Masayuki Fujihara

Adaptive Q‐tree Godunov‐type scheme for shallow water equations

SUMMARY This paper presents details of a second-order accurate, Godunov-type numerical model of the twodimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe’s flux function is used for the convection terms and a non-linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non-uniform, unlike other formulations given in the literature based on Roe’s approximate Riemann solver. The model is based on hierarchical quadtree (Q-tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth-averaged vorticity. Validation tests include windinduced circulation in a dish-shaped basin, two-dimensional frictionless rectangular and circular dambreaks, an oblique hydraulic jump, and jet-forced flow in a circular reservoir. Copyright

An optimal management strategy for stochastic population dynamics of released Plecoglossus altivelis in rivers

Excessive predation pressure from the waterfowl Phalacrocorax carbo (Great Cormorant) on Plecoglossus altivelis (Ayu) has recently been a severe problem of river environment in Japan. Local fishery cooperatives are currently suffering from economic difficulties due to decrease of the fish catch of P. altivelis. Local fishery cooperatives and municipalities have been enthusiastically trying to develop countermeasures that can effectively reduce the predation pressure; however, their effectiveness and efficiency have not been systematically quantified well. This aim can be achieved with the help of an appropriate mathematical model. In this paper, based on a pure death process, a practical stochastic control model for population dynamics of released P. altivelis in river environment under predation pressure from P. carbo, harvesting by human, and environmental fluctuations is proposed. Finding an optimal management strategy ultimately reduces to solving a 2D Hamilton–Jacobi–Bellman equation, which is performed with a finite element scheme. Its application to a Japanese river environment successfully computes the optimal management strategy that is consistent with the reality. Numerical sensitivity analysis of the presented mathematical model is also performed for comprehension of dependence of the optimal strategy on the model parameters.

Numerical Simulation of a Hamilton-Jacobi-Bellman Equation for Optimal Management Strategy of Released Plecoglossus Altivelis in River Systems

A stochastic differential equation model for population dynamics of released Plecoglossus altivelis (Ayu) in a river system subject to feeding damage by Phalacrocorax carbo (Great Cormorant) and fishing activity by human is proposed. A stochastic optimal control problem to maximize the sum of the cost of countermeasure to prevent the feeding damage and the benefit of harvesting the fish is formulated, which ultimately reduces to solving a Hamilton-Jacobi-Bellman equation. Application of a Petrov-Galerkin finite element scheme to the equation successfully computes the optimal management strategies for the population dynamics of P. altivelis in a real river system and ecological and economical indices to verify them.

Verifying optimality of rainfed agriculture using a stochastic model for drought occurrence

It may be paradoxical but subsistence rainfed agriculture is the predominant source of food in Sub-Saharan Africa where the production uncertainty is associated with the stochastic nature of rainfall. This paper attempts to comprehend the rationale of this situation by a mathematical approach. Considering the level of drought severity as the zero-reverting Ornstein–Uhlenbeck process, optimality of rainfed agriculture is investigated in the context of stochastic control theory. Occurrence of drought terminating growth of crops is modelled with the concept of first exit time. A stochastic control problem allowing for virtual cost of irrigation, water stress to crops, and benefits of farming is formulated with irrigation effort as the control variable. The Hamilton–Jacobi–Bellman equation governing the optimal control is studied to identify the set of cost functions optimizing rainfed agriculture in an inverse problem approach. Data and information were collected in the coastal savanna agro-ecological zone of Ghana, to identify model parameters, formulate the stochastic control problem, solve the inverse problem, and then verify optimality of rainfed agriculture. The results indicated that rainfed agriculture is not optimal when the crop is more tolerant to water stress.

Application of Stochastic Control Theory to Biophysics of Fish Migration Around a Weir Equipped with Fishways

A weir installed along a river cross-section potentially serves as a physical barrier that prevents fishes from migrating toward upstream. Many rivers in the world encounter this severe and ubiquitous ecological issue. The objective of this paper is to present a biophysical application of stochastic control theory to upstream fish migration in river reaches where movements of individual fishes are considered as horizontally 2-D controlled processes. Identifying the biological and ecological objective function to be maximized with the dynamic programming principle leads to a 2-D nonlinear elliptic equation referred to as the Hamilton-Jacobi-Bellman Equation (HJBE). Solving the HJBE leads to an optimal swimming velocity field of individual fishes in water flows. Utilizing appropriate differential equations associated with the HJBE enables us to efficiently and consistently compute attraction ability of fishways installed at a weir from a statistical viewpoint. An application of the present mathematical model to upstream migration of juvenile Plecoglossus altivelis (Ayu) around a recently renovated weir in Hii River, San-in area, Japan is carried out in order to assess attraction ability of its associated fishways.

Robust optimal diversion of agricultural drainage water from tea plantations to paddy fields during rice growing seasons and non-rice growing seasons

We investigated seasonal water quality changes on different land uses in a small agricultural watershed dominated with rice and tea crops. Observed NO3–N in tea plantations and tea drainage channels was sufficiently high enough to cause downstream water pollution throughout the year. On contrary, significant NO3–N reductions were observed in abandoned and active paddy fields, and were highest in active paddy fields during rice growing seasons. Accordingly, a question is posed whether deliberately diverting NO3–N contaminated drainage water from tea plantations to paddy fields would be effective a strategy for reducing downstream water pollution. NO3–N ratios reduced in paddy fields are however neither distinct nor known in advance, and this uncertainty could consequently lead to further environmental risks like NO3–N leaching and production of greenhouse N2O gas. We assume that NO3–N reduction rates of paddy fields and drainage canals are constrained in an ellipsoidal set of uncertainty. A portfolio optimisation problem is formulated to determine optimal rates of diversion from tea drainage channel into paddy fields. The problem is numerically solved in robust optimisation approach by maximising minimum NO3–N reduced under uncertainty. An application demonstrated that robust approach allocates different optimal rates of diversion to hedge the risk of possible malfunctioning of NO3–N reduction processes. Overall worst-case NO3–N reduction is highest in rice growing season, due to increased drainage water allocations to active paddy fields. The robust optimisation approach has potential to support decision-making processes for reducing NO3–N pollution and water demand pressure on water resources, and cost of rice production thereof.

SUB-DARCY-SCALE MODELING OF NON-UNIFORM FLOW THROUGH POROUS MEDIA WITH MIXED WETTABILITIES

A physically-based conceptual model is developed using spatially distributed sub-Darcy-scale clusters in a regular grid to reproduce both the hydraulic properties and the non-uniform wetting and drainage fronts through porous media mixed with hydrophilic and hydrophobic grains. In the model, cellular automaton-like algorithm is employed to route water- and air-intrusion paths through the mixed porous media. Water retention characteristics, which are ones of the macroscopic properties of porous media, are estimated by accumulating a certain numbers of clusters after reaching equilibrium states in a wetting or drainage process. In the experimental part of this study, normal (hydrophilic) and artificially hydrophobized glass spheres, whose diameter is about 0.2 mm, are used as materials, and hydraulic properties of homogeneously mixed glass spheres with various rates are measured. The measured data are compared to model estimation, and the availability of the model is shown.

Active Water Quality Management in Rural Small Watersheds

Diverse land and water uses are found in rural areas of Monsoon regions. Based on field studies in Japan and Bangladesh, water quality management problems are considered at small watershed levels. A study site in Japan is chosen in a hilly area. Upland tea plantations in the site are the major source of nitrate pollution to the downstream water bodies because of heavy fertilizer application. Utilizing paddy fields and irrigation tanks as buffer of the polluted water is feasible, but it risks vulnerable aquatic ecosystems. Another study site is in the floodplains of Bangladesh. Countless ponds are used in the rural communities for domestic, animal watering and fishery purposes. Introduction of comparatively warm groundwater into the ponds during winter seasons may accelerate the growth of fish. However, groundwater depletion is the major concern. The concept of Markov decision processes is applied to find the optimal actions for the two different water quality management problems.

ESTIMATION OF PERMEABILITY OF POROUS MEDIA WITH MIXED WETTABILITIES USING PORE-NETWORK MODEL

Water flow through porous media is involved with physical factors such as size and configuration of pores, contact angle of grain surface, and connectivity of pores. To deal with these factors, pore-scale modeling is essential. Pore aggregate referred to as pore-network is extracted from randomly packed spherical grains with the modified Delaunay tessellation method. Water flow in a pore-network is formulated in terms of a network flow problem in hydraulics, in which friction and local losses such as contraction and enlargement of pipes are explicitly treated, while those are implicitly included in the hydraulic conductance of capillary tubes in existing works. Through the numerical experiments, it was confirmed that the effect of the local loss on the hydraulic conductivity could be negligible. Furthermore, the relative permeabilities of porous media with mixed wettabilities along the primary drainage and imbibition processes were estimated.

A Stochastic Impulse Control Model for Population Management of Fish-Eating Bird Phalacrocorax Carbo and Its Numerical Computation

Feeding damage from a fish-eating bird Phalacrocorax carbo to a fish Plecoglossus altivelis is severe in Japan. A stochastic impulse control model is introduced for finding the cost-effective and ecologically conscious population management policy of the bird. The optimal management policy is of a threshold type; if the population reaches an upper threshold, then taking a countermeasure to immediately reduce the bird to a lower threshold. This optimal policy is found through solving a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). We propose a numerical method for HJBQVIs based on a policy iteration approach. Its accuracy on numerical solutions and the associated free boundaries for the management thresholds of the population, is investigated against an exact solution. The computational results indicate that the proposed numerical scheme can successfully solve the HJBQVI with the first-order computational accuracy. In addition, it is shown that the scheme captures the free boundaries subject to errors smaller than element lengths.