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(c) MSc Module MTMW14 : Numerical modelling of atmospheres and oceans Week 3: Modelling the Real World 3.1 Staggered time schemes (semi-implicit) 3.2 Physical parameterisations 3.3 Ocean modelling

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2 Course content Week 1: The Basics 1.1 Introduction 1.2 Brief history of numerical weather forecasting 1.3 Dynamical equations for the unforced fluid Week 3: Modelling the real world 3.1 Physical parameterisations 3.2 Ocean modelling 3.3 Staggered time schemes and the semi-implicit method Week 4: More advanced spatial methods 4.1 Staggered horizontal and vertical grid discretisations 4.2 Lagrangian and semi-lagrangian schemes 4.3 Series expansion methods Week 5: Final thoughts 5.1 Revision 5.2 Test 5.3 Survey of state-of-the-art coupled climate models

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3 3.2 Physical parameterisations Q is the fluid dynamics term ( dynamical core ) F is forcing and dissipation due to physical processes such as: radiation clouds convection horizontal and vertical mixing/transport soil moisture and land surface processes etc. The atmosphere and oceans are FORCED DISSIPATIVE fluid dynamical systems:

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4 “Earth System” Science, NASA 1986 From Earth System Science – Overview, NASA, 1986

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5 In the Hollow of a Wave off the Coast at Kanagawa", Hokusai 3.1 Unresolved process: horizontal mixing

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6 3.1 Small-scale mixing Two approaches: Direct Numerical Simulation (DNS) try to resolve all spatial scales using a very high resolution model. Large-Eddy Simulation (LES) model the effects of sub-grid scale eddies as functions of the large-scale flow ( closure parameterisation ).

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7 3.1 Horizontal mixing in vorticity eqn Or more correctly model fluxes due to unresolved scales as f(.) + random noise rather than just f(.). (STOCHASTIC PHYSICS)

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8 1.3 Some mixing parameterisations Diffusion Assume that vorticity flux due to eddies acts to reduce the large-scale vorticity gradient: Note: Mixing isn’t always down-gradient. For example, vorticity fluxes due to mid- latitude storms act to strengthen the westerly jets!

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9 1.3 Some mixing parameterisations Hyper-diffusion Diffusion is found to damp synoptic features too much. More scale-selective damping can be obtained using: Note: Widely used in spectral models.

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Some mixing parameterisations Strain-dependent viscosity (Smagorinsky 1963) Put more diffusion in regions that have more large-scale strain: Note: Diffuses more strongly in high strain zones (e.g. between cyclones) but is computationally expensive to implement.

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Spectral blocking and instability In 2-d turbulence and the atmosphere and oceans, enstrophy cascades to ever smaller scales: This cascade can’t continue for ever in numerical models and so in the absence of any mixing parameterisations, energy builds up on the smallest scales. This is known as spectral blocking and can lead to non-linear instability: Phillips, N. A., 1959: An example of nonlinear computational instability. In: B. Bolin (Editor), The Atmosphere and the Sea in Motion. Rockefeller Inst. Press in association with Oxford Univ.Press, New York,

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Convection schemes Why parameterise cumulus convection? Precipitation is caused by rising of air due to: Local convective instability Large-scale ascent (slantwise convection) Orographic influences

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Convection schemes Convection occurs on small spatial scales (less than 10km) not explicitly resolvable by weather and climate models. It is important for producing precipitation and releasing latent heat (diabatic heating of atmosphere). producing clouds that affect radiation (e.g. cumulonimbus Cb, cumulus Cu, and stratocumulus Sc) vertical mixing of heat and moisture (and horizontal momentum) Further reading: J. Kiehl’s Chapter 10 in Climate System Modelling (Editor: K. Trenberth) Emanuel, K.A. 1994: Atmospheric convection, Oxford University Press.

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Summary of main points Modelling physical processes is an important aspect of weather prediction and climate simulation. Not as simple as modelling the fluid dynamics parts. Physical schemes take roughly the same amount as computer time as the fluid dynamics schemes. Forcing (e.g. radiation) and dissipative processes (e.g. mixing) need to be correctly modelled. Unresolved processes that feedback onto the large-scale flow need to be parameterised in terms of the the large- scale flow.

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Summary of main points

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23 1 Basic idea: Represent time derivative by centered time finite difference (CT) and then put different terms of the equation on different time levels. (mixed scheme) Why? Different equations have different stability properties (e.g. diffusion equation is unstable for CT and advection equation is unstable for FT scheme). Treat some of the terms implicitly in order to slow down simulated waves and get round CFL criterion (semi-implicit)

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Staggered time schemes (semi-implicit) Example: stable leapfrog (centred) for advection stable forward for diffusion Note: both are conditionally stable

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Semi-implicit gravity wave schemes Key concept: slow down fast gravity waves by treating them implicitly while treating other terms explicitly Example: shallow water equations

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Semi-implicit gravity wave schemes Split the operator: Centered scheme (explicit) Backward scheme (fully implicit) Mixed scheme (semi-implicit)

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Semi-implicit gravity wave schemes The sparse operator 1+hG is a lot easier to invert than is 1+2hQ – obtain a simple Helmholtz equation for height field. Rank of matrix to be inverted is 1/3 that of fully implicit scheme. Fast sparse matrix techniques can be used to invert (1+hG) Causes gravity waves to be treated implicitly and so larger time steps can be used without running into CFL problems ECMWF example: Ritchie …

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Summary of main points

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30 Winnighoff 1968, Arakawa and Lamb 1977.

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32 ECMWF algorithm

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Summary of main points

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36 If the lapse rate exceeds dry adiabatic, a convective adjustment restores the lapse rate to dry adiabatic, with conservation of dry static energy in the vertical. A moist convective adjustment scheme after Manabe et al. (1965) also operates when the lapse rate exceeds moist adiabatic and the air is supersaturated. (For supersaturated stable layers, nonconvective large-scale condensation takes place--see Precipitation). In moist convective layers it is assumed that the intensity of convection is strong enough to eliminate the vertical gradient of potential temperature instantaneously, while conserving total moist static energy. It is further assumed that the relative humidity in the layer is maintained at 100 percent, owing to the vertical mixing of moisture, condensation, and evaporation from water droplets. Shallow convection is not explicitly simulated. Manabe, S., J. Smagorinsky, and R. F. Strickler, 1965: Simulated climatology of a general circulation model with a hydrologic cycle. Mon. Wea. Rev., 93, Arakawa, A. and W.H. Schubert, 1974: Interaction of cumulus cloud ensemble with the largescale environment, Part I. J. Atmos. Sci., 31, Betts, A.K. and M.J. Miller, 1986: A new convective adjustment scheme. Part I: Observational & theoretical basis. QJRMS, 112, Kuo, H.L., 1974: Further studies of the parameterization of the influence of cumulus convection on large scale flow. J. Atmos. Sci., 31, Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117,

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