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# Proof of the Monotonicity of Grid Size and Its Application in Grid-Size Selection for Mesoscale Models

• Terrain characteristics can be accurately represented in spectrum space. Terrain spectra can quantitatively reflect the effect of topographic dynamic forcing on the atmosphere. In wavelength space, topographic spectral energy decreases with decreasing wavelength, in spite of several departures. This relationship is approximated by an exponential function. A power law relationship between the terrain height spectra and wavelength is fitted by the least-squares method, and the fitting slope is associated with grid-size selection for mesoscale models. The monotonicity of grid size is investigated, and it is strictly proved that grid size increases with increasing fitting exponent, indicating that the universal grid size is determined by the minimum fitting exponent. An example of landslide-prone areas in western Sichuan is given, and the universal grid spacing of 4.1 km is shown to be a requirement to resolve 90% of terrain height variance for mesoscale models, without resorting to the parameterization of subgrid-scale terrain variance. Comparison among results of different simulations shows that the simulations estimate the observed precipitation well when using a resolution of 4.1 km or finer. Although the main flow patterns are similar, finer grids produce more complex patterns that show divergence zones, convergence zones and vortices. Horizontal grid size significantly affects the vertical structure of the convective boundary layer. Stronger vertical wind components are simulated for finer grid resolutions. In particular, noticeable sinking airflows over mountains are captured for those model configurations.
•  Ansoult M. M., 1989: Circular sampling for fourier analysis of digital terrain data . Mathematical Geology,21, 401-410, doi: 10.1007/BF00897325. Boer G. J., T. G. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci. ,40, 164-184, doi:10.1175/1520-0469(1983)040<0164: LSTDTI>2.0.CO;2. Booth A. M., J. J. Roering, and J. T. Perron, 2009: Automated landslide mapping using spectral analysis and high-resolution topographic data: Puget Sound lowlands, Washington, and Portland Hills, Oregon. Geomorphology , 109, 132-147, doi:10.1016/j.geomorph.2009.02.027. Bretherton F. P., 1969: Momentum transport by gravity waves. Quart. J. Roy. Meteor. Soc.,95, 213-243, doi: 10.1002/qj. 49709540402. Denis B., J. C\ot\'e, and R. Laprise, 2002: Spectral decomposition of two-dimensional atmospheric fields on limited-area domains using the Discrete Cosine Transform (DCT). Mon. Wea. Rev.,130, 1812-1829, doi: 10.1175/1520-0493(2002)130<1812: SDOTDA>2.0.CO;2. Goff J. A., B. E. Tucholke, 1997: Multiscale spectral analysis of bathymetry on the flank of the Mid-Atlantic Ridge: Modification of the seafloor by mass wasting and sedimentation. J. Geophys. Res.,102, 15 447-15 462, doi: 10.1029/97JB 00723. Hanley J. T., 1977: Fourier analysis of the Catawba Mountain knolls, Roanoke county, Virginia. Mathematical Geology, 9, 159-163, doi: 10.1007/BF02312510. Hough S. E., 1989: On the use of spectral methods for the determination of fractal dimension. Geophys. Res. Lett.,16, 673-676, doi: 10.1029/GL016i007p00673. Hsu H. M., M. W. Moncrieff, W. W. Tung, and C. H. Liu, 2006: Multiscale temporal variability of warm-season precipitation over North America: Statistical analysis of radar measurements. J. Atmos. Sci. ,63, 2355 -2368, doi:10.1175/JAS3752. 1. Kain J. S., J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 2784- 2802. Kain J. S., Coauthors, 2008: Some practical considerations regarding horizontal resolution in the first generation of operational convection-allowing NWP. Wea.Forecasting, 23, 931- 952. Perron J. T., J. W. Kirchner, and W. E. Dietrich, 2008: Spectral signatures of characteristic spatial scales and nonfractal structure in landscapes. J. Geophys. Res. , 113,F04003, doi:10.1029/2007JF000866. Pielke R. A., 1981: Mesoscale numerical modeling. Advances in Geophysics, B. Saltzman, Ed. , Academic Press Inc.,New York, 185- 344. Pielke R. A., 1984: Mesoscale Meteorological Modeling. Academic Press, San Diego,599 pp. Poulos G. S., R. A. Pielke, 1994: A numerical analysis of Los Angeles Basin pollution transport to the Grand Canyon under stably stratified, southwest flow conditions. Atmos. Environ., 28, 3329- 3357. Ramanathan N., K. Srinivasan, 1995: An estimation of optimum grid size for Kashmir Valley by spectral method. J. Appl. Meteor., 34( 12), 2783- 2786. Rayner J. 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Ramanathan, 1994: Terrain variance spectra for Indian Western Ghats. Proceedings-Indian National Science Academy Part A, 60A, 133- 138. Trenberth K. E., A. Solomon, 1993: Implications of global atmospheric spatial spectra for processing and displaying data. J. Climate,6, 531-545, doi: 10.1175/1520-0442(1993)006 <0531:IOGASS>2.0.CO;2. Wang W. T., Y. Wang, 2004: A spectral analysis of satellite topographic profile: A coincident pattern between latitudinal topographic and westerly perturbation on the lee side of Qinghai-Tibet Plateau. Journal of Nanjing University, 40( 4), 304- 317. (in Chinese) Young G. S., R. A. Pielke, 1983: Application of terrain height variance spectra to mesoscale modeling. J. Atmos. Sci., 40, 2555- 2560. Young G. S., R. A. Pielke, and R. C. Kessler, 1984: A comparison of the terrain height variance spectra of the Front Range with that of a hypothetical mountain. J. Atmos. Sci.,41(8), 1249-1252, doi: 10.1175/1520-0469(1984)041<1249:ACOTTH> 2.0.CO;2. Zhao K., M. 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## Manuscript History

Manuscript revised: 06 November 2014
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Proof of the Monotonicity of Grid Size and Its Application in Grid-Size Selection for Mesoscale Models

• 1. Laboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
• 2. University of Chinese Academy of Sciences, Beijing 100049
• 3. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100049
• 4. Public Weather Service Center, China Meteorological Administration, Beijing 100081
• 5. National Meteorological Center, China Meteorological Administration, Beijing 100081

Abstract: Terrain characteristics can be accurately represented in spectrum space. Terrain spectra can quantitatively reflect the effect of topographic dynamic forcing on the atmosphere. In wavelength space, topographic spectral energy decreases with decreasing wavelength, in spite of several departures. This relationship is approximated by an exponential function. A power law relationship between the terrain height spectra and wavelength is fitted by the least-squares method, and the fitting slope is associated with grid-size selection for mesoscale models. The monotonicity of grid size is investigated, and it is strictly proved that grid size increases with increasing fitting exponent, indicating that the universal grid size is determined by the minimum fitting exponent. An example of landslide-prone areas in western Sichuan is given, and the universal grid spacing of 4.1 km is shown to be a requirement to resolve 90% of terrain height variance for mesoscale models, without resorting to the parameterization of subgrid-scale terrain variance. Comparison among results of different simulations shows that the simulations estimate the observed precipitation well when using a resolution of 4.1 km or finer. Although the main flow patterns are similar, finer grids produce more complex patterns that show divergence zones, convergence zones and vortices. Horizontal grid size significantly affects the vertical structure of the convective boundary layer. Stronger vertical wind components are simulated for finer grid resolutions. In particular, noticeable sinking airflows over mountains are captured for those model configurations.

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