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An analytical model of an urban heat island circulation in calm conditions

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Abstract

We obtain an analytical model solution to an idealization of an urban heat island (UHI) circulation—the steady shallow convective flow of a viscous stably stratified fluid over a differentially heated lower boundary without system rotation (no Coriolis force) or background wind. The coupled linearized equations of motion and thermal energy are solved for flows in two-dimensional Cartesian (slab-symmetric) and axisymmetric geometries forced by laterally varying surface buoyancies considered as Gaussian and parabolic arch functions. The scaled problem is free of governing parameters, and the solutions are universal. In all cases the flow is dominated by an in-up-out circulation. The updraft is stronger in the axisymmetric case than in the slab-symmetric case, while the surrounding downdraft, and the inflow and outflow branches of the circulation are stronger in the slab-symmetric case. These differences are explained in terms of the response of the perturbation pressure to the thermal forcing in the different geometries. The scalings themselves provide insight into observations that daytime UHI circulations can be stronger than nighttime circulations despite the relative weakness of urban–rural temperature contrasts during the daytime.

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Notes

  1. Also known as an urban breeze (e.g., [49]), urban breeze circulation (UBC; e.g., [66]), country breeze (e.g., [5]), country wind (e.g., [27]), or buoyancy and turbulence driven atmospheric circulation (BTDAC; e.g., [24]).

  2. Defined as flows in which the frictional terms in the equations of motion are larger than the Coriolis and acceleration terms.

  3. Of course, for \( \nu \) to be large during the day while Pr is small (less than 1), the daytime thermal diffusivity κ must be very large (i.e., exceed the value of \( \nu \)).

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Acknowledgements

The authors thank the anonymous reviewers for their helpful comments.

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Authors and Affiliations

  1. School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK, USA

    Alan Shapiro & Evgeni Fedorovich

  2. Center for Analysis and Prediction of Storms, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK, USA

    Alan Shapiro

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Correspondence to Alan Shapiro.

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Shapiro, A., Fedorovich, E. An analytical model of an urban heat island circulation in calm conditions. Environ Fluid Mech 19, 111–135 (2019). https://doi.org/10.1007/s10652-018-9621-9

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