Coastal Lagoons - Chapter 6 ppsx

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 6 L1686_C06.fm Page 231 Monday, November 1, 2004 3:39 PM © 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 L1686_C06.fm Page 232 Monday, November 1, 2004 3:39 PM © 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 C OMPUTATIONAL G RID 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 L1686_C06.fm Page 233 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press 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 C ONTROL V OLUME A PPROACH 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 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. L1686_C06.fm Page 234 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press 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 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 where while the diffusive flux, using the approach of Chapter 3, is given by FIGURE 6.2 Example of one-dimensional (1D) grid. V i V i−1 V i+1 Accumulation Rate = − + () ()VC VC t ii tt ii t∆ ∆ QC QC ii i i tt −− ++ = − () 1 2 1 2 1 2 1 2 * QuA iii−−− = 1 2 1 2 1 2 − () − +       + () − +       − − − − = ++ + + = νν i i ii ii tt ii ii ii tt A CC xx A CC xx 1 2 1 2 1 2 1 2 1 1 2 1 1 1 2 1 () () ** ∆∆ ∆∆ L1686_C06.fm Page 235 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press In these equations, t * is a time interval between t and t + ∆ t , to be defined according to criteria outlined in the next paragraph. is the concentration on the interface between elements i and i – 1 and will be specified later. Combining the three equations, we obtain: (6.1) In order to introduce the Taylor series discretization methods and to analyze stability and accuracy concepts, let us consider a simplified version of Equation (6.1). Consider the particular case of a channel with uniform and permanent geometry and regular discretization. The cross section ( A ), volume ( V ), and discharge are constant. Assume that diffusivity can be considered constant. Under these conditions, Equation (6.1) becomes (6.2) where U is the constant cross-section average velocity and ∆ x is the ratio between the volume and the average cross section. This is the most popular form of the transport equation but, as shown above, it is applicable only to particular conditions. Additional approaches are required to calculate the advective flux, because the concentration is defined at the center of the control volumes and not at the faces. These approaches and their numerical consequences are described in the next sections. FIGURE 6.3 Generic control volume in a 1D discretization. C i−1 C i C i+1 V i+1 V i V i−1 Q i− 1 / 2 Q i+ 1 / 2 ν i− 1 / 2 ν i− 1 / 2 A i− ∆x i−1 A i+ 1 / 2 ∆x i+1 ∆x i 1 / 2 C i− 1 2 () () () ( * * VC VC t QC QC A CC xx A CC x ii tt ii t ii ii tt i i ii ii tt ii ii i + −− ++ = − − − − = ++ + − =− () − () − +       + () − ∆ ∆ ∆∆ ∆ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 1 1 2 νν ++       + = ∆x i tt 1 ) * CC t U CC x CCC x i tt i t ii tt iii tt + −+ = −+ = − = −       + −+       ∆ ∆∆ ∆ 1 2 1 2 11 2 2 * * ν L1686_C06.fm Page 236 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press 6.2.3 N UMERICAL C ALCULATION OF A DVECTION 6.2.3.1 Spatial Approach Three common approaches are used to estimate concentration values at control volume faces: • Linear approach • Upstream stepwise approach • Quadratic upwind approach (QUICK) 6.2.3.1.1 Linear Approach In the linear approach it is assumed that: Assuming a discretization where the grid size is uniform, it is easily seen that this approach generates central differences as obtained using the Taylor series (see Section 6.2.4). 6.2.3.1.2 Upstream Stepwise Approach In this case, it is assumed that the concentration at the left face is This discretization respects the transportivity property of advection. This property states that advection can transport properties only downstream or that information comes only from upstream. The linear approach does not respect this property because volume i will get information of downstream concentration through the average process. The violation of this property can generate instabilities and will create conditions to obtain negative values of the concentration. The upstream discretization avoids this limitation but, as shown in the following paragraphs, it can introduce unrealistic numerical diffusion. 6.2.3.1.3 Quadratic Upwind Approach (QUICK) The quadratic upwind approach, or QUICK scheme, is an attempt at a compromise between respecting the transportivity property and keeping numerical diffusion at low values. In this case, it is assumed that the concentration distribution around a point follows a quadratic distribution centered on the upstream side of the face C Cx C x xx i ii i i ii − −− − = + + 1 2 11 1 ∆∆ ∆∆ QCC QCC iii iii >⇒ = () <⇒ = () − − − 0 0 1 2 1 2 1 L1686_C06.fm Page 237 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press being calculated. For the left face, we obtain Using the Taylor series discretization described in the next paragraph, it can be seen that, in the case of a regular discretization, advection calculated using this approach is third-order accurate, 1 while pure upstream discretization is first-order accurate and the linear approach (central differences) is second-order accurate. The inconvenience of the QUICK discretization is that it requires additional approaches close to the boundaries. This is not a very limiting factor in 1D calculation but it is in 2D or 3D calculations, especially when the geometry is irregular. 6.2.3.2 Temporal Approach In previous paragraphs, spatial discretization was analyzed. A solution was described for the diffusion term and three discretizations were suggested for the advection term but nothing was said about the time level at which the variables used to calculate advection or diffusion are evaluated. Figure 6.4 shows an example of a time evolution of a property C at a point. The curved line shows the continuous evolution and filled circles show values at each time step. Vertical arrows show C values at the beginning and end of a particular time step ∆ t . The flux in that time step is proportional to the product ∆ C ∆ t . Values at the beginning and end of a time step are shown, as well as concentration variation during that time step. The rate of accumulation at this point is proportional to the slope of this line. The slope of this line also gives an idea of the errors associated with the choice of t * . FIGURE 6.4 Visualization of the consequences of temporal discretization. Property evolves within a time step, but values used to calculate flux do not. Time Property value ∆C ∆t 0 140 QCCCC QCCCC iiiii iiiii >⇒ = + − () <⇒ = + − () − −− − −+ 0 0 1 2 1 2 6 8 1 3 8 1 8 2 6 8 3 8 1 1 8 1 L1686_C06.fm Page 238 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press Models with explicit numerical schemes use t * = t , while models with implicit schemes consider t * = t + ∆ t. It can be seen from the figure that when the slope of the curve is positive, explicit models underestimate the advective fluxes, † while when the slope is negative, they overestimate them, introducing (at least) a phase error. Implicit schemes, on the other hand, underestimate or overestimate the fluxes by a value of the same order. The consideration of an intermediate value between t and t + ∆ t generates more accurate fluxes. The next subsection shows that t * = t + 1 / 2 ∆ t (semi-implicit method) gives the maximum accuracy. Values at t * = t + 1 / 2 ∆ t can be obtained by averaging the values of the properties calculated at time t and time t + ∆ t. An increasing number of calculations to perform is the price to pay for accuracy improvement. The next subsection shows that implicit methods have better stability properties than explicit methods. It can be shown that stability properties of the semi-implicit methods are similar to those of implicit methods. Because of their stability and accu- racy properties, semi-implicit methods are the most efficient numerical methods. 6.2.4 TAYLOR SERIES APPROACH Traditionally, discretized equations are obtained from partial differential equations by replacing derivatives with finite-differences obtained using the Taylor series. The Taylor series provides information on the truncation errors arising when replacing derivatives by finite-differences. In contrast, the control volume introduced in the previous subsection gives information about physical approaches used during dis- cretization. When applied correctly, both methods must produce the same discretized equations. In order to introduce the Taylor series discretization methods and to analyze stability and accuracy concepts, let us consider the differential equation correspond- ing to Equation (6.2): (6.3) This equation describes the advection–diffusion transport in a channel with uniform velocity, a permanent geometry, and diffusivity. 6.2.4.1 Time Discretization The Taylor series relates the value of a property in a point (or time instant) with the values of the property in another point and the derivatives in the same point: † In explicit methods the flux during a time step is proportional to the area of the rectangle with side lengths ∆t and C t , while in implicit methods it is proportional to ∆t and C t+∆t . ∂ ∂ + ∂ ∂ = ∂ ∂ C t U C x C x ν 2 2 CC t C t tC t tC t t n C t t i tt i t i t i t i t nn n i t n+ + =+ ∂ ∂       + ∂ ∂       + ∂ ∂       ++ ∂ ∂       + ∆ ∆ ∆∆ ∆ ∆ 22 2 33 3 1 23 0 !! ()L L1686_C06.fm Page 239 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press Truncating this series at the first derivative, we obtain (6.4) This equation states that the resolution of all the terms of the equation at time t allows the calculation of the variable at time t +∆t with first-order precision because the first missing term in the series is multiplied by ∆t. Similarly, we can relate the concentration at time t with the concentration at time t + ∆t: Truncating this series after the first derivative as before, we obtain (6.5) This equation shows that in implicit methods the truncation error is also of the first order, as in explicit methods, although processes are computed at time t + ∆t. The difference between implicit and explicit methods is their stability properties, as described in the following. From the above paragraph, it is expected that explicit and implicit methods should have the same truncation error and it is also expected that the calculation of the derivatives (or fluxes) at the center of the time step must have a smaller truncation error. To demonstrate this, let us use the Taylor series to relate properties at time and with variables at . (6.6) Subtracting the second equation from the first equation, we obtain ∂ ∂       = − + + C t CC t Ot i t i tt i t∆ ∆ ∆() CC t C t tC t tC t t n C t t i t i tt i tt i tt i tt nn n i tt n =− ∂ ∂       + ∂ ∂       − ∂ ∂       ++ ∂ ∂       + + + ++ + + ∆ ∆ ∆∆ ∆ ∆ ∆∆ ∆ ∆ 22 2 33 3 1 23 0 ! ! ()L ∂ ∂       = − + + + C t CC t t i tt i tt i t ∆ ∆ ∆ ∆0( ) t tt+∆ tt+∆/2 CC t C t t C t t CC t C t t C i tt i tt i tt i tt i t i tt i tt ++ + + + + =+ ∂ ∂       + () ∂ ∂         + () =− ∂ ∂       + () ∂ ∆∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ / / / / / 2 2 2 2 2 2 3 2 2 2 2 2 2 2 0 2 2 2 2 ∂∂         + () + t t i tt 2 2 3 0 2 ∆ ∆ / ∂ ∂       = − + () + + C t CC t t i tt i tt i t ∆ ∆ ∆ ∆ /2 2 0 2 L1686_C06.fm Page 240 Monday, November 1, 2004 3:39 PM © 2005 by CRC Press [...]... ∂x 3  i * (6. 7) (6. 8) Subtracting Equation (6. 8) from Equation (6. 7), we get the so-called central difference for the first-order spatial derivative of C: C* − Ci*−1  ∂C  + 0( ∆x )2   = i +1  ∂x  i 2∆ x * (6. 9) From Equation (6. 7), we obtain an expression for a noncentered derivative (right side derivative), while from Equation (6. 8), we obtain a left-side derivative, both with a first-order truncation... 1. 56 1.08 –0.33 –2.29 –3. 76 –3. 26 0.31 0 –0.50 –1.00 –1.13 –0.50 0.97 2.81 3.93 2.94 –1.00 –7.14 –12.54 1 1.00 0.5 –0.50 –1 .63 –2.13 –1. 16 1 .66 5.59 8.52 7.52 0.38 i + 1 i 0 0.50 1.00 1.13 0.50 –0.97 –2.81 –3.93 –2.94 1.00 7.14 12.54 + 2 0 0.00 0.25 0.75 1.31 1. 56 1.08 –0.33 –2.29 –3. 76 –3. 26 0.31 i +3 0 0 0 0 0 0 0 0 0 0 0 0 Total Amount 1 1 1 1 1 1 1 1 1 1 1 1 diffusion In that case Equation (6. 2)... Ratio (pr) 1 2 3 4 5 6 Curonian Lagoon Odra Lagoon Vistula Lagoon Grande-Entrée Lagoon Ria Formosa Lagoon Mar Menor Lagoon 0.007 0.2 86 0.529 1 1 1.081 0.008 0.012 0.004 0.021 0.0 46 0.005 © 2005 by CRC Press L 168 6_C 06. fm Page 2 56 Monday, November 1, 2004 3:39 PM influenced by river run-off The Odra Lagoon is in an intermediate position Both the Grande-Entrée and the Ria Formosa lagoons are totally under... Grande-Entrée Lagoon FIGURE 6. 10 Main gradations of salting factor and its annual average values for some selected lagoons © 2005 by CRC Press L 168 6_C 06. fm Page 257 Monday, November 1, 2004 3:39 PM Salting factor for Darss-Zingst Bodden Chain and Vistula Lagoons Salting factor, dimensionless 1.1 1 0.9 0.8 0.7 0 .6 0.5 Jan Feb Mar Apr May June July Aug Sept Oct Vistula Lagoon Nov Dec Year Darss-Zingst... of its hydrographic features According to this classification, lagoons are divided into three types: choked lagoons, restricted lagoons, and leaky lagoons The type of lagoon is determined by the water exchanges with the adjacent coastal sea, in the presence of tides and wind-driven circulation.3 Related geomorphic © 2005 by CRC Press L 168 6_C 06. fm Page 250 Monday, November 1, 2004 3:39 PM n Pl ai d ie... hemisphere The Venice and Mar Menor lagoons can be represented by lagoons of type C (see Chapter 9.3 for details) Other lagoons may feature a network of channels (Figure 6. 8D), which become dry during hot seasons or during low tidal phases These lagoons can be represented by a number of nodes ( m = 1, M ) connected by links Each link has a length (Lkm ), © 2005 by CRC Press L 168 6_C 06. fm Page 252 Monday, November...  i ∆x (6. 10) C* − Ci*−1  ∂C  + 0( ∆x )   = i  ∂x  i ∆x (6. 11) * * © 2005 by CRC Press L 168 6_C 06. fm Page 242 Monday, November 1, 2004 3:39 PM If Equation (6. 10) is used when the velocity is negative and Equation (6. 11) is used when the velocity is positive, the first derivative is computed using an “upstream method,” since in both cases no downstream information is used Adding Equation (6. 7) and... Popen [m2/km2] 5.97 2.34 36. 5 1.8 51 3.2 L 168 6_C 06. fm Page 255 Monday, November 1, 2004 3:39 PM Leaky lagoon with high fresh run-off influence Leaky lagoon with high marine influence 0.1 5 4 2 0.01 1 6 3 Choked lagoon with high fresh run-off influence Choked lagoon with high marine influence Hypersaline lagoon (Slag>Ssea) Restriction ratio (Pr), dimensionless 1.0 0.001 0 0.2 0.4 0 .6 0.8 Salt ratio (Slag/Ssea),... new time level, C t + ∆t = © 2005 by CRC Press 1 Ct 1 + α ∆t L 168 6_C 06. fm Page 244 Monday, November 1, 2004 3:39 PM Implicit scheme, a = 1 120 concentration 100 analytical solution time step 0.1 time step 0.5 time step 1.0 time step 1.5 time step 2.1 80 60 40 20 0 0 2 4 6 time 8 10 12 FIGURE 6. 6 Solution of the decay equation (Equation (6. 13)) with the implicit scheme with different time steps As can... 2005 by CRC Press L 168 6_C 06. fm Page 251 Monday, November 1, 2004 3:39 PM Land a V, H d, l, h Coastal line b1 b b2 A b m Lkm k a D a bi a Qi b1 Land area b V, H b hi, li,di bi V, H Qi b2 qi Coastal line Q2 d, l, h Q3 Vi bN+1 Qi B C E FIGURE 6. 8 Simple basic descriptions of lagoon shapes during the pre-modeling analysis of the lagoon For example, the Vistula, Curonian, and Kara Bagaz Gol lagoons may be approximated . time level: (6. 7) (6. 8) Subtracting Equation (6. 8) from Equation (6. 7), we get the so-called central differ- ence for the first-order spatial derivative of C: (6. 9) From Equation (6. 7), we obtain. –0.50 –1 .63 0.50 1.31 0 1 5 0 1. 56 0.97 –2.13 –0.97 1. 56 0 1 6 0 1.08 2.81 –1. 16 –2.81 1.08 0 1 7 0 –0.33 3.93 1 .66 –3.93 –0.33 0 1 8 0 –2.29 2.94 5.59 –2.94 –2.29 0 1 9 0 –3. 76 –1.00 8.52. 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