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- ItemA scale invariance criterion for les parametrizations(Stuttgart : Gebrüder Bornträger Verlagsbuchhandlung, 2014) Schaefer-Rolffs, U.; Knöpfel, R.; Becker, E.Turbulent kinetic energy cascades in fluid dynamical systems are usually characterized by scale invariance. However, representations of subgrid scales in large eddy simulations do not necessarily fulfill this constraint. So far, scale invariance has been considered in the context of isotropic, incompressible, and three-dimensional turbulence. In the present paper, the theory is extended to compressible flows that obey the hydrostatic approximation, as well as to corresponding subgrid-scale parametrizations. A criterion is presented to check if the symmetries of the governing equations are correctly translated into the equations used in numerical models. By applying scaling transformations to the model equations, relations between the scaling factors are obtained by demanding that the mathematical structure of the equations does not change. The criterion is validated by recovering the breakdown of scale invariance in the classical Smagorinsky model and confirming scale invariance for the Dynamic Smagorinsky Model. The criterion also shows that the compressible continuity equation is intrinsically scale-invariant. The criterion also proves that a scaleinvariant turbulent kinetic energy equation or a scale-invariant equation of motion for a passive tracer is obtained only with a dynamic mixing length. For large-scale atmospheric flows governed by the hydrostatic balance the energy cascade is due to horizontal advection and the vertical length scale exhibits a scaling behaviour that is different from that derived for horizontal length scales.
- ItemCorrelation between traffic density and particle size distribution in a street canyon and the dependence on wind direction(München : European Geopyhsical Union, 2006) Voigtländer, J.; Tuch, T.; Birmili, W.; Wiedensohler, A.Combustion of fossil fuel in gasoline and diesel powered vehicles is a major source of aerosol particles in a city. In a street canyon, the number concentration of particles smaller than 300 nm in diameter, which can be inhaled and cause serious health effects, is dominated by particles originating from this source. In this study we measured both, particle number size distribution and traffic density continuously in a characteristic street canyon in Germany for a time period of 6 months. The street canyon with multistory buildings and 4 traffic lanes is very typical for larger cities. Thus, the measurements also are representative for many other street canyons in Europe. In contrast to previous studies, we measured and analyzed the particle number size distribution with high size resolution using a Twin Differential Mobility Analyzer (TDMPS). The measured size range was from 3 to 800 nm, separated into 40 size channels. Correlation coefficients between particle number concentration for integrated size ranges and traffic counts of 0.5 were determined. Correlations were also calculated for each of the 40 size channels of the DMPS system, respectively. We found a maximum of the correlation coefficients for nucleation mode particles in the size range between 10 and 20 nm in diameter. Furthermore, correlations between traffic and particles in dependence of meteorological data were calculated. Relevant parameters were identified by a multiple regression method. In our experiment only wind parameters have influenced the particle number concentration significantly. High correlation coefficients (up to 0.8) could be observed in the lee side of the street canyon for particles in the range between 10 and 100 nm in diameter. These values are significantly higher than correlation coefficients for other wind directions and other particle sizes. A minimum was found in the luff side of the street. These findings are in good agreement with theory of fluid dynamics in street canyons.
- ItemSimilitude of ice dynamics against scaling of geometry and physical parameters(München : European Geopyhsical Union, 2016) Feldmann, Johannes; Levermann, AndersThe concept of similitude is commonly employed in the fields of fluid dynamics and engineering but rarely used in cryospheric research. Here we apply this method to the problem of ice flow to examine the dynamic similitude of isothermal ice sheets in shallow-shelf approximation against the scaling of their geometry and physical parameters. Carrying out a dimensional analysis of the stress balance we obtain dimensionless numbers that characterize the flow. Requiring that these numbers remain the same under scaling we obtain conditions that relate the geometric scaling factors, the parameters for the ice softness, surface mass balance and basal friction as well as the ice-sheet intrinsic response time to each other. We demonstrate that these scaling laws are the same for both the (two-dimensional) flow-line case and the three-dimensional case. The theoretically predicted ice-sheet scaling behavior agrees with results from numerical simulations that we conduct in flow-line and three-dimensional conceptual setups. We further investigate analytically the implications of geometric scaling of ice sheets for their response time. With this study we provide a framework which, under several assumptions, allows for a fundamental comparison of the ice-dynamic behavior across different scales. It proves to be useful in the design of conceptual numerical model setups and could also be helpful for designing laboratory glacier experiments. The concept might also be applied to real-world systems, e.g., to examine the response times of glaciers, ice streams or ice sheets to climatic perturbations.