A novel high resolution model without open boundary conditions applied to the China Seas: first investigation on tides

We developed a Global Ocean Circulation and Tide Model （GOCTM） with coarse grids in the open deep ocean degrading ‘smoothly’ into the highly resolved China Seas （CS） of refined grids to study the tides and circulation there.GOCTM is based on the framework of the Finite Volume approach for better mass conservation through improved transports across the discrete individual control volume.It also takes a full advantage of the geometric flexibility of unstructured mesh using a realistic global topography including the Arctic Ocean.The CS are given a special focus by refining the unstructured grids,but they are embedded into global domain naturally.Furthermore,GOCTM not only successfully avoids the treatment of the open boundaries,but also optimizes the trade-off between computational cost and model accuracy.Meanwhile,GOCTM is driven by the astronomical tide-generating potential and the secondary tide-generating potential directly,together with the wind stress and heat flux.GOCTM succeeds in reproducing the global eight principal tidal harmonic constants.Particularly,the simulated tidal results in the CS are improved compared to some other regional models with the discrepancy of 3.9 cm for M 2 tide.This idea of GOCTM can also be referred for other regional ocean study.

Inversion of two-dimensional tidal open boundary conditions of M_2 constituent in the Bohai and Yellow Seas

Two-dimensional tidal open boundary conditions of the M2 constituent in the Bohai and Yellow Seas(BYS) have been estimated by assimilating T/P altimeter data.During inversion,independent point(IP) strategy was used,in which several IPs on the open boundary is assumed,values at these IPs can be optimized with an adjoint method,and those at other grid points are determined by linearly interpolating the values at IPs.The reasonability and feasibility of the model are tested by ideal twin experiments.In the practical experiment(PE) after assimilation,the cost function may reach 1% or less of its initial value.Mean absolute errors in amplitude and phase can be less than 5 cm and 5°,respectively,and the obtained co-chart can show the character of the M2 constituent in the BYS.The results of the PE indicate that using only two IPs on the open boundary can yield better simulated results.

Integrability of N-Component Bariev Model Under Open Boundary Conditions

N-component Bariev model for correlated hopping under open boundary conditions in one dimension is studied in the framework of Bethe ansatz method. The energy spectrum and the related Bethe ansatz equations are obtained.

ON THE EXACT HAUSDORFF MEASURE OF A CLASS OF SELF-SIMILAR SETS SATISFYING OPEN SET CONDITION

In this paper, we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition （OSC）. As applications, we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.

Ocean Model Open Boundary Conditions with Volume, Heat and Salinity Conservation Constraints

Open boundary conditions （OBCs） for a regional ocean model that can be integrated stably over a long timeframe, as well as satisfy the volume, heat and salinity conservation constraints, were developed. First, the idea that the inward and outward flux information can be treated separately in the OBCs was adopted. Second, in order to maintain the property that the volume, heat and salinity remains conserved in the simulation domain, conservation constraints were added to the OBCs, and an inverse method utilized to solve the constraint equations. Ideal experiments were designed to investigate the conservation property, and the OBCs were found to work efficiently to maintain the volume, heat and salinity conservation. It was found that simulations were comparable to observations when the OBCs were applied to a regional ocean model.

AN ADAPTIVE OPEN BOUNDARY CONDITION

A significant improvement of the open boundary condition which was originally studied by Engquist and Majda is given.The method given here is applicable without limitation of the angle between the direction of the incident wave and the normal to the boundary.It also improves the precision of the method from order one to two.The test examples show that this method is much better than the method mentioned above.

The High Precision Open Boundary Conditions Designed for Transient Waves

In Rels. [2-4] there is an Adaptive. Open Boundary Condition (AOBC) designed for transient waves which overcomes the limitation of the existing Open BoundaryCondition (OBC) and can be used for the cases of waves with arbitrary incidentangles. In this article a new family of high order AOBC has been designed on the basisof the above mentioned AOBC with the first order.In comparison with all other OBC with the same order, this new family of AOBC has the highest precision.

Ferroresonance at Open Phase Operating Conditions of Power Transformers

The reliability of electric supply to consumers is one of the most important factors that determine the requirements imposed on modem utility companies. This paper presents the results of investigation by computer software of the overvoltages resulting from a ferroresonance conditions in MV networks at open phase operating condition with and without connection to earth on source and load sides of distribution transformer. This overvoltage may reach 4.2 pu on one of the HV side of transformer unswitched phases. The results of the study show that ferroresonance overvoltage may be controlled by replacing fuses with circuit breakers on HV side to ensure switching-off all phases. Insertion of resistor or reactor in the neutral of source and loadsides of the transformer with 5% active load will help in suppressing overvoltages.

Entanglement in an anisotropic spin-1 Heisenberg chain

The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy △ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearestneighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.

A Negative Answer to a Conjecture on Upper Convex Density

Let E be a self-similar set satisfying the open set condition. Professor Xu conjectures in his doctoral degree thesis that if H^8（E） 〈｜E｜^8, then for any x ∈ E, the inequality ^-D^3C（E,x）〉H^8（E）/｜E｜^8holds, where 3 = dimH（E）. The above conjecture is negatively answered in this＇paper.

STUDY ON THE ADJOINT METHOD IN DATA ASSIMILATION AND THE RELATED PROBLEMS

It is not reasonable that one can only use the adjoint of model in data assimilation. The simulated numerical experiment shows that for the tidal model, the result of the adjoint of equation is almost the same as that of the adjoint of model: the averaged absolute difference of the amplitude between observations and simulation is less than 5.0 cm and that of the phase-lag is less than 5.0°. The results are both in good agreement with the observed M2 tide in the Bohai Sea and the Yellow Sea. For comparison, the traditional methods also have been used to simulate M2 tide in the Bohai Sea and the Yellow Sea. The initial guess values of the boundary conditions are given first, and then are adjusted to acquire the simulated results that are as close as possible to the observations. As the boundary conditions contain 72 values, which should be adjusted and how to adjust them can only be partially solved by adjusting them many times. The satisfied results are hard to acquire even gigantic efforts are done. Here, the automation of the treatment of the open boundary conditions is realized. The method is unique and superior to the traditional methods. It is emphasized that if the adjoint of equation is used, tedious and complicated mathematical deduction can be avoided. Therefore the adjoint of equation should attract much attention.

A Note on Upper Convex Density

For a self-similar set E satisfying the open set condition,upper convex density is an important concept for the computation of its Hausdorff measure,and it is well known that the set of relative interior points with upper convex density 1 has a full Hausdorff measure.But whether the upper convex densities of E at all the relative interior points are equal to 1？ In other words,whether there exists a relative interior point of E such that the upper convex density of E at this point is less than 1？ In this paper,the authors construct a self-similar set satisfying the open set condition,which has a relative interior point with upper convex density less than 1.Thereby,the above problem is sufficiently answered.

A THEORETICAL FRAMEWORK FOR THE CALCULATION OF HAUSDORFF MEASURE-SELF—SIMILAR SET SATISFYING OSC

A theoretical framework for the calculation of Hausdorff measure of self-similar sets satisfying OSC has been established.

Pointwise dimensions of general Moran measures with open set condition

Recently Lou and Wu obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.In this paper,we prove that the result is still true under the open set condition.Due to the lack of the strong separation condition,our approach is essentially different from that used by Lou and Wu.We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.

2-D CHARACTERISTICS LIKE MODEL AND OPEN BOUNDARY CONDITION FOR UNSTEADY OPEN CHANNEL FLOW MODELING

A characteristic like model is presented to calculate the two dimensional flow in open channels with finite or infinite dimensions. The comparisons between the computational results and experimental data show the model to be robust in simulating two dimensional unsteady flows. The comparisons also show the effectiveness of the open boundary condition in simulation of wave propagation process in an infinite domain with computational domain of finite dimension.

ANALYTIC METHOD FOR INVERSING THE OPEN BOUNDARY CONDITIONS IN A SEMI-CLOSED SEA AREA—CASE I FOR INVERSING THE TIDAL LEVELS AT OPEN BOUNDARY

In this paper, a computational case was employed to describe thecomputational procedure for inversing the tidal level open boundary conditions using an analyticmethod. The area for finding the solution is a circular area with a circular arc with the openingangle 60°-being the open boundary and the other circular arc being the solid wall boundary.Proceeding from the reestablished elliptic partial differential equation satisfied by the tidallevel function, the extended spectrum method was used to derive the general solution of the equationfor the sea of constant depth, and the impermeable solid wall condition (the second class boundarycondition) and the adequately specified open boundary conditions were then applied to determine theundetermined coefficients of the general solution, thus obtaining the tidal level distributionfunction. In this way, both the first and second class boundary values at the solid wall boundarywere obtained. With the above boundary values as the boundary conditions, the tidal level values atthe open boundary were then inversed by means of the general solution of tidal wave equation. Thevalidity of inversion method could be verified by comparing the inversed tidal level distributionswith the originally specified open boundary values.

An SPH wave-current flume using open boundary conditions

In this paper, a novel numerical wave-current flume is implemented based on the Smoothed Particle Hydrodynamics (SPH) method. The proposed flume is more stable and computationally efficient than conventional ones because it cancels wave paddles or sponge layers while using a non-reflective open boundary condition to simultaneously generate and absorb waves and currents. Numerical tests of nonlinear regular waves interacting with steady currents in the same/opposite directions indicate that the wave-current interactions mainly affect the horizontal velocity distributions in the water body, with little influence on the vertical movements. Water particle trajectories are stretched, and the mass transportation effects are enhanced when the waves and currents are in the same direction. When the directions are opposed, the trajectories, as well as the mass transportation, are compressed, suppressed and even reversed. The applications of solitary wave-current interaction over a submerged bump further show that the relative direction between the wave and the current has remarkable influence on the wave breaking pattern over an underwater obstacle.

An Equivalent Condition for the Self Similar Sets on the Real Line to Have Best Coverings

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.： Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17（2）, 493-502 （2004）].

ON THE HIGH PRECISION ADAPTIVE OPEN BOUNDARY CONDITIONS FOR TRANSIENT WAVES

The adaptive open boundary conditions (AOBC) designed by Chen and Zou for transient waves overcome the limitation of the existing open boundary conditions (OBC) and can be used for the cases of waves with arbitrary incident angles. In this paper a new family of AOBC has been designed on the basis of the AOBC with first order mentioned above. In comparing with all other OBC with the same order, this new family of AOBC has the highest precision. It can be generalized into 3D problems without difficulty and its forms in different curvilinear coordinate systems can be got very easily. The distinguished advantages above mentioned of the AOBC will be discussed in this paper.

The Hausdorff Dimension of Sections

The notion of finite-type open set condition is defined to calculate the Hausdorff dimensions of the sections of some self-similar sets, such as the dimension of intersection of the Koch curve and the line x=α with α∈ Q.