Approximated Equations of Urban Climate by Scale Analysis

Simplification of basic fluid equations is the first step to provide a sensible boundary for an urban climate simulation problem. In this paper, we tried to provide a suitable framework for ongoing researches in that field. Current research is a proposition for climate simulation models of urban areas, which have large growth in size and remarkable population increase in them, and results can be applied for other purposes like avionic science, too. Basic conservation equations are converted to characteristic form, and scale analysis was performed based on desired data to calculate their values. Desired terms to be retained in the conservation of mass, momentum, were horizontal velocities terms, and in the conservation equations of heat and mass of other atmospheric contaminants was the rate of change of potential temperature and the mass of air contaminant, respectively. In order to evaluate the results of simplified equations, observational data for 14 different cities (Atlanta, Chicago, Dallas, Houston, Miami, New Orleans, New York, Los Angeles, Philadelphia, Phoenix, Washington DC, Toronto, Vancouver and Montreal), which are selected based on a parent project, gathered. Observational data are provided by National Renewable Energy Laboratory (NREL) named Typical Meteorological Year (TMY3) for US cities and Environment Canada named Canadian Weather for Energy Calculations (CWEC) for Canadian cities on hourly basis. Data are analyzed statistically on monthly and yearly basis to find out mean, maximum, minimum, mode and standard variation of important parameters in fluid equations; wind speed, temperature and pressure. Data showed that pressure and density change is negligible for different urban areas and different time of year, they both have maximum standard deviation occurred in Montreal no more than 0.06 kg/m3 and 8.66 mbar which with respect to mean value of 1.25 kg/m3 and 1010 mbar have less than 5% and 1% change, respectively. Wind and temperature were highly variable as expected by sunlight availability during days and its absence during nights. Montreal had the most unstable weather condition between those urban areas and it had maximum error for this simplification. Vancouver and New Orleans had lowest wind speed mode which made them more stable than other selected urban areas and less generated error. The most important result of this paper is negligible variation of pressure in horizontal surface. We verified it by simulation of Montreal, as the most unstable city, using WRF (version 3.3). Simulation also verified that the maximum variation of the hydrostatic pressure in the modeling domain is less than 1%. Although there is no preferred geometric shape exists for urban areas, scale of interest was assumed to be less than a square with 30km×30km sides which applies for most of metropolitan around the world. Resulted equations are simplest form that should be treated for parameterization before using numerical methods for climate simulation.