Abstract
A steady-state, one-dimensional, mesoscale, non-linear problem of the airflow over a mountain ridge is considered in the framework of the bulk theory in an attempt to develop a meteorological generalisation of the hydraulic problem formulated and solved previously by Houghton and Kasahara. In essence, only three modifications of their formulation are made in the present paper: (a) inversion strength is introduced and considered as an additional dependent variable; (b) the atmosphere above the flow is assumed to be stably stratified with a constant vertical potential temperature gradient; and (c) the condition requiring the flow to return to its initial state after descending and moving away from the ridge is removed.
The problem is reduced to a transcendental algebraic equation, the solution of which describes all the possible types of flow, including a discontinuous one. A classification and meteorological criterion graphically presented in the form of a map is proposed for the realisation of each type of flow.
If the ridge is higher than some threshold, a jump arises above the windward slope and propagates against the flow. As a result, the initial parameters change to new values for which a secondary steady-state solution exists. An iterative method has been developed to calculate these new initial parameters.
The general features of the different types of flow, including those with a jump and secondary flows, are discussed. It is shown that if the ridge height exceeds some second threshold there are no solutions at all, which means that the flow is totally blocked. Concrete calculated examples are presented of all possible types of steady-state flows including secondary ones.
It is shown that after introduction of these modifications, the hydraulic model of Houghton and Kasahara acquires more real meteorological meaning.
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References
Bacmeister, J. T. and Pierrehumbert, R. T.: 1988, ‘On High-Drag States of Nonlinear Stratified Flow over an Obstacle’, J. Atmos. Sci. 45, 63–80.
Ball, F. K.: 1956, ‘The Theory of Strong Katabatic Winds’, Aust. J. Phys. 9, 373–386.
Berkofsky, L.: 1982, ‘The Behaviour of the Atmosphere in the Desert Planetary Boundary Layer’, Final Scientific Report, AFOSR-82-0285 to U.S. Air Force Office of Scientific Research, Boiling AFB, DC, 30 pp.
Bower, J. B. and Durran, D. R.: 1986, ‘A Study of Wind Profiler Data Collected Upstream during Windstorms in Boulder, Colorado’, Mon. Wea. Rev. 114, 1491–1500.
Brinkmann, W. A. R.: 1974, ‘Strong Downslope Winds at Boulder, Colorado’, Mon. Wea. Rev. 102, 592–602.
Colson, DeVer: 1954, ‘Meteorological Problems in Forecasting Mountain Waves’, Bull. Amer. Meteorol. Soc. 35, 363–371.
Durran, D. R.: 1986, ‘Another look at Downslope Windstorms. Part I: The Development of Analogs to Supercritical Flow in an Infinitely Deep, Continuously Stratified Fluid’, J. Atmos. Sci. 43, 2527–2543.
Durran, D. R. and Klemp, J. B.: 1987, ‘Another Look at Downslope Winds. Part II: Nonlinear Amplification Beneath Wave-Overturning Layers’, J. Atmos. Sci. 44, 3402–3412.
Gutman, L. N.: 1972, Introduction to the Nonlinear Theory of Mesoscale Meteorological Processes, Israel Program for Scientific Translations, Jerusalem 1972, p. 224.
Gutman, L. N.: 1991, ‘Downslope Windstorms. Part I: Effect of Air Density Decrease with Height’, J. Atmos. Sci. 48, 2545–2551.
Gutman, L. N. and Apterman, I.: 1992, ‘Downslope Windstorms. Part II: Effect of External Wind Shear’, J. Atmos. Sci. 49, 1173–1180.
Gutman, L. N. and Berkofsky, L.: 1985, ‘On the Theory of the Well-Mixed Layer Containing a Zero-Order Jump’, Boundary-Layer Meteorol. 31, 287–301.
Gutman, L. N. and Khain, A. P.: 1975, ‘Meso-Meteorological Processes in the Free Atmosphere Governed by Orography’, Atmospheric and Oceanic Physics (Izvestiya Academy of Sciences, USSR) 9(2), 65–70.
Houghton, D. D. and Kasahara, A.: 1968, ‘Nonlinear Shallow Fluid Flow over an Isolated Ridge’, Communications on Pure and Applied Mathematics 21, 1–23.
Lalaurett, F. and Andre, T. C.: 1985, ‘On the Integral Modelling of Katabatic Flow’, Boundary-Layer Meteorol. 33, 135–149.
Long, R. R.: 1954, ‘Some Aspects of the Flow of Stratified Fluids: II’, Tellus 6, 97–115.
Manins, P. C. and Sawford, B. L.: 1979, ‘A Model of Katabatic Winds’, J. Atmos. Sci. 36, 619–630.
Narimousa, S., Long, R. R., and Kitaigorodskii, S. A.: 1986, ‘Entrainment Due to Turbulent Shear Flow at the Interface of a Stably Stratified Fluid’, Tellus 38A, 76–87.
Nieuwstadt, F. T. M. and Glendening, I. W.: 1989, ‘Mesoscale Dynamics of the Depth of a Horizontally Non-Homogeneous Well-Mixed Boundary Layer’, Beitr. Phys. Atmosph. 62(4), 275–288.
Qi, Ying, Zhou Jingnan, and Fu Baopu: 1994, ‘Airflow over a Mountain and the Convective Boundary Layer’, Boundary-Layer Meteorol. 68, 301–318.
Ramenskiy, A. U., Kononenko, S. U., and Gutman, L. N.: 1976, ‘The Stationary Nonlinear One-Dimensional Mesoscale Problem of the Influence of Orography on the Motion of an Air Mass’, Atmospheric and Oceanic Physics (Izvestiya Academy of Sciences, USSR) 12(8), 875–877.
Tennekes, H.: 1973, ‘A Model for the Dynamics of the Inversion above a Convective Boundary Layer’, J. Atmos. Sci. 30, 558–567.
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Gutman, L.N., Burde, G.I. & Apterman, I.Z. A bulk theory of airflow over a mountain ridge allowing for a stable overlying atmosphere. Boundary-Layer Meteorol 80, 375–402 (1996). https://doi.org/10.1007/BF00119424
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DOI: https://doi.org/10.1007/BF00119424