SHYFEM Finite Element Model for Coastal Seas User Manual

SHYFEM Finite Element Model for Coastal Seas User Manual
Finite Element Model for Coastal Seas
User Manual
Georg Umgiesser
ISDGM-CNR, S. Polo 1364
30125 Venezia, Italy
Version 4.93
December 2, 2005
Equations and resolution techniques
2.1 Equations and Boundary Conditions . . . . . . . . . . . . .
2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Discretization in Time - The Semi-Implicit Method .
2.2.2 Discretization in Space - The Finite Element Method
2.2.3 Mass Conservation . . . . . . . . . . . . . . . . . .
2.2.4 Inter-tidal Flats . . . . . . . . . . . . . . . . . . . .
3.1 The pre-processing routine vp . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Optimization of the bandwidth . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Internal and external node numbering . . . . . . . . . . . . . . . . . . . .
The Model
4.1 The Parameter Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.1.1 The General Structure of the Parameter Input File . . . . . . . . . . 10
4.1.2 The Single Sections of the Parameter Input File . . . . . . . . . . . 11
5.1 Plotting of maps with plotmap . . . . . . . . . . . . . . . . . . . . . . . . 22
5.1.1 The parameter input file for plotmap . . . . . . . . . . . . . . . . 22
The Water Quality Module
6.1 General Description . . . . . . . . . . . . . . . . . . . .
6.2 The coupling . . . . . . . . . . . . . . . . . . . . . . .
6.3 Light limitation . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Light attenuation formula by Steele and Di Toro
6.3.2 Light attenuation formula by Smith . . . . . . .
6.4 Initialization . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Post processing . . . . . . . . . . . . . . . . . . . . . .
6.6 The Sediment Module . . . . . . . . . . . . . . . . . . .
6.6.1 Parameters for the Water Quality Module . . . .
Copyright (c) 1992-1998 by Georg Umgiesser
Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the
above copyright notice appear in all copies and that both that copyright notice
and this permission notice appear in supporting documentation.
This file is provided AS IS with no warranties of any kind. The author shall
have no liability with respect to the infringement of copyrights, trade secrets
or any patents by this file or any part thereof. In no event will the author be
liable for any lost revenue or profits or other special, indirect and consequential
Comments and additions should be sent to the author:
Georg Umgiesser
S. Polo 1364
30125 Venezia
: ++39-41-5216875
: ++39-41-2602340
E-Mail : [email protected]
Chapter 1
The finite element program SHYFEM is a program package that can be used to resolve the hydrodynamic equations in lagoons, coastal seas, estuaries and lakes. The program uses finite
elements for the resolution of the hydrodynamic equations. These finite elements, together
with an effective semi-implicit time resolution algorithm, makes this program especially
suitable for application to a complicated geometry and bathymetry.
This version of the program SHYFEM resolves the depth integrated shallow water equations.
It is therefore recommended for the application of very shallow basins or well mixed estuaries. Storm surge phenomena can be investigated also. This two-dimensional version of
the program is not suited for the application to baroclinic driven flows or large scale flows
where the the Coriolis acceleration is important.
Finite elements are superior to finite differences when dealing with complex bathymetric
situations and geometries. Finite differences are limited to a regular outlay of their grids.
This will be a problem if only parts of a basin need high resolution. The finite element
method has an advantage in this case allowing more flexibility with its subdivision of the
system in triangles varying in form and size.
This model is especially adapted to run in very shallow basins. It is possible to simulate
shallow water flats, i.e., tidal marshes that in a tidal cycle may be covered with water during
high tide and then fall dry during ebb tide. This phenomenon is handled by the model in a
mass conserving way.
Finite element methods have been introduced into hydrodynamics since 1973 and have
been extensively applied to shallow water equations by numerous authors [3, 9, 5, 4, 6].
The model presented here [10, 11] uses the mathematical formulation of the semi-implicit
algorithm that decouples the solution of the water levels and velocity components from
each other leading to smaller systems to solve. Models of this type have been presented
from 1971 on by many authors [7, 2, 1].
Chapter 2
Equations and resolution
Equations and Boundary Conditions
The equations used in the model are the well known vertically integrated shallow water
equations in their formulation with water levels and transports.
+ gH + RU + X = 0
+ gH + RV +Y = 0
∂ζ ∂U ∂V
where ζ is the water level, u, v the velocities in x and y direction, U,V the vertical integrated
velocities (total or barotropic transports)
Z ζ
Z ζ
u dz
v dz
g the gravitational acceleration, H = h + ζ the total water depth, h the undisturbed water
depth, t the time and R the friction coefficient. The terms X,Y contain all other terms that
may be added to the equations like the wind stress or the nonlinear terms and that need not
be treated implicitly in the time discretization. following treatment.
The friction coefficient has been expressed as
g u2 + v2
C2 H
with C the Chezy coefficient. The Chezy term is itself not retained constant but varies with
the water depth as
C = ks H 1/6
where ks is the Strickler coefficient.
In this version of the model the Coriolis term, the turbulent friction term and the nonlinear
advective terms have not been implemented.
At open boundaries the water levels are prescribed. At closed boundaries the normal velocity component is set to zero whereas the tangential velocity is a free parameter. This
corresponds to a full slip condition.
The Model
The model uses the semi-implicit time discretization to accomplish the time integration.
In the space the finite element method has been used, not in its standard formulation, but
using staggered finite elements. In the following a description of the method is given.
Discretization in Time - The Semi-Implicit Method
Looking for an efficient time integration method a semi-implicit scheme has been chosen.
The semi-implicit scheme combines the advantages of the explicit and the implicit scheme.
It is unconditionally stable for any time step ∆t chosen and allows the two momentum
equations to be solved explicitly without solving a linear system.
The only equation that has to be solved implicitly is the continuity equation. Compared
to a fully implicit solution of the shallow water equations the dimensions of the matrix are
reduced to one third. Since the solution of a linear system is roughly proportional to the
cube of the dimension of the system the saving in computing time is approximately a factor
of 30.
It has to be pointed out that it is important not to be limited with the time step by the CFL
criterion for the speed of the external gravity waves
∆t < √
where ∆x is the minimum distance between the nodes in an element. With the discretization
described below in most parts of the lagoon we have ∆x ≈ 500m and H ≈ 1m, so ∆t ≈ 200
sec. But the limitation of the time step is determined by the worst case. For example, for
∆x = 100 m and H = 40 m the time step criterion would be ∆t < 5 sec, a prohibitive small
The equations (1)-(3) are discretized as follows
ζn+1 − ζn 1 ∂(U n+1 +U n ) 1 ∂(V n+1 +V n )
U n+1 −U n
∂(ζn+1 + ζn )
+ gH 21
+ RU n+1 + X = 0
V n+1 −V n
∂(ζn+1 + ζn )
+ gH 12
+ RV n+1 +Y = 0
With this time discretization the friction term has been formulated fully implicit, X,Y fully
explicit and all the other terms have been centered in time. The reason for the implicit
treatment of the friction term is to avoid a sign inversion in the term when the friction
parameter gets too high. An example of this behavior is given in Backhaus [1].
If the two momentum equations are solved for the unknowns U n+1 and V n+1 we have
n+1 + ζn )
1 ∂(ζ
U − ∆tgH 2
− ∆tX
1 + ∆tR
n+1 + ζn )
1 ∂(ζ
V − ∆tgH 2
− ∆tY
1 + ∆tR
If ζn+1 were known, the solution for U n+1 and V n+1 could directly be given. To find ζn+1
we insert (2.9) and (2.10) in (2.6). After some transformations (2.6) reads
− (∆t/2)
) + (H
1 + ∆tR ∂x
= ζn + (∆t/2)2
) + (H
1 + ∆tR ∂x
2 + ∆tR
− (∆t/2)
1 + ∆tR
∂X ∂Y
2(1 + ∆tR) ∂x
The terms on the left hand side contain the unknown ζn+1 , the right hand contains only
known values of the old time level. If the spatial derivatives are now expressed by the finite
element method a linear system with the unknown ζn+1 is obtained and can be solved by
standard methods. Once the solution for ζn+1 is obtained it can be substituted into (2.9)
and (2.10) and these two equations can be solved explicitly. In this way all unknowns of
the new time step have been found.
Note that the variable H also contains the water level through H = h + ζ. In order to avoid
the equations to become nonlinear ζ is evaluated at the old time level so H = h + ζn and H
is a known quantity.
Discretization in Space - The Finite Element Method
While the time discretization has been explained above, the discretization in space has still
to be carried out. This is done using staggered finite elements. With the semi-implicit
method described above it is shown below that using linear triangular elements for all
unknowns will not be mass conserving. Furthermore the resulting model will have propagation properties that introduce high numeric damping in the solution of the equations.
For these reasons a quite new approach has been adopted here. The water levels and the
velocities (transports) are described by using form functions of different order, being the
standard linear form functions for the water levels but stepwise constant form functions
for the transports. This will result in a grid that resembles more a staggered grid in finite
difference discretizations.
Let u be an approximate solution of a linear differential equation L. We expand u with the
help of basis functions φm as
u = φm um
m = 1, K
where um is the coefficient of the function φm and K is the order of the approximation. In
case of linear finite elements it will just be the number of nodes of the grid used to discretize
the domain.
To find the values um we try to minimize the residual that arises when u is introduced into L
multiplying the equation L by some weighting functions Ψn and integrating over the whole
domain leading to
ψn L(u) dΩ =
ψn L(φm um ) dΩ = um
ψn L(φm ) dΩ
If the integral is identified with the elements of a matrix anm we can write (2.13) also as a
linear system
n = 1, K m = 1, K
anm um = 0
Once the basis and weighting functions have been specified the system may be set up and
(2.14) may be solved for the unknowns um .
L aa
Hf L
C [email protected] L
E @
f @
i C E PP
S S A A S S A n S AX
L aa
t L
C j QQ L
i t
t @
Jt A
Figure 2.1: a) form functions in domain
b) domain of influence of node i
Staggered Finite Elements
For decades finite elements have been used in fluid mechanics in a standardized manner.
The form functions φm were chosen as continuous piecewise linear functions allowing a
subdivision of the whole area of interest into small triangular elements specifying the coefficients um at the vertices (called nodes) of the triangles. The functions φm are 1 at node m
and 0 at all other nodes and thus different from 0 only in the triangles containing the node
m. An example is given in the upper left part of Fig. 1a where the form function for node i
is shown. The full circle indicates the node where the function φi take the value 1 and the
hollow circles where they are 0.
The contributions anm to the system matrix are therefore different from 0 only in elements
containing node m and the evaluation of the matrix elements can be performed on an element basis where all coefficients and unknowns are linear functions of x and y.
This approach is straightforward but not very satisfying with the semi-implicit time stepping scheme for reasons explained below. Therefore an other way has been followed in
the present formulation. The fluid domain is still divided in triangles and the water levels
are still defined at the nodes of the grid and represented by piecewise linear interpolating
functions in the internal of each element, i.e.
ζ = ζm φm
m = 1, K
However, the transports are now expanded, over each triangle, with piecewise constant (non
continuous) form functions ψn over the whole domain. We therefore write
U = Un ψn
n = 1, J
where n is now running over all triangles and J is the total number of triangles. An example
of ψn is given in the lower right part of Fig. 1a. Note that the form function is constant 1
over the whole element, but outside the element identically 0. Thus it is discontinuous at
the element borders.
Since we may identify the center of gravity of the triangle with the point where the transports Un are defined (contrary to the water levels ζm which are defined on the vertices of
the triangles), the resulting grid may be seen as a staggered grid where the unknowns are
defined on different locations. This kind of grid is usually used with the finite difference
method. With the form functions used here the grid of the finite element model resembles
very much an Arakawa B-grid that defines the water levels on the center and the velocities
on the four vertices of a square.
Staggered finite elements have been first introduced into fluid mechanics by Schoenstadt
[8]. He showed that the un-staggered finite element formulation of the shallow water equations has very poor geostrophic adjustment properties. Williams [12, 13] proposed a similar
algorithm, the one actually used in this paper, introducing constant form functions for the
velocities. He showed the excellent propagation and geostrophic adjustment properties of
this scheme.
The Practical Realization
The integration of the partial differential equation is now performed by using the subdivision of the domain in elements (triangles). The water levels ζ are expanded in piecewise
linear functions φm , m = 1, K and the transports are expanded in piecewise constant functions ψn , n = 1, J where K and J are the total number of nodes and elements respectively.
As weighting functions we use ψn for the momentum equations and φm for the continuity
equation. In this way there will be K equations for the unknowns ζ (one for each node) and
J equations for the transports (one for each element).
In all cases the consistent mass matrix has been substituted with the their lumped equivalent. This was mainly done to avoid solving a linear system in the case of the momentum
equations. But it was of use also in the solution of the continuity equation because the
amount of mass relative to one node does not depend on the surrounding nodes. This was
important especially for the flood and dry mechanism in order to conserve mass.
Finite Element Equations
If equations (2.9,2.10,2.11) are multiplied with their weighting functions and integrated
over an element we can write down the finite element equations. But the solution of the
water levels does actually not use the continuity equation in the form (2.11), but a slightly
different formulation. Starting from equation (2.6), multiplied by the weighting function
ΦM and integrated over one element yields
Z ∂(U n+1 +U n )
∂(V n+1 +V n )
ΦN (ζ
− ζ ) dΩ + ( 2 )
dΩ = 0
+ ΦN
If we integrate by parts the last two integrals we obtain
Z ∂ΦN n+1
∂ΦN n+1
ΦN (ζn+1 − ζn ) dΩ − ( ∆t2 )
+U n ) +
+V n ) dΩ = 0
plus two line integrals, not shown, over the boundary of each element that specify the
normal flux over the three element sides. In the interior of the domain, once all contributions of all elements have been summed, these terms cancel at every node, leaving only
the contribution of the line integral on the boundary of the domain. There, however, the
boundary condition to impose is exactly no normal flux over material boundaries. Thus,
the contribution of these line integrals is zero.
If now the expressions for U n+1 ,V n+1 are introduced, we obtain a system with again only
the water levels as unknowns
ζn+1 dΩ
∂ΦN ∂ζn+1 ∂ΦN ∂ζn+1
) dΩ
∂x ∂x
∂y ∂y
∂ΦN ∂ζn ∂ΦN ∂ζn
ΦN ζn dΩ + (∆t/2)2 αg H(
) dΩ
∂x ∂x
∂y ∂y
∂ΦN n ∂ΦN n
+ (∆t/2)(1 + α) (
U +
V ) dΩ
Ω ∂x
− (∆t 2 /2)α (
Y ) dΩ
Ω ∂x
+ (∆t/2) αg
Here we have introduced the symbol α as a shortcut for
1 + ∆tR
The variables and unknowns may now be expanded with their basis functions and the complete system may be set up.
Mass Conservation
It should be pointed out that only through the use of this staggered grid the semi-implicit
time discretization may be implemented in a feasible manner. If the Galerkin method is
applied in a naive way to the resulting equation (2.11) (introducing the linear form functions for transports and water levels and setting up the system matrix), the model is not
mass conserving. This may be seen in the following way (see Fig. 1b for reference). In the
computation of the water level at node i, only ζ and transport values belonging to triangles
that contain node i enter the computation (full circles in Fig. 1b). But when, in a second
step, the barotropic transports of node j are computed, water levels of nodes that lie further
apart from the original node i are used (hollow circles in Fig. 1b). These water levels have
not been included in the computation of ζi , the water level at node i. So the computed transports are actually different from the transports inserted formally in the continuity equation.
The continuity equation is therefore not satisfied.
These contributions of nodes lying further apart could in principle be accounted for. In this
case not only the triangles Ωi around node i but also all the triangles that have nodes in
common with the triangles Ωi would give contributions to node i, namely all nodes and
elements shown in Fig. 1b. The result would be an increase of the bandwidth of the matrix
for the ζ computation disadvantageous in terms of memory and time requirements.
Using instead the approach of the staggered finite elements, actually only the water levels
of elements around node i are needed for the computation of the transports in the triangles
Ωi . In this case the model satisfies the continuity equation and is perfectly mass conserving.
Inter-tidal Flats
Part of a basin may consist of areas that are flooded during high tides and emerge as islands
at ebb tide. These inter-tidal flats are quite difficult to handle numerically because the
elements that represent these areas are neither islands nor water elements. The boundary
line defining their contours is wandering during the evolution of time and a mathematical
model must reproduce this features.
For reasons of computer time savings a simplified algorithm has been chosen to represent
the inter-tidal flats. When the water level in at least one of the three nodes of an element
falls below a minimum value (5 cm) the element is considered an island and is taken out
of the system. It will be reintroduced only when in all three nodes the water level is again
higher then the minimum value. Because in dry nodes no water level is computed anymore,
an estimate of the water level has to be given with some sort of extrapolation mechanism
using the water nodes nearby.
This algorithm has the advantage that it is very easy to implement and very fast. The
dynamical features close to the inter-tidal flats are of course not well reproduced but the
behavior of the method for the rest of the lagoon gave satisfactory results.
In any case, since the method stores the water levels of the last time step, before the element is switched off, introducing the element in a later moment with the same water levels
conserves the mass balance. This method showed a much better performance than the one
where the new elements were introduced with the water levels taken from the extrapolation
of the surrounding nodes.
Chapter 3
The pre-processing routine vp is used to generate an optimized version of the file that describes the basin where the main program is to be run. In the following a short introduction
in using this program is given.
The pre-processing routine vp
The main routine hp reads the basin file generated by the pre-processing routine vp and uses
it as the description of the domain where the hydrodynamic equations have to be solved.
The program vp is started by typing vp on the command line. From this point on the
program is interactive, asking you about the basin file name and other options. Please
follow the online instructions.
The routine vp reads a file of type GRD. This type of file can be generated and manipulated
by the program grid which is not described here. In short, the file GRD consists of nodes
and elements that describe the geometrical layout of the basin. Moreover, the elements
have a type and a depth.
The depth is needed by the main program hp to run the model. The type of the element is
used by hp to determine the friction parameter on the bottom, since this parameter may be
assigned differently, depending on the various situations of the bottom roughness.
This file GRD is read by vp and transformed into an unformatted file BAS. It is this file that
is then read by the main routine hp. Therefore, if the name of the basin is lagoon, then the
file GRD is called lagoon.grd and the output of the pre-processing routine vp is called
The program vp normally uses the depths assigned to the elements in the file GRD to
determine the depth of the finite elements to use in the program hp. In the case that these
depth values are not complete, and that all nodes have depths assigned in the GRD file, the
nodal values of the depths are used and interpolated onto the elements. However, if also
these nodal depth values are incomplete or are missing altogether, the program terminates
with an error.
Optimization of the bandwidth
The main task of routine vp is the optimization of the internal numbering of the nodes and
elements. Re-numbering the elements is just a mere convenience. When assembling the
system matrix the contribution of one element after the other has to be added to the system
matrix. If the elements are numbered in terms of lowest node numbers, then the access
of the nodal pointers is more regular in computer memory and paging is more likely to be
However, re-numbering the nodes is absolutely necessary. The system matrix to be solved
is of band-matrix type. I.e., non-zero entries are all concentrated along the main diagonal
in a more or less narrow band. The larger this band is, the larger the amount of cpu time
spent to solve the system. The time to solve a band matrix is of order n · m2 , where n is the
size of the matrix and m is the bandwidth. Note that m is normally much smaller than n.
If the nodes are left with the original numbering, it is very likely that the bandwidth is
very high, unless the nodes in the file GRD are by chance already optimized. Since the
bandwidth m is entering the above formula quadratically, the amount of time spent solving
the matrix will be prohibitive. E.g., halving the bandwidth will speed up computations by
a factor of 4.
The bandwidth is equal to the maximum difference of node numbers in one element. It is
therefore important to re-number the nodes in order to minimize this number. However,
there exist only heuristic algorithms for the minimization of this number.
The two main algorithms used in the routine vp are the Cuthill McGee algorithm and the
algorithm of Rosen. The first one, starting from one node, tries to number all neighbors in
a greedy way, optimizing only this step. From the points numbered in this step, the next
neighbors are numbered.
This procedure is tried from more than one node, possibly from all boundary nodes. The
numbering resulting from this algorithm is normally very good and needs only slight enhancements to be optimum.
Once all nodes are numbered, the Rosen algorithm tries to exchange these node numbers,
where the highest difference can be found. This normally gives only a slight improvement
of the bandwidth. It has been seen, however, that, if the node numbers coming out from the
Cuthill McGee algorithm are reversed, before feeding them into the Rosen algorithm, the
results tend to be slightly better. This step is also performed by the program.
All these steps are performed by the program without intervention by the operator, if the
automatic optimization of bandwidth is chosen in the program vp. The choices are to not
perform the bandwidth optimization at all (GRD file has already optimized node numbering), perform it automatically or perform it manually. It is suggested to always perform
automatic optimization of the bandwidth. This choice will lead to a nearly optimum numbering of the nodes and will be by all means good results.
If, however, you decide to do a manual optimization, please follow the online instructions
in the program.
Internal and external node numbering
As explained above, the elements and nodes of the basin are re-numbered in order to optimize the bandwidth of the system matrix and so the execution speed of the program.
However, this re-numbering of the node and elements is transparent to the user. The program keeps pointers from the original numbering (external numbers) to the optimized numbering (internal numbers). The user has to deal only with external numbers, even if the
program uses internally the other number system.
Moreover, the internal numbers are generated consecutively. Therefore, if there are a total
of 4000 nodes in the system, the internal nodes run from 1 to 4000. The external node
numbers, on the other side, can be anything the user likes. They just must be unique. This
allows for insertion and deletion of nodes without having to re-number over and over again
the basin.
The nodes that have to be specified in the input parameter file use again external numbers.
In this way, changing the structure of the basin does not at all change the node and element
numbers in the input parameter file. Except in the case, where modifications actually touch
nodes and elements that are specified in the parameter file.
Chapter 4
The Model
In the following an overview is given on running the model SHYFEM. The model needs a
parameter input file that is read on standard input. Moreover, it needs some external files
that are specified in this parameter input file. The model produces several external files
with the results of the simulation. Again, the name of this files can be influenced by the
parameter input file
The Parameter Input File
The model reads one input file that determines the behavior of the simulation. All possible
parameters can and must be set in this file. If other data files are to be read, here is the place
where to specify them.
The model reads this parameter file from standard input. Thus, if the model binary is called
hp and the parameter file param.str, then the following line starts the simulation
hp < param.str
and runs the model.
The General Structure of the Parameter Input File
The input parameter file is the file that guides program performance. It contains all necessary information for the main routine to execute the model. Nearly all parameters that
can be given have a default value which is used when the parameter is not listed in the file.
Only some time parameters are compulsory and must be present in the file.
The format of the file looks very like a namelist format, but is not dependent on the compiler
used. Values of parameters are given in the form : name = value or name = ’text’. If
name is an array the following format is used :
name = value1 , value2, ... valueN
The list can continue on the following lines. Blanks before and after the equal sign are
ignored. More then one parameter can be present on one line. As separator blank, tab and
comma can be used.
Parameters, arrays and data must be given in between certain sections. A section starts
with the character $ followed by a keyword and ends with $end. The $keyword and $end
must not contain any blank characters and must be the first non blank characters in the line.
Other characters following the keyword on the same line separated by a valid separator are
Several sections of data may be present in the input parameter file. Further ahead all sections are presented and the possible parameters that can be specified are explained. The
sequence in which the sections appear is of no importance. However, the first section must
always be section \$title, the section that determines the name of simulation and the
basin file to use and gives a one line description of the simulation.
Lines outside of the sections are ignored. This gives the possibility to comment the parameter input file.
Figure 4.1 shows an example of a typical input parameter file and the use of the sections
and definition of parameters.
The Single Sections of the Parameter Input File
Section $title
This section must be always the first section in the parameter input file. It contains only
three lines. An example is given in figure 4.2.
The first line of this section is a free one line description of the simulation that is to be carried out. The next line contains the name of the simulation (in this case name_of_simulation).
All created files will use this name in the main part of the file name with different extensions. Therefore the hydrodynamic output file (extension out) will be named name_of_simulation.out.
The last line gives the name of the basin file to be used. This is the pre-processed file of the
basin with extension bas. In our example the basin file name_of_basin.bas is used.
The directory where this files are read from or written to depends on the settings in section
$name. Using the default the program will read from and write to the current directory.
Section $para
This section defines the general behavior of the simulation, gives various constants of parameters and determines what output files are written. In the following the meaning of all
possible parameters is given.
Note that the only compulsory parameters in this section are the ones that chose the duration
of the simulation and the integration time step. All other parameters are optional.
Compulsory time parameters This parameters are compulsory parameters that define
the period of the simulation. They must be present in all cases.
Start of simulation. (Default 0)
End of simulation.
Time step of integration.
Output parameters The following parameters deal with the output frequency and start
time to external files. The content of the various output files should be looked up in the
appropriate section.
The default for the time step of output of the files is 0 which means that no output file is
written. If the time step of the output files is equal to the time step of the simulation then at
every time step the output file is written. The default start time of the output is 0.
Time step and start time for writing to file OUT, the file containing the
general hydrodynamic results.
Time step and start time for writing to file EXT, the file containing hydrodynamic data of extra points. The extra points for which the data is
written to this file are given in section extra of the parameter file.
Time step and start time for writing the restart
benchmark test for test lagoon
itanf = 0 itend = 86400
ireib = 2 ilin = 0
href = 0.23 iczv = 1
idt = 300
kbound = 73 74 76
ampli = 0.50 period = 43200. phase = 10800. zref = 0.
kbound = 353,350, 349
iqual = 1
kbound = 1374 1154 1160 1161
iqual = 1
-------------- MAREOGRAFI PER TARATURA-------------13,133,99,259,328,772,419,1141,1195,1070,1064,942,468,1154
----------------------------------------------------$extra ------- MAREOGRAFI + SEZIONI PER TARATURA---------13,133,99,259,328,772,419,1141,1195,1070,1064,942,468,1154
$area ------- old chezy values ---------0 33.
1 25. 27.
2 21. 23.
3 20. 25. 1154 1153
4 27.
5 27.
6 27.
Figure 4.1: Example of a parameter input file
free one line description of simulation
Figure 4.2: Example of section $title
file (extension RST). No restart file is written with idtrst equal to 0. itrst Time to use
for the restart. If a restart is performed, then the file name containing the restart data has to
be specified in restrt and the time record corresponding to itrst is used in this file.
Time step and start time for writing to file RES, the file containing residual hydrodynamic data.
Time step and start time for writing to file RMS, the file containing
hydrodynamic data of root mean square velocities.
Time step and start time for writing to file FLX, the file containing discharge data through defined sections. The transects for which the discharges are computed are given in section flux of the parameter file.
Model parameters The next parameters define the inclusion or exclusion of certain terms
of the primitive equations.
Linearization of the momentum equations. If ilin is different from 0
the advective terms are not included in the computation. (Default 1)
This parameter decides how the advective (non-linear) terms are computed. The value of 0 (default) uses the usual finite element discretization over a single element. The value of 1 choses a semi-lagrangian
approach that is theoretically stable also for Courant numbers higher
than 1. It is however recommended that the time step is limited using
itsplt and coumax described below. (Default 0)
Linearization of the continuity equation. If iclin is different from 0 the
depth term in the continuity equation is taken to be constant. (Default
Compute system matrix every nrand iterations. This parameter can be
used to speed up computations if the system matrix does not depend
crucially on the varying water depth (e.g., in deep waters). It is recommended to leave it to the default value of 1 (compute at every iteration).
(Default 1)
The next parameters allow for a variable time step in the hydrodynamic computations. This
is especially important for the non-linear model (ilin=0) because in this case the criterion
for stability cannot be determined a priori and in any case the time integration will not be
unconditionally stable.
The variable time steps allows for longer basic time steps (here called macro time steps)
which have to be set in idt. It then computes the optimal time step (here micro time step)
in order to not exceed the given Courant number. However, the value for the macro time
step will never be exceeded.
Type of variable time step computation. If this value is 0, the time
step will kept constant at its initial value. A value of 1 devides the
initial time step into (possibly) equal parts, but makes sure that at the
end of the micro time steps one complete macro time step has been
executed. The last mode itsplt = 2 does not care about the macro
time step, but always uses the biggest time step possible. In this case it
is not assured that after some micro time steps a macro time step will
be recovered. Please note that the initial macro time step will never be
exceeded. (Default 0)
Normally the time step is computed in order to not exceed the Courant
number of 1. However, in some cases the non-linear terms are stable even for a value higher than 1 or there is a need to achieve a
lower Courant number. Setting coumax to the desired Courant number
achieves exactly this effect. (Default 1)
In case of itsplt = 2 this parameter makes sure that after a time of
idtsyn the time step will be syncronized to this time. Therefore, setting
idtsyn = 3600 means that there will be a time stamp every hour, even
if the model has to take one very small time step in order to reach that
time. This parameter is useful only for itsplt = 2 and its default value
of 0 does not make any syncronization.
These parameters define the weighting of time steps in the semi-implicit algorithm. With
these parameters the damping of gravity waves can be controlled. Only modify them if you
know what you are doing.
Weighting of the new time level of the transport terms in the continuity
equation. (Default 0.5)
Weighting of the new time level of the pressure term in the momentum
equations. (Default 0.5)
Coriolis parameters
The next parameters define the parameters to be used in the Coriolis
If this parameter is 0, the Coriolis terms are not included in the computation. A value of 1 uses a beta-plane approximation with a variable
Coriolis parameter f , whereas a value of 2 uses an f-plane approximation where the Coriolis parameter f is kept constant over the whole
domain. (Default 0)
Average latitude of the basin. This is used to compute the Coriolis parameter f . If not given the latitude in the basin file is used. If given the
value of dlat in the input parameter file effectively substitues the value
given in the basin file.
Depth parameters The next parameters deal with various depth values of the basin.
Reference depth. If the depth values of the basin and the water levels
are referred to mean sea level, href should be 0 (default value). Else
this value is subtracted from the given depth values. For example, if
href = 0.20 then a depth value in the basin of 1 meter will be reduced
to 80 centimeters.
Minimum total water depth that will remain in a node if the element
becomes dry. (Default 0.01 m)
Total water depth at which an element will be taken out of the computation because it becomes dry. (Default 0.05 m)
Total water depth at which a dry element will be re-inserted into the
computation. (Default 0.10 m)
Minimum water depth (most shallow) for the whole basin. All depth
values of the basin will be adjusted so that no water depth is shallower
than hmin. (Default is no adjustment)
Maximum water depth (deepest) for the whole basin. All depth values
of the basin will be adjusted so that no water depth is deeper than hmax.
(Default is no adjustment)
Bottom friction The friction term in the momentum equations can be written as Ru and
Rv where R is the variable friction coefficient and u, v are the velocities in x, y direction
respectively. The form of R can be specified in various ways. The value of ireib is
choosing between the formulations. In the parameter input file a value λ is specified that is
used in the formulas below.
Type of friction used (default 0):
0 No friction used
1 R = λ is constant
2 λ is the Strickler coefficient. In this formulation R is written as
1/6 and λ = k is the Strickler coefficient.
R = Cg2 |u|
H with C = ks H
In the above formula g is the gravitational acceleration, |u| the
modulus of the current velocity and H the total water depth.
3 λ is the Chezy coefficient. In this formulation R is written as R =
g |u|
and λ = C is the Chezy coefficient.
C2 H
4 R = λ/H with H the total water depth
5 R = λ |u|
The default value for the friction parameter λ. Depending on the value
of ireib the coefficient λ is describing linear friction or a Chezy or
Strickler form of friction (default 0).
Normally R is evaluated at every time step (iczv = 1). If for some
reason this behavior is not desirable, iczv = 0 evaluates the value of
R only before the first time step, keeping it constant for the rest of the
simulation. (default 1)
The value of λ may be specified for the whole basin through the value of czdef. For more
control over the friction parameter it can be also specified in section area where the friction
parameter depending on the type of the element may be varied. Please see the paragraph
on section area for more information.
Physical parameters
if needed.
The next parameters describe physical values that can be adjusted
Average density of sea water. (Default 1025 kg m−3 )
Average density of air. (Default 1.225 kg m−3 )
Average gravitational acceleration. (Default 9.81 m s−2 )
Wind parameters The next two parameters deal with the wind stress to be prescribed at
the surface of the basin.
The wind data can either be specified in an external file (ASCII or binary) or directly in the
parameter file in section wind. The ASCII file or the wind section contain three columns,
the first giving the time in seconds, and the others the components of the wind speed. Please
see below how the last two columns are interpreted depending on the value of iwtype. For
the format of the binary file please see the relative section. If both a wind file and section
wind are given, data from the file is used.
The wind stress is normally computed with the following formula
τx = ρa cD |u|ux
τy = ρa cD |u|uy
where ρa , ρ0 is the density of air and water respectively, u the modulus of wind speed and
ux , uy the components of wind speed in x, y direction. In this formulation cD is a dimensionless drag coefficient that varies between 1.5 ·10−3 and 3.2 ·10−3 . The wind speed is
normally the wind speed measured at a height of 10 m.
The type of wind data given (default 1):
0 No wind data is processed
1 The components of the wind is given in [m/s]
2 The stress (τx , τy ) is directly specified
3 The wind is given in speed [m/s] and direction [degrees]. A direction
of 0o specifies a wind from the north, 90o a wind from the east
4 As in 3 but the speed is given in knots
Drag coefficient used in the above formula. The default value is 0 so it
must be specified. Please note also that in case of iwtype = 2 this value
is of no interest, since the stress is specified directly.
Concentrations The next parameters deal with the transport and diffusion of a conservative substance. The substance is dissolved in the water and acts like a tracer.
Flag if the computation on the tracer is done. A value different from 0
computes the transport and diffusion of the substance. (Default 0)
Reference (ambient) concentration of the tracer in any unit. (Default 0)
Type of advection scheme used for the transport and diffusion equation.
Normally an upwind scheme is used (0), but setting the parameter itvd
to 1 choses a TVD scheme. This feature is still experimental, so use
with care. (Default 0)
Horizontal diffusion parameter (general). (Default 0)
Horizontal diffusion parameter for the tracer. (Default 0)
Vertical turbulent diffusion parameter for the tracer. (Default 0)
Vertical molecular diffusion parameter for the tracer. (Default 1.0e-06)
Temperature and salinity The next parameters deal with the transport and diffusion of
temperature and salinity.
Flag if the computation on the temperature is done. A value different
from 0 computes the transport and diffusion of the temperature. (Default 0)
Flag if the computation on the salinity is done. A value different from
0 computes the transport and diffusion of the salinity. (Default 0)
Reference (ambient) temperature of the water in centigrade. (Default 0)
Reference (ambient) salinity of the water in psu (practical salinity units,
per mille). (Default 0)
Horizontal diffusion parameter for temperature. (Default 0)
Horizontal diffusion parameter for salinity. (Default 0)
Output for concentration, temperature and salinity The next parameters define the
output frequency of the computed concentration (temperature, salinity) to file. They also
define the internal time step to be used with the time integration.
Time step and start time for writing to file CON (concentration), TEM
(temperature) and SAL (salinity).
Frequency of internal time step of the solution of the transport and diffusion equation. Normally at every (external) time step of the hydrodynamic equations one transport-diffusion (internal) time step is executed. If the external time step is too long, the solution of the transportdiffusion equations with the same time step may lead to instabilities.
These instabilities can be avoided if more internal time steps are executed in one external step. istot gives the number of internal time
steps to be executed in one external step. (Default 1)
Section $name
In this sections names of directories or input files can be given. All directories default to
the current directory, whereas all file names are empty, i.e., no input files are given.
Directory specification This parameters define directories for various input and output
Directory where basin file BAS resides. (Default .)
Directory where output files are written. (Default .)
Directory for temporary files. (Default .)
Default directory for other files. (Default .)
File names The following strings enable the specification of files that account for initial
conditions or forcing.
File with initial water level distribution. This file must be constructed
by the utility routine zinit.
File with wind data. The file may be either formatted or unformatted.
For the format of the unformatted file please see the section where the
WIN file is discussed. The format of formatted ASCII file is in standard
time-series format, with the first column containing the time in seconds
and the next two columns containing the wind data. The meaning of the
two values depend on the value of the parameter iwtype in the para
File with rain data. This file is a standard time series with the time in
seconds and the rain values in mm/day.
File with heat flux data. This file must be in a special format to account
for the various parameters that are needed by the heat flux module to
run. Please refer to the information on the file qflux.
restrt Name of the file if a restart is to be performed. The file has to be produced by a
previous run with the parameter idtrst different from 0. The data records used in the file
for the restart must be given by time itrst.
Lagrangian trajectories !LAGR
Section $bound
These parameters determine the open boundary nodes and the type of the boundary: level
or flux boundary. At the first the water levels are imposed, on the second the fluxes are
There may be multiple sections bound in one parameter input file, describing all open
boundary conditions necessary. Every section must therefore be supplied with a boundary
number. The numbering of the open boundaries must be increasing. The number of the
boundary must be specified directly after the keyword bound, such as bound1 or bound 1.
Array containing the node numbers that are part of the open boundary.
The node numbers must form one contiguous line with the domain (elements) to the left. This corresponds to an anti-clockwise sense. At least
two nodes must be given.
Type of open boundary.
0 No boundary values specified
1 Level boundary. At this open boundary the water level is imposed and
the prescribed values are interpreted as water levels in meters.
2 Flux boundary. Here the discharge in m3 s−1 has to be prescribed.
3 Internal flux boundary. As with ibtyp = 2 a discharge has to be
imposed, but the node where discharge is imposed can be an internal node and need not be on the outer boundary of the domain. For every node in kbound the volume rate specified will
be added to the existing water volume. This behavior is different from the ibtyp = 2 where the whole boundary received the
discharge specified.
4 Momentum input. The node or nodes may be internal. This feature
can be used to describe local acceleration of the water column.
The unit is force / density [m4 s−2 ]. In other words it is the rate
of volume [m3 s−1 ] times the velocity [m/s] to which the water is
If the boundary conditions for this open boundary are equal to the ones
of boundary i, then setting iqual = i copies all the values of boundary
i to the actual boundary. Note that the value of iqual must be smaller
than the number of the actual boundary, i.e., boundary i must have been
defined before.
The next parameters give a possibility to specify the file name of the various input files
that are to be read by the model. Values for the boundary condition can be given at any
time step. The model interpolates in between given time steps if needed. The grade of
interpolation can be given by intpol.
All files are in ASCII and share a common format. The file must contain two columns, the
first giving the time of simulation in seconds that refers to the value given in the second
column. The value in the second column must be in the unit of the variable that is given.
The time values must be in increasing order. There must be values for the whole simulation,
i.e., the time value of the first line must be smaller or equal than the start of the simulation,
and the time value of the last line must be greater or equal than the end of the simulation.
File name that contains values for the boundary condition. The value
of the variable given in the second column must be in the unit determined by ibtyp, i.e., in meters for a level boundary, in m3 s−1 for a flux
boundary and in m4 s−2 for a momentum input.
Factor with which the values from boundn are multiplied to form the
final value of the boundary condition. E.g., this value can be used to
set up a quick sensitivity run by multiplying all discharges by a factor
without generating a new file. (Default 1)
File name that contains values for the respective boundary condition,
i.e., for concentration, temperature and salinity. The format is the same
as for file boundn. The unit of the values given in the second column
must the ones of the variable, i.e., arbitrary unit for concentration, centigrade for temperature and psu (per mille) for salinity.
Order of interpolation for the boundary values read through files. Use
for 1 for stepwise (no) interpolation, 2 for linear interpolation. The
default is cubic interpolation (4)
The next parameters can be used to impose a sinusoidal water level (tide) or flux at the open
boundary. These values are used if no boundary file boundn has been given. The values
must be in the unit of the intended variable determined by ibtyp.
Amplitude of the sinus function imposed. (Default 0)
Period of the sinus function. (Default 43200, 12 hours)
Phase shift of the sinus function imposed. A positive value of one quarter of the period reproduces a cosine function. (Default 0)
Reference level of the sinus function imposed. If only zref is specified
(ampli = 0) a constant value of zref is imposed on the open boundary.
With the next parameters a constant value can be imposed for the variables of concentration,
temperature and salinity. In this case no file with boundary values has to be supplied. The
default for all values is 0, i.e., if no file with boundary values is supplied and no constant is
set the value of 0 is imposed on the open boundary.
Constant boundary values for concentration, temperature and salinity
respectively. If these values are set no boundary file has to be supplied.
(Default 0)
The next two values are used for constant momentum input. This feature can be used to
describe local acceleration of the water column. The values give the input of momentum
in x and y direction. The unit is force / density (m4 s−2 ). In other words it is the rate of
volume (m3 s−1 ) times the velocity (m/s) to which the water is accelerated.
These values are used if boundary condition ibtyp = 4 has been chosen and no boundary
input file has been given. If the momentum input is varying then it may be specified with
the file boundn. In this case the file boundn must contain three columns, the first for the
time, and the other two for the momentum input in x, y direction.
Constant values for momentum input. (Default 0)
Section $wind
In this section the wind data can be given directly without the creation of an external file.
Note, however, that a wind file specified in the name section takes precedence over this
section. E.g., if both a section wind and a wind file in name is given, the wind data from
the file is used.
The format of the wind data in this section is the same as the format in the ASCII wind file,
i.e., three columns, with the first specifying the time in seconds and the other two columns
giving the wind data. The interpretation of the wind data depends on the value of iwtype.
For more information please see the description of iwtype in section para.
Section $extra
In this section the node numbers of so called “extra” points are given. These are points
where water level and velocities are written to create a time series that can be elaborated
later. The output for these “extra” points consumes little memory and can be therefore
written with a much higher frequency (typically the same as the integration time step) than
the complete hydrodynamical output. The output is written to file EXT.
The node numbers are specified in a free format on one ore more lines. An example can be
seen in figure 4.1. No keywords are expected in this section.
Section $flux
In this section transects are specified through which the discharge of water is computed by
the program and written to file FLX. The transects are defined by their nodes through which
they run. All nodes in one transect must be adjacent, i.e., they must form a continuous line
in the FEM network.
The nodes of the transects are specified in free format. Between two transects one or more
0’s must be inserted. An example is given in figure 4.3.
1001 1002 1004 0
35 37 46 0 0 56 57 58 0
435 0 89 87
Figure 4.3: Example of section $flux
The example shows the definition of 5 transects. As can be seen, the nodes of the transects
can be given on one line alone (first transect), two transects on one line (transect 2 and 3),
spread over more lines (transect 4) and a last transect.
Chapter 5
There are several routines that do a post-processing of the results of the main routine. The
most important are described in this chapter. Note that in the model framework no program
is supplied to do time series plots. However, there are utility routines that will extract data
from the output files. These data will be written in a way that it can be imported into a
spreadsheet or any other plotting program that does the nice plotting.
Plotting of maps with plotmap
The parameter input file for plotmap
The format of the parameter input file is the same as the one for the main routine. Please
see this section for more information on the format of the parameter input file.
Some sections of the parameter input file are identical to the sections used in the main
routine. For easier reference we will repeat the possible parameters of these section here.
Section $title
This section must be always the first section in the parameter input file. It contains only
three lines. An example has been given already in figure 4.2.
The only difference respect to the $para section of the main routine is the first line. Here
any description of the output can be used. It is just a way to label the parameter file. The
other two line with the name of simulation and the basin are used to open the files needed
for plotting.
Section $para
These parameters set generic values for the plot.
Note that the only compulsory parameter in this section is iwhat, a parameter that determines what to plot. All other parameters are optional.
Flag that determines what to plot (default 0):
0 Nothing plotted
1 Plot basin (grid and isolines of depth)
2 Plot velocities
3 Plot transports
4 Plot water levels
5 Plot concentration
6 Plot temperature
7 Plot salinity
8 Plot rms-velocity
Lower left corner of the plotting area. (Default is whole area)
Upper right corner of the plotting area. (Default is whole area)
Lower left corner of the area where the legend is plotted.
Upper right corner of the area. where the legend is plotted.
Grid size if the results are interpolated on a regular grid. A value of
0 does not use a regular grid but the original finite element grid for
plotting. (Default 0)
Not used anymore.
Typical length scale to be used when scaling velocity or transport arrows. If dxygrd is given this length is used and typls is not used. If
not given it is computed from the basin parameters. (Default 0)
Additional factor to be used with typls to determine the length of the
maximum or reference vector. This is the easiest way to scale the velocitiy arrows with an overall factor. (Default 1)
Reference value to be used when scaling arrows. If given, a vector with
this value will have a length of typls*typlsf on the map, or, in case
dxygrd is given, dxygrd*typlsf. If not set the maximum value of the
velocity/transport will be used as velref,traref. (Default 0)
Minimum value for which an arrow will be plotted. With this value you
can eliminate small arrows in low dynamic areas. (Default 0)
Name of file that gives the boundary line that is not part of the finite
element domain. The file must be in BND format. You can use the
program to create the file from a GRD file. (Default is no
Create overlay of velocity vectors on scalar value. With the value of
0 no overlay is created, 1 creates an overlay with the velocity speed.
The value of 2 overlays vertical velocities 3 water levels and 4 overlays
bathymetry.(Default 0)
Normally the horizontal velocities are plotted in scale. The value of
inorm can change this behavior. A value of 1 normalizes velocity vectors (all vectors are the same length), whereas 2 scales from a given
minimum velocity velmin. Finally, the value of 3 uses a logarithmic
scale. (Default 0)
Section $color
The next parameters deal with the definition of the colors to be used in the plot. A color
bar is plotted too.
Flag that determines the type of color table to be used. 0 stands for gray
scale, 1 for HSB color table. (Default 0)
Array that defines the values for the isolines and colors that are to be
plotted. Values given must be in the unit of the variable that will be
plotted, i.e., meters for water levels etc.
Array that gives the color indices for the plotting color to be used.
Ranges are from 0 to 1. The type of the color depends on the variable icolor. For the gray scale table 0 represents black and 1 white.
Values in between correspond to tones of gray. For the HSB color table
going from 0 to 1 gives the color of the rainbow. There must be one
more value in color than in isoval. The first color in color refers
to values less than isoval(1), the second color in color to values between isoval(1) and isoval(2). The last color in color refers to
values greater than the last value in isoval.
Lower left corner of the area where the color bar is plotted.
Upper right corner of the area where the color bar is plotted.
Difference of values between isolines. If this value is greater then 0 the
values for isolines and the respective colors are created automatically
without need to specify the single values in arrays isoval and color.
(Default 0)
Factor for the values that are written to the color bar legend. This enables you, e.g., to give water level results in mm (faccol = 1000).
(Default 1)
Decimals after the decimal point for the values written to the color bar
legend. Use the value -1 to not write the decimal point. (Default -1)
Text for the description of the color bar. This text is written above the
color bar.
Section $arrow
The next parameters deal with the reference arrow that is plotted in a legend. The arrow
regards the plots where the velocity or the transport is plotted.
Lower left corner of the area where the reference arrow is plotted.
Upper right corner of the area where the reference arrow is plotted.
Factor for the value that are written to the arrow legend for the velocity
and transport. This enables you, e.g., to give velocities in mm/s (facvel
= 1000). (Default 1)
Decimals after the decimal point for the values written to the arrow legend (velocity and transport). Use the value -1 to not write the decimal
point. (Default 2)
Text for the description of the arrow legend (velocity and transport).
This text is written above the arrow legend.
Length of arrow in legend (in velocity or transport units). If not given
the arrow length will be computed automatically. (Default 0)
Section $legend
In this section annotations in the plots can be given. The section consists of a series of lines
that must contain the following information:
The first two values (x, y) give the position of the annotation. The third value gives the
font size in points for the text to be written and the last entry on the line is the text for the
annotation. A small example of an annotation would be:
-3000. -2000. 12
3200. 2500. 12
’Adriatic Sea’
’Lido Inlet’
where the two annotations are written with a font size of 12 points. The text Adriatic Sea
is written at (-3000,-2000), so that the lower left corner of the A of Adriatic is at the specified
Section $name
Directory specification This parameters define directories for various input and output
Directory where basin file BAS resides. (Default .)
Directory where output files are written. (Default .)
Directory for temporary files. (Default .)
Default directory for other files. (Default .)
Chapter 6
The Water Quality Module
by Donata Melaku Canu, Georg Umgiesser, Cosimo Solidoro
The coupling between EUTRO and FEM constitute a structure which is meant to be a
generic water quality for full eutrophication dynamics. The Water Quality model is described fully in Umgiesser et al. (2003).
General Description
The water quality model has been derived from the EUTRO module of WASP (released
by the U.S. Environmental Protection Agency (EPA) (Ambrose et al., 1993) and modified.
It simulates the evolution of nine state variables in the water column and sediment bed,
including dissolved oxygen (DO), carbonaceous biochemical oxygen demand (CBOD),
phytoplankton carbon and chlorophyll a (PHY), ammonia (NH3), nitrate (NOX), organic
nitrogen (ON), organic phosphorus (OP), orthophosphate (OPO4) and zooplankton (ZOO).
The interacting nine state variables can be considered as four interacting systems: the carbon cycle, the phosphorous cycle, the nitrogen cycle and the dissolved oxygen balance
(Fig. ??). Different levels of complexity can be selected by switching the eight variables
on and off, in order to address the specific topics.
The evolution of phytoplankton concentration (Reaction 1, Table 6.1) is described by the
anabolic and the catabolic terms, plus a grazing term related to zooplankton concentration
(Reaction 10, 11 and 12, Table 6.2), which however is treated as a constant in the original
version. The anabolic term (Reaction 10, Table 6.2) is related to light intensity, temperature
and concentration of nutrients in water, while the catabolic term (Reaction 11, Table 6.2)
depends on temperature.
Phytoplankton growth is described by combining a maximum growth rate under optimal
conditions, and a number of dimensionless factors, each ranging from 0 to 1, and each one
referring to a specific environmental factor (nutrient, light availability), which reduces the
phytoplanktonic growth insofar as environmental conditions are at sub-optimal levels. Phytoplankton stochiometry is fixed at the user-specified ratio, so that no luxury uptake mechanisms are considered, and the uptake of nutrients is directly linked to the phytoplankton
growth, and described by the same one-step kinetic law. More specifically, the influence of
inorganic phosphorous and nitrogen availability on phytoplankton growth/nutrients uptake
is simulated by means of Michealis-Menten-Monod kinetics (Reactions 42 and 43, Table
6.2). Phytoplankton uptakes nitrogen both in the forms of ammonia and nitrate, but ammonia is assimilated preferentially, as indicated in the ammonia preference relation (Reaction
38, Table 6.2). The influence of temperature is given by an exponential relation (Reaction
13, Table 6.2), while the functional forms for the limitation due to sub-optimal light con26
dition can be chosen between three alternative options, namely the formulation proposed
by Di Toro et al. (1971) and the one proposed by Smith (1980) (Di Toro and Smith subroutines, Reaction 44, Table 6.2) and the Steele formulation (Steele, 1962) that can use
hourly light input values. The choice between different available functional forms (Ditoro,
Smith, and Steele) is made by setting the index LGHTSW equal to 1, 2 or 3. The new version
is therefore able to simulate diurnal variations depending on light intensity, such as night
anoxia due to phytoplankton respiration during nighttime.
Finally, the two frequently used models for combining maximum growth and limiting factors, the multiplicative and the minimum (or Liebig’s) model, are both implemented, and
the user can choose which one to adopt (Reaction 41, Table 6.2).
Nitrogen and phosphorous are then returned to the organic compartment (ON, OP) via phytoplankton and zooplankton respiration and death. After mineralization, the organic form
is again converted into the dissolved inorganic form available for phytoplankton growth.
The DO mass balance is influenced by almost all of the processes going on in the system.
The reaeration process acts to restore the thermodynamic equilibrium level, the saturation
value, while respirations activities and mineralization of particulated and dissolved organic
matter consume DO and, of course, photosynthetic activity produces it. Other terms included in the DO mass balance are the ones referring to redox reactions such as nitrification
and denitrification. The reaeration rate is computed from the model in agreement with either the flow-induced rate or the wind-induced rate, whichever is larger. The wind-induced
reaeration rate is determined as a function of wind speed, water and air temperature, in
agreement with O’Connor (1983), while the flow-induced reaeration is based on the Covar method (Covar, 1976), i.e., it is calculated as a function of current velocity, depth and
The dynamic of a generic herbivorous zooplankton compartment (ZOO), meant to be representative of the pool of all the herbivorous zooplankton species, is followed and accordingly
the subroutines relative to phytoplankton, organic matter, nutrients, and dissolved oxygen,
which were influenced by such a modification.
The grazing has been described by means of a type II functional relationship, as it is usually done for aquatic ecosystems. However, the possibility to select a type III relationship,
as well as to maintain the original parameterisation of constant zooplankton, has been included.
The zooplankton assimilates the ingested phytoplankton with an efficiency EFF, and the
fraction not assimilated, ecologically representative of faecal pellets and sloppy feeding, is
transferred to the organic matter compartments (dotted lines Fig. ??). Finally, zooplankton
mortality is described by a first order kinetics. The code has been written by adopting
the standard WASP nomenclature system, and the choice between the different available
functional forms is performed by setting the index IGRAZ. A choice of 0 (the default value)
corresponds to the original EUTRO version, giving the user the ability to chose easily
between the extended version or revert to the original one.
The coupling
Mathematical models usually describe the coupling between ecological and physical process by suitable implementation of an advection/diffusion equation for a generic tracer,
∂ Θi
∂ Θi
s ∂ Θi
+U · ∇ Θi − wi
= Kh ∇H Θi +
+ F (Θ , T, I , ..)
where U is the (average components of the) velocity, the Θi are the tracers which compose
the entire vector of the biological state variable Θ and F is a source term. T and I indicate, respectively, water temperature and Irradiance level, while wsi represent the downward
= Q(S)
General Reactor Equation
Q(ZOO) = GZ − DZ
Q(NH3) = Nalg1 + ON1 − Nalg2 − N1
Q(NOX) = N1 − NOalg − NIT 1
Q(ON) = ONalg − ON1
Phytoplankton PHY [mg C/L]
Zooplankton ZOO [mg C/L]
Ammonia NH3 [mg N/L]
Nitrate NOX [mg N/L]
Organic Nitrogen ON [mg N/L]
Q(OPO4) = OPalg1 + OP1 − OPalg2
Inorganic Phosphorous OPO4
[mg P/L]
Q(OP) = OPalg3 − OP1
Organic Phosphorous OP [mg P/L]
Q(CBOD) = C1 − OX − NIT 2
Carbonaceous Biological Oxygen
Demand CBOD [mg O2 /L]
Q(DO) = DO1 + DO2 + DO3
− DO4 − N2 − OX − SOD
Dissolved Oxygen DO [mg O2 /L]
Table 6.1: Mass balances
flux rates (sinking velocity) for the tracer Θi , and Kh and Kv are the eddy coefficients for
horizontal and vertical turbulent diffusion.
The term F includes the contributions of the biological/biogeochemical activities, and the
whole biological state vector Θ is explicitly considered in the last term of equation 6.1,
without a spatial operator. As far as the biologically induced variations are regarded, the
fate of each tracer in every location x, y, z is tightly coupled to other tracers in the same
location, but is not directly influenced by processes going on elsewhere.
Therefore, in this approximation the global temporal variation of any tracer (state variable,
conservative or not) can be split into the sum of two independent contributions:
∂ Θi
∂ Θi 
∂ Θi 
∂t  phys
∂t biol
and it might be convenient, in writing a computer code, to devote independent modules
to computation of each of them. Indeed, most of the modern water quality programs do
have, at least conceptually, a modular structure. In this way the same code can be used for
simulating different situations: by switching off the module referring to the reactor term
the transport of a purely passive tracer is reproduced, while a 0D, close and uniformly
stirred biological system is simulated if the module referring to the physical term is not
included. Finally, the inclusion of both modules gives the evolution of tracers subjected to
both physical and biogeochemical transformation, in a representation that, depending upon
the parameterisation of the physical module, can be 1, 2 or 3 dimensional.
The whole water quality module is contained in a file weutro.f and the call to EUTRO is
made through a subroutine call that is done from the main program through an appropriate
interface. There is a clean division between the hydrodynamic motor, parameters used
by the model and the resolution of the differential equations and the ecological model as
evidenced by the overall structure of the modules.
It is the responsibility of the main module to implement the time loop administration, the
advective and diffusive transport of the state variables, both in the horizontal and vertical
direction and the application of the boundary conditions.
The typical use of the new EUTRO module is as follows: the main program first sets all
parameters needed in EUTRO through the call to EUTRO_INI. These parameters are the
kinetic constants of the reactions that are described in EUTRO and are considered constant
phytoplankton growth
phytoplankton death
grazing rate coefficient
GP1 = Lnut ∗ Llight ∗ K1C ∗ K1T (T −T0 )
phytoplankton growth rate with nutrient and light limitation
DP1 = RES + K1D
Zine f f = (1 − EFF) ∗ GRZ
Zsink = Zine f f + DZ
Nalg1 = NC ∗ DPP ∗ (1 − FON)
Nalg2 = PN ∗ NC ∗ GPP
NOalg = (1. − PN) ∗ NC ∗ GPP
ONalg = NC ∗ (DPP ∗ FON + Zsink )
source of organic nitrogen from
phytoplankton and zooplankton
ON1 = KNCmin ∗ KNTmin 0 ∗ ON
(T −T )
OP1 = KPCmin ∗ KPTmin 0 ∗ OP
mineralization of ON
mineralization of OP
OPalg1 = PC ∗ DPP ∗ (1. − FOP)
source of inorganic phosphorous
from algal death
OPalg2 = PC ∗ GPP
sink of inorganic phosphorous for
algal growth
OPalg3 = PC ∗ (DPP ∗ FOP + Zsink )
source of organic phosphorous
from phytoplankton and zooplankton death
KDC ∗ KDT (T −T0 )
oxidation of CBOD
(T −T0 )
N1 = KCnit ∗ KTnit
∗ KnitDO
∗ NH3
(T −T )
KCdenit KTdenit 0
∗ NOX ∗ Kdenit
(T −T )
phytoplankton respiration and
death rate
zooplankton growth rate
zooplankton death rate
grazing inefficiency on phytoplankton
sink of zooplankton
source of ammonia from algal death
sink of ammonia for algal growth
sink of nitrate for algal growth
Table 6.2: Functional Expression Description
C1 = OC ∗ (K1D ∗ PHY + Zsink )
NIT 2 = 45 ∗ 14
∗ NIT 1
DO1 = KA ∗ (Osat − DO)
source of CBOD from phytoplankton and zooplankton death
sink of CBOD due to denitrification
reareation term
DO2 = PN ∗ GP1 ∗ PHY ∗ OC
dissolved oxygen produced by phytoplankton using NH3
DO3 = (1 − PN) ∗ GP1 ∗ PHY
+ 1.5 ∗ NC
∗ 32 ∗ 12
growth of phytoplankton using NOX
DO4 = OC ∗ RES
∗ N1
N2 = 14
respiration term
oxygen consumption due to nitrification
ammonia preference
RES = K1RC ∗ K1RT (T −T0 )
(T −T0 )
algal respiration
sediment oxygen demand
Lnut = min(X1, X2) , mult(X1, X2)
minimum or multiplicative nutrient
limitation for phytoplankton growth
PN =
X1 =
X2 =
Llight =
(−KE∗H) )
∗ e−(KE∗H) ∗ e(1− Is ∗e
KA = F(Wind,Vel, T, Tair , H)
nitrogen limitation for phytoplankton growth
phosphorous limitation for phytoplankton growth
light limitation for phytoplankton
re-areation coefficient
Table 6.2: (continued) Functional Expression Description
K1D = 0.12 day−1
KGRZ = 1.2 day−1
phytoplankton death rate constant
grazing rate constant
KPZ = 0.5 mg C/L
half saturation constant for phytoplankton in grazing
KDZ = 0.168 day−1
K1C = 2.88 day−1
zooplankton death rate
phytoplankton growth rate constant
K1T = 1.068
phytoplankton growth rate temperature constant
KN = 0.05 mg N/L
nitrogen half saturation constant for phytoplankton growth
KP = 0.01 mg P/L
phosphorous half saturation constant for phytoplankton growth
KCnit = 0.05 day−1
KTnit = 1.08
Knit = 2.0 mg O2 /L
KCdenit = 0.09 day−1
KTdenit = 1.045
Kdenit = 0.1 mg O2 /L
KNCmin = 0.075 day−1
nitrification rate constant
nitrification rate temperature constant
half saturation constant for nitrification
denitrification rate constant
denitrification rate temperature constant
half saturation constant for denitrification
mineralization of dissolved ON rate constant
KNTmin = 1.08
mineralization of dissolved ON rate temperature
oxidation of CBOD rate constant
oxidation of CBOD rate temperature constant
N/C ratio
C P/C ratio
O/C ratio
grazing efficiency
fraction of ON from algal death
fraction of OP from algal death
fraction of dissolved inorganic phosphorous
mineralization of dissolved OP rate constant
KDC = 0.18 day−1
KDT = 1.047
NC = 0.115 mg N/mg C
PC = 0.025 mg P/mg
OC = 32/12 mg O2 /mg C
EFF = 0.5
FON = 0.5
FOP = 0.5
FOPO4 = 0.9
KPCmin = 0.0004 day−1
KPTmin = 1.08
KBOD = 0.5 mg O2 /L
K1RC = 0.096 day−1
K1RT = 1.068
mineralization of dissolved OP rate temperature
CBOD half saturation constant for oxidation
algal respiration rate constant
algal respiration rate temperature constant
Is = 1200000 lux/day
optimal value of light intensity for phytoplankton
KE = 1.0 m−1
light extinction coefficient
SOD1 = 2.0 mg O2 /L
day−1 m
sediment oxygen demand rate constant
SODT = 1.08
T0 = 20 o C
sediment oxygen demand temperature constant
optimal temperature value
Table 6.3: Parameters
[o C]
[o C]
[m3 ]
water temperature
air temperature
DO concentration value at saturation
incident light intensity at the surface
current speed
wind speed
Table 6.4: Variables
for a site. They have to be set therefore only once at the beginning of the simulation. Once
set, these parameters are available to the EUTRO module as global parameters.
For every box in the discretized domain (horizontal and vertical), and for every time step,
the main program calls the subroutine EUTRO0D. Inside EUTRO0D the differential equations
that describe the bio-chemical reactions are solved with a simple Euler scheme.
The values passed into EUTRO0D can be roughly divided into 4 groups. The first group is
made out of the aforementioned constants that represent the kinetic constants and other parameters that do not vary in time and space. The second group represents the state variables
that are actually modified by EUTRO0D through the bio-chemical reactions. These variables
are transported and diffused by the main routine and are just passed into EUTRO0D for the
description of the processes. After the call no memory remains in EUTRO0D of these state
variables. They must therefore be stored away by the main routine to be used in the next
time step again. The third and fourth groups of values have to do with the forcing terms.
They have been divided in order to account for the different nature of the forcing terms.
The third group consists of the hydrodynamic forcing terms that are directly computed by
the hydrodynamic model and parameters related to the box discretization. They consist of
water temperature, salinity, current velocity, and depth and type of the box. Here the type
identifies the position of the box (surface, water column, sediment), which is needed for
some of the forcings to be applied. These variables are passed directly into EUTRO through
a parameter list. The last group contains other forcing terms that are not directly related
to the hydrodynamic model. These consist of the meteorological forcings (wind speed,
air temperature, ice cover), light climate (surface light intensity, day length) and sediment
fluxes. These parameters are set through a number of commodity functions that are called
by the main routine. The reason why the last two parameter groups are handled differently
from each other has also to do with the fact that the third group is highly variable in time
and space. Variables like current velocity change with every time step and are normally
different from element to element. The fourth group is very often only slowly variable in
time (light, wind) and can very often be set constant in space. Therefore these values can
be set at larger intervals, and do not have to be changed when looping over all the elements
in the domain.
The overall flow of information during one time step is the following: First the hydrodynamic model resolves the momentum and continuity equation to update the current velocities and water levels. After that the physical (temperature and salinity) and bio-chemical
scalars are advected and diffused. Once this advection step has been handled the new loadings and forcing terms are set-up and then EUTRO0D is called for the bio-chemical reactions.
Note that the operator splitting technique, which decouples the advective and diffusive
transport from the source term, allows for different time steps of the two processes for a
more efficient use of the computer resources.
Default values of the water quality parameters are already set in the code. Owner specific
parameters for the water quality model should be written in the subroutine param_user.
Light limitation
Light attenuation formula by Steele and Di Toro
The well known light limitation function proposed by Steele is given as:
I 1− II
e o
where I is the light intensity and Io the optimal light intensity. P is the limiting function
and takes values between 0 and 1.
In this form, P is a function of depth z and of time t (P = P(z,t)) since the light intensity
depends on depth and time (I = I(z,t)). The depth dependence of I can be written as
I(z) = Ii e−kz
where Ii is the incident light intensity on the surface (still dependent on time) and k is an
extinction coefficient. Inserting (6.4) into (6.3) gives the equation
Ii −kz
Ii −kz 1− IIi e−kz
e e o
= e e−kz e Io e .
P(z,t) =
We now compute the average of P over the water column. This gives
P(t) =
P(z,t)dz =
e IIoi Z
Ii −kz
e−kz e− Io e
With the substitution x = e−kz and therefore dx/dz = −ke−kz = −kx the integral can be
transformed into
P(t) =
e IIoi Z
xe− Io x
e IIi Z H − Ii x
dx = − o
e Io dx.
kH 0
Solving this integral gives finally
− Ioi x
− Ioi e−kz
− Ioi e−kH
− Ioi
P(t) =
This is the depth integrated form of the Steele limiting function for instantaneous light.
If we want to work with the average light over one day, then equation (6.6) can be easily
averaged over one day. If T is the averaging period (one day), f the fraction of the day with
daylight and Ii is approximated with a step function, 0 at night and Ia during daytime, then
we can write
e h − IIa e−kH
e h − IIa e−kH
− e− Io =
− e− Io .
f e o
P(t)dt = f T
e o
Equation (6.7) represents the limiting function given by Di Toro and used in the EUTRO
program of WASP. Therefore, equation (6.7) is just the Steele limiting function, but using
Ia , the average incident light intensity over daylight hours instead the instantaneous incident
light intensity Ii used in the original Steele formula (6.6).
Light attenuation formula by Smith
EUTRO uses also a light limitation formula by Smith. In the manual this is given as
− Isi e−kH
− Isi
P(t) =
where now IS is the optimal light intensity which is not a constant but is continuously
adjourned by the program. Again, Ii is the instantaneous light intensity at the surface.
Ii is given as
Ii =
sin( )
between 0 and f (daylight) and Ii = 0 otherwise. Averaging (6.9) over one whole day (0–1)
Z f
Ii =
sin( )dt = It
0 2f
and averaging only over daylight hours gives
Z f
)dt = = Ia .
This shows that It in equation (6.9) is the average light intensity at the surface over one
whole day (ITOT in Eutro) and that Ia = It / f is the average light intensity at the surface
over daylight hours. This must be taken into account when the Di Toro formulation is used.
Note that in EUTRO the actual formula used is Ia = 0.9It / f where the parameter 0.9 probably accounts for some losses during the integration. The corresponding variables in EUTRO are FDAY for f and IAV for Ia .
This section describes the interpolation of data for the initialisation of the model.
To create a file with initial conditions the program laplap can be used. The program is
called as laplap < namefile.dat. This makes a laplacian interpolation of specified data
contained in the namefile.dat. This data file should have the first line empty and shold
contain two colums containing, respectively, node number and data values for the node.
It generates two files, laplap.nos and laplap.dat. The first one can be used to check
if the procedure has been conducted well, creating a map with the plotting procedure (see
Postprocessing section). The .dat file name should be given in the section $name of the
.str file to initialize the model.
You can create initialisation files for temperature, salinity, wind field and biological variables. If you want to initialise the biological model with biological data you should create
a single data file merging the 9 data files (one for each variable) using the inputmerge.f
Post processing
This section shows how to generate derivate variables.
The post processing routines elaborate the water quality outputs to generate derivate variables. They allow to generate variables such as averages, (both, in time and in space), sum
differences, and water quality variables such us Vismara, TRIX and BOD5.
The routines and their usage are the following:
nosmaniav.f It generates a file containing, for each node of the spatial domani, average, minimum and maximum values of the specified variable of the whole simulation.
nosmaniqual.f It generates a water quality file from the elaboration of the state variable. It computes, for each node, and at each time step a water quality index that can be
chosen between two suggested indexes: Vismara QualityV and TRIX a well known quality
index applied to the water quality definition at coastal seas and estuaries.
70-90 or 110-120
50-70 or 120-130
<30 or >130
Table 6.5: Classification of Water Quality Indices
These indices can be computed using the definitions in Table 6.5 and the following equations:
QualityV = class(O2sat) + class(BOD5) + class(NH3)
auxt1 = (phyto/30.)*1000
auxt2 = (nh3+nox)*1000
auxt3 = (opo4+op)*1000
TRIX = log10(auxt1*o2satp*auxt2*auxt3+1.5)/1.2
generates a file of total inorganic Nitrogen as sum of NH3 and NOx
nosdif.f computer for the chosen state variable, the difference between the values at
two times step.
computer the difference between the variable outputs of two simulations
computes the BOD5 values from the CBOD outputs as:
bod5 = cbod*(1. - exp( -5. * par1 ))
+ (64./14.) * nh3 * (1. - exp( -5. * par2 ))
To run one of the postprocessing routine write the name of the routine and enter.
The Sediment Module
The sediment buffer action on the biogeochemical cycles could be very important, especially in the shallow water basins and during the storm surge events.
The routine wsedim (introduced in April 2004) aims to address the resuspention/sinking
dynamics of nitrogen and phosphorous. This routine can be switched on and off as needed
by the user, setting the bsedim parameter true or false in the bio3d routine. It is called
after the the eutrophication subroutine.
It allows to follow the dynamics of two additional variables, OPsed and ONsed that simulate the evolution of Nitrogen and Phosphorous detritus in sediment that are not subjected
to advection-diffusion processes.
These two variables interact with the Nitrogen and Phosphorous cycle as decribed by the
equations in Table 6.6. When the wsedim subroutine is switched on, OP, ON, NH3 and
OPO4 are updated at each time step in agreement with those equations.
The resuspension is a linear function of the water velocity calculated by the hydrodynamic
model at each box, as written in Table 6.7. The amount of the sinking nutrients depends
on specific prossess parameters, as given in Table 6.7, and on the depth of the underlying
= Q(S)sed
General Reactor Equation
Q(NH3)sed = NH3res
Q(ON)sed = (ONres − ONsink )
Ammonia NH3 [mg N/L]
Organic Nitrogen ON [mg N/L]
Q(OPO4)sed = OPO4res
Inorganic Phosphorous
OPO4 [mg P/L]
Q(OP)sed = (OPres − OPsink )
Organic Phosphorous OP [mg P/L]
Q(ONsed )
= ONsink − ONres
− NH3res
Sediment Organic Nitrogen
ONsed [mg N/L]
Q(OPsed )
= OPsink − OPres
− OPO4res
Sediment Organic Phosphorous
OPsed [mg N/L]
Table 6.6: Sediment Mass balances
= Volsed ∗ KNCsed
∗ KNT (T −T0 ) ∗ ONsed
Mineralization of organic nitrogen in
ONres = Volsed ∗ KNres ∗ Fvel ∗ ONsed
Resuspention of organic nitrogen in
Sink of organic nitrogen from the water column
= Volsed ∗ KPCsed
∗ KPT (T −T0 ) ∗ OPsed
Mineralization of organic phosphorous in sediment
OPres = Volsed ∗ KPres ∗ Fvel ∗ OPsed
Resuspention of organic phosphorous in sediment
P ))
= Vol ∗ (1−exp(−dt/τ
Sink of organic phosphorous from
the water column
τN = H/ws
Time scale for sinking processes of
organic N
τP = H/ws
Time scale for sinking processes of
organic P
N ))
= Vol ∗ (1−exp(−dt/τ
Table 6.7: Sediment functional expressions
KNCsed = 0.075
KNT = 1.08
KPCsed = 0.22
KPT = 1.08
KNres = 0.1
KPres = 0.1
FPON = 0.5
FPOP = 0.5
Fvel = 1
ws = 10 m/day
T0 = 20 o C
Mineralization of sediment ON rate constant
Mineralization of sediment ON rate temperature constant
Mineralization of sediment OP rate constant
Mineralization of sediment OP rate temperature constant
Fraction of sediment depth resuspended/day
Fraction of sediment depth resuspended/day
Fraction of particulate organic N
Fraction of particulate organic P
Velocity coefficient
Sinking velocity
Optimal temperature value
Table 6.8: Sediment parameters
[m3 ]
[m3 ]
sediment volume
total depth of water column
time step
Table 6.9: Sediment variables
Parameters for the Water Quality Module
Compulsory bio parameters These parameters are compulsory parameters that define
if the water quality module is run and what kind of output is written.
Flag if the computation on the temperature is done. The model writes
at each time step the state variable values in the the .bio output file
Flag if the average, minimum, maximum file of variables bio, salinity,
temperature is done. if itsmed=1 the model writes .sav, .tav output
files of the corresponding variables.
Boundary conditions Boundary conditions have to be given in a file in every section
File name that contains boundary conditions for concentration of the
water quality state variables. The format is the same as for the file
boundn. The unit of the values given in the second and following column (9 data columns for EUTRO) must the ones of the variable.
Initial conditions Initialization of variables are done by file. The files can be created by
the progam laplap. They have to be given in section $name.
File with concentration values of water quality variable to be used for
the initialization.
Files with salinity concentration values [psu] and Temperature values
[deg C] for the initialization.
Files with tracer concentration values [for the initialization.
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