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The dynamics of modulated point-vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). For sufficiently close vortices (dipoles) the trajectories of their center-of-mass are shown to correspond to those of a point particle moving freely on a rotating sphere. For finite size vortex pairs, a qualitative similarity to the geodesic dynamics is found. The advection dynamics generated by vortex pairs on a rotating sphere is found to be chaotic. In the short time dynamics we point out a transition from closed to open chaotic advection, which implies that the transport properties of the flow might drastically be altered by changing the initial conditions of the pair on the sphere. Due to spherical topology, for long times, even the open advection patterns are found to gradually cross over to that corresponding to a homogeneous closed mixing. This pattern extends along a zonal band, whereas short term closed mixing remains always bounded to the moving pair.

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This paper is devoted to the analysis of the effectiveness of the use of arable land. This is an issue, which is important for national-level decision makers. The particular calculations are carried out for Hungary, but similar analysis can be made for each country having several parts with different geographical conditions.In general the structure of the use of arable land has been developed in an evolutionary manner in each country. This paper is devoted to the evaluation of the effectiveness of this structure. Some main crops must be included in the analysis such that the land used for their production is a high percentage in the total arable land of the country. From agricultural point of view the question to be answered is whether or not the same level of supply is achievable with high probability on a smaller area. As the agriculture is affected by stochastic factors via the weather, no supply can be guaranteed up to 100 per cent. Thus each production structure provides the required supply only with a certain probability. One inequality corresponding to each crop must be satisfied at the same time with a prescribed probability. The main theoretical difficulty here is that the inequalities are not independent from one another from stochastic point of view as the yields of the crops are highly correlated. The problem is modeled by a chance constrained stochastic programming model such B. Vizvári ( ) Ann Oper Res (2011) 190:325-337 that the stochastic variables are on the left-hand side of the inequalities, while the right-hand sides are constants. Kataoka was the first in 1963 who solved a similar problem with a single inequality in the probabilistic constraint. The mathematical analysis of the present problem is using the results of Kataoka. This problem is solved numerically via discretization.Numerical results for the optimal structure of the production are presented for the case of Hungary. It is shown that a much higher probability, i.e. a more safe supply, can be achieved on a smaller area than what is provided by the current practice.

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