Two numerical models, described in detail herein, have been constructed and used to investigate the formation of secondary vortices in axisymmetrically-forced rotating flows. The particular type of vortex flow examined is that developed in a laboratory vortex simulator where secondary vortices have been produced and extensively studied. The first numerical model generated a collection of steady state, axisymmetric vortex flows based on a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small amplitude, axially asymmetric, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability whereas damping indicated stability. For those flows found to be unstable, the linear perturbations of various azimuthal wavenumbers were analyzed in detail, and from the perturbation growth rates, structures, phase speeds, and energetics, the nature of the instability could be studied. The results of the instability study show that the vortex is stable for the lowest swirl ratios but that above a certain value, instability persists indefinitely. The most rapidly growing wavenumber shifts steadily with increasing swirl from 1 to around 5 in the swirl range investigated. Growth rates were found to be high enough for secondary vortices to form in the laboratory simulator in just a few seconds. Structurally, the perturbation fields were found to have a helical tilt and to be centered near the radius of maximum vertical vorticity in the axisymmetric vortex. They propagated in the same azimuthal direction as the rotation of the axisymmetric flow, but at a somewhat lower angular velocity at the surface. These linear results are all consistent with observed laboratory behavior. From this, it was concluded that linear theory is capable of explaining many important aspects of secondary vortices. An analysis of the perturbation energy equation revealed that at the higher swirl ratios, the perturbation received most of its energy from the deformation of the axisymmetric flow due to the radial distribution of azimuthal velocity, while for low swirl the primary source was from the radial distribution of the vertical velocity. No other component of the axisymmetric vortex ever contributed more than about 25% of these terms.

One result of the search for inexpensive alternative sources of protein has been the rise in interest in aquaculture, the rearing of aquatic organisms under controlled conditions. In this dissertation we examine several management approaches to the efficient rearing of aquatic animals, using mathematical modeling to discover optimal production decisions. In addition we demonstrate the feasibility of simultaneous decision and performance control, providing empirical support for a theoretical extension of traditional variance analysis techniques. The results of three studies are included. In the first we model a situation in which the manager of an aquaculture system must decide when and how many animals to stock initially, how many animals to harvest each period, and when to restock an enclosure in order to maximize contribution. We consider both limited and unlimited growing seasons, solving mixed-integer and linear programs. We examine the effects of technological improvements on production strategies. Consistent improvement in contribution is noted, along with some variation in strategy. In the second study we introduce seasonal variation in revenues and lengthen the growing season. The resulting large-scale real-world mixed-integer problem necessitates the use of a heuristic and two strategies, selective expansion and sieve, in order to achieve a near-optimal solution within a reasonable length of time. In the third study we focus on the uncertainty inherent in the aquaculture environment. We provide empirical evidence of the feasibility of a performance evaluation system which gives explicit consideration to the effects of environmental uncertainty and incorporates intraperiod adaptive behavior on behalf of the individual responsible for implementation of model-specified activities. The system we describe may be used in the simultaneous evaluation of individual and model performances, thus clarifying responsibilities for variances and improving production control.

The central goal of this dissertation is to develop a simple but powerful theory to handle a problem which arises in management situations where an optimally exploited, system at steady state is subjected to a set of continuous, persistent and unpredictable disturbances emanating from the system's environment. Such disturbances drive the system out of steady state. The question that arises in such a situation is whether there exists any additional control which can be imposed on the disturbed system in order to drive it back to the steady state and maintain it there for all future time? We show in this dissertation that such a control is possible provided bounds for the disturbances are known. We develop the additional control using concepts from reachability and the so-called Liapunov's "second method". We further develop some theory concerning certain problems which arise in generating the boundary of the reachable set, ∂R(•) using the controllability maximum principle. In generating ∂R(•) several boundary controls may be used to generate different parts of ∂R(•). We show that all the parts of ∂R(•) are polygonally connected. We also show that for a second-order system if an equilibrium point under constant control is hyperbolic and lies on ∂R(•), it is asymptotically stable. Further, in persistently disturbing a system, it is desirable to have some idea about the boundedness of the disturbed system. If the system is bounded then a boundary can be generated using controllability maximum principle. We give some theory and discussion on how to test such boundedness for linear, quasilinear and some cases of nonlinear systems. The last two chapters of this dissertation show how the theory is applied to a second-order system; in particular to a second-order grazing system.

A model for the hot slab ignition problem is analyzed to determine critical conditions based on the parameters of the system. Activation energy asymptotics, a singular perturbation approach, is applied to the governing equation resulting in a Volterra integral equation of the second kind whose solution represents the temperature perturbation at the surface of the hot slab. The system is said to be supercritical for given parameter values when the temperature perturbation blows up in small finite time, an indication of ignition, or subcritical when the blow up time is large, indicating that heat loss effects overcome the hot slab ignition mechanisms. Comparison principles for integral equations are used to construct upper and lower solutions of the equation. The exact solution as well as the upper and lower solutions depend on two parameters ε, the Zeldovich number a measure of the heat release and λ, the scaled hot slab size. Upper and lower bounds on the transition region, delineating the super-critical from the sub-critical region, are derived based upon the lower and upper solution behavior. The product integration method is used to compute solutions of the Volterra equation for values of ε and λ in the transition region. The computations indicate that a critical curve, λ(c) lying between the analytic bounds, exists.

A procedure for estimating runoff from convective storms in the semiarid Southwest is needed for the design of small hydraulic structures. The aim of this study was to develop and test rainfall and runoff relationships based on the analysis of 12 years of hydrologic data for an 18-square mile experimental watershed. rhe experimental area is divided into four subwatersheds ranging in size from 0.5 to 7.8 square miles, Vegetation and soils are typical of what is encountered in the valley floors of southern Arizona. Rainfall is measured at 29 locations. Isohyetal maps were prepared for all of the storms which lead to the development of a rainfall model that describes the distribution of rainfall in space. An exponential relationship was found to adequately represent the spatial variation of each storm. A single equation for all storms was developed by using a parameter that is related to the storm center depth. The Kolmogorov-Smirnov procedure was used to test the hypothesis that storm center location is governed by chance in areas not influenced by topographic changes. It was found that the assumption which states that convective storm cells are randomly located within valley floors is acceptable. An equation was derived for calculating point rainfall probabilities from raingage network data, The results were based on the random location of storm centers and on an extremal distribution function fitted to storm center depths. The calculated probabilities were found to be significantly higher than the observed probabilities determined from a nearby, long-term U. S. Weather Bureau station. The volume of runoff from small, semiarid watersheds was found to be a function primarily of mean rainfall. In a multiple linear regression model, mean rainfall accounted for 67 to 82 percent of the variance. The use of a time distribution factor which includes the maximum 15-minute intensity reduced the unexplained variance to 11 to 16 percent. Inserting a space distribution variable into the model indicated that storm center location on the watershed was not a significant factor in predicting runoff. An antecedent rainfall index did not produce any significant correlation with runoff from convective storms. For winter frontal storms, however, a four-day antecedent rainfall index was found to be an important factor in oxplaining runoff. It appears that the commonly used Soil Conservation Service method underestimates convective storm runoff for most storm center depths below about three inches. A direct comparison with the multiple regression equation was not possible as this method does not take into account the variability of convective rainfall in time and space. As a means for estimating runoff volumes for ungaged watersheds, a runoff coefficient was defined as the ratio of runoff to effective rainfall (mean rainfall less initial abstractions). It appears that as a first approximation, the runoff coefficient can be considered as being equal to the coefficient in the well known rational formula. There is some evidence to the belief that the runoff coefficient is affected by a storm's time distribution factor. It was demonstrated that runoff volume recurrence intervals can be determined adequately from the rainfall and runoff relationships developed in this study.

The purpose of this study was to develop a stochastic model of the snowfall, snow accumulation and ablation process. Snow storms occurring in a fixed interval were assumed to be a homogeneous Poisson process with intensity X. The snow storm magnitudes were assumed to be independent and identically distributed random variables. The magnitudes were independent of the number of storms and concentrated at the storm termination epochs. The snow water equivalent from all storms was a compound Poisson process. In the model, storms then occurred as positive jumps whose magnitudes equaled the storm amounts. Between storms, the snowpack ablated at a constant rate. Random variables characterizing this process were defined. The time to the occurrence of the first snowpack, generated by the first storm, was a random variable, the first snow-free period. The snowpack lasted for a random duration, the first snowpack duration. The alternating sequence of snow-free periods followed by snowpacks of random duration continued throughout the fixed interval. The snow-free periods were independent and identically distributed random variables as were the snowpack durations. The sum of each snow-free period and the immediately following snowpack duration formed another sequence of independent and identically distributed random variables, the snow-free, snow cycles. The snow-free, snow cycles represented the interarrival times between epochs of complete ablation, and thus defined a secondary renewal process. This process, called the snow renewal process, gave the number of times the snowpacks ablated in the interval. Distribution functions of the random variables were derived. The snow-free periods were exponentially distributed. The distribution function of the snowpack durations was obtained using some results from queueing theory. The distribution function of the first snow-free, snow cycle was derived by convoluting the density function of the first snowfree period and the first snowpack duration. The distribution of the sum of n snow-free, snow cycles was then the n-fold convolution of the first snow-free, snow cycle with itself. The probability mass function of the snow renewal process was evaluated numerically, from a known relationship with the sum of snow-free, snow cycles. The snowpack ablation rate was considered to be a random variable, constant within a season, but varying between seasons. The snowpack durations and snow-free, snow cycles were conditioned on the ablation rate, then unconditional distributions derived. An application of the model was made in the case where snow storm magnitudes were exponentially distributed. Specific expressions for the distribution functions of the random variables were obtained. These distributions were functions of the Poisson parameter X, the exponential parameter of storm magnitudes, Ne l, and the snowpack ablation rate. The snow model was compared with data from the climatological station at Flagstaff, Arizona. Snow storms were defined as sequences of days receiving 0.01 inch or more of snow water equivalent separated from other storms by one or more dry days. Snow storms occurred approximately as a homogeneous Poisson process. Storm magnitudes were exponentially distributed. Empirical distributions of snowpack ablation rates were obtained as the coefficients of a regression analysis of snowpack ablation. Two methods of estimating the Poisson parameter were used. The theoretical distribution functions were compared with the observed. The method of moments estimate generally gave more satisfactory results than the second estimate.

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No description available.

The objective of this study is to develop a statistical model to calculate the effectiveness of an airborne jammer on analog communication and broadcast receivers, such as AM and FM Broadcast Radio and Television receivers. During the development the required power margin in dB, or equivalently, the required linear ratio, between the jammer power and the carrier power at the target receiver input was first determined. Subsequently, using probabilities that the jammer power will exceed the target signal's carrier power, the required power margin was calculated. This power margin was determined by statistical techniques to predict the propagation characteristics of communication and broadcast signals, such as Log-Normal Shadowing, and Small-Scale Fading. From the model, it was determined that it is difficult to achieve high probabilities of exceeding the required jamming margins with a single jammer. Hence, the use of spatial diversity jamming is recommended, that is, using two or more jammers spaced sufficiently far apart from each other, such that their jamming signals at the targeted area are de-correlated due to the differences in their respective angles of arrival.

Recent Chairman of the Joint Chiefs of Staff Instructions (CJCSI) direct the development of new concepts to prioritize linking budgeting and programming for near- and mid-term resource allocation planning. For the Department of Defense (DoD), planning that falls within the Future Years Defense Plan (FYDP) or 0-7 years may be viewed as mid-term. This thesis raises the question of whether these new concepts will work best for the Department of the Navy (DON) or whether another methodology would better fit the Navy's near- and mid-term needs. Further, the thesis asks whether the implementation of the recently promulgated CJCSI instructions would be more disruptive to the DON instead of an alternative modification of what the DON is already using successfully. This thesis also explores the potential value of the newly formed capability planning initiative within the DON. The Navy initiative is compared to private-sector practices to define similarities and to develop additional potentially useful methods. The thesis also explores the potential usefulness of linear programming or mathematical decision modeling for the application of weights and values to relate input variables and relationships to desired outputs. Finally, procurement narrative statements in the FY07 DON budget are analyzed and results, in terms of use of the capability concept, are reported.