U.S. Geological Survey

U.S. Department of the Interior

A Semi-Implicit, Three-Dimensional Model 

for Estuarine Circulation

Open-File Report 2006–1004

In cooperation with the Interagency Ecological Program 

for the San Francisco Bay–Delta Estuary

A Semi-Implicit, Three-Dimensional Model 

for Estuarine Circulation

By Peter E. Smith

U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2006–1004

Prepared in cooperation with the

Interagency Ecological Program for the San Francisco Bay−Delta Estuary
U.S. Department of the Interior

Dirk Kempthorne, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2006

For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal Center

Denver, CO 80225

For more information about the USGS and its products:

Telephone: 1-888-ASK-USGS

World Wide Web: http://www.usgs.gov/

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report.

Suggested citation: Smith, P.E., 2006, A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation: 

U.S. Geological Survey Open-File Report 2006–1004, 176 p.
Preface

This report mostly is a republication of a Ph.D. thesis prepared by the author for the University of 

California, Davis, in 1997. Only minor changes in the original thesis were made; these changes did 

not change the substance of the report. The reference list and the discussions of the state of 

knowledge in 3-D estuarine modeling have not been upgraded beyond 1997. The intent of 

republishing this work is to make its content available in an online format as part of the U.S. 

Geological Survey Open-File Report series. The version of the computer model described in this 

report is applicable only to simple rectangular geometries. Since 1997, it has been entirely 

recoded using the latest (Fortran 95) programming enhancements and generalized for applications 

to estuaries that have arbitrary geometries, including islands and barriers, and multiple open 

boundaries. The model has been used successfully in several applications to different parts of the 

San Francisco Bay–Delta Estuary, California. Although the model was designed mostly to apply to 

estuaries, the University of California, Davis, has successfully applied it to studies of lakes (Clear 

Lake and Lake Tahoe, California) also. Additional publications describing the model and its 

applications will be forthcoming.
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v

Contents

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective, Scope, and Organization of this Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Progress on Three-Dimensional Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 The Present [1997] Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2. Governing Equations and Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Instantaneous-Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Dissolved Species Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.4 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Mean-Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Turbulence Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1.1 Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1.2 Eddy Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Treatment of the Pressure Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1 Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1.1 Kinematic Surface Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1.2 Dynamic Surface Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.2 Bottom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2.1 Kinematic Bottom Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.2.2 Dynamic Bottom Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.3 Shoreline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Scaling Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.1 Horizontal Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6.2 Vertical Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3. Layer Averaging the Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Salt Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Equation Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4. Finite-Difference Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
vi

4.2 Semi-Implicit One-Dimensional Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Two-Level Semi-Implicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

4.2.2 Three-Level Semi-Implicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

4.3 Semi-Implicit Three-Dimensional Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1 Semi-Implicit Leapfrog Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80

4.3.1.1 Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.1.3 Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1.4 Salt Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Semi-Implicit Trapezoidal Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92

5. Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 One-Dimensional Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 Hydrograph Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94

5.2.1.1 Two-Level Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.1.2 Three-Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.1.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.2 Tidal Flow in a Closed Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.2.1 Two-Level Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2.2.2 Three-Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3 Three-Dimensional Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.1 Seiching in a Rectangular Basin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.1.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Appendix A - Coriolis Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Appendix B - Equation of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Appendix C - The σ Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Appendix D - Expansion of Explicit Finite-Difference Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Appendix E - Computation of the Vertical Eddy Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
vii

Figures

Figure 1.1. Sketch of density-driven (gravitational) circulation in an estuary 

caused by the longitudinal density gradient.................................................................... 4

Figure 2.1. Sketch showing orientation of Cartesian coordinate system with 

corresponding velocity components................................................................................. 18

Figure 2.2. Sketch defining the free-surface elevation ............................................................... 31

Figure 2.3. Sketch showing velocities for mass flux balance on a surface control 

volume in three dimensions................................................................................................ 34

Figure 2.4. Sketch showing viscous stresses on a surface layer of a water body 

affected by a wind stress with components τxs and τys................................................ 35

Figure 2.5. Sketch showing flow over the bottom boundary of an estuary with 

roughness elements of varying sizes................................................................................ 37

Figure 2.6. Sketch showing velocities for mass flux balance on a bottom control volume 

in three dimensions.............................................................................................................. 38

Figure 2.7. Sketch showing viscous stresses on a bottom layer of a water body with 

boundary stress components τxb and τyb ........................................................................ 39

Figure 2.8. Grids showing horizontal stretching of coordinates using two regions in both 

the x and y directions............................................................................................................. 47

Figure 2.9. Plan view of an estuary shown in (A) physical space and (B) computational space 

using the transformation of Boericke and Hall (1974) .................................................... 48

Figure 2.10. Orthogonal curvilinear grid used by Blumberg and Herring (1987) to model a 

portion of the California coast............................................................................................ 49

Figure 2.11. Nonorthogonal curvilinear grid used by Johnson and others (1991, 1993) to 

model Chesapeake Bay....................................................................................................... 50

Figure 2.12. Sketch of a multilayered system........................................................................................ 52

Figure 2.13. Three-dimensional model layering schemes using 10 layers to resolve the 

deep water channel ............................................................................................................. 54

Figure 3.1. Multilevel model computational grid showing the stacked layers and the 

location of variables............................................................................................................. 56

Figure 3.2. Diagram of a 3-dimensional horizontal velocity profile approximated by layeraveraged values ................................................................................................................... 58

Figure 4.1. Definition sketch for 1-dimensional channel ................................................................... 70

Figure 4.2. Two-time-level staggered grid for the 1-dimensional model........................................ 71

Figure 4.3. Three-time-level staggered grid for the 1-dimensional model ..................................... 75

Figure 4.4. Computational stencils for (A) explicit leapfrog differencing method and 

(B) implicit (Crank-Nicolson type) leapfrog differencing method used in the 

three-time-level 1-dimensional scheme........................................................................... 76

Figure 4.5. The space-staggered horizontal grid used in the 3-dimensional model showing 

the location of variables...................................................................................................... 80

Figure 4.6. Multilevel staggered grid for the 3-dimensional model. ................................................ 81

Figure 4.7. Representation of five-diagonal coefficient matrix ........................................................ 89

Figure 4.8. Illustration of the "natural" ordering scheme used in forming the matrix system 

of equations for the free-surface elevation........................................................................ 89

Figure 5.1. Channel characteristics and initial flow conditions for the 1-dimensional 

hydrograph test problem..................................................................................................... 95

Figure 5.2. Hydrographs computed with the two-level, semi-implicit scheme using a 

large spatial step size of 5,000 feet and time step size of 5 minutes............................ 96

Figure 5.3. Base solution of the hydrograph problem shown at 125-minute intervals during 

the simulation........................................................................................................................ 98

Figure 5.4. Graphs showing error measures using the two-level semi-implicit scheme on the 

hydrograph test problem.....................................................................................................100

Figure 5.5. Graph showing two-level scheme solutions for the hydrograph problem using 

three different values for space steps and a time step of 5 minutes...........................102
viii

Figure 5.6. Graph showing solution using the two-level scheme on the verge of going 

unstable when using only one iteration............................................................................103

Figure 5.7. Graphs showing error measures using the two-level semi-implicit scheme on the 

hydrograph test problem.....................................................................................................105

Figure 5.8. Graph showing solution using two iterations with the two-level scheme 

that is stabilized, and greatly improved, with an additional iteration ..........................107

Figure 5.9. Graphs showing error measures using the three-level semi-implicit scheme 

on the hydrograph test problem for different values of the computational

space interval........................................................................................................................108

Figure 5.10. Graphs showing error measures using the three-level semi-implicit scheme 

on the hydrograph test problem for different values of the computational

time interval ...........................................................................................................................110

Figure 5.11. Graph showing comparison of computed hydrographs with the leapfrog 

semi-implicit scheme and the two-level semi-implicit scheme ...................................112

Figure 5.12. Graph showing comparison of solutions to the hydrograph problem with the 

leapfrog semi-implicit scheme and the four-point model of Schaffranek and 

others (1981) using a space step of 5,000 feet and a time step of 5 minutes 

for all simulations .................................................................................................................114

Figure 5.13. Channel characteristics and initial flow conditions for the 1- dimensional 

tidal test problem..................................................................................................................115

Figure 5.14. Graph showing water surface elevations at six locations along the

tidal channel..........................................................................................................................116

Figure 5.15. Graph showing solution of the tidal problem for water surface elevation and 

velocity at the location x = 180,000 feet ............................................................................117

Figure 5.16. Graphs showing solutions for water surface elevation and velocity computed 

with the two-level semi-implicit scheme using a space step of 6,000 feet 

and a time step of (A) 60 minutes and (B) 30 minutes ....................................................118

Figure 5.17. Graph showing solution for volumetric transport using the two-level 

semi-implicit scheme...........................................................................................................119

Figure 5.18. Graphs showing solutions using one iteration of the two-level scheme with 

(A) advection terms turned on and (B) advection terms turned off .............................122

Figure 5.19. Graphs showing solutions using the three-level semi-implicit, leapfrog scheme 

(no trapezoidal step) (A) with no smoothing and (B) with smoothing..........................124

Figure 5.20. Graphs showing solutions using the three-level semi-implicit scheme for the 

tidal test problem..................................................................................................................125

Figure 5.21. Graphs showing comparison of error measures between the two-level scheme 

with two iterations and the three-level leapfrog-trapezoidal scheme with one 

iteration of the trapezoidal step .........................................................................................128

Figure 5.22. Diagram of (A) the test problem for an oscillating wave (seiche) in a rectangular 

basin and (B) the horizontal computational grid for the numerical solution ..............130

Figure 5.23. Graphs showing computed and analytical solutions for the 3-dimensional (3-D)

seiching test problem assuming an ideal (non-viscous) fluid and using

the 3-D, explicit, leapfrog scheme for the numerical solution......................................132

Figure 5.24. Graphs showing computed and analytical solutions for the 3-dimensional (3-D)

seiching test problem assuming an ideal (non-viscous) fluid and using

the 3-D, semi-implicit, leapfrog scheme for the numerical solution............................133

Figure 5.25. Graphs showing 20-hour time history of computed and analytical solutions 

for the seiching of an ideal fluid.........................................................................................134

Figure 5.26 Graphs showing mid-basin solutions for the 3-dimensional seiching problem 

using an ideal fluid and three different time steps for the semi-implicit 

leapfrog scheme...................................................................................................................135

Figure 5.27. Graphs showing mid-basin solutions for the 3-dimensional seiching problem 

using an ideal fluid and three different time steps for the semi-implicit 

leapfrog-trapezoidal scheme .............................................................................................136

Figure 5.28. Sample velocity distribution after 0.6 hour of simulation for the 3-dimensional 

seiching problem using a viscous fluid.............................................................................138
ix

Figure 5.29. Graphs showing computed solution for the 3-dimensional seiching problem 

using a viscous, homogeneous fluid .................................................................................138

Figure 5.30. Graphs showing simulated 20-hour time history for seiching of a viscous, 

homogeneous fluid...............................................................................................................139

Figure 5.31. Graph showing velocity profiles for the 3-dimensional seiching problem 

using a viscous, homogeneous fluid in a 44-kilometer basin and a

vertical eddy viscosity defined by a mixing length formula ..........................................141

Figure 5.32.Graph showing velocity profiles for the 3-dimensional seiching problem 

using a viscous, homogeneous fluid in a 44-kilometer basin and a

constant vertical eddy viscosity .......................................................................................141

Figure 5.33. Graphs showing computed solution of the 3-dimensional seiching problem 

using a viscous fluid of nonhomogeneous density.........................................................142

Figure A.1. Estuarine reference frame shown fixed to the earth’s surface at latitude φ

and rotating about the earth’s axis with angular velocity Ω.........................................161

Figure A.2. The earth’s rotation causes a centripetal acceleration that affects the local 

acceleration of gravity at the earth’s surface .................................................................162

Figure C.1. Definition sketch for the estuarine free surface, bottom, and total depth in a 

z-coordinate system.............................................................................................................167

Figure C.2. Schematic of a σ-grid shown in (A) z-coordinate space and 

(B) σ -coordinate space ......................................................................................................168
x

Tables

Table 2.1. Dimensionless numbers in 3-dimensional equations ........................................................ 44

Table 2.2. Relative magnitude of the terms in the momentum equation........................................... 45

Table 2.3. Relative magnitude of the terms in the salt equation ........................................................ 45

Table 5.1. Definition of error measures for the hydrograph test problem........................................ 97

Table 5.2. Ranges of Courant numbers for hydrograph problem simulations using 

5 minutes as a time step .................................................................................................... 99

Table 5.3. Error measures using the two-level semi-implicit scheme on the hydrograph problem 

with the number of iterations ranging from 1 to 10 .......................................................102

Table 5.4. Ranges of Courant numbers for hydrograph problem simulations using a 

space step of 2,000 feet .....................................................................................................104

Table 5.5. Definition of error measures for the tidal test problem.....................................................120

Table 5.6. Ranges of Courant numbers for tidal problem simulations using 6,000 feet as a 

space step............................................................................................................................121
xi

Acronyms and Abbreviations

ADI, alternating-direction implicit

CFL criterion, Courant-Friedrich-Lewy criterion

ELM, Eulerian-Lagrangian Method

H.O.T., higher order terms

MICCG, modified incomplete Cholesky conjugate-gradient 

NSPCG, non-symmetric preconditioned conjugate-gradient 

PCGM, preconditioned conjugate-gradient method

RMS, root mean square 

RMSe, root mean square error

UCD, University of California at Davis

USGS, U.S. Geological Survey

1-D, one-dimensional

2-D, two-dimensional

3-D, three-dimensional

cm, centimeter

cm/sec, cm/s, centimeter per second

cm

2

/sec, centimeter squared per second

ft, foot

ft

3

/s, cubic foot per second

km, kilometer

kg/m, kilogram per meter

kg/m

3

/cm, kilogram per cubic meter per centimeter

m, meter

m/s, m/sec, meter per second m3

/s, cubic meter per second

s, second

sec-1, per second
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Abstract

A semi-implicit, finite-difference method for the numerical solution of the three-dimensional equations for circulation in estuaries is presented and tested. The method uses a three-time-level, leapfrog-trapezoidal scheme that is essentially second-order

accurate in the spatial and temporal numerical approximations. The three-time-level scheme is shown to be preferred over a twotime-level scheme, especially for problems with strong nonlinearities. The stability of the semi-implicit scheme is free from any 

time-step limitation related to the terms describing vertical diffusion and the propagation of the surface gravity waves. The scheme 

does not rely on any form of vertical/horizontal mode-splitting to treat the vertical diffusion implicitly. At each time step, the 

numerical method uses a double-sweep method to transform a large number of small tridiagonal equation systems and then uses 

the preconditioned conjugate-gradient method to solve a single, large, five-diagonal equation system for the water surface elevation. The governing equations for the multi-level scheme are prepared in a conservative form by integrating them over the height 

of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent variables so that the depthintegrated continuity equation that is used in the solution for the water surface elevation is linear. Volumetric transports are computed explicitly from the momentum equations. The resulting method is mass conservative, efficient, and numerically accurate. 

1. Introduction

1.1 Background

Estuaries are a valuable natural resource. Each benefits the economy of the surrounding region, is important as habitat for 

living creatures, and is cherished for scenic and recreational qualities. The exhaustible nature of estuarine resources requires that 

they be afforded a high level of environmental protection. As centers of population growth, commerce, and industrial development, 

estuaries frequently are subjected to change related to the human activities that take place within and around their shorelines. Activities such as altering freshwater inflow, dredging channels, reclaiming tidal wetlands, discharging municipal and industrial wastes, 

and accidentally spilling toxic chemicals can degrade the quality of estuarine waters and adversely affect the biological populations 

that are part of the estuarine ecosystem. An evaluation of the environmental effects of an estuarine change usually requires an

understanding of the physical processes of water circulation and mixing. The distribution and transport of estuarine salts, sediments, contaminants, and certain biological organisms (for example, plankton and larval fish) are governed by circulation and mixing. Water replacement rates (flushing times) in estuaries are determined by the cumulative effects of long-term circulation patterns 

and mixing processes.

A Semi-Implicit, Three-Dimensional Model for

Estuarine Circulation

By Peter E. Smith
2 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The application of three-dimensional (3-D), hydrodynamic models for studies of circulation and mixing in estuaries and 

coastal seas is a relatively recent development. Over the last quarter century, advances in the design of numerical methods and the 

availability of more powerful computers have stimulated an increase in the use and development of 3-D models. Studies of Chesapeake Bay by Blumberg and Goodrich (1990) and Johnson and others (1993), Hudson-Raritan Estuary by Oey and others (1985a, 

b, c) and Blumberg and Galperin (1990), Delaware Bay by Galperin and Mellor (1990a, b) and Walters (1992, 1997), Galveston 

Bay by Wang (1994) and Berger and others (1994, 1995), Tampa Bay by Hess (1994), Puget Sound by Chu and others (1989), 

Massachusetts Bay by Blumberg and others (1993), San Francisco Bay by Smith and Cheng (1990) and Cheng and Casulli (1996), 

and Upper Narragansett Bay by Muin and Spaulding (1997b) are examples of applying a 3-D model to an estuary or a semienclosed bay in the United States. Many additional applications have taken place on estuaries in Europe, Asia, and Canada where

efforts toward developing and improving multidimensional, numerical models for simulating shallow-water hydrodynamics have 

been even more significant than in the United States. With no real slowing in sight to the continued improvement and availability 

of low-cost computing facilities, the trend toward increasing use of 3-D models is likely to continue.

A significant trend, accompanying that in modeling, has taken place in estuarine hydrography toward the increased use and 

development of improved field instruments for measuring currents and salinity in estuaries. Instruments, such as shallow-water 

acoustic Doppler current profilers and fast-response conductivity-temperature-depth profilers, now make it possible to obtain field 

data on the vertical structure of velocity and salinity in estuaries that are more accurate and have greater vertical resolution than 

could be obtained previously for a reasonable cost. These advances in field measurements fulfill a critical need for data appropriate 

for skill assessment of 3-D models; many of the existing applications of 3-D models have been based on insufficient data to supply 

reasonable confidence in the predictive ability of the model.

Because there exists considerable temporal and spatial variability in the currents and water properties of most estuaries, field 

measurements alone are rarely dense enough in space and time to describe the circulation and mixing processes adequately; therefore, models are needed to interpret sparse observations. For studies of long-term transport in estuaries, one of the greatest needs 

is for predictions of water currents that are averaged in such a way as to filter out the tidal oscillations; these tidally averaged currents are referred to as residual currents. Three-dimensional models can be particularly useful for deriving complete distributions 

of residual currents (residual circulation) and can employ either Eulerian (averaging at a fixed point) or Lagrangian (averaging 

based on particle tracking) methods. The only other practical alternative to numerical modeling for estuarine investigations is physical scale modeling, which has well-known shortcomings involving both flexibility and cost.

Water motions in estuaries result primarily from the combined forcing of the tides, wind stress at the free surface, rotation of 

the Earth, and density gradients, caused primarily by salinity gradients. These driving forces are balanced by internal and bottom 

frictional forces. The simulation of 3-D estuarine flows is based on partial differential equations that describe the conservation of 

mass, momentum, and salt1 for a turbulent, incompressible flow with a time-dependent free surface. Water depths in estuaries are 

generally much smaller than the wave length of the tidal oscillations on the free surface, so the flow is classified as shallow water. 

In such flows, the change of vertical velocities with time (vertical accelerations) is extremely small, and the vertical equation of 

motion is replaced by the hydrostatic approximation.

1

If temperature simulations are considered, then an additional statement for conservation of thermal energy is required.
1. Introduction 3

In the past, some hydrodynamic studies of estuaries have used two-dimensional (2-D) models in which the 3-D governing 

equations are averaged over either the vertical dimension (vertically averaged models) or the lateral dimension (laterally averaged 

models). The operation of 2-D models is less computer intensive than 3-D models, and 2-D models generally are easier to program. 

Studies using 2-D models also can be less time consuming than studies using 3-D models because the need to parameterize and 

calibrate either the vertical or horizontal mixing processes is avoided. The parameterization of vertical mixing, in particular, is one 

of the most challenging problems in the application of 3-D models to density stratified flows. Although 2-D models are useful in 

some applications, they are generally limited by fairly restrictive assumptions. Vertically averaged models require the estuarine 

density profile to be well mixed vertically. Estuaries typically undergo significant periods of time when they are stratified by freshwater inflows or weak tidal mixing. A vertically averaged model applied to a stratified estuary can compute fictitious horizontal 

density gradients between waters of different depths that cause significant errors in the model predictions and are difficult to detect; 

for this reason, applications of vertically averaged models, even to weakly stratified systems, should be avoided. Laterally averaged 

models require hydrodynamic variables that are homogenous across the estuarine cross section, which in a strict sense rarely exists 

in real estuaries. Only in estuaries of very small aspect ratio (the ratio of estuarine width to length) should the assumption of lateral 

homogeneity be considered, and even then the validity of the assumption should be checked.

Because significant variations in currents and salinity occur in all three spatial dimensions in most estuaries, 3-D models are

needed. For example, tidal currents in estuaries that flow in opposite directions at different depths or travel in one way near a shoreline and in the opposite way near the center of a channel are not uncommon. The density gradients that occur in estuaries are severe 

enough to affect the longitudinal and transverse circulations and the vertical mixing. Today hydraulic engineers increasingly are 

being asked to use models as part of chemical or biological studies to analyze and to predict the transport and mixing of contaminants or biological organisms. In these studies, a model is most valuable when it correctly simulates the actual physical processes 

causing the transport and mixing of substances and organisms in the estuary under investigation. Because of the different length 

scales that typically characterize the vertical and horizontal dimensions in most estuaries, the processes of transport and mixing 

can be separated into those components that vary horizontally but are vertically uniform, and those that vary vertically as well as 

horizontally. The first category includes residual flows driven by the tides interacting with the estuarine geometry, river inflows, 

depth-integrated wind-driven flows, and tidal trapping by shoreline irregularities or side embayments (Fischer and others, 1979). 

The second category includes vertical variations in wind-driven flows, vertical turbulent diffusion of mass and momentum, mixing 

caused by differential velocities over the water column depth (shear flow dispersion), and the vertical density-driven (or gravitational) circulation (fig. 1.1). If both horizontal and vertical processes of transport and mixing are important in an estuary, a 3-D 

model must be used. If the density-driven circulation is important, the 3-D model equations for the circulation and the salt distribution must be coupled and solved simultaneously. The coupling of equations results in significantly increased running costs of a 

3-D model, but usually coupling is required in estuarine modeling. Ultimately, constructing a model that is capable of correctly 

simulating all the important mechanisms causing transport and mixing in any real estuary is a formidable challenge, but recent 

progress in 3-D modeling is moving closer to that goal.
4 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

ρ2 ρ1 >

q

u

z1

z2

p = Pressure

Water surface

EXPLANATION

u = Velocity

A

Water surface

p1 z1 p ( ) 2 z2 ( )

ρ = Water density g = Acceleration of gravity

q = Flow rate z = Distance measured

 vertically downward

 from the water surface

B

Landward

Seaward

p = ρ gz

ρ1

ρ2

Figure 1.1. Density-driven (gravitational) circulation in an estuary caused by the longitudinal density 

gradient (ρ2 - ρ1). 

The pressure distribution on the faces of a control volume are shown in (A) based on the hydrostatic law 

(p = ρgz). In the surface layers of the water column, the water surface slope causes the net pressure 

force to be seaward. In the bottom layer, the effect of the longitudinal density gradient on the pressure 

distribution causes the net pressure force to be landward. The resulting velocity profile is two-layered, 

as shown in (B). The importance of the density-driven circulation as a transport mechanism in many estuaries is one reason why a 3-dimensional model is needed. (All quantities shown are tidally-averaged.)
1. Introduction 5

1.2 Objective, Scope, and Organization of this Work

The objective of this report is to describe the development and testing of a new finite-difference method for the numerical 

computation of 3-D flow in estuaries. The numerical formulation is believed to be an advance over techniques used for existing 

3-D hydrodynamic models for estuarine and coastal seas. Because field-scale simulations of estuaries using 3-D models are still

constrained by the limitations of modern-day computers, special effort was made to develop an efficient computational scheme 

that will be useful for lengthy (seasonal) simulations, but without loss of numerical accuracy. 

Referring to the computer program here as a model is somewhat premature, despite the use of that term in the title and elsewhere in this report. A model in the engineering sense usually implies a computer program that is ready for applications to real 

systems. The present computer program is not that far advanced.2 The program is tested in three dimensions on a closed rectangular 

basin of constant depth only; the nontrivial task of generalizing the coding to treat estuaries of irregular geometry with open boundaries remains to be accomplished. The speedy accomplishment of this task, however, is planned, and no real difficulties are anticipated.3

 The computer program is intended for eventual use on real systems and was designed and coded with that end result in 

mind.

The motivation for developing the numerical model described here is eventual use in 3-D (three-dimensional) hydrodynamic 

simulations of San Francisco Bay and Delta to study the effects of varying amounts of freshwater inflow on circulation and mixing 

in the estuary. Because the emphasis of this research is applied, more discussion is included that is related to the equation development and the assumptions that are built into the model than would typically be included in a strictly mathematical treatise.

The report is organized into seven chapters. In the next section of this chapter, the progress in 3-D modeling over the last 

quarter century is reviewed; following that is a discussion of the new 3-D model formulation and its advantages over those already 

in use. In Chapter 2, the governing equations, boundary conditions, and assumptions and approximations for the estuarine model 

are discussed thoroughly. In Chapter 3, the model equations are developed for a horizontally layered grid system by averaging the 

3-D governing equations over each layer. The layer-averaged formulation of the equations helps to ensure that mass and momentum are conserved in the model. Chapter 4 details the finite-difference formulation used in the model, which is based on a semiimplicit method. The semi-implicit numerical method is introduced in section 4.2 using two different schemes to solve equations

for unsteady flow in one dimension; then the complete details for the semi-implicit discretization of the equations in three dimensions are given beginning in section 4.3. Chapter 5 presents the numerical experiments used to test the model, starting with two 

cases in one dimension followed by one case in three dimensions. The report concludes with a summary (Chapter 6) and the list 

of references (Chapter 7).

1.3 Acknowledgments

The research for this report was supported under a cooperative agreement between the U.S. Geological Survey and the 

Interagency Ecological Program for the San Francisco Bay−Delta Estuary. Special thanks are due Randy Brown (retired) of the 

California Department of Water Resources and Jim Arthur (retired) of the U. S. Bureau of Reclamation for their role in providing 

the support from their respective agencies. I also thank Henry Wong of the U.S. Bureau of Reclamation who helped with some of 

the computer graphics and original typing of the report.

2

Note that "present" refers to 1997 (see Preface).

3

As of 2006, this task has been completed. The model has been applied to several real estuaries and lakes and tested using a suite of analytical solutions.
6 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

As noted in the preface, this report is mostly a republication of the author’s Ph.D. thesis prepared for the University of 

California at Davis (UCD) in 1997. I thank my thesis committee chair at UCD, Professor Bruce E. Larock, for his advice and 

encouragement during the course of the research and for his valuable reviews of drafts of the manuscript. Professor Ian P. King

also provided valuable review comments on the final draft. 

I thank Ralph Cheng of the USGS for introducing me to the field of estuarine hydrodynamics and for generously sharing his 

knowledge of numerical modeling. It was through Ralph Cheng that I met Professor Vincenzo Casulli (University of Trento, Italy) 

while Vincenzo was on sabbatical leave to the USGS. From Vincenzo I learned much about the semi-implicit numerical scheme; 

I thank him very much for his generous advice and help. I also thank John Donovan of the USGS for his skillful help with some 

of the typing and preparation of the figures for the original thesis. 

1.4 Progress on Three-Dimensional Modeling

The first 3-D models used to simulate tidal motions in estuaries and coastal seas appeared in the early 1970s (Heaps, 1972; 

Leendertse and others, 1973; Sündermann, 1975). In the quarter century since that time, significant advances have been made in 

the fundamental numerical algorithms and equation formulations used in these models. The models of today are more realistic and

numerically efficient than those from a quarter century ago.

The earliest 3-D circulation models actually were developed in the 1960s for applications to oceans and lakes. The well known 

ocean model by Bryan (1969) and the lake model by Liggett (1969) were based on an assumption known as the rigid-lid approximation in which the free surface is held constant and the surface gravity waves (including ocean tides and lake seiches) are filtered 

out of the solutions. By eliminating the fast-moving gravity waves, these models were able to employ large time steps so that longterm simulations of large regions could be done economically. For studying large-scale oceanic circulation on a coarse numerical 

grid, rigid-lid models generally are adequate and are still being developed for modern applications (for example, Haidvogel and

others, 1991; Gerdes, 1993). For lakes, rigid-lid models should only be used in studies focusing on time periods much longer than 

the dominant seiching period of the lake (Sheng, 1986b). Sheng and others (1978) compared a rigid-lid model with a model that 

solves directly for the variable free surface (free-surface model) and showed the rigid-lid model gave poor results for periods of 

active seiching of Lake Erie under spatially and temporally varying winds. A study by Huang and Sloss (1981) used a rigid-lid 

model for Lake Ontario and obtained reasonably good results for the mean monthly circulation.
1. Introduction 7

In many estuarine modeling studies, the free-surface tidal motions are themselves of primary interest, and the rigid-lid approximation clearly is not suitable. In modeling studies for which the interest is in residual motions, Nihoul and Ronday (1975) and 

Durance (1976) pointed out that the tidal motions must be taken into account because of the significant driving force they produce 

on the residual circulation through nonlinear interactions. Thus in nearly all estuarine and coastal modeling studies, a variable free 

surface is an essential feature of the model.

Leendertse and others (1973) and Leendertse and Liu (1975, 1977) were the first to develop a 3-D model for estuaries and 

coastal seas incorporating a fully time-varying free-surface location and using standard finite-difference grid boxes in all three 

dimensions. This model was noteworthy, not only because it was developed first, but because it was very complete in the equation 

formulation, including nonlinear friction and advection, the effects of the Earth’s rotation, horizontal shear stresses, arbitrary bottom topography, coupled simulation of salt and hydrodynamics, density-gradient forcing, and a physically realistic vertical turbulence parameterization (Leendertse and Liu, 1977). The details of the model formulation were thoroughly described in the series

of Rand Corporation reports, which benefitted future investigators in the field. The significant drawback of the Rand model was

its explicit,4 finite-difference scheme that limited the size of the time step to the time a surface gravity wave takes to travel between 

two adjacent horizontal grid points—a limitation referred to as the Courant-Friedrich-Lewy (CFL or Courant) stability condition

for the gravity waves. When using a high resolution numerical grid in an estuary with areas of deep water, this limitation can be 

very severe. As an example, in an application of the Rand model to San Francisco Bay using a horizontal grid size of 

Δx = Δy = 1 km, Leendertse and Liu (1975) used a time step of only 15 seconds. Because the 12.4-hour (M2) tide is the dominant 

period in San Francisco Bay, a time step of up to one-half hour would theoretically be adequate for temporal resolution of the 

physics.

4

An explicit numerical scheme is one in which the unknown hydrodynamic variables at any spatial point at time level n+1 may be computed entirely from 

known values at neighboring points from one or more previous time levels. In these schemes, each unknown value can be related to known values by a single 

algebraic equation and found directly (explicitly). In contrast, implicit schemes are ones in which the unknown variables at an entire row or mesh of points are 

obtained by the simultaneous solution of a system of algebraic equations. In the first version of the Rand model (Leendertse and others, 1973), all terms in the 

finite-difference equations were treated explicitly. In a later version of the Rand model (Leendertse and Liu, 1977), the vertical diffusion terms were converted to 

an implicit treatment for stability reasons.
8 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Freeman and others (1972) presented a free-surface model for modeling the Great Lakes. This model generally resembled the 

Rand model except that the vertical coordinate z was transformed into a new “normalized” coordinate called sigma (σ).5 The time 

step for this model was limited by the same gravity-wave CFL stability condition as the Rand model and therefore was not practical 

for long-term simulations, especially on the computers available at that time.

The model by Heaps (1972, 1974) differed significantly from the other early models in that it avoided the use of grid boxes 

in the vertical dimension and instead used a spectral method. In the spectral approach, the horizontal components of current are 

expanded in terms of a set of functions (called the basis set) through the water depth. A 3-D model using the spectral method is 

reduced essentially to a 2-D model that determines the variation in the coefficients of the basis functions over the horizontal dimension and through time. The spectral method does account directly for the temporal variation in the free surface and improves computational efficiency by reducting the dimension of the problem by one. Because the spectral method can represent the vertical 

profiles of horizontal currents as continuous functions, it can give superior numerical accuracy over the grid-box method (Davies 

and Stephens, 1983; Davies, 1991; Davies and Lawrence, 1994). Although efficient and numerically accurate, the early spectral 

models were not nearly as far advanced in equation formulation as the finite-difference models. The model by Heaps (1972) solved 

only a linear form of the governing equations and neglected advection, horizontal shear stresses, and density forcing terms. The 

vertical, internal shear stresses were parameterized with an eddy viscosity that remained constant over the depth and with time. 

Years later, strategies for obtaining solutions with nonlinearities, stratified density, and variations in the vertical eddy coefficients 

were devised. Good review papers on the progress in using spectral methods for representing the vertical current structure in 3-D 

models are by Heaps (1980), Davies (1987), and Davies and others (1997a).

The spectral method also can be used for the horizontal dimension (for example, Krauss and Wubber, 1982), but this approach 

generally is attractive only if the region being modeled has regular boundaries or can be transformed into one having regular boundaries (Davies, 1987). The set of basis functions in the spectral method can be chosen as piece-wise polynomial functions (either 

2-D functions in the horizontal domain or 1-D functions in the vertical domain), in which case the method actually can be regarded 

as a finite-element method (for example, Koutitas and O’Connor, 1982).

A third category of spectral models uses a spectral approach in the time domain. These 3-D models are usually called harmonic models (for example, Lynch and Werner, 1987; Walters, 1992) and use a frequency-domain numerical scheme to replace 

the traditional time-stepping approach. Harmonic models also have previously been developed in two dimensions by Pearson and 

Winter (1977), Le Provost and Poncet (1978), Snyder and others (1979), Walters (1988), and Burau and Cheng (1989) among others. Because models that are spectral in time represent the unknowns of water level and velocity by summing harmonic functions 

with astronomical tidal frequencies, they are particularly well suited for studying the propagation and interactions of the astronomical constituents of the tide; these models are not particularly well suited for studies that deal with strong nonlinearities and aperiodic forcing.

Although there are many applications for which spectral models are useful, they are still not ideally suited for the most dynamically complex problems involving stratified estuaries with nonlinear effects that require an advanced parameterization of the turbulent processes. Therefore the discussion here is limited mostly to finite-difference models that are considered because of their 

generality.

5The σ transformation is widely used in 3-D models and is discussed in more detail in section 2.6.2 and Appendix C of this report.
1. Introduction 9

In attempting to reduce the computation time of 3-D, explicit, finite-difference models while retaining free-surface effects, a

so-called mode-splitting technique has been used by many investigators. Several variations of mode-splitting are used, but in general, mode-splitting is a separation of the 3-D governing equations into a set of equations describing the 2-D depth-mean flow (the 

external mode) and a set describing the vertical structure of flow (the internal mode). The time discretization for the gravity-wave 

terms in the external-mode equations can be either explicit or implicit; the internal-mode equations are treated explicitly except for 

the vertical diffusion terms, which usually are treated implicitly to avoid a time-step limitation in shallow water (Davies, 1985). 

By using an explicit time discretization for the external-mode gravity-wave terms, the time step must be small enough to satisfy 

the gravity-wave CFL condition. By applying a time-splitting algorithm, however, the internal-mode solution can be integrated by 

using a much larger time step than the external mode. Because the internal-mode solution can be expensive in terms of computation 

time, significant economies are gained by solving the internal mode explicitly with a large time step (usually at least an order of 

magnitude larger than the time step for the external mode). By using an implicit time discretization for the external-mode gravitywave terms, the CFL limitation can be avoided, although this advantage is partially offset by the additional complexities involved 

in an implicit formulation (including the need to solve a matrix system of algebraic equations each time step). Implicit mode-splitting, however, allows the external and internal modes to be solved using the same long time step. Depending on the physical situation and the computer being used (parallel or sequential), either the explicit or the implicit time discretization method may be 

more computationally efficient with mode-splitting. In either case the use of mode-splitting will usually lead to solutions that are 

far more computationally efficient than those that solve the primitive 3-D equations directly using a fully explicit scheme.

Mode-splitting refers to models that solve the equations of both modes numerically and retain the three-dimensionality of the 

governing equations. Some models, such as early ones by Jelesnianski (1970), Forristall (1974), and Nihoul (1977), numerically 

solve a depth-integrated, 2-D model and then analytically solve a locally 1-D or “point” model for the vertical profiles of currents 

based on the classic Ekman theory (Welander, 1957); a model proposed by Tee (1979) is similar to these models although somewhat more flexible in the vertical formulation. These simple 3-D models (Tee, 1987) are not mode-splitting models in the sense 

used here. Because the simple models are constrained by assumptions, they are not suited to shallow-water estuaries that are stratified and involve nonlinear effects. 

Simons (1973, 1974, 1980) is credited as the first to introduce the mode-splitting technique for use on modeling the Great 

Lakes. Later, Madala and Piacsek (1977) used it to develop an ocean model. This method has since been adopted by numerous 

other investigators for use in 3-D shallow-water circulation modeling (for example, Blumberg and Mellor, 1980, 1987; Wang, 

1982; Sheng, 1983; Hess, 1985; Lardner and Smoczynski, 1990; Blumberg, 1991; Johnson and others, 1991; Hamrick, 1992; Jin, 

1993; Chapman and others, 1996; Muin and Spaulding, 1997a). In fact, most 3-D shallow-water models now being used are modesplitting models. The mode-splitting technique is also being implemented for modeling the large-scale circulation of the ocean 

(Killworth and others, 1991; Dukowicz and Smith, 1994; Semtner, 1995).
10 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Although mode-splitting has become widely accepted in 3-D modeling, it has several important drawbacks that are often overlooked. If an explicit time discretization is used for both modes of a time-splitting scheme, the external mode (2-D) velocities must 

be the exact depth average of the internal mode (3-D) velocities; otherwise, the computations will become unstable. Dukowicz and 

Smith (1994) point out that it is necessary when using the explicit-leapfrog, finite-difference scheme to integrate the external-mode 

equations over a time interval corresponding to twice the internal mode time step to ensure that the time-averaged 2-D variables 

are properly centered in time and satisfy the continuity equation. Splitting methods, in general, have an increased amount of poorly 

understood errors associated with them; to keep these errors small, the maximum allowable time step for the internal mode calculations may have to be limited in certain cases. If an implicit time discretization is used for the external-mode gravity waves in a 

mode-splitting model, time-splitting errors can be eliminated if the external and internal mode time steps are chosen to be equal; 

however, the separate calculations of the 2-D and 3-D variables lead to difficulties in consistently representing the magnitude of 

the bottom frictional stress between the external and internal modes. The problem arises because, for consistency (and for true

three-dimensionality), the 2-D external-mode equations must include bottom stress as a nonlinear function of the 3-D bottom 

velocity. The 3-D bottom velocity cannot be used implicitly in the external-mode computations without implementing some form 

of iteration involving the internal mode. To circumvent this problem, the bottom stress term must remain explicit in the external 

mode, which in shallow water and in the presence of strong currents can lead to a rather restrictive limitation on the time step for 

stability.

Instead of mode-splitting, some 3-D modelers have used other forms of splitting methods. For example, the latest versions of 

the 3-D Rand Corporation model (Leendertse, 1989) and the 3-D TRISULA model from Delft Hydraulics of the Netherlands (Uittenbogaard and others, 1992) are based on one of the best known splitting techniques—the alternating-direction-implicit (ADI) 

method. These models are basically 3-D extensions of ADI methods successfully used in two dimensions (Leendertse, 1967, 1987; 

Stelling, 1984). In each of these models, the vertical diffusion is treated implicitly. Leendertse (1989) is especially critical of the 

use of mode-splitting because it “degrades the accuracy of computation” to first order. The ADI models are formulated using essentially second-order accuracy numerics and are computationally efficient. These models also are unconditionally stable, which 

allows the use of large time steps. However, for large time steps, the ADI models may cause inaccuracies when dealing with flow

domains where the bathymetry is complex, especially where one or more narrow channels within a modeled region are not aligned 

with the principal directions of the numerical grid. To eliminate these inaccuracies, either the time step or horizontal grid size of a 

simulation must be decreased, which can have a significant effect on model efficiency. This source of inaccuracy in ADI models 

is known as the ADI effect (Stelling and others, 1986) and is more likely to be significant in water bodies like estuaries, where 

typically there are deep-water shipping channels crossing through shoal areas, rather than in the coastal ocean. 
1. Introduction 11

A 3-D model presented by de Goede (1991) uses an implicit, time-splitting, finite-difference scheme that does not employ 

ADI methods so that inaccuracies caused by the ADI effect are absent, even for large time steps. The 3-D scheme has a strong 

resemblance to the 2-D scheme described in Wilders and others (1988). The 3-D scheme, which neglects advection, density-forcing, and horizontal shear stress terms, is based on a two-stage splitting procedure in which the first stage requires the solution of a 

large number of independent tridiagonal systems of equations involving the implicit treatment of vertical diffusion. In the second 

stage, the terms describing the propagation of the surface gravity waves (that is, the water surface pressure gradient in the momentum equations and the velocity divergence in the continuity equation) are treated implicitly. This stage results in a five-diagonal 

matrix system to be solved for the water surface elevation. Once the water- surface elevation is known, the velocities are computed 

explicitly. A later version of the model (de Goede, 1992, Chapter 7) includes advective terms, but the time-splitting procedure then 

is reduced to only first-order accuracy in time for the advection and bottom friction calculations, and the computational efficiency 

is reduced.

The special approach used by de Goede (1991) for the implicit treatment of the gravity-wave terms is also used in the 3-D 

finite-difference models of Backhaus (1985) and Casulli and Cheng (1992). These latter two models are referred to as semi-implicit 

and are appealing because they involve no mode-splitting, ADI, or other splitting procedure and yet still include implicit vertical 

diffusion. In three dimensions, semi-implicit schemes almost always are preferable, for reasons of computational efficiency, to the 

alternative of a fully implicit scheme (without splitting). When the 3-D equations are approximated using a fully implicit scheme, 

a coupled system of nonlinear equations for velocity and surface elevation formed at each time step requires enormous computer 

resources to solve.6 For many practical problems, however, there simply is not enough benefit from using an implicit scheme for 

the advection, Coriolis, baroclinic (density-driven) pressure gradient, and horizontal shear stress terms of the governing equations 

to justify the extra computational expense (which is considerable). These terms are associated with slower phase speeds than the 

gravity waves and often do not result in a too-restrictive time step limitation for linear stability when calculated explicitly. Thus, 

in a semi-implicit, 3-D scheme, these terms are treated explicitly, and only the gravity-wave and vertical diffusion terms are treated 

implicitly. 

Some authors have used the term semi-implicit loosely to refer to any method in which one or more terms in the governing 

equations are treated implicitly, and one or more terms are treated explicitly. Using this definition, all of the ADI methods can be 

classified as semi-implicit. Li and Zhan (1993) call their finite-element model semi-implicit, even though only the vertical diffusion and Coriolis terms are treated implicitly. Here, in the current report, semi-implicit is used specifically to refer to the basic 

approach originating with Kwizak and Robert (1971) in atmospheric modeling (see also Mesinger and Arakawa, 1976; Haltiner 

and Williams, 1980). The approach is based on treating two groups of terms differently: the gravity-wave terms in the model equations are treated implicitly to avoid the time-step limitation due to the gravity-wave CFL condition and other terms are treated 

explicitly. For multidimensional models, the gravity-wave terms should be treated implicitly in both horizontal coordinate directions at once, in contrast to the alternating-direction, semi-implicit scheme used by Wolf (1983). 

6Fully implicit schemes are widely used in one-dimensional (1-D) hydrodynamic models, but only because 1-D problems generally require only minimal 

computer resources.
12 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

For shallow-water modeling, the great benefit of the semi-implicit approach is that the solution for the water surface elevation 

can be uncoupled from the solution for the velocity by inserting the equations of motion into the divergence terms of the depthintegrated continuity equation. This step results in a single system of linear equations that is solved at each time step for the new 

surface elevation over the entire domain. The coefficient matrix for the system is symmetric and positive-definite so that the equations can be solved efficiently using an iterative technique, such as the preconditioned conjugate-gradient method. Velocities in 

this approach are computed explicitly using the newly updated surface elevation. Compared with the fully implicit method (without 

splitting), the semi-implicit method requires not only considerably less computer time but also less memory and storage and is of 

similar accuracy. Because the semi-implicit method does not involve any ADI technique, it is not affected by any time-step limitation or inaccuracy related to the ADI effect.

The semi-implicit method has been successfully implemented by many authors in 2-D finite-difference models (for example, 

Backhaus, 1983; Duwe and others, 1983; Casulli, 1990; Muin and Spaulding, 1996). Benqué and others (1982) and Wilders and 

others (1988) have used the semi-implicit approach for the propagation of the gravity waves in the context of fully implicit splitting 

methods in two dimensions. The semi-implicit method has also been used for the 2-D external-mode computations in 3-D modesplitting models (for example, Madala and Piacsek, 1977; Wang, 1982; Blumberg, 1991; Hamrick, 1992; Muin and Spaulding, 

1997a). Muin and Spaulding (1996) comment that the semi-implicit method was preferred over an ADI method in their 2-D applications using nonorthogonal (boundary-fitted) grids; the ADI method gave accurate solutions only for orthogonal or slightly nonorthogonal grid systems. 

Casulli and Cheng (1992) made a significant contribution to the implementation of the semi-implicit method in three dimensions. They devised a concise and accurate procedure to include implicit vertical diffusion in the 3-D scheme without resorting to 

any form of splitting method. Their procedure avoids the inconsistency in the treatment of the bottom frictional-stress condition 

referred to by Stronach and others (1993, p. 19) regarding an updated version of the Backhaus (1985) 3-D model. Using two executions of a tridiagonal-matrix (double-sweep) algorithm beneath each grid box in the horizontal plane, Casulli and Cheng (1992) 

formally transform each vertical set of x- and y-momentum equations into a matrix equation for the discrete horizontal velocities 

in each layer, expressed in terms of the unknown water surface pressure gradients. The matrix equations are then formally substituted into the depth-integrated continuity equation to eliminate the unknown vertical integrals of the 3-D horizontal velocities. The 

depth-integrated continuity equations for all horizontal grid boxes form a five-diagonal system of equations for the unknown water 

surface elevations over the entire domain. Once the matrix system for water surface elevation is solved using a conjugate-gradient 

method, the 3-D velocities are solved explicitly from the momentum equations with the newly determined water surface gradients 

inserted. The overall computational scheme is quite efficient. 
1. Introduction 13

The finite-difference scheme used by Casulli and Cheng (1992) is based on two time levels and uses an Eulerian-Lagrangian 

method (ELM) with linear interpolation for the treatment of the advective terms in the momentum equations. The scheme uses no 

coordinate transformations and solves the governing momentum equations in a nonconservative form that is consistent with the 

use of an ELM. The scheme is basically an extension of an earlier 2-D model by Casulli (1990) that has been used successfully 

(Cheng and others, 1993; Signell and Butman, 1992). The 3-D scheme has been used successfully for San Francisco Bay (Cheng 

and Casulli, 1996). A drawback of the Casulli and Cheng (1992) scheme is that it is only first-order accurate in the time discretization, which can cause artificial damping of tidal amplitudes and velocities. Casulli and Cattani (1994) present a modified version 

of the scheme in which the time discretization for the gradient of the free-surface elevation in the momentum equations and the

velocity in the continuity equation can be centered in time, according to Crank and Nicolson (1947), to achieve second-order accuracy for these terms. This modification does improve the overall accuracy of the scheme, but for fully nonlinear problems, the 

finite-difference momentum equations are still first-order accurate in the advection and Coriolis terms and the bottom friction condition. The depth-integrated continuity equation also is first-order accurate in the representation used for the total depth of flow, 

which is evaluated backward-in-time. All terms in the two-level scheme could conceivably be centered in time if an iterative procedure was implemented, as will be discussed later. The scheme from Casulli and Cheng (1992) also has been implemented by 

Stansby and Lloyd (1995).

In this discussion, relatively little has been said regarding 3-D models that use the finite-element method (Zienkiewicz, 1977)

for the numerical discretization in the horizontal plane. In general, the research on 3-D modeling has concentrated far more on the 

finite-difference method than the finite-element method.7 This emphasis on the finite-difference method may in part be due to a 

perception that, at least until recently, finite-element models were not fully competitive with finite-difference models in terms of 

computational efficiency. Because research on the finite-element method for solving free-surface flow problems started much later 

than research on the finite-difference method, there has been a period in which finite-element models have been “catching up” with 

the finite-difference models in terms of both accuracy and efficiency. Of course, the advantage of the finite-element method has 

always been its flexibility in allowing unstructured grid networks that can be employed to represent shoreline curvature better and 

to increase grid-point density where needed. This advantage has become somewhat less significant, however, with the recent 

advances in curvilinear-coordinate, finite-difference models.

The progress on 3-D finite-element models followed most of the same trends as for 3-D finite-difference models. Two early 

3-D models (Kawahara and others, 1982; Shubinski and Walton, 1982) use explicit time discretizations with simple elements, even

though the finite-element method does not lead naturally to explicit procedures; both these models use a multilevel, finite-difference approach in the vertical dimension. The model by Koutitas and O’Connor (1982) uses a time-splitting method (fractional step 

approach) to separate the horizontal computations from the vertical computations, which are both done using finite elements. The 

model by King (1985) uses the rather costly approach (in terms of computer time) of solving the nonlinear, 3-D equations using a 

fully implicit, time-stepping scheme without any form of splitting procedure; to improve computational efficiency of simulations, 

the model allows the convenient coupling of 1-, 2-, and 3-D elements so that the user can limit the 3-D analysis to the region in 

7

Of seventeen 3-D models surveyed in a review paper by Cheng and Smith (1990), only three used the finite-element method for the numerical 

discretization in the horizontal plane; the other models used the finite-difference method.
14 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

which it is needed. Lynch and Werner (1987) and Walters (1992) chose an efficient spectral (harmonic) approach in the time 

domain for their 3-D finite-element models. These models separate the horizontal and vertical components of the solution much 

like mode-splitting models except that no time-stepping is employed. The model by Lynch and Werner (1987) is linear and allows 

either analytical or numerical (finite- element) solutions in the vertical dimension. The model by Walters (1992) includes nonlinear 

terms. Lynch and Werner (1991) use mode-splitting in a nonlinear, time-stepping, 3-D finite-element model in which the external

mode is solved by way of a semi-implicit method with a wave-equation rearrangement of the governing equations. The wave-equation rearrangement has been prevalent in 2-D modeling (for example, Lynch and Gray, 1979; Kinnmark, 1986) and suppresses 

short wavelength solution oscillations that have long been the bane of finite-element modelers. In the external mode of the model 

by Lynch and Werner (1991), the friction and nonlinear part of the water surface slope term must be treated explicitly. A new 

approach for 3-D modeling has been demonstrated by Luettich and others (1994): velocity is replaced with shear stress as the 

dependent variable in the internal-mode computations of a mode-splitting, finite-element model. The rationale for changing variables is that, because shear stress normally varies more slowly near sheared boundaries, it can be resolved more readily than velocity. Luettich and others (1994) solve the external mode by using a harmonic model. Additional references on finite-element models 

can be found in the review papers by Westerink and Gray (1991) and Davies and others (1997). 

1.5 The Present [1997] Model

Numerous factors were considered in the design of the 3-D model described herein. Because the model will be used for calculations of the circulation in real estuaries under varying conditions, the equation formulation must be complete, incorporating a 

time-varying free surface, nonlinearities, and horizontal density-gradient forcing for prognostic purposes (a baroclinic model). The 

numerical scheme must be flexible enough to accept an advanced turbulence parameterization suitable for stratified flows later and 

be capable of providing accurate simulations under both periodic forcing from tides and aperiodic forcing by freshwater inflows

and the wind. The shallowness of estuarine basins also requires using a quadratic (rather than linear) stress law for the bottom frictional-boundary condition. Owing to these very general requirements, the use of any of the various spectral approaches in time or 

space was not considered. Because the new 3-D finite-difference techniques (semi-implicit and ADI) obviate the need for modesplitting to deal efficiently with gravity-wave propagation, the finite-difference approach was chosen for the numerical approximations in the new model.

The objectives in the design of the numerical scheme for the model were that it be efficient for long-term (at least seasonal) 

simulations of estuaries and second-order accurate in the truncation errors to ensure that solutions are not overly damped or diffused by artificial viscosity. Implementing the numerical scheme using the conservation form of the governing equations for both 

the flow and salt transport also was considered desirable. Mass and momentum are more readily conserved by the conservation 

form of the basic equations, and the depth-integrated continuity equation is linear.
1. Introduction 15

Presently, the differences in the computational efficiencies of the new finite-difference, the ADI (Leendertse, 1989; Uittenbogaard and others, 1992), and the semi-implicit (Casulli and Cheng, 1992) methods are difficult to assess and depend on the 

geometry and hydrodynamic conditions of the particular water body under study. As de Goede (1992) points out, semi-implicit 

methods require solving a pentadiagonal matrix system using a two-dimensional structure, whereas the ADI schemes solve only 

simple tridiagonal systems. The ADI schemes, however, may be limited because of geometry to a certain maximum time step for 

accuracy (the so-called ADI effect). The semi-implicit methods do not suffer from the ADI effect, but they may have a known 

time-step limitation related to the explicit treatment of other terms. The dependence on geometry and bathymetry by the ADI 

scheme is a disadvantage because errors due to the ADI effect can go undetected by an unaware user and cause misinterpretation 

of model results. Ultimately the decision was made to implement the semi-implicit approach in the new model. To simulate the 

hydrodynamics of a geometrically complex estuary, such as the Suisun Bay region of San Francisco Bay, the semi-implicit scheme 

may indeed be the most efficient approach.

The semi-implicit scheme implemented herein closely follows the approach outlined by Casulli and Cheng (1992) for the 

implicit inclusion of vertical diffusion in a 3-D calculation without recourse to mode-splitting. The details of the overall scheme, 

however, differ from those in Casulli and Cheng (1992) in many significant ways. The time integration used here is a three-timelevel, semi-implicit, leapfrog-trapezoidal method. Except for a small amount of first-order error introduced by the uncentered (in 

time) treatment of horizontal diffusion for stability considerations, the scheme is second-order accurate in the truncation errors of 

the finite-difference approximations in both space and time. All terms (other than horizontal diffusion), as well as nonlinear coefficients, are centered in time during the leapfrog and trapezoidal steps of the scheme to achieve second-order accuracy in time. The 

accurate evaluation of the nonlinearities in the equation of motion is important, because nonlinearities have a significant effect on 

the tidally averaged circulation in shallow estuaries. The governing equations for the multilevel scheme are prepared in a conservative form by integrating them over the height of each layer. The layer-integrated, volumetric transports replace velocities as the 

dependent variables so that the depth-integrated continuity equation that is used in the solution for the water surface elevation is 

linear. The advection terms in the momentum and salt-transport equations are solved using explicit, leapfrog-trapezoidal integration rather than an ELM as used by Casulli and Cheng (1992). The leapfrog-trapezoidal approach does very well with the conservation of salt, which can be a problem with the ELM approach. The global matrix solution for water surface elevation in the new

model is implemented using the conjugate-gradient method with a modified incomplete Cholesky preconditioner (Axelsson and 

Lindskog, 1986). This method was selected after testing a number of other preconditioners and acceleration methods.
16 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

In the initial design of the 3-D model, it was not obvious whether a two- or three-time-level time-integration scheme would 

be the better choice for optimal computational efficiency, accuracy, and stability. Both approaches are used in existing semiimplicit models. The nonlinear terms can be time centered in the context of a two-level scheme by iterating to a solution and averaging the old and new time levels. Duwe and others (1983) suggested a two-iteration (two-step) procedure in a 2-D, two-level, 

semi-implicit model and found it improved accuracy and stability for the simulation of markedly nonlinear flows. Of course, iteration significantly increases the computer time of a model simulation and therefore is costly in three dimensions. A three-level, 

semi-implicit, leapfrog scheme is attractive because all terms can be readily centered in time without iteration. However, leapfrog 

schemes have been used extensively in atmospheric modeling and are known occasionally to exhibit instabilities that are thought

to be caused by strong nonlinearities; a widely cited example is Lilly (1965, p. 23). To suppress nonlinear instabilities, the addition 

of an intermittent two-level, trapezoidal step, following a leapfrog step, has been successful (Mesinger and Arakawa, 1976). The 

combined scheme is referred to as leapfrog-trapezoidal. A semi-implicit, leapfrog-trapezoidal scheme was tested in an early version of the model described herein (Smith and Larock, 1993) and is also used in the 3-D model by Hamrick (1992).

One goal of the research in this report was to investigate two- and three-time-level integration schemes for use in the 3-D, 

semi-implicit model. The two types of integration schemes were thoroughly tested and compared using numerical experiments 

with the 1-D equations representing open channel flow; the results are presented in Chapter 5. A similar comparison using the full 

3-D equations would have proven too laborious, so it was not pursued. The 1-D testing shows in many instances that the threelevel, leapfrog scheme can be used without a trapezoidal step and can provide second-order accuracy with excellent efficiency. 

With strong nonlinearities present in the bottom friction term, the two-level scheme without iteration gives poor results owing to 

the first-order approximation of that term. Both schemes are comparable in efficiency and accuracy when iteration is used. Iteration 

extends the stability of both schemes. On the basis of the 1-D model testing it was decided to use the three-level scheme in the 3-D 

model. A numerical experiment using the 3-D model is included in Chapter 5.

The 3-D equations solved herein use standard Cartesian coordinates, which were simplest for developing the numerical 

scheme. A variety of coordinate transformations from the Cartesian system are prevalent today in multidimensional modeling, and

these are discussed in some detail in Chapter 2.

The horizontal location of variables on the 3-D numerical grid is the staggered arrangement known as an Arakawa C-grid 

(Mesinger and Arakawa, 1976). Most of the finite-difference schemes that have been developed for hydrodynamic modeling of 

estuaries and coastal seas have been based on this grid (Lardner and Song, 1992). It has the advantage that difference quotients are 

easily centered in space to obtain second-order accuracy. Stelling (1984, p. 102) discusses some of the other advantages of using 

a staggered grid, versus a nonstaggered grid, in terms of efficiency, minimizing spurious oscillations, and simplicity of implementing boundary conditions.
2. Governing Equations and Boundary Conditions 17

2. Governing Equations and Boundary Conditions

2.1 Introduction

This chapter develops the governing equations and boundary conditions used in the 3-D circulation model. The assumptions 

and approximations inherent in the development of the equations are highlighted to ensure that the model is applied properly. In 

section 2.2, the basic 3-D equations for the instantaneous flow that describe exactly the fluctuating turbulent variables are presented. Because the instantaneous-flow equations have long been recognized as intractable for applications to rivers and estuaries, 

the time-averaged 3-D equations for the mean-flow (turbulent-averaged) variables are introduced in section 2.3. In this section, the 

closure problem resulting from the equation-averaging process is discussed, and the concepts of an eddy viscosity and eddy diffusivity are introduced. In section 2.4, the boundary conditions for an estuarine flow are specified. In section 2.5, a scaling analysis 

is used to estimate the relative magnitude of the various terms in the differential equations for an estuary, using San Francisco Bay 

as an example. Finally, section 2.6 discusses the horizontal and vertical coordinate transformations that are used in some existing 

3-D models. These coordinate transformations are offered only as background because they are not used in the present model.

2.2 Instantaneous-Flow Equations

In estuaries, the flows are almost always turbulent. The exact equations describing the instantaneous turbulent motion originate from the conservation laws for mass, momentum, and salt. These equations are introduced in the following three sections.

The reference frame is a right-handed, Cartesian coordinate system with axes xi (i = 1, 2, 3) oriented such that the positive 

x1-axis is directed horizontally to the east, the positive x2-axis is directed horizontally to the north, and the positive x3-axis is 

directed vertically upward along the line of action of gravity (fig. 2.1). In an estuary, the boundary is continuously in motion 

because of the earth’s rotation. It is convenient to choose a reference frame at rest relative to the boundaries of the estuary rather 

than one that is fixed in space (relative to the fixed stars). The reference frame is therefore chosen to be fixed on the earth’s surface 

and rotating with the earth’s axis at an angular velocity Ω equal to 1 revolution per sidereal8 day (7.29 x 10−5 radian/second). 

The horizontal scale of many estuaries is sufficiently large that the earth’s rotation will introduce a significant term in the equation of motion. The area is not so large, however, that the curvature of the earth’s surface must be considered. The plane x3 = 0 is 

therefore considered to be a tangent plane to the geoid and coincides with mean sea level.

8“One sidereal day = 23 hours 56 minutes 4 seconds = 86,164 seconds is the time required for the earth to rotate once about its axis, relative to the fixed 

stars. Since the earth revolves about the sun it must turn a little further to point back to the sun and complete one solar day—hence the solar day is a little longer 

than the sidereal day” (Pond and Pickard, 1986, p. 38). 
18 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

2.2.1 Mass Conservation

The principle of conservation of mass requires the net flux of mass through a control volume fixed to the rotating reference 

frame to be balanced by the net rate of change of the mass within the volume. The basic differential equation which expresses this 

principle is (Currie, 1974)

∂ρ ̃

∂t

------ ∂ ρ ̃ u ̃

i ( )

∂xi

+ ---------------- = 0 , (2.1)

where

u ̃

i are the instantaneous velocity components,

ρ

 ̃ is the instantaneous density,

xi are the Cartesian coordinates, and

t is time.

This equation is written using Einstein’s summation convention (Spain, 1960) in which a repeated subscript in any term indicates 

a summation over the three coordinate directions. This convention is used elsewhere in this chapter when it affords greater brevity 

than other notation.

Because equation 2.1 is written as a partial differential equation, it is assumed that the velocity is a continuous function. For 

this reason, equation 2.1 is often referred to as the equation of continuity, or simply the continuity equation. 

In discussing the continuity equation, it is convenient to introduce the concept of the substantial derivative, which for density 

is expressed as

Dρ ̃

Dt

------- ∂ρ ̃

∂t

------ u ̃

i

∂ρ ̃

∂xi

= + ------- . (2.2)

Figure 2.1. Orientation of Cartesian coordinate system with corresponding 

velocity components u ̃

i (i = 1, 2, 3).

Vertically upward

North

East

u1 ( )  ̃

u2 ( )  ̃

u3 ( )  ̃

x1

x2

x3
2. Governing Equations and Boundary Conditions 19

This expression describes the rate of change in density of a given mass of fluid as it is followed through the flow. Using the 

substantial derivative, equation 2.1 can be rewritten as

1

ρ ̃

--

Dρ ̃

Dt

------- ∂u ̃

i

∂xi

+ ------- = 0 . (2.3)

Because the compressibility of water is small, it is usually assumed thatDρ ̃ ⁄ Dt = 0 for water flows of constant density 

everywhere, so that equation 2.3 can be simplified to

∂u ̃

i

∂xi

------- = 0 . (2.4)

When the estuarine water density is a function of salinity, temperature, and pressure, it is not correct to equate Dρ ̃ ⁄ Dt to zero 

simply by invoking incompressibility. The formal definition of incompressibility requires only that the fluid density not be affected 

by changes in pressure, which is not equivalent to the statement that Dρ ̃ ⁄ Dt = 0 if the density of the moving fluid element is 

changed by the molecular conduction of heat or the exchange of water molecules for salt ions. However, for estuaries, these effects 

are miniscule and can be safely ignored when considering mass or volume conservation. Therefore, equation 2.4 can in fact be used 

as a good approximation to the instantaneous equation for mass conservation in estuaries.

2.2.2 Momentum Conservation

The principle of conservation of linear momentum arises from Newton’s second law of motion and requires that the change 

of momentum in a control volume fixed to the rotating reference frame be equal to the net influx of momentum into the control 

volume and the sum of the forces acting on it. In the form of a differential equation, this principle results in the following instantaneous equation of motion for estuaries (Pritchard, 1971a):

∂ ρ ̃ u ̃

i ( )

∂t ---------------- ∂ ρ ̃ u ̃

i

u ̃

j ( )

∂x

j

+ --------------------- ∂p ̃

∂x

i

– ------- 2ε

ijkΩj

ρ ̃ u ̃

k – ρ ̃ gi ρ ̃ ν

∂2u ̃

i

∂x

j

∂x

j

= + + --------------- , (2.5)

where

gi is the local acceleration of gravity in the direction xi,

p ̃ is the instantaneous pressure,

Ωj is the component of the angular velocity of the earth in the direction xj,

ν is the coefficient of (molecular) kinematic viscosity (assumed constant), and

εijk is the alternating tensor defined by the relations ε123 = ε231= ε312 = 1, ε213 = ε132 = ε321 = –1, and εijk = 0 if two or more 

subscripts are equal.

Equation 2.5 is the famous Navier-Stokes equation in tensor notation including gravitational and Coriolis body-force terms; the

tensor equation represents three scalar equations corresponding to the three possible values of the free subscript i.
20 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

It is common to simplify the two terms on the left side of equation 2.5 by first expanding, using the product rule for 

differentiation, into the form

ρ ̃

∂u ̃

i

∂t ------- u ̃

i

∂ρ ̃

∂t

------ u ̃

i

∂ ρ ̃ u ̃

j ( )

∂xj

---------------- ρ ̃ uj

∂u ̃

i

∂xj

++ + ------- . (2.6)

The second and third terms of this expression sum to zero, because they represent the continuity equation (eq. 2.1) multiplied by 

the velocity. Using this simplification, the equation of motion becomes

ρ ̃

∂u ̃

i

∂t ------- ρ ̃ u ̃

j

∂u ̃

i

∂xj

+ ------- ∂p ̃

∂xi

– ------- 2ε

ijk

Ωjρ ̃ u ̃

k – ρ ̃ gi ρ ̃ ν

∂2u ̃

i

∂xj∂xj

= + + --------------- . (2.7)

On the left side of equation 2.7 are the inertia or acceleration terms that represent the rate of change of linear momentum of a unit 

volume of the fluid. The first term is a temporal acceleration, and the second is the advective acceleration. The right side of equation 

2.7 lists the forces causing the acceleration. The first term is the pressure force term, which in an estuary can result from horizontal 

gradients in the water surface, density, and atmospheric pressure. The second term is the Coriolis body-force term, which is derived 

in detail in Appendix A. The Coriolis force is actually a pseudoforce that arises solely as a result of the rotating reference frame. 

(In an estuary in the northern hemisphere, the Coriolis force acts perpendicular to, and to the right of, the velocity vector; it is 

proportional to the magnitude of the velocity.) The third term on the right side of equation 2.7 is the gravity body-force term, which 

in the chosen reference system is defined by ρ ̃ gi = ( ) 0 0, , – ρ ̃ g . The fourth term is the viscous-shear term and represents internal 

forces in the fluid resulting from the fluid viscosity.

A widely used approximation to equation 2.7, first introduced by Boussinesq (1903), is known as the Boussinesq approximation. It consists essentially of neglecting variations in density insofar as they affect the mass of the fluid but retaining them where 

they affect the weight.

To clarify the Boussinesq approximation, the relations ρ ̃ ρ0 ρ ̃ = + ′ and p ̃ p0 p ̃ = + ′ can be substituted into equation 2.7, 

where ρ0 and p0 are the reference values of density and pressure that satisfy hydrostatic equilibrium such that ∂p0 ∂xi ⁄ ρ0gi = – , 

and ρ

 ̃ ′ and p ̃′ are small deviations from the reference values caused by the fluid motion and stratification in combination. The 

result, after cancelling terms and dividing by ρ0, is (similar to Turner [1973], p. 9)

1 ρ ̃ ′

ρ0

+ ----- ⎝ ⎠ ⎛ ⎞ ∂u ̃

i

∂t ------- u ̃

j

∂u ̃

i

∂xj

+ -------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1

ρ0

----- ∂p ̃′

∂xi

– ------- 2ε

ijk

Ωj 1 ρ ̃ ′

ρ0

+ ----- ⎝ ⎠ ⎛ ⎞ u ̃

k – ρ ̃ ′

ρ0

-----gi 1 ρ

 ̃ ′

ρ0

+ ----- ⎝ ⎠ ⎛ ⎞ ν

∂2u ̃

i

∂xj∂xj

= + + --------------- . (2.8)

Here the density ratio ρ

 ̃ ′ ρ0 ⁄ appears as only a small correction to the inertia, Coriolis, and viscous-shear terms, but it is of great 

importance to the gravity term where it appears alone. The Boussinesq approximation neglects the effect of density variations in 

each term except for the gravity term. This simplification allows an average value for the density of the flow to be used in the 

horizontal momentum equations (x1- and x2-directions) where the gravity term is zero; the actual density must be considered only 

in the x3-equation. Equation 2.7 with the Boussinesq approximation becomes

ρ0

∂u ̃

i

∂t ------- ρ

0 u ̃j

∂u ̃

i

∂xj

+ ------- ∂p ̃

∂xi

– ------- 2ε

ijk

Ωjρ0u ̃

k – ρ

 ̃ gi ρ0ν

∂2u ̃

i

∂xj∂xj

= + + --------------- . (2.9)
2. Governing Equations and Boundary Conditions 21

In a stratified estuarine flow, the lighter (lower density) water near the surface will actually tend to accelerate faster under an applied 

pressure gradient than the heavier (higher density) water near the bottom. This influence of the vertical density variation on the 

fluid inertia is neglected in the Boussinesq approximation. For the normal range of stratification in estuaries this is equivalent to 

less than a 2 percent correction. It therefore appears reasonable to neglect this factor in calculating the inertial response of the flow. 

A recent evaluation of the Boussinesq approximation in ocean models is given by Mellor and Ezer (1995).

2.2.3 Dissolved Species Conservation

The principle of conservation of salt in an estuary requires the time rate of change in salt mass within a control volume to be

balanced by the net flux of salt from the control volume. For a control volume at rest relative to the estuary, the salt flux is composed of both advection (transport by the motion of the fluid) and molecular diffusion (transport by molecular motions). The differential equation expressing the conservation principle for the instantaneous salt concentration s ̃ is known as the advection-diffusion equation and may be written as (Fischer and others, 1979)

∂s ̃

∂t

----- ∂ u ̃

i

s ̃ ( )

∂xi

+ --------------- λ ∂2s ̃

∂xi

∂xi

= --------------- . (2.10)

The coefficient λ is called the molecular diffusivity (assumed constant) and has units of (length)2/time. The molecular-diffusion term involving λ is based on Fick’s law of diffusion (Fischer and others, 1979, p. 30−34), which relates the mass flux to the 

concentration gradient of the solute. Molecular diffusion is a result of the random scattering of solute particles by the constant 

motion and collision of molecules in the fluid; in equation 2.10, the rate of molecular diffusion in a moving fluid is assumed to 

equal the rate for a fluid at rest. In turbulent estuarine flows, the contribution of molecular diffusion to the salt flux is almost always 

insignificant, and the advection by the instantaneous velocity is the dominant mechanism. In section 2.3.1, the concept of turbulent 

diffusion is introduced; it is many times more efficient in promoting the mixing of salt than is molecular diffusion.

2.2.4 Equation of State

To close the set of governing equations 2.4, 2.9, and 2.10, an equation of state in the form

ρ ̃ f s ̃ Θ

 ̃ = ( ) , (2.11)

is needed to relate the density to salinity and temperature Θ

 ̃ ( ) The density represented by ρ ̃ is normally chosen to be the potential 

density (Pond and Pickard, 1986, p. 8), which is computed as a function of salinity and potential temperature at atmospheric 

pressure rather than actual temperature. Potential density is used to eliminate any effect of the compressibility of water when 

comparing water densities at significantly different depths.9 The equation of state used in the model is given in Appendix B. 

9

The distinction of using potential density instead of actual density is not important in estuaries unless the water depths exceed about 100 m. For a change 

in pressure of 100 m of water, the density of sea water (s ~ 35) changes by approximately 0.5 kg/m3, which is roughly equivalent to the effect caused by a change 

in salinity at constant pressure of 0.6. Because a vertical density distribution caused by the compressibility of water does not affect the water-column stability, the 

potential density is the correct indicator of stability.
22 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

In some cases, the temperature variations in an estuary may be small, in which case the effect of temperature variations on 

density will be minimal. Under these conditions it is acceptable to specify either a constant or slowly varying temperature 

distribution rather than compute the temperature distribution using the model. In the model described herein, the temperature distribution is assumed to be constant for the entire water body. Temperature predictions can easily be added to the model by including 

an equation for the conservation of thermal energy similar in form to equation 2.10, but with Θ

 ̃ substituted for s ̃ and appropriate 

terms for heat sources and sinks. 

2.3 Mean-Flow Equations

Although equations 2.4, 2.9, and 2.10 describe the fluid motion and salt transport for both laminar and turbulent flows, it is 

not practical to use them in numerical calculations of turbulent flow. Turbulence in estuaries occurs over a wide range of eddy

sizes; the largest scales are comparable to the basin dimensions and the smallest scales are determined by the fluid viscosity. The 

length, time, and velocity scales of the smallest eddies are referred to as the Kolmogorov microscales; for the open ocean, Fischer 

and others (1979, p. 58) estimate these scales to be about 0.1 cm, 1 s, and 0.1 cm/s, respectively. In an estuarine flow, these scales 

should be close to the same values. For numerical predictions, it is not practical on even the fastest computers to use such small 

spatial and temporal increments to determine the motions of the smallest three-dimensional eddies. Fortunately, the details of the 

fluctuating turbulent motion are rarely of interest in estuarine modeling studies. It is therefore common practice to average the 

instantaneous-flow and -transport quantities to separate the essentially stochastic (random) motions of the turbulence from the

deterministic motions of the mean flow.

One rigorous approach to the averaging of instantaneous quantities is to resort to the statistical method of ensemble averaging

as described by Pritchard (1971a) for the 3-D estuarine model equations and as used in formal textbooks on the theory of turbulence 

(Monin and Yaglom, 1971, 1975). Ensemble averages have the advantage of encompassing all time scales of the stochastic motion, 

but they have the disadvantage that they cannot be measured in the field or laboratory. Bedford and others (1987) have introduced 

generalized filtering methods that incorporate high-order averages of the instantaneous quantities over both the space and time

dimensions. These generalized filtering methods are theoretically appealing, but they introduce many additional terms into the governing equations that are not conveniently included in a model formulation; further research demonstrating the improvements to 

real predictions by using generalized filtering methods is needed before their use can be justified.

The averaging that is used here is the conventional time averaging suggested in the late nineteenth century by Osborne Reynolds (Reynolds, 1894). It represents a field situation in which measurements are made at a fixed point in a water body and time 

averaged over some period (usually minutes) to determine a mean quantity. Reynolds suggested decomposing the instantaneous 

quantities into mean and fluctuating parts. For the instantaneous velocity, pressure, density, and salinity, the decomposition is

u ̃

i ui u′i = + , p ̃ = p p + ′ , ρ ̃ = ρ ρ′ + , s ̃ = s s + ′ , (2.12)
2. Governing Equations and Boundary Conditions 23

where the mean quantities are defined by these time averages:

ui

1

T

-- u ̃

i dt

t

t T +

∫ = , p

1

T

-- p ̃ dt

t

t + T

∫ = , ρ

1

T

-- ρ ̃ dt

t

t T +

∫ = , s 1

T

-- s ̃dt

t

t T +

∫ = . (2.13)

The size of the averaging interval T is difficult to prescribe exactly, but in general it must be long compared to the time scale 

of the individual fluctuations but small compared to the time scale at which the unsteady mean flow is varying. In estuaries where 

the astronomical tides are a primary source of unsteadiness, the mean flow can vary significantly in just 15 minutes. The minimum 

required value for T will vary between flows and can truly be determined only by experiment. Indeed, a criticism of the time average is that T cannot always be chosen to include all of the time scales of the stochastic (turbulent) motion. An argument also can 

be made that an average should be taken in space as well as time to average the instantaneous quantities over the spatial turbulent 

eddy structure. The Reynolds time average, however, still is used extensively in practice and has not yet been shown to be 

inadequate in a significant way.

Substituting the relations 2.12 into equations 2.4, 2.9, and 2.10 and time averaging each term in the way indicated by 2.13 

yields the mean-flow and transport equations (For details, see most texts on fluid mechanics or turbulence such as Sabersky, 

Acosta, and Hauptmann, 1971; Hinze, 1975; and Tennekes and Lumley, 1972):

Continuity equation,

ui ∂

xi ∂

------- = 0 ; (2.14)

Momentum equation,

ρ0

ui ∂

∂t ------- ρ0 uj

ui ∂

xj ∂ + ------- ∂p

xi ∂

------ 2ε

ijkΩj

ρ0 uk ρgi ρ0 xj ∂

∂

ν

ui ∂

xj ∂

------- ui

′ uj – ′ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ = – – + + ; (2.15)

Salt transport equation,

∂s

∂t

---- ui ∂ s

xi ∂

+ --------- x

i ∂

∂ λ ∂s

xi ∂

------ ui – ′ s′ ⎝ ⎠ ⎛ ⎞ = ; (2.16)

Equation of state,

ρ = f( ) s, Θ . (2.17)

These equations are identical in form to the instantaneous equations except for the extra terms ui′ uj′ and ui′ s′ that have been 

introduced by the Reynolds averaging process (the overbar represents the Reynolds time average). These terms originate with the

nonlinear advection terms and are non-zero because correlations exist between the fluctuating velocities and between the velocities 

and salt fluctuations. In a physical sense, these terms, when multiplied by density, represent fluxes of momentum and salt 

associated with the turbulent fluctuations. The momentum flux acts as an effective stress on the fluid that is analogous to the

viscous stress in gases caused by momentum flux associated with molecular fluctuations. The presence of the heretofore 

undetermined turbulent stress and salt transport terms in the mean flow equations make the number of unknowns greater than the 

number of equations. Resolving this imbalance between equations and unknowns is the problem of turbulence closure.
24 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

2.3.1 Turbulence Closure

Turbulence closure is achieved by introducing additional equations, either algebraic or differential, governing the turbulent 

stress and salt transport terms. It is possible to use the instantaneous equations to develop additional differential transport equations 

that describe exactly the behavior of the individual turbulent stress and salt transport terms, but these equations contain new

unknowns involving averages of the triple products (triple correlations) of the fluctuating quantities. Transport equations for the 

triple correlations have also been derived, but quadruple correlations are then introduced and closure is still not obtained. The process of introducing additional higher-order correlation equations can go on indefinitely, but the number of unknowns will always 

exceed the number of equations until some further approximation is introduced to provide the necessary additional equations to 

close the system. The infinite system of equations is referred to as the Friedmann-Keller system.

The suite of possible turbulence closure schemes or models is large, and the interested reader is referred to state-of-the-art 

reviews by Reynolds (1976), Rodi (1980a), and the American Society of Civil Engineers (1988). Reviews of models pertaining 

specifically to turbulence closure in stratified environmental flows and estuaries are given by Rodi (1980b, 1987), Blumberg 

(1986), Viollet (1988), and Davies and others (1995, 1997b). A paper by Nunes Vaz and Simpson (1994) compares seven different 

turbulent closure schemes for the modeling of estuarine stratification.

The most basic classification scheme for turbulence models is based on whether the model employs Boussinesq’s (1877) wellknown eddy viscosity concept. The models that do not use an eddy viscosity are usually based on approximations to the exact transport equations for the turbulent stresses and are called second-order-closure schemes (Rodi, 1980a, p. 33). Although second-orderclosure schemes have great potential and are gaining in popularity, the eddy-viscosity approach is still the most widely used 

approach and is the approach used here. Among eddy-viscosity models there is a great difference from the simplest to the most 

complex models. In this report, a complex turbulence model is not introduced because the primary purpose is to present a numerical 

scheme for the 3-D model equations. Nothing in the numerical scheme, however, presents any conceptual difficulty in introducing

a more complex turbulence model later.

The turbulent processes themselves in estuaries are very complex, especially when density stratification occurs. Turbulence 

can originate near the estuary bed, in the interior of the fluid where density-gradient effects and internal waves are important, and 

near the free surface caused by wind waves and wind-generated surface drift. For a discussion of the physical processes related to 

turbulence and mixing in estuaries see, for example, Bowden (1977) and Abraham (1988).

2.3.1.1 Eddy Viscosity

The eddy-viscosity concept is based on an analogy with Stokes’ viscosity law for viscous stress in laminar flow. The turbulent 

stress is equated to a product of the eddy viscosity and a fluid-deformation tensor. For three-dimensional flows this is expressed 

as (Rodi, 1980a, p.10)

u′

i u′

j – νt

ui ∂

xj ∂

------- uj ∂

xi ∂

+ ------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 2

3

--Kδij = – , (2.18)

where

vt is the eddy viscosity,

δij is the Kronecker delta (defined as δij = 1 for i = j and δij = 0 for i ≠ j), and

Κ is the kinetic energy per unit mass of the fluctuating turbulent motion and is equal to ui

′ ui ( ) ′ ⁄ 2 .
2. Governing Equations and Boundary Conditions 25

Because in estuaries the turbulent stresses per unit mass are almost always much larger than the laminar stresses per unit mass

(v∂ui/∂xj in eq. 2.15), the laminar stresses generally are neglected.

Using equation 2.18 to replace the turbulent stress in equation 2.15, the turbulent stress term in the x1-direction becomes

u1′ u1 ∂( ) – ′

x1 ∂ ------------------------- u1

′ u2 ∂( ) – ′

x2 ∂ ------------------------- u1

′ u3 ∂( ) – ′

x3 ∂

+ + -------------------------

∂

x1 ∂

------- 2νt

∂u1

∂x1

-------- 2

3 – --K ∂x2 ∂

------- νt

u1 ∂

x2 ∂

-------- u2 ∂

x1 ∂

+ -------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- νt

u1 ∂

x

3 ∂

-------- u3 ∂

x

1 ∂

+ -------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ = + + .

(2.19)

The kinetic energy per unit mass K is a scalar sum of normal turbulent stresses and therefore acts similar to a pressure. Because 

the gradient of 2

3

--K generally is small compared to the pressure-gradient term in equation 2.15, it can be neglected. It also is 

common to neglect the horizontal gradient of the vertical velocity, ∂u3/∂x1, in 2.19 because it too is small. The stress terms in the 

x1-momentum equation then appear as

∂

x1 ∂

------- 2νt

∂u1

∂x1

-------- ∂

x2 ∂

------- νt

u1 ∂

x2 ∂

-------- u2 ∂

x1 ∂

+ -------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- νt

u1 ∂

x3 ∂ + + -------- . (2.20)

For the x2-momentum equation, the corresponding result is

∂

∂x1

-------- νt

∂u2

∂x1

-------- u1 ∂

x2 ∂

+ -------- ⎝ ⎠ ⎛ ⎞ ∂

x2 ∂

------- 2νt

u2 ∂

x2 ∂

-------- ∂

x3 ∂

------- νt

u2 ∂

x3 ∂ + + -------- . (2.21)

Because the gradients of the turbulent stresses in the x3-momentum equation are all small compared with the gravitational 

acceleration g, they generally are neglected entirely.

The expressions 2.20 and 2.21 are one form of the stress terms. Another form for these terms can be obtained by first rearranging the right side of 2.19 (omitting ∂ 2

3

--K ⎝ ⎠ ⎛ ⎞ ∂x1 ⁄ ) as follows:

∂

x1 ∂

------- 2νt

∂u1

∂x1

-------- ∂

x2 ∂

------- νt

u1 ∂

x2 ∂

-------- u2 ∂

x1 ∂

+ -------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- νt

u1 ∂

x3 ∂

-------- u3 ∂

x1 ∂

+ -------- ⎝ ⎠ ⎛ ⎞ + +

∂

∂xj

------- νt

∂u1

∂xj

-------- ⎝ ⎠ ⎛ ⎞ ∂

∂xj

------- νt

∂uj

∂x1

-------- ⎝ ⎠ ⎛ ⎞ +

∂

xj ∂

------ νt

u1 ∂

xj ∂

-------- ⎝ ⎠ ⎛ ⎞ νt

∂

x1 ∂

------- uj ∂

xj ∂

------- ⎝ ⎠ ⎛ ⎞ uj ∂

x1 ∂

------- νt ∂

xj ∂ + + ------- .

=

=

(2.22)

In this last expression, the second term equals zero by the continuity equation (eq. 2.14), and the third term can be neglected if the 

gradients of the eddy viscosity are assumed to be small.10 After these simplifications, the stress terms in the x1- and x2-momentum 

equations, respectively, can be written as

10In an estuary of large horizontal extent, horizontal gradients of the eddy viscosity generally are small. The vertical gradient of the eddy viscosity, 

∂vt/∂x3, may not be small, but it is multiplied in equation 2.22 by ∂u3/∂x1, the horizontal gradient of the vertical velocity, which as noted previously can be 

neglected.
∂

x1 ∂

------- νt

u1 ∂

x1 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x2 ∂

------- νt

u1 ∂

x2 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- νt

u1 ∂

x3 ∂

-------- ⎝ ⎠ ⎛ ⎞ + +

26 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

(2.23)

and

∂

x1 ∂

------- νt

u2 ∂

x1 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x2 ∂

------- νt

u2 ∂

x2 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- νt

u2 ∂

x3 ∂

-------- ⎝ ⎠ ⎛ ⎞ + + . (2.24)

The form of 2.23 and 2.24 does not preserve the symmetry of the turbulent stress tensor because the right sides of 

u1′ u2 – ′ νt u1 ∂ x2 = ( ) ⁄ ∂ and u2′ u1 – ′ νt u2 ∂ x1 = ( ) ⁄ ∂ are no longer equal as they should be. For this reason, the form of 

2.20 and 2.21 sometimes is preferred. It turns out, however, that the loss of symmetry in the stress tensor generally is not a serious 

concern for estuarine modeling.

When 2.20 and 2.21 or 2.23 and 2.24 is introduced into equation 2.15, the problem of turbulence closure is shifted to that of 

defining the distribution of the eddy viscosity. Unlike the molecular viscosity, the eddy viscosity is not a fluid property but instead 

varies throughout the flow field as a function of the length and velocity scales of the turbulent eddies. Because the largest-scale 

eddies are primarily responsible for the turbulent transport of momentum and salt, these largest eddies influence most the magnitude of vt

. The geometry of the estuary determines to a great extent the size of the largest eddies. Because of the striking disparity 

between the length scales of the horizontal and vertical dimensions of most estuaries, the intensity and size of the horizontal eddies 

are much greater than those of the vertical eddies. The turbulent motions under these conditions are said to be anisotropic (direction 

dependent). It is typical to define separate horizontal and vertical eddy viscosities, AH and AV, to account for the anisotropy. The 

turbulent stress terms in the form of 2.23 and 2.24 then become

∂

x1 ∂

------- AH

u1 ∂

x1 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x2 ∂

------- AH

u1 ∂

x2 ∂

-------- ⎝ ⎠ ⎛ ⎞ ∂

x3 ∂

------- AV

u1 ∂

x3 ∂

-------- ⎝ ⎠ ⎛ ⎞ + + (2.25)

and

∂

∂x1

-------- AH

∂u2

∂x1

-------- ⎝ ⎠ ⎛ ⎞ ∂

∂x2

-------- AH

∂u2

∂x2

-------- ⎝ ⎠ ⎛ ⎞ ∂

∂x3

-------- AV

∂u2

∂x3

-------- ⎝ ⎠ ⎛ ⎞ + + . (2.26)

For estuaries, a possible range of magnitudes is 103 to 106 cm2/s for AH and 1 to 500 cm2/s for AV.

Because of the large horizontal extent of flow in most estuaries, the horizontal variations in mean-flow quantities generally 

are much more gradual than the vertical variations. As a result, the four terms in 2.25 and 2.26 involving horizontal gradients of 

mean velocities ordinarily are much smaller than the two terms involving vertical gradients. (This will be demonstrated also in

Section 2.5 using scale analysis.) Each of the terms in 2.25 and 2.26 represent momentum diffusion due to turbulence in addition 

to their interpretation as stresses. Horizontal momentum diffusion generally is less significant in estuaries than vertical momentum 

diffusion.
2. Governing Equations and Boundary Conditions 27

 Some amount of horizontal momentum diffusion must be present in numerical calculations to reproduce horizontal circulations in side bays, as shown theoretically by Flokstra (1976), and through numerical experiments by Ponce and Yabusaki (1980, 

1981). For this reason, the horizontal, momentum-diffusion terms should not be omitted from a model even if they are relatively

small. In some models, horizontal diffusive transport is introduced through either a diffusive numerical scheme or a smoothing 

procedure. It is better to choose a numerical scheme that is free of numerical diffusion and introduce a proper amount of physical 

diffusion with the appropriate terms in the governing equations. Horizontal diffusion terms are easily included in an estuarine

model because refined modeling of the distribution of AH generally is not warranted. For flows with constant or slowly varying 

geometries, a constant value for AH often is satisfactory. For flows with more complex geometries, approaches such as those proposed in papers by Smagorinsky and others (1965) and Leith (1968) that relate the magnitude of AH to the size of the largest eddies 

being resolved by a model and to the local deformation field or vorticity, respectively, can be used.

The assumption of a constant vertical eddy viscosity generally is not adequate for a realistic description of the vertical, turbulent-momentum diffusion in estuaries. In open channel flows that are steady and neutrally buoyant (unstratified), theory and 

experiment show that AV has a nearly parabolic distribution over the depth with the maximum value occurring near mid-depth 

(Nezu and Rodi, 1986). In a density stratified flow, AV is reduced near the stratification so that the vertical distribution becomes 

bimodal, with maxima above and below the density gradient and a minimum at the gradient itself (Bowden, 1977). The effect of 

stratification can be represented by empirical stability functions ΦΜ of the type

AV AV0ΦM = ( ) Ri , (2.27)

where AV0is the neutral value of the eddy viscosity without stratification and Ri is the gradient Richardson number defined as

Ri g

ρ

-- ∂ρ

∂x3

-------- ∂u1

∂x3

-------- ⎝ ⎠ ⎛ ⎞ 2 ∂u2

∂x3

-------- ⎝ ⎠ ⎛ ⎞ 2

+

–1

= – . (2.28)

The Richardson number represents a ratio of gravity to inertial forces and characterizes the importance of buoyancy effects. The 

subscript M in equation 2.27 identifies the stability function for momentum diffusion which can be expressed in the form

ΦM( ) Ri 1 β1 ( ) + Ri α1 = (2.29)

as proposed by Munk and Anderson (1948).11 In equation 2.29, β1 and α1 are coefficients that generally are chosen to fit whatever 

data is available. A review by Delft Hydraulics Laboratory (1974) reports that values of β1 = 10 and α1 = −0.5, used by Munk and 

Anderson (1948), best fit most of the experimental data available for large Ri (Ri ≥ 0.7); for Ri < 0.7, the scatter in the data was so 

great that no best fit could be selected. Because values of Ri below 0.7 are common in environmental flows, it is understandable 

that numerous values for β1 and α1 have been reported in the literature. Blumberg (1986, p. 87, table 4.1) gives a partial summary 

of previously reported values for β1 and α1.

The vertical distribution of AV0in equation 2.27 can be determined by using the well-known mixing-length model that originated with Prandtl (1925). The mixing-length model relates the eddy viscosity under unstratified conditions to the gradients of 

velocity in the mean flow and a single parameter known as the mixing length Λ. For general 3-D flows, the mixing-length model 

is (Rodi, 1980, p. 18)

11Other functional forms for ΦM have been given by Blumberg (1977) and Henderson-Sellers (1982), but these have not been as widely used as 2.29.
AV0

Λ2 ∂ui

∂xj

------- ∂uj

∂xi

+ ------- ⎝ ⎠ ⎛ ⎞ ∂ui

∂xj

-------

1⁄

2

=

28 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

. (2.30)

For flows where the most significant velocity gradients are the vertical gradients, ∂u1/∂x3 and ∂u2/∂x3, the expression can be 

approximated by

AV0Λ2 ∂u1

∂x3

-------- ⎝ ⎠ ⎛ ⎞ 2 ∂u2

∂x3

-------- ⎝ ⎠ ⎛ ⎞ 2

+

1⁄

2

= . (2.31)

The mixing-length Λ can be a parameter adjusted during model calibration or defined beforehand by using the following formula 

for unstratified, open channel flow:

Λ κz' 1 z'

H– ---- ⎝ ⎠ ⎛ ⎞

1⁄

2

= , (2.32)

where

κ is the von Kármán constant (~ 0.41),

z′ is the vertical distance measured from the estuarine bottom to the point considered, and

H is the total water depth.

Equation 2.32 is derived by assuming the logarithmic velocity law for steady, open-channel flow (Keulegan, 1938) applies over the 

entire depth of flow.12 Even without density stratification, the velocity profiles in unsteady, estuarine flows are often nonlogarithmic owing to the effects of accelerations and secondary currents that are induced by a non-uniform channel geometry or 

transverse density gradients. Velocity profiles are also affected if flows are carrying sufficiently high sediment loads that sediment 

grains in suspension damp the vertical components of turbulent fluctuations. For these reasons, equation 2.32 must be considered 

as approximate for use in estuaries.

Wind blowing across an estuary can affect the magnitude of the eddy viscosity by inducing vertical shear in the horizontal 

velocities (∂u1/∂x3 and ∂u2/∂x3) from surface drift and by causing surface waves that produce oscillatory water motions. The effect 

of velocity shear on the eddy viscosity is incorporated in equation 2.31 as the shear is developed from the wind stress boundary 

condition imposed at the free surface. Wind-wave effects, however, must be simulated by a separate wave-current interaction 

model such as those discussed by Grant and Madsen (1979, 1986), Spaulding and Isaji (1987), Signell and others (1990), and 

Davies and others (1993). Wind waves in estuaries generally have periods below 10 seconds so their effect on water movements 

is within the time scale considered as turbulence. Water movements are influenced by surface waves to a depth below the free 

surface of about one-half their wave length. In shallow water, if wave-induced oscillatory water motions are non-zero at the estuarine bottom, the turbulence is enhanced, which in turn effects bottom friction and ultimately the wind-driven flow field (Davies 

and others, 1993). Wave conditions are a function of wind speed, duration, and fetch, so a separate wave model is needed for hydrodynamic simulations to predict wave conditions from meteorological data and information on the geometry of the water body. 

Because the formulations for wave and wave-current interaction models are rather specialized subjects, they are not considered 

here. However, for realistic applications of a 3-D estuarine model during wind conditions (especially in shallow waters), a 

wind-wave component should be included in the model.

12Nezu and Rodi (1986, p. 338, eq. 10) have refined equation 2.32 by including a wake function in the region away from the boundary (z′/H > 0.2) to 

account for deviations from the logarithmic law in steady, open-channel flow. The Nezu and Rodi refinement is not needed here, considering the approximate 

nature of equation 2.32 for estuarine flows.
2. Governing Equations and Boundary Conditions 29

2.3.1.2 Eddy Diffusivity

The turbulent transport of mass (in this case salt) in estuaries is closely analogous to the turbulent transport of momentum. 

The analogy, referred to as the Reynolds analogy, is indicated by the similar roles played by the mass flux term u′

i s′ in 

equation 2.16 and the momentum flux term u′

i u′

j in equation 2.15 (Sayre, 1975). Similar to the way in which the eddy viscosity 

is introduced with equation 2.18 by relating the turbulent transport of momentum to the gradients of mean velocity, an eddy diffusivity is introduced to relate the turbulent transport of salt to the gradient of the salt concentration

u′

i s′ D ∂s

∂x

i

= ------- , (2.33)

where D is the eddy diffusivity for salt mass.

The numerical value of D, like the eddy viscosity, depends on the direction of the turbulent transport, and therefore horizontal 

and vertical eddy diffusivities, DH and DV, are defined.13 The turbulent salt flux terms in equation 2.16 then become

∂

x

i ∂

------ u′

i ( ) – s′ ∂

∂x1

-------- DH

∂s

∂x1

-------- ⎝ ⎠ ⎛ ⎞ ∂

∂x2

-------- DH

∂s

∂x2

-------- ⎝ ⎠ ⎛ ⎞ ∂

∂x3

-------- DV

∂s

∂x3

-------- ⎝ ⎠ ⎛ ⎞ = + + . (2.34)

For estuaries, a possible range of magnitudes is 103 to 106 cm2/s for DH and 10−1 to 500 cm2/s for DV. Because the molecular 

diffusivity for salt (λ ~ 10−5

 cm2/s) is at least four orders of magnitude smaller than the smallest possible eddy diffusivity, the 

molecular diffusion term in equation 2.16 can safely be neglected.

The values of the horizontal and vertical eddy diffusivities can be related to the corresponding values of the eddy viscosities

by defining horizontal and vertical turbulent Schmidt numbers

ScH

AH

DH

= ------- and ScV

AV

DV

= ------- . (2.35)

When momentum transfer is unaffected by mass transfer, it can be assumed that the Reynolds analogy is complete and the Schmidt 

numbers equal unity. For estuaries, ScH = 1.0 is generally a reasonable assumption, but the vertical mass transfer of salt can cause 

significant vertical density stratification that alters the momentum transfer and the value of ScV. To account for the variation in 

ScV , a separate stability function for salt diffusion can be introduced as

DV DV0Φs = ( ) Ri , (2.36)

13D is actually a second rank, symmetric tensor, Dij. If the coordinate axes are chosen to coincide with the principal axes of the flow (and therefore the 

principal axes of the diffusion tensor), the tensor is diagonal, although not isotropic, in which case

Dij

DH 0 0

0 DH 0

0 0 DV

= .
30 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

where DV0is the neutral value of the eddy diffusivity without stratification. By using the Reynolds analogy, DV0can be equated 

to AV0as defined by equation 2.31. The form of Φs(Ri), as proposed by Munk and Anderson (1948), is

Φ

s

( ) Ri 1 β2 ( ) + Ri α2 = , (2.37)

where β2 and α2 are coefficients chosen by Munk and Anderson: β2 = 3.33 and α2 = −1.5. The numerical values for β2 and α2 are 

not universal constants, and in all cases they should be checked against available data and adjusted as necessary. According to 2.29

and 2.37 (and using the coefficients chosen by Munk and Anderson [1948]), the vertical turbulent Schmidt number ScV increases 

with Ri and with the amount of stable stratification. Officer (1977, p. 17, table 1.2) published values from six estuaries for the 

inverse of the vertical turbulent Schmidt number ( = DV/AV) which, when inverted into Schmidt numbers, lie in the range of 10 to 

0.83.14 For a Richardson number of 0.5, the Munk and Anderson formulas predict that ScV = 1.8.

2.3.2 Treatment of the Pressure Term

The shallow-water flows considered herein are assumed to be essentially horizontal and, therefore, vertical accelerations and 

velocities are negligible compared to gravity. As a result of this assumption, only the pressure and gravity terms are retained in the 

x3-momentum equation, which then reduces to the hydrostatic pressure equation.15 The Coriolis terms in the horizontal momentum 

equations that involve the vertical velocity u3 can also be neglected, as discussed in Appendix A. By incorporating these simplifications and representing the turbulence fluxes as shown previously, equations 2.14 to 2.16 can be written in coordinates 

(x, y, z) = (x1, x2, x3) as

∂u

∂x

----- ∂v

∂y

----- ∂w

∂z + + ------ = 0 , (2.38)

∂u

∂t

----- ∂uu

∂x -------- ∂uv∂y -------- ∂uw∂z +++ --------- – fv 1

ρ0

------∂p

∂x

----- ∂

∂x

----- AH

∂u

∂x

----- ⎝ ⎠ ⎛ ⎞ ∂

∂y

----- AH

∂u

∂y

----- ⎝ ⎠ ⎛ ⎞ ∂

∂z

----- AV

∂u

∂z

----- ⎝ ⎠ ⎛ ⎞ = – +++ , (2.39)

∂v

∂t

----- ∂uv

∂x -------- ∂vv∂y -------- ∂vw∂z +++ + --------- fu 1

ρ0

----- ∂p

∂y

----- ∂

∂x

----- AH

∂v

∂x

----- ⎝ ⎠ ⎛ ⎞ ∂

∂y

----- AH

∂v

∂y

----- ⎝ ⎠ ⎛ ⎞ ∂

∂z

----- AV

∂v

∂z

----- ⎝ ⎠ ⎛ ⎞ = – + + + , (2.40)

0 1

ρ0

-----∂p

∂z – ----- ρρ0

= – -----g , and (2.41)

∂s

∂t

---- ∂us

∂x -------- ∂vs∂y ------- ∂ws∂z +++ --------- ∂

∂x

----- DH

∂s

∂x

----- ⎝ ⎠ ⎛ ⎞ ∂

∂y

----- DH

∂s

∂y

----- ⎝ ⎠ ⎛ ⎞ ∂

∂z

----- DV

∂s

∂z

----- ⎝ ⎠ ⎛ ⎞ = + + , (2.42)

where u, v, and w are the velocities in the x, y, and z directions and f is the Coriolis parameter (see Appendix A). The hydrostatic 

pressure equation is 2.41, which has been substituted for the z-momentum equation. The advective acceleration terms (∂uu/∂x, 

∂uv/∂y, ∂uw/∂z, and so forth) in equations 2.39 and 2.40 are written in a conservative form (or divergence from) similar to that used 

in equation 2.5. This form is obtained by adding the continuity equation (eq. 2.38), multiplied by the appropriate velocity 

component, to the left side of the momentum equations (similar to eq. 2.6).

14The values for AV and DV reported by Officer (1977) were for tidally averaged conditions.

15Casulli and Stelling (1996) present a 3-D model in which the hydrostatic pressure assumption is not introduced and give examples showing when the 

non-hydrostatic pressure component should not be neglected.
2. Governing Equations and Boundary Conditions 31

Using the hydrostatic equation, the horizontal pressure gradient terms in the momentum equations, (∂p/∂x)/ρ0 and (∂p/∂y)/ρ0, 

can be reformulated in terms of the gradients of the water surface elevation and density. Integrating 2.41 from a depth z to the free 

(water) surface, where z = ζ(x, y, t) (fig. 2.2), gives

∂p

∂z

----- dz′

z

ζ

∫ g ρ dz′

z

ζ

∫ = – . (2.43)

Then carrying out the left-side integration and assuming the pressure at the free surface equals pa, the atmospheric pressure, yields

pa – p g ρ dz′

z

ζ

∫ = – . (2.44)

Now differentiating 2.44 with respect to x and y and applying Leibnitz’ rule to move the differentiation inside the integral term 

leaves

∂p

∂x

----- pa ∂

∂x -------- gρs

∂ζ

∂x

----- g

∂ρ

∂x

----- dz′

z

ζ

∫ = + + (2.45)

and

∂p

∂y

----- pa ∂

∂y -------- gρs

∂ζ

∂y

----- g

∂ρ

∂y

----- dz′

z

ζ

∫ = + + , (2.46)

where ρs is the surface density. If one assumes ρs = ρ0 to the same order of approximation as the Boussinesq approximation, then 

the pressure gradient terms become

1

ρ0

----- ∂p

∂x

----- 1

ρ0

----- pa ∂

∂x -------- g

∂ζ

∂x

----- g

1

ρ0

----- ∂ρ

∂x

----- dz′

z

ζ

∫ = + + (2.47)

Figure 2.2. Sketch defining the free-surface elevation ζ.

Water surface

ζ ( ) x, , y t x

z

y

Datum
32 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

and

1

ρ0

----- ∂p

∂y

----- 1

ρ0

----- pa ∂

∂y -------- g

∂ζ

∂y

----- g

1

ρ0

----- ∂ρ

∂y

----- dz′

z

ζ

∫ = + + . (2.48)

The integral terms in equations 2.47 and 2.48 are referred to as baroclinic terms and increase with depth16 under most conditions. Because the baroclinic terms require a complete specification of the density field, they couple the hydrodynamic solution to 

the solution for the salt field. The importance of these terms accounts for the primary difference between modeling an estuary and 

a river of homogeneous density. In an estuary, the horizontal density gradients often are an important driving force for the flow, 

as illustrated for the longitudinal density gradients in figure 1.1.

The water surface slope terms in equations 2.47 and 2.48 are referred to as the barotropic terms and are constant with depth. 

These incorporate the important driving force of the tide. 

The terms in equations 2.47 and 2.48 involving the gradients of atmospheric pressure generally are neglected in estuarine 

models, using the rationale that no significant gradients of atmospheric pressure are produced because of the relatively small size 

of estuaries. Changes in atmospheric pressure cause the water level in most estuaries to adjust “coherently” (uniformly) over the 

entire estuary. Normal variations in atmospheric pressure of ±20 millibars cause variations in estuarine water levels of approximately 20 cm; these variations are introduced into a numerical simulation through the open boundary conditions rather than 

through the governing equations themselves. Gradients of atmospheric pressure generally are important only in wide-area models 

of the coastal or open ocean when, for example, storm-surge predictions are attempted. If the atmospheric pressure terms are 

neglected, it then is common to assume the pressure at the free surface corresponds to zero “gage” pressure.

Substituting equations 2.47 and 2.48, without the atmospheric pressure gradient terms, into equations 2.39 and 2.40 gives

∂u

∂t

----- ∂uu

∂x -------- ∂uv∂y -------- ∂uw∂z +++ --------- – fv g

∂ζ

∂x – -----

g

1

ρ0

----- ∂ρ

∂x

----- dz′

z

ζ

∫ ∂

∂x

----- AH

∂u

∂x

----- ⎝ ⎠ ⎛ ⎞ ∂

∂y

----- AH

∂u

∂y

----- ⎝ ⎠ ⎛ ⎞ ∂

∂z

----- AV

∂u

∂z

----- ⎝ ⎠ ⎛ ⎞ – + + +

= (2.49)

and

∂v

∂t

----- ∂uv

∂x -------- ∂vv∂y -------- ∂vw∂z +++ + --------- fu g

∂ζ

∂y – -----

g

1

ρ0

----- ∂ρ

∂x

----- dz′

z

ζ

∫ ∂

∂x

----- AH

∂v

∂x

----- ⎝ ⎠ ⎛ ⎞ ∂

∂y

----- AH

∂v

∂y

----- ⎝ ⎠ ⎛ ⎞ ∂

∂z

----- AV

∂v

∂z

----- ⎝ ⎠ ⎛ ⎞ – + + + .

= (2.50)

These are the forms of the x- and y-momentum equations that apply to estuarine tidal flows influenced by density variations.

16For the case of a vertically well-mixed estuary, the density is independent of z and the baroclinic terms increase linearly with depth.
2. Governing Equations and Boundary Conditions 33

2.4 Boundary Conditions

To complete the system of mean-flow equations introduced in section 2.3, it is necessary to specify the boundary conditions 

for an estuarine flow. For a 3-D model application, boundary conditions must be specified at the free surface, bottom, shoreline, 

and open boundaries of the estuary. In this report, it is assumed that all boundaries other than the free surface are fixed and do not 

change with time. This eliminates any consideration of vertical movement of the bottom profile caused by sediment transport or 

the lateral movement of shoreline boundaries caused by wetting and drying of tidal flats. The treatment of these moving boundaries 

involves topics that are outside the scope of this report. It also is assumed that the estuary bottom is impermeable and that the effects 

of precipitation and evaporation at the free surface are negligible. The 3-D model test cases included in this report do not involve 

open boundaries, so the formulation of a generalized open boundary condition is not discussed here. It is recognized, 

however, that the proper specification of open boundary conditions is a challenging problem in modeling.

2.4.1 Free Surface

Hydrodynamic boundary conditions at the free surface require that kinematic and dynamic conditions be satisfied. These conditions are discussed below. The boundary condition for the salt transport equation requires zero salt flux (∂s/∂z = 0) at the free 

surface.

2.4.1.1 Kinematic Surface Condition

The kinematic surface condition is derived by balancing the mass fluxes into and out of a control volume enclosing a thin 

layer of fluid mass immediately below the free surface. Figure 2.3 illustrates the velocities on the faces of the control volume at 

one instant in time. The change in mass within the control volume during a time interval Δt (∂ζ/∂t × ρΔxΔyΔt) must equal the net 

inflow of mass to the control volume minus the net outflow. The flow through a fluid surface is the product of the velocity normal 

to the surface, and the area. Mathematically, the mass balance (after canceling density17) can be expressed as

∂ζ

∂t

-----ΔxΔyΔt u

1

2

--∂u

∂x – -----Δx ⎝ ⎠ ⎛ ⎞ Δz 12

--∂ζ

∂x – -----Δx ⎝ ⎠ ⎛ ⎞ ΔyΔt v

1

2

--∂v

∂y – -----Δy ⎝ ⎠ ⎛ ⎞ Δz 12

--∂ζ

∂y – -----Δy ⎝ ⎠ ⎛ ⎞ = + ΔxΔt

u

1

2

--∂u

∂x

+ -----Δx ⎝ ⎠ ⎛ ⎞ Δz 1

2

--∂ζ

∂x

+ -----Δx ⎝ ⎠ ⎛ ⎞ – ΔyΔt v

1

2

--∂v

∂y

+ -----Δy ⎝ ⎠ ⎛ ⎞ Δz 1

2

--∂ζ

∂y

+ -----Δy ⎝ ⎠ ⎛ ⎞ – ΔxΔt

w

1

2

--∂w

∂z – ------Δz ⎝ ⎠ ⎛ ⎞ + ΔxΔyΔt . 

(2.51)

Dividing through the above equation by ΔxΔyΔt, then cancelling terms and neglecting the small term involving ∂w/∂z which 

vanishes as Δz → 0, gives the kinematic condition for the free surface:

∂ζ

∂t

----- u∂ζ

∂x

----- v

∂ζ

∂y

+ + ----- – w = 0 on z = ζ( ) xyt , , . (2.52)

This form of the kinematic condition requires that evaporation and precipitation at the free surface be negligible and also ignores 

the effects from any mass of water that may break away from the free surface as spray during especially rough seas.

17Any effects of density variations on the mass balance are neglected using similar arguments made earlier in developing the continuity equation. The 

flow also is assumed incompressible.
2. Governing Equations and Boundary Conditions 34

2.4.1.2 Dynamic Surface Condition

The derivation of the dynamic surface condition is similar in method to the derivation of the kinematic condition. The equation of motion (ΣF = ma) is applied to a control volume enclosing a thin fluid layer at the surface (fig. 2.4). Viscous normal and 

shearing stresses act on the vertical and bottom faces of the fluid layer, and the free surface is subject to a horizontal wind stress 

vector τs with components τxs and τys acting in the plane of the free surface.18 The pressure at the free surface is constant, corresponding to atmospheric pressure (zero gage pressure). The free surface is assumed to be smooth, so that the wind transfers stress 

to the water through viscous shear. Because the terms in the equation of motion involving the mass of the water in the control volume are of higher order in smallness than the stresses themselves, they will vanish in the limit as the dimensions of the control 

18The free surface is assumed to be sufficiently close to horizontal that the cosines of the vertical angles between the free surface and the x- and y-axes 

equal unity.

Figure 2.3. Velocities for mass flux balance on a surface control volume in three dimensions. 

Expressions in the form represent the velocities on each control volume face. Expressions in the form

represent the heights of each vertical control volume face.

u 1

2

--∂u

∂x + -----Δx

Δz 1

2

--∂ζ

∂x + -----Δx

w

1

2

-- ∂w

∂z – ------ Δ z

z

v

1

2

-- ∂v

∂y – -----Δ y

x

u

1

2

-- ∂u

∂x

+ ----- Δ x

v

1

2

-- ∂v

∂y

+ -----Δ y

y

u

1

2

-- ∂u

∂x – ----- Δ x

Δ z

1

2

-- ∂ζ

∂x – ----- Δ x

Δ z

1

2

-- ∂ζ

∂y

+ ----- Δ y

Δ z

1

2

-- ∂ζ

∂x

+ ----- Δ x

Δ z

Control

volume

Δ x

Δ y

ζ ( ) x, y

Datum

Δ z

1

2

-- ∂ζ

∂y – ----- Δ y
2. Governing Equations and Boundary Conditions 35

volume approach zero. The application of the equation of motion then reduces to a stress balance on the control volume. Summing

the forces (stress × area) in the x-direction from figure 2.4 gives

τxz

1

2

--

∂τxz

∂z – ----------Δz ⎝ ⎠ ⎛ ⎞ – ΔxΔy = 0.

τ

xy

1

2

--

∂τ

xy

∂y

+ ----------Δy ⎝ ⎠ ⎛ ⎞ Δz 1

2

--∂ζ

∂y

+ -----Δy ⎝ ⎠ ⎛ ⎞ Δx τ

xy

1

2

--

∂τ

xy

∂y – ----------Δy ⎝ ⎠ ⎛ ⎞ Δz 12

--∂ζ

∂y – -----Δy ⎝ ⎠ ⎛ ⎞ + – Δx

τ

xsΔxΔy τxx

1

2

--

∂τxx

∂x

+ ----------Δx ⎝ ⎠ ⎛ ⎞ Δz 1

2

--∂ζ

∂x

+ -----Δx ⎝ ⎠ ⎛ ⎞ Δy τ

xx

1

2

--

∂τ

xx

∂x – -----------Δx ⎝ ⎠ ⎛ ⎞ Δz 12

--∂ζ

∂x – -----Δx ⎝ ⎠ ⎛ ⎞ + – Δy

(2.53)

Solving this equation for τxs, then cancelling terms and eliminating the terms which vanish as Δz → 0, yields

τxs τ

xx

∂ζ

∂x – ----- τ

xy

∂ζ

∂y – ----- τ

xz = + . (2.54)

For the y-direction, the corresponding relation is

τ

ys

τ

yx

∂ζ

∂x – ----- τ

yy

∂ζ

∂y – ----- τ

yz = + . (2.55)

τxz

1

2

-- τxz ∂

∂z – -------- Δz

τx y

1

2

-- τx y ∂

∂y – ---------Δ y

τxx

1

2

-- τxx ∂

∂x

+ ---------Δ x

τyy

1

2

-- τy y ∂

∂y

+ -------- Δ y

τxx

1

2

-- τxx ∂

∂x – ---------Δ x

Control

volume

τyz

1

2

-- τyz ∂

∂z – -------- Δz

τy y

1

2

-- τy y ∂

∂y – ---------Δ y

τyx

1

2

-- τyx ∂

∂x – -------- Δ x

τx y

1

2

-- τx y ∂

∂y

+ ---------Δ y

τyx

1

2

-- τyx ∂

∂x

+ ---------Δ x

τ ys

τ xs

z

x

y

Datum

Figure 2.4. Viscous stresses on a surface layer of a water body affected by a wind stress with components 

τxs and τys .

Expressions in the form represent the viscous stresses on each vertical control volume face. τxy

1

2

--

∂τ

xy

∂y

+ ----------Δy
36 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The above expressions are generally simplified one step further to facilitate the formulation of a numerical boundary condition. By 

neglecting the slope of the free surface, the first two terms on the right sides of equations 2.54 and 2.55 are eliminated. Then the 

dynamic condition equations become

τ

xs τxz = and τys

τ

yz = , (2.56)

which is the form used in most estuarine models. If the atmospheric pressure-gradient terms in equations 2.47 and 2.48 are included 

in a model application, the dynamic condition must also include the specification of atmospheric pressure, p = pa(x, y, t), at the 

free surface. When a wind is blowing over a large water body, it should also be understood that the free surface will quickly change 

from smooth conditions, as assumed above, to rough conditions as a result of wave generation. The wind will create pressure 

differences across the waves that impart stresses to the water column as form drag rather than viscous shear stress alone. Because 

the net effect of the wind on the water body during either smooth or rough surface conditions is a momentum transfer (flux) from 

the air to the water flow, it is not really important how the transfer occurs physically except to understand the processes involved.

The fluid stresses (τxz, τyz) in equations 2.56 can be defined using the eddy viscosity concept (eq. 2.18) as

τ

xz ρ0AV

∂u

∂z = ----- and τ

yz ρ0AV

∂v

∂z = ----- , (2.57)

where it has been assumed that the horizontal gradients of vertical velocity can be neglected. The wind stresses (τxs, τys) customarily 

are defined using a quadratic stress law

τ

xs cwρaWa

2 = sinφ and τ

ys

cwρaWa

2 = cosφ , (2.58)

where

ρa is the atmospheric density (~1.2 kg/m3), 

cw is a dimensionless drag coefficient,

Wa is the wind speed 10 m above the water surface, and

Φ is the angle between the direction of the wind and the positive y-axis (measured positive in the clockwise direction from 

the y-axis to the wind vector).

The value for the drag coefficient cw is influenced by several factors, including wind speed, wave height, the stability of the 

meteorological boundary layer over the water surface, and the degree of variability of the wind speed and direction. Much research 

has been done on determining the value of cw. Blake (1991) discusses well the dependence of cw on wind speed and wave height 

and includes a reference list of previous work. Estimates of the drag coefficient range between 1.0 × 10-3 and 1.5 × 10-3 for wind 

speeds between about 4 and 15 m/s. For light winds over a smooth water surface or high winds over a rough water surface, higher

values may be required.
2. Governing Equations and Boundary Conditions 37

2.4.2 Bottom

At a solid boundary of an estuarine flow, the formal hydrodynamic condition is zero velocity. However, for numerical modeling of real estuaries, this boundary condition, known as no-slip, generally is not appropriate. The bottom boundary of an estuary 

is composed of roughness elements of varying sizes, as represented in figure 2.5

z

x

Figure 2.5. Flow over the bottom boundary of an estuary with roughness elements of varying sizes.

. The roughness elements can be due to individual 

sediment grains, bedforms, vegetation, or mounds and burrows related to biological activity. The bottom of most estuaries is classified as hydraulically rough in the sense used in hydraulic engineering to imply the boundary elements protrude above the thickness of the viscous sublayer (Tennekes and Lumley, 1972, p. 158-160). The no-slip condition is satisfied only at the exact boundary 

surface itself, while along an average boundary, defined as z = 0 in figure 2.5, the mean velocity obtained by integrating along an 

arbitrary segment of the x-axis will usually not be zero. The roughness elements at the boundary generate turbulent wakes that are 

responsible for form drag on the bottom that by far is the largest part of the resistance to flow (the other part is caused by the viscous 

shearing stresses that act locally on the actual boundary surface in the tangential direction). The no-slip condition applied at z = 0

does not account for the influence of the form drag or its possible variations in space. Moreover, implementing the no-slip condition exactly at the boundary surface, even for a smooth boundary, is only meaningful if the very steep velocity gradients that prevail 

in the viscous sublayer are captured by the numerical model. In the viscous sublayer, the effect of fluid viscosity is important, so 

the viscous stress terms in the momentum equations must be included. Resolving the viscous sublayer requires a numerical grid 

size smaller than 1 millimeter, which is expensive in terms of computer resources. Because the flow in the immediate vicinity of 

the bottom boundary is important only insofar as it determines the overall stress exerted on the boundary, it is preferable to base 

the bottom boundary condition on the specification of the stress directly as a function of a velocity at some point away from the 

boundary (normally at the lowest model grid point, but well outside the viscous sublayer). This concept is used in formulating the 

dynamic boundary condition introduced below.

A kinematic condition at the bottom must also be satisfied, similar to that at the surface. The boundary condition at the bottom 

for the salt transport equation is zero salt flux.
38 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

2.4.2.1 Kinematic Bottom Condition

The kinematic bottom condition is derived by balancing mass fluxes on a thin fluid layer immediately above the bottom of 

the estuary (fig. 2.6). The result is

u

∂h

∂x

----- v

∂h

∂y

+ + ----- w = 0 on z h = – ( ) x y, , (2.59)

which is similar to the result at the surface except that the location of the bottom z = – h (x, y), unlike the free surface, is not a 

function of time and therefore ∂h/∂t = 0. In this case, h represents a vertical distance measured positive downward from a datum 

(such as a mean sea level height) above the bottom. Equation 2.59 is sometimes referred to as the tangential flow condition because 

it requires the flow on the boundary to be tangent to the boundary and not to separate from it.

w

1

2

-- ∂w

∂z

+ ------ Δz

v

1

2

-- ∂v

∂y – ----- Δ y

u

1

2

-- ∂u

∂x

+ ----- Δ x

v

1

2

-- ∂v

∂y

+ ----- Δ y

u

1

2

-- ∂u

∂x – ----- Δ x

Δ z

1

2

-- ∂h

∂x – ----- Δ x

Δ z

1

2

-- ∂h

∂y

+ ----- Δ y

Δ z

1

2

-- ∂h

∂y – ----- Δ y

Δ z

1

2

-- ∂h

∂x

+ ----- Δ x

Δ z

Control 

volume

Δ x

z

x

y

h(x,y)

Datum

Δ y

Figure 2.6. Sketch showing velocities for mass flux balance on a bottom control volume in three dimensions.

Expressions in the form represent the velocities on each control volume face. Expressions in the form 

represent the heights of each vertical control volume face.

u 1

2

--∂u

∂x + -----Δx

Δz 1

2

--∂h

∂x + -----Δx
2. Governing Equations and Boundary Conditions 39

2.4.2.2 Dynamic Bottom Condition

The dynamic bottom condition is derived by applying the equation of motion to the thin fluid layer on the bottom (fig. 2.7). 

The equation of motion reduces to a balance between the stresses exerted by the boundary on the fluid and the stresses in the fluid 

exerted on the boundary. The stress vector exerted by the boundary is τb and has components τxb and τyb acting along the boundary. 

The boundary slope in any direction is assumed sufficiently small so that the cosine of the vertical angle with the horizontal plane 

is unity. The balance of stresses results in

τ

xb τxx

∂h

∂x – ----- τ

xy

∂h

∂y – ----- τ

xz = + (2.60)

and

τ

yb τyx

∂h

∂x – ----- τ

yy

∂h

∂y – ----- τ

yz = + (2.61)

for the x- and y-directions, respectively. It is common practice to neglect entirely the slopes of the bottom boundary, ∂h/∂x and 

∂h/∂y, so the first two terms on the right sides of the above equations can be eliminated. Then the dynamic equations become

τ

xb τxz = and τyb τyz = , (2.62)

which are the counterparts to equations 2.56 used at the surface. To evaluate equations 2.62, the fluid stresses (τxz, τyz) can be 

defined using equations 2.57 applied at the bottom of the estuary. It remains then to define the boundary stresses (τxb, τyb).

Control

volume

τxz

1

2

-- τxz ∂

∂ z

+ -------- Δz

τxy

1

2

-- τxy ∂

∂ y – --------- Δ y

τxx

1

2

-- τxx ∂

∂ x

+ -------- Δ x

τyy

1

2

-- τyy ∂

∂ y

+ --------- Δ y

τyz

1

2

-- τyz ∂

∂ z

+ -------- Δz

τyy

1

2

-- τyy ∂

∂ y – --------- Δ y

τyx

1

2

-- τyx ∂

∂ x – --------- Δ x

τxy

1

2

-- τxy ∂

∂ y

+ --------- Δy

τyx

1

2

-- τyx ∂

∂ x

+ --------- Δ x

z

x

y

Datum

h(x,y)

τxx

1

2

-- τxx ∂

∂ x – --------- Δ x

τyb

τxb

Figure 2.7. Viscous stresses on a bottom layer of a water body with boundary stress components τxb and τyb. 

Expressions in the form represent the viscous stresses on each control volume face. τ

yz

1

2

--

∂τ

yz

∂z + -----------Δz
40 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

As noted previously, the strategy in defining the boundary stresses is to relate them to the velocity at a point above the boundary. For this purpose, the well-known logarithmic velocity law generally is adequate so long as the bottom topography is not highly 

nonuniform and the boundary layer is not stratified.19 It is assumed that the boundary layer is unaffected by wave-generated currents. Gross and Nowell (1983) found that the dynamic effects from temporal variability (unsteadiness) in a tidal boundary layer 

did not measurably affect the accuracy of predictions with the logarithmic law in the near-bed region. The appropriate logarithmic 

law for rough boundary flow is (Tennekes and Lumley, 1972, p. 165, eq. 5.2.49)

ures

u

--------- 1

κ

--ln z'

z0

= ----

* 

, (2.63)

where

z′ is a vertical distance measured outward from an average position among the roughness elements,

ures isthe resultant velocity parallel to the boundary at a height z' ,

u* isthe resultant shear velocity defined as τb ρ0 ⁄ ,

z0 is a characteristic length scale related to the height of the roughness elements and corresponding with the point 

of zero velocity (ures = 0 at z′ = z0), and

κ is von Kármán’s constant (~ 0.41).

Equation 2.63 is theoretically valid outside the region of boundary roughness and within a layer of constant (Reynolds) stress.

Although a “constant stress layer” exists in channel flow only in the immediate neighborhood of the boundary, the logarithmic 

profile typically represents the flow well outside this layer. In fact, for steady, uniform channel flow, the logarithmic law 

traditionally has been used to describe the velocity profile over the entire depth of flow (Keulegan, 1938). Under many ordinary 

conditions of tidal flow in estuaries, the logarithmic law is a reasonable representation of the velocity profile for the first few meters 

above the bed.

Equation 2.63 can be solved for the boundary shear stress to obtain a quadratic stress law of the form

τb ρ0Cdures ures = , (2.64)

where Cd is a dimensionless drag coefficient. The quadratic term in 2.64 involving the velocity is written with an absolute value 

sign so the frictional stress always will oppose the flow. In practice, equation 2.64 is rearranged to represent the two components 

of shear stress with

τ

xb ρ0Cd ub

2 v

b

2

+ ⎝ ⎠ ⎛ ⎞1⁄2ub and τ

yb ρ0Cd ub

2 v

b

2

+ ⎝ ⎠ ⎛ ⎞1⁄2v

b = = , (2.65)

where ub, vb are the horizontal velocity components at a point z'

b above the bottom, and

Cd κ

2 lnz'

b

z0

----- ⎝ ⎠ ⎛ ⎞ –2

= . (2.66)

19Refinements of the logarithmic law that incorporate the effect of density stratification are available (Turner, 1973, Chapter 5; Adams and Weatherly, 

1981).
2. Governing Equations and Boundary Conditions 41

Equations 2.65 allow the shear stress components (τxb, τyb) to be estimated from the velocities (ub, vb) determined by the model 

and an estimate of either z0 or Cd.

The value of z0 depends greatly on the grain size, bottom shape, and near-bed sediment transport (Grant and Madsen, 1982) 

and can vary widely. Pritchard (1956) suggested the values

z0 = 0.02 cm for a mud bottom,

z0 = 0.13 cm for a sand-gravel bottom, and

z0 = 0.16 cm for a sand-mud bottom.

The higher value suggested for the sand-mud bottom was attributed to the possible presence of ripples. In the presence of high 

sediment bedload transport, much higher values of z0 are often used, such as 1 cm (Adams and Weatherly, 1981).

Some modelers prefer to specify the drag coefficient Cd instead of z0 because it does not vary as widely as z0. Values of Cd

derived from fitting the logarithmic law to field data are usually reported using z'

b = 100 cm. In 3-D model studies, the vertical 

location of the lowest model grid point above the boundary normally defines the value for z'

b. In 2-D model studies, Cd values are 

sometimes reported that are based on the use of depth-averaged velocity components in equations 2.65. In cases where the bed 

roughness z0 is uniform, Cd should not remain constant if the value of z'

b is varying significantly. In general, the use of Cd in the 

range 0.0025 to 0.004 is fairly typical among 3-D coastal and estuarine modelers; this range of Cd corresponds to a range in z0, for 

z'

b = 100 cm, of 0.027 to 0.15 cm. Walters (1992), however, required values for Cd in excess of 0.03 when modeling the tides in 

the upper Delaware Estuary where channel constrictions and bars are prevalent.

2.4.3 Shoreline

Along a shoreline boundary, in theory, either a no-slip or quadratic-stress-law boundary condition can be applied. When modeling large water bodies, however, the typical horizontal grid spacing is so large (>0.25 km) that these boundary conditions yield 

a distorted horizontal velocity distribution near the boundary. Instead, a “perfect-slip” boundary condition is normally applied that 

permits the flow to move parallel to the boundary without any boundary frictional resistance. The perfect-slip condition is a reasonable choice so long as the shoreline boundary layer has little influence on horizontal velocity distributions, which is true in most 

estuaries. The perfect-slip condition can be implemented by adding a line of model nodes outside the boundary and defining the 

velocity at those outside nodes to be equal to the velocity at the interior nodes adjacent to the boundary. 

The mass flux of salt at a shoreline boundary must be set to zero. This is normally achieved by letting no flow cross the boundary, although care must be exercised so that the numerical method does not inadvertently transport salt across the boundary by 

diffusion.
42 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

2.5 Scaling Analysis

It is useful to recast the governing equations in a dimensionless form so that the magnitude of various terms or groups of terms 

can be estimated for a typical estuary (in this case San Francisco Bay). Cheng and others (1976) and Lynch (1986) have presented 

similar dimensionless analyses of the 3-D equations, but they selected scaling parameters representative of large lakes. Sheng 

(1983) presents a 3-D coastal ocean model where the governing equations are solved in a dimensionless form.

The dimensionless variables, denoted by an asterisk (*), are chosen as follows:

x* x

L = -- , y* y

L = -- , z* z

H = --- , t* = ωt ,

u* u

ur

= ---- , v* v

ur

= ---- , w* wL

urH = --------- ,

ζ* ζ

D = --- , ρ* ρ ρ1 –

ρ2 ρ1 – = ----------------- , and s* s s1 –

s2 s1 – = --------------- , (2.67)

where

L is a horizontal length scale,

H is a vertical length scale representing the water depth,

D is a vertical length scale representing the tidal amplitude,

ur is a horizontal reference velocity,

ρ1 is a minimum density,

ρ2 is a maximum density,

s1 is a minimum salinity,

s2 is a maximum salinity, and

ω is a frequency of the tidal forcing.

Introducing the dimensionless variables into equations 2.38, 2.49, 2.50, and 2.42 transforms the 3-D equations into the following 

dimensionless forms:

Continuity equation,

∂u*

∂x* -------- ∂v*∂y* -------- ∂w*∂z* + + --------- = 0 ; (2.68)
2. Governing Equations and Boundary Conditions 43

Horizontal momentum equations,

β∂u*

∂t* -------- Ro ∂u*u*

∂x* --------------- ∂u*v*∂y* --------------- ∂u*w*∂z* + + + ---------------- – v* Ro

Fr2 – --------D

H

---∂ζ*

∂x* --------

Ro

Fri

2 -------- ∂ρ*

∂x* ---------dz*

z*

D

H

--- ζ*

∫ – EH

∂

∂x* -------- ∂u*∂x* ---------- ⎝ ⎠ ⎛ ⎞ ∂∂y* -------- ∂u*∂y* ---------- ⎝ ⎠ ⎛ ⎞ + EV

∂

∂z* -------- ∂u*∂z* ---------- ⎝ ⎠ ⎛ ⎞ + +

= (2.69)

and

β∂v*

∂t* -------- Ro ∂u*v*

∂x* --------------- ∂v*v*∂y* -------------- ∂v*w*∂z* + + + + ---------------- u* Ro

Fr2 – --------D

H

---∂ζ*

∂y* --------

Ro

Fri

2 -------- ∂ρ*

∂y* ---------dz*

z*

D

H

--- ζ*

∫ – EH

∂

∂x* -------- ∂v*∂x* ---------- ⎝ ⎠ ⎛ ⎞ ∂∂y* -------- ∂v*∂y* ---------- ⎝ ⎠ ⎛ ⎞ + EV

∂

∂z* -------- ∂w*∂z* ----------- ⎝ ⎠ ⎛ ⎞ + + ;

= (2.70)

Salt transport equation,

β

Ro

------∂s*

∂t* -------- ∂u*s*

∂x* -------------- ∂v*s*∂y* -------------- ∂w*s*

∂z* + + + ---------------

1

PeH

--------- ∂

∂x* -------- ∂s*∂x* -------- ⎝ ⎠ ⎛ ⎞ ∂∂y* -------- ∂s*∂y* -------- ⎝ ⎠ ⎛ ⎞ +

1

PeV

--------- ∂

∂z* -------- ∂s*∂z* -------- ⎝ ⎠ ⎛ ⎞ +

=

. (2.71)

The coefficients of most terms now form recognizable dimensionless numbers that are identified in table 2.1. In arriving at the two 

horizontal momentum equations (2.69 and 2.70), the eddy coefficients were assumed to be constant. Also, the coefficients of the 

Coriolis terms were arranged to be unity in these two equations. The salt transport equation (2.71) was arranged so that the 

coefficient of the advection terms was unity.

Estimates of the scaling parameters and coefficients that are typical of the northern reach of San Francisco Bay are

L = 100 km H = 10 m

D = 1 m ur = 0.25 m/sec

f = 10−4 sec−1 ω = 5 × 10−5 sec−1

ρ1 = 103 kg/m3 ρ2 = 1. 025 × 103 kg/m3

s1 = 1 s2 = 34

ρ0 = 103 kg/m g = 10 m/sec2

AH = 102 m2/sec AV = 10–2 m2/sec

DH = 102 m2/sec DV = 10–2 m2/sec
44 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Table 2.1.Dimensionless numbers in 3-D equations.

Name Symbol Definition Ratio between Estimate

Rossby

number Ro

ur

fL

---- fluid acceleration

Coriolis acceleration

------------------------------------------------- 2.5 10– 2 ×

Froude

number Fr

ur

( ) gH 1

⁄2

--------------- fluid velocity

surface wave speed

---------------------------------------------- 2.5 10– 2 ×

Internal

Froude

number

Fri

ur

gHρ2 ρ1 –

ρ0

----------------- ⎝ ⎠ ⎛ ⎞ 1

⁄2

----------------------------------- fluid velocity

internal wave speed

----------------------------------------------- 0.5

Unsteadiness

number β ω

f

--- inertial period

tidal period

--------------------------------- 0.5

Depth

ratio — D

H

--- tidal amplitude

depth ----------------------------------- 10– 1

Horizontal

Ekman

number

EH

AH

fL2 ------- horizontal momentum diffusion

Coriolis acceleration ---------------------------------------------------------------------------- 10– 4

Vertical

Ekman

number

EV

AV

fH2 -------- vertical momentum diffusion

Coriolis acceleration --------------------------------------------------------------------- 1

Horizontal

Péclet

number

PeH

urL

DH

-------- mass advection

horizontal mass diffusion

------------------------------------------------------------ 2.5 102 ×

Vertical

Péclet

number

PeV

urH

DV

--------- mass advection

vertical mass diffusion

------------------------------------------------------ 2.5 102 ×
2. Governing Equations and Boundary Conditions 45

Because the northern reach of San Francisco Bay is influenced by freshwater inflow from the Sacramento−San Joaquin River Delta, 

significant density gradients can occur there. The forcing period (1/ω) was chosen to be approximately six hours, or one-quarter 

of the diurnal tidal cycle in San Francisco Bay, and corresponds with roughly the full range of local velocity change (from flood 

to ebb tide).

For the above set of parameters, the corresponding magnitudes of the dimensionless numbers are shown in table 2.1. In the 

dimensionless equations, the chosen parameters scale each partial derivative term to approximately equal size and of order unity. 

The coefficients for each term then provide an order-of-magnitude estimate of the relative importance of the terms and groups of 

terms with respect to others. The relative magnitude of the terms in the momentum balance equation are shown in table 2.2 and in 

the salt balance equation in table 2.3.

Owing to the strong tidal forcing in San Francisco Bay, the water surface slope terms are the largest in the momentum equation 

(table 2.2). The baroclinic, Coriolis, vertical momentum diffusion, and temporal acceleration terms also are very significant. The 

advective acceleration and horizontal momentum diffusion terms are the smallest terms in the estuarine momentum equation and 

generally are not of major significance. It is not recommended, however, that these terms be neglected. The need for some amount 

of horizontal momentum diffusion to reproduce horizontal circulation in side bays was discussed in section 2.3.1.1. The scaling 

analysis here also does not consider changes in the estuarine bathymetry or lateral geometry. Bathymetric and (or) geometric 

changes can induce significant horizontal velocity shear that may cause the advective acceleration and horizontal momentum diffusion terms to become locally significant. In San Francisco Bay, regions of high velocity gradients occur near the sharp bathymetric gradients at the edges of the deepwater navigation channels, near flow constrictions, and behind islands and headlands 

where eddies form.

Table 2.2. Relative magnitude of the terms in the momentum equation.

Term Coefficient Relative magnitude

Water surface slope 4

Baroclinic 1

Coriolis acceleration 1.0 1

Vertical momentum diffusion EV 1

Temporal acceleration β 0.5

Advective acceleration Ro 0.025

Horizontal momentum diffusion EH 0.0001

Table 2.3. Relative magnitude of the terms in the salt equation.

Term Coefficient Relative magnitude

Advective transport 1.0 1

Vertical salt diffusion 0.004

Horizontal salt diffusion 0.004

Ro D

Fr2H -------------

Ro

Fri

2 ----------

1 PeV ⁄

1 PeH ⁄
46 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

In contrast to the momentum equation, the advective transport terms in the salt equation are very significant (table 2.3) and 

should be evaluated carefully in a numerical model to avoid artificial diffusion or spurious oscillations of the salt concentration 

field. Péclet numbers as large as those in table 2.1 indicate that estuarine flows are “advection dominated,” meaning that the transport of salt (and generally other solutes) is mostly by advection by the water currents and not by turbulent diffusion. The diffusion 

terms in the salt transport equation, however, should not be neglected. The accurate modeling of the vertical diffusion of salt is 

essential in a stratified estuary if the vertical profile of salinity is to be reproduced accurately. The modeling of horizontal salt diffusion may be less highly refined than the vertical diffusion in most estuaries, but it is still required to determine the proper horizontal distribution of salt. Quite often the size of the horizontal numerical grid used in estuarine modeling is large. All the horizontal advective motions acting over smaller scales than the grid are not resolved directly by the model but are considered to be 

mixing, and they must be represented by the horizontal diffusion terms. For this reason, the horizontal eddy diffusion coefficient, 

DH, is partly based on the size of the smallest eddies that are resolved by a model.

2.6 Coordinate Transformations

The 3-D equations presented so far are written in standard Cartesian or rectangular coordinates.20 Thus, the equations have a 

relatively simple form. When using the finite-difference method, the Cartesian equations generally are solved on a square or rectangular grid. The calculations can be very efficient because the simple form of the equations and the uniform grid system are convenient for the implementation of most numerical schemes. Over the last two decades, however, it has become increasingly common to transform the governing equations into other coordinate systems that allow variable or curvilinear grids to be used with

greater flexibility in representing curved shoreline and bottom boundaries smoothly, or for resolving interior regions of interest 

using a higher density of grid points. The transformations can involve the horizontal coordinates or the vertical coordinate or both. 

Horizontal coordinate transformations are used almost exclusively in transforming the governing equations for finite-difference

solution methods. Finite-element and finite-volume methods allow one to work with irregular grid systems without transforming 

the coordinates of the governing equations. It is also possible to use irregular (usually triangular) grids with finite-difference methods (see, for example, Thacker, 1977), but coordinate transformations are more commonly used. A brief review of the horizontal 

and vertical coordinate transformations in use today [1997] for estuarine and coastal modeling follows.

20If a circulation model is to be applied to regions of the ocean having substantial variations in latitude (generally greater than a few tens of degrees), then 

the governing equations in rectangular coordinates are not appropriate. Spherical coordinates must be used to consider the curvature of the earth’s surface. For the 

relatively small horizontal scale of most estuaries, the use of rectangular coordinates introduces negligible errors.
2. Governing Equations and Boundary Conditions 47

2.6.1 Horizontal Coordinate Transformations

The simplest of horizontal coordinate transformations are those that stretch coordinates to form a numerical grid that is rectangular but not uniform. The transformation used by Butler (1978, 1980), and also implemented in the 3-D model by Sheng (1983),

takes the form

x ax bxxˆ cx = + , y ay byyˆ cy = + , (2.72)

where ax, bx, cx, ay, by, and cy are arbitrarily chosen stretching coefficients. The governing equations are usually solved on a 

rectangular grid with square grid boxes after being transformed into the new coordinates xˆ , yˆ . Several regions within the entire 

computational domain can be defined by using different sets of stretching coefficients for each region (fig. 2.8); the grid at the 

transition between regions should vary smoothly to prevent computational problems. When the stretching coefficients are constants 

within each region, the transformation by equations 2.72 does not add any extra terms to the transformed governing equations; the 

transformation does introduce the stretching coefficients into the horizontal derivative terms, however.

Boericke and Hall (1974) used a form of equations 2.72 with ax = 0, bx = cx = cy = 1, and ay and by as geometric (width) 

variables to map the shoreline of an irregular estuary to a rectangular computational space (fig. 2.9). The grid system is one form 

of a so-called boundary-fitted grid in which the boundaries of the water body are transformed to coincide with the coordinate lines 

of the grid system. The transformation positions the y-coordinate grid points uniformly across the width of the estuary along each 

x-coordinate grid line. The y grid spacing then varies with the changing width of the estuary along the longitudinal (x) direction. 

Different transformations can be applied to different regions to have greater flexibility in positioning points.

x

y

y region 2

x region 1

x

region

2

y region 1

yˆ

xˆ

Figure 2.8. Horizontal stretching of coordinates using two regions in both the x and y directions. (Modified from Sheng, 1983, fig. 2.2.)
48 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Noye and Stevens (1987) used the same horizontal transformation as Boericke and Hall but solved the transformed governing 

equations by using a nonuniform grid spacing in order to increase the grid resolution in regions of interest within the original coordinate system. The nonuniform grid spacing in the computational space must be chosen according to certain criteria presented by

Noye and Stevens to retain the accuracy of the numerical approximations that are achievable for uniformly spaced grids. Because

the stretching coefficients are not constant in the method of Boericke and Hall, an additional term is introduced into the expression 

for the transformed x-derivative.

To achieve even greater flexibility for the horizontal grid point placement than with stretched-coordinate models, curvilinear 

coordinate models can be used. These models solve forms of the governing equations that are transformed into curvilinear coordinates. The transformed equations can be solved on a rectangular grid, which is then transformed into a curvilinear grid in the 

original coordinate system. The curvilinear grid is generally fitted to the physical boundary to avoid the stair-step patterns that 

appear when curved boundaries are represented by conventional rectangular grids. Clustering grid points is possible in regions 

where increased grid resolution is needed, such as navigation channels (Thompson and Johnson, 1986).

Curvilinear coordinate models have been developed that are based on horizontal grid transformations that are conformal (Reid 

and others, 1977), generalized orthogonal (Willemse and others, 1985; Blumberg and Herring, 1987), and nonorthogonal (Johnson, 

1982; Johnson and others, 1991; Sheng, 1986a; Spaulding, 1984; Spaulding and others, 1990; Muin and Spaulding, 1996). Conformal transformations are fully defined by analytic functions of a complex variable by using the techniques of conformal mapping 

(Vallentine, 1967). Except at relatively few singular points, conformal grids are orthogonal: families of grid lines intersect perpendicularly to one another. By the definition of a conformal transformation, the angles of the rectangular grid in computational space 

are preserved in magnitude and sense under the transformation, which assures the orthogonality of the grid in the physical space. 

For estuarine modeling, a generalized orthogonal curvilinear grid is usually preferred over a conformal grid because it allows a 

more flexible spacing of grid lines; thus, for example, it is easy with a generalized orthogonal grid to choose grid lines that are

y

x

yˆ

xˆ

yˆ = 1

yˆ = 0

x x = ˆ

y b x = ( ) + B x( ) yˆ

A B

b x( )

B x( )

Figure 2.9. Plan view of an estuary shown in (A) physical space and (B) computational space using the transformation of 

Boericke and Hall (1974).
2. Governing Equations and Boundary Conditions 49

more closely spaced near a shoreline (fig. 2.10). The determination of a suitable mapping function for a conformal transformation 

can be a difficult task even for some relatively simple transformations. Nonorthogonal curvilinear grids are the most general of the 

curvilinear grids because they relax the strict requirement of orthogonality.21 An accurate fitting of a grid to the boundary of an 

irregularly shaped estuary often requires a nonorthogonal grid; conformal or generalized orthogonal grids generally must represent 

an irregular shoreline by some form of smoothed mean boundary profile. An example of a nonorthogonal curvilinear grid, used by 

Johnson and others (1991, 1993) for hydrodynamic modeling of Chesapeake Bay, is shown in figure 2.11.

The price for the greater flexibility of using curvilinear grids is that the form of the curvilinear governing equations is much 

more complicated than the Cartesian form, particularly the form in nonorthogonal curvilinear coordinates. The nonorthogonal curvilinear equations presented by Johnson and others (1991)22 list thirty additional terms appearing in each horizontal momentum 

equation resulting only from the transformation of the two Cartesian terms representing horizontal momentum diffusion; approximately double the number of original terms is also the result of transforming from the Cartesian form of the advection, pressure 

21Nonorthogonal grids generally should not deviate far from being orthogonal if numerical errors for the equations to be solved on the grid are to be kept 

under control.

22The equations presented by Johnson and others (1991) are in a form in which the velocity components are transformed to be everywhere perpendicular 

to the local curvilinear coordinate lines. This velocity component transformation is referred to as contravariant.

San Francisco

Bay

38o

123o 122o 121o 120o

36o

34o

Figure 2.10. Orthogonal curvilinear grid used by Blumberg and Herring (1987) to model a 

portion of the California coast. 

The grid spacing in the offshore direction varies from ~4 kilometers near the coast to 

~25 kilometers on the west. (Modified from Blumberg and Herring, 1987, fig. 11B.)
50 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

gradient, and Coriolis terms. The governing equations for use with orthogonal grids are significantly less complex that those for 

nonorthogonal grids, but they still include extra terms that represent the curvature of the coordinate system. Owing to the extra 

terms, the computational time that is required in solving the nonorthogonal curvilinear equations is approximately double that 

required for solving the Cartesian equations (Spaulding, 1984). Solving the orthogonal curvilinear equations requires approximately 25 percent greater computational time than the Cartesian equations (A.F. Blumberg, oral commun., 1995). Ideally the 

savings from any reduction in the number of grid points that is made possible by using a curvilinear model should be sufficient to 

compensate for the greater cost of the calculations for individual grid points. For a nonorthogonal curvilinear model in particular, 

that is not always possible. In these instances, the benefits from the greater flexibility in the placement of grid points using a nonorthogonal curvilinear model must outweigh the greater computational cost of the model, or a Cartesian or orthogonal-curvilinear 

model should be used.

James R

York R

Potomac R

Rappahannock R

Patuxent R

Patapsco R

Susquehanna R

Choptank R

C&D Canal

N

Figure 2.11. Nonorthogonal curvilinear grid used by Johnson and others (1991, 1993) to model Chesapeake Bay. 

(Modified from Johnson and others, 1991, fig. 2.)
2. Governing Equations and Boundary Conditions 51

Besides the increased complexity of the governing equations, the following additional drawbacks of models that are based on 

the curvilinear forms of the governing equations should be considered:

1. Considerable effort must be expended in defining the curvilinear grid transformation from computational to physical 

space. Generally, a numerical grid generator like the computer program EAGLE (Thompson and others, 1985) is needed to 

assign the spatial locations of the curvilinear grid points. For an orthogonal grid, the grid generation process is often iterative, 

continuing until a grid is found that reasonably approximates all boundaries of the domain while also satisfying the constraint

equations for the orthogonality of intersecting grid lines. Reviews of grid generation techniques for fluid mechanics computations are given by Eiseman (1985) and Hoffmann (1989).

2. Curvilinear models are not ideally suited for situations when the wetting and drying of boundary grid points occurs, as is 

common in estuarine tidal flats. In these situations, the estuarine boundary is in motion, and there is little likelihood that the 

boundary will be represented smoothly through all phases of the wetting and drying cycle.

3. When using curvilinear grids (especially strongly nonorthogonal ones), one must ensure that numerical dispersion errors 

are not introduced as waves propagate through regions of varying grid size and aspect ratio. Further studies of these errors 

are needed.

4. According to Wang (1992), a guaranteed conservative form of the advection terms does not exist for the horizontal 

momentum equations expressed with contravariant velocity components in a curvilinear, orthogonal or nonorthogonal, 

coordinate system. Although using a conservative form of the momentum advection terms in an estuarine model 

formulation is not always essential, it is usually desirable for the best possible behavior of the numerical model.

The model described in this report will use Cartesian horizontal coordinates only. As can be concluded from this discussion, 

for cases in which a reasonably uniform grid resolution is appropriate and when precise boundary fitting is not important (because, 

for example, boundary areas have very low velocities or are subject to wetting and drying cycles), Cartesian models are preferred 

for their efficiency, convenience, and generally superior numerical behavior. In any case, solving the Cartesian form of equations 

should always be the first step in developing and testing a new model, as is the case herein.

2.6.2 Vertical Coordinate Transformations

Three-dimensional models that are based upon a layered or grid-box approach in the vertical direction generally use one of 

three types of vertical coordinate systems. The first is a standard z-coordinate, or level-plane, system that has been used in the equations derived so far. Using z-coordinates, the governing equations are solved on a vertical grid system that is composed of horizontal layers with exchanges of mass, momentum, and salt taking place between the layers. The position of the layers is fixed vertically. The details of schematizing the governing equations using a z-coordinate layering system are presented in Chapters 3 and 4.
52 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The second type of vertical coordinate system uses density to replace depth (z) as the vertical coordinate, making density an 

independent variable in the transformed system (x, y, ρ, t). The coordinate z becomes a dependent variable that identifies the depth 

at which the density is defined for a given x, y location and time t. The governing equations are schematized by using stacked layers 

of uniform density separated by moving, impermeable interfaces. The locations of the interfaces are defined as 

zk = ζk(x, y, t), k = 1, km for a system of km layers, as illustrated in figure 2.12. In the general case, the layers are coupled at the 

interfaces through internal friction (momentum transfer), although layered models of seiching in lakes and ocean flows having 

small vertical Ekman numbers are often used without friction and give satisfactory results. In layered models, a new function P is 

a substitute for the hydrostatic pressure p. It is called the Montgomery potential, defined as (Cushman-Roisin, 1994, p. 171)

P p = + ρgz .

The horizontal pressure gradients ∂p/∂x and ∂p/∂y, evaluated at constant z, become ∂P/∂x and ∂P/∂y when evaluated at constant ρ.

The density-coordinate models are generally referred to as multilayer models (for example, Laevastu, 1975; Liu and Leendertse, 1978) or isopycnic models (for example, Bleck and Boudra, 1981, 1986; Boudra and others, 1987). They are appealing for 

their relatively simplified mathematics and conceptually realistic approach. They are not, however, well suited to estuarine modeling, which includes a density structure and estuarine geometry that are typically very complicated. In estuaries, the pycnoclines 

(constant density surfaces) deviate considerably from the horizontal and tend to converge toward one another and to intersect regularly with the surface and bottom boundaries. Upwelling and downwelling flows can also occur. These situations cause severe 

numerical difficulties for multilayer models, as the layers must then be permitted to converge, collapse, and reappear. Bleck and 

Boudra (1986) have made progress in dealing with the impediments of multilayer modeling by developing an isopycnic model 

x

z

y

ζ

ζ 1

ζ 2

ζ km–1

ζ km = – h ( ) x ,y

ρ1

ρ2

ρkm

k = 1

k = 2

k km =

.

.

.

Figure 2.12. Sketch of a multilayered system.
2. Governing Equations and Boundary Conditions 53

with a special flux-corrected-transport (Zalesak, 1979) algorithm to prevent the collapsing of coordinate layers, but their approach 

is still not tractable for application to most estuaries. In a few instances, multilayer models have been used with some success in 

studies of estuarine density fronts.

The third type of coordinate system is based on a vertically “normalized” or “stretched” coordinate that is derived by transforming the z-coordinate to create a constant depth flow domain. This coordinate system is called the σ-coordinate system; its use 

generally requires a constant number of grid boxes in the vertical at all horizontal grid points in a model, independent of the water 

depth. This type of transformation was first proposed by Phillips (1957) for use in numerical meteorological forecasting and was 

later introduced for lake modeling by Freeman and others (1972) and Haq and others (1974). Several years later, applications to

ocean modeling (Durance, 1976; Davies, 1980; Owen, 1980) and bay modeling (Sen Gupta and others, 1981) first appeared. 

Because the σ-coordinate system is so widely used in 3-D models of water bodies having variable topography, the details for the 

transformation of the governing equations are included in Appendix C.

The chief advantage of the σ-coordinate system is that it maps the surface and bottom into horizontal coordinate surfaces.23

The σ numerical grid is not fixed in space but actually moves up and down with any oscillation of the free surface; in this way, it 

treats the dynamic free surface in a fairly straightforward manner. Its use also eliminates the need to add layers of small vertical 

extent to correctly resolve the currents in shallow areas. It is particularly convenient, therefore, to use this system for coastal ocean 

modeling when it is desirable for the vertical grid spacing to be small in the shallow waters of the continental shelf and large in the 

deep waters of the ocean. It is possible to use coordinate stretching within the σ-coordinate system to vary the spacing of grid points 

to achieve higher resolution at some point in the water column such as near the free surface or bottom (Noye and Stevens, 1987;

Huang and Spaulding, 1995). Some authors (Spall and Robinson, 1990; Gerdes, 1993) have developed models using hybrid coordinate systems between the σ-coordinate and z-coordinate systems, which attempt to draw upon the best features of each.

In estuaries, σ-coordinate models are not always desirable because the large number of vertical grid layers needed to resolve 

the flow in a deep water estuarine channel may not be warranted in adjacent shallow zones which are often well-mixed vertically; 

in these cases, the σ grid may lead to “over-resolution” in the shallow zones (fig. 2.13) and significantly increase the computational 

cost of running a σ-coordinate model over a z-coordinate model.

Another drawback of the σ-transformation is that the transformation can lead to severe numerical errors in regions of rapidly 

changing depth such as are common in estuaries. Haney (1991) gives examples of numerical errors that a σ-coordinate ocean model 

produces when computing the pressure gradient force near steep topography and inadequate vertical and horizontal grid resolution. 

The errors are due to spatial truncation errors and a problem of “hydrostatic inconsistency” discussed by Janjic (1977). Evaluating 

the pressure gradients near steep slopes in σ-coordinate models involves taking a difference between two relatively large terms 

that often are nearly equal. The truncation errors from the approximation of each term can become greatly magnified after the terms 

23King (1985) presented a modified σ-transformation scheme that maps only the free surface onto a horizontal surface and preserves the bottom profile. 

The modified transformation, although giving up some of the mathematical elegance of the original transformation, was recommended for cases where the slope 

of the bottom profile varies sharply.
10

9

8

6

5

4

3

2

Layer 1

Water surface

7

Level plane grid

10

9

8

6

5

4

3

2

Layer 1

Water surface

7

σ Grid

Figure 2.13. Three-dimensional model layering schemes using 10 layers to resolve the deep water channel.

The σ grid has more layers than are needed in the shallow water, which generally is well mixed.

2. Governing Equations and Boundary Conditions 54

are combined. Deleersnijder and Beckers (1992) noted that truncation errors in σ models apply to the space derivatives of any variable in the 3-D equations, not just pressure. If the full approximation of the horizontal diffusive fluxes is included in a σ-coordinate 

model, truncation errors of the numerical approximation of these terms also can become large. Several studies have shown that a

σ-coordinate model must have a high level of grid refinement to produce accurate results near irregular topography. Walters and 

Foreman (1992) were unable to choose a grid fine enough to obtain acceptable results using a finite-element σ-coordinate model 

of the Vancouver Island continental shelf. Johnson and others (1989) were forced to convert from using a σ to a z coordinate in 

their 3-D model of Chesapeake Bay after finding the grid resolution using the σ-coordinate model was inadequate to maintain the 

proper amount of salt stratification in the deep channels of the bay during lengthy model simulations. In estuaries, therefore, a zcoordinate model may be better suited for hydrodynamic modeling for reasons of both accuracy and efficiency.
3. Layer Averaging the Governing Equations 55

3. Layer Averaging the Governing Equations

3.1 Introduction

The present model is based on a z-coordinate system, as discussed in section 2.6.2. The grid system that is used in solving the 

model equations is composed of layers, as is illustrated in figure 3.1. Because the interfaces between the layers are level planes, 

this kind of model is sometimes called a multilevel model (Liu and Leendertse, 1978) to distinguish it from a multilayer model 

which is more commonly used to identify a density-coordinate model. Because both multilevel and multilayer models are based 

on the concept of layers, Cheng and others (1976) referred to both types of models as multilayer models but labeled them as Type I 

and Type II models, respectively.

The hydrodynamic variables usually change considerably over the depth of an estuarine flow. In order to base the model computations on mass and momentum fluxes in the different layers, it is necessary to prepare the governing equations for the finitedifference method by integrating them over the height of each layer. By choosing the layer-integrated, volumetric transports as

dependent variables, this approach will insure that the model equations are in a conservative form. One advantage of this form is 

that the depth-integrated continuity equation is linear. If primitive variables were used, terms involving products of the layer thicknesses and the horizontal velocity components would appear in the discrete form of the depth-integrated continuity equation; when 

the time-varying surface layer thickness and the surface velocity components are out of phase (as is typical of a standing wave), 

the calculations for these terms are susceptible to phase errors and amplitude errors. Another benefit of integrating the governing 

equations over layers is that the equations for a single-layer system reduce to a conservative form of the 2-D vertically averaged 

equations; thus in an area of shallow water where only one layer is required, the 3-D model becomes automatically a 2-D model 

without any modifications to the model coding.

The procedure for layer averaging the 3-D governing equations of Chapter 2 is defined in some detail in Chapter 3. Very few 

approximations are made; the set of model equations for a layered system that is the result is fully three-dimensional. The form of 

the layer-averaged equations is similar to that first presented by Leendertse and others (1973).

The starting point is the 3-D equations in approximately the form of equations 2.38 to 2.42 with the turbulent stresses represented by τxx ρ0 = – u′u′ , τxy ρ0 = – u′v′ , τxz ρ0 = – u′w′ , τyx ρ0 = – v′u′ , τyy ρ0 = – v′v′ , and τyz ρ0 = – v′w′ , 

and the turbulent salt fluxes represented by Jx ρ0 = – u′s′, Jy ρ0 = – v′s′, and Jz ρ0 = – w′s′. The introduction of the eddy viscosity and eddy diffusivity will come in a later step. The 3-D equations in the appropriate form are
h1

h2

h3

h4

h5

h6

z 1

⁄2

z 3

⁄2

z 5

⁄2

z 7

⁄2

z 9

⁄2

z 11⁄2

z 13⁄2

k = 1

k = 2

k = 3

k = 4

k = 5

]

x

z

y

Datum

u3 u3

w3

p3 s3 ,

w4

Figure 3.1. Multilevel model computational grid showing the stacked layers and the location of variables.

∂u

∂x

----- ∂v

∂y

----- ∂w

∂z + + ------ = 0

∂u

∂t

----- ∂uu

∂x -------- ∂uv∂y -------- ∂uw∂z +++ --------- – fv 1

ρ0

-----∂p

∂x – ----- 1ρ0

----- τxx ∂

∂x ---------- τxy ∂∂y ---------- τxz ∂∂z + + --------- ⎝ ⎠ ⎛ ⎞ = +

∂v

∂t

----- ∂uv

∂x -------- ∂vv∂y -------- ∂vw∂z +++ + --------- fu 1

ρ0

-----∂p

∂y – ----- 1ρ0

----- τyx ∂

∂x ---------- τyy ∂∂y ---------- τyz ∂∂z + + --------- ⎝ ⎠ ⎛ ⎞ = +

0 1

ρ0

-----∂p

∂z – ----- gρ

ρ0

= – ------

∂s

∂t

---- ∂us

∂x -------- ∂vs∂y

------- ∂ws

∂z +++ --------- 1

ρ0

----- Jx ∂

∂x

------- Jy ∂

∂y

------- Jz ∂

∂z + + ------- ⎝ ⎠ ⎛ ⎞ =

3. Layer Averaging the Governing Equations 56

(3.1)

(3.2)

(3.3)

(3.4)

. (3.5)

Figure 3.1 illustrates a multilevel grid for a specific case where the number of layers at each vertical location is either five or 

six. The layers are counted downward from the surface and are of height hk. In general, the height of each layer is constant; the 

exceptions are those at the surface and the bottom. The surface layer height h1varies as a function of both time and space because 

of the propagation of the tides and other waves. The bottom layer height varies with the changing bathymetry as a function of space 

only.24 The layer types are designated as either surface, middle, or bottom. In the special case of a single layer representing the 

entire depth of flow, the layer will be classified as a surface layer, but the position of the bottom of this layer will vary. In the 

general case, the number of layers at each horizontal location is km. Because no wetting and drying of nodal points is considered, 

it is assumed that the free surface never drops below the bottom of the first layer.

24Because the test case included in Chapter 5 of this report is for a constant-depth basin, the number of layers (km) does not vary with horizontal location 

and the bottom layer is a constant height.
3. Layer Averaging the Governing Equations 57

For present purposes, the heights of all middle layers are equal, although this is not a requirement; for generality, the designation of a height (hk) that could vary from layer to layer will be retained for these layers. The use of equal layer heights permits a 

convenient implementation of a vertical finite-difference approximation that is second-order accurate in space. If uneven layer

heights are chosen, the vertical numerical approximation will become only first-order accurate. The changing heights of the surface 

and bottom layers will actually introduce some first-order error into the approximation of the vertical mixing terms, but the error 

in most cases is minor.

The average of any 3-D variable over a layer k will be indicated by a subscript k. The layer averages for the model variables 

of horizontal velocity, pressure, density, and salinity are defined as

uk

1

hk

---- u zd ,

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ = vk

1

hk

---- v zd ,

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ = pk

1

hk

---- p zd ,

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ =

ρk

1

hk

---- ρ dz ,

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ = and sk

1

hk

---- s zd ,

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ =

(3.6)

where the integration limits zk–1⁄2 and zk+1⁄2 define the z coordinates of the layer interfaces. The coordinate z1⁄2 = ζ represents the free 

surface elevation, and zkm+1⁄2 = zb represents the bottom elevation measured from the datum.

Over the height of each layer, the 3-D variable can be represented as the sum of the average value for that layer and a deviation 

from that average at the point (x, y, z) (fig. 3.2): 

uxyzt ( ) ,,, uk( ) xyt , , uk = + ′′( ) xyzt ,,, ,

vxyzt ( ) ,,, vk( ) xyt , , vk = + ′′( ) xyzt ,,, ,

pxyzt ( ) ,,, pk( ) xyt , , pk = + ′′( ) xyzt ,,, , (3.7)

ρ( )ρ xyzt ,,, k( )ρ xyt , , k = + ′′( ) xyzt ,,, , and

sxyzt ( ) ,,, sk( ) xyt , , sk = + ′′( ) xyzt ,,, .

The integral of the deviation term over a layer will by definition equal zero; for example, ∫

zk - 1/

2

zk + 1/

2

uk’’ dz = 0. The integral of 

a variable that already is a layer average yields a product of the layer-average variable and the layer thickness; for example, 

∫

z k - 1/

2

zk + 1/

2

uk dz = ukhk. The quantities ukhk and vkhk are volumetric transports and are represented by the symbols Uk and Vk, 

respectively. The volumetric transports are dependent variables in the model; the velocities uk and vk are computed from the 

volumetric transports by dividing by hk. 
Figure 3.2. Diagram of a 3-dimensional horizontal velocity profile u(z) approximated by layer-averaged 

values uk (eq. 3.6). The deviation of uk from u is . uk

′′

DEPTH

u VELOCITY

k = 1

=

k = 2

k = 3

k = 4

k = 5

k = 6

-z

x

y

u z( )

u z( ) ( )

( )z

u6

u6 u5

u4

u3

u2

uk + uk z

u1

“

“

Figure 3.2. Diagram of a 3-dimensional horizontal velocity profile u(z) approximated by layer-averaged 

values uk (eq. 3.6). The deviation of uk from u is . uk

′′

58 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Leibnitz’ rule relates the derivative of a 3-D variable to its layer average. For a generic variable F(x, y, t), the Leibnitz rule, 

applied to an integral over a layer, produces

∂

∂x

----- F zd

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂F

∂x

------ dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ Fk 1 – ⁄2

zk 1 – ⁄2 ∂

∂x --------------- Fk 1 + ⁄2

zk 1 + ⁄2 ∂

∂x = + – ---------------- (3.8)

or, after rearranging and inserting Fk,

∂F

∂x

------ dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂x

----- hkFk ( ) Fk 1 – ⁄2

zk 1 – ⁄2 ∂

∂x – --------------- Fk 1 + ⁄2

zk 1 + ⁄2 ∂

∂x = + ---------------- . (3.9)

The integral over a layer for a product of two variables, F and G, is related to the product of the two layer-averaged variables as 

follows:

<FG> < Fk Fk ( ) + ′′ Gk Gk ( ) + ′′ >

<FkGk> <FkGk

′′> <Fk′′Gk> <Fk′′Gk +++ ′′>

hkFkGk <Fk

′′Gk + ′′> .

=

=

= (3.10)
3. Layer Averaging the Governing Equations 59

Here < > is shorthand notation for the layer integral ∫

zk - 1/2

zk + 1/

2

( ) dz. The integral of a term involving a derivative of z can be 

simplified by using the fundamental theorem of calculus. Thus 

∂F

∂z

------ dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ Fk 1 – ⁄2 Fk 1 + ⁄2 = – , (3.11)

where the fractional subscripts k 1 – ⁄2 and k 1 + ⁄2 indicate that the variable F is evaluated at the interface between two layers. 

Formally, an interface value is defined by Fk 1 + ⁄2 [ ] Fxyzt ( ) ,,, z zk 1 + ⁄2

= = . For computational purposes, the interface values can 

be evaluated as the simple average of the adjacent layer-averages of the same variables, such as

Fk 1 + ⁄2

1

2

-- Fk Fk + 1 = ( ) + (3.12)

or, if unequal layer heights are used, the formula

Fk 1 + ⁄2 θFk + 1 ( ) 1 – θ Fk = + (3.13)

can be applied with θ hk hk hk + 1 = ⁄( ) + to improve the accuracy of interpolations.

3.2 Continuity Equation

Integration of the continuity equation over a layer yields

∂u

∂x

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂v

∂y

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂w

∂z

------ dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ + + = 0 (3.14)

or, after substitution of equations 3.9 and 3.11,

Uk ∂

∂x --------- Vk ∂∂y -------- uk 1 – ⁄2

zk 1 – ⁄2 ∂

∂x --------------- vk 1 – ⁄2

zk 1 – ⁄2 ∂

∂y ---------------- wk 1 – ⁄2 – + –

uk 1 + ⁄2

zk 1 + ⁄2 ∂

∂x --------------- vk 1 + ⁄2

zk 1 + ⁄2 ∂

∂y --------------- wk 1 + ⁄2 + –

+

+ = 0 . (3.15)

The bracketed terms in equation 3.15 are simplified by applying the boundary conditions for each layer. With the exception of the 

free surface and the bottom of the estuary, the layer interfaces are horizontal; thus,

zk 1 ± ⁄2 ∂

∂x --------------- = 0 . (3.16)

At the free surface and the bottom, the kinematic boundary conditions presented in Chapter 2 (eqs. 2.52 and 2.59) are applicable. 

These conditions, written in the present notation, are

z1

⁄2 ∂

∂t -------- u1

⁄2

z1

⁄2 ∂

∂x -------- v1

⁄2

z1

⁄2 ∂

∂y -------- w1

⁄2 + + – = 0 (3.17)

and
ukm 1 + ⁄2

zkm 1 + ⁄2 ∂

∂x ------------------- vkm 1 + ⁄2

zkm 1 + ⁄2 ∂

∂y ------------------- wkm 1 + ⁄2 + – = 0

60 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

, (3.18)

where the subscripts 1

⁄2 and km 1 + ⁄2 refer to the free surface and the bottom, respectively. Applying the boundary conditions to 

equation 3.15 results in the following forms of the continuity equations for surface, middle, and bottom layers:

Surface layer,

∂ζ

∂t

----- w3

⁄2 – U1 ∂

∂x --------- V1 ∂∂y

+ + -------- = 0 ; (3.19)

Middle layers,

wk 1 – ⁄2 wk 1 + ⁄2 – Uk ∂

∂x --------- Vk ∂∂y

+ + -------- = 0 k = 2 3 ... ,, , km – 1 ; (3.20)

Bottom layer,

wkm 1 – ⁄2

Ukm ∂

∂x ------------- Vkm ∂∂y

+ + ------------ = 0 . (3.21)

In equation 3.19, the substitution ζ z1 2⁄ = has been made to use conventional notation. By combining equations 3.20 and 3.21, 

it is possible to write an expression for the vertical velocity component at an interface k 1 – ⁄2 as

wk 1 – ⁄2

Uk ∂

∂x --------- Vk ∂∂y

+ --------

⎩ ⎭ ⎨ ⎬ ⎧ ⎫

k = 2

km

= – ∑ . (3.22)

This expression can be evaluated for w3

⁄2 and substituted into equation 3.19 to yield this expression for the water surface elevation:

∂ζ

∂t

----- Uk ∂

∂x --------- Vk ∂∂y

+ --------

⎩ ⎭ ⎨ ⎬ ⎧ ⎫

k = 1

km

+ ∑ = 0 . (3.23)

A variation of equation 3.23 is obtained by integrating the continuity equation over the entire depth of flow. This is written

∂ζ

∂t

----- ∂

∂x

----- Uk

k = 1

km

∑⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ∂

∂y

----- Vk

k = 1

km

∑⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞

+ + = 0 , (3.24)

where the quantities Uk

k = 1

km

∑ and Vk

k = 1

km

∑ represent the vertically-integrated components of the volume-transport velocity. 

Equation 3.24 is the actual form of the continuity equation that is used in the model in determining the water surface elevation.
3. Layer Averaging the Governing Equations 61

3.3 Momentum Equations

In developing the layer-averaged form for the horizontal momentum equations, consider first integrating the acceleration 

terms over a layer for the x-component equation:

∂u

∂t

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂uu

∂x -------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂uv

∂y -------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂uw

∂z --------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ +++ . (3.25)

Applying equations 3.9 and 3.11 gives

uk 1 + ⁄2 0 uk 1 + ⁄2

zk 1 + ⁄2 ∂

∂x --------------- vk 1 + ⁄2

zk 1 + ⁄2 ∂

∂y --------------- wk 1 + ⁄2 + + + – .

Uk ∂

∂t --------- ∂∂x

----- uu dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂y

----- uv dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ uk 1 – ⁄2

zk 1 – ⁄2 ∂

∂t --------------- uk 1 – ⁄2

zk 1 – ⁄2 ∂

∂x --------------- vk 1 – ⁄2

zk 1 – ⁄2 ∂

∂y --------------- wk 1 – ⁄2 + + – + + –

(3.26)

At the free surface and bottom, the bracketed terms disappear because of the kinematic boundary conditions. At all other layer 

interfaces, the derivatives of zk 1 – ⁄2 and zk 1 + ⁄2 equal zero. Thus, the simplified form of 3.26 applicable to all layers is

Uk ∂

∂t --------- ∂

∂x

----- uu dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂y

----- uv dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ( ) uw k 1 – ⁄2 ( ) uw k 1 + ⁄2 +++ – , (3.27)

subject to ( ) uw 1

⁄2 ( ) uw km 1 + ⁄2 = = 0 25 . Substituting equation 3.10 for the integral expressions in 3.27 and noting that 

uk′′ u uk = – and vk′′ v vk = – (see eq. 3.2) gives

∂

∂x

----- u uk ( ) – 2 dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂y

----- u uk ( ) – v vk ( ) – dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ + +

Uk ∂

∂t --------- ( ) Uu k ∂∂x ----------------- ( ) Vu k ∂∂y ---------------- ( ) uw k 1 – ⁄2 ( ) uw k 1 + ⁄2 +++ –

, (3.28)

25The quantities ( ) uw 1

⁄2 and ( ) uw km 1 + ⁄2 are not literally zero for the general case of a sloping free surface and bottom boundary (with slip 

allowed). These quantities are set to zero in equation 3.27 to force the bracketed terms in expression 3.26 to vanish at the free surface and bottom as required by 

the kinematic boundary conditions.
62 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

which is the desired form for the x-component, layer-averaged advection terms. The corresponding expression for the y-momentum 

equation is

∂

∂x

----- u uk ( ) – v vk ( ) – dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂y

----- v vk ( ) – 2 dz .

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ + +

Vk ∂

∂t -------- ( ) Uv k ∂∂x ----------------- ( ) Vv k ∂∂y ---------------- ( ) vw k 1 – ⁄2 ( ) vw k 1 + ⁄2 +++ –

(3.29)

The two integral expressions in each of 3.28 and 3.29 account for the variation of the 3-D velocity over a layer and sometimes 

are referred to as momentum dispersion terms. These terms, although typically quite small, are theoretically zero only if the velocity does not vary with z over a layer. These terms can be formally represented by introducing momentum correction coefficients 

into the advection terms in a way which is similar to the coefficients introduced into a two-dimensional model by Pritchard (1971b, 

p. 31). Because little information is available to estimate momentum coefficients for a layered model, this approach is not followed 

here and the terms will be dropped. If necessary, one could add these terms into the horizontal stresses and adjust the value assigned 

to the horizontal eddy viscosity.

The integration of the Coriolis term in the x- and y-momentum equations over a layer is straightforward and results in

f v dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ – f Vk = – (3.30)

and

+ f u dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ + f Uk = . (3.31)

Integration of the pressure-gradient term over the depth of a layer proceeds as follows. Considering the x-direction gradient, 

after integration and application of Leibnitz’ rule, the result is

1

ρ0

----- ∂p

∂x

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ 1

ρ0

----- hkpk ∂

∂x ------------- pk 1 – ⁄2

zk 1 – ⁄2 ∂

∂x – --------------- pk 1 + ⁄2

zk 1 + ⁄2 ∂

∂x = + --------------- . (3.32)

At the free surface, it is assumed that the atmospheric pressure is a zero gage pressure so p1

⁄2 = 0 . After again recalling that 

zk 1 ± ⁄2 ∂ ⁄ ∂x = 0 at layer interfaces, the right-side expression for the surface and middle layers reduces to

1

ρ0

----- hkpk ∂

∂x ------------- , (3.33)
3. Layer Averaging the Governing Equations 63

and for the bottom layer to

1

ρ0

----- hkmpkm ∂

∂x --------------------- pkm 1 + ⁄2

ρ0

----------------- zkm 1 + ⁄2 ∂

∂x

+ ------------------- , (3.34)

where pkm 1 + ⁄2 represents the pressure at the bottom of the bottom layer. Whenever hk is a constant, it can be moved outside the 

differentiation sign. Thus, for a middle layer the term can be rewritten as

hk

ρ0

----- ∂x

∂pk k = 2 3 ... ,, , km – 1 . (3.35)

For a surface layer, hk is not constant, so the expression for 3.33 expands into

1

ρ0

----- h1p1 ∂

∂x -------------- h1

ρ0

----- ∂x

∂p1 p1

ρ0

----- ∂ζ

∂x = + ----- ,

which, after substitution for p1 in the second term on the right side, becomes26

1

ρ0

----- h1p1 ∂

∂x -------------- h1

ρ0

----- ∂x

∂p1 h1

ρ0

----- gρ1

2

-------- ∂ζ

∂x

----- g

ρ0h1

----------- ∂ζ

∂x

----- ρ ρ1 ( ) – dz′dz .

z

ζ

∫ z3

2

⁄

ζ

∫ = + + (3.36)

26The average pressure p1 for the surface layer is defined by integrating the hydrostatic pressure p = g ∫

ζ

z ρ dz′ over the surface layer as follows:

p1

1

h1

----- p zd

z3

2

⁄

ζ

∫ g

h1

----- ρ dz′dz

z

ζ

∫z3

2

⁄

ζ

∫ = = ,

then ρ ρ1 ρ1 = + ′′ is substituted to obtain

p1

gρ1

h1

-------- dz′dz g

h1

----- ρ′′dz′dz

z

ζ

∫

z3

2

⁄

ζ

∫ +

z

ζ

∫z3

2

⁄

ζ

∫ =

1

2

--gρ1h1

g

h1

----- ρ ρ1 ( ) – dz′dz .

z

ζ

∫

z3

2

⁄

ζ

∫ = +
64 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

For a bottom layer, the expression using 3.34 can be shown to be

1

ρ0

----- hkmpkm ∂

∂x --------------------- pkm 1 + ⁄2

ρ0

----------------- zkm 1 + ⁄2 ∂

∂x

+ ------------------- hkm

ρ0

-------- pkm ∂

∂x ----------- hkm

ρ0

-------- gρkm

2 ------------ hkm ∂

∂x – ----------- gρ0hkm

--------------- hkm ∂

∂x ----------- ρ ρkm ( ) – dz′dz .

z

z

km 1 – ⁄2

∫ z

km 1 + ⁄2

z

km 1 – ⁄2

∫ = + (3.37)

The two double integral terms in equations 3.36 and 3.37, containing the three-dimensional density ρ, are zero if the density is 

assumed to be uniform vertically across the surface and bottom layers. An approximation is introduced if these terms are neglected 

when the vertical density varies continuously within the surface and bottom layers. The error from the approximation is small if 

the surface and bottom layer heights are small enough to avoid a large change in density across the layers. If the near-surface and 

bottom waters of an estuary are vertically mixed by surface wind and bottom friction, respectively, the error of the approximation 

is eliminated. These terms are neglected here, as is customary. In summary, the x-momentum pressure gradient term for the surface, 

middle, and bottom layers is approximated by the following expressions:

Surface layer,

 1

ρ0

----- ∂x

∂p dz

z3

2

⁄

ζ

∫ h1

ρ0

----- ∂x

∂p1 gρ1

2

-------- ∂ζ

∂x

+ ----- ⎝ ⎠ ⎛ ⎞ = ; (3.38)

Middle layers,

1

ρ0

----- ∂p

∂x

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ hk

ρ0

----- pk ∂

∂x = ------- k = 2 3,, , ... km – 1 ; (3.39)

Bottom layer,

1

ρ0

----- ∂x

∂p dz

z

km 1 + ⁄2

z

km 1 – ⁄2

∫ hkm

ρ0

-------- pkm ∂

∂x ----------- gρkm

2 ------------ hkm ∂

∂x – ----------- ⎝ ⎠ ⎛ ⎞ = . (3.40)

For the special case of a single layer representing the entire depth of flow, the result is

1

ρ0

----- ∂x

∂p dz

z3

2

⁄

ζ

∫ h1

ρ0

----- ∂x

∂p1 gρ1

2

-------- ∂ζ

∂x

----- gρ1

2

-------- z3

2

⁄ ∂

∂x

+ + ------- ⎝ ⎠ ⎛ ⎞ = , (3.41)

where z

3 2⁄ represents the location of the bottom. Similar expressions are available for the pressure gradient term in the ymomentum equation.
3. Layer Averaging the Governing Equations 65

The average pressure in any layer can be related to the average pressure in the layer above through the hydrostatic pressure 

equation, which is approximated by

pk pk – 1 gρk 1 – ⁄2hk 1 – ⁄2 = + .

Here hk 1 – ⁄2 is the average of the heights for layers k - 1 and k and represents the vertical distance between the centers of the two 

layers. By differentiating this equation with respect to x and y, the pressure gradients for one layer can be determined from the layer 

above. For the x-direction pressure gradient, the result is

∂x

∂pk pk – 1 ∂

∂x --------------- g

hk 1 – ⁄2

ρk 1 – ⁄2 ∂

∂x = + -------------------------------

pk – 1 ∂

∂x --------------- ghk 1 – ⁄2

ρk 1 – ⁄2 ∂

∂x ---------------- gρk 1 – ⁄2

hk 1 – ⁄2 ∂

∂x = + + ----------------

pk – 1 ∂

∂x --------------- ghk – 1

2 --------------- ρk – 1 ∂

∂x --------------- ghk

2

-------- ρk ∂

∂x -------- gρk – 1

2 --------------- hk – 1 ∂

∂x --------------- gρk

2

-------- hk ∂

∂x = + ++ + ------- . (3.42)

In the surface layer, the x pressure gradient is approximated by

∂x

∂p1 gρ1

2

-------- ∂ζ

∂x

----- gh1

2

-------- ρ1 ∂

∂x = + -------- . (3.43)

Successively substituting equations 3.43 and 3.42 into equations 3.38 to 3.40, a general formula for the x pressure gradient term 

can be seen by induction to be 

1

ρ0

----- ∂p

∂x

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ hk

ρ0

----- gρ1

∂ζ

∂x

----- gh1

2

-------- ρ1 ∂

∂x -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂x ---------------- ghm

2

--------- ρm ∂

∂x

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

= + + ∑ (3.44)

where the summation is omitted for k = 1. The corresponding formula for the y pressure gradient term is

1

ρ0

----- ∂p

∂y

----- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ hk

ρ0

----- gρ1

∂ζ

∂y

----- gh1

2

-------- ρ1 ∂

∂y -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂y ---------------- ghm

2

--------- ρm ∂

∂y

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

= + + ∑ . (3.45)

Now consider the integration of the stress terms in equations 3.2 and 3.3 over a layer. After integration and application of 

equations 3.9 and 3.11 to the x-component terms, the result is

τxx –( )k 1 + ⁄2

zk 1 + ⁄2 ∂

∂x --------------- τxy ( )k 1 + ⁄2

– zk 1 + ⁄2 ∂

∂y --------------- τxz ( )k 1 + ⁄2

– + ⎠

⎞

1

ρ0

----- hτxx ( )k ∂

∂x -------------------- hτxy ( )k ∂∂y -------------------- τxx –( )k 1 – ⁄2

zk 1 – ⁄2 ∂

∂x --------------- τxy ( )k 1 – ⁄2

– zk 1 – ⁄2 ∂

∂y --------------- τxz ( )k 1 – ⁄2

+ + + ⎝

⎛

1

ρ0

----- τxx ∂

∂x ---------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ 1

ρ0

----- τxy ∂

∂x ---------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ 1

ρ0

----- τxz ∂

∂x --------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ++ =

. (3.46)
66 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

At the free surface and bottom, the dynamic boundary conditions from Chapter 2 (eqs. 2.54 and 2.60) can be applied to equation 

3.46 in the form

τxs τxx ( )1

⁄2

z 1

⁄2 ∂

∂x – ---------- τxy ( )1

⁄2

z 1

⁄2 ∂

∂y – ---------- τxz ( )1

⁄2

= + (3.47)

and

τxb τxx ( )km 1 + ⁄2

zkm 1 + ⁄2 ∂

∂x – ------------------- τxy ( )km 1 + ⁄2

zkm 1 + ⁄2 ∂

∂y – ------------------- τxz ( )km 1 + ⁄2

= + . (3.48)

At all other layer interfaces, equation 3.16 again applies so that the right side of 3.46 reduces to

1

ρ0

----- hτxx ( )k ∂

∂x -------------------- hτxy ( )k ∂∂y -------------------- τxz ( )k 1 – ⁄2

τxz ( )k 1 + ⁄2+ + – ⎝ ⎠ ⎛ ⎞ , (3.49)

subject to the redefinition that τxz ( )1

⁄2

τxs = and τxz ( )km 1 + ⁄2τxb = . It is now convenient to introduce the concept of the 

horizontal eddy viscosity in the form

τxx ( )k ρ0 AH

∂u

∂x

----- ⎝ ⎠ ⎛ ⎞

k

= τxy ( )k ρ0 AH

∂u

∂y

----- ⎝ ⎠ ⎛ ⎞

k

= , (3.50)

so that equation 3.49 becomes

∂

∂x

----- AHh∂u

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- AHh∂u

∂y

----- ⎝ ⎠ ⎛ ⎞

k

τxz ( )k 1 – ⁄2

ρ0

---------------------- τxz ( )k 1 + ⁄2

ρ0

+ + – ---------------------- , (3.51)

which is the final form of the layer-averaged stress terms. The expression for the y-momentum stress terms is

∂

∂x

----- AHh∂v

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- AHh∂v

∂y

----- ⎝ ⎠ ⎛ ⎞

k

τyz ( )k 1 – ⁄2

ρ0

---------------------- τyz ( )k 1 + ⁄2

ρ0

+ + – ---------------------- (3.52)

subject to τyz ( )1

⁄2

τys = and τyz ( )km 1 + ⁄2τyb = . 

3.4 Salt Transport Equation

The steps in deriving the layer-averaged salt transport equation are similar to those used on the momentum equations. Integration of the advection terms in equation 3.5 results in

∂s

∂t

---- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂us

∂x -------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂vs

∂y

------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂ws

∂z --------- dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ +++ =

( ) hs k ∂

∂t --------------- ( ) uhs k ∂∂x ---------------- ( ) vhs k ∂∂y ---------------- ( ) ws k 1 – ⁄2 ( ) ws k 1 + ⁄2 +++ –

∂

∂x

----- u uk ( ) – s sk ( ) – dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ ∂

∂y

----- v vk ( ) – s sk ( ) – dz

z

k 1 + ⁄2

z

k 1 – ⁄2

∫ + + , (3.53)
3. Layer Averaging the Governing Equations 67

which is very similar in form to the momentum acceleration terms expressed by 3.28 and 3.29. Equation 3.53 is valid for all layers 

subject to the boundary conditions of ( ) ws 1

⁄2 ( ) ws km 1 + ⁄2 = = 0 , which cause the kinematic conditions at the free surface and the 

bottom of the water column to be satisfied; the kinematic conditions prevent the advective flux of salt through the boundaries.

Integration of the turbulent diffusion terms in equation 3.5 results in

1

ρ0

----- hJx ( )k ∂

∂x ------------------ hJy ( )k ∂∂y ------------------ Jz ( )k 1 – ⁄2

Jz ( )k 1 + ⁄2+ + – ⎝ ⎠ ⎛ ⎞ , (3.54)

which is subject to the boundary conditions of no turbulent salt flux at the water column free surface and bottom; this condition is 

enforced by setting Jz ( )1

⁄2

Jz ( )km 1 + ⁄2= = 0 27 . Substituting the concept of a horizontal eddy diffusivity in the form

Jx ( )k ρ0 DH

∂s

∂x

----- ⎝ ⎠ ⎛ ⎞

k

= Jy ( )k ρ0 DH

∂s

∂y

----- ⎝ ⎠ ⎛ ⎞

k

= (3.55)

into 3.54 results in

∂

∂x

----- DHh∂s

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- DHh∂s

∂y

----- ⎝ ⎠ ⎛ ⎞

k

Jz ( )k 1 – ⁄2

ρ0

-------------------- Jz ( )k 1 + ⁄2

ρ0

+ + – -------------------- (3.56)

which is the final form of the layer-averaged turbulent salt flux terms. The form of 3.56 is close to that of the layer-averaged 

turbulent stress terms in the momentum equations expressed by 3.51 and 3.52.

The two integral terms in equation 3.53 are salt dispersion terms representing advective fluxes of salt in the x- and y-directions 

caused by vertical non-uniformities of the velocity and salinity over a layer. If the 3-D velocity and salt concentration do not vary 

with z over a layer, these terms will vanish. In general, if layer heights are small enough to avoid a large change in velocity and 

salinity across the layers, these terms will be small. Here these terms will be dropped. If for a certain application these terms are 

significant, the magnitude of DH can be adjusted.

27The conditions of no turbulent flux of salt at the water column free surface and bottom are formally represented by 

0 Jx ( )1

⁄2

z 1

⁄2 ∂

∂x – ---------- Jy ( )1

⁄2

z 1

⁄2 ∂

∂y – ---------- Jz ( )1

⁄2

= +

and

0 Jx ( )km 1 + ⁄2

zkm 1 + ⁄2 ∂

∂x – ------------------- Jy ( )km 1 + ⁄2

zkm 1 + ⁄2 ∂

∂y – ------------------- Jz ( )km 1 + ⁄2

= + .

Because the terms involving the boundary slopes are already eliminated in 3.54, the conditions are satisfied simply by requiring that 

Jz ( )1

⁄2

Jz ( )km 1 + ⁄2= = 0 in 3.56.
68 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

3.5 Equation Summary

A recapitulation of the layer-averaged form of the governing equations follows:

Continuity equations,

∂ζ

∂t

----- ∂

∂x

----- Uk

k = 1

km

∑⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ∂

∂y

----- Vk

k = 1

km

∑⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞

+ + = 0 and (3.57)

( ) w k 1 + ⁄2

k 1 – ⁄2 Uk ∂

∂x – --------- Vk ∂∂y = – -------- k = 2 3 ... ,, , km ; (3.58)

Momentum equations,

Uk ∂

∂t --------- ( ) Uu k ∂∂x ----------------- ( ) Vu k ∂∂y ---------------- ( ) uw k 1 + ⁄2

k 1 – ⁄2 fVk – hk

ρk

-----gρ1

∂ζ

∂x

+++ + = -----

hk

ρk

----- gh1

2

-------- ρ1 ∂

∂x -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂x ---------------- ghm

2

--------- ρm ∂

∂x

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

– +∑ ∂

∂x

-----AHh∂u

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

-----AHh∂u

∂y

----- ⎝ ⎠ ⎛ ⎞

k

τxz

ρ

------ ⎝ ⎠ ⎛ ⎞

k 1 + ⁄2

k 1 – ⁄2

+++

(3.59)

and

Vk ∂

∂t -------- ( ) Uv k ∂∂x ----------------- ( ) Vv k ∂∂y ---------------- ( ) vw k 1 + ⁄2

k 1 – ⁄2 fUk

hk

ρk

-----gρ1

∂ζ

∂y

+ + + ++ = -----

hk

ρk

----- gh1

2

-------- ρ1 ∂

∂y -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂y ---------------- ghm

2

--------- ρm ∂

∂y

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

– +∑ ∂

∂x

----- AHh∂v

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- AHh

∂v

∂y

----- ⎝ ⎠ ⎛ ⎞

k

τyz

ρ

------ ⎝ ⎠ ⎛ ⎞

k 1 + ⁄2

k 1 – ⁄2

+++ ;

(3.60)

Salt transport equation,

( ) hs k ∂

∂t --------------- ( ) uhs k ∂∂x ---------------- ( ) vhs k ∂∂y ---------------- ( ) ws k 1 + ⁄2

k 1 – ⁄2 +++

∂

∂x

----- DHh∂s

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- DHh∂s

∂y

----- ⎝ ⎠ ⎛ ⎞

k

Jz

ρ

---- ⎝ ⎠ ⎛ ⎞

k 1 + ⁄2

k 1 – ⁄2

= + + . (3.61)

The notation ( ) k 1 + ⁄2

k 1 – ⁄2 represents the difference between interface values for a layer. The layer-averaged density ρk has been 

substituted for ρ0 in the denominator of the pressure, vertical stress, and vertical salt flux terms; this substitution requires little 

effort to implement in the numerical model and reduces any error caused by the Boussinesq approximation. Except as noted for 

equation 3.58, this set of equations applies for all layers; the free surface and the bottom boundary conditions are satisfied by 

defining wkm 1 + ⁄2 = 0 , ( ) uw 1

⁄2 ( ) uw km 1 + ⁄2 = = 0, ( ) vw 1⁄2 ( ) vw km 1 + ⁄2 = = 0, τxz,τyz ( )1

⁄2

τxs,τys = ( ), 

τxz,τyz ( )km 1 + ⁄2τxb,τyb = ( ) , ( ) ws 1

⁄2 ( ) ws km 1 + ⁄2 = = 0, and Jz ( )1

⁄2

Jz ( )km 1 + ⁄2= = 0 . As noted, the summation term in 

equations 3.59 and 3.60 is omitted for a surface layer (k = 1). Equations 3.57 to 3.61 are the 3-D equations that are discretized by 

using a finite-difference method in the next chapter.
4. Finite-difference Formulation 69

4. Finite-difference Formulation

4.1 Introduction

This chapter describes the semi-implicit finite-difference formulation used on the 3-D governing equations of Chapter 3. The 

details of the 3-D formulation begin in section 4.3.

The semi-implicit method is a general time-integration technique that can be implemented by various choices for a finitedifference scheme. One of the fundamental choices is between schemes involving either two or three time levels in the finite-difference equations. There are 3-D, semi-implicit, estuarine and coastal ocean models based on both two-level (Backhaus, 1985; 

Casulli and Cheng, 1992) and three-level (Hamrick, 1992; Muin and Spaulding, 1997a) finite-difference schemes. The three-level 

models use the semi-implicit approach in the 2-D, external-mode portion of a mode-splitting scheme. Three-level schemes sometimes are not favored for use in hydrodynamic codes owing to the extra storage they require and the extra difficulty they entail by 

requiring more than one initial condition to start the computation. Yet three-level schemes have the advantage of being easily centered in time to achieve second-order accuracy for the time integration.

In the research leading to this report, both a two-level scheme and a three-level scheme were investigated to implement the 

semi-implicit method in three dimensions. The two schemes were first coded using the 1-D equations for open channel flow and 

tested on a few nonlinear cases of flows in a rectangular channel with homogeneous density. On the basis of this testing, the threelevel scheme was chosen for extension to three dimensions.

The details of the two 1-D schemes are given in the next section of this chapter before proceeding with the details of the 3-D 

scheme. The presentation of the 1-D schemes is a useful basis for comparing the two- and three-level approaches, and it also demonstrates the straightforward implementation of the semi-implicit method in one dimension before becoming involved with the 

greater mathematical complexity of implementing the method in three dimensions. The testing of the 1-D schemes is included in 

Chapter 5.

4.2 Semi-Implicit One-Dimensional Schemes

The 1-D equations to be solved are the equations of continuity and x-momentum for a rectangular channel with friction. 

Because the water density is considered to be homogeneous, no salt transport equation is needed and no longitudinal densitygradient term is in the x-momentum equation. The equations are the following:28

28The 1-D equations for unsteady open-channel flow are derived in numerous references such as Liggett (1975) and Cunge and others (1980). Yen (1973) 

presents a derivation by integrating the full 3-D governing equations. The form of equations 4.1 and 4.2 are taken from equations 2.28 in Cunge and others (1980, 

p.16) after simplifying them by assuming a constant channel width.
70 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Continuity,

∂ζ

∂t

------ ∂U

∂x

+ ------- = 0 ; (4.1)

x-Momentum,

∂U

∂t

------- ∂uU

∂x ----------- gH∂ζ

∂x

+ + ----- gHSf = – . (4.2)

Here, U = uH is the discharge per unit width (or volumetric transport); u is the average channel velocity; H = h + ζ is the total 

depth of flow, where h is the location of the channel bottom measured positive downward from a datum (fig. 4.1); ζ is the water 

surface elevation measured positive upward from a datum; and x and t are the independent variables. The variable Sf is a friction 

slope defined by29

Sf

n2U U

H 2

R4

⁄3

= = ------------------ γU U , (4.3)

where n is Manning’s resistance coefficient and the hydraulic radius of the channel is R = BH⁄(B + 2H) where B is the width of the 

channel. For a wide channel (B >> H), the hydraulic radius in equation 4.3 can be replaced by the total depth H.

The two-level and the three-level schemes use a Crank-Nicolson time-averaging approach (Crank and Nicolson, 1947) for the 

implicit treatment of the pressure gradient term gH(∂ζ ⁄∂x) in the 1-D x-momentum equation and for the volumetric transport gradient term ∂U⁄∂x in the 1-D continuity equation. The two-level scheme is sometimes referred to as a trapezoidal scheme because 

the Crank-Nicolson averaging is centered between the values at the beginning and end of a time step, similar to the well-known 

trapezoidal rule for numerical integration. Friction is treated implicitly in both 1-D models, and the advection term ∂(uU)⁄∂x in the 

x-momentum equation is treated explicitly. By this choice of implicit terms the finite-difference equations are free from the time 

step limitation due to the Courant-Friedrich-Lewy (CFL) criterion30 based on the gravity-wave speed (Casulli and Cheng,

29The form of the friction slope defined by Manning’s formula in equation 4.3 assumes metric units are used. For English units an additional coefficient 

of 2.208 would appear in the denominator.

30The Courant-Friedrich-Lewy stability criterion can be expressed as 

Cr ≤ 1

in which Cr is known as a Courant (or CFL) number. The Courant number definition is based either on the speed of surface gravity waves or the advective 

velocity u. These two Courant numbers are referred to herein as the gravity-wave Courant number and the advection Courant number and in one dimension are 

defined by Crg = ( )Δ u gH + ⋅ t ⁄ Δx and Cra = u ⋅ Δ Δt ⁄ x , respectively.

Figure 4.1. Definition sketch for 1-dimensional channel.

h(x)

ζ ( ) x t,

H = ( ) x t, h + ζ

Datum

Water surface

/

x
4. Finite-difference Formulation 71

1990). If no iteration process is used, the equations are still subject to a CFL stability limitation on Δt based on the velocity u

if, for example, upwind differencing is used for the explicit treatment of the advection term (Casulli and Cheng, 1990); however, 

because velocity generally is an order of magnitude smaller than the surface wave speed in a typical estuary, potentially there can 

be a ten-fold increase in the time step for the semi-implicit scheme beyond that required for a fully explicit scheme. For accuracy 

considerations it is usually desirable to time-center the nonlinear coefficients in the two-level scheme by introducing an iteration 

process. The iteration also guarantees the unconditional linear stability of the scheme if centered differences for the advection term 

are updated with new-time-level information within the iteration process. The three-level scheme is already centered in time without iteration, but it includes an option for inserting one or more iterations of a two-level (trapezoidal) scheme to stabilize the threelevel scheme and suppress any high frequency oscillations that may appear in solutions due to strong nonlinearities or steep 

gradients.

In both the two-level and the three-level schemes, the two dependent variables (in this case U and ζ) are computed at different 

points on a staggered grid; the two-time-level grid, involving time levels t = nΔt and t = (n + 1)Δt, is shown in figure 4.2. The grid 

is uniform in space, so there are IM-1 equal space intervals Δx with IM being the number of U or ζ grid points. The discretization 

in time is defined by the set of points tn = nΔt, n = 0, 1, 2, ... where Δt = constant. The computation starts from a known condition 

at time t = 0, and subsequent time step solutions advance from the previously known solution at time tn to the new time tn + 1. 

A specification of known upstream and downstream boundary conditions is required for a problem to be solved when the flow 

conditions are subcritical.

Figure 4.2. Two-time-level staggered grid for the 1-dimensional model.

Δt

U i 3

2

⁄ 3

2 – ⁄

 i-1 i i+1

n+ 1

n

U i 1

2

⁄ 1

2 – ⁄ ζ i –1 U i + ζ i ζ i + 1

x

Δ x

U i +
72 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

4.2.1 Two-Level Semi-Implicit Scheme

The two-level semi-implicit scheme closely resembles the iterative scheme first published by Abbott and Ionescu (1967) and 

later used by Verwey (1971) in a model for the Danish Hydraulic Institute. The scheme here is most similar to the version of the 

Abbott-Ionescu scheme described by Cunge and others (1980, p. 97−98). One change in the scheme here is in the manner in which 

the system of finite-difference equations are formed for each time step. The momentum equations are substituted into the continuity 

equations so that a linear system of equations is formed involving only the unknown water surface elevations {ζi

}. This reduces 

the size of the matrix to be solved to half the size of one involving both dependent variables. For typical 1-D problems the savings 

in storage space for the reduced matrix generally is not important, but the savings for 3-D problems can be significant.

The equations are solved by using separate explicit and implicit stages to keep the computer code modular. In a solution for 

one dimensional flow with homogeneous density, only the advection term in the momentum equation is treated explicitly. Owing 

to the staggered grid, the continuity and momentum equations are applied to different control volumes shifted one-half cell relative 

to one another. Continuity is applied at the ζ-points and momentum is applied at the U-points.

The equation for the explicit (or advection) stage is

U

ˆ

i 1 – ⁄2 Ui 1

⁄2 –

n –

Δt ---------------------------------- ( ) uU i

n 1 + ⁄2 ( ) uU i – 1n 1 + ⁄2 –

Δx

+ ------------------------------------------------------ = 0 (4.4)

which is solved directly for U

ˆ

i 1 – ⁄2 to give

U

ˆ

i 1 – ⁄2 Ui 1 – ⁄2

n

 – Δt

Δx

------ ( ) uU i

n 1 + ⁄2 ( ) uU i – 1

n 1 + ⁄2 = ( ) – .

U

ˆ

i 1 – ⁄2

(4.5)

 is a temporary variable that represents a partial value for Ui 1

⁄2 –

n+ 1

 after application of the advection operator. The equations 

for the implicit (or wave-propagation) stage are as follows:

Continuity,

ζi

n + 1 ζi

n –

Δt ---------------------- 1

2

-- Ui 1 + ⁄2

n + 1 Ui 1 – ⁄2n + 1 –

Δx ---------------------------------- Ui 1 + ⁄2

n Ui 1 – ⁄2n –

Δx

+ ---------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ = 0 ; (4.6)

Momentum,

Ui 1 – ⁄2

n + 1 U

ˆ

i 1 – ⁄2 –

Δt ---------------------------------- g

2

--Hi 1 – ⁄2

n 1 + ⁄2 ζ i

n + 1 ζ i – 1n + 1 –

Δx ----------------------------- ζ i

n ζ i –1n –

Δx

+ --------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ = gHi 1 – ⁄2

n 1 + ⁄2 – γ i 1 – ⁄2

n 1 + ⁄2 χi 1 – ⁄2 Ui 1 – ⁄2

n Ui 1 – ⁄2

n + 1 1 χi 1 – ⁄2

( ) – Ui 1 – ⁄2

n Ui 1 – ⁄2n ( ) + . (4.7)

Here χ i 1 – ⁄2 is a weighting coefficient for the frictional resistance term. The use of time level n 1 + ⁄ 2 for the variables in equations 

4.5 and 4.7 is symbolic since no such time level exists; it means that the variables are evaluated mid-way between time levels nΔt

and ( )Δ n + 1 t ; for example,

Ui 1 + ⁄2

n 1 + ⁄2 1

2

-- U

 ̃ i 1 + ⁄2

n + 1

Ui 1 + ⁄2

n

= ( ) + . (4.8)
4. Finite-difference Formulation 73

The solution scheme involves iteration; the symbol (~) in equation 4.8 denotes an approximate solution at time t = (n + 1)Δt taken 

from the preceding iteration. For the first (starting) iteration, it is assumed that U

 ̃ i 1 + ⁄2

n + 1

Ui 1 + ⁄2

n

= , u ̃ i 1 + ⁄2

n + 1 u i 1 + ⁄2

n = , and 

H

 ̃ i 1 + ⁄2

n + 1

H i 1 + ⁄2

n

= ; in this way, the variables superscripted with n 1 + ⁄ 2 are always known. One should use caution if this scheme 

is used with only one iteration each time step; then the scheme is only first-order accurate because the coefficients Hi 1 – ⁄2

n 1 + ⁄2 and 

γ i 1 – ⁄2

n 1 + ⁄2 in equation 4.7 and the advection term ( ) uU i

n 1 + ⁄2 ( ) uU i – 1

n 1 + ⁄2 ( )Δ – ⁄ x in equation 4.4 are evaluated backward-in-time at the nΔt

time level instead of being centered at the n 1 ( )Δ + ⁄ 2 t time level. In Chapter 5 it will be demonstrated that a significant loss of 

accuracy can result if the scheme is used with only one iteration. 

The weighting coefficient χ was suggested by Cunge and others (1980) to improve accuracy in evaluating the friction term 

when a rapid variation and reversal of the discharge can occur; in the first iteration, χ i 1 – ⁄2 is equal to 1.0, and in each subsequent 

iteration, it is defined by

χi 1 – ⁄2

Ui 1 – ⁄2

n 1 + ⁄2 Ui 1 – ⁄2n 1 + ⁄2 Ui 1 – ⁄2

n

Ui 1 – ⁄2

n

–

Ui 1 – ⁄2

n Ui 1 – ⁄2

n + 1 Ui 1 – ⁄2

n ( ) –

= ----------------------------------------------------------------- . (4.9)

Because the water surface elevation ζi and the volumetric transport Ui 1 – ⁄2 are computed at different nodal points, the values 

for (uU)i, (uU)i − 1 and Hi 1 – ⁄2 in equations 4.5 and 4.7 must be obtained by some form of interpolation. The usual procedure is to 

calculate the average of the two adjacent values on the grid. For improved accuracy here, a center-weighted average of the four

adjacent values is used. Thus, the total depth H at the point i 1 + ⁄2 is

Hi 1 + ⁄2

9

16

----- Hi + 1 + Hi ( ) 1

16

----- Hi + 2 Hi – 1 = – ( ) + , (4.10)

which applies at all points except those immediately adjacent to the boundary, where a two-point average is used.

Before actually solving the implicit stage of the two-level scheme, the momentum equation (eq. 4.7) is first rewritten in the 

form

Ui 1 – ⁄2

n + 1 Ri 1 – ⁄2

n 1 + ⁄2 U

ˆ

i 1 – ⁄2

g

2 – --( ) Δt ⁄ Δx Hi 1 – ⁄2

n 1 + ⁄2 ζi

n + 1 ζi – 1n + 1 ζi

n ζi – 1

n ( ) – + – gΔx 1 χi 1 – ⁄2 ( ) – Hi 1 – ⁄2n 1 + ⁄2

γ

i 1 – ⁄2

n 1 + ⁄2

Ui 1 – ⁄2

n Ui 1 – ⁄2

n

– ) ⎝

⎛ = (4.11)

where Ri 1 – ⁄2

n 1 + ⁄2 1 gΔt χi 1 + – ⁄2Hi 1 – ⁄2

n 1 + ⁄2 γi 1 – ⁄2

n 1 + ⁄2 Ui 1 – ⁄2

n

( ) –1 = . After substituting equation and a similar expression for Ui 1 + ⁄2

n + 1 into the 

continuity equation (eq. 4.6), Ui 1 + ⁄2

n + 1 and Ui 1 – ⁄2n + 1

 are eliminated, and an equation of the following form is obtained:

Ai ζ i – 1

n + 1 Bi ζ i

n + 1 Ci ζ i + 1

n + 1

+ + = Di , (4.12)
74 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

where the coefficients Ai, Bi, Ci, and Di, are known functions of the flow variables defined by

Ai

g

4

– --( ) Δt ⁄ Δx

2

Ri 1 – ⁄2

n 1 + ⁄2 Hi 1 – ⁄2

n 1 + ⁄2 = ,

Bi 1 g

4

--( ) Δt ⁄ Δx

2 Ri 1 + ⁄2

n 1 + ⁄2 Hi 1 + ⁄2

n 1 + ⁄2 Ri 1 – ⁄2

n 1 + ⁄2

Hi 1 – ⁄2

n 1 + ⁄2 + ⎝ ⎠ ⎛ ⎞ = + ,

Ci

g

4

– --( ) Δt ⁄ Δx

2

Ri 1 + ⁄2

n 1 + ⁄2 Hi 1 + ⁄2

n 1 + ⁄2 = , and

Di ζ i

n g

4

+ --( ) Δt ⁄ Δx

2 Ri 1 + ⁄2

n 1 + ⁄2 Hi 1 + ⁄2

n 1 + ⁄2 ζi + 1n ζin ( ) – Ri 1 – ⁄2

n 1 + ⁄2

Hi 1 – ⁄2

n 1 + ⁄2 ζin ζ i – 1n + ( ) – ⎝ ⎠ = ⎛ ⎞

1

2

--( ) Δt ⁄ Δx Ri 1 + ⁄2

n 1 + ⁄2 U

ˆ i 1 + ⁄2 Ri 1 – ⁄2

n 1 + ⁄2

U

ˆ i 1 – ⁄2 – –

1

2

--( ) Δt ⁄ Δx Ui 1 + ⁄2

n Ui 1 – ⁄2n – ( ) –

1 χi 1 – ⁄2 –( ) – Ri 1 – ⁄2

n 1 + ⁄2 Hi 1 – ⁄2

n 1 + ⁄2

γi 1 – ⁄2

n 1 + ⁄2 Ui 1 – ⁄2

n Ui 1 – ⁄2n ) .

g

2

-- ( ) Δt

2 ( ) ⁄ Δx 1 χi 1 + ⁄2 ( ) ( – Ri 1 + ⁄2

n 1 + ⁄2 Hi 1 + ⁄2

n 1 + ⁄2

γi 1 + ⁄2

n 1 + ⁄2 Ui 1 + ⁄2

n Ui 1 + ⁄2n +

(4.13)

Applying equation 4.12 to each of the interior points of the computational grid results in a system of IM-2 linear algebraic equations 

involving IM unknown values of ζi

n + 1

. The matrix form of the equations is tridiagonal (elements occur only on the main diagonal 

and on the first subdiagonals above and below). After adding two boundary conditions to complete the equation set, the tridiagonal 

system can be rapidly solved using the double-sweep method (Cunge and others, 1980, p. 106−108).31 Once the water surface 

elevations are known, the volumetric transports are found explicitly through the momentum equation in the form of equation 4.11. 

The scheme generally gives satisfactory results while using no more than two iterations per time step (that is, two solutions of the 

tridiagonal system of equations).

A linear analysis of stability for the two-level, semi-implicit scheme without an iteration procedure is discussed by Casulli 

and Cheng (1990). They considered three choices for the explicit approximation of the advection term: space-centered differencing, upwind differencing, and an Eulerian-Lagrangian method (ELM) using linear interpolation. Only the ELM method was found 

to be unconditionally linearly stable. The linear stability of the upwind differencing formulation requires that the Courant number 

based on the material velocity u not exceed unity. The linear stability of the centered differencing formulation requires the flow to 

be strictly subcritical and the time step to satisfy the inequality

k1Δt

1

Fr2 < -------- – 1,

where Fr u gH = ⁄ is the Froude number and k1 gSf = ⁄ u is a friction parameter taken as constant. Without using any iteration, 

the centered differencing of the advection term must be evaluated at the backward (nΔt) time level; the resulting advection scheme 

is only first-order accurate in time. Using iteration, the advection term in equation 4.4 is centered in both space and time, thus 

gaining full second-order accuracy. The centering in time is achieved in the context of an explicit calculation by weighting the 

forward time level solution from the previous iteration with the backward time level solution (eq. 4.8). This equal weighting was 

originally shown to be unconditionally linearly stable by Abbott and Ionescu (1967). Here the centered differencing approach (with 

iteration) is preferred over the ELM approach of Casulli and Cheng (1990) because it is more efficiently implemented and generally 

will result in greater accuracy for most typical choices of the grid parameters Δt and Δx. The ELM approach also is not easily 

implemented for the strictly conservative form of the governing equations used here.

31In the mathematics literature the double-sweep method is sometimes called the Thomas algorithm (Ames, 1977, p. 52).
4. Finite-difference Formulation 75

Of course a proof of unconditional or conditional stability for the linear (constant coefficient) shallow water equations will 

not necessarily apply to the full nonlinear equations. The existence of a phenomenon called nonlinear instability (Phillips, 1959) 

is well known in the field of computational fluid dynamics. As will be demonstrated in Chapter 5, nonlinear instabilities seem to 

arise in the semi-implicit model, most likely because of the lack of a built-in damping mechanism in the scheme. When these instabilities occurred in test runs using either one or two iterations, they were in all cases suppressed and eliminated by additional iterations (usually one), resulting in a smooth solution. This stabilizing effect from the iteration procedure ought to be useful for real 

modeling situations. Duwe and others (1983) recognized this effect and added a second step (iteration) to their 2-D semi-implicit 

model to remove nonlinear instabilities.

4.2.2 Three-Level Semi-Implicit Scheme

The three-level scheme uses the same staggered spatial grid as the two-level scheme, but it involves an additional time level 

at t = (n − 1)Δt (fig. 4.3). The scheme is based on the well-known explicit leapfrog differencing method (fig. 4.4A) and uses an 

implicit (Crank-Nicolson type) modification to leapfrog differencing for the discretization of implicit terms (fig. 4.4B).

Δt

Δt

Ui 3

2 – ⁄

 i- 1 i i+1

n+ 1

n

n- 1

U i 1

2 – ⁄ U i 1

2 + ⁄ ζ i –1 ζ i ζ i+1

t

x

Δ x

U i 3

2⁄ +

Figure 4.3. Three-time-level staggered grid for the 1-dimensional model.
n+1

n

n -1

U i 1

2 – ⁄

i 1

2 – ⁄

ζ i U i 1

2 + ⁄

i 1

2 + ⁄

Δ t

Δ t

Δx

t

x

U

x

-------

n

t ------

n+1

n

U ζ

ζ

ζ

i U

Δ t

Δ t

Δx

t

x

n - 1

t ------

∂ U

∂ x

-------

n

1

2

-- ∂ U

∂ x

-------

n+1

∂ U

∂ x

-------

n– 1

= +

∂ U

∂

∂

∂ x

-------

n+ 1

∂ U

∂ x

∂

∂

∂

∂

-------

n– 1

A

B

( (

76 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 4.4. Computational stencils for (A) explicit leapfrog differencing method and (B) implicit (CrankNicolson type) leapfrog differencing method used in the three-time-level 1-dimensional scheme.
4. Finite-difference Formulation 77

The equation for the explicit stage is

U

ˆ

i 1 – ⁄2 Ui 1 + ⁄2

n – 1 –

2Δt ------------------------------------------ ( ) uU i

n ( ) uU i – 1

n –

Δx + --------------------------------------------- = 0 , (4.14)

which is solved directly for U

ˆ

i 1 – ⁄2 to give

U

ˆ

i 1 – ⁄2 Ui 1 + ⁄

2

n – 1 – 2Δt

Δx -------- ( ) uU i

n ( ) uU i – 1n –

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ = , (4.15)

where U

ˆ

i 1 – ⁄2 , again, is a temporary variable. The equations for the implicit stage are the following:

Continuity,

ζi

n + 1 ζin – 1 –

2Δt ------------------------------------ 1

2

--

U i 1 + ⁄2

n + 1 Ui 1– ⁄

2

n + 1 ( ) –

Δx -----------------------------------------------

U i 1 + ⁄

2

n –1 Ui 1– ⁄

2

n –1 ( ) –

Δx + -----------------------------------------------

⎝ ⎠ ⎜ ⎟

⎛ ⎞

+ = 0 ; (4.16)

Momentum,

Ui 1– ⁄

2

n + 1 U

ˆ

i 1 – ⁄

2 –

2Δt ---------------------------------- g

2

--Hi 1 – ⁄

2

n ζ i

n + 1 ζ i – 1

n + 1 –

Δx ------------------------------ ζ i

n– 1 ζ i –1

n– 1 –

Δx

+ ----------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ gHi 1– ⁄

2

n – γ i 1– ⁄

2

n Ui 1– ⁄

2

n – 1 Ui 1– ⁄

2

n + 1 = . (4.17)

In this scheme, the nonlinear coefficients Hi 1 – ⁄2

n

 and γ i 1 – ⁄2

n in the momentum equation and the advection term in equation 4.15 are 

easily centered in time by evaluating them at time level n. Therefore, no iteration is required to achieve second-order accuracy in 

the time integration, in contrast to the two-level scheme. Spatial interpolations in the form of equation 4.10 also can be used in the 

three-level scheme.

The solution of the implicit stage requires rewriting the momentum equation in the form

Ui 1 – ⁄2

n + 1 Ri 1 – ⁄2

n U

ˆ

i 1 – ⁄2 – g( ) Δt ⁄ Δx Hi 1 – ⁄2

n ζi

n + 1 ζi – 1

n + 1 ζi

n – 1 ζi – 1n – 1 = ( ) ( ) – + – , (4.18)

where Ri 1 – ⁄2

n

1 + 2gΔtHi 1 – ⁄2

n γi 1 – ⁄2

n Ui 1 – ⁄2

n – 1 ( ) –1 = . Then the substitution of equation 4.18 and a similar expression for Ui 1 + ⁄2

n + 1

 into 

the continuity equation (eq. 4.16) results in an equation identical in form to equation 4.12,

Ai ζ i – 1

n + 1 Bi ζ in + 1 Ci ζ i + 1n + 1

+ + = Di , (4.19)

but with the coefficients Ai, Bi, Ci, and Di, defined as follows:

Ai g( ) Δt ⁄ Δx

2 – Ri 1 – ⁄2

n Hi 1 – ⁄2

n = ,

Bi 1 g( ) Δt ⁄ Δx

2 (Ri 1 + ⁄2

n Hi 1 + ⁄2

n Ri 1 – ⁄2

n Hi 1 – ⁄2

n

= + + ),

Ci g( ) Δt ⁄ Δx

2 – Ri 1 + ⁄2

n Hi 1 + ⁄2

n = , and

Di ζ i

n – 1

g( ) Δt ⁄ Δx

2 Ri 1 + ⁄2

n Hi 1 + ⁄2

n ζi + 1n – 1 ζi

n – 1 ( ) – Ri 1 – ⁄2

n – Hi 1 – ⁄2

n ζi

n – 1 ζ i – 1

n – 1 = + ( ) ( ) –

( ) Δt ⁄ Δx Ri 1 + ⁄2

n U

ˆ i 1 + ⁄2 Ri 1 – ⁄2

n U

ˆ i 1 – ⁄2 – ( ) –

( ) Δt ⁄ Δx Ui 1 + ⁄2

n – 1 Ui 1 – ⁄2n – 1 – ( ) – (4.20)

The system of equations that results from applying equation 4.19 to each of the computational grid points is solved for ζi

n +1

⎩ ⎭ ⎨ ⎬ ⎧ ⎫

using the double-sweep method. Equation 4.18 is then used to calculate the volumetric transports Ui 1 – ⁄2

n + 1

⎩ ⎭ ⎨ ⎬ ⎧ ⎫ explicitly.
78 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

As noted previously, it is sometimes helpful to follow the time-step solution of the semi-implicit leapfrog scheme with one 

or (occasionally) more iterations of a two-level semi-implicit scheme to stabilize the solution and suppress any two-grid-interval 

oscillations in time. The two-level scheme from section 4.2.1 can be used, but with the variables superscripted with n 1 + ⁄ 2 defined 

by

Hi 1 – ⁄2

n 1 + ⁄2 1

2

-- H

 ̃ i 1 – ⁄2

n + 1

Hi 1 – ⁄2

n

+ ⎝ ⎠ ⎛ ⎞ = ,

U i 1 – ⁄2

n 1 + ⁄2 1

2

-- U

 ̃ i 1 – ⁄2

n + 1

Ui 1 – ⁄2

n + ⎝ ⎠ ⎛ ⎞ = , and

ui 1 – ⁄2

n 1 + ⁄2 1

2

-- u ̃

i 1 – ⁄2

n + 1 ui 1 – ⁄2n + ⎝ ⎠ = ⎛ ⎞ .

In this case, the symbol (~) denotes the estimates of the unknowns from the semi-implicit leapfrog solution. The two-level scheme 

also is convenient for starting the computations of the three-level scheme from the initial condition at t = 0. The combination of a 

three-level scheme followed by a two-level scheme is referred to here as a semi-implicit, leapfrog-trapezoidal scheme. Explicit

leapfrog-trapezoidal schemes commonly are used in atmospheric modeling (Mesinger and Arakawa, 1976) and other areas of 

computational physics (Zalesak, 1979, p. 361−362).

The semi-implicit three-level scheme can easily be converted to a standard explicit leapfrog scheme with relatively minor 

changes to the coding. The explicit leapfrog scheme executes more rapidly32 than the semi-implicit scheme if identical time and 

space step sizes are used in both schemes; therefore, when accuracy considerations restrict the time step so that the CFL criterion 

for surface waves does not exceed unity, the explicit scheme is preferred. Using the explicit scheme, each nodal value of the water 

surface elevation at the (n + 1)Δt time level is computed directly from the continuity equation by

ζi

n + 1 ζi

n – 1 2Δt Ui 1 + ⁄2

n Ui 1 – ⁄2n = – ( )Δ – ⁄ x . (4.21)

The volumetric transport is obtained directly from the momentum equation written in the form

Ui 1 – ⁄2

n + 1 Ri 1 – ⁄2

n U

ˆ

i 1 – ⁄2 2gΔtHi 1 – ⁄2

n ζi

n ζi – 1

n = ( ) – ( )Δ – ⁄ x , (4.22)

where U

ˆ

i 1 – ⁄2 is still defined in equation 4.15, and Ri 1 – ⁄2

n is defined in equation 4.18. It also is easy to include a step with an explicit 

trapezoidal scheme after applying the explicit leapfrog scheme.

4.3 Semi-Implicit Three-dimensional Scheme

The 3-D finite-difference scheme is applied to the layer-averaged governing equations derived in Chapter 3 (eqs. 3.57−3.61). 

The semi-implicit, leapfrog-trapezoidal scheme just presented in one dimension can be extended to three dimensions. The finitedifference equations are described in detail below.

32The speed-up from using the explicit versus the semi-implicit leapfrog scheme (with identical time and space steps) is not really significant in one 

dimension. In three dimensions the speed-up is indeed significant.
4. Finite-difference Formulation 79

The premise of the semi-implicit method in three-dimensions is identical to that in one dimension: certain key terms in the 

governing equations are treated implicitly to ensure that the stability of the scheme does not depend on the CFL criterion for the 

surface-wave celerity. The terms treated implicitly are the two horizontal divergence terms involving the summation of layer volumetric transports in the continuity equation (eq. 3.57), and the barotropic pressure gradient terms in the x- and y-momentum equations (eqs. 3.59 and 3.60). In addition to these terms, the vertical stress (or vertical momentum diffusion) terms in the momentum 

equations are treated implicitly to avoid a time-step limitation in shallow water, as discussed in Davies (1985).33 The remaining 

terms in the momentum equations (advection, Coriolis, horizontal momentum diffusion, and the baroclinic pressure gradient) are 

treated explicitly. In the salt transport equation (eq. 3.61), only the vertical salt diffusion term is treated implicitly.

The choice for the horizontal location of variables on the numerical grid is the staggered arrangement known as a C-grid 

(Mesinger and Arakawa, 1976) (fig. 4.5). The C-grid has the advantage that difference quotients are easily centered in space to 

obtain second-order accuracy. The horizontal dimensions of each cell are Δx and Δy, considered to be constant and equal here. The 

vertical locations of variables are pictured in figure 4.6 for the multilevel grid (also refer to fig. 3.1). The center of each 3-D cell 

is numbered with indices i, j, k. Pressure (pi, j, k), salinity (si, j, k), and density (ρi, j, k) are defined at the center. The x-direction 

velocity component (ui 1 + ⁄2, ,j k ) and volumetric transport (Ui 1 + ⁄2, ,j k ) are located at half-integers of i and whole integers of j and 

k; the y-direction velocity component (vi j 1 , + ⁄2, k ) and volumetric transport (Vi j 1 , + ⁄2, k ) are located at half-integers of j and whole 

integers of i and k; the vertical velocity component (wijk 1 – ⁄2 , , ) is located at each layer interface between cell centers. The water 

surface elevation (ζi j , ) is a two-dimensional variable defined at the center of each horizontal grid cell with integer values of i and 

j. As discussed in Chapter 3, the height of a surface or bottom layer in the present model can vary in space; in the surface layer, 

the height can vary with both space and time. For generality here, the layer height variable is designated with a subscript including 

all three spatial indices and a superscript indicating the time step (for example, hijk , ,

n ); the height is defined at each cell center. 

The computer code has been developed with the layer height variable as a 3-D, time-dependent array so that arbitrary geometries

and wetting and drying of nodal points can eventually be easily incorporated. For the development here, the indices on h which 

pertain to layers of constant height are superfluous.

The overall shape of the horizontal grid is rectangular with dimensions (imax, jmax). The grid boxes that fall within the boundaries of a water body are labeled as “wet cells” and are indexed, by row, between the values of i = i1, im, and, by column, between

the values of j = j1, jm. For a rectangular water body, i1, j1, im, and jm are constants that are independent of the row or column. 

For a water body of arbitrary shape, the indices i1 and im vary between rows and the indices j1 and jm vary between columns; these 

indices are used to define the boundaries of the water body. Points exterior to the boundary are dry cells. In order to simplify the 

computer code, at least one dry cell must always be adjacent to the boundary. Thus for a rectangular domain of wet cells, it is typical that i1 = j1 = 2 and im = imax − 1 and jm = jmax − 1.

33The implicit treatment of the vertical stress terms in the 3-D formulation is roughly analogous to treating the friction term implicitly in the 1-D 

formulation. 
80 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

In describing the computations, it is assumed that the integration process has progressed to a time level n and that all model 

variables are defined for that time and also for the previous time level n − 1. The computations advance the solution variables one 

time step to time level n + 1. Thereafter, the computation cycle is repeated for time level n + 2 and again for each subsequent time 

level until a simulation is completed. To start a computation, the information needed at the n − 1 time level usually is not available. 

For this case, the trapezoidal scheme is used by itself to obtain the solution at the second time level; because the trapezoidal scheme 

involves only two time levels, it does not require starting information at the n − 1 time level. In starting the computation, the coefficients of the trapezoidal scheme, which are normally defined at time level n

1 + ⁄ 2 , are evaluated backward-in-time at time level n.

4.3.1 Semi-Implicit Leapfrog Scheme

In this section, the finite-difference forms of the equations of momentum, continuity, and salt transport are introduced using 

the semi-implicit leapfrog scheme. The steps in the numerical integration procedure are outlined. The discretization of the momentum equations is discussed first, followed by the continuity equation and the salt transport equation.

u U,

v V,

ζ , , , , s ρ h w

Δ x

y

x

j Δ y

j – 1

i + 1 i +

j 3

+ ⁄ 2

i +

j 1 + ⁄2

i 1

⁄2 1 – ⁄2

j 1 – ⁄ 2

i 3⁄ 2 3 – ⁄ 2

j 3 – ⁄ 2

i – 1 i

j + 1

Figure 4.5. The space-staggered horizontal grid (C-grid) used in the 3-dimensional model showing the location 

of variables.
h1

Δx

Δy

hkm

ui + , j, km

zkm + 

zkm–

vi, j – , km wi, j, km + 

wi, j, km –

pi, j, km

pi, j, k

hk

pi, j, 1

Layer km

(bottom layer) 

Layer k

Layer 1

(surface layer)

u

i – , j, km

, k vi, j + , km

ui + , j, k

zk–

vi, j – wi, j, k +

wi, j, k – 

ui – j, k

vi, j, + , k

ui + , j, 1

z

z

vi, j – , 1

wi, j, 

wi, j, 12

⁄

12

⁄

1⁄2

3

2

⁄

12

⁄

12

⁄

12

⁄

12

⁄

12

⁄

ui – , j, 1

vi, j + ,1

ζ

z

y (north)

x (east)

Datum (normally mean

water surface)

zk +1⁄2

12

⁄

12

⁄

12

⁄

12

⁄

12

⁄

3 2

⁄

12

⁄

1

2

⁄

12

⁄

1

2 ⁄

12

⁄

12

⁄

12

⁄

12

⁄

4. Finite-difference Formulation 81

Figure 4.6. Multilevel staggered grid for the 3-dimensional model.
82 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

4.3.1.1 Momentum equations

The solution of the momentum equations is separated into two stages in which first the explicit terms and then the implicit 

terms are evaluated. The separation is not a form of time-splitting but simply a grouping of terms for convenience in making the 

computer program modular and facilitating the presentation of equations. The finite-difference equation for the x-direction 

momentum equation is written so it is centered within layer k at the horizontal point i 1

⁄ 2 ( )Δ + x , jΔy ; the explicit stage is represented by

U

ˆ

i 1 + ⁄2, ,j k Ui 1 + ⁄2, ,j k

n – 1

2Δt ADVx ( )n CORx ( )n [– + BCLINICx ( )n – HDIFFx ( )n – 1 ]

i 1 + ⁄2, ,j k = + + , (4.23)

where the terms in brackets refer to the discretized form of the corresponding groupings of terms in equation 3.59. The bracketed 

terms are expanded fully using the leapfrog scheme and displayed in Appendix D (eqs. D.1–D.4). The symbol ( )ˆ in equation 4.23

denotes a solution for the layer volumetric transport which includes only the contribution from the explicit terms. The 

corresponding finite-difference equation for the explicit stage of the y-direction momentum equation is centered within layer k at 

the point iΔx, j 1 ( )Δ + ⁄ 2 y and is represented by 

U

ˆ

i j , 1 + ⁄2, k Ui j , 1 + ⁄2, k

n – 1

2Δt ADVy ( )n CORy ( )n BCLINICy ( )n – HDIFFy ( )n – 1 ]

i j , 1 + ⁄2, k = + [– – + . (4.24)

Here the bracketed terms refer to groupings of terms in equation 3.60, and they are expanded fully in Appendix D (eqs. D.5–D.8). 

All terms in equations 4.23 and 4.24, except horizontal diffusion (HDIFF), are centered in time at time level n to achieve secondorder numerical accuracy. The horizontal diffusion is written backward-in-time at time level n − 1 because the centering of that 

term can result in a weak instability; although formally this uncentered treatment of the horizontal diffusion introduces first-order 

truncation error, the actual size of the error should be miniscule. Presently in the model, the horizontal momentum exchange (or 

eddy viscosity) coefficient AH is defined at the center of each cell with whole integers of the indices i, j, k.

The shear stress terms in equations 3.59 and 3.60 can be replaced by introducing the concept of the vertical momentum 

exchange (or eddy viscosity) coefficient:34

τxz

ρ

------ AV

∂u

∂z = ----- , τyz

ρ

------ AV

∂v

∂z = ----- . (4.25)

Then the finite-difference equation for the implicit stage of the x-momentum equation is written as

Δt AVi 1 + ⁄2 j k 1 – ⁄2 , ,

n

⎝

⎜

⎛ ( ) U h⁄ i 1 + ⁄2, ,j k – 1

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j kn + 1 –

hi 1 + ⁄2 j k 1 – ⁄2 , ,

n + 1 ---------------------------------------------------------------------------- ui 1 + ⁄2, ,j k – 1

n – 1

ui 1 + ⁄2, ,j k

n – 1 –

hi 1 + ⁄2 j k 1 – ⁄2 , ,

n – 1

+------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ ⋅

Ui 1 + ⁄2, ,j k

n + 1 U

ˆ

i 1 + ⁄2, ,j k g

Δt

Δx – ------hi 1

⁄2 + , ,j k n ρi 1 + ⁄2, ,j 1 n

ρi 1 + ⁄2, ,j k n

---------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi + 1, j n + 1 ζi j ,

n + 1 ζi + 1, j n – 1 ζi j ,n – 1 = ⋅ ( ) – + –

 AVi 1 + ⁄2 j k 1 + ⁄2 , ,

n ( ) U h⁄ i 1 + ⁄2, ,j k

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j k + 1n + 1 –

hi 1 + ⁄2 j k 1 + ⁄2 , ,

n + 1 ---------------------------------------------------------------------------- ui 1 + ⁄2, ,j k

n – 1

ui 1 + ⁄2, ,j k + 1

n – 1 –

hi 1 + ⁄2 j k 1 + ⁄2 , ,

n – 1

+ ------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

⎠

⎟

⎞ – ⋅ , (4.26)

34The z-derivatives of u and v in equations 4.25 are mathematically undefined at the layer interfaces because the layer-averaged velocities are 

discontinuous there. A z-derivative at an interface is therefore understood to imply a finite-difference quotient involving the layer variables above and below the 

interface.
4. Finite-difference Formulation 83

where the overbar ( ) on a layer height or density variable is used to represent a spatial average in the x-direction between adjacent 

values; for example, hi 1 + ⁄2, ,j k hijk , , hi + 1, ,j k = ( ) + ⁄ 2. Also, hi 1 + ⁄2 j k 1 – ⁄2 , , is defined to be the average of layer heights 

hi 1

⁄2 + , ,j k – 1 and hi 1

⁄2 + , ,j k . The average values of the layer heights only are needed in the computations involving the surface and 

bottom layers where the heights are permitted to vary horizontally. The substitution ui 1 + ⁄2, ,j k n + 1 ( ) U h⁄ i 1 + ⁄2, ,j k

n + 1 = has been made in 

the vertical diffusion term because the dependent variable used in the model is the layer volumetric transport Uk

n + 1 rather than 

the average layer velocity uk

n + 1

. The layer velocity uk is available in the computer code at time level n − 1 and therefore is used 

directly in the diffusion term.

The finite-difference equation similar to 4.26 for y-momentum is

Δt AVi j 1 + ⁄2 k 1 – ⁄2 , ,

n

⎝

⎜

⎛ ( ) V h⁄ i j 1 , , + ⁄2 k – 1

n + 1 ( ) V h⁄ i j 1 , , + ⁄2 kn + 1 –

hi j 1 + ⁄2 k 1 – ⁄2 , ,

n + 1 ------------------------------------------------------------------------------ vi j 1 , , + ⁄2 k – 1

n – 1

vi j 1 , , + ⁄2 k

n – 1 –

hi j 1 + ⁄2 k 1 – ⁄2 , ,

n – 1

+------------------------------------------------------ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ ⋅

Vi j 1 , , + ⁄2 k

n + 1 V

ˆ

i j 1 , , + ⁄2 k g

Δt

Δy – ------hi j 1 , , + ⁄2 k n ρi j 1 , , + ⁄2 1 n

ρi j 1 , , + ⁄2 k n

---------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi j , + 1 n + 1 ζi j ,

n + 1 ζi j , + 1 n – 1 ζi j ,n – 1 = ⋅ ( ) – + –

 AVi j 1 + ⁄2 k 1 + ⁄2 , ,

n ( ) V h⁄ i j 1 , , + ⁄2 k

n + 1 ( ) V h⁄ i j 1 , , + ⁄2 k + 1n + 1 –

hi j 1 + ⁄2 k 1 + ⁄2 , ,

n + 1 ------------------------------------------------------------------------------ vi j 1 , , + ⁄2 k

n – 1

vi j 1 , , + ⁄2 k + 1

n – 1 –

hi j 1 + ⁄2 k 1 + ⁄2 , ,

n – 1

+ ------------------------------------------------------ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

⎠

⎟

⎞ – ⋅ . (4.27)

Here the overbar represents a spatial average in the y-direction between adjacent values; for example, 

hi j 1 , , + ⁄2 k hijk , , hi j , , + 1 k = ( ) + ⁄ 2 ; also hi j 1 + ⁄2 k 1 – ⁄2 , , hi j 1 , , + ⁄2 k – 1 hi j 1 + , , + ⁄2 k = ( ) ⁄ 2 . Hereinafter the use of overbars is dropped 

for convenience, and a layer height or density variable possessing a half-integer subscript in any one of the three spatial indices is 

considered to be an average of the nearest adjacent values.35

Because the surface layer height is time dependent, hi j , , 1

n + 1

 is unknown for the evaluation of equations 4.26 and 4.27. This 

value is estimated in the computations by extrapolating in time using the second-order formula

h

ˆ

i j , , 1 n + 1 3 hi j , , 1

n hi j , , 1n – 1 ( ) – hi j , , 1n – 2 = + . (4.28)

Using this estimate of hi j , , 1

n + 1

 in equations 4.26 and 4.27 has worked well in numerical testing of the model and is more accurate 

than, say, choosing hi j , , 1

n + 1 hi j , , 1n ≈ . The estimate of h

ˆ

i j , , 1

n + 1

 is made first during each time step calculation immediately before the 

array that stores values of hi j , , 1

n – 2 is rewritten; therefore, the storage of an extra 3-D array is not required to evaluate equation 4.28.

In equations 4.26 and 4.27, the boundary shear stress terms for the wind and bottom friction are boundary conditions for the 

surface and bottom layers. The wind stress is specified as a forcing function at the free surface by substituting

AVi 1 + ⁄2 j 1

⁄2 , ,

n 1

2

-- ( ) U h⁄ i 1 + ⁄2, ,j 0

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j 1n + 1 –

hi 1 + ⁄2 j 1

⁄2 , ,

n + 1 -------------------------------------------------------------------- ui 1 + ⁄2, ,j 0

n – 1

ui 1 + ⁄2, ,j 1

n – 1 –

hi 1 + ⁄2 j 1

⁄2 , ,

n – 1

+ ------------------------------------------------ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ ⋅ τxs ( )i 1 + ⁄2 j 1

⁄2 , ,

n

ρi 1 + ⁄2 , ,j 1 n = ----------------------------

and

35In the 3-D model, higher order averages such as the one in equation 4.10 in the 1-D model are not used.
AVi j 1 + ⁄2 1

⁄2 , ,

n 1

2

--

( ) V h⁄ i j 1 , , + ⁄2 0

n + 1 ( ) V h⁄ i j 1 , , + ⁄2 1n + 1 –

hi j 1 + ⁄2 1

⁄2 , ,

n + 1 ----------------------------------------------------------------------- vi j 1 , , + ⁄2 0

n – 1

vi j 1 , , + ⁄2 1

n – 1 –

hi j 1 + ⁄2 1

⁄2 , ,

n – 1

+ ------------------------------------------------ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ ⋅ τys ( )i j 1 + ⁄2 1

⁄2 , ,

n

ρi j 1 , , + ⁄2 1 n = ----------------------------

84 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

,

where τxs and τys are defined by equations 2.58. The use of the zero subscript refers to a fictitious layer that does not enter into the 

computations. The frictional stress at the bottom is specified by substituting

AVi 1 + ⁄2 j km 1 + ⁄2 , ,

n 1

2

--

( ) U h⁄ i 1 + ⁄2, ,j km

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j km +1n + 1 –

hi 1 + ⁄2 j km 1 + ⁄2 , ,

n + 1 ---------------------------------------------------------------------------------- ui 1 + ⁄2, ,j km

n – 1

ui 1 + ⁄2, ,j km +1

n – 1 –

hi 1 + ⁄2 j km 1 + ⁄2 , ,

n – 1

+ ------------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ ⋅ τxb ( )i 1 + ⁄2 j km 1 , , + ⁄2

n

ρi 1 + ⁄2 , ,j km n = -----------------------------------------

and

AVi j 1 + ⁄2 km 1 + ⁄2 , ,

n 1

2

--

( ) V h⁄ i j 1 , , + ⁄2 km

n + 1 ( ) V h⁄ i j 1 , , + ⁄2 km + 1n + 1 –

hi j 1 + ⁄2 km 1 + ⁄2 , ,

n + 1 ------------------------------------------------------------------------------------ vi j 1 , , + ⁄2 km

n – 1

vi j 1 , , + ⁄2 km +1

n – 1 –

hi j 1 + ⁄2 km 1 + ⁄2 , ,

n – 1

+ ------------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ ⋅ τyb ( )i j 1 + ⁄2 km 1 + ⁄2 , ,

n

ρi j 1 , , + ⁄2 km n = --------------------------------------- ,

where the subscript km + 1 refers to a fictitious layer. The frictional stresses τxb and τyb are determined using equations 2.65 written 

in terms of the horizontal volumetric transports computed in the bottom layer. Thus 

τxb ( )i 1 + ⁄2 j km 1 + ⁄2 , ,

n

ρi 1 + ⁄2, ,j km

n ------------------------------------------ Cdi 1 + ⁄2, j

n Ui 1 + ⁄2, ,j km

n – 1

( )2 Vi 1 + ⁄2, ,j km

n – 1 ( )2 +

hi 1 + ⁄2, ,j km

n ( )2 ------------------------------------------------------------------------------Ui 1 + ⁄2, ,j km

n + 1 = (4.29)

and

τyb ( )i j 1 + ⁄2 km 1 + ⁄2 , ,

n

ρi j 1 , , + ⁄2 km

n ------------------------------------------ Cdi j 1 + ⁄2 ,

n Ui j 1 , , + ⁄2 km

n – 1 ( )2 Vi j 1 , , + ⁄2 km

n – 1

( )2 +

hi j 1 , , + ⁄2 km

n ( )2 ------------------------------------------------------------------------------Vi j 1 , , + ⁄2 km

n + 1 = , (4.30)

where

Vi 1 + ⁄2, ,j km

n – 1

Vi 1 j 1 + , + ⁄2, km

n – 1 Vi j 1 , + ⁄2, kmn – 1 Vi j 1 , – ⁄2, kmn – 1 Vi + 1 j 1 , – ⁄2, kmn – 1 = ( ) +++ ⁄ 4 ,

Ui j 1 , , + ⁄2 km

n – 1

Ui 1

⁄2 + , ,j km

n – 1 Ui 1 + ⁄2, , j + 1 kmn – 1 Ui 1 – ⁄2, ,j kmn – 1 Ui 1 + ⁄2, , j + 1 kmn – 1 = ( ) + ++ ⁄ 4, and

Cdi j 1 + ⁄2 ,

n κ

2 hi j 1 , + ⁄2 , km

n ( ) ⁄ 2 z0 ( )i j 1 + ⁄2 , [ ] ln( ) ⁄ –2 = . 

The use of an overbar to represent a four-point spatial average for a volumetric transport variable will be retained in subsequent 

equations.
4. Finite-difference Formulation 85

The vertical momentum exchange (or eddy viscosity) coefficient AV in equations 4.26 and 4.27 is a variable that must be computed as the flow field evolves. First the eddy viscosity under neutral (unstratified) conditions is computed using the Prandtl mixing 

length model in the form of equation 2.31; then the neutral value is adjusted for the effect of stratification by the stability function 

represented by equation 2.27. The individual values of AV are computed at the layer interfaces lying vertically between pressure 

points (in the same location as the vertical component of velocity w in figure 4.6); they are then interpolated horizontally to the 

layer interfaces lying vertically between the velocity components where they are required to evaluate equations 4.26 and 4.27. The 

discretized equations for the computation of the vertical distribution of AV are given in Appendix E. In the equations hereafter, AV

is considered known.

Because the x- and y-momentum equation treatments are similar, only the x-momentum equation is discussed. From this point, 

the strategy of Casulli and Cheng (1992) is followed by rewriting the system of momentum equations for node i 1 ( ) + ⁄ 2, j in the 

compact matrix form36

Ai 1 + ⁄2, jUi 1 + ⁄2, j

n + 1

Gi 1 + ⁄2, j g

Δt

Δx

------ρi 1 + ⁄2, ,j 1

n ζi + 1, j

n + 1 ζi j ,n + 1 ( ) – Ri 1 + ⁄2, j = – , (4.31)

where U, R, G, and A are

Ui 1 + ⁄2, j

n + 1

U1

n + 1

U2

n + 1 ... Ukmn + 1

[ ] , ,, i 1 + ⁄2, j

T = , Ri 1 + ⁄2, j

h1

n

ρ1

n ----- h2

n

ρ2

n ----- ... hkm

n

ρkm

n , ,, ---------

i 1 + ⁄2, j

T

= ,

Gi 1 + ⁄2, j =

U

ˆ

1 g

Δt

Δx

------h1

n ρ1

n

ρ1

n -----

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi + 1, j n – 1 ζi j ,

n – 1 – ( ) – 0 ΔtAV

n

3

⁄2

u1

n – 1

u2

n – 1 ( ) –

h3

⁄2

n – 1 – --------------------------------- 2Δt

τxs ( )1

⁄2

n

ρ1

n + + ---------------

U

ˆ

2 g

Δt

Δx

------h2

n ρ1

n

ρ2

n -----

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi + 1, j n – 1 ζi j ,

n – 1 – ( ) – ΔtAV

n

3

⁄2

u1

n – 1

u2

n – 1 ( ) –

h3

⁄2

n – 1 --------------------------------- ΔtAV

n

5

⁄2

u2

n – 1

u3

n – 1 ( ) –

h5

⁄2

n – 1 + – ---------------------------------

•

•

•

U

ˆ

km g

Δt

Δx

------hkm

n ρ1

n

ρkm

n ---------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi + 1, j n – 1 ζi j ,

n – 1 – ( ) – ΔtAV

n

km 1 – ⁄2

ukm – 1

n – 1

ukm

n – 1 ( ) –

hkm 1 – ⁄2

n – 1 + + ------------------------------------- 0

i 1 + ⁄2, j , and

36Bold-faced variables are matrices.
1

ΔtAV

n

3

⁄2

h1

n + 1 h3

⁄2

n + 1 ⋅

+ ------------------------------ 

ΔtAV

n

3

⁄2

h2

n + 1 h3

⁄2

n + 1 ⋅

– ------------------------------

 

ΔtAV

n

3

⁄2

h1

n + 1 h3

⁄2

n + 1 ⋅

– ------------------------------ 1

ΔtAV

n

3

⁄2

h2

n + 1 h3

⁄2

n + 1 ⋅

------------------------------ ΔtAV

n

5

⁄2

h2

n + 1 h5

⁄2

n + 1 ⋅

+ + ------------------------------ ΔtAV

n

5

⁄2

h3

n + 1 h5

⁄2

n + 1 ⋅

– ------------------------------

 

ΔtAV

n

5

⁄2

h2

n + 1 h5

⁄2

n + 1 ⋅

– ------------------------------

ΔtAV

n

km 1 – ⁄2

hkm

n + 1 hkm 1 – ⁄2

n + 1 ⋅

– ------------------------------------

ΔtAV

n

km 1 – ⁄2

hkm – 1

n + 1 hkm 1 – ⁄2

n + 1 ⋅

– --------------------------------------- 1

ΔtAV

n

km 1 – ⁄2

hkm

n + 1 hkm 1 – ⁄2

n + 1 ⋅

----------------------------------- 2ΔtCd

n U2 ( )km

n – 1 V2 ( )km

n – 1

+

h2 ( )km

n + + ----------------------------------------------------------------

i 1 + ⁄2, j

Ai 1 + ⁄2, j =

0

0

86 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

.

The matrix A is tridiagonal and clearly has all positive elements on the main diagonal and all nonpositive elements on the offdiagonals; it has the properties referred to in mathematics as those of an M-matrix.37 In general, A is non-symmetric38 but 

diagonally dominant.39 Since the system of equations 4.31 also should be irreducible,40 it can be uniquely transformed with a 

double-sweep algorithm to the form

Ui 1 + ⁄2, j

n + 1

[ ] A–1G i 1 + ⁄2, j g

Δt

Δx

------ρi 1 + ⁄2, ,j 1

n ζi + 1, j

n + 1 ζi j ,n + 1 ( ) – [ ] A–1R i 1 + ⁄2 = – , j . (4.32)

In executing the double-sweep algorithm, the results actually sought are only the two matrix products A−1G and A−1R, which are 

column vectors equal in order to the number of model layers; the inverse of A is never computed by itself. Equation 4.32 is useful 

because it is an expression that can be formally substituted into the continuity equation.

37A matrix A = (αij) is an M-matrix if (αij ≤ 0) for i ≠ j, A is nonsingular (determinant A ≠ 0), and A−1 ≥ 0. (Meijerink and van der Vorst, 1977)

38A matrix A = (αij) is symmetric if αij = αji and non-symmetric otherwise. In the two-level model formulated by Casulli and Cheng (1992), the matrix A

is symmetric because the model is formulated with the dependent variable of velocity rather than volumetric transport. Here the matrix A is symmetric only when 

the layer heights are all equal. This rarely occurs because the surface layer height generally varies with time. There is, however, very little computational penalty 

for executing the double sweep algorithm on the non-symmetric rather than symmetric system.

39A km × km matrix, A = (αij), is diagonally dominant if αii

αij

j = 1, i j ≠

km

≥ ∑ and for some i strict inequality holds.

40An irreducible system of km equations means that the equations cannot be rearranged so that some of the unknown variables can be found by solving 

fewer than km equations.
4. Finite-difference Formulation 87

4.3.1.2 Continuity Equation

The finite-difference analog of the continuity equation is

ζi j ,

n + 1 ζi j ,n – 1 Δt

Δx

------ Ui 1 + ⁄2, ,j k

n + 1

Ui 1 – ⁄2, ,j k

n + 1

Ui 1 + ⁄2, ,j k

n – 1

Ui 1 – ⁄2, ,j k

n – 1

( ) – + –

k = 1

km

– ∑

Δt

Δy

------ Vi j 1 , , + ⁄2 k

n + 1

Vi j 1 , , + ⁄2 k

n + 1

Vi j 1 , , + ⁄2 k

n – 1

Vi j 1 , , – ⁄2 k

n – 1

( ) – + –

k = 1

km

– ∑

=

. (4.33)

By defining a row matrix Ξ = [1, 1,... , 1] with km elements equal to unity, equation 4.33 can be rewritten in matrix notation as

ζi j ,

n + 1 ζi j ,n – 1 Δt

Δx

------ ΞUi 1 + ⁄2, j

n + 1

ΞUi 1 – ⁄2, j

n + 1 – – Δt

Δy

------ ΞVi j 1 + ⁄2 ,

n + 1 ΞVi j 1 – ⁄2 ,n + 1 – – di j ,

n – 1 = – , (4.34)

where

di j ,

n – 1 Δt

Δx

------ Ui 1 + ⁄2, ,j k

n – 1

k = 1

km

∑ Ui 1 – ⁄2, ,j k

n – 1

k = 1

km

– ∑ Δt

Δy

------ Vi j 1 , , + ⁄2 k

n – 1

k = 1

km

∑ Vi j 1 , , + ⁄2 k

n – 1

k = 1

km

= + – ∑ .

After substituting equation 4.32 for U and a similar equation for V into equation 4.34, an equation involving only the unknown 

surface elevation is obtained:

ζi j ,

n + 1

g

Δt

Δx

------ ⎝ ⎠ ⎛ ⎞ 2

ρi 1 + ⁄2, ,j 1 n [ ] ΞA–1R i 1 + ⁄2, j ζi + 1, j

n + 1 ζi j ,n + 1 – { ( ) –

ρi 1 – ⁄2, ,j 1 n [ ] ΞA–1R i 1 – ⁄2, j ζi j ,

n + 1 ζi j , – 1n + 1 – ( )} –

g

Δt

Δx

------ ⎝ ⎠ ⎛ ⎞ 2

ρi j 1 , , + ⁄2 1 n [ ] ΞA–1R i j 1 + ⁄2 , ζi j , + 1

n + 1 ζi j ,n + 1 – { ( ) –

ρi j 1 , , – ⁄2 1 n [ ] ΞA–1R i j 1 – ⁄2 , ζi j ,

n + 1 ζi j , – 1n + 1 – ( )} –

ζi j ,

n – 1 Δt

Δx

------ ΞA–1G

i 1 + ⁄2, j

ΞA–1G

i 1 – ⁄2, j –

⎩ ⎭ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎧ ⎫

– =

Δt

Δy – ------ ΞA–1G

i j 1 + ⁄2 , ΞA–1Gi j 1 – ⁄2 , –

⎩ ⎭ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎧ ⎫

– di j ,

n – 1

 .

(4.35)
88 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Because the matrix A is an M-matrix, its matrix inverse A−1 has all non-negative elements (for proof see Lancaster and 

Tismenetsky, 1985). Therefore, the matrix products ΞA−1R and ΞA−1

G in equation 4.35 are each a single non-negative number. 

Rearranging equation 4.35, the following form can be obtained:

sxi 1 – ⁄2, j ζ i – 1, j

n + 1 – syi j 1 – ⁄2 , ζi j , – 1

n + 1 – r

i j , ζi j ,

n + 1

+ syi j 1 + ⁄2 , ζi j , + 1

n + 1 – sx i 1 + ⁄2 , j ζi + 1, jn + 1 – qi j , = (4.36)

where

sxi 1 ± ⁄2 , j g

Δt

Δx

------ ⎝ ⎠ ⎛ ⎞ 2

ρi 1 ± ⁄2 , ,j 1

n

ΞA–1R i 1 ± ⁄2 , j = ,

syi j 1 ± ⁄2 , g

Δt

Δx

------ ⎝ ⎠ ⎛ ⎞ 2

ρi j 1 , ± ⁄2 , 1

n

ΞA–1R i j 1 ± ⁄2 , = ,

ri j , 1 sxi 1

⁄2 + , j

 sxi 1

⁄2 – , j + + syi j 1 + ⁄2 , syi j 1 – ⁄2 , = + + ,

qi j , ζ i j ,

n – 1 Δt

Δx – ------ ΞA–1G i 1 + ⁄2, j ΞA–1G i 1 – ⁄2, j –

⎝ ⎠ ⎜ ⎟ = ⎛ ⎞

Δt

Δy – ------ ΞA–1G i j 1 + ⁄2 , ΞA–1G i j 1 – ⁄2 , –

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ di j ,

n – 1 – ,

and di j ,

n – 1 is as defined for equation 4.34. Equation 4.36 can be written at each of the interior nodal points of the rectangular grid 

(excluding the fictitious row and column along each boundary) to form N = (imax − 2) × (jmax − 2) simultaneous linear equations 

in the unknowns ζi j , { } . If the set of equations is written in matrix form, the coefficient matrix is five-diagonal with a tridiagonal 

band along the main diagonal and two additional diagonals displaced an equal amount above and below the main diagonal (fig.4.7); 

the amount the outer diagonals are displaced is referred to as the matrix bandwidth and is dependent on the dimensions of the grid 

and the ordering of the unknowns. Here the most common “natural” ordering is used in which the numbering is done along the 

smallest dimension of the finite-difference grid. For example, if the smallest dimension is the y (north-south) direction, the natural 

ordering scheme numbers from bottom to top (south to north) on a column starting with the first (westernmost) column that is not 

fictitious (fig. 4.8). Other orderings such as the “ordering along the diagonals” and the “red-black” ordering also are mathematically 

consistent (see Young, 1971, p. 159). For problems involving irregularly shaped regions, the form of the coefficient matrix does 

not change. Dry point equations are represented with a value of 1.0 on the main diagonal and zeros for the other elements of the 

equation row; the continuity equation for a dry point therefore is reduced to ζi j ,

n + 1 ζi j ,n = where the assigned water surface 

elevation is artificial. Because all the dry point equations can be made identical, only one must be stored.

The coefficient matrix for the system of water surface elevation equations 4.36 is both symmetric and positive definite,41 a 

fortuitous circumstance. Therefore, the equations can be solved efficiently by iteration using the preconditioned conjugate-gradient 

method discussed in the next section. Once the ζi j ,

n + 1 are determined, equation 4.32 and the corresponding equation for Vi j 1 + ⁄2 ,

n + 1

can be solved explicitly for the new layer volumetric transports.

41For a formal discussion of positive definite equation systems, see Golub and Van Loan (1989, p. 139ff). In general terms, if A is symmetric with nonnegative diagonal elements and diagonally dominant, A is positive definite.
0

0

0

0

Bandwidth

{

4. Finite-difference Formulation 89

Fictitious column

Fictitious column

Fictitious row

Fictitious row

imax = 10, jmax = 6

Finite-difference grid

4

3

2

1

8

7

6

5

11

10

9

16

15

14

13

20

19

18

17

24

23

22

21

28

27

26

25

32

31

30

29

12

y

x

Figure 4.7. Representation of five-diagonal coefficient matrix.

Figure 4.8. Illustration of the "natural" ordering scheme used in forming the matrix system of equations for the freesurface elevation. 

In the above example, the bandwidth would be five. imax is the total number of columns in the finite-difference grid 

(including fictitious columns). jmax is the total number of rows in the finite-difference grid (including fictitious rows).
90 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Finally, the vertical velocity w in the water column can be derived from the continuity equation as

wijk 1 – ⁄2 , ,

n + 1

wijk 1 + ⁄2 , ,

n + 1 Ui 1 + ⁄2, ,j kn + 1 Ui 1 – ⁄2, ,j kn + 1 – ⎝ ⎠ – ⎛ ⎞ ⁄ Δx Vi j 1 , , + ⁄2 kn + 1 Vi j 1 , , – ⁄2 kn + 1 = – ( )Δ – ⁄ y

( ) k = 2 3 4 ... ,,, , km . 

(4.37)

This equation is solved explicitly, starting from the bottom layer in which wi,j, km + 1⁄2 is assigned a zero value. 

4.3.1.3 Matrix Solution

The matrix ζi j ,

n + 1 { } can be solved by either direct (noniterative) or iterative techniques. The primary difference is that direct 

techniques (such as Gaussian elimination and its variants) produce a solution after a predetermined number of arithmetic operations, whereas iterative methods produce a sequence of approximate solutions which converge to the correct value. For large systems of linear equations where the coefficient matrix is sparsely populated (few nonzero elements), iterative techniques generally 

represent the best approach since they require less storage42 and are less susceptible to round off errors. If a good first estimate for 

the unknowns in an iterative solution is available, it can also require less computer time to obtain a solution of comparable accuracy 

to a direct solution. The preconditioned conjugate-gradient method (PCGM) (Meijerink and van der Vorst, 1977) is an especially 

attractive iterative technique that is ideally suited for matrix solutions that have symmetric and positive definite coefficient matrices; convergence of the iterative process in those solutions is generally rapid. Various forms of the PCGM have been tested and

compare favorably with other iterative methods when applied to ground-water flow modeling problems where the matrix system 

of equations is similar in form to that solved here (see for example, Kuiper, 1981, 1987; Meyer and others, 1989; Hill, 1990).

The preference for direct or iterative solution techniques may well depend on the size of the problem being solved. Estuarine 

and coastal modeling problems can easily involve large matrix solutions with 100,000 or more unknowns. Because iterative techniques can efficiently solve these large systems with considerably lower storage requirements than direct techniques, they are usually the best choice. However, there is a trend toward solving larger and larger systems by direct methods, so further research

directly comparing the two approaches for a variety of actual 3-D estuarine modeling problems is still needed. Here the PCGM 

was adopted without doing any comparison with a direct technique. For the small problem tested in this report, a direct solution 

technique certainly could have been chosen.

The matrix solution for ζi j ,

n + 1 { } is implemented by using the non-symmetric preconditioned conjugate-gradient (NSPCG) 

software package developed at the Center for Numerical Analysis at the University of Texas at Austin (Kincaid and others, 1989). 

This package provides many preconditioning and acceleration methods to aid in selecting an iterative technique that is optimal for 

a particular matrix structure, computer architecture (vector or scalar), and computer storage limitations. The reader is referred to 

Oppe and others (1988) for the preconditioners and acceleration methods that are available. Without testing the 3-D estuarine 

model on actual large scale field problems, it is not possible to judge fairly the iterative techniques available in the NSPCG package. A few comparisons of techniques were made for the simple problem in this report, and the conjugate-gradient accelerator 

clearly was more efficient than the next best choice, which was successive over relaxation. The best preconditioner was modified 

incomplete Cholesky (see Axelsson and Lindskog, 1986), which was used for all test runs in this report. Also promising, however, 

was using a red-black (or odd-even) reduction scheme on the matrix system and then solving the reduced matrix system using the 

42In direct solution techniques, many of the elements of a sparse coefficient matrix that are originally zero will become nonzero in the course of the 

computation, and they will require storage space. In computations with iterative techniques, no new nonzero elements are introduced in the coefficient matrix.
4. Finite-difference Formulation 91

modified incomplete Cholesky conjugate-gradient (MICCG) scheme; this approach demands greater storage than using MICCG 

alone because the reduced system matrix must be stored. All comparisons were made on a scalar workstation. The NSPCG package 

is coded specifically for comparing iterative techniques, but it may not be ideal for achieving maximum efficiency in an operational 

code. Once further experience is gained with iterative matrix schemes for the model, an efficient coding of the best overall scheme 

will be implemented in the model.

In order to speed convergence of the iterative solution at each time step, the initial estimates for the unknown array of ζi j ,

n + 1

are taken from the surface layer heights h

ˆ

i j , , 1

n + 1

 that are estimated by extrapolation using equation 4.28 (ζ is obtained from the surface layer height by subtracting the depth of the layer below the model datum). Using these initial estimates (which are usually 

quite accurate), the iterative solution typically required no more than one or two iterations to converge to a solution within a strict 

tolerance. It is likely that larger and more complex problems will require more iterations.

4.3.1.4 Salt Transport Equation

The salt transport equation is solved after the hydrodynamic variables are determined. The vertical salt flux term in equation 

3.61 is replaced by introducing the vertical salt exchange (or eddy diffusion) coefficient:

Jz

ρ

---- DV

∂s

∂z = ---- . (4.38)

In the finite-difference equation, the vertical diffusion of salt is treated implicitly, and the advection and horizontal diffusion 

terms are treated explicitly. The finite-difference equation is centered within a layer k at the horizontal point iΔx, jΔy: 

hijk , ,

n + 1

sijk , ,

n + 1 hijk , ,n – 1

sijk , ,

n – 1 –

2Δt ---------------------------------------------------------- DV

n

ijk 1 + ⁄2 , ,

sijk , ,

n + 1

sijk , , + 1

n + 1 – sijk , ,n – 1

sijk , , + 1

n – 1 ( ) + –

hijk , ,

n hijk , , + 1n ( ) +

+ ----------------------------------------------------------------------------------------

DV

n

ijk 1 – ⁄2 , ,

sijk , , – 1

n + 1

sijk , ,

n + 1 – sijk , , – 1n – 1

sijk , ,

n – 1 ( ) + –

hijk , , – 1

n hijk , ,n ( ) + – --------------------------------------------------------------------------------------- Fijk , ,

n = , (4.39)

where Fijk , ,

n represents the explicit terms and is defined by

Fijk , ,

n ADVs ( )ijk , ,

n – HDIFFs ( )ijk , ,n – 1 = + . (4.40)

The expanded form of the two terms in Fijk , ,

n is included in Appendix D (eqs. D.9 and D.10). Equation 4.39 can be rearranged 

into the following form

Aijk , , sijk , , – 1

n + 1 Bijk , , sijk , ,n + 1 Cijk , , sijk , , + 1n + 1

+ + Dijk , , = (4.41)
92 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

where

Aijk , , – DV

n

ijk 1 – ⁄2 , , hijk , , – 1

n hijk , ,n = ⁄ , ( ) +

Bijk , , hijk , ,

n + 1 ⁄ ( ) 2Δt DVn

ijk 1 + ⁄2 , , hijk , ,

n hijk , , + 1n ⁄ ( ) + DVn

ijk 1 – ⁄2 , , hijk , , – 1

n hijk , ,n = + + ⁄ , ( ) +

Cijk , , – DVijk 1 + ⁄2 , ,

n hijk , ,

n hijk , , + 1n = ⁄ , ( ) + and

Dijk , , Fijk , ,

n hijk , ,

n – 1

sijk , ,

n – 1 = + ( ) ⁄ ( ) 2Δt

DV

n

ijk 1 + ⁄2 , , sijk , ,

n – 1 sijk , , + 1 n – 1 ( ) – hijk , ,

n hijk , , + 1n – ⁄ ( ) +

DV

n

ijk 1 – ⁄2 , , sijk , , – 1 n – 1 sijk , ,

n – 1 ( ) – hijk , , – 1

n hijk , ,n ( ) +

· + ⁄

(4.42)

Applying equation 4.41 to each of the layers at a computational point iΔx, jΔy results in a system of km equations involving km

unknown values of sijk , ,

n + 1 . The boundary conditions are satisfied by choosing Ai,j, 1 = 0, Ci,j, km = 0 and setting the diffusion 

coefficients to zero at the free surface and bottom. The matrix form of the equations is tridiagonal, which can be efficiently solved 

with the double sweep algorithm. Once the new salinities are computed, they are used to update the density field using the equation 

of state presented in Appendix B.

4.3.2 Semi-Implicit Trapezoidal Scheme

 The finite-difference equations for the semi-implicit trapezoidal scheme are nearly identical to those of the semi-implicit 

leapfrog scheme, except the time interval over which the scheme is applied is halved to Δt from 2Δt. The integration procedure is 

centered at the time level (n + 1⁄2)Δt and no longer involves the time level (n - 1)Δt. The scheme, as used here, involves three time 

levels (n, n + 1⁄2, n + 1) but is referred to as a two-level scheme since it is executed over a single time step. The variables needed 

in the scheme at the (n + 1⁄2)Δt time level are determined by averaging; for example,

Ui 1 + ⁄2 , ,j k

n 1 + ⁄2 1

2

-- U

 ̃

i 1 + ⁄2 , ,j k

n + 1 Ui 1 + ⁄2 , ,j k

n

+ ⎝ ⎠ = ⎛ ⎞ ,

where the symbol (  ̃ ) denotes the estimate of the unknown from the semi-implicit leapfrog solution. The form of the explicit stage 

for the x-momentum finite-difference equation is

U

ˆ

i 1 + ⁄2, ,j k U i 1 + ⁄2, ,j k

n Δt A DVx ( )n 1 + ⁄2 CORx ( )n 1

[– + + ⁄2 BCLINICx ( )n 1– + ⁄2 HDIFFx ( )n ]

i 1 + ⁄2, ,j k = + + , (4.43)
5. Numerical Experiments 93

which is very similar to equation 4.23 for the leapfrog scheme. The bracketed terms are identical to those expanded in Appendix

D but are evaluated at the time level indicated. The finite-difference equation for the implicit stage of the x-momentum equation is

U i 1 + ⁄2, ,j k

n + 1 U

ˆ

i 1 + ⁄2, ,j k – g

2

--- Δt

Δx

------ hi 1

⁄2 + , ,j k

n 1 + ⁄2

ρi 1 + ⁄2, ,j 1 n 1 + ⁄2

ρi 1 + ⁄2, ,j k n 1 + ⁄2

---------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

ζi + 1, j

n + 1 ζi j ,n + 1 ζi + 1, j

n ζi j ,n = ⋅ ( ) – + –

Δt

2

----- AVi 1 + ⁄2 j k 1 – ⁄2 , ,

n 1 + ⁄2

⎝

⎜

⎛ ( ) U h⁄ i 1 + ⁄2, ,j k – 1

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j kn + 1 –

h i 1 + ⁄2 j k 1 – ⁄2 , ,

n + 1 ---------------------------------------------------------------------------- ui 1 + ⁄2, ,j k – 1 n ui 1 + ⁄2, ,j k n –

h i 1 + ⁄2 j k 1 – ⁄2 , ,

n + ------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

+ ⋅

 AVi 1 + ⁄2 j k 1 + ⁄2 , ,

n 1 + ⁄2 ( ) U h⁄ i 1 + ⁄2, ,j k

n + 1 ( ) U h⁄ i 1 + ⁄2, ,j k + 1n + 1 –

hi 1 + ⁄2 j k 1 + ⁄2 , ,

n + 1 ---------------------------------------------------------------------------- ui 1 + ⁄2, ,j k

n ui 1 + ⁄2, ,j k + 1n –

hi 1 + ⁄2 j k 1 + ⁄2 , ,

n + ------------------------------------------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞

⎠

⎟

⎞ – ⋅ , (4.44)

which is analogous to the leapfrog equation 4.26. The continuity equation for the trapezoidal scheme is

ζi j ,

n + 1 ζi j ,

n 1

2

-- Δt

Δx

------ Ui 1 + ⁄2, ,j k

n + 1

Ui 1 – ⁄2, ,j k

n + 1

Ui 1 + ⁄2, ,j k

n Ui 1 – ⁄2, ,j k

n ( ) – + –

k = 1

km

– ∑

1

2

-- Δt

Δx

------ Vi j 1 , , + ⁄2 k

n + 1

Vi j 1 , , + ⁄2 k

n + 1

Vi j 1 , , + ⁄2 k

n Vi j 1 , , – ⁄2 kn ( ) – + –

k = 1

km

– ∑ .

=

(4.45)

It is straightforward to develop all the equations for the trapezoidal step from those already presented in section 4.3.1.

The iteration for the matrix solution in the trapezoidal step is started with the earlier estimates for ζi, j

n +1

⎩ ⎭ ⎨ ⎬ ⎧ ⎫ from the leapfrog 

step. The accuracy of these estimates causes the iterative convergence to be rapid.

Experience with the 3-D model has shown that the trapezoidal step is not always needed. For example, the 3-D test case in 

this report could be solved accurately with just the leapfrog step. The trapezoidal step is needed mostly to stabilize the solution for 

markedly nonlinear problems and to improve the accuracy of the time integration when large time steps are used. If necessary to

stabilize a solution, more than one iteration of the trapezoidal step can also be used; a sparing use of additional iterations is advisable, however, to keep the computer run time of the model from becoming excessively long.

5. Numerical Experiments

5.1 Introduction

Numerical experiments are useful in first verifying that the computer coding of a computational scheme can accurately solve 

the governing equations under at least some combinations of computational grid intervals and wave conditions. For this purpose,

the test problems used in experiments must have solutions that are available either analytically or from another independently 

developed and verified computer model. Once the computer code is verified, numerical experiments are then useful in studying 

the stability and convergence properties of a numerical scheme by comparing solutions that are computed by using varying time 

and space steps and wave conditions.
94 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The von Neumann (or Fourier) method (O’Brien and others, 1950) is a widely used tool for studying the stability of numerical 

schemes that are used in solving linear differential equations. It is based on representing the solution of a difference equation as a 

finite Fourier series and studying the decay or amplification of each mode individually to determine stability or instability. By 

introducing the concept of a complex propagation factor, Leendertse (1967) extended the von Neumann method for use in analyzing the convergence of linear numerical schemes. Using two-time-level, finite-difference discretizations of the linear shallow water 

equations, Casulli and Cheng (1990) and Casulli and Cattani (1994) used the von Neumann method to study the stability of 1-D 

and 3-D semi-implicit methods, respectively. Cunge and others (1980) present a von Neumann convergence analysis in the form 

of amplitude and phase error portraits for the 1-D semi-implicit scheme of Abbott-Ionescu. Using a finite-element discretization 

of the 1-D linear shallow-water equations, Gray and Lynch (1977) applied the von Neumann method to analyze the convergence 

of both two- and three-time-level semi-implicit schemes. Although the von Neumann method is indeed useful, it is fundamentally 

incomplete as it does not take into account the full effects of nonlinearity, iteration schemes, and interpolations on the properties 

of stability and convergence.

In this chapter, numerical experiments are used to investigate the stability and convergence of the full nonlinear form of the 

1-D and 3-D semi-implicit schemes presented in Chapter 4. An emphasis is placed on studying the effect of iteration.43 Instances 

of nonlinear instability are shown to occur in the 1-D experiments that are not revealed by the linear von Neumann analysis. Iteration is shown to be effective in suppressing the instabilities and obtaining a smooth solution. Because adding iterations does

increase the computer run time of a model simulation, finding the minimum number of iterations that are needed to achieve a stable 

and accurate solution is an important goal.

5.2 One-Dimensional Experiments

The two-level and three-level semi-implicit schemes for one dimension are tested below using two numerical experiments. 

The first experiment routes a hydrograph down a rectangular channel with an initially shallow depth of flow. The experiment represents an inland river modeling problem that is strongly nonlinear and has high frictional resistance. The second experiment simulates the propagation of a repeating semidiurnal tide into a closed basin of constant depth. The tidal flow experiment is influenced 

only a little by frictional resistance but is affected by nonlinear advection and the rapidly changing bidirectional flow that is typical 

in estuaries.

5.2.1 Hydrograph Routing

This experiment was first designed by the U.S. Geological Survey (USGS) in the early 1980s to test and compare three different 1-D finite-difference flow models, including the two widely used models developed by the National Weather Service (Fread, 

1978) and the U.S. Geological Survey (Schaffranek and others, 1981).44 The experiment has since been used by USGS researchers 

(for example, DeLong, 1993, fig. 2) to test other flow models that are based on numerical schemes of varying accuracies. The 

43In the case of the three-time-level scheme, iteration is used here to mean the addition during each time step of one or more trapezoidal-scheme steps 

after the leapfrog- scheme step.

44The 1-D models by Fread (1978) and Schaffranek and others (1981) are based on the fully implicit four-point scheme by Preissmann (1961); there are a 

few differences between the two models in the details of how the four-point scheme is implemented.
5. Numerical Experiments 95

solution to the fully nonlinear test problem is well known from these past model applications; the convergence characteristics of 

these previously tested models vary widely, depending on their numerical accuracy.

The channel characteristics and initial flow conditions for the hydrograph experiment are given in figure 5.1. The channel is 

rectangular; top width B = 100 ft (30.5 m),45 bed slope S0 = 0.001, and length L = 150,000 ft (45,720 m). The Manning resistance 

coefficient for the channel is 0.045. The initial steady flow is Q0 = 250 ft3/s (7.1 m3/s), and the initial flow depth according to 

Manning’s formula is H0 = 1.688 ft (0.51 m). The channel is considered wide enough so that the hydraulic radius in equation 4.3

can be approximated by the flow depth.

For the test problem, the discharge at the upstream boundary of the channel (x = 0) is described by

Q t( ) 250 750

π

-------- 1 πt

75 – cos----- ⎝ ⎠ ⎛ ⎞ = + 0 < <t 150

and

Q t( ) = 250 t ≥ 150 ,

where the time t is expressed in minutes from the start of the simulation and discharge is in cubic feet per second (ft3/s). At the 

downstream boundary of the channel (x = 150,000), the depth of flow is constant at H0. The simulation is run for 500 minutes. The 

results of the hydrograph routing are examined at x = 50,000 ft (15,240 m) to avoid any effect from the downstream boundary 

condition on the computed hydrograph. (The water level at the downstream boundary does not rise until the hydrograph passes 

50,000 ft.)

Q = Q (t )

Channel characteristics

B = 100 feet

L = 150,000 feet

S0 = 0.001

 Manning’s n = 0.045

Initial flow conditions

Q0 = 250 cubic feet per second

H0 = 1.688 feet

L

B

Datum

Q0 Δ

Δ

1

Q = Q0

1 S0 ⁄

x

H0

45The 1-D numerical experiments, as originally designed, were posed in English units; these units are adopted here for consistency. The metric unit 

conversions are shown in parentheses following the English units.

Figure 5.1. Channel characteristics and initial flow conditions for the 1-dimensional 

hydrograph test problem.
96 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

100

0 100 200 300 400 500

200

300

400

500

600

700

800

TIME, IN MINUTES

Upstream

Boundary

50,000 feet

Δt = 5 seconds, Δ x = 50 feet, niter = 5

Δt = 5 minutes, Δ x = 5,000 feet, niter = 1

Δt = 5 minutes, Δ x = 5,000 feet, niter = 2

RMS error

niter = 1 7.6 percent

niter = 2

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

2.7 percent

5.2.1.1 Two-Level Scheme

Figure 5.2 shows three computed hydrographs using the two-level, semi-implicit scheme described in section 4.2.1. Two of 

the solutions are based on computations using a coarse numerical grid size of Δx = 5,000 ft (1,524 m) and Δt = 5 minutes; the third 

solution is considered a “base solution” and is computed using an extremely small grid size that keeps the numerical error in the 

solution negligible. The base solution agrees to a high degree of accuracy with the solutions to this problem obtained using the 

other flow models tested by the USGS. All three solutions were computed with the number of iterations (niter) of the numerical 

scheme fixed throughout the simulation; the coarse grid solutions used values of niter = 1 and niter = 2, the base solution used 

niter = 5.

Figure 5.2 indicates that a significant improvement in accuracy is gained by solving the coarse grid case using two iterations 

rather than one. The root mean square error (RMSe in table 5.1) of the solution measured against the base solution improved from 

7.6 percent using niter = 1 to 2.7 percent using niter = 2. The improvement is due to a centering in time of the nonlinear coefficients 

of the finite-difference momentum equation by the iteration process, as discussed in section 4.2.1. Because the frictional attenuation of the routed hydrograph is significant, the backward-in-time evaluation of the frictional coefficient γ in equation 4.7 is mostly 

responsible for the large underestimation of the wave attenuation in the solution using niter = 1; also affecting the accuracy, but to 

a lesser degree, is the backward-in-time evaluation of the coefficient H in the pressure (water surface slope) and friction terms in 

equation 4.7. The centering in time of both γ and H by the iteration process raises the accuracy of the time integration scheme from 

first to second order.

Figure 5.2. Hydrographs computed with the two-level, semi-implicit scheme using a large spatial step size of 

Δx = 5,000 feet (1,524 meters) and a time step size of Δt = 5 minutes. 

Solutions using one iteration (niter = 1) and two iterations (niter = 2) are compared against a base solution using a very small time and space step and five iterations (niter = 5). The hydrograph at the upstream boundary 

is also plotted. RMS, root mean square.
Table 5.1. Definition of error measures (expressed as percentages) for the hydrograph test problem.

Description Definition

Root mean square error: Root mean square error 

of the computed hydrograph, normalized on 

the peak discharge of the base solution 

hydrograph

Amplitude error: Error in the peak discharge of 

the computed hydrograph, normalized on the 

peak discharge of the base solution hydrograph

Phase error: Error in the time associated with the 

center of gravity of the computed hydrograph, 

normalized on the time associated with the center of gravity of the base solution hydrograph where

Mass preservation error: Error in mass preservation for the computed solution, normalized on 

the total mass from the base solution

nts = number of time steps in 500 minute simulation

= discharge for the base solution at location x and time 

= peak discharge for the base solution at location x (= 510.25 cubic feet per second at x = 50,000 feet)

Tb(x) = time associated with center of gravity of base solution at location x (= 362.85 minutes for x = 50,000 feet)

Hb(x, t) = depth of flow for the base solution at location x and time t

Q0 = initial steady discharge (= 250 cubic feet per second)

H0 = initial water depth determined by Manning’s formula for a wide rectangular channel (= 1.688464 feet)

TL = time of simulation (= 500 minutes)

L = length of channel (= 150,000 feet)

Integrations were computed numerically using Simpson’s Rule.

RMSe

100 Qn( ) x Qb

n ( ) – ( ) x 2

n = 1

nts

∑

1

⁄2

( ) nts 1

⁄2 Qb

max( ) x -------------------------------------------------------------------------

x = 50,000 feet

=

Ae

100 Qmax( ) x Qb

max ( ) – ( ) x

Qb

max( ) x --------------------------------------------------------------

x = 50,000 feet

=

Pe

100 T x( ) Tb ( ) – ( ) x

Tb( ) x --------------------------------------------

x = 50,000 feet

=

T x( ) ∫0

TL

tQxt ( ) , Q0 ( ) – dt

∫0

TL

Qxt ( ) , Q0 ( ) – dt

= ---------------------------------------------------

Me 100 1 ∫0

L

Hxt ( ) , H0 ( ) – dx

∫0

L

Hb( ) x t , H0 ( ) – dx

– ---------------------------------------------------

⎝ ⎠ ⎜ ⎟

⎛ ⎞

t = 500 minutes

=

Qb

n( ) x t n = Δt

Qb

max( ) x

5. Numerical Experiments 97
98 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The solution shown in figure 5.2 using two iterations is only slightly improved if additional iterations are included in the simulation. The error that remains in the solution when two (or more) iterations are used is attributable to the coarse Δx space step 

chosen. In figure 5.3, the base solution for discharge is plotted at 125-minute intervals against distance along the channel to indicate 

the spatial scale of the wave as it propagates down the channel. A space step of 5,000 ft (1,524 m) represents the upstream wave 

over only six or seven grid points. The coarse spatial resolution causes a decrease in the speed of the solution gravity waves, which 

manifests itself as a time lag between the center of gravity of the computed hydrograph and the base solution. This tendency for 

coarsely resolved wave components of a numerical solution to be delayed (or accelerated) compared to nature is known as phase 

error.

The effect of space step size on the solution convergence and stability is evaluated systematically by testing the two-level 

scheme using increasing values for Δx, starting with a small value of 100 ft (30.5 m); the time step is fixed at 5 minutes. The errors 

in each computed solution relative to the base solution are quantified in various ways by computing the root mean square error 

(RMSe), amplitude error (Ae), phase error (Pe), and mass preservation error (Me) that are summarized in table 5.1. The first three 

of these error measures are based on a comparison between the computed hydrograph and the base solution hydrograph at x = 

50,000 ft (15,240 m). The mass preservation error is computed from the error (relative to the base solution) in total mass of water 

simulated within the channel at the termination of the simulation (t = 500 minutes). Each of the error measures are expressed as a 

percentage. The parameters of the base solution and the definitions of the variables used in defining the error measures are listed 

at the bottom of table 5.1.

100

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

200

300

400

500

600

700

800

t = 125 minutes

t = 250 minutes

t = 375 minutes

t = 500 minutes

DISTANCE, IN FEET x 1,000

DISCHAR

GE, IN 

CUBI

C FEET PEER 

SECOND

Figure 5.3. Base solution of the hydrograph problem shown at 125-minute intervals 

during the simulation.
5. Numerical Experiments 99

Table 5.2 lists the values of Δx used in the numerical experiment and the ranges of the Courant numbers occurring during each 

simulation. Both the gravity-wave (Crg) and the advection (Cra) Courant numbers are listed because both are relevant to the stability and accuracy of the semi-implicit schemes.

The four error measures that are computed from solutions using the two-level model are graphed in figure 5.4 using three 

curves that each represent a solution for a different value of niter. The results clearly show an improvement in the solution accuracy 

when the number of iterations are increased from one to two. There is no choice for Δx that yields a solution with acceptable 

accuracy when only one iteration is used and the time step is 5 minutes. The solutions using two iterations converge to the base 

solution within acceptable error tolerances for spatial step sizes of 2,500 ft (762 m) or less. 

The solutions with the three largest values of Δx (= 2,500, 5,000, and 10,000) and niter = 2 are plotted in figure 5.5 to illustrate 

the improvement in accuracy with decreasing Δx. It is apparent that the spatial step sizes of 5,000 ft (1,524 m) and 10,000 ft 

(3,048 m) are much larger than should ever be employed in practice; these solutions show that the magnitude of errors can be large, 

even when iteration is used if Δx is not chosen properly.

The error measures plotted in figure 5.4 for solutions using five iterations are not significantly different from those using two 

iterations and in fact are mostly larger. Table 5.3 lists the error measures computed for ten solutions using increasing values of 

niter when Δx = 2,500 ft (762 m) and Δx = 5 minutes; the iterative process of convergence to a solution actually “overshoots” on 

the second iteration, and the solution using two iterations does indeed have smaller error measures overall than the solutions using 

additional iterations. While this will not be true in general, it is significant that for this difficult test problem, the addition of only 

a second iteration to the scheme is sufficient to raise the accuracy to an acceptable level.

Table 5.2. Ranges of Courant numbers for hydrograph problem simulations using 

5 minutes as a time step (Δt).

[Δx, size of the computational space interval]

Δx, in feet Crg Cra

100 26.7−37.5 4.4-6.8

500 5.3−7.5 0.89-1.4

1,000 2.7−3.8 0.44-0.68

2,000 1.3−1.9 0.22-0.34

2,500 1.1−1.5 0.18-0.27

5,000 0.53−0.75 0.09-0.14

10,000 0.27−0.38 0.04-0.07

Crg = gravity-wave Courant number = 

Cra = advection Courant number = 

(u gH + ) Δ⋅ t ⁄ Δx

u ⋅ Δ Δt ⁄ x
1

2

3

4

5

6

7

8

9

0

1 iteration

5 iterations

2 iterations

-1

10

Δ x, IN FEET x 1,000

-5

0

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

-10

-15

AMPLITUDE ERROR, IN PER

CENT ROOT MEAN 

SQUARE ERROR, IN PER

CENT

100 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.4. Error measures using the two-level semi-implicit scheme on the hydrograph test problem. 

Δt = 5 minutes for all simulations. Δt is the size of the computational time interval. Δx is the size of the 

computational space interval.
0

1

2

3

4

5

6

7

8

-1

-2

1 iteration

5 iterations

2 iterations

Δ x, IN FEET x 1,000

1

2

3

4

5

0

-1

-2

0 1 2 3 4 5 6 7 8 9 10

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

5. Numerical Experiments 101

Figure 5.4. Continued.
102 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Table 5.3. Error measures using the two-level semi-implicit scheme on the hydrograph problem with the number of iterations (niter) ranging from 1 to 10. 

[Note: the error measures are expressed in percent. All simulations used Δx = 2,500 feet and Δt = 5 minutes] 

Number of iterations Root mean square 

error Amplitude error Phase error Mass preservationerror

1 7.37 25.6 0.53 0.50

2 0.41 0.56 0.05 – 0.04

3 0.79 0.37 0.19 0.23

4 0.81 0.50 0.19 0.23

5 0.81 0.49 0.18 0.23

6 0.81 0.49 0.18 0.23

7 0.81 0.49 0.18 0.23

8 0.81 0.49 0.18 0.23

9 0.81 0.49 0.18 0.23

10 0.81 0.49 0.18 0.23

Δ x = 2,500 0.4 percent

Δ x = 5,000 2.7 percent

Δ x = 10,000 8.6 percent

Δ t = 5 minutes, Δ x = 2,500 feet, niter = 2

Δ t = 5 minutes, Δ x = 5,000 feet, niter = 2

Δ t = 5 minutes, Δ x = 10,000 feet, niter = 2

Upstream

boundary

50,000 feet

100

0 100 200 300 400 500

200

300

400

500

600

700

800

TIME, IN MINUTES

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

RMS error

Figure 5.5. Two-level scheme solutions for the hydrograph problem using three different values for space steps (Δx) and a 

time step (Δt) of 5 minutes. RMS, root mean square.
5. Numerical Experiments 103

The iteration process has a definite effect on the stability of the two-level scheme. The solutions attempted using a single iteration (with Δx = 5 minutes) remained stable and free of significant oscillations only for values of Δx greater than or equal to 

1,000 ft (305 m). The solution using Δx = 1,000 ft (305 m) corresponds to a maximum gravity-wave Courant number of 3.8 and 

an advection Courant number of 0.68 (table 5.2). For smaller values of Δx and increasing Courant numbers, the solutions using 

one iteration exhibited either severe oscillations or complete instability. Figure 5.2 shows the solution using a single iteration with 

Δx = 500 ft (152 m) and Δt = 5 minutes. The solution has severe oscillations and is on the verge of going completely unstable 

at the termination of the 500 minute simulation. The addition of a second iteration stabilizes the solution and removes most of

the error. For the space-centered differencing of the advection term used here, Casulli and Cheng (1990) found, using the 

von Neumann analysis, the requirement that the flow be subcritical to ensure linear stability of the two-level scheme without 

implementing iterations. Because the flow in the test problem remains subcritical throughout the simulation, the instabilities that 

occur for high Courant numbers are a form of nonlinear instability. No instabilities were encountered in any of the solutions 

attempted using two or five iterations for Courant numbers as high as Crg = 37.5 and Cra = 6.8.

100

0 100 200 300 400 500

200

300

400

500

600

700

800

Upstream

boundary

50,000 feet

Base solution

Δ t = 5 minutes, Δ x = 500 feet, niter = 1 

Δ t = 5 minutes, Δ x = 500 feet, niter = 2 

RMS error

niter = 1 7.4 percent

niter = 2 0.2 percent

TIME, IN MINUTES

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

Figure 5.6. Solution using the two-level scheme on the verge of going unstable when using only one iteration (niter = 1). 

A second iteration (niter = 2) stabilizes the solution and removes most of the error. The solution using two iterations is almost indistinguishable from the base solution. Δx is the size of the computational space interval. Δt is the size of the 

computational time interval. RMS, root mean square
104 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

All of the error measures presented in figure 5.4 for solutions with Δx ≥ 1,000 ft (305 m) were obtained using the four-point 

interpolation scheme given by equation 4.10. For solutions with Δx < 1,000 ft (305 m), two-point interpolation was used because 

it gave results identical to those from the higher-order interpolation. Only the solutions using the two largest values of Δx (5,000 

and 10,000) were affected significantly when the low-order interpolation was used; for these solutions there was additional 

attenuation in the routed hydrograph.

The two-level model was tested again to determine the behavior of errors as the time step Δt was varied over a wide range 

while keeping the space step fixed at 2,000 ft (610 m). The list of time steps and the ranges of Courant numbers for the test runs 

are given in table 5.4. The graphs of the four error measures from table 5.1 are presented in figure 5.7 using three curves that are 

labeled by the number of iterations used in the solutions (similar to fig. 5.4). Because the time base of the input hydrograph is 150 

minutes, a time step of 15 minutes corresponds to ten points in time to resolve the hydrograph.

An inspection of the graphs in figure 5.7 reveal the following information regarding the numerical errors:

1. The magnitude of the errors generally increases with Δt.

2. There is much less error in the solutions using two iterations than in those using one iteration. The time step must 

be reduced to 1 minute or less to keep the errors for solutions using one iteration to a reasonable level; a time step 

of 10 minutes or less is acceptable if two or more iterations are used.

3. Mass preservation and phase errors are reasonably small (< ± 1.4 percent) for all runs using Δx = 2,000 ft (610 

m). Mass preservation errors are negative (indicating a loss of mass) for the solutions using two iterations, but 

positive otherwise; phase errors generally are positive (lagging) except for the three solutions using two iterations 

and Δt ≥ 5 minutes, which have negative (leading) phase errors. In general, mass preservation and phase errors 

are affected more by a large Δx than by a large Δt, as suggested by comparing results in figure 5.7 with those in 

figure 5.4.

Regarding stability, the following information was found in preparing the test simulations for figure 5.7:

1. No stable solutions were obtained using one iteration with a time step greater than 5 minutes and gravity-wave 

and advection Courant numbers greater than 1.9 and 0.34, respectively.

2. The solutions using two iterations were stable up to a time step of 15 minutes, but the solution for Δt = 15 minutes 

was spoiled with severe oscillations (see fig. 5.8).

3. The solutions using five iterations were stable up to a time step of 20 minutes, the largest attempted.

Table 5.4. Ranges of Courant numbers for hydrograph problem simulations using a 

space step (Δx) of 2,000 feet.

Δt, in minutes Crg Cra

0.5 0.13–0.19 0.02–0.03

1 0.27–0.38 0.04–0.07

2.5 0.67–0.95 0.11–0.18

5 1.3–1.9 0.22–0.34

10 2.7–3.8 0.44–0.68

15 4.0–5.6 0.67–1.0

20 5.3–7.5 0.89–1.4
2

4

6

8

0

-1

10

1 iteration

5 iterations

2 iterations

Δ t, IN MINUTES

-5

0

5

10

15

20

25

30

-10

-15

0 5 10 15 20

ROOT MEAN 

SQUARE ERROR, IN PER

CENT AMPLITUDE ERROR, IN PER

CENT

5. Numerical Experiments 105

Figure 5.7. Error measures using the two-level, semi-implicit scheme on the hydrograph test problem.

Δx is the size of the computational space interval; Δx = 2,000 feet for all simulations. Δt is the size of 

the computational time interval.
0

2

4

6

8

-2

1 iteration

5 iterations

2 iterations

Δ t, IN MINUTES

1

2

3

4

5

0

-1

-2

0 5 10 15 20

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

106 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.7. Continued.
5. Numerical Experiments 107

Overall the results for this numerical experiment indicate the two-level scheme benefits in terms of both accuracy and stability 

from the use of iterations. The improvements gained by a second iteration outweigh the costs associated with the increase (by 

nearly twofold) in the computer run time. Use of more than two iterations may not be cost effective unless the extra iterations are 

needed to stabilize a particular solution for which reducing the size of the time step is not desirable. It is common in iterative solution schemes to control the number of iterations internally in the computer code by allowing the iterations to continue until some 

predefined error tolerance is satisfied, on the basis of the change in solution variables between iterations. This process is not recommended here as it would typically require a minimum of three iterations, which often will be more than are needed. 

5.2.1.2 Three-Level Scheme

The graphs of error measures for solutions to the hydrograph problem using the three-level scheme are plotted in figures 5.9

and 5.10. The four error measures are plotted against Δx in figure 5.9 and against Δt in figure 5.10. The grid sizes used for the 

solutions in these figures correspond exactly with those for the two-level solutions in figures 5.4 and 5.7 and are listed in tables 

5.2 and 5.4. The three curves plotted in each graph of figures 5.9 and 5.10 correspond to solutions using (1) the semi-implicit leapfrog scheme, without a trapezoidal step; (2) the semi-implicit leapfrog-trapezoidal scheme, with one iteration of the trapezoidal 

step; and (3) the semi-implicit leapfrog-trapezoidal scheme, with four iterations of the trapezoidal step. These three solution methods are similar in terms of computational effort to the two-level solutions using one, two, and five iterations, respectively. This 

similarity allows the two- and three-level schemes to be compared directly.

100

0 100 200 300 400 500

200

300

400

500

600

700

800

Upstream

boundary

50,000 feet

Base solution

Δ t = 15 minutes, Δ x = 2000 feet, niter = 3

Δ t = 15 minutes, Δ x = 2000 feet, niter = 2

RMS error

niter = 2 6.8 percent

niter = 3 2.9 percent

TIME, IN MINUTES

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

Figure 5. 8. Solution using two iterations (niter = 2) with the two-level scheme that is stabilized, 

and greatly improved, with an additional iteration (niter = 3). 

Δx is the size of the computational space interval. Δt is the size of the computational time interval. 

RMS, root mean square
1

2

3

4

5

6

7

8

9

0

-1

10

Leapfrog

Leapfrog-trapezoidal, 4 iterations

Leapfrog-trapezoidal, 1 iteration

Δ x, IN FEET x 1,000

-5

0

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

-10

-15

AMPLITUDE ERROR, IN PERCENT ROOT MEAN SQUARE ERROR, IN PERCENT

108 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.9. Error measures using the three-level semi-implicit scheme on the hydrograph test problem for different values of the computational space interval (Δx). 

Δt = 5 minutes for all simulations. Δt is the size of the computational time interval. 

On each of these graphs, the error measures plotted for the leapfrog-trapezoidal 

solutions using 1 and 4 iterations are virtually indistinguishable.
0

1

2

3

4

5

6

7

8

-1

-2

Leapfrog

Leapfrog-trapezoidal, 4 iterations

Leapfrog-trapezoidal, 1 iteration

Δ x, IN FEET x 1,000

1

2

3

4

5

0

0 1 2 3 4 5 6 7 8 9 10

-1

-2

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

5. Numerical Experiments 109

Figure 5.9. Continued.
Δ t, IN MINUTES

2

4

6

8

9

7

5

3

1

0

−1

10

Leapfrog

Leapfrog-trapezoidal, 4 iterations

Leapfrog-trapezoidal, 1 iteration

AMPLITUDE ERROR, IN PER

CENT ROOT MEAN 

SQUARE ERROR, IN PER

CENT

0

10

20

25

15

5

−5

30

−10

−15

0 5 10 15 20

110 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.10. Error measures using the three-level semi-implicit scheme on the hydrograph 

test problem for different values of the computational time interval (Δt). 

Δx = 2,000 feet for all simulations. Δx is the size of the computational space interval.
5. Numerical Experiments 111

0

2

4

6

8

−2

Leapfrog

Leapfrog-trapezoidal, 4 iterations

Leapfrog-trapezoidal, 1 iteration

Δ t, IN MINUTES

1

2

3

4

5

0

−1

−2

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

0 5 10 15 20

Figure 5.10. Continued.
112 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

 As seen in figures 5.9 and 5.10, the leapfrog scheme46 solution to the hydrograph problem produces errors much smaller than 

those produced by the two-level scheme without iteration (niter = 1) shown in figures 5.4 and 5.7. By involving three time levels 

in the finite-difference equations, the leapfrog scheme achieves second-order accuracy in the time integration by evaluating the 

nonlinear coefficients γ and H in equation and the advection term in equation 4.14, at the time level n, which is centered between 

levels n + 1 and n – 1 (fig.4.3). As pointed out previously, the solutions to this test problem are particularly sensitive to the 

evaluation of the frictional coefficient γ. A direct comparison of computed hydrographs using the leapfrog and two-level 

(niter = 1) schemes is shown in figure 5.11 for a grid size of Δx = 2,500 ft (762 m) and Δt = 5 minutes. The results show that the 

leapfrog scheme gives a much better estimation of the frictional attenuation in the hydrograph.

The error measures for the leapfrog scheme solutions plotted in figures 5.9 and 5.10 are mostly smaller47 than those for the 

leapfrog-trapezoidal solutions, although only slightly. As the next 1-D numerical experiment will demonstrate, this is not always 

true; in some instances, the addition of a trapezoidal scheme step will improve the accuracy of the leapfrog scheme significantly. 

For the hydrograph test problem, however, the primary benefit of the trapezoidal step is not to improve accuracy but to extend the 

range of stability of the scheme. The improvement in stability is the same as that gained through the addition of iterations to the 

46As used here, the phrase leapfrog scheme is meant to imply the semi-implicit version of the scheme rather than the explicit version. 

47The mass preservation errors, however, are mostly larger in the leapfrog solutions than in the leapfrog-trapezoidal solutions.

Semi-implicit two-level, niter = 1

Upstream

boundary

50,000 feet

Semi-implicit leapfrog

RMS error

Leapfrog 0.3 percent

Two-level 7.4 percent

100

0 100 200 300 400 500

200

300

400

500

600

700

800

TIME, IN MINUTES

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

Figure 5.11. Comparison of computed hydrographs with the leapfrog semi-implicit scheme and 

the two-level semi-implicit scheme (niter = 1). 

The leapfrog solution has very little error. For both runs, Δx = 2,500 feet, Δt = 5 minutes. 

Δx is the size of the computational space interval. Δt is the size of the computational time 

interval. RMS, root mean square. 
5. Numerical Experiments 113

two-level scheme. In the simulations for figure 5.9, the basic leapfrog scheme yielded stable solutions using a Δt of 5 minutes only 

for spatial step sizes that are greater than or equal to 1,000 ft (305 m). In figure 5.10, the leapfrog solutions were stable only up to 

a time step of 5 minutes for the space step of 2, 000 ft (610 m). The leapfrog-trapezoidal solutions in figure 5.10 with one iteration 

of the trapezoidal step were stable up to a time step of 15 minutes; with four iterations, the solutions were stable up to a time step 

of 20 minutes.

In comparing the error measures in figure 5.10 with those in figure 5.7, it is significant that the addition of the trapezoidal step 

to the leapfrog scheme gave solutions that generally have equal or smaller errors than solutions using two iterations of the twolevel scheme. In particular, for the large time steps of 10 and 15 minutes, the error measures are noticeably smaller in figure 5.10

for the solutions using the leapfrog-trapezoidal scheme (1 iteration) than in figure 5.7 for the solutions using the two-level scheme 

(niter = 2). The leapfrog-trapezoidal scheme solution for Δt = 15 minutes and Δx = 2,000 ft (610 m) also did not exhibit the oscillations shown in figure 5.8 for the solution using the two-level scheme (niter = 2).

5.2.1.3 Remarks

Based on the results of the hydrograph numerical experiment, the three-level scheme is more accurate than the two-level 

scheme. The two-level scheme without using iterations is only first-order accurate and requires an extremely small numerical grid 

size to reduce the error in the hydrograph solutions to an acceptable level. Using two iterations, it is possible to raise the accuracy 

of the two-level scheme to second order, but the same order of accuracy can be achieved at approximately half the computational

cost by using the three-level, leapfrog scheme. The use of iteration in the solutions for either scheme extends the range of computational intervals and Courant numbers for which the solutions are stable. If a single iteration of the trapezoidal step is used in the 

three-level scheme (so that, for example, a stable solution is obtained for a large time step), generally the scheme still is more accurate than the comparable two-level scheme using two iterations. The use of more than one iteration of the trapezoidal step in the 

three-level scheme, or more than two iterations in the two-level scheme, was not worthwhile for the hydrograph problem, considering the extra computational cost incurred for the minimal improvement in accuracy that is gained; if the extra iterations are

needed to stabilize a particular solution, then the time step and Courant numbers may be too large on the basis of accuracy 

considerations anyway.

As noted previously, the hydrograph routing problem has been used by the USGS to test three 1-D models (Fread, 1978; 

Schaffranek and others, 1981; and Delong, 1993) that are based on the popular four-point finite-difference scheme. A few comments can be made regarding how the semi-implicit, three-level scheme presented here compares with the four-point scheme. The 

four-point scheme, first introduced by Preissmann (1961), is a fully implicit scheme in which both of the independent variables are 

computed at the same grid points rather than at different ones as in the case of a staggered grid. The scheme is nearly second-order 

accurate in the space and time integration methods so long as a finite-difference weighting factor θ is chosen close to 0.5 (for 

unconditional linear stability, θ must be greater than or equal to 0.5). The overall accuracy (convergence characteristics) of the 

four-point and semi-implicit schemes are similar for solutions with gravity-wave Courant numbers greater than one (see Fread, 

1974, for a detailed report on the convergence characteristics of the four-point scheme). For gravity-wave Courant numbers less

than one, the high frequency solution components of the four-point scheme have leading phase errors; these errors are the cause
114 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

of the 2Δx-wavelength oscillation in the four-point solution shown in figure 5.12 using the model of Schaffranek and others (1981). 

Solutions obtained using the two other four-point models included oscillations similar to those in figure 5.12. No oscillations, either 

leading or lagging, are present in the semi-implicit, leapfrog solution using the staggered grid. Solution oscillations are a common 

problem in modeling estuaries of complex bathymetry, so a numerical scheme is usually sought in which the 2Δx waves are 

damped.

The semi-implicit, leapfrog scheme is considerably more computationally efficient than the four-point scheme when both use 

the same values for Δt and Δx. Because the four-point scheme is fully implicit, it involves the solution of a system of nonlinear 

algebraic equations at each time step for the unknown water surface elevations and volumetric transports using a functional iterative procedure such as the Newton-Raphson method (Amein and Fang, 1970). The semi-implicit leapfrog scheme only involves 

the solution of a tridiagonal system of linear equations at each time step for the water surface elevation using the double-sweep 

method; the volumetric transports are computed explicitly. When a trapezoidal step is added to the leapfrog scheme, the computational time is still significantly lower than that for the four-point scheme.48 Because in one dimension the computational time for 

48A carefully controlled comparison of the computer run times for the leapfrog-trapezoidal and four-point schemes was not done because the four-point 

computer codes are only a part of complete modeling systems for natural rivers that have many features and options that affect model run time; the leapfrogtrapezoidal scheme was programmed only as a test code for rectangular channels exclusively.

100

0 100 200 300 400 500

200

300

400

500

600

700

800

TIME, IN MINUTES

Upstream

boundary

50,000 feet

Semi-implicit leapfrog

Base solution

4-point implicit (θ = 0.6)

Crg = 0.53 – 0.75

DISCHAR

GE, IN 

CUBI

C FEET PER 

SECOND

Figure 5.12. Comparison of solutions to the hydrograph problem with the leapfrog semiimplicit scheme and the four-point model of Schaffranek and others (1981) using a 

space step (Δx) of 5,000 feet and a time step (Δt) 0f 5 minutes for all simulations. 

Crg is the gravity-wave Courant number.
5. Numerical Experiments 115

a numerical scheme generally is not too important in choosing a scheme, the four-point scheme is sometimes preferred over other

schemes for the convenience of the fully dense grid and the ease with which the compact four-point computational stencil allows

the space interval Δx to be varied between node points without affecting the accuracy of the approximation. In one dimension, 

staggered grid schemes are in general not as well suited to irregular Δx distance intervals as the four-point scheme. In three dimensions, however, the computational run time of a multidimensional four-point scheme would be prohibitively long for most real 

applications, and the semi-implicit scheme clearly is a better choice for reasons of computational efficiency. Flexibility in the horizontal grid point placement in a 3-D model can be achieved using a coordinate transformation, which is easily accommodated 

with a semi-implicit scheme, as demonstrated in the 3-D, curvilinear-coordinate model of Muin and Spaulding (1997a).

5.2.2 Tidal Flow in a Closed Channel

The second 1-D numerical experiment uses a variation of a problem suggested by Alan F. Blumberg (written commun., 1992) 

to test the numerical damping characteristics of a hydrodynamic model. It is the propagation of a tidal wave from the sea into a 

wide rectangular channel that is closed at the landward boundary (fig. 5.13). The channel is 315,000 ft (96 km) long and filled with 

water initially at rest (Q0 = 0) to a uniform depth H0 = 33 ft (10 m). The channel is wide in comparison with the depth of water, so 

the simulations are based on a unit width. The Manning resistance coefficient for the channel is n = 0.020.

Channel characteristics

B = 1 foot (unit width channel)

L = 315,000 feet

S0 = 0.00

 Manning’s n = 0.020

Initial flow conditions

Q0 = 0.0

H0 = 33 feet

L

Δ

Datum

H0

Tidal

forcing

ζ ζ = (t)

Wall

Sea

x

EXPLANATION

Figure 5.13. Channel characteristics and initial flow conditions for the 1-dimensional tidal test problem. 

The water in the channel is initially at rest at a uniform depth of H0. Tidal forcing is introduced at the seaward boundary 

causing the water surface to oscillate. Two successive positions of the water surface are shown. At the seaward 

boundary, water moves up and down and horizontally. At the wall boundary, water moves up and down only.
116 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The driving force for this problem is the variation in water surface elevation ζ at the seaward boundary that is caused by a 

simple harmonic tide

ζ( )t A 2πt

t

p

-------- ⎝ ⎠ ⎛ ⎞ = sin , (5.1)

where the time t is expressed in hours from the start of the simulation. The tidal amplitude is A = 3 ft (0.9 m) and the tidal period 

is tp = 12.4 hours. A period of 12.4 hours corresponds with the principal lunar constituent of the astronomical tide known as the 

M2 tide.

The dimensions of the channel length and depth were chosen so that the wavelength49 of the M2 tide is close to four times the 

length L of the tidal channel. Under these conditions, the traveling wave entering the channel interferes constructively (adds 

together) with the wave reflected from the wall boundary, thereby setting up a resonant oscillation in the channel. The result is a 

tidal range of water surface elevation that is greater at the closed boundary than at the open boundary. This effect is illustrated in 

figure 5.14 by using a solution to the problem showing water surface elevation at six locations along the channel against time.

49The wavelength of the tide can be estimated as Lw t

p gH0 = ⋅ with tp expressed in seconds.

TIME, IN DAYS

EXPLANATION

0 1 2

−2

0

−1

−3

−4

2

1

3

5

4

−5

0 feet

120,000 feet

60,000 feet

180,000 feet

300,000 feet

240,000 feet

WATER 

SURFA

CE ELEVATION, IN FEET

Figure 5.14. Water surface elevations at six locations along the tidal channel. 

The distances are measured landward from the seaward boundary.
5. Numerical Experiments 117

Each test simulation for the tidal problem is run for a total of six days (144 hours). This allows sufficient time for any solution 

instabilities to appear. The results of each simulation are presented at the location x = 180,000 ft (55 km) from the seaward boundary of the tidal channel. A solution for water surface elevation and velocity is shown in figure 5.15. After simulating ten cycles of 

the M2 tide, results from the 11th cycle are examined in detail and used in the computation of error measures. 

5.2.2.1 Two-Level Scheme

Figure 5.16 shows three solutions of the tidal problem using the two-level scheme; one is the base solution included in both 

graphs. The solid lines present the base solution obtained by using the two-level scheme with a very small grid size of Δt = 0.5 

minutes, Δx = 400 ft (122m) and with five iterations (niter = 5). The base solution is assumed to be free of numerical error for the 

purposes of computing error measures for the solutions with coarser grid sizes. The two solutions indicated by the dashed curves 

in figure 5.16 are computed using large time steps of 60 minutes (fig. 5.16A) and 30 minutes (fig. 5.16B.); both solutions use a 

space step (Δx) of 6,000 ft (1,828 m) and two iterations (niter = 2). Figure 5.17 shows a plot of the volumetric transport (U = uh) 

from the solution using Δt = 60 minutes compared with the volumetric transport of the base solution. The root mean square error 

(RMSe) of the solution is computed relative to the base solution using the volumetric transport variable defined in table 5.5. The 

RMSe is 8.7 percent for the 60-minute solution and 2.3 percent for the 30-minute solution. The other error measures defined in are 

computed as Ae = 5.2 percent, Pe = 0.35 percent, and Me = 4.7 percent for the solution using Δt = 60 minutes, and 

Ae = – 0.8 percent, Pe = 0.16 percent, and Me = 1.0 percent for the solution using Δt = 30 minutes. 

TIME, IN DAYS

0 3 1 2 45 6

−3

−2

−1

0

1

2

3

−4 Water surface elevation

Velocity

−5

4

5

11th cycle

WATER 

SURFA

CE ELEVATION, IN FEET

VELO

CITY, IN FEET PER 

SECOND

Figure 5.15. Solution of the tidal problem for water surface elevation and velocity at the location x = 180,000 feet. 

The results of simulations for the 11th cycle of the M2 tide are examined in detail in figure 5.16.
u

11th M2 cycle

x = 180,000 feet

-3

-1

1

3

-5

-5

5

Δ t = 0.5 minutes, Δ x = 400 feet, niter = 5

Δ t = 60 minutes, Δ x = 6,000 feet, niter = 2

Δ t = 0.5 minutes, Δ x = 400 feet, niter = 5

Δ t = 30 minutes, Δ x = 6,000 feet, niter = 2

ζ

A

u

-3

-1

1

3

124 126 128 130 132 134 136

5

ζ

B

TIME, IN HOURS

WATER 

SURFA

CE ELEVATION, IN FEET

VELO

CITY, IN FEET PER 

SECOND

118 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.16. Solutions for water surface elevation (ζ) and velocity (u) computed with the two-level semi-implicit 

scheme using a space step (Δx) of 6,000 feet and a time step (Δt) of (A) 60 minutes and (B) 30 minutes. 

The solutions using two iterations (niter = 2) are compared against a base solution that used small time and space 

steps and five iterations (niter = 5). The solutions are shown for the 11th M2 cycle at a location 18,000 feet from the 

seaward boundary of the tidal channel. 
5. Numerical Experiments 119

Considering that time steps of 60 minutes and 30 minutes are quite large relative to the 12.4 hour period of the tidal forcing,

the solutions in figure 5.16 are reasonably good. Simulations using these two time steps were made using the two-level scheme, 

trying both larger and smaller values of Δx, and the results generally were only slightly changed. Because of the long tidal wavelength, Δx, up to the largest value of Δx = 30,000 ft (9,144 m), had relatively little effect on the solutions.

The two-level scheme was tested further by keeping Δx constant at 6,000 ft (1,828 m) and varying Δt over a wide range. The 

values of Δt and the corresponding Courant numbers are given in table 5.6. The test simulations revealed that only for the smallest 

choice, Δt = 1 minute, is a stable solution obtained when the two-level scheme is used with one iteration (niter = 1); the 1 minute 

solution had very little error (RMSe = 0.05 percent). Each simulation using niter = 1 and Δt ≥ 5 minutes ran for only several tidal 

cycles before oscillations developed that eventually grew to destroy the solutions entirely. The oscillations could be traced back to 

the nonlinear advection term in the momentum equation and appeared first in the solutions at the times just prior to the high and 

low water points in the tidal cycle. As a consequence of the standing wave character of the solution, at the times just before high 

and low water, the volumetric transport and water surface elevation are changing in opposite directions and the magnitude of the 

advection term is oscillating near zero.

11th M2 cycle

RMS error = 8.7 percent

x = 180,000 feet

-50

0

50

-100

124 128 130 132 134 136 126

100

base solution

Δt = 60 minutes, = 6,000 feet, Δx niter = 2

TIME, IN HOURS

 VOLUMETRI

C TRAN

SPORT, IN 

SQUARE FEET PER 

SECOND

Figure 5.17. Solution for volumetric transport (U = uh) using the two-level semi-implicit scheme. 

The solution for time step Δt = 60 minutes and space step Δx = 6,000 feet is compared with the base solution. 

RMS, root mean square. 
Table 5.5. Definition of error measures (expressed as percentages) for the tidal test problem. 

[The first three error measures are based on comparisons between the computed solution and the base solution during the 11th tidal 

cycle at x =180,000 feet. The mass preservation error is computed as the maximum error (relative to the base solution) in total mass 

of water within the channel during each simulated time step]

Description Definition

Root mean square error: Root mean square error of 

the computed volumetric transport, normalized on the 

maximum volumetric transport of the base solution

RMSe

100 Un( ) x Ub

n ( ) – ( ) x 2

n n = 1

nts11

∑

1

⁄2

( ) nts11 1

⁄2 Ub

max( ) x ---------------------------------------------------------------------------

x = 180,000

=

Amplitude error: Error in the maximum water 

surface elevation, normalized on the maximum water 

surface elevation of the base solution

Ae

100 ζmax() ζ x b

max ( ) – ( ) x

ζb

max( ) x -----------------------------------------------------------

x = 180,000

=

Phase error: Error in the time associated with the center of gravity of the computed water surface distribution, normalized on the time associated with the 

center of gravity of the base solution

Pe

100 T x( ) Tb ( ) – ( ) x

Tb( ) x --------------------------------------------

x = 180,000

=

where

T x( ) ∫t1

t2

tHxt ( ) , H0 ζ

min [ ] – ( ) + ( ) x dt

∫t1

t2

Hxt ( ) , H0 ζ

min [ ] – ( ) + ( ) x dt

= -----------------------------------------------------------------------------------

Mass preservation error: Maximum error in mass 

preservation for the computed solution, normalized 

on the total mass from the base solution

Me max En = n = 1 2 ... ,, , nts

where

t1 = 7,440 minutes (= 124 hours)

t

2 = 8,190 minutes (= 136.5 hours)

n1 = t

1 ⁄ Δt

n2 = t

2 ⁄ Δt

nts11 = n2 n1 –

nts = number of time steps in a 6 day simulation

Ub

n( ) x = volumetric transport for the base solution at location x and time t n = Δt

Ub

max( ) x = maximum volumetric transport for the base solution at location x (= 82.34 square feet per second at 

180,000 feet)

ζb

max( ) x = maximum water surface elevation for the base solution at location x (= 4.00 feet)

ζmin ( ) x = minimum water surface elevation at location x

H0 = initial water depth in tidal channel (= 33.0 feet)

L = length of channel (= 315,000 feet)

Integrations were computed numerically with Simpson’s Rule.

En 100 1 ∫0

L

Hxt ( ) , H0 ( ) –( ) – 5 dx

∫0

L

Hb( ) x t , H0 ( ) –( ) – 5 dx

– ---------------------------------------------------------

⎝ ⎠ ⎜ ⎟

⎛ ⎞

t n = Δt

=

120 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
5. Numerical Experiments 121

An example of an unstable solution is shown in figure 5.18A using Δt = 10 minutes. The first 1.5 days of the solution are 

plotted before the solution eventually fails completely. In figure 5.18B, the same solution is shown with the advection term “turned 

off;” a smooth solution is obtained. Because the base solution shown in figure 5.18B includes the advection term, the effect of 

neglecting this term is visible; advection increases the mean (tidally averaged) value of the volumetric transport by a small, but not 

necessarily insignificant, amount. The solution results for other values of Δt are similar to those in figure 5.18 although the oscillations tended to occur slightly later in the simulations for larger Δt. In all cases, however, when advection was neglected, the oscillations disappeared. By trial and error it was discovered that stable solutions could be obtained with larger values of Δx. In particular, solutions using Δx = 30,000 ft (9,144 m) gave stable and accurate solutions up to a time step of 15 minutes with niter = 1; for 

time steps larger than 15 minutes, the solutions were stable but overdamped. Of course, by using larger values of Δx, the Courant 

numbers were decreased, which may have contributed to the improved stability. 

It should be noted that no attempt was made to obtain stable results by smoothing the solutions using niter = 1. In all cases, 

however, the solutions were made stable by adding a second iteration (or more) to the scheme. The error measures for these solutions are presented later.

5.2.2.2 Three-Level Scheme

Using a space step of Δx = 6,000 ft (1,829 m), the semi-implicit, leapfrog scheme gave more stable solutions to the tidal problem than the two-level scheme without iteration. Of the time steps listed in table 5.6, the two-level scheme (niter = 1) gave a stable 

solution only for Δt = 1 minute; the leapfrog scheme gave stable solutions for Δt = 1, 5, and 10 minutes. All the solutions with the 

leapfrog scheme using Δt ≥ 15 minutes developed oscillations and eventually became unstable. Similar to the results observed for 

the two-level scheme, the unstable leapfrog scheme solutions were stabilized when the advection term was turned off; there was 

an increasing tendency, however, for the numerical solution to amplify the range of the tidal oscillation as the time step increased.

Δt, in minutes Crg Cra

1 0.31–0.38 0.04

5 1.6–1.9 0.21

10 3.1–3.8 0.43

15 4.6–5.7 0.66

30 9.2–11.3 1.33

45 13.8–17.0 2.00

60 18.4–22.6 2.66

Crg = gravity-wave Courant number =

Cra = advection Courant number =

( )Δ u gH + ⋅ t ⁄ Δx

u ⋅ Δ Δt ⁄ x

Table 5.6. Ranges of Courant numbers for tidal problem simulations using 

 = 6000 feet as a space step.

[Δx, size of the computational space interval]

Δx
x = 180,000 feet, Δ t = 10 minutes, Δ x = 6,000 feet, niter = 1, advection on

x = 180,000 feet, Δ t = 10 minutes, Δ x = 6,000 feet, niter = 1, advection off

Base solution

Base solution

-50

0

50

-100

100 A

-50

0

50

-100

0 0.5 1.0 1.5

100

B

TIME, IN DAYS

VOLUMETRI

C TRAN

SPORT, IN 

SQUARE FEET PER 

SECOND

122 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.18. Solutions using one iteration (niter = 1) of the two-level scheme with 

(A) advection terms turned on and (B) advection terms turned off. 

The instability is a nonlinear one that appears to come from the advection terms. 

Δt is the size of the computational time interval. Δx is the size of the computational 

space interval.
5. Numerical Experiments 123

To stabilize the leapfrog solutions without neglecting the advection term, a weak time filter proposed by Asselin (1972) was 

added to the computational scheme. Because this filter requires three consecutive values in time of the function to be filtered, it 

could be applied only to the three-level scheme. The filter is essentially a smoothing 

procedure that replaces the center value of the function to be smoothed using the relation

Fn Fn β

2

-- Fn – 1 2Fn – Fn + 1 = + ( ) + , (5.2)

where F is a smoothed value and β is a smoothing coefficient, usually chosen to be 0.05. Prior to smoothing Fn, the value of Fn – 1

in equation 5.2 is already smoothed from the previous time step.

Solutions using the leapfrog scheme with and without the smoothing procedure are shown in figure 5.19. The simulation without smoothing becomes unstable during the eighth tidal cycle. With smoothing, the solution is both stable and accurate. A solution 

with Δt = 30 minutes is similarly stabilized with smoothing by using β = 0.05; stable solutions could not be obtained, however, for 

the two largest time steps (t = 45 and 60 minutes), even with a larger value for β.

Because very little computer time is required to apply the time filter, using it is better than adding one or more trapezoidal 

steps to the scheme if it can be used successfully to stabilize a solution. However, the question naturally arises: Does the time filter 

introduce too much damping (smoothing) into the solution? It was found that the filter damping is minimal so long as the β coefficient is kept small (β ≤ 0.10). In addition, because the unsmoothed leapfrog solutions to the tidal problem tended to be slightly 

over-amplified, the small amount of damping introduced by the filter when β = 0.05 actually reduced the solution errors slightly.

The four error measures from table 5.5 are graphed against Δt in figure 5.20 for solutions using the three-level scheme and 

Δx = 6,000 ft (1, 829 m). The three curves in each graph again represent the three different levels of iteration in the scheme beginning with leapfrog only (no iteration). The solutions using both the leapfrog scheme and the leapfrog-trapezoidal scheme (one iteration) included smoothing (β = 0.05) for Δt ≥ 10 minutes; the solutions using the leapfrog-trapezoidal scheme (four iterations) 

included no smoothing. Without smoothing, three of the solutions using one iteration of the trapezoidal step were found to be unstable; these used Δt = 10, 15, and 45 minutes. It was apparent that one iteration of the trapezoidal step was not always sufficient to 

stabilize a leapfrog solution; in one case, a stable leapfrog solution (using Δx = 6,000 ft, Δt = 10 minutes, no smoothing) became 

unstable only after a trapezoidal step was added. In general, iteration had a stabilizing effect since all solutions were stable without 

smoothing when two or more iterations of the trapezoidal step were added to the leapfrog scheme. The error measures for solutions 

using the leapfrog-trapezoidal scheme with one and four iterations are nearly equal. The effects from the smoothing in the solutions 

using one iteration are very small; the only noticeable effect of the smoothing is that most of the amplitude errors in figure 5.20

are more negative (damped) for the solutions in which the smoothing is used.
Base solution

x= 180,000 feet, Δt = 15 minutes, Δx = 6,000 feet, no smoothing

-50

0

50

-100

100

A

x = 180,000 feet, Δt = 15 minutes, Δx = 6,000 feet, smoothing = 0.05 

Base solution

-50

0

50

-100

0 1.0 2.0 3.0 4.0 5.0 6.0

100

B

RMS error = 2.8 percent

TIME, IN DAYS

VOLUMETRI

C TRAN

SPORT, IN 

SQUARE FEET PER 

SECOND

124 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.19. Solutions using the three-level semi-implicit, leapfrog scheme (no trapezoidal 

step) (A) with no smoothing and (B) with smoothing (β = 0.05). 

Without smoothing the run becomes unstable after 4 days. With only minimal smoothing a 

very good solution is obtained. Δt is the size of the computational time interval. Δx is the 

size of the computational space interval. RMS, root mean square.
0

5

10

15

20

-5

Leapfrog

Leapfrog-trapezoidal, 4 iterations

Leapfrog-trapezoidal, 1 iteration

0

5

10

15

20

-5

Δt, IN MINUTES

0 40 10 20 30 50 60

AMPLITUDE ERROR, IN PER

CENT ROOT MEAN 

SQUARE ERROR, IN PER

CENT

5. Numerical Experiments 125

Figure 5.20. Solutions using the three-level semi-implicit scheme for the tidal test problem. 

All simulations used space step Δx = 6,000 feet. The solutions with the leapfrog scheme and 

the leapfrog-trapezoidal scheme (1 iteration) used smoothing (β = 0.05) for time step 

Δt ≥ 10 minutes. 
0

6

8

10

-1

leapfrog

leapfrog-trapezoidal, 4 iterations

leapfrog-trapezoidal, 1 iteration

4

2

0

6

8

10

-1

0 10 20 30 40 50 60

Δ t, IN MINUTES

4

2

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

126 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.20. Continued.
5. Numerical Experiments 127

5.2.2.3 Remarks

The comparison between the error measures for the two-level scheme (niter = 2) and the leapfrog-trapezoidal scheme (one 

iteration of the trapezoidal step) are given in figure 5.21. The two schemes are very similar in terms of accuracy. The main difference between the two schemes using this test problem was in the performance of the schemes without iteration. Although both 

schemes suffered from stability problems when iteration was not used, the problems were less troublesome with the leapfrog 

scheme. The availability of a time filter is an advantage of the three-level scheme.50 Two of the simulations attempted using the 

leapfrog scheme could be made stable with this filter without introducing too much damping into the solution. The leapfrog solutions that were stable without smoothing actually improved slightly (had reduced errors) with smoothing because the unsmoothed 

leapfrog scheme has a slight tendency to amplify the solution to this tidal problem. Several of the unsmoothed leapfrog-trapezoidal 

solutions with one iteration were unstable; these solutions could be stablized by the filter. Because the computational time required 

to use the filter is minimal, and the effects on the solution accuracy generally are minimal (and in some cases are beneficial), there 

is little reason not to employ it routinely in the three-level solutions.

Based on the results of both 1-D numerical experiments, it was decided to use the three-level scheme in the 3-D model.

5.3 Three-Dimensional Experiment

The 3-D numerical experiment uses a problem presented in a report by Leendertse and Liu (1977) to test the early Rand Corporation 3-D model. It simulates a uninodal standing oscillation or seiche (Wilson, 1972) in a rectangular basin. The simple geometry and boundary conditions of the problem make it a good test case for the basic computational scheme of a 3-D model.

5.3.1 Seiching in a Rectangular Basin

The test basin for the seiching experiment is 46 km long, 14 km wide, and 12 m deep (fig. 5.22). Initially the fluid in the basin 

is at rest but displaced so that the free surface profile is a half-cosine wave. At the moment the free surface is released, the fluid 

begins an oscillatory motion that continues until the motion is damped out by frictional resistance. The initial amplitude of the 

seiche is 25 cm displaced vertically upward at the west (left) end of the basin. The horizontal computational grid for the numerical 

solution is shown in figure 5.22B; the grid size is Δx = Δy = 2 km. The vertical grid consists of six layers; each layer is 2 m thick. 

50 The time filter also is easily added to a 3-D, three-level scheme.
0

3

6

8

10

-1

Two-level scheme, niter = 2

Three-level, leapfrog-trapezoidal scheme, niter = 1

9

7

5

4

2

1

0

5

10

15

-5

0 10 20 30 40 50 60

Δ t, IN MINUTES

AMPLITUDE ERROR, IN PER

CENT ROOT MEAN 

SQUARE ERROR, IN PER

CENT

128 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.21. Comparison of error measures between the two-level scheme with two iterations (niter = 2) and the three-level leapfrog-trapezoidal scheme with one iteration (niter = 1) 

of the trapezoidal step. 

All simulations were with space step Δx = 6,000 feet. The leapfrog-trapezoidal solutions 

were smoothed (β = 0.05) for time step Δt ≥ 10 minutes.
0

1

3

4

5

-1

2

Two-level scheme, niter = 2

Three-level, leapfrog-trapezoidal scheme, niter = 1

0

3

6

8

10

-1

0 10 20 30 40 50 60

Δ t, IN MINUTES

9

7

5

4

2

1

PHA

SE ERROR, IN PER

CENT MASS PRE

SERVATION ERROR, IN PER

CENT

5. Numerical Experiments 129

Figure 5.21. Continued.
j = 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

2

3

4

5

6

7

8

9

i = 

Δ

Datum

x

z

Basin boundary

B

L

y

ζ (x,0) = ζ0 cos π x

L

------

12

meters

ζ (x, t) ζ0

Initial seiche amplitude

ζ0 = 25 centimeters

Δ x

Δ y

u, U

v, V

ζ, s, ρ, h, w Δ x = Δ y = 2 kilometers

B = 14 kilometers

EXPLANATION

L = 46 kilometers

A

B

130 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.22. Diagram of (A) the test problem for an oscillating wave (seiche) in a rectangular basin and 

(B) the horizontal computational grid for the numerical solution. 

The vertical grid uses six layers, each 2 meters thick. 
5. Numerical Experiments 131

An analytical solution to the seiching problem exists if the Coriolis and advection terms in the governing equations are 

neglected and the fluid in the basin is assumed to be frictionless (see Neumann and Pierson, 1966, p. 291). The first test of the 3-D 

numerical scheme was the simulation of this ideal fluid motion. The computed and analytical solutions for the initial 0.675 hours 

of the simulation are shown in figure 5.23 at the computational nodal points indicated; the computations use the simplest formulation of the leapfrog scheme in which all terms in the finite-difference equations are treated explicitly (explicit leapfrog). For this 

computation, the time step was restricted to 45 seconds so that the gravity-wave Courant number did not exceed unity. In 

figure 5.23A, the solution for the horizontal velocity component u is plotted for the middle of the basin. Because the internal friction in the fluid is zero, no shear in the velocity profile develops and the solution for u is identical within each vertical layer. In 

figures 5.23B and 5.23C, the solution for the water surface elevation ζ and the vertical velocity component w are plotted for the 

west end of the basin. The velocity w, which varies with the vertical location in the water column, is plotted only for the third layer 

from the surface (k = 3). The curves representing the computed and analytical solutions in figure 5.23 are indistinquishable on the 

graphs, indicating negligible error in the computed results. The computed solution at all other nodes also was in virtually perfect 

agreement with the analytical solution.

The results shown in figure 5.24 are identical to those shown in figure 5.23 except that the numerical solution is computed 

using the semi-implicit, leapfrog scheme. The numerical results again agree with the analytical solution and in this case provide 

confidence that the computations for the matrix solution that is a part of the semi-implicit scheme are working correctly in the 3-D 

computer code.

Figure 5.25 shows the time history of the solutions for u, ζ, and w from figure 5.24 extended to twenty hours. Owing to the 

absence of friction, the oscillation continues indefinitely without damping at a period of T 2L gH ( )1

⁄2 = ⁄ . The agreement between 

the computed and analytical solutions is excellent over the entire 20 hours. 

To test the semi-implicit, leapfrog scheme under conditions having gravity-wave Courant numbers of one and above, three 

simulations were conducted using time steps of 3, 5, and 10 minutes. These time steps correspond to Courant numbers (Crg) equalling 1.0, 1.6, and 3.3, respectively. The 20-hour computed solutions for the horizontal velocity component u at mid-basin are plotted in figure 5.26 for the three time steps; the analytical solution also is plotted on each graph. The results indicate that a significant 

lagging phase error can occur in the numerical solution when the Courant number is increased. This tendency for lagging phase 

error in the semi-implicit scheme at Courant numbers greater than one is well known (Mesinger and Arakawa, 1976) and was identified earlier in the results of the 1-D hydrograph-routing numerical experiment. In the seiching experiment, the phase error 

becomes more prevalent as the simulation runs longer because it is cumulative.

The magnitude of the phase error can be reduced, although not eliminated, by the addition of the trapezoidal step to the semiimplicit, leapfrog scheme. Figure 5.27 presents the same graphs as figure 5.26 but with numerical computations using the semiimplicit, leapfrog-trapezoidal scheme. The phase error is reduced significantly (by more than half) with the addition of the trapezoidal step. Using more than one iteration of the trapezoidal step failed to make any significant difference in the computed solution 

results.
28

24

20

16

12

8

4

0

-4

30

25

20

15

10

5

0

-5

-10

0.005

0.000

-0.005

-0.010

-0.015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

TIME, IN HOURS

Computed

Analytical

Node i = 13, j = 5 (middle of basin) 

Node i = 2, j = 5 (west side of basin)

Node i = 2, j = 5, k = 3 (west side of basin, third layer)

A

B

C WATER SURFACE ELEVATION,

IN CENTIMETERS

u VELOCITY COMPONENT,

IN CENTIMETERS PER SECOND

w VELOCITY COMPONENT, 

IN CENTIMETERS PER SECOND

132 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.23. Computed and analytical solutions for the 3-dimensional (3-D) seiching test problem assuming 

an ideal (non-viscous) fluid and using the 3-D, explicit, leapfrog scheme for the numerical solution. 

A computational time step of 45 seconds was used for the numerical solution. 
5. Numerical Experiments 133

28

24

20

16

12

8

4

0

-4

u VELO

CITY 

COMPONENT,

IN 

CENTIMETER

S PER 

SECOND

30

25

20

15

10

5

0

-5

-10

WATER 

SURFA

CE ELEVATION,

IN 

CENTIMETER

S

0.005

0.000

-0.005

-0.010

-0.015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

w VELO

CITY 

COMPONENT,

IN 

CENTIMETER

S PER 

SECOND

TIME, IN HOURS

Computed

Analytical

Node i = 13, j = 5 (middle of basin) 

Node i = 2, j = 5 (west side of basin) 

Node i = 2, j = 5, k = 3 (west side of basin, third layer) 

Figure 5.24. Computed and analytical solutions for the 3-dimensional (3-D) seiching test problem assuming 

an ideal (non-viscous) fluid and using the 3-D, semi-implicit leapfrog scheme for the numerical solution. 

A computational time step of 45 seconds was used for the numerical solution. 
Node i = 2, j = 5, k = 3

Node i = 2, j = 5

30

20

10

0

-10

-20

-30

w VELOCITY COMPONENT,

IN CENTIMETERS PER SECOND

u VELOCITY COMPONENT,

IN CENTIMETERS PER SECOND

Computed Analytical

Node i = 13, j = 5

0 5 1 0 1 5 2 0

TIME, IN HOURS

30

20

10

0

-10

-20

-30

0.015

0.010

0.005

0

-0.005

-0.010

-0.015

WATER SURFACE ELEVATION, 

IN CENTIMETERS

134 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.25. Twenty-hour time history of computed and analytical solutions for the seiching of an ideal fluid. 

The numerical solution is computed using the 3-dimensional semi-implicit leapfrog scheme and a time step 

of 45 seconds.
Δ t = 3 minutes

30

20

10

0

-10

-20

-30

Computed Analytical

Node i = 13, j = 5

Node i = 13, j = 5 Δ t = 5 minutes

Node i = 13, j = 5 Δ t = 10 minutes

u VELO

CITY 

COMPONENT, IN 

CENTIMETER

S PER 

SECOND

0 5

30

20

10

0

-10

-20

-30

30

20

10

0

-10

 10 15 20

-20

-30

TIME, IN HOURS

5. Numerical Experiments 135

Figure 5.26. Mid-basin solutions for the 3-dimensional seiching problem using an 

ideal fluid and three different time steps (Δt) for the semi-implicit leapfrog scheme. 

The analytical solution is shown for comparison.
Δ t = 3 minutes

30

20

10

0

-10

-20

-30

u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND

Computed Analytical

Node i = 13, j = 5

Node i = 13, j = 5 Δ t = 5 minutes

Node i = 13, j = 5 Δ t = 10 minutes

0 5 10 15 20

TIME, IN HOURS

30

20

10

0

-10

-20

-30

30

20

10

0

-10

-20

-30

136 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.27. Mid-basin solutions for the 3-dimensional seiching problem using an ideal fluid 

and three different time steps (Δt) for the semi-implicit leapfrog-trapezoidal scheme. 

The analytical solution is shown for comparison.
5. Numerical Experiments 137

After testing the 3-D scheme for the seiching problem using a non-viscous (ideal) fluid, it was tested for a viscous (real) fluid. 

For the viscous fluid problem, all terms in the finite-difference equations were included except the Coriolis term.51 The fluid was 

assumed to be of homogeneous density, and the bottom drag coefficient (used in eq. 2.64) was set to Cd = 0.008. The horizontal 

eddy viscosity AH was set to zero.52

The vertical turbulence parameterization used to simulate the seiching problem by Leendertse and Liu (1977) was based on 

a one-equation turbulence model (Launder and Spaulding, 1972; Rodi, 1980a). In the one-equation model, a differential transport

equation is solved for the distribution of the turbulent kinetic energy per unit mass (K), and a simple empirical relation (such as eq. 

2.32) is used to specify the distribution of the turbulent mixing length (Λ); the turbulent eddy viscosity is computed from K and Λ

using the Kolmogorov-Prandtl expression: 

AV = Λ K . (5.3)

As an independent check on the full nonlinear calculations of the semi-implicit, 3-D scheme, this code was used to reproduce 

as nearly as possible the predictions for the viscous seiching problem presented by Leendertse and Liu (1977). For this purpose the 

mixing length turbulence parameterization of the semi-implicit model was modified to more closely represent the one-equation 

parameterization used in Leendertse and Liu (1977). As discussed in Rodi (1980a, p. 22), the mixing length parameterization is a 

special case of the one-equation parameterization if the turbulence is in a state of local equilibrium. Local equilibrium means that 

the production and dissipation of the turbulence energy are in balance at each point in the flow and that the transport influence, by 

advection or diffusion, of turbulent kinetic energy from other points in the flow is negligible. By equating the production and dissipation terms used by Leendertse and Liu (1977), the following mixing length formula is obtained:

AV03.16Λ2 ∂u

∂z

----- ⎝ ⎠ ⎛ ⎞ = . (5.4)

This equation is identical to the form of equation 2.31 in this report (assuming ∂v / ∂z = 0), except for the introduction of the 

coefficient 3.16. Thus, in the simulation of the seiching problem, the coefficient 3.16 was inserted into the turbulence coding of the 

semi-implicit scheme. The initial value of the turbulent kinetic energy per unit mass was chosen to be 0.4 cm2/s2 (Leendertse and 

Liu, 1977) and was used in equation 5.4 to estimate a background value for the eddy viscosity; the background value was then 

added to the value calculated by equation 5.4. The value for the horizontal eddy viscosity used by Leendertse and Liu (1977) was 

zero.

A sample velocity distribution from the seiching problem is shown in figure 5.28 after 0.6 hour of simulation. The gradual 

development of the vertical velocity profile at the center of the basin is shown in figure 5.29A. The viscous effects in the computed 

solution cause these velocities to be lower than those in the analytical solution for a non-viscous fluid. The computed water surface 

elevation for the viscous fluid is shown at the west end of the basin in figure 5.29B and is damped slightly from the ideal fluid 

solution. The computed results in figure 5.29 are in close agreement with those presented by Leendertse and Liu (1977, figures 7 

and 8). The 20-hour time history of the viscous solution is plotted in figure 5.30, illustrating the solution damping. The solution 

was checked for mass preservation in the surface layer of the model and the error was virtually zero.

51The Coriolis term was set to zero so that there was no y-direction variation in the computed velocity field.

52Several test simulations were done using increasing values for AH. No noticeable diffusion appeared in solutions for AH less than about 100 m2

/s, which 

is large.
20 cm/sec

Velocity scale

Δ

138 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Figure 5.28. Sample velocity distribution after 0.6 hour of simulation for the 3-dimensional seiching problem using 

a viscous fluid. 

cm/sec, centimeter per second.

Δ t = 3 minutes

Semi-implicit, leapfrog-trapezoidal

u VELO

CITY 

COMPONENT,

IN 

CENTIMETER

S PER 

SECOND

Node i = 13, j = 5 (middle of basin)

Node i = 2, j = 5 (west end of basin)

WATER 

SURFA

CE ELEVATION,

IN 

CENTIMETER

S

TIME, IN HOURS

28

24

20

16

12

8

4

0

-4

k = 1

k = 2

k = 3

k = 4

k = 5

k = 6

Computed solution

(viscous fluid)

Analytical solution

(non-viscous fluid)

Analytical solution

(non-viscous fluid)

Computed solution

(viscous fluid)

A

B

30

25

20

15

10

5

0

-5

-10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 5.29. Computed solution for the 3-dimensional seiching problem using a viscous, homogeneous fluid. 

Analytical solution for an ideal (non-viscous) fluid. Δt, time step; k, layer number
5. Numerical Experiments 139

u VELO

CITY 

COMPONENT,

IN 

CENTIMETER

S PER 

SECOND

Node i = 13, j = 5

Node i = 2, j = 5

WATER 

SURFA

CE ELEVATION,

IN 

CENTIMETER

S

TIME, IN HOURS

30

20

10

0

-10

-20

-30

30

20

10

0

-10

-20

-30

0 5 10 15 20

Δ t = 3 minutes

semi-implicit, leapfrog-trapezoidal

k = 1

k = 2

k = 3

k = 4

k = 5

k = 6

Figure 5.30. Simulated 20-hour time history for seiching of a viscous, homogeneous fluid. 
140 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Continuing with comparisons of test simulations with those of Leendertse and Liu (1977), a second simulation of the viscous 

seiching problem was computed using a slightly shorter basin length of 44 km. The simulation covered 20 hours using the same 

model parameters as those used to simulate the longer basin. The calculations used the semi-implicit, leapfrog scheme with a time 

step of 45 seconds. The vertical profiles of the u velocity component at mid-basin were saved to a file at a constant time interval 

of 3.125 hours (250 time steps). A plot of these profiles is shown in figure 5.31. The same solution profiles, digitized from the 

report by Leendertse and Liu (1977, fig. 12), are also shown. Although the two sets of profiles are in reasonably close agreement, 

the semi-implicit solution predicts larger absolute magnitudes for the velocities in each profile than was computed by Leendertse 

and Liu. It was suspected that the cause of the difference was that equation 5.4 underestimated the magnitude of the eddy viscosities 

computed by the turbulence model of Leendertse and Liu. Leendertse and Liu (1977, fig. 6D) graphically present a distribution 

of the magnitudes of the turbulent kinetic energy per unit mass calculated from their turbulence model after 60 time steps of 

simulation; using equation 5.3 to translate these values into a distribution of the eddy viscosity, a depth-averaged eddy viscosity 

at mid-basin was calculated as approximately 300 cm2/s.53 The depth-averaged eddy viscosity calculated during the semi-implicit 

simulation after 60 time steps (using eq. 5.4) was only 200 cm2/s. To decrease the differences between the two sets of velocity 

profiles in figure 5.31, another simulation with the semi-implicit, leapfrog scheme was made using a constant value of 300 cm2/s 

for the eddy viscosity. The two sets of profiles are in closer agreement (fig. 5.32); some differences still exist but are most likely 

attributable to the difference in turbulence parameterizations or to inexact digitizing of profiles from the report by Leendertse and 

Liu. Note that the computed profiles using the constant eddy viscosity exhibit less vertical variation in velocity (shear) than in the 

previous solution in which the eddy viscosity varied over the depth.

One final simulation of the seiching problem was made to test the semi-implicit scheme for conditions using variable density. 

A linear vertical density gradient of Δρ / Δz = 0.035 kg/m3/cm was introduced with freshwater at the free surface. Equation 5.4 

was used to estimate the eddy viscosity under homogeneous density conditions, and the effects of stratification were incorporated 

using the stability functions (eqs. 2.27 and 2.36) proposed by Munk and Anderson (1948). The longer basin length of 46 km was 

used. The oscillation in this test showed greater vertical shear in the velocity profile than the test using the homogeneous density 

because the vertical mixing was reduced by the stratification (fig. 5.33). The results were very similar to those of Leendertse and 

Liu (1977, fig. 9), but they were not identical because of differences in the turbulence parameterizations and stability functions 

used in the two models.

53A depth-averaged eddy viscosity of 300 cm2

/s is quite high for a basin 12 meters deep. A standard estimate for the depth-averaged eddy viscosity in 

constant density flow in which the turbulence is generated by bottom shear stress is 0.067Hu* (Fischer and others, 1979). This estimate corresponds to a 

maximum of about AV0

= 100 cm2/s. A simulation using AV0= 100 cm2/s did not agree well with the solution of Leendertse and Liu (1977).
t = 3.125 hours 

t = 6.250 hours 

t = 9.375 hours t = 18.750 hours 

t = 15.625 hours 

t = 12.500 hours 

Node i = 13, j = 5 (mid-basin)

Leendertse and Liu (1977)

Semi-implicit, leapfrog scheme

u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND

-12 -9 -6 -3 0 3 6 9 12

0

2

4

6

8

10

12

DI

STAN

CE FROM BOTTOM, IN METER

S

Surface

Bottom

5. Numerical Experiments 141

Figure 5.31. Velocity profiles for the 3-dimensional (3-D) seiching problem using a viscous, homogeneous fluid in a 44-kilometer basin and a vertical eddy viscosity defined by a mixing length formula. 

The profiles were computed using the 3-D, semi-implicit, leapfrog scheme and a vertical eddy 

viscosity defined by equation 5.4. For comparison, the velocity profiles from the report by Leendertse 

and Liu (1977, fig. 12) also are shown.

-12 -9 -6 -3 0 3 6 9 12

0

2

4

6

8

10

12 Surface

Bottom

u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND

DI

STAN

CE FROM BOTTOM, IN METER

S

Node i = 13, j = 5 (mid-basin) Semi-implicit, leapfrog scheme

Leendertse and Liu (1977)

t = 3.125 hours 

t = 6.250 hours 

t = 9.375 hours t = 18.750 hours 

t = 15.625 hours 

t = 12.500 hours 

Figure 5.32. Velocity profiles for the 3-dimensional (3-D) seiching problem using a viscous, homogeneous fluid in 

a 44-kilometer basin and a constant vertical eddy viscosity. 

The profiles were computed using the 3-D, semi-implicit, leapfrog scheme and a vertical eddy viscosity of 300 

cm2

/sec. For comparison, the velocity profiles from the report by Leendertse and Liu (1977, fig. 12) also are shown.
142 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

28

24

20

16

12

8

4

0

-4

u VELOCITY COMPONENT,

IN CENTIMETERS PER SECOND

30

25

20

15

10

5

0

-5

-10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

WATER SURFACE ELEVATION,

IN CENTIMETERS 0.005

0.000

-0.005

-0.010

-0.015

w VELOCITY COMPONENT,

IN CENTIMETERS PER SECOND

TIME, IN HOURS

Node i = 2, j = 5, k = 3 (west side of basin, third layer) 

Node i = 13, j = 5 (middle of basin)

Nonhomogeneous fluid

k = 1

k = 2

k = 3

k = 4

k = 5

k = 6

Computed solution

(viscous fluid)

Analytical solution

(non-viscous, homogeneous fluid)

Node i = 2, j = 5 (west end of basin) 

Analytical solution

(non-viscous, homogeneous fluid)

Computed solution

(viscous, nonhomogeneous fluid)

Analytical solution

(non-viscous, homogeneous fluid)

Computed solution

(viscous, nonhomogeneous fluid)

Δ t = 3 minutes

Semi-implicit, leapfrog-trapezoidal

Figure 5.33. Computed solution of the 3-dimensional (3-D) seiching problem using a viscous fluid of nonhomogeneous density. 

For comparison, the analytical solution for an ideal (non-viscous, homogeneous) fluid is shown. The computed solution is determined 

using the 3-D, semi-implicit, leapfrog-trapezoidal scheme and a time step of 3 minutes. k, layer number; Δt, time step. 
6. Summary and Conclusions 143

5.3.1.1 Remarks

Although more testing is necessary, the 3-D solutions look very good and establish confidence that the computer coding of 

the numerical scheme is working as expected. Phase error is more apparent in the seiching experiment than in the 1-D test experiments because the phase error is cumulative during the seiching calculation. Phase error is a well known feature of implicit 

schemes when the time step is relatively large and the CFL condition significantly exceeds unity. The trapezoidal step is a useful 

device to reduce the phase error, although at additional computational expense. Solutions for the 1-D tidal experiment generally 

contained very small phase errors, because an open boundary condition in the experiment prevents any accumulation of error. 

Consequently, in the modeling of real estuaries, it is likely that phase error will not be a problem. 

6. Summary and Conclusions

A new 3-D, finite-difference model has been developed and tested. The model uses a semi-implicit, leapfrog-trapezoidal 

numerical scheme that is efficient and essentially second-order accurate in the spatial and temporal numerical approximations. The 

numerical scheme is based on treating the gravity-wave and vertical diffusion terms implicitly and other terms in the governing

equations explicitly. The model does not rely on any form of vertical/horizontal mode-splitting to incorporate implicit vertical diffusion into the semi-implicit scheme. The governing equations for the multilevel scheme are arranged in conservation form by integrating them over the height of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent 

variables so that the depth-integrated continuity equation used in the solution for the water surface elevation is linear. The advantage of the semi-implicit approach is that the solution for the water surface elevation is uncoupled in the model from the solution 

for volumetric transport. A five-diagonal system of linear equations is solved at each time step for the water surface elevation using 

an efficient preconditioned conjugate-gradient method. Volumetric transports are computed explicitly from the momentum 

equations.

Prior to developing the 3-D model, two 1-D models described in sections 4.2.1 and 4.2.2 were tested to compare two-timelevel versus three-time-level approaches for the time integration. On the basis of the 1-D numerical results presented in Chapter 5, 

the three-time-level (semi-implicit, leapfrog-trapezoidal) scheme was extended to three dimensions. The advantage of leapfrogtrapezoidal integration is the ease with which nonlinear terms can be centered in time to achieve second-order accuracy during both 

of the leapfrog and trapezoidal steps. The 1-D hydrograph test experiment described in Chapter 5 shows that the centering in time 

of the coefficient for the nonlinear bottom friction term can be important for the overall accuracy of a numerical solution. Nonlinear 

terms in the two-level scheme can be centered in time only by averaging the old and new time levels, which makes iteration necessary. The three-level, leapfrog scheme can occasionally be used very efficiently without a trapezoidal step (no iteration) to

achieve second-order accuracy. The three-level-scheme solutions to the hydrograph problem that used one trapezoidal step after 

every leapfrog step generally had smaller errors than the two-level-scheme solutions using two iterations in each time step owing 

to the benefit of centering in time both steps of the leapfrog-trapezoidal scheme. The 1-D tidal test experiment described in Chapter 

5 illustrates that nonlinear advection can cause instabilities in both the two-level and three-level schemes, especially when iteration 

is not used. The instabilities were less troublesome for the three-level, leapfrog scheme, and in some instances they could be eliminated by adding a simple time filter (Asselin, 1972) that introduced only minimal damping into a solution. The range of stability 
144 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

for both the two-level and three-level schemes can be extended by using iteration, although the number of iterations should be kept 

to a minimum because iteration in three dimensions is expensive. Using more than one iteration of the trapezoidal step in the threelevel scheme or more than two iterations in the two-level scheme was not worthwhile for the 1-D test experiments, considering the 

extra computational expense incurred for minimal improvement in accuracy; in the few instances when extra iterations were 

needed to stabilize a particular solution, the values for the time step and Courant numbers were too large anyway to retain needed 

accuracy.

Although more testing of the 3-D, semi-implicit scheme is necessary, the results from the seiching experiment look promising. 

Accurate results are obtained for time steps that exceed the CFL condition for the gravity waves. Mass preservation is excellent. 

Because of the implicit nature of the numerical scheme, some phase errors do appear in solutions having large time steps. However, 

it is unlikely that this source of error will be prevalent in real tidal simulations of estuaries. If it is, it can at least be minimized by 

the use of the trapezoidal step in the scheme.

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Appendixes
160 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Appendix A - Coriolis Acceleration

This section presents a derivation of the Coriolis acceleration term appearing in the equation of motion. A general treatment 

of the Coriolis acceleration for particle motion in a rotating reference frame can be found in many standard textbooks on mechanics 

(for example, Beer and Johnston, 1962). The derivation here is specifically for tidally-driven flows in estuaries and discusses the 

form of the Coriolis term that is in common use. Vector notation is used in this section, although not elsewhere in this report, 

because it is an especially natural and compact notation for deriving the Coriolis acceleration. Vectors are identified with boldface type. In particular, heavy use is made of the vector cross product which is defined in any standard textbook on vector analysis, 

such as Wills (1958).

Consider the earth’s center as the origin of a reference frame Ex’y’z’ (fig. A.1) that is fixed in space (neglecting the motion of 

the earth’s center as it travels through space). The estuarine reference frame Oxyz is fixed to the surface of the earth and is rotating 

with angular velocity Ω about the earth’s axis. Let P be the position of a material element of fluid that is moving with the flow in 

the estuary. The position of P is defined at any instant by the vector r in the rotating frame and the vector r′ in the fixed frame. By 

simple vector addition one can write

r′ = R r + , (A.1)

where R is the position vector of the point O at the origin of the rotating frame and has a magnitude equal to the earth’s radius.

The absolute velocity v′ of the point P relative to the fixed frame is obtained by differentiating A.1, 

v′ dr′

dt

------- dR

dt

------- dr

dt = = + ----- . (A.2)

The first term on the right side of this expression is the velocity of the origin of the rotating reference frame Oxyz. From elementary 

mechanics, the velocity of rigid-body rotation about a fixed axis is defined by the vector cross product

dR

dt

------- = Ω × R . (A.3)

The second term on the right side of A.2 is the velocity of P relative to a nonrotating frame having the same orientation as Ex’y’z’, 

but with the origin at the point O. In this case, the velocity is defined by

dr

dt

----- Ω r

dr

dt

------- ⎝ ⎠ ⎛ ⎞

r = × + , (A.4)

where (dr/dt)r is the rate of change of r with respect to the rotating frame Oxyz and Ω × r is induced by the rotation. Substituting 

A.3 and A.4 into A.2 gives the absolute velocity as

v′ Ω × R + Ω × r

dr

dt

------- ⎝ ⎠ ⎛ ⎞

r = + . (A.5)

Differentiating this expression again using A.3 and the operator (d/dt) defined by A.4 gives an expression for the absolute 

acceleration
R

y ‘

x ‘

z‘

z

y (north)

Estuary

(size greatly

exaggerated)

Earth’s surface

Equator

r

Ω

P

O

x-axis is east

(into page)

r ‘

E

Earth’s

center

N

Axis of rotation



Appendix A - Coriolis Acceleration 161

Figure A.1. Estuarine reference frame Oxyz shown fixed to the earth’s surface at latitude φ and rotating about 

the earth’s axis with angular velocity Ω. 

The earth’s center is the origin of a reference frame Ex′y′z′ that is considered to be fixed in space. 
162 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

a′ dv′

dt

------- d( ) Ω × R

dt ----------------------- d( ) Ω × rdt --------------------- d

dt

---- dr

dt

------- ⎝ ⎠ ⎛ ⎞

r + +

dΩ

dt

-------× Ω R dR

dt ×------- dΩdt

-------× Ω r

dr

dt × Ω ----- drdt

------- ⎝ ⎠ ⎛ ⎞

r

d2r

dt2 -------- ⎝ ⎠ ⎛ ⎞

r

+ +++ × +

dΩ

dt

------- × ΩΩ R ( ) × R dΩ

dt × + ------- × ΩΩ r ( ) × r 2Ω dr

dt

------- ⎝ ⎠ ⎛ ⎞

r

× + × d2r

dt2 -------- ⎝ ⎠ ⎛ ⎞

r

++ +

= =

=

= .

(A.6)

Now setting dΩ / dt = 0 for steady rotation of the earth, and defining (dr/dt)r = v and (d2r/dt2)r = a as the velocity and acceleration, 

respectively, relative to the rotating frame, one obtains the following expression relating the absolute acceleration in the fixed frame 

to the relative acceleration in the rotating frame:

a′ = a + Ω Ω ×( )ΩΩ × R + ×( ) × r + 2Ω × v . (A.7)

For motion in even the largest estuary, r is much less than R and the term Ω × (Ω ×r) can safely be neglected. The term 

Ω × (Ω ×R) is the centripetal acceleration directed towards the earth’s axis at the point O′ (fig. A.2). The true gravitational 

acceleration at the earth’s surface is g = g0 – Ω × (Ω ×R) where g0 is the nonrotating gravitational acceleration. The centripetal 

acceleration is therefore already incorporated into the equation of motion in the gravitational body force, as normally defined, so 

long as the z-axis is assumed to lie along the line of action of g.

Equation A.7 has now been reduced to

a′ = a + 2Ω × v (A.8)

In terms of the Eulerian velocity field, u(x,t), specified in coordinates relative to the rotating frame, this expression becomes

a′ ∂u

∂t = ------ + ( ) u .∇ u + 2Ω × u (A.9)

Ω × (Ω × R) 

N

S

E Equator

O‘

g0

R

φ

g0 g

– Ω × (Ω × R)

Earth

Ω

g = g0 – Ω × (Ω × R)

Figure A.2 The earth’s rotation causes a centripetal acceleration Ω × (Ω ×R) that affects the local acceleration of gravity g at the earth’s surface. 

The nonrotating gravitational acceleration is g0.
Appendix A - Coriolis Acceleration 163

where the advective acceleration (u • ∇)u is introduced by the Eulerian specification of the flow. Equation A.9 states that the 

equation of motion in the rotating frame is equivalent to that in an absolute frame so long as it is assumed that a pseudo body force 

per unit mass represented by (–2Ω × u)54 acts upon the fluid. This pseudo body force is known as the Coriolis acceleration 

(force/unit mass) and is often significant in estuarine modeling.

In Cartesian coordinates using u = ui + vj + wk and Ω = Ωxi + Ωyj + Ωzk, the acceleration terms are

∂u

∂t

----- u

∂u

∂x

----- v

∂u

∂y

----- w

∂u

∂z

----- 2Ωz – v 2Ωy +++ + w , (A.10)

∂v

∂t

----- u

∂v

∂x

----- v

∂v

∂y

----- w

∂v

∂z

----- 2Ωzu 2Ωx +++ + – w , and (A.11)

∂w

∂t

------ u

∂w

∂x

------ v

∂w

∂y

------ w

∂w

∂z

------ 2Ωxv 2Ωy +++ + – u . (A.12)

For a reference frame chosen using the z-axis vertically-upward, but using the horizontal axes rotated by an angle θ, the components 

of Ω are dependent on both the latitude φ and the rotation θ by 

Ωx = Ω φθ cos sin ,

Ωy = Ω φθ cos cos , and (A.13)

Ωz = Ω φ sin .

When the x-axis is toward the east and the y-axis is toward the north (θ = 0°), these reduce to

Ωx = 0 ,

Ωy = Ω φ cos , and (A.14)

Ωz = Ω φ sin .

In a shallow estuary, the vertical velocity component w is typically very small, and therefore the two Coriolis terms involving w in 

the horizontal momentum equations (A.10 and A.11) are usually neglected. Frequently the term (Ωxv – Ωyu) in the z-momentum 

equation is also neglected because it is considered to be small in comparison with g. Pond and Pickard (1986, p. 40) point out, 

however, that, although (Ωxv – Ωyu) may be small compared with g, it may not always be small compared with the difference 

between the pressure term in the z-momentum equation and g. Evces and Raney (1990) formulated a 2-D vertically-averaged 

hydrodynamic model that included the z-component Coriolis term and estimated it may correct the total Coriolis term for a high 

tidal range estuary near latitude φ = 45° by10 percent. The correction will be most pronounced at high latitudes (φ > 45°) and in 

some instances may warrant inclusion in a model applied at lower latitudes. For the model discussed herein, however, the term is 

neglected, as is customary. In this case, the form of the acceleration terms in the rotating reference frame are most commonly 

expressed as

∂u

∂t

----- u

∂u

∂x

----- v

∂u

∂y

----- w

∂u

∂z +++ ----- – fv (A.15)

54Using the subscript notation from Chapter 2 of this report, the Coriolis acceleration 2Ω × u is written as 2εijkΩj

uk.
∂v

∂t

----- u

∂v

∂x

----- v

∂v

∂y

----- w

∂v

∂z +++ + ----- fu

∂w

∂t ------ u

∂w

∂x

------ v

∂w

∂y

------ w

∂w

∂z +++ ------ ,

164 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

(A.16)

(A.17)

where f = 2Ωz = 2|Ω| sin φ is called the Coriolis parameter and is equal to 1.03 × 10–4 sec-1 at φ = 45°. Because the area of study 

is usually less than a few degrees of latitude for estuarine problems, the Coriolis parameter is held fixed at its value at the point of 

contact with a tangent plane to the earth. The tangent plane is called the f-plane, and this approximation is referred to as the f-plane 

approximation.
Appendix B - Equation of State 165

Appendix B - Equation of State

To evaluate the pressure gradient terms in the equations of motion and to assess the water column stability under stratified 

conditions, the density field must be determined. An equation of state is needed to define the dependence of density on salinity (s) 

and temperature (Θ). Because water is slightly compressible, the density also depends to a small degree on pressure; this dependence is eliminated by considering the density as a potential density evaluated at atmospheric pressure.

Many of the pre-1980 formulas for the equation of state are based on the well-known hydrographic tables prepared by Knudsen (1901). In 1980 a new International Equation of State (IES 80)(Millero and Poisson, 1981; Unesco, 1981) was introduced using 

the Practical Salinity Scale, 1978 (Unesco, 1979), which is based on electrical conductivity measurements. The use of IES 80 is

now recommended over the Knudsen tables.

Two options are available in the 3-D model for the equation of state. For both, density is computed in kg/m3, salinity is 

expressed according to the Practical Salinity Scale, and temperature is expressed in degrees Celsius. The first option is IES 80 as 

given by (rearranged from the formula presented by Pond and Pickard [1986], p. 310)

ρ( ) s, Θ = a s b s c sd + ( ) × ( ) + × ( ) + × , (B.1)

where

a a0 Θ a1 Θ a2 + Θ a3 Θ a4 Θ a5 = + ( ) × ( ) + × ( ) × ( ) + × ( ) + × ,

b b0 Θ b1 Θ b2 + Θ b3 Θ b4 = + ( ) × ( ) + × ( ) × ( ) + × ,

c c0 Θ c1 Θ c2 = + ( ) × ( ) + × ,

d d0 = ,

and the coefficients are

a0 = 999.842594, a1 = 6.793952 10–2 × , a2 = –9.095290 10–3 × ,

a3 = 1.001685 a4 = –1.120083 10–6 × , a5 = 6.536332 10–9 10 × , –4 × ,

b0 = 8.24493 b1 = –4.0899 10–3 × , b2 = 7.6438 10–5 10 × , –1 × ,

b3 = –8.2467 b4 = 5.3875 10–9 × , c0 = –5.72466 10–3 10 × , –7 × ,

c1 = 1.0227 c2 = –1.6546 10–6 × , d0 = 4.8314 10–4 10 × . –4 × ,
166 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

The second option is for the case when density can be taken as a function of salinity only (temperature is uniform). In this 

case, a simple linear relation is used, as proposed by Hansen and Rattray (1965):

ρ() ρ s r = ( ) 1 + ks , (B.2)

where ρr and k are coefficients. For the 3-D test case in this report, the two coefficients ρr and k were determined by fitting equation 

2 to IES 80 for a specified temperature and range of salinity. For example, for Θ = 17 οC and salinity in the range 10−20, the 

optimized coefficients are

ρr = 998.7959 and k = 0.000763 .

The coefficient ρr is close to the density of fresh water in kg/m3. Because density is computed often in the model computations, it 

is advantageous to use equation 2 whenever possible to speed up model performance.
Appendix C - The σ Transformation 167

Appendix C - The σ Transformation

In this section, the transformation of the 3-D governing equations from a z-coordinate system (x, y, z, t) to a vertically normalized, σ-coordinate system xˆ yˆ σ t

ˆ ( ) ,, , is outlined. By choosing the free surface as σ = 0 and the bottom as σ = −1, the transformation between coordinate systems is defined as

xˆ = x , yˆ = y , σ z – ζ( ) xyt , ,

hxy ( )ζ , + ( ) xyt , , ------------------------------------------- z – ζH = = ---------- , t

ˆ = t , (C.1)

where H = h + ζ (fig. C.1). An example of a σ grid in z-coordinate space and σ-coordinate space is shown schematically in 

figure C.2. Because the σ axis is vertical while the constant σ surfaces are sloping, the coordinate transformation is nonorthogonal.

The dependent variables u, v, ρ, and s transform straightforwardly between coordinate systems as

G

ˆ xˆ yˆ σ t

ˆ ( ) ,, , = = G x( , , y z Hσ ζ + , t)

where G represents any of the four variables. Under the transformation, the new velocities uˆ and vˆ are still the horizontal 

(Cartesian) components of the velocity vector and are not measured along the sloping σ surfaces. The dependent variables H and 

ζ are not functions of the transformed vertical coordinate, so they can be used interchangeably with H

ˆ and ζ

ˆ . For the transformed 

vertical velocity, the following definition is adopted:

ζ (x, y, t)

x

z

y

Datum

Water surface (z = ζ)

h (x, y) H(x, y, t) = h + ζ

Bottom (z = – h)

/

55

55Several authors, such as Blumberg and Mellor (1987) and Stelling and van Kester (1994), choose to define the transformed vertical velocity 

as ω = H(Dσ / Dt) so that it has units of (length/time). The definition of ω in equation C.2 has units of (1/time).

Figure C.1. Definition sketch for the estuarine free surface, bottom, and total 

depth (H) in a z-coordinate system.
ω

Dσ

Dt = -------

168 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

. (C.2)

The original vertical velocity w is related to ω by

w

Dz

Dt

------ D

Dt

-----( ) Hσ ζ + Hω σ

DH

Dt

-------- Dζ

Dt == = + + ------- . (C.3)

Expanding the operators D / Dt in the above expression leads to

w Hω u σ

∂H

∂x

------ ∂ζ

∂x

+ ------- ⎝ ⎠ ⎛ ⎞ v σ

∂H

∂y

------ ∂ζ

∂y

+ ------- ⎝ ⎠ ⎛ ⎞ σ

∂H

∂t

------ ∂ζ

∂t

+ ------- ⎝ ⎠ ⎛ ⎞ = +++ . (C.4)

An expression for ∂w / ∂σ can be shown to be

∂w

∂σ

------ H∂ω

∂σ

------ ∂ζ

∂t

----- u

∂H

∂x

------ v

∂H

∂y

+ + -------- ⎝ ⎠ ⎛ ⎞ σ

∂H

∂x

------ ∂u

∂σ

------ ∂H

∂y

------ ∂v

∂σ

+ ------ ⎝ ⎠ ⎛ ⎞ ∂ζ

∂x

----- ∂u

∂σ

------ ∂ζ

∂y

-----∂v

∂σ

+ ------ ⎝ ⎠ ⎛ ⎞ = + ++ . (C.5)

By applying the familiar chain rule of calculus, the first-order derivatives in z coordinates are expressed in terms of σ coordinates by the following operators:

∂

∂x

----- ∂

∂xˆ ----- σ

H

--- ∂H

∂xˆ ------ ∂∂σ

------ 1

H

--- ∂ζ

∂xˆ ----- ∂∂σ = – – ------ , (C.6)

∂

∂y

----- ∂

∂yˆ ----- σ

H

--- ∂H

∂yˆ ------ ∂∂σ

------ 1

H

--- ∂ζ

∂yˆ ----- ∂∂σ = – – ------ , (C.7)

∂

∂z

---- 1

H

--- ∂

∂σ = ------ , and (C.8)

∂

∂t

---- ∂

t

ˆ ∂

---- σ

H

--- ∂H

t

ˆ ∂

------ ∂

∂σ – ------ 1H

--- ∂ζ

t

ˆ ∂

----- ∂

∂σ = – ------ . (C.9)

σ = – 1

x

z

y

z = 0

z = – h ( x, y )

σ = 0

σ

A B

xˆ

yˆ

Figure C.2. Schematic of a σ-grid shown in (A) z-coordinate space and (B) σ -coordinate space.
Appendix C - The σ Transformation 169

Differential operators for the second-order, spatial derivatives are also needed in transforming the diffusion terms. In complete form 

these are (Sheng, 1983, p. 224)

∂

∂x

----- ∂

∂x

----- ⎝ ⎠ ⎛ ⎞ 1

H

--- ∂

∂xˆ ----- H ∂∂xˆ ----- ⎝ ⎠ ⎛ ⎞ ∂∂σ

------ σ

∂H

∂xˆ ------ ∂∂xˆ ----- ⎝ ⎠ ⎛ ⎞ – σ

∂

∂xˆ ----- ∂H∂xˆ ------ ∂∂σ

------ ⎝ ⎠ ⎛ ⎞ –

1

H

--- ∂H

∂xˆ -------- ⎝ ⎠ ⎛ ⎞ 2 ∂

∂σ

------ σ2 ∂

∂σ

------ ⎝ ⎠ ⎛ ⎞ ∂

∂xˆ ----- ∂ζ∂xˆ ----- ∂∂σ

------ ⎝ ⎠ ⎛ ⎞ – ∂ζ

∂xˆ ----- ∂2

∂σ∂xˆ – ------------

∂ζ

∂xˆ ------- ⎝ ⎠ ⎛ ⎞ 2 1

H

--- ∂2

σ2 ∂

-------- 2

H

--- ∂ζ

∂xˆ -----∂H∂xˆ ------ ∂∂σ

------ σ

∂

∂σ

------ ⎝ ⎠ ⎛ ⎞

+

+ + ,

=

∂

∂y

----- ∂

∂y

----- ⎝ ⎠ ⎛ ⎞ 1

H

--- ∂

∂yˆ ----- H ∂∂yˆ ----- ⎝ ⎠ ⎛ ⎞ ∂∂σ

------ σ

∂H

∂yˆ ------ ∂∂yˆ ----- ⎝ ⎠ ⎛ ⎞ – σ

∂

∂yˆ ----- ∂H∂yˆ ------ ∂∂σ

------ ⎝ ⎠ ⎛ ⎞ –

1

H

--- ∂H

∂yˆ -------- ⎝ ⎠ ⎛ ⎞ 2 ∂

∂σ

------ σ2 ∂

∂σ

------ ⎝ ⎠ ⎛ ⎞ ∂

∂yˆ ----- ∂ζ∂yˆ ----- ∂∂σ

------ ⎝ ⎠ ⎛ ⎞ – ∂ζ

∂yˆ ----- ∂2

∂σ∂yˆ – ------------

∂ζ

∂yˆ ------- ⎝ ⎠ ⎛ ⎞ 2 1

H

--- ∂2

σ2 ∂

-------- 2

H

--- ∂ζ

∂yˆ -----∂H∂yˆ ------ ∂∂σ

------ σ

∂

∂σ

------ ⎝ ⎠ ⎛ ⎞

+

+ + , and 

=

∂

∂z

---- ∂

∂z

---- ⎝ ⎠ ⎛ ⎞ 1

H

--- ∂

∂σ

------ 1

H

--- ∂

∂σ

------ ⎝ ⎠ ⎛ ⎞ = .

(C.10)

(C.11)

(C.12)

In deriving operators C.6 to C.12, use was made of the identities

∂σ

∂x

------ σ

H

--- ∂H

∂x

------ 1

H

--- ∂ζ

∂x

+ ------- ⎝ ⎠ ⎛ ⎞ = – ,

∂2σ

x2 ∂

--------- σ

H

--- ∂2H

x2 ∂

--------- 2σ

H2 ------ ∂H

∂x -------- ⎝ ⎠ ⎛ ⎞ 2 2

H2 ------∂ζ

∂x

-----∂H

∂x

------ 1

H

--- ∂2ζ

x2 ∂ = – + + – -------- , and

∂σ

∂x

------- ⎝ ⎠ ⎛ ⎞ 2 σ2

H2 ------ ∂H

∂x -------- ⎝ ⎠ ⎛ ⎞ 2 2σ

H2 ------∂ζ

∂x

-----∂H

∂x

------ 1

H2 ------ ∂ζ

∂x

------- ⎝ ⎠ ⎛ ⎞ 2

= + + ,

and similar identities for the gradient of σ with respect to y.

Using these operators, the 3-D equations in z coordinates presented in Chapter 2 (equations 2.38, 2.49, 2.50, and 2.42) are 

transformed into the σ-coordinate system:

Continuity equation,

∂ζ

∂t

----- ∂Huˆ

∂xˆ ---------- ∂Hvˆ∂yˆ --------- H∂ω

∂σ

+++ ------ = 0 ; (C.13)
170 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Momentum equations,

∂

∂xˆ ----- HAH

∂uˆ

∂xˆ ----- ⎝ ⎠ ⎛ ⎞ ∂∂yˆ ----- HAH

∂uˆ

∂yˆ ----- ⎝ ⎠ ⎛ ⎞ H.O.T. ∂∂σ

------ AV

H

------- ∂uˆ

∂σ

------ ⎝ ⎠ ⎛ ⎞ + + ++ and

gH

ρ0

------- H ∂

∂xˆ ----- ρ

ˆ dσ′

σ

0

∫ ∂H

∂xˆ ------ σ ρ

ˆ ∂

∂σ

------dσ′

σ

0

∫ – +

∂Huˆ

t

ˆ ∂

---------- ∂Huˆ uˆ

∂xˆ ------------- ∂Huˆ vˆ∂yˆ ------------- H∂uˆω

∂σ

+++ ---------- – fHvˆ gH∂ζ

∂xˆ = – -----

∂

∂xˆ ----- HAH

∂vˆ

∂xˆ ----- ⎝ ⎠ ⎛ ⎞ ∂∂yˆ ----- HAH

∂vˆ

∂yˆ ----- ⎝ ⎠ ⎛ ⎞ H.O.T. ∂∂σ

------ AV

H

------ ∂vˆ

∂σ

------ ⎝ ⎠ ⎛ ⎞ + + ++ ;

gH

ρ0

------- H ∂

∂yˆ ----- ρ

ˆ dσ′

σ

0

∫ ∂H

∂yˆ ------ σ ρ

ˆ ∂

∂σ

------dσ′

σ

0

∫ – +

∂Hvˆ

t

ˆ ∂

--------- ∂Huˆ vˆ

∂xˆ ------------- ∂Hvˆ vˆ∂yˆ ------------ H∂vˆω

∂σ

+++ + --------- fHuˆ gH∂ζ

∂yˆ = – -----

(C.14)

(C.15)

Salt transport equation,

Hs ∂ ˆ

t

ˆ ∂

--------- Huˆs ∂ ˆ

∂xˆ ------------ Hvˆs ∂ ˆ∂yˆ ------------ H ωs ∂ ˆ

∂σ

+++ = ---------

∂

∂xˆ ----- HDH

s∂ˆ

∂xˆ ----- ⎝ ⎠ ⎛ ⎞ ∂∂yˆ ----- HDH

s∂ˆ

∂yˆ ----- ⎝ ⎠ + + ⎛ ⎞ H.O.T. ∂∂σ

------ DV

H

------- s∂ˆ

∂σ

------ ⎝ ⎠ ⎛ ⎞ + . (C.16)

The higher order terms (H.O.T.) in equations C.14, C.15, and C.16 are those due to all but the first term on the right sides of 

operators C.10 and C.11. The H.O.T. contain horizontal gradients of the water depth and water surface slope and often are neglected 

in practice (Mellor and Blumberg, 1985). Neglecting these terms is convenient because their inclusion can add considerably to the 

cost of running a model and can lead to a less stable and accurate numerical formulation. In regions of steep bottom slopes, 

however, the omission of the H.O.T. in solving the salt transport equation can cause spurious vertical diffusion of salt, especially 

in cases where significant salt stratification is present. In these cases the complete transformation must be included or a special 

numerical solution strategy, such as the finite-volume method applied to a σ grid by Stelling and van Kester (1994), should be used.
Appendix D - Expansion of Explicit Finite-Difference Terms 171

Appendix D - Expansion of Explicit Finite-Difference Terms

The explicit finite-difference terms in the 3-D momentum and salt transport equations are represented in abbreviated form in 

equations 4.23, 4.24, and 4.43. The expanded form of these terms using the leapfrog scheme is presented here. For the x-momentum 

equation (eq. 4.23), the finite-difference expressions for the advection, Coriolis, baroclinic density gradient, and horizontal

diffusion terms are

ADVx ( )i 1 + ⁄2, ,j k

n ( ) Uu k ∂

∂x ----------------- ( ) Vu k ∂∂y ---------------- ( ) uw k 1 + ⁄2

k 1 – ⁄2 + + ⎝ ⎠ ⎛ ⎞

i 1 + ⁄2, j

n

= =

1

4Δx --------- Ui 3 + ⁄2 , ,j k

n Ui 1 + ⁄2 , ,j kn ( ) + ui 3 + ⁄2 , ,j kn ui 1 + ⁄2 , ,j kn + ( ⋅ ( ) +

 Ui 1 + ⁄2 , ,j k

n Ui 1 – ⁄2 , ,j kn ( ) + ui 1 + ⁄2 , ,j kn ui 1 – ⁄2 , ,j kn – ⋅ ( )) +

1

4Δy --------- Vi 1 j 1 + , + ⁄2 , k

n Vi j 1 , + ⁄2 , kn ( ) + ui 1 + ⁄2 , , j + 1 kn ui 1 + ⁄2 , ,j kn + ( ⋅ ( ) +

 Vi + 1 j 1 , – ⁄2 , k

n Vi j 1 , – ⁄2 , kn ( ) + ui 1 + ⁄2 , ,j kn ui 1 + ⁄2 , , j – 1 kn – ⋅ ( )) +

1

4

-- wi 1 j k 1 – ⁄2 + , ,

n wijk 1 – ⁄2 , ,n (( ) + ui 1 + ⁄

2 , ,j k – 1

n ui 1 + ⁄

2 , ,j k

n + ⋅ ( ) +

 wi 1 j k 1 + ⁄2 + , ,

n wijk 1 + ⁄2 , ,n ( ) + ui 1 + ⁄

2 , ,j k

n ui 1 + ⁄

2 , ,j k + 1

n – ⋅ ( )) + , (D.1)

CORx ( )i 1 + ⁄2, ,j k

n fVk ( )i 1 + ⁄2, ,j kn = =

f

4

-- Vi 1 j 1 + , + ⁄2 , k

n Vi j 1 , + ⁄2 , kn Vi 1 j 1 + , – ⁄2 , kn Vi j 1 , , – ⁄2 kn + ( ) ++ +

, (D.2)

BCLINICx ( )i 1 + ⁄2, ,j k

n hk

ρk

----- gh1

2

-------- ρ1 ∂

∂x -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂x ---------------- ghm

2

--------- ρm ∂

∂x

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

+ ∑ ⎝ ⎠ ⎜ ⎟

⎛ ⎞

⎝ ⎠ ⎜ ⎟

⎛ ⎞ n

i 1 + ⁄2 , j

= =

g

2

-- hi + 1, ,j k

n hijk , ,n ( )ρ + i + 1, ,j kn ρijk , ,n ( ) +

–1

+ ( ) ⋅

1

2Δx --------- hi + 1, ,j 1

n hi j , , 1n ( ρ ( ) + i + 1, ,j 1n ρi j , , 1n × ⋅ ( ) –

hi + 1, ,j m – 1

n hijm , , – 1n ( ρ ( ) + i + 1, ,j m – 1n ρijm , , – 1n ⋅ ( ) –

m = 2

k

+ ∑

hi + 1, ,j m

n hijm , ,n ( )ρ + i + 1, ,j mn ρijm , ,n + ⋅ ( ))) – , and (D.3)
HDIFFx ( )i 1 + ⁄2, ,j k

n – 1 ∂

∂x

----- AHh∂u

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- AHh∂u

∂y

----- ⎝ ⎠ ⎛ ⎞

k

+ ⎝ ⎠ ⎛ ⎞

i 1 + ⁄2, j

n – 1

= =

1

( ) Δx 2 ------------- hi + 1, ,j k

n – 1 AHn – 1

i + 1, ,j k ui 3 + ⁄2 , ,j k

n – 1

ui 1 + ⁄2, ,j k

n – 1

+ ( ( ) –

hijk , ,

n – 1 AHn – 1

ijk , , ui 1 + ⁄2, ,j k

n – 1

ui 1 – ⁄2, ,j k

n – 1 – ( )) –

1

8( ) Δy 2 ----------------- hi + 1, , j + 1 k

n – 1 hi j , , + 1 kn – 1 ( ) + AHn – 1

i + 1, , j + 1 k AH

n – 1

i j , , + 1 k + (( ⋅ ( ) +

hi + 1, ,j k

n – 1 hijk , ,n – 1 ( ) + AHn – 1

i + 1, ,j k AH

n – 1

ijk , , + ⋅ ( )) +

ui 1 + ⁄2 , , j + 1 k

n – 1

ui 1 + ⁄2 , ,j k

n – 1 × ( ) –

hi + 1, ,j k

n – 1 hijk , ,n – 1 ( ) + AHn – 1

i + 1, ,j k AH

n – 1

ijk , , – ( ⋅ ( ) +

hi + 1, , j – 1 k

n – 1 hi j , , – 1 kn – 1 ( ) + AHn – 1

i + 1, , j – 1 k AH

n – 1

i j , , – 1 k + ⋅ ( )) +

ui 1 + ⁄2 , ,j k

n – 1

ui 1 + ⁄2 , , j – 1 k

n – 1 × ) ( ) – .

172 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

(D.4)

For the y-momentum equation (eq. 4.24), the finite-difference expressions are

ADVy ( )i j 1 , , + ⁄2 k

n ( ) Uv k ∂

∂x ----------------- ( ) Vv k ∂∂y ---------------- ( ) vw k 1 + ⁄2

k 1 – ⁄2 + + ⎝ ⎠ ⎛ ⎞

i j 1 + ⁄2 ,

n

= =

1

4Δx --------- Ui 1 + ⁄2, j + 1, k

n Ui 1 + ⁄2 , ,j kn ( ) + vi + 1 j 1 , , + ⁄2 kn vi j 1 , , + ⁄2 kn + ( ⋅ ( ) +

 Ui 1 – ⁄2 , , j + 1 k

n Ui 1 – ⁄2 , ,j kn ( ) + vi j 1 , , + ⁄2 kn vi – 1 j 1 , , + ⁄2 kn – ⋅ ( )) +

1

4Δy --------- Vi j 3 , + ⁄2 , k

n Vi j 1 , + ⁄2 , k

n ( ) + vi j 3 , , + ⁄2 k

n vi j 1 , , + ⁄2 k

n + ( ⋅ ( ) +

 Vi j 1 , , + ⁄2 k

n Vi j 1 , – ⁄2 , kn ( ) + vi j 1 , , + ⁄2 kn vi j 1 , , – ⁄2 kn – ⋅ ( )) +

1

4

-- wi j + 1 k 1 – ⁄2 , ,

n wijk 1 – ⁄2 , ,n (( ) + vi j 1 , , + ⁄2 k – 1n vi j 1 , , + ⁄2 kn + ⋅ ( ) +

 wi j + 1 k 1 + ⁄2 , ,

n wijk 1 + ⁄2 , ,n ( ) + vi j 1 , , + ⁄2 kn vi j 1 , , + ⁄2 k + 1n – ⋅ ( )) + , (D.5)

CORy ( )i j 1 , , + ⁄2 k

n fUk ( )i j 1 , , + ⁄2 kn = =

f

4

-- Ui 1

⁄2 + , j + 1, k

n Ui 1 – ⁄2 , j + 1, kn Ui 1 – ⁄2 , ,j kn Ui 1 + ⁄2 , ,j kn + ( ) + ++ , (D.6)
BCLINICy ( )i j 1 , + ⁄2 , k

n hk

ρk

----- gh1

2

-------- ρ1 ∂

∂y -------- ghm – 1

2 ---------------- ρm – 1 ∂

∂y ---------------- ghm

2

--------- ρm ∂

∂y

+ --------- ⎝ ⎠ ⎛ ⎞

m = 2

k

+ ∑ ⎝ ⎠ ⎜ ⎟

⎛ ⎞

⎝ ⎠ ⎜ ⎟

⎛ ⎞ n

i j 1 + ⁄2 ,

= =

g

2

-- hi j , , + 1 k

n hijk , ,n ( )ρ + i j , , + 1 kn ρijk , ,n ( ) +

–1

+ ( ) ⋅

1

2Δy --------- hi j , , + 1 1

n hi j , , 1n ( ρ ( ) + i j , , + 1 1n ρi j , , 1n × ⋅ ( ) –

hi j , , + 1 m – 1

n hijm , , – 1n ( ρ ( ) + i j , , + 1 m – 1n ρijm , , – 1n ⋅ ( ) –

m = 2

k

+ ∑

hi j , , + 1 m

n hijm , ,n ( )ρ + i j , , + 1 mn ρijm , ,n + ⋅ ( ))) –

Appendix D - Expansion of Explicit Finite-Difference Terms 173

, (D.7)

HDIFFy ( )i j 1 , , + ⁄2 k

n – 1 ∂

∂x

----- AHh∂v

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- AHh∂v

∂y

----- ⎝ ⎠ ⎛ ⎞

k

+ ⎝ ⎠ ⎛ ⎞

i j 1 + ⁄2 ,

n – 1

= =

1

8( ) Δx 2 ----------------- hi + 1, , j + 1 k

n – 1 hi + 1, ,j kn – 1 ( ) + AHn – 1

i + 1, , j + 1 k AH

n – 1

i + 1, ,j k + (( ⋅ ( ) +

hi j , , + 1 k

n – 1 hijk , ,n – 1 ( ) + AHn – 1

i j , , + 1 k AH

n – 1

ijk , , + ⋅ ( )) +

vi + 1 j 1 , , + ⁄2 k

n – 1

vi j 1 , , + ⁄2 k

n – 1 × ( ) –

hi j , , + 1 k

n – 1 hijk , ,n – 1 (( ) + AHn – 1

i j , , + 1 k AH

n – 1

ijk , , – ⋅ ( ) +

hi – 1, , j + 1 k

n – 1 hi – 1, ,j kn – 1 ( ) + AHn – 1

i – 1, , j + 1 k AH

n – 1

i – 1, ,j k + ⋅ ( )) +

vi j 1 , , + ⁄2 k

n – 1

vi – 1 j 1 , , + ⁄2 k

n – 1 × ( )) –

1

( ) Δy 2 ------------- hi j , , + 1 k

n – 1 AHn – 1

i j , , + 1 k vi j 3 , , + ⁄2 k

n – 1

vi j 1 , , + ⁄2 k

n – 1

+ ( ( ) –

hijk , ,

n – 1 AHn – 1

ijk , , vi j 1 , , + ⁄2 k

n – 1

vi j 1 , , – ⁄2 k

n – 1 – ( )) – . (D.8)

For the salt transport equation (eq. 4.43), the finite-difference expressions for the advection and horizontal diffusion terms are

ADVs ( )ijk , ,

n ( ) uhs k ∂

∂x ------------------ ( ) vhs k ∂∂y ------------------ ( ) ws k 1 + ⁄2

k 1 – ⁄2 + + ⎝ ⎠ ⎛ ⎞

i j ,

n

= =

1

2Δx --------- ui 1 + ⁄2 , j k,

n ( ) hs i + 1, ,j k

n ( ) hs i, ,j kn

( ) + ui 1

⁄2 – , ,j k

n ( ) hs i, ,j k

n ( ) hs i – 1, ,j kn

+ ( ) – ( ) +

1

2Δy --------- vi j 1 , + ⁄2 , k

n ( ) hs i, j + 1, k

n ( ) hs i, ,j kn

( ) + vi j 1 , – ⁄2 , k

n ( ) hs i, ,j k

n ( ) hs i, , j – 1 kn

+ ( ) – ( ) +

1

2

-- wi j k 1 – ⁄2 , ,

n sijk , , – 1

n sijk , ,n ( ) + wi j k 1 + ⁄2 , ,

n sijk , ,

n sijk , , +1n + ( ) – ( ) + , and (D.9)
HDIFFs ( )ijk , ,

n ∂

∂x

----- DHh∂s

∂x

----- ⎝ ⎠ ⎛ ⎞

k

∂

∂y

----- DHh∂s

∂y

----- ⎝ ⎠ ⎛ ⎞

k

+ ⎝ ⎠ ⎛ ⎞

i j ,

n – 1

= =

1

4( ) Δx 2 ----------------- hi + 1, ,j k

n – 1 hijk , ,n – 1 ( ) + DHn – 1

i + 1, ,j k DH

n – 1

ijk , , + ( ⋅ ( ) +

si + 1, ,j k

n – 1

sijk , ,

n – 1 × ( ) –

hijk , ,

n – 1 hi – 1, ,j kn – 1 – ( ) + DHn – 1

ijk , , DH

n – 1

i – 1, ,j k ⋅ ( ) +

sijk , ,

n – 1

si – 1, ,j k

n – 1

( ) – ) ×

1

4( ) Δy 2 ----------------- hi j , , + 1 k

n – 1 hijk , ,n – 1 ( ) + DHn – 1

i j , , + 1 k DH

n – 1

ijk , , + ( ⋅ ( ) +

si j , , + 1 k

n – 1

sijk , ,

n – 1 × ( ) –

– hijk , ,

n – 1 hi j , , – 1 kn – 1 ( ) + DHn – 1

ijk , , DH

n – 1

i j , , – 1 k ⋅ ( ) +

sijk , ,

n – 1

si j , , – 1 k

n – 1

( ) – ) × .

174 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

(D.10)
Appendix E - Computation of the Vertical Eddy Coefficients 175

Appendix E - Computation of the Vertical Eddy Coefficients

This appendix defines the discretized equations used in the model to compute the vertical distribution of the eddy viscosity 

(AV) for diffusion of momentum and the eddy diffusivity (DV) for diffusion of salt mass.

The eddy viscosity under neutral (unstratified) conditions is computed using the Prandtl mixing length formula (eq. 2.31) in 

the discretized form

AV0 ( )ijk 1 – ⁄2 , ,

n Λ2 ( )ijk 1 – ⁄2 , ,

n ui 1 + ⁄2 , ,j k – 1 n ui 1 + ⁄2 , ,j k n – ui 1 – ⁄2 , ,j k – 1 n ui 1 – ⁄2 , ,j k n + –

hijk , , – 1 n hijk , ,

n + -------------------------------------------------------------------------------------------------------------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 2

⎝

⎜

⎛ = ⋅

vi j 1 , , + ⁄2 k – 1 n vi j 1 , , + ⁄2 k n – vi j 1 , , – ⁄2 k – 1 n vi j 1 , , – ⁄2 k n + –

hijk , , – 1 n hijk , ,

n + -----------------------------------------------------------------------------------------------------------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 2

⎠

⎟

⎞

1

⁄2

+ . (E.1)

The mixing length Λ is estimated at the layer interfaces for location i, j using equation 2.32. Equation E.1 is evaluated for AV0 at 

each layer interface except at the free surface and bottom where the wind and frictional stresses are applied.56

To introduce the effect of stratification on the eddy viscosity, a stability function (eq. 2.27) is used:

AV

n

ijk 1 – ⁄2 , , AV0 ( )ijk 1 – ⁄2 , ,

n 1 β1Riijk 1 – ⁄2 , ,

n ( ) + α1 = . (E.2)

The coefficients β1 = 10.0 and α1 = –0.5 were used by Munk and Anderson (1948); these coefficients are variables in the model 

and can be adjusted for best fit of the available data. The Richardson number Ri

 is discretized from equation 2.28 as

Riijk 1 – ⁄2 , ,

– g

ρijk , , – 1 n ρijk , ,

n ( ) + ⁄ 2 --------------------------------------------------- ρijk , , – 1 n ρijk , ,

n ( ) –

hijk , , – 1 n hijk , ,

n ( ) + ⁄ 2 = ⋅ ---------------------------------------------------

ui 1 + ⁄2 , ,j k – 1 n ui 1 + ⁄2 , ,j k n – ui 1 – ⁄2 , ,j k – 1 n ui 1 – ⁄2 , ,j k n + –

hijk , , – 1 n hijk , ,

n + -------------------------------------------------------------------------------------------------------------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 2

⎝

⎜

⎛ ×

vi j 1 , , + ⁄2 k – 1 n vi j 1 , , + ⁄2 k n – vi j 1 , , – ⁄2 k – 1 n vi j 1 , , – ⁄2 k n + –

hijk , , – 1 n hijk , ,

n + -----------------------------------------------------------------------------------------------------------------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 2

⎠

⎟

⎞

–1

+ . (E.3)

56For conditions where a wind stress generates significant wave action on the free surface, the magnitude of the eddy viscosity may be larger than 

estimated by equation E.1 because of the turbulence associated with oscillatory water motions. In those cases a wind-wave component, as mentioned in section 

2.3.1.1, should be included in the model.
176 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

Because the eddy viscosities are needed in the momentum equations at the layer interfaces lying vertically between the u- and 

v-points, these are determined by horizontal interpolation as follows:

AV

n

i 1 + ⁄2 j k 1 – ⁄2 , ,

1

2

-- AV

n

ijk 1 – ⁄2 , , AV

n

i + 1 j k 1 – ⁄2 , , = ( ) +

AV

n

i j 1 + ⁄2 k 1 – ⁄2 , ,

1

2

-- AV

n

ijk 1 – ⁄2 , , AV

n

i j + 1 k 1 – ⁄2 , , = ( ) +

.

To avoid model instabilities, the eddy viscosity within the interior of the water column is not permitted to take a value below some 

preassigned minimum, usually about 1 cm2/s.

The eddy diffusivity is computed from equation 2.36 as

DV

n

ijk 1 – ⁄2 , , AV0 ( )ijk 1 – ⁄2 , ,

n 1 β2Riijk 1 – ⁄2 , ,

n ( ) + α2 = . (E.4)

The Munk-Anderson coefficients in this case are β2 = 3.33 and α2 = –1.5 .