than one physics evaluation per time step significantly improves the accuracy in a model problem. 4. Although they state that “efficient integration of the dynamical equations in atmospheric models is well explored” this has to be read appreciating the historical context 5.Nevertheless, the coupling problem remains. 6. An attempt is made to classify slow and fast processes. 7.short-time variability is reduced 8.transfer from convective to dynamic precipitation. 2.2004: “The numerics of physical parameterization” [10]. Beljaars et al. 1.sequential splitting (tendencies of the explicit processes are computed first and are used as input to the subsequent implicit fast process) is preferable over parallel splitting (tendencies of all the parametrized processes are computed independently of each other) for problems with multiple time scales, because a balance between processes is obtained during the time integration.

7. 1.2004/5/6: “Analysis of Parallel versus Sequential Splittings for Time-Stepping Physical Parameterizations” [11], “Mixed Parallel/Sequential-Split Schemes for Time-Stepping Multiple Physical Parameterizations” [12] and “Some numerical properties of approaches to physics dynamics coupling for NWP”[13]. Dubal, Wood and Staniforth 1.shed light on the parallel versus sequential (and several more flavors) debate using analysis. 2.some advantages exist for parallel splitting over sequential splitting 3.(e.g., parallel computation and not requiring an ordering of physical processes), 4.sequential-split more flexible in eliminating splitting errors. 2.Termonia and Hamdi (2007) 1. impact of coupling on the numerical properties of the entire scheme of the model. 3.Williamson (2007) The evolution of dynamical cores for global atmospheric models. J. Meteorol. Soc. Jpn 85B: 241–269. 1.Reiterates that problema is not well understood 2.Coupling often technically convenient 3.Probably a better review of the literature (in model context) than the above 4.2007:”An iterative time-stepping scheme for the Met Office’s semi-implicit semi-Lagrangian non-hydrostatic model” [14]. Diamantakis, Davies and Wood 1.iterative cycling of dynamical core. 2.here in the NewDynamics, the same iterative loop exists in ENDGame, 3.option for different coupling strategies. 4.Effective resolution and stability are improved.

8. 1.2008: “Convergence of aqua-planet simulations with increasing resolution in the Community Atmospheric Model, Version 3” [15]. Williamson 1.convergence runs with resolution varying from T42-T340 and 40-5 minutes. 2.Convergence in larger scales of the zonal average equatorial precipitation 3.and equatorial wave propagation, 4.non convergent mass shift from polar to equatorial 5.nc zonal average cloud fraction decrease 6.simulations show sensitivity to the parametrization time step 7.as well as horizontal resolution 8.time step is fixed: global averages do not converge with increasing resolution for all fields. 9.no indication that either precipitable water or precipitation converges with increasing resolution. 2.2013: “Numerical issues associated with compensating and competing processes in climate models: an example from ECHAM-HAM”, [16]. Wan et al. 1.effect of time stepping scheme on sulfuric acid (H2 SO4 ) gas evolution 2.highlights the challenges involved as climate models (and the connectivity of the parameterizations) gain in complexity. 3.2013: “The effect of time steps and time-scales on parametrization suites” [17]. Williamson 1.interaction of parameterization schemes 2.how non-atmospheric states can be created, then further aggregated by sequentially coupled parameterizations, which do no benefit from the “equilibrating action” of the dynamical core

9. [1] J. Lander and B. J. Hoskins. Believable scales and parameterizations in a spectral transform model. Monthly Weather Review, 125(2):292– 303, Feb 1997. [2] Alain Caya, René Laprise, and Peter Zwack. Consequences of using the splitting method for implementing physical forcings in a semi-implicit semi-lagrangian model. Monthly Weather Review, 126(6):1707–1713, Jun 1998. [3] DAVID L. WILLIAMSON. Convergence of atmospheric simulations with increasing horizontal resolution and fixed forcing scales. Tellus A, 51(5):663–673, 1999. [4] ANDREA MOLOD. Running gcm physics and dynamics on different grids: algorithm and tests. Tellus A, 61(3):381–393, 2009. [5] Nils Wedi. The numerical coupling of the physical parameterizations to the "dynamical" equations in a forecast model. ECMWF, tm 274 edition. [6] Andrew Staniforth, Nigel Wood, and Jean Côté. Analysis of the numerics of physics-dynamics coupling. Quarterly Journal of the Royal Meteorological Society, 128(586):2779–2799, 2002. [7] Andrew Staniforth, Nigel Wood, and Jean Côté. A simple comparison of four physics dynamics coupling schemes. Monthly Weather Review, 130(12):3129–3135, Dec 2002. [8] David L. Williamson. Time-split versus process-split coupling of parameterizations and dynamical core. Monthly Weather Review, 130(8):2024–2041, Aug 2002. [9] M. J. P. Cullen and D. J. Salmond. On the use of a predictor-corrector scheme to couple the dynamics with the physical parametrizations in the ecmwf model. Quarterly Journal of the Royal Meteorological Society, 129(589):1217–1236, 2003. [10] A Beljaars, P Bechtold, M Köhler, J J Morcrette, A Tompkins, P Viterbo, and N Wedi. The numerics of physical parameterization. In Proc. ECMWF Workshop on Recent Developments in numerical methods for atmosphere and ocean modelling. Eur. Cent. For Medium-Range Weather Forecasts, 2004. [11] Mark Dubal, Nigel Wood, and Andrew Staniforth. Analysis of parallel versus sequential splittings for time-stepping physical parameterizations. Monthly Weather Review, 132(1):121–132, Jan 2004. [12] Mark Dubal, Nigel Wood, and Andrew Staniforth. Mixed parallel-sequential-split schemes for time-stepping multiple physical parameterizations. Monthly Weather Review, 133(4):989–1002, Apr 2005. [13] Mark Dubal, Nigel Wood, and Andrew Staniforth. Some numerical properties of approaches to physics-dynamics coupling for nwp. Quarterly Journal of the Royal Meteorological Society, 132(614):27–42, 2006. [14] Michail Diamantakis, Terry Davies, and Nigel Wood. An iterative time-stepping scheme for the met office’s semi-implicit semi-lagrangian non- hydrostatic model. Quarterly Journal of the Royal Meteorological Society, 133(625):997–1011, 2007. [15] David Williamson. Convergence of aqua-planet simulations with increasing resolution in the community atmospheric model, version 3. Tellus A, 60(5), 2008. [16] H. Wan, P. J. Rasch, K. Zhang, J. Kazil, and L. R. Leung. Numerical issues associated with compensating and competing processes in climate models: an example from echam-ham. Geoscientific Model Development, 6(3):861–874, 2013. [17] David L. Williamson. The effect of time steps and time-scales on parametrization suites. Quarterly Journal of the Royal Meteorological Society, 139(671):548–560, 2013. WIKI!

10. Analysis with more complex systems

11. Sequential coupling

12. Parallel coupling

14. Theta convergence

21. Summary Idealised GCMs have a very different response Internal variabillity has to be considered in forecast like experiments The steady state error should show up in the climate signal but difficult to predict where

23. Many thanks! Questions?