Scaling/ scaled mesoscale modelling

Fio. 4. Types of multiscale modeling and solution strategies. Hybrid models (one model at each scale) apply well when there is separation of scales (onion or nested-type models). When there is lack of separation of scales, mesoscale models need to be developed where the same technique (e.g., MD or MC) is accelerated. Alternatively, multigrid (heterogeneous) hybrid models can be employed where the unresolved degrees of freedom are determined from a finer scale model and passed to a coarser scale model. 

The linkage of microscopic and macroscopic properties is not without challenges, both theoretical and experimental. Statistical mechanics and thermodynamics provide the connection between molecular properties and the behavior of macroscopic matter. Coupled with statistical mechanics, computer simulation of the structure, properties, and dynamics of mesoscale models is now feasible and can handle the increase in length and time scales. 

Lenz C-J, Muller F, Schliinzen KH (2000) The sensitivity of mesoscale chemistry transport model results to boundary values. Environ Monit Assess 65 287-298 L6pez SD, Liipkes C, Schliinzen KH (2005) The effects of different k-e-closures on the results of a micro-scale model for the flow in the obstacle layer. Meteorol Z 14 839-848 Muller F, Schliinzen KH, Schatzmann M (2000) Test of numerical solvers for chemical reaction mechanisms in 3D air quality models. Environ Model Softw 15 639-646 Schliinzen KH (1990) Numerical studies on the inland penetration of sea breeze fronts at a coastline with tidally flooded mudflats. Beitr Phys Atmos 63 243-256 Schliinzen KH, Katzfey JJ (2003) Relevance of subgrid-scale land-use effects for mesoscale models. Tellus 55A 232-246... 

As the structure, dynamics and properties are determined by phenomena on many length and time scales physical modelling is subdivided into the quantum mechanical, atomistic, mesoscale, microscale and continuum levels, while research into the way in which these levels are linked is known as hierarchical or multiscale modelling. The typical structural levels arising in the polymer field are shown Figure 1. 

Despite these successes, molecular modelling will never be able to address phenomena, which involve length scales of hundreds of nanometers and more, such as the phase separation of block-copolymers, or very long timescales, such as mechanical properties related to sub Tg relaxations. Therefore more and more effort is devoted to the development of mesoscale models linked in a well defined way to the atomic level. 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

More often, three-dimensional gridded meteorological forecast data provided by mesoscale meteorological models such as MM5 will be available. The resolution of the grid will typically be on the order of 1 km, which is too large to capture microscale surface features. Mesoscale model results can be coupled with fine-scale models by enforcing conservation of mass to provide enhanced fine-scale detail. 

We have determined that nudging the models with the WTM surface observations alone appeared to have very limited impact on the mesoscale weather and dispersion forecasts. It is basically three-dimensional (two horizontal dimensions plus time) data assimilation over a small area. The impulses caused by the data insertion dissipated rapidly within 3 h after the turnoff of data nudging regardless of the length of relaxation time scale for model adjustment and nudging period. However, impulses caused by FDDA... 

In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

Figure 13.2 Multi-scale modelling hierarchy [35], showing the approximate regimes of time and length scales over which atomistic modelling techniques (quantum and molecular mechanics) can usefully be applied and how these link with process methods via mesoscale modelling.

The subgrid-scale fluxes in mesoscale models are parametrized in terms of resolvable variables. Turbulence theory, as observed fi om atmospheric field campaigns over horizontally homogeneous landscapes and for undisturbed atmospheric conditions, is the basis for all mesoscale model representations of the vertical subgrid-scale flux... 

In contrast to the vertical fluxes, horizontal subgrid-scale fluxes in mesoscale models are not realistically represented. They are included only to smooth the model calculations horizontally. 

The parametrization of cumulus cloud rainfall utilizes some form of one-dimensional cloud model. These are called cumulus cloud parametrization schemes. Then-complexity ranges from instantaneous readjustments of the temperature and moisture profile to the moist adiabatic lapse rates when the relative humidity exceeds saturation, to representations of a set of one-dimensional cumulus clouds with a spectra of radii. These parametrizations typically focus on deep cumulus clouds, which produce the majority of rainfall and diabatic heating associated with the phase changes of water. Cumulus cloud parametrizations remain one of the major uncertainties in mesoscale models since they usually have a number of tunable coefficients, which are used to obtain the best agreement with observations. Also, since mesoscale-model resolution is close to the scale of thunderstorms, care must be taken so that the cumulus parametrization and the resolved moist thermodynamics in the model do not double count this component of the and Sq.. 

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