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Melvin E. Stern

Abstract

When air flows over terrain where the surface temperature varies with position, non-adiabatic heat will be added or subtracted. We consider the time-dependent disturbances induced in a uniform basic current as the result of differential heating on a flat rotating earth.

The problem of obtaining a plausible mathematical-physical description of the heating function is then discussed. A method which was proposed in a previous paper, Stern and Malkus (1953), is re-examined and restated, to clarify the underlying assumptions and the point of departure from the classical theory of the eddy conduction of heat.

With this description of the heating function, it is shown that the mean motions may be specified in terms of an “equivalent mountain.” This depends, in general, on the Coriolis parameter as well as the surface temperature, undisturbed wind speed, and the eddy conductivity of the heated region. The theory is applied to the small-scale sea-breeze problem, and it is shown that, by retaining the linearized advective term in the momentum equation, it is possible to explain the frequently observed phase relation between the diurnal temperature wave and the sea breeze without introducing friction. Additional derived relations for the hodograph suggest tests by means of future observations.

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Joanne Starr Malkus
and
Melvin E. Stern

Abstract

The effect upon a stable atmosphere of a heat source (5–10 km width), which is a specified function of the coordinates, is investigated theoretically. A two-dimensional problem is chosen, with the x-coordinate in the direction of the mean undisturbed surface wind U, and the z-direction vertical. The heat source, a finite-width pulse in x, has maximum amplitude at the ground (z = 0) and decays with height. A steady state, in which the heat supplied by the source is continuously carried away downstream, is assumed, and the equations of motion are linearized by the method of perturbations. By Fourier analysis of the pulse function, the perturbation equations are made separable, and solutions for the streamline flow are obtained. It is shown that “lee waves,” i.e., extended downstream oscillations in the streamlines, occur only if the undisturbed wind or stability undergoes a change in the vertical. The results of the analysis are compared with observations made over Nantucket Island, a flat sandy strip about 5 km in width.

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Melvin E. Stern
and
Joanne Starr Malkus

Abstract

In Part I, the convective motions produced by the flow of a stable air stream over a small flat island were studied when the distribution of heating as a function of the coordinates was assumed. In Part II, the mechanism of the heating is examined. It is found that the heat source obeys an eddy-conduction equation and is established by turbulent eddying in the mixed ground layer. The streamline displacement may be divided into two components, one obeying the equation for air flow over a mountain ridge and the other obeying a heat-conduction equation, the latter component being important only in the near vicinity of the island. An “equivalent mountain” corresponding to the heated island may be specified analytically; it depends only upon the temperature distribution along the surface, the wind speed, the eddy conductivity in the ground layer, and the undisturbed stability. Its amplitude is related to the maximum streamline displacement. The “equivalent mountain” for Nantucket Island is calculated for two extreme observational cases. The complete streamline picture is constructed for several examples of an air stream whose properties remain unaltered to great heights. The basic current possessing a change in stability or wind speed at an upper level is also discussed, and the forecasting of lee waves is related to the development of the mixed ground layer via the height of the “equivalent mountain.” An expression for the sea breeze is also derived.

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