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Harrison E. Cramer

Abstract

The principle of dynamic entrainment is applied to the turbulent-mixing problem. With use of the velocity fluctuations characteristic of turbulent flow and the equation of continuity, an expression is derived for the mass exchange between an accelerated fluid and its environment, under steady-state conditions. A two-stage diffusion mechanism is postulated, in which it is held that the dilution and spreading of aerosols occur in consequence of the inflow and outflow required by continuity in regions of accelerated motion. It is assumed that the inflow and outflow are orderly, that the atmosphere is incompressible, that changes in density are negligible, and that the entrained air is uniformly mixed with the aerosol. Several elementary models are described, and the results obtained by numerical integration in selected cases are discussed. It is indicated that the turbulent-mixing process depends upon the scale of the velocity fluctuations, and upon the ratio of the amplitude of these fluctuations to the mean wind speed.

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Harrison E. Cramer
and
Frank A. Record

Abstract

Power spectra of the eddy-velocity components have been determined at four levels within the layer from 2 to 12 meters under varying conditions of mean wind speed, trajectory and thermal stability. A filtering technique suggested by J. W. Tukey has been used to obtain rough estimates of contributions to the total variance for seven consecutive frequency intervals within the range from about 0.5 to 0.005 cycles per second. At the higher frequencies studied, variances for all three components are approximately equal and equipartition of turbulent energy is thus indicated. Spectra for the u- and v-components appear to be invariant with frequency at the lowest level, and tend to increase slowly with decreasing frequency at the higher levels. The w-spectra at all levels fall off sharply with decreasing frequency, contributions to the vertically-directed energy becoming almost negligible at the lowest frequencies investigated.

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Harrison E. Cramer
and
Frank A. Record

Abstract

Direct measurements have been made of the vertical flux of heat and momentum in the layer from 2 to 12 meters. Eddy velocities were obtained from hot-wire anemometers and light bivanes, mounted at four levels; temperature fluctuations were measured with fast-response thermocouples, mounted at three levels. Data were recorded by taking photographs of indicating dials, at the rate of one exposure per second. Six sets of data have been analyzed, each set corresponding to one period of observation approximately 10 minutes in length. In four sets of data, the flow was over a rough land surface; in one set, the flow came directly from a water (ocean) surface; in the remaining set, the flow was principally over water except for a short land trajectory immediately upwind from the point of observation.

The flux data show a maximum variation from two- to four-fold within the layer. Over land, the shearing stress tends to decrease with height during the day and to increase with height at night; over water, both the heat flux and the momentum flux tend to decrease with height and also are significantly smaller at all levels than over a land surface. Values are presented for the coefficients of eddy viscosity K m, eddy conductivity K h, surface drag C d, and for von Kámán's constant K. For a land surface, the eddy coefficients are approximately equal near the ground; K m increases with height at a slower rate than K h during the day, while at night the reverse is true. Over a water surface, K m is considerably larger than K h at all levels. The values for c d are in good agreement with previous estimates, based on less direct measurements. The rather complicated variation of k with stability and height is discussed; the average of all determinations is approximately 0.4.

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Henry G. Houghton
and
Harrison E. Cramer

Abstract

It is held that entrainment is a necessary dynamic consequence of the vertical stretching of an accelerated convective column. On this basis, equations are developed for the rate of entrainment, the vertical divergence and the lapse rate, for both unsaturated air and cloud air. It is assumed that a steady state exists, the cross section of the rising column is invariant with height, the entrained air is uniformly mixed with the rising air and the environment is at rest. The equations are integrated numerically over height in a number of selected cases. In unsaturated air, entrainment results in a lapse rate which is always greater than the dry adiabatic; if the environmental lapse-rate is superadiabatic, the lapse rate of the rising air is intermediate between the lapse rate of the environment and the dry adiabatic lapse-rate. In a cloud, entrainment results in a lapse rate intermediate between the environmental lapse-rate and the moist adiabatic lapse-rate. The lapse rate of the rising air increases as the relative humidity of the environment decreases. As a result of entrainment, the cloud liquid-water content increases with height at a significantly slower rate than would result from a simple lifting process. A decrease in the humidity of the environment reduces the rate of increase of liquid water with height, but it does not appear possible to “dry out” the cloud even in a dry environment. The horizontal velocity-convergence is found to be of the order of 10−3 sec−1, and the computed vertical velocities in the cloud are in general agreement with those observed by the Thunderstorm Project. It is pointed out that the entrained air may be added through an ordered inflow, by turbulent exchange or by a combination of the two. It is assumed here that an ordered inflow occurs.

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