A two-person, noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem. In this paper, we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron. The decision variables are variously partitioned into continuous and discrete sets. The leader goes first, and through his choice may influence but not control the responses available to the follower. For two reasons the resultant problem is extremely difficult to solve, even by complete enumeration. First, it is not possible to obtain tight upper bounds from the natural relaxation; and second, two of the three standard fathoming rules common to branch and bound cannot be applied fully. In light of these limitations, we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations. A series of heuristics are then proposed in an effort to strike a balance between accuracy and speed. The computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables.

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Twenty-one warm-season heavy-rainfall events in the central United States produced by mesoscale convective systems (MCSs) that developed above and north of a surface boundary are examined to define the environmental conditions and physical processes associated with these phenomena. Storm-relative composites of numerous kinematic and thermodynamic fields are computed by centering on the heavy-rain-producing region of the parent elevated MCS. Results reveal that the heavy-rain region of elevated MCSs is located on average about 160 km north of a quasi-stationary frontal zone, in a region of low-level moisture convergence that is elongated westward on the cool side of the boundary. The MCS is located within the left-exit region of a south-southwesterly low-level jet (LLJ) and the right-entrance region of an upper-level jet positioned well north of the MCS site. The LLJ is directed toward a divergence maximum at 250 hPa that is coincident with the MCS site. Near-surface winds are light and from the southeast within a boundary layer that is statically stable and cool. Winds veer considerably with height (about 140) from 850 to 250 hPa, a layer associated with warm-air advection. The MCS is located in a maximum of positive equivalent potential temperature e advection, moisture convergence, and positive thermal advection at 850 hPa. Composite fields at 500 hPa show that the MCS forms in a region of weak anticyclonic curvature in the height field with marginal positive vorticity advection. Even though surface-based stability fields indicate stable low-level air, there is a layer of convectively unstable air with maximum-e CAPE values of more than 1000 J kg 1 in the vicinity of the MCS site and higher values upstream. Maximum-e convective inhibition (CIN) values over the MCS centroid site are small (less than 40 J kg 1) while to the south convection is limited by large values of CIN (greater than 60 J kg 1). Surface-to-500-hPa composite average relative humidity values are about 70%, and composite precipitable water values average about 3.18 cm (1.25 in.). The representativeness of the composite analysis is also examined. Last, a schematic conceptual model based upon the composite fields is presented that depicts the typical environment favorable for the development of elevated thunderstorms that lead to heavy rainfall.

A 30-yr climatology of the snow-to-liquid-equivalent ratio (SLR) using the National Weather Service (NWS) Cooperative Summary of the Day (COOP) data is presented. Descriptive statistics are presented for 96 NWS county warning areas (CWAs), along with a discussion of selected histograms of interest. The results of the climatology indicate that a mean SLR value of 13 appears more appropriate for much of the country rather than the often-assumed value of 10, although considerable spatial variation in the mean exists. The distribution for the entire dataset exhibits positive skewness. Histograms for individual CWAs are both positively and negatively skewed, depending upon the variability of the in-cloud, subcloud, and ground conditions.

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