Coastal Lagoons - Chapter 6

6 Modeling Concepts Boris Chubarenko, Vladimir G. Koutitonsky, Ramiro Neves, and Georg Umgiesser CONTENTS 6.1 Introduction 6.2 Numerical Discretization Techniques 6.2.1 Computational Grid 6.2.2 Control Volume Approach 6.2.3 Numerical Calculation of Advection 6.2.3.1 Spatial Approach 6.2.3.1.1 Linear Approach 6.2.3.1.2 Upstream Stepwise Approach 6.2.3.1.3 Quadratic Upwind Approach (QUICK) 6.2.3.2 Temporal Approach 6.2.4 Taylor Series Approach 6.2.4.1 Time Discretization 6.2.4.2 Spatial Discretization 6.2.5 Stability and Accuracy 6.2.5.1 Introductory Example 6.2.5.2 Stability 6.2.5.3 The Need for a Fine Resolution Grid 6.3 Pre-Modeling Analysis and Model Selection 6.3.1 Hydrographic Classification 6.3.1.1 Morphometric Parameters 6.3.1.2 Hydrological Parameters 6.3.2 Description of Forcing Factors 6.3.2.1 General Hierarchy of Driving Forces 6.3.2.2 Water Budget Components 6.3.2.2.1 Surface Evaporation Budget 6.3.2.2.2 Ocean–Lagoon Exchange Budget 6.3.2.3 Heat Budget 6.3.3 Pre-Estimation of Spatial and Temporal Scales 6.3.3.1 Flushing Time 6.3.3.1.1 Integral Flushing Time 6.3.3.1.2 Local Flushing Time 6.3.3.2 Surface and Bottom Friction Layers 6.3.3.3 Time Scales of Current Adaptation © 2005 by CRC Press 6.3.3.3.1 Wind Driven Current 6.3.3.3.2 Equilibrium Current Structure 6.3.3.3.3 Gradient Flow Development 6.3.3.4 Wind Surge 6.3.3.5 Seiches or Natural Oscillations of a Lagoon Basin 6.3.3.6 Wind Waves 6.3.3.7 Coriolis Force Action 6.3.4 Objectives of Modeling 6.3.5 Recommendations for Model Selection 6.3.5.1 Selection Possibilities for Hydrodynamic and Transport Models 6.3.5.2 Possible Simplifications in Spatial Dimensions 6.3.5.3 Possible Simplification in the Physical Approach 6.3.5.4 Possible Simplification According to the Task To Be Solved 6.3.5.5 Computer, Data, and Human Resources 6.4 Model Implementation 6.4.1 Bathymetry and the Computational Grid 6.4.1.1 Laterally Integrated Models 6.4.1.2 Horizontal Resolution Models 6.4.2 Initial Conditions 6.4.3 Boundary Conditions 6.4.4 Internal Coefficients: Calibration and Validation 6.5 Model Analysis 6.5.1 Model Restrictions 6.5.1.1 Physical Restrictions 6.5.1.2 Numerical Restrictions 6.5.1.3 Subgrid Processes Restrictions 6.5.1.4 Input Data Restrictions 6.5.2 Sensitivity Analysis 6.5.3 Calibration 6.5.4 Validation Acknowledgments References Note: The term modeling is used in this chapter in the sense of “numerical modeling.” Physical modeling, conceptual modeling, or numerical model-ing will only be used explicitly in relevant cases. 6.1 INTRODUCTION In Chapter 3, the concept of transport equation was introduced, starting from the concepts of control volume and accumulation rate of a property inside this control volume. Diffusive and advective fluxes were also defined to account for exchanges between the control volume and its neighborhood, and the concept of evolution equation was introduced by adding sources and sinks to the transport equation. A “model” is © 2005 by CRC Press built on the same concepts. Its implementation requires the definition of at least one control volume, the calculation of the fluxes across its boundary, and the calculation of the source and sinks using values of the state variables inside the volume. The number of dimensions of the model depends on the importance of relevant property gradients. The simplest model is the “zero-dimensional” model. In this model, there is no spatial variability, and only one control volume needs to be considered. At the other extreme of complexity is the three-dimensional (3D) model, which is required when properties vary along the three spatial dimensions. Whatever the number of its dimensions, a model must include the following elements: · Equations · Numerical algorithm · Computer code The order of the items in this list can also be considered the order of their chrono-logical development. Hydrodynamic equations are based on mass, momentum, and energy conservation principles, which were presented in Chapter 3. These have been known for more than 100 years. Actually, numerical algorithms used to solve hydro-dynamic models were attempted even before the existence of computers. The analytical equations and the numerical algorithms developed before the existence of computers allowed the rapid development of modeling starting in the 1960s, when computers were made available to a small scientific community. Since that time, models and the mod-eling community have evolved exponentially. Modern integrated computer codes have done more for interdisciplinarity than 100 years of pure field and laboratory work. The number of implementations of a model to solve various problems increases the knowledge of the range of validity of the model equations. The accuracy of the numerical algorithm is better known and confidence in the results increases. At that time, the major source of errors in the results is the existence of mistakes in the data files. Once the model equations, algorithms, and results are validated, the next priority is the development of a user-friendly graphical interface that simplifies the use of the model by nonspecialists. This reduces the errors of input files and simplifies the checking of those files. This chapter presents the concepts and methodologies used to build models and to understand their functioning. 6.2 NUMERICAL DISCRETIZATION TECHNIQUES Computers can solve only algebraic equations. Analytic equations, integral or dif-ferential, must be discretized into algebraic forms. The procedure followed depends on the form of the analytical equation to be solved. The control volume approach is best for the integral form of evolution equations, while the Taylor series is best suited for differential equations. 6.2.1 COMPUTATIONAL GRID The calculation of fluxes across a control volume surface is simpler if the scalar product of the velocity by the normal to each elementary area (face) composing that © 2005 by CRC Press FIGURE 6.1 Example of a grid for a three-dimensional (3D) computation. Two vertical domains are used. The upper domain uses a sigma coordinate. The lower one uses a Cartesian. surface remains constant in each of them. The control volume that makes that calculation simpler must have faces perpendicular to the reference axis. If rectangular coordinates are used, the control volume generating the simpler discretization is a parallelepiped. In the case of a large oceanic model, a suitable control volume will have faces laying on meridians and parallels. In depth-integrated models, also called two-dimensional or 2D horizontal mod-els, the upper face of the control volume is the free surface and the lower face is the bottom. In three-dimensional or 3D models, a control volume occupies only part of the water column and its shape depends on the vertical coordinate used. In coastal lagoons, Cartesian and sigma-type coordinates (or a combination of both) are the most commonly used coordinates. The ensemble of all control volumes forms the computational grid. In finite-difference-type grids, control volumes are organized along spatial axes and a struc-tured grid is obtained. In contrast, typical finite-element grids are nonstructured. The latter are more difficult to define, but they are more flexible, thus allowing some variability in the spatial resolution. Figure 6.1 shows an example of a very general finite-difference-type grid using several discretizations in the vertical direction. A system can be considered one-dimensional (1D) if properties change only along one physical dimension. In this case, control volumes can be aligned along the line of variation and one spatial coordinate is enough to describe their locations. Properties are considered as being constants across control volume faces perpendic-ular to that axis. Fluxes across the faces not perpendicular to that axis are null or have no net resultant. 6.2.2 CONTROL VOLUME APPROACH Control volumes used in numerical models have the same meaning as the derivation of the evolution equation in Chapter 3. A discretization is adequate if it generates a simple calculation algorithm while maintaining the accuracy of the results. The © 2005 by CRC Press Vi−1 Vi Vi+1 FIGURE 6.2 Example of one-dimensional (1D) grid. simpler calculation is obtained if properties can be considered as being constant inside the control volume and along parts of its surface. To make this possible without com-promising accuracy, the control volume must be as small as possible; a fine-resolution grid is needed. In a 1D model, properties can be stored into 1D arrays (vectors). Adjacent elements of a generic element i are i – 1 on the left side and i + 1 on the right side (Figure 6.2). The length of a control volume must be small enough to allow properties in its interior to be represented by the value at its center. In that case, equations deduced in Section 3.2 apply and the rate of accumulation in volume i will be given by Accumulation Rate = (VC )t+ t − (VC )t where t is the time step of the model. This equation is simplified if the volume remains constant in time. This is not the case in most coastal lagoons subjected to changing winds and it is certainly not the case in tidal lagoons. Exchanges between i volume and neighboring ones are accounted for by advec-tive and diffusive fluxes. Their calculation requires some hypotheses. Let us consider Figure 6.2 and define the distances between the faces (spatial step) and the location points where other auxiliary variables are defined as shown in Figure 6.3. The net advective gain of matter to volume i is given by ( i−12 i−12 − Q+12 i+12 )t=t* where Q = u A while the diffusive flux, using the approach of Chapter 3, is given by 2 2 2 t=t* t=t* i i−1 i+1 i 2 i−12  12( xi + xi−1) i+12 i+12  12( xi + xi+1) © 2005 by CRC Press ... - tailieumienphi.vn