Reply by author to Dr. Naidu’s discussion

Geophysics ◽  
1971 ◽  
Vol 36 (3) ◽  
pp. 618-618
Author(s):  
D. J. Gendzwill
Keyword(s):  
Density Distribution ◽  
Gravity Model ◽  
Fourier Transformation ◽  
Exact Expression ◽  
Inverse Solution ◽  
Density Contrast ◽  
Closed Expression ◽  
One Dimensional ◽  
Particular Solution ◽  
Density Pattern

I appreciate Dr. Naidu’s interest in my paper. He has omitted some detail in the equations (such as the factor G) but the sense of his argument is perfectly clear, and I agree that the method of Fourier transformation is more general than my particular solution. In fact, Novosolitskii (1965) has presented the inverse solution for any horizontal density distribution in a slab. Nevertheless, it seems to me that the existence of a closed exact expression for the model I discussed is of some interest in that it presents a unique formula for an elementary gravity model. The closed expression may be manipulated to derive characteristic interpretation curves such as those presented. The gradational density contrast models may be superimposed to produce almost any arbitrary one‐dimensional density pattern in a slab.

scholarly journals An Example of a Plane Shock of Variable Strength

1963 ◽  
Vol 13 (4) ◽  
pp. 297-302 ◽  
Author(s):  
P. Smith
Keyword(s):  
Density Distribution ◽  
Plane Shock ◽  
One Dimensional ◽  
Variable Strength ◽  
Polytropic Gas ◽  
Particular Solution ◽  
Inhomogeneous Gas

AbstractA particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.

scholarly journals Revisiting Twomey's approximation for peak supersaturation

2014 ◽  
Vol 14 (19) ◽  
pp. 25901-25930
Author(s):  
B. J. Shipway
Keyword(s):  
Lookup Table ◽  
Accurate Calculation ◽  
Cloud Droplets ◽  
Closed Expression ◽  
One Dimensional ◽  
Additional Information ◽  
Droplet Nucleation ◽  
Physically Based ◽  
Thermodynamic Conditions ◽  
Independent Variable

Abstract. Twomey's seminal 1959 paper provided lower and upper bound approximations to the estimation of peak supersaturation within an updraft and thus provides the first closed expression for the number of nucleated cloud droplets. The form of this approximation is simple, but provides a surprisingly good estimate and has subsequently been employed in more sophisticated treatments of nucleation parametrization. In the current paper, we revisit the lower bound approximation of Twomey and make a small adjustment which can be used to obtain a more accurate calculation of peak supersaturation under all potential aerosol loadings and thermodynamic conditions. In order to make full use of this improved approximation, the underlying integro-differential equation for supersaturation evolution and the condition for calculating peak supersaturation are examined. A simple rearrangement of the algebra allows for an expression to be written down which can then be solved with a single lookup table with only one independent variable for an underlying lognormal aerosol population. Multimode aerosol with only N different dispersion characteristics require only N of these one-dimensional lookup tables. No additional information is required in the lookup table to deal with additional chemical, physical or thermodynamic properties. The resulting implementation provides a relatively simple, yet computationally cheap and very accurate physically-based parametrization of droplet nucleation for use in climate and NWP models.

IV.—Expansion of a Finite One-dimensional Gas Cloud into a Vacuum

1953 ◽  
Vol 64 (1) ◽  
pp. 57-70
Author(s):  
A. G. Mackie
Keyword(s):  
Density Distribution ◽  
Analytical Solutions ◽  
Initial Density ◽  
Initial Distribution ◽  
One Dimensional ◽  
Gas Streams ◽  
Subsequent Motion ◽  
Gas Cloud ◽  
Initial Density Distribution ◽  
Stationary Period

SynopsisAn investigation is made of the motion of a one-dimensional finite gas cloud which is initially at rest and is allowed to expand into a vacuum in both directions. The density of the gas at rest is assumed to rise steadily and continuously from zero at the boundaries to a maximum in the interior of the cloud.If the subsequent motion is continuous, it is completely specified by analytical solutions in seven different regions of the x-t plane joined together along characteristics. The motion of one of the boundaries is discussed, and conditions found for it to have (i) an initial stationary period or (ii) a final constant velocity of advance into the vacuum. The gas streams in both directions from a dividing point at zero velocity. This point ultimately tends to the mid-point of the initial distribution.The possible breakdown of the continuity of the motion is discussed, and a condition on the initial density distribution found for shock-free flow to be maintained.

scholarly journals The constrained gravity model with power function as a cost function

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
Bablu Samanta ◽  
Sanat Kumar Mazumder
Keyword(s):  
Cost Function ◽  
Gravity Model ◽  
Power Function ◽  
Transportation Problem ◽  
Trip Distribution ◽  
Total Cost ◽  
Numerical Example ◽  
Particular Solution ◽  
Generalized Cost ◽  
The Given

A gravity model for trip distribution describes the number of trips between two zones, as a product of three factors, one of the factors is separation or deterrence factor. The deterrence factor is usually a decreasing function of the generalized cost of traveling between the zones, where generalized cost is usually some combination of the travel, the distance traveled, and the actual monetary costs. If the deterrence factor is of the power form and if the total number of origins and destination in each zone is known, then the resulting trip matrix depends solely on parameter, which is generally estimated from data. In this paper, it is shown that as parameter tends to infinity, the trip matrix tends to a limit in which the total cost of trips is the least possible allowed by the given origin and destination totals. If the transportation problem has many cost-minimizing solutions, then it is shown that the limit is one particular solution in which each nonzero flow from an origin to a destination is a product of two strictly positive factors, one associated with the origin and other with the destination. A numerical example is given to illustrate the problem.

One-Dimensional Density Distributions of Powder Media in Dies Subjected to Multishock Compactions

1988 ◽  
Vol 110 (4) ◽  
pp. 355-360 ◽  
Author(s):  
Y. Sano
Keyword(s):  
Shock Wave ◽  
Density Distribution ◽  
Friction Force ◽  
The Other ◽  
Wall Friction ◽  
Uniform Density ◽  
Simple Type ◽  
One Dimensional ◽  
Density Distributions ◽  
The One

A theoretical attempt to clarify the reason why the compacts of powder media have uniform density distributions as the density of the compacts becomes high, is made for the compaction of the copper powder medium of a simple type by punch impaction. Based on the one-dimensional equation of motion including the effect of die wall friction force, there are two main factors which influence the density distribution of the medium during the compaction process; one is the propagation of the shock wave passing through the medium, while the other is the friction force between the circumferential surface of the medium and the die wall. The equation reveals that the effect of the force increases little as the density becomes high as a result of the repetitive traveling of the shock wave between the punch and plug. The propagation or more definitely the repetitive traveling, on the other hand, increasingly unformalizes the density distribution during the process as the number of the traveling increases. Owing to the aforementioned effects of the two factors on the density distribution during the process, the high density compacts become uniform.

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