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Stratified Flow Past a Hill: Dividing Streamline Concept Revisited

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Abstract

The Sheppard formula (Q J R Meteorol Soc 82:528–529, 1956) for the dividing streamline height \(H_\mathrm{s}\) assumes a uniform velocity \(U_\infty \) and a constant buoyancy frequency N for the approach flow towards a mountain of height h, and takes the form \(H_\mathrm{s}/h=\left( {1-F} \right) \), where \(F=U_{\infty }/Nh\). We extend this solution to a logarithmic approach-velocity profile with constant N. An analytical solution is obtained for \(H_\mathrm{s}/h\) in terms of Lambert-W functions, which also suggests alternative scaling for \(H_\mathrm{s}/h\). A ‘modified’ logarithmic velocity profile is proposed for stably stratified atmospheric boundary-layer flows. A field experiment designed to observe \(H_\mathrm{s}\) is described, which utilized instrumentation from the spring field campaign of the Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) Program. Multiple releases of smoke at \(F\approx 0.3\)–0.4 support the new formulation, notwithstanding the limited success of experiments due to logistical constraints. No dividing streamline is discerned for \(F\approx 10\), since, if present, it is too close to the foothill. Flow separation and vortex shedding is observed in this case. The proposed modified logarithmic profile is in reasonable agreement with experimental observations.

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Acknowledgments

This research was funded by Office of Naval Research Award # N00014-11-1-0709, Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) Program. The authors wish to thank Brent Fields, Kevin Wilcock and Jeff Wintle (Dugway Proving Ground Explosive Division); without their technical support and expertise, the Materhorn smoke-visualization experiments would have not been possible. In addition, the authors wish to thank Michael Carston and Paul Broderick (Dugway Proving Ground Meteorological Division) for their support and invaluable technical contributions to the set-up and maintenance of the large suite of instruments deployed during the MATERHORN Program. We are also grateful for helpful comments provided by three anonymous reviewers, which led to considerable improvement of the paper.

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Authors and Affiliations

  1. Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame, Notre Dame, IN, USA

    Laura S. Leo, Michael Y. Thompson, Silvana Di Sabatino & Harindra J. S. Fernando

  2. Department of Physics and Astronomy, University of Bologna, Bologna, Italy

    Silvana Di Sabatino

  3. Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA

    Harindra J. S. Fernando

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  1. Laura S. Leo
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  4. Harindra J. S. Fernando
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Correspondence to Laura S. Leo.

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Leo, L.S., Thompson, M.Y., Di Sabatino, S. et al. Stratified Flow Past a Hill: Dividing Streamline Concept Revisited. Boundary-Layer Meteorol 159, 611–634 (2016). https://doi.org/10.1007/s10546-015-0101-1

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