The Ministry of National Infrastructures
GEOLOGICAL SURVEY OF ISRAEL
Formulating a Dynamic Limnological Model for the Dead Sea
Selection of a computer code and preliminary simulations
Gavrieli I., Dvorkin Y., Lyakhovsky V., Lensky N. and Gazit-Yaari N.
Jerusalem, March 2003
Report GSI/3/2003
The Ministry of National Infrastructures
GEOLOGICAL SURVEY OF ISRAEL
Formulating a Dynamic Limnological Model for the Dead Sea
Selection of a computer code and preliminary simulations
Gavrieli I., Dvorkin Y., Lyakhovsky V., Lensky N. and Gazit-Yaari N.
Jerusalem, March 2003
Report GSI/3/2003
Abstract
The Princeton Oceanographic Model (POM) was chosen as the appropriate code for the
formulation of the dynamic limnological model for the Dead Sea. The model, when
completed, will simulate seawater mixing in the Dead Sea, as expected once the "Peace
Conduit" is constructed. The POM code was developed in the late 1970's by Blumberg
and Mellor (1987), with subsequent contributions from other people. It has been used
for modeling of open oceans, coastal regions, estuaries and lakes. It is a public domain
code, which has numerous available versions that can be combined for the purpose of
simulating seawater inflow to Dead Sea. Additional advantages include the fact that
Princeton University provides free on line consulting services, up to date information
and supports a discussion group. These have already been used by us and shown to be
valuable resources.
The DTM map of the Dead Sea basin was used as the basis for the construction of the
bathymetric input file. Most simulations were run using seawater for which the physical
characteristics are well established and therefore allow verifying the validity of the
changes and modification introduced by us to the code. Initial runs indicated that the
more common σ-grid version of POM introduces pressure gradient errors and results in
stability loss and unrealistically large velocities. These problems were eliminated when
the Z-grid version was used. The next runs were stability tests on a stratified lake with
no external forcing. These showed that the molecular diffusion coefficient, which is
based on the smoothing role of the diffusion mechanism in ocean-scale, are too large
and leads to the rapid erosion of stratification. These tests demonstrated that for the
purpose of modeling the expected stratification in the Dead Sea, the molecular diffusion
coefficient in the code should have the same order of magnitude as real ion diffusion
coefficients, i.e., 4 orders of magnitude smaller than that used in the POM.
Following the above modification, constant wind forcing was applied on a box filled
with seawater and Dead Sea brine in order to determine the slope of the water and
observe the development of the Ekman layer. These were compared with known
analytical results and excellent agreements were found. The impact of wind forcing on
the stratified lake (still filled with seawater) was then simulated with real wind data. The
layered structure was preserved all over the year resulting in minor change in the
thermo- and halo- cline depth after a whole year of real wind.
Finally, thermal forcing was introduced to the code and simulations were run on both
seawater and Dead Sea brine, the latter using the equation of state of the Dead Sea.
Other parameters were similar and are based on known or assumed seawater
parameters. This also required incorporating a module to calculate the change in water
mass and the impact on salinity (due to evaporation). Simulations on seawater with
constant heat flux and no wind results in gradual formation of a stable thermocline,
while loss of fresh water due to evaporation leads to formation of an unstable halocline,
as expected. Introducing real, one year, meteorological data resulted in the development
of a stabilizing thermocline and destabilizing halocline during the spring and summer
months. During the following fall months, cooling sets in and both thermo- and haloclines deepen significantly. Simulations done with Dead Sea brine resulted with similar
general trends, but the maximum temperature at the end of the summer is slightly higher
and absolute increase in salinity is much higher. The difference in salinity increase is
explained by the effect of the relative water content in the brine, which is smaller than
in seawater, and therefore the impact of evaporation is greater. This also explains the
deeper stratification in the Dead Sea brine as compared with seawater at the end of the
year because the destabilizing effect of the halocline in the brine is greater.
The preliminary simulations runs using the POM indicate that the POM is a good
platform for the development of the dynamic limnological model for the Dead Sea.
They also emphasize the importance of step by step examination and modification of
the code and the need to determine the appropriate and unique parameters for the Dead
Sea and Dead Sea – seawater mixtures. Future work will include re-evaluation of the
Boussinesq approximation, introduction of surface and groundwater inflows.
parameterization of the code against known data, introduction to the code of the new
equation of state developed for Dead Sea –seawater mixtures, saturation and mineral
precipitation calculations, true light penetration coefficients etc. These tasks will be the
next stage in the development of the model.
Table of Content
Introduction .......................................................................................................................1
Model Modification – Principals.......................................................................................2
The Princeton Oceanographic Model (POM)....................................................................3
Testing + Initial Modification of POM to the Dead Sea system .......................................5
1... Model geometry and stability ..........................................................................5
2... Wind forcing....................................................................................................7
3... Impact of wind forcing on stratification ..........................................................8
4... Thermal forcing ...............................................................................................9
5... Water mass balance (evaporation).................................................................12
6... Heat- and wind-driven stratification build up................................................13
7... A new equation of state for Dead Sea- seawater mixtures ............................15
Gypsum precipitation experiments..................................................................................17
Future Work.....................................................................................................................19
References .......................................................................................................................21
Appendix A: Publications related to lake studies............................................................22
Appendix B: The Basics of POM ....................................................................................25
List of Figures:
Fig. 1. Number of POM users over years.
Fig. 2. A digital terrain map (DTM) of the Dead Sea
Fig. 3. Bathymetry and horizontal grid of the Dead Sea.
Fig. 4. The vertical grid used in the model.
Fig. 5. Flow pattern and surface elevation generated due to
pressure gradient errors in σ-grid POM.
Fig. 6. Comparison between z- and s-grids.
Fig 7. Maximal surface velocity with zero initial velocity and no forcing.
Fig. 8. Profiles obtained with the POM default molecular diffusion
(D = 10-5) with no external forcing
Fig. 9. Model simulation and analytical solution for wind-driven surface elevation
Fig. 10. Slopes of the wind-driven surface elevation of seawater vs Dead Sea brine.
Fig. 11. Model simulation vs. analytical solution for Ekman layer.
Fig. 12. Surface elevation and flow velocities for seawater and
constant wind simulation.
Fig. 13. N-S temperature cross-section for seawater and constant wind simulation.
Fig. 14. Surface elevation with seawater and real wind simulation
Fig. 15. N-S temperature cross-section with initial seawater
stratification and real wind simulation
Fig. 16. Initial and vertical temperature and salinity profiles for one year
simulation with real wind and initial seawater stratification
Fig. 17. Temperature and salinity distribution after 1 year simulation with seawater,
constant heat flux of 400 W/m2 and zero wind.
Fig. 18. Monthly vertical temperature and salinity distribution during 1 year
simulation with seawater, constant heat flux and zero wind.
Fig. 19. Monthly vertical temperature and salinity distribution during 1 year
simulation with seawater and real wind.
Fig. 20. Monthly vertical temperature and salinity distribution during 1 year
simulation with Dead Sea brine and real wind.
Fig. 21. Calculated densities; Pitzer’s vs. the new equation of state.
Fig. 23. The σ coordinate system
Introduction
The present report summarizes the opening phase of the formulation of a dynamic
limnological model for the Dead Sea. The necessity for such a model rose following
renewed talks between Israel and Jordan about the construction of the "Peace Conduit"
that will convey seawater from the Red Sea to the Dead Sea. The proclaimed aims of
the "Peace Conduit", as stated jointly by the two nations during the Johannesburg 2002
World Summit, are to save the Dead Sea and its environment, and to utilize the 400
meters elevation difference between the Seas to desalinize seawater. Yet, the mixing of
seawater and/or reject brine from the desalinization plant with Dead Sea brine will
result in changes in the limnology of the lake that over the long run will alter its
characteristics and uniqueness. The dynamic limnological model should provide insight
into the long-term evolution of the Dead Sea with continuous seawater inflow and be
the basis for economical and environmental assessment of the proposed project. The
opening phase of the formulation of the model was jointly financed by the Ministry of
Regional Cooperation and the Geological Survey of Israel - Ministry of National
Infrastructure. .
An interim report, titled "The Impact of the Proposed “Peace Conduit” on the Dead SeaEvaluation of Current Knowledge on Dead Sea – Seawater Mixing" (Gavrieli et al.,
2002) was submitted in 2002. As its title suggests, the report summarizes the expected
impact of the "Peace Conduit" on the Dead Sea based on of available data and studies
conducted in the past. The report highlighted the sound understanding of the present
limnology of the Dead Sea but pointed out the difficulties in providing insight into the
evolution of the lake once seawater mixes with the brine. Thus, based on current
knowledge it is not possible at this stage to quantify expected changes in the Dead Sea
and therefore their environmental, industrial and economical outcome cannot be
assessed. The dynamic limnological model should provide some of the answers and
tools to deal with these questions.
The opening phase of the model formulation lasted about a year. The major aim of this
phase was to identify the oceanographic/limnological code that would be most suitable
for modeling the mixing of seawater in the Dead Sea. Once a model was chosen, first
2
modifications were made in order to establish whether that code can be adjusted to the
Dead Sea system. Contemporaneous with the work on the model, the opening phase
included the identification of information gaps relevant for the development of the
model and to the understanding of the future of the Dead Sea. Some of these
information gaps were pointed out in the interim report. Some initial research into these
topics has begun, though no major funding was allocated for this purpose.
Model Modification - Principals
The modification of any model for a specific lake is a time consuming, difficult and
expensive work (for instance, the development of the model for Lake Kinneret has
began several years ago with an estimated budget of 7,000,000 NIS). Such adaptations
make use of the large physical, chemical and environmental data base and common
knowledge already available for similar water bodies. However, because of its unique
brine, no similar data, experience and understanding exist for the Dead Sea. In addition,
to the best of our knowledge, no code was adopted to model the mixing between two
such very different fluids. These differences impact on the physical characteristics of
the mixed product (e.g. equation of state), its chemistry (e.g. precipitation of minerals
due to outsalting) and the water-air interaction (e.g. rate of evaporation). Therefore,
because of the uniqueness of the problem at hand, the code that we choose to modify
and adopt for the Dead Sea must be clearly written, well documented, have proven
capabilities, and open for major modification, if proven to be necessary.
As outlined below, the code adopted for the Dead Sea is the Princeton Oceanographic
Model (POM). Because of the numerous unknowns related to the Dead Sea brine,
modifications must first be validated using seawater, i.e. running the code as if the Dead
Sea basin is filled with seawater instead of Dead Sea brine. In addition, empirical
parameters in the code, which are well established for seawater, must be re-evaluated
for their validity for the Dead Sea brine. During the opening phase some sensitivity tests
were carried out with regard to the various measured and empirical parameters. The
purpose was to define which of these needs to be determined most precisely for the
Dead Sea and the seawater-Dead Sea mixing products. Preliminary, short runs with true
3
meteorological data for the Dead Sea basin were also conducted. These data were
provided by Dr. I. Gertman from the Israel Limnologic and Oceanographic Research.
It should be emphasized that the final aim of the dynamic limnological model is to
provide a long-term (decades) forecast rather than exact, short-term calculation for the
behavior of the lake. The major questions are the thickness of stratification, composition
of mixed layer, rate of change in salinity/density of the surface water, rate of
evaporation and the predicted duration of stratification before overturn. The model
stability and reliability needs to be verified with respect to these questions and time
scale. Calibration and model-tuning are possible only with respect to the conditions of
the Dead Sea as they are known today and based on available data. Once inflow of
seawater is included in the model, there is no data to calibrate the model with. This
situation requires that numerous sensitivity tests be carried out and the shaping
parameters be identified. Once recognized, these parameters can be studied in depth to
provide the best possible forecast for the evolution of the Dead Sea.
The Princeton Oceanographic Model (POM)
Over the last two decades several criteria have emerged by which oceanographers
classify their models. These include: (1) Geography i.e. models designed for the world
oceans or lakes or both. (2) the physical processes described: models can be
hydrodynamic, thermodynamic or both; (3) the condition one puts on the surface (4) the
way the vertical degree of freedom is handled, including modal vertical decomposition
(5) the presence of density variations. Detailed review of the existing Ocean Models
could be found at Internet site of the Computational Science Education Project (CSEP)
http://csep1.phy.ornl.gov/CSEP/OM/OM.html
Following a literature search and consultations with oceanographers in Israel (Dr.
Brenner, Israel Oceanographic and Limnological Institute and Prof. Paldor, the Hebrew
University) and abroad (Dr. Ezer, Princton University; Prof. Peltier, Univ. of Toronto,
Canada; Prof. Wells, Portland University; Prоf. Yuen, Minnesota University and
4
Minnesota Super-computer Institute) it became apparent that the optimal model for our
purpose is the Princeton Ocean Model, known to the international community as POM.
POM is a σ-coordinate, free surface, primitive equation ocean model, which includes a
turbulence sub-model. It was developed in the late 1970's by Blumberg and Mellor
(1987), with subsequent contributions from other people. The model has been used for
modeling of open oceans, coastal regions, estuaries and lakes,. POM simulations have
been used in various applications in oceanography and limnology, such as:
•
PROFS-Princeton Regional Ocean Forecasting System
•
Coast Survey Development Laboratory Forecasting Systems (NOAA/NOS)
•
U.S. East Coast Coastal Ocean Forecast System, COFS (NCEP)
•
Gulf of Mexico (Dynalysis)
•
The Great Lakes Forecasting System (GLERL)
•
North Pacific Ocean Forecast System, NPAC (NRL)
•
Tampa Bay Physical Oceanographic Real-Time System, PORTS (USF)
•
Mediterranean Forecasting Pilot Program
•
Ice-Ocean Forecasts for the East Coast of Canada (BIO)
•
GOTM- General Ocean Turbulence Model (Burchard et al.,)
•
ECOM-si version of POM (A. Blumberg, HydroQual)
•
MI-POM at the Norwegian Meteorological Institute (A. Melsom)
•
POM in Virtual Reality (G. Wheless, ODU)
The POM web site (http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/index.html)
provides free on line consulting services, up to date information, and supports a
discussion group. Based on the information presented at the POM web site, POM user
group presently includes 1523 Registered Users (Fig. 1) from 61 countries. Total
number of POM related publications is 518, among them 22 are related to lake studies
(Appendix A). The basic equations of the POM are presented in Appendix B.
5
Testing and Adaptation of POM to the Dead Sea system
Our working strategy in testing and modifying the POM to the Dead Sea system was to
first build a proper grid for the Dead Sea, fill the basin with seawater (for which the
physical/chemical/limnological coefficients are well established) and check the validity
of the various parameters and assumptions used in the POM, to our system. Once the
validity has been verified, the basin can be "filled" with the Dead Sea brine and the
model parametric search can begin.
1. Model geometry and stability
The very first step was to build a computational grid based on the digital terrain map of
Israel (DTM by John Hall, 1996; Fig. 2). We use a horizontal computational grid of
19x51 points (cell size is 1 x 1 km) based on the smoothed bathymetry (Fig. 3). The
vertical resolution was chosen to describe properly the Ekman layer (see below) and
includes 25 layers of different width, starting form 0.5m at the surface to 50m in the
deeper parts of the lake (Fig. 4). The inclusion of this real bathymetry into the σ-grid
POM version resulted in stability loss and unrealistically large velocities when the
initial velocities were set to zero and no external forcing was applied (Fig. 5). This loss
of stability was due to the σ-grid related pressure gradient errors discussed by Kliem
and Pietrzak (1999). To avoid these pressure gradient errors, we chose to switch to the
z-grid version of POM (see Mellor et al, 2002). The difference between σ- and z- grids
is illustrated in Fig. 6. The new, z-grid version of POM with real bathymetry is free of
the pressure gradient errors. A test run of 10-year simulations with zero initial velocities
and no external forcing resulted in maximal velocity that does not exceed 3x10-4 m/s for
single precision run, while for the run with double precision it was on the order of 10-15
m/s (Fig. 7). Consequently we decided to use the z-grid version of POM with double
precision.
The next step was to check the stability of pre-existing stabilizing thermo- and haloclines with no external forcing (no wind, no heat exchange). This tested the ability of
the model to maintain existing profile without any forcing, theoretically leaving only the
effect of molecular diffusion on erasing the profile.
6
In molecular diffusion mechanism, the diffusive length, L, is proportional to square root
of time t:
L=(D t)1/2
where D is the diffusion coefficient. The default value of POM is D =10-5 m2/s, is based
on the smoothing role of the diffusion mechanism in ocean-scale, short-time
calculations. However, this value is much larger than true molecular diffusion
coefficient and yields diffusive length on the order of 17 m/year. Correspondingly, for a
50-year period the diffusive distance is on the order of 125m. Fig. 8 presents numerical
simulations which demonstrate the erasing process of the initial stratification during a
10-year period. In agreement with analytical estimations, both the thermocline and
halocline are significantly smoothed after 3 years and totally erased after 10 years.
Thus, no real profile will survive the required time period of the dynamic limnological
model even without any external forcing.
Following the above calculations, we decreased the molecular diffusion coefficient to
D=10-9 m2/s, which corresponds to the real magnitude of the molecular diffusion
coefficients for ions in water (e.g., Ingebritsen and Sanford, 1998). In this case the
diffusion distance for a 50-year period will be only on the order of 1 m and will not
interfere with the mixing processes caused by wind and heat (not shown). This value
was used throughout the following simulations. This choice may lead to some numerical
artifacts related to the Aulerian numerical scheme, and might lead us in the future to
reduce the gridlines spacing at the expected depth of stratification.
The above stability analyses provides the preliminary estimated CPU time for running
the z-grid POM version with the above-described bathymetry and grid geometry of the
Dead Sea. Running the code on the computer available at the GSI (Silicon Graphics Inc.
with one MIPS R12000 processor) takes ~10 min for a 1 day simulation. To decrease
computation time we also make use of the Compaq Alpha with 4 processors available to
us in the Hebrew University. With this computer, 1 day simulation takes ~ 1 min of
CPU time, which means that 1 year takes ~ 6 hours and 50 years simulation requires ~ 2
weeks of the CPU time. Clearly an even stronger computer such as SUN Fire 280R
7
server with two 1 GHz UltraSPARC-III processors is required for efficient work
on the model.
2.
Wind forcing
Adaptation and examination of the model began with two runs that included constant
wind forcing acted on seawater in a box, excluding and including the Coriolis force.
The advantage of this testing is that numerical solution for the surface elevation and
flow velocities can be compared with the analytical solutions.
a). A constant Northern wind of 15 m/s was applied on a box of 100m depth for the first
test that ignores the Coriolis force. From the basic flow equations (see, e.g. Gill, 1982)
it can be derived that in the open ocean the surface elevation is
η ( y ) = h02 +
2Ys
y − h0 ,
ρw g
where h0 is the basin depth; Ys is the wind stress (Ys=CD|v|vρa, where CD is the drag
coefficient; v is the wind velocity ρa is the air density); ρw – water density.
Fig. 9 presents the numerical solution and the model results. The difference between the
lines is due to the physical boundaries imposed in the model calculations. Fig. 10
presents the results of the same model simulation, but with the Dead Sea brine instead
of the regular seawater. As expected, the slope of the line corresponding to the Dead
Sea brine is less than that of regular seawater due to the higher density of the Dead Sea
brine.
b) The second test examined the development of the Ekman layer under constant wind
forcing. In this case the seawater viscosity and the Coriolis force result in velocity
change with depth, according to the following equation (see, e.g. Gill, 1982)
u(z)=U0 exp{-z/k}sin(z/k),
v(z)=U0 exp{-z/k} cos(z/k),
8
with k=(2ν/f)1/2, where ν is the kinematic viscosity; f – the Coriolis parameter and u0 is
the velocity of the surface layer.
The wind direction was chosen in such a way that the East-West velocity at the water
surface would be zero (u(z=0)=0). The results are presented in Fig. 11, where the solid
lines represent analytical expressions and the dashed lines the model simulations. The
main difference between the two solutions is that the analytically calculated velocities
vanish with depth while simulation shows non-zero velocity at all depths. This
phenomenon is due to the influence of the basin boundaries in the simulation. In this run
we used constant wind of 15 m/s on a box of 300m depth.
3. Impact of wind forcing on stratification
The first test of wind forcing on pre-existing stratification was through the introduction
of constant northern wind of 15 m/s on stabilizing thermo- and halo- clines using real
Dead Sea bathymetry and regular seawater. The results of the numerical simulations are
presented in Figs 12-13. Simulation began with initial stable stratification; the upper 25
meters were set with temperature of 2oC above the temperature of the deep water and
with salinity lower by 2 units compared with the salinity of the deep water. Fig. 12
shows the surface elevation and the horizontal surface velocities after 10 and 30 days.
The surface elevation reaches its steady state at the given constant wind within several
days and does not change much after 10 days. Thus, there is no significant difference
between surface elevations shown in Fig. 12a and 12b. In contrast, the temperature and
depth of the upper water body change gradually (Fig. 13). The hot water is driven
southward and are somewhat replaced by colder water, slightly increasing the depth of
the thermocline. The horizontal N-S temperature gradient is also increasing.
The numerical results shown in Figs. 14-16 demonstrate the effect of real wind on the
same (as mentioned above) initial water stratification. Unlike the previous tests with
constant wind, here there is no steady-state surface elevation. Thus, there is a significant
variability between elevation maps shown after one, two and three days (Fig. 14). The
changing wind does not produce significant horizontal gradient in temperature or
9
salinity distribution and there is no significant difference between vertical profiles
shown in Fig. 15 after one, two and three days. The layered structure is preserved all
over the year resulting in minor change in the thermo- and halo- cline depth after a
whole year. Fig. 16 compares initial temperature and salinity profiles and the respective
profiles after 1 year of true winds (year 1998, data from I. Gertman).
4. Thermal forcing
A major part of the dynamic limnological model for the Dead Sea is the quantification
of the changes in heat stored in the upper layers of the Dead-Sea that result from the
balance between input and output of heat through the sea surface. The following
formulation of the heat balance equations is based on the EPA report (1984). Overall,
heat budget calculations require separate calculation of each of the heat budget terms.
The net heat flux Qn [W/m2] is given by:
Qn = QSN + Q AN − QBR − QE + QC
where QSN is net solar (short wave) radiation; QAN is net atmospheric (long wave)
radiation; QBR is back (long wave) radiation from the water surface; QE is evaporative
heat flux and QC is the conductive heat flux.
Solar radiation is given as input to the model based on measured meteorological data,
which is measured directly on the meteorological raft at the Dead Sea (QM) (I. Gertman,
IOL). Data is available both as hourly values and as monthly averages. The net solar
(short wave) radiation is calculated as:
QSN = 0.94 QM
The factor 0.94 accounts for average reflectance at the water surface following the
recommendation of Ryan and Harleman (1973).
10
Net atmospheric (long wave) radiation is calculated from the measured air temperature,
Ta ºC, and the vapor pressure, ea (mb). Both should be measured two meters above the
water surface.
(
)
(
QAN = 2.89 ⋅ 10−8 ⋅ 1 + 0.129 ea (Ta + 273.15) 1 + 0.17C 2
4
)
where: C is the fraction of the sky covered by clouds. A value of C=0.2 was used for
the presented simulations.
Steffan-Boltzmann equation corrected by a factor of 0.97 to account for the deviation of
the water body from an ideal black body is used for back (long wave) radiation from the
water surface:
QBR = 0.97 ⋅ 5.67 ⋅ 10−8 (Ts + 273.15)
4
where: Ts ºC, the sea surface temperature, is calculated in the model and will be checked
and calibrated against the Dead Sea measured surface temperature.
Evaporative heat flux is calculated as follows:
Q E = (1.01 − 9.1 ∗ 10 −4 Ts ) f (W )(e s − e a )
where: f(W) is heat flux wind speed function (see separate discussion at the end of the
section); es is saturation vapor pressure at the water surface, calculated according to the
equation:
es = 6.47 + 0.219Ts + 0.032Ts
2
Atmospheric vapor pressure, ea, is calculated using the measured relative humidity (RH)
and air temperature:
(
ea = RH 6.47 + 0.219Ta + 0.032Ta
Conductive heat flux is calculated as follows:
Q c = 0.6 f (W )(T s − Ta )
2
)
11
where f(w) is the heat flux wind speed function (the same function as above) allows to
account for the wind effect on evaporation rate and conductive heat flux. Following
Ryan and Harleman (1973) and Edinger et al. (1974) most models use the power-law
relation:
f (W ) = a + bW c
where a, b, c are constants that should be defined for every region. Values a=9.2, b=
0.46, c=2 were used for the Lake Hefner model (Cole and Wells, 2002).
In the series of preliminary simulations present hereafter, a constant value for the heat
flux wind speed function was used: f(W)=2.
The incoming short wave (solar) radiation, QSN, penetrates the surface of the water and
is distributed between different vertical levels according to the equation:
Q P (z k ) = Q SN ⋅ TR ⋅ exp(ε ⋅ ∆z k ),
where: TR is the share of the solar radiation that is not absorbed in the uppermost layer;
ε − is the extinction coefficient (m-1) and ∆zk is the depth of the kth layer. To provide for
different water types, the classification of Jerlov (1976) was used. The constants TR and
ε were set according to the following table:
NTP
TYPE
TR
ε (m-1)
1
I
.32
.037
2
IA
.31
.042
3
IB
.29
.056
4
II
.26
.073
4
II
.24
.127
In all the following simulations water type IB (NTP=3) was used.
12
5. Water mass balance (evaporation)
Evaporation from the water surface leads to increase of salinity. This aspect is totally
absent from the POM code and was introduced as follows:
For the calculation purposes we define:
S and S* – Salinity [g/kg] before and after evaporation respectively,
A and h – Area and thickness of the upper layer cells
α (<0) – rate of the level change due to evaporation,
∆t – length of time step.
By definition, the salinity, S, is the mass (in grams) of salt per 1 kg of solution:
S=
M water
M salt
M
+ salt
1000
where: Mwater =ρwater A h is the water mass in kg,
Msalt – mass of salt, in grams, in 1 kg of solution.
We express the ratio Msalt / Mwater as
R≡
M salt
S
.
=
M water 1 − S
1000
Following evaporation, water level changes and the new mass of water is defined by:
M*water = Mwater + ∆Mwater,
where ∆Mwater = ρwater A α ∆t. Since there is no change in the mass of salts, the new
salinity is:
13
S* =
M
*
water
M salt
M
+ salt
=
1000
M salt
M water + ∆M water +
M salt
=
1000
M salt
=
∆M water
1+
M water
M water
M
+ salt
( M water * 1000 )
Thus we have,
S* =
R
R
α∆t
1+
+
h
1000
In the case of multi component system (N salts) the corresponding formulae are:
Ri =
S i* =
Si
1 N
1−
∑ Si
1000 i =1
1+
α∆t
h
Ri
+
1 N
∑ Ri
1000 i =1
6. Heat- and wind- driven stratification build up
In this section we present a series of numerical results demonstrating the combined
effect of surface heat exchange, penetrative heat, water evaporation and wind forcing. In
all cases we use heat penetration of NTP=3 (see section 4) and latent and specific heat
of seawater.
We started simulations with uniform initial temperature, 25oC and salinity. First we
studied the effect of constant solar (short wave) radiation of 400 W/m2, zero wind
forcing and regular seawater. Under these conditions the equilibrium surface water
14
temperature is 39oC. The constant heating results in gradual formation of a stable
thermocline, while loss of fresh water due to evaporation leads to formation of an
unstable halocline. The temperature and salinity distributions along the N-S cross
section after one year are shown in Fig. 17. As expected, due to the uniformity of the
spatial distribution of heating, there are no horizontal gradients of temperature or
salinity. Therefore, in the following figures we demonstrate the vertical profiles of a
single point that represent the whole reservoir. During the first month the surface
temperature increases by more then 10oC (Fig. 18a) and the thermocline is very shallow.
The temperature changes for more then 6oC in the upper 0.5 m. The salinity of the upper
0.5 m also increases very fast up to 2.5 units during this period (Fig. 18b). This leads to
stability loss and active mixing of the upper layers. As a result of the mixing of the
uppermost layers, stable stratification develops and the temperature in the upper water
mass continues to increase, but much slower than before. In contrast to temperature
increase, after the first month and following the mixing described above, the salinity of
the surface layer decreases by more then 1.5 unit and remains nearly constant
throughout the year. With time, a salinity front with a sharp interface develops and
propagates downward, reaching a maximal depth of about 20 meters at the end of the
year. Due to the penetrative radiation, deeper layers are heated and the temperature
distribution turns out to be smoother and deeper than the salinity distribution.
The second set of tests (Fig. 19) comprises the use of real meteorological data that
includes measured solar radiation, atmospheric temperature, and wind with the same
uniform initial distribution of the salinity and temperature. Simulation still uses regular
seawater and its equation of state. Because of wind-driven mixing we do not expect the
initial formation of thin surface layer with the high temperature and salinity discussed
above. Furthermore, due to the variability in wind direction we expect to find a general
uniformity in surface temperature distribution. During winter months the solar radiation
is relatively low and the surface temperature decreases by about 0.1°C (Fig. 19a). The
salinity remains constant because there is very limited evaporation (Fig. 19b).
Stratification develops during spring and is maintained during the hot summer months.
By the end of the summer a stable thermocline and unstable halocline have developed.
During the following fall months, cooling sets in and both thermo- and halo- clines
deepen significantly.
15
Finally, simulations were done using Dead Sea salinity (taken as 277 g/kg) and the
Dead Sea equation of state suggested by Anati (1999). The general trend observed is
similar (Fig. 20), but the maximum temperature at the end of the summer is slightly
higher and absolute increase in salinity is much higher, as compared with the simulation
for seawater. The difference in salinity increase is due to the effect of the relative water
content in the brine, which is smaller than in seawater, and therefore the impact of
evaporation is greater. This also explains the deeper stratification in the Dead Sea brine
as compared with seawater at the end of the year because the destabilizing effect of the
halocline in the brine is greater.
It should be noted that all the above calculations were carried out to compare the impact
of radiation and evaporation on the depth and stability of stratification in seawater and
Dead Sea brine. Clearly we do not yet have the correct parameters (such as heat
capacity, true heat penetration, wind drag coefficient, etc) to compare simulations with
real limnological data. This will be done in the following phases of the formulation of
the model.
7. A new equation of state for Dead Sea- seawater mixtures
(The density calculations needed for the development of the equation of state were done
by Prof. B. Krumgalz)
Introduction of seawater into the Dead Sea poses a modeling challenge due to the
extreme wide range of densities (1.028-1.238 kg/lit) and composition of the two end
members. The equation of state of the Dead Sea (Anati 1999) has therefore to be
expanded to include compositions and densities following mixing with seawater. The
new equation of state has to account for the major variables affecting the density of the
expected mixed brine: temperature range of 15-40°C and the different expected ion
compositions. As a first approximation we assume that the brine is incompressible.
High variation of ion composition is expected due to (i) different mixing ratios between
Dead Sea and Red Sea waters, (ii) precipitation of different salts and (iii) evaporation.
The equation of state was established by fitting calculated densities of different brines
that span the expected range of conditions and compositions. For the calculations, Dead
Sea brine and Red Sea water were used as end member to calculate the expected
compositions of the solutions derived from different mixing ratios (10:1 to 10:1). Some
16
of these mixtures were then theoretically "evaporated", to attain more concentrated
solutions, while taking into consideration salt precipitation. The densities of the
solutions were then calculated by Prof. B. Krumgalz using a model he developed
(Krumgalz et al., 1995) which is based on the Pitzer’s approach. As discussed hereafter,
most of the above mixtures should be oversaturated with respect to gypsum. The effect
of gypsum precipitation on the final density of the mixture was also examined. This was
done by calculating the amount of Ca and SO4 that needs to be removed from the
solution to re-attain saturation. It was found that the effect of gypsum precipitation on
the density of the mixture is rather small as compared to the effect of its composition;
the density of supersaturated brines is very similar to the density of the same brines after
removing the gypsum crystals and attainment of saturation. Figure 21 shows the
agreement between densities that were calculated with the results of the fitted equation
of state. The equation of state agrees with density measurements of brines with known
density composition and temperature.
In order to use the new equation of state in the POM, there is need to modify the code to
track the concentrations of all the major chemical components instead of salinity alone,
as the POM does. This modification is also required in order to calculate the degree of
saturation of the mixed solution with respect to different salts, mainly gypsum and
halite, and the amount of salts that will precipitate from the brine. These calculations
will be done using the Pitzer's approach (Krumgalz et al., 1995) once Pitzer's equations
are incorporated into the code.
The equation of state and the degrees of saturation are calculated based on the
concentration of ions in the mixtures, in molalities (m- mole/Kg H2O). However, the
POM does its physical calculations based on salinity, (S- g/Kg). Therefore, during each
time step, there is need to switch between these units. Following is the equations
introduced to the POM for this purpose.
A)
Switching from molality to salinity
Let Cm be salt concentration in molality (mol/Kg H2O). Accordingly, solution
containing 1Kg of H2O weights 1000 + Cm*µ grams (where µ is the molecular weight
of the salt). Since in 1000 + Cm*µ grams of solution there are Cm*µ grams of salt, the
salinity, S, of this solution is:
S=
Cm µ
* 1000
1000 + C m µ
17
If we have a solution containing N salts, then
S=
C m,i * µ i
N
1000 + ∑ (C m,i * µ i )
* 1000
i =1
B)
Switching from salinity to molality:
Let S be the salinity of a solution in g/Kg. Then there are S grams of salts for (1000-S)
grams of water. Accordingly, for 1000 grams of water there are x grams of salt. Then
x=
x S 1000
1000S
; Cm = =
.
µ µ 1000 − S
1000 − S
If there are N salts in the solution, the concentration of each salt i, in molality, is
C m ,i =
Si
1000
µi
N
1000 − ∑ S i
.
i =1
Gypsum precipitation experiments
(The experimental work is done in collaboration with Dr. J. Ganor, Ben Gurion
University, under contract with GSI)
Mixing of seawater in the Dead Sea brine will lead to gypsum crystallization. This has
been shown by numerous studies done during the 1970s and 80s as part of the feasibility
study of the Med-Dead canal. The major question remaining open with this respect is if
the Dead Sea will turn "white" because the gypsum crystals will remain suspended in
the surface water, or will they precipitate to the bottom soon after crystallization. A
third possibility is that the gypsum will crystallize only after critical oversaturation has
been achieved. Under this scenario, "whitening" events can be expected. The fate of the
gypsum crystals in the water column is of outmost importance in evaluating the future
of the Dead Sea: If the gypsum crystals remain suspended in the surface water, they will
not only change the scenery of the lake, but may lead to enhanced light reflectance,
18
different light/heat penetration, and thus change the rate of evaporation and surface
temperature. These factors will not only impact on the limnology of the Dead Sea and
its environment but will also determine the volume of seawater required to raise the lake
level and later maintain it at the desired level.
As part of the opening phase, the GSI has initiated some preliminary experimental work
to better understand the mechanism of gypsum precipitation. Experiments and their
interpretations are done by Dr. Ganor et al. (2003) at the Ben Gurion University. Unlike
the work done in the pass, the new study involves continuous and controlled mixing of
seawater with Dead Sea brine, at carefully monitored ratio. Experiments were
conducted using a well-stirred flow-through apparatus in which the sulfate input and
output are carefully monitored. The amount of gypsum that precipitates during mixing
in the apparatus can be calculated accordingly, and the rate of gypsum crystallization
determined. Addition of crystallization seeds will eventually allow to determine the
impact of crystal size on the rate of crystallization, and if this can be controlled.
Precipitation rate (R) is calculated at steady state from the consumption of SO4:
R = − q(Cout − Cinp )
where Cout and Cinp are the concentrations of SO4 in the output and the input solutions,
respectively (mol kg-1), and q is the fluid volume flux through the system (kg s-1). As
the input solution is mixed inside the reaction cell, its concentration is calculated by:
Cinp =
C DS * q DS + C RS * q RS
q DS + q RS
where the subscripts DS and RS refer to Dead Sea and Red Sea, respectively. Seed
material was prepared from a transparent crystal of gypsum (selenite), which was
crushed and sieved to retain the 53-149 µm size fraction.
Preliminary results show that gypsum precipitation kinetics are controlled by the
amount of gypsum. The rate increases linearly with the mass of gypsum at steady state
(Fig. 22). Precipitation rate at steady state was 2±0.6.10-9 mol s-1 g-1 (0.03 g gypsum
day-1 per each g of seed). One of the experiments was conducted with no initial mineral
19
seeding. After less than 24 hours, the amount of gypsum precipitated is sufficient so
that further precipitation becomes controlled by the mass of gypsum available.
Future work
As in the opening phase, most of the work in the future needs to concentrate on further
adaptation of the POM to the Dead Sea system, and the derivatives of Dead Sea –
seawater mixing. The development of the model needs to include the following:
1. Changing the POM from a salinity-based model to a multi-chemical component
model. This will require substituting the scalar advection equation for salinity
transport by a system of advection equations for each of the chemical
components. These will include the following ions: Na, K, Mg, Ca, Cl, HCO3
and SO4 and minerals (CaSO4·2H2O and NaCl) .
2. Incorporating Pitzers equations to calculate the saturation degrees of the
mixtures with respect to different minerals. This will include calculation of the
required weight of the mineral that needs to precipitate out from the mixture in
order to bring it back to saturation.
3. Density calculations based on chemical composition. Here we will check the
impact of theoretical over-saturation vs. mineral crystallization and removal of
minerals from the brine on the long-term stability of stratification.
4. Impact of maintaining the crystallized minerals in the water column.
5. Simulations of surface inflows. Initially this will include freshwater inflow from
the major sources: the Jordan River, Arnon River, Arava River and En Fescha.
This module will be later used to introduce the seawater/reject brine into the
Dead Sea.
6. Examination of the impact of inflow of groundwater at depth.
7. Re-evaluation of the Boussinesq approximation for the momentum equation,
used in oceanographic and limnological modeling where the density variation is
relatively small (of few sigma units at most), for systems with a large range of
density variations (of the order of hundreds of sigma-units, as is the case
between Dead Sea brine and seawater). This examination might lead to the
conclusion that there is need to use the POM-nb (non- Boussinesq) version
20
instead of the POM-gcs. (Z-grid vesion). Such modification would require that
the Z-grid be incorporated into the POM-nb.
8. Re-evaluation of the assumption of incompressibility for the fluid continuity
equation
9. Determination of light reflectivity at the Dead Sea water surface.
10. Determination of light penetration in the Dead Sea, and the expected change due
to changing water composition. In addition the impact of biological blooming
and gypsum precipitation needs to be examined to determine their potential
impact on the heat balance and the long-term stability of the water column.
11. Determination of latent and specific heats for the Dead Sea and Dead Sea –
seawater mixtures.
12. Determination of the wind stress function and drag-coefficient in the Dead Sea.
13. Calibration of the wind effect on the evaporative heat balance. This will aid in
resolving one of the major issues concerning the rate of evaporation from the
Dead Sea surface.
14. Parametric search study and tuning/calibration of the model against existing
meteorological and limnological records of the Dead Sea between years 1991
and 2000.
15. Introduction of seawater and simulations of different scenarios.
21
References
Anati A. 1999. The salinity of hypersaline brines: concepts and misconceptions. Int. J.
Salt Lake Res. 8, p. 55-70.
Blumberg A.F and Mellor G.L. 1987. A description of a three-dimensional coastal
ocean circulation model. Three-dimensional shelf models, coastal and estuarine
sciences, 5, N, Heaps, Ed., American Geophysical Union
Cole and Wells S. 2002. CE-QUAL-W2 users manual.
Edinger J.E., Brady D.K. and Geyer J.C. 1974.Heat Exchange and Transport in the
Environment. Report No.14,Research Project RP-49.Electric Power Research
Institute, Palo Alto, CA.
EPA report (1984) Proceedings of Stormwater and Water Quality Model Users Group
Meeting April 12-13,1984. EPA-600/9-85-003, January, 1985.
Ganor, J., Avital A., Talby R. 2003. Gypsum precipitation rate in Red Sea – Dead Sea
mixtures – preliminary results. Isr. Geol. Soc. Annu. Meet.
Gavrieli I., Lensky N., Gazit-Yaari N. and Oren A. 2002. The Impact of the Proposed
“Peace Conduit” on the Dead Sea- Evaluation of Current Knowledge on Dead
Sea – Seawater Mixing. Isr. Geol. Surv., Rep. GSI/23/02.
Gill A.E. 1982 Atmosphere-Ocean Dynamics. Academic Press, 662pp
Hall J.K. 1996. Topography and bathymetry of the Dead Sea Depression.
Tectonophysics, 266 (1-4):177-185.
Ingebritsen S.E and Sanford W.E. 1998. Groundwater in Geologic Processes,
Cambridge University Press, 341 pp.
Jerlov N.G.1976. Marine Optics, 14, 231 pp. Elsvier Sci. Pub. Co. Amsterdam.
Kliem N. and Pietrzak J.D. 1999. On the pressure gradient error in sigma coordinate
ocean models: A comparison with a laboratory experiment. J. Geophys. Res.,
104(C12):29781-29800.
Krumgalz B.S., Pogorelsky R. and Pitzer K.S. 1995. Ion interaction approach to
calculations of volumetric properties of aqueous multiple-solute electrolyte
solutions, Journal of Solution Chemistry 1995, 24, 1025-1038.
Mellor G.L. and Blumberg A.F. 1985. Modeling vertical and horizontal diffusivities
with sigma coordinate system. Mon. Wea. Rev. 113, 1380-1383.
Mellor G.L., Hakkinen S., Ezer T. and Patchen R. 2002. A generalization of a sigma
coordinate ocean model and an intercomparison of model vertical grids. Ocean
Forecasting: Conceptual Basis and Applications, N. Pinardi and J. D. Woods
(Eds.), Springer, 55-72,
Ryan,P.J. and Harleman D.R.F. 1973. An Analytical and Experimental Study of
Transient Cooling Pond Behavior. Report No.161. Ralph M. Parsons
Laboratory, Department of Civil Engineering, Massachusetts Institute of
Technology, Cambridge, MA. January 1973.
22
Appendix A: 22 POM applications for lakes
1.
Ahsan, A. K. M. Q. and A. F. Blumberg, Three-dimensional hydrothermal model
of Onondaga Lake, New York. J. Hyd. Eng., 125(9), 912-923, 1999
2.
Bedford, K. and D. Schwab, The Great Lakes Forecasting System - Lake Erie
Nowcasts/Forecasts. Marine Technology Society Annual conference (MTS '91),
Marine Technology Society, Washington, DC, pp. 260-264, 1991
3.
Bedford, K. and D. Schwab, Recent developments in the Great Lakes Forecasting
System (HLFS) performance evaluation. Second Conference on Coastal
Atmospheric and Oceanic Prediction and Processes, Jan. 11-16, 1998, Phoenix,
AZ, American Meteorological Society, Boston, MA, 45-50, 1998
4.
Bedford, K., C.-C. Yen, J. Kempf, D. Schwab,R. Marshall and C. Kuan, A 3DStereo Graphics Interface for Operational Great Lakes Forecasts. Ed. M.
Spaulding, Amer. Soc. of Civil Engrs., New York, NY., pp. 248-257, 1990
5.
Beletsky, D. and D. J. Schwab, Modeling circulation and thermal structure in Lake
Michigan: Annual cycle and interannual variability. J. Geopys. Res., 106(C9),
19,745-19,771, 2001
6.
Beletsky, D., W. P. O'Connor, and D. J. Schwab, Hydrodynamic Modeling for the
Lake Michigan Mass Balance Project. G. Delic and M.F. Wheeler (eds.), Next
Generation Environmental Models and Computa- tional Methods, Proceedings of
the
Next
Generation
Environmental
Models
Computational
Methods
(NGEMCOM) Workshop at the National Environmental Supercomputing Center,
1997
7.
Blumberg
A.F.,
Turbulent
mixing
processes
in
lakes,
reservoirs
and
impoundments Physics- based Modeling of Lakes, Reservoirs and Impoundments.
Ed.Gray W.G. Amer. Soc. Civ. Eng. New York, 79-104, 1986
8.
Blumberg, A. F. and D. M. Di Toro, Effects of climate warming on dissolved
oxygen concentrations in Lake Erie. Trans. Amer. Fisheries Soc., 119, 210-223,
1990
9.
Bowyer, P. A., Topographically controlled circulation and mixing in a lake. J.
Geophys. Res., 106(C4), 7065-7080, 2001
10. Chen, C., J. Zhu, E. Ralph, S. A. Green, J. W. Budd and F. Y. Zhang, Prognostic
modeling studies of the Keweenaw Current in Lake Superior. Part I: Formation
and Evolution. J. Phys. Oceanogr., 31, 379-395, 2001
23
11. Chen, C., J. Zhu, K. Kang, H. Liu, E. Ralph, S. A. Green and J. W. Budd, Crossfrontal transport along the Keweenaw coast in Lake Superior: a Lagrangian model
study. Dyn. Atmos. Ocean, 36, 83-102, 2002
12. D. J. Schwab, D. J. , D. Beletsky and J. Lou, The 1998 Coastal Turbidity Plume in
Lake Michigan. Est. Coastal and Shelf Sci.,50, 49-58, 2000
13. Kelley, J. G. W., J. S. Hobgood, K. W. Bedford, and D. Schwab, Generation of
three-dimensional lake model forecasts for Lake Erie. Weather & Forecasting, 13,
659-687, 1998
14. Kelley, J., C.-C. Yen, K. Bedford, J. Hobgood and D. Schwab, Coupled Lake Erie
Air-Sea, Storm Resolving Forecasts and Predictions, the Vien to Project. Proc. of
the 3rd International Conference of Estuarine and Coastal Modeling, Oak Brook,
Illinois, September 8-10, 1993, Amer. Soc. of Civil Engrs., New York, NY. pp.
202-215, 1993
15. Lou, J., D. J. Schwab, D. Belsky and N. Hawley, A model of sediment
resuspension and transport dynamics in southern Lake Michigan. J. Geophys.
Res., 105(C3), 6591-6610, 2000
16. O'Connor, W.P., and Schwab, D.J., Sensitivity of Great Lakes Forecasting System
Nowcasts to Meteorological Fields and Model Parameters, in: M.L. Spaulding, K.
Bedford, A. Blumberg, R. Cheng, and C. Swanson (eds.), Estaurine and Coastal
Modeling III. Proceedings of the 3rd International Conference, ASCE, Sep. 8-10,
1993, Oak Brook, IL, 149-157, 1994
17. O'Connor, W.P., Schwab, D.J., and Lang, G.A., Forecast Verification for Eta
Model Winds Using Lake Erie Storm Surge Water Levels. Weather and
Forecasting, 14(1), 119-133, 1999
18. Schwab, D. J., W. P. O'Connor and G. L. Mellor, On the net cyclonic circulation
in large stratified lakes. J. Phys. Oceanogr., Vol. 25, No. 6, Part II, 1516-1520,
1995
19. Schwab, D.J. and K.W. Bedford, Initial Implementation of the Great Lakes
Forecasting System: A Real-Time System for Predicting Lake Circulation and
Thermal Structure. Can. J. Water Poll. Control., 1994
20. Schwab,D. J. and K. W. Bedford, The Great Lakes forecasting system. Coastal
Ocean Prediction, Coastal and Estuarine Studies, Vol. 56, C. N. K. Mooers (Ed.),
American Geophysical Union, Washington, DC, 157-173, 1999
24
21. Taguchi, K., and K. Nakata, Analysis of water quality in Lake Hamana using a
coupled physical and biochemical model. J. Mar. Sys., 16, 107-132, 1998
22. Zhu, J., Chen, C., E. Ralph, S. A. Green, J. W. Budd and F. Y. Zhang, Prognostic
modeling studies of the Keweenaw Current in Lake Superior. Part II: Simlation. J.
Phys. Oceanogr., 31, 396-410, 2001
25
Appendix B: The Basics of POM
We present here the basic equations of the POM based on the USER GUIDE, which can
be found at the POM-internet site:
http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/index.html
The sigma coordinate equations are based on the transformation,
x * = x , y* = y, σ =
z- η *
,t = t
H +η
(1a, b, c, d)
where x,y,z are the conventional cartesian coordinates; D ≡ H + η where H (x, y ) is
the bottom topography and η(x, y, t) is the surface elevation. Thus, σ ranges from σ= 0
at z = η to σ = -1 at z = H. Sigma levels are illustrated in Fig.23. After conversion to
sigma coordinates and deletion of the asterisks, the basic equations may be written (in
horizontal cartesian coordinates),
∂ DU
∂ DV
∂ω
∂η
+
+
+
= 0
∂x
∂y
∂σ
∂t
(2)
∂ UD
∂ U2D
∂ UVD
∂U ω
∂η
+
+
+
- fVD + gD
∂t
∂x
∂y
∂σ
∂x
+
gD2
ρo
o  ∂ρ ′
∫σ  ∂ x
−
σ ′ ∂ D ∂ρ ′ 
∂  K M ∂U 
dσ ′ =
+ F x (3)

D ∂ x ∂ σ′ 
∂σ  D ∂σ 
∂ VD
∂UVD
∂ V2 D
∂ Vω
∂η
+
+
+
+ fUD + gD
∂t
∂x
∂y
∂σ
∂y
+
gD2
ρo
o
∫σ
 ∂ρ ′
σ ′ ∂ D ∂ρ ′ 
∂  KM ∂V 
 ∂ y − D ∂y ∂ σ ′  dσ ′ = ∂σ  D ∂σ  + F y (4)
26
∂ TD
∂ TUD
∂TVD
∂ Tω
∂  KH ∂T 
∂R
+ FT −
+
+
+
=
∂t
∂x
∂y
∂σ
∂σ  D ∂σ 
∂z
∂SUD
∂Sω
∂SD
∂SVD
∂  K H ∂S 
+ FS
+
+
+
=
∂x
∂t
∂y
∂σ
∂σ  D ∂σ 
(5)
(6)
∂q 2 D
∂Uq2 D
∂Vq 2 D
∂ωq2
∂  K q ∂q2 
+
+
+
=
∂σ
∂t
∂x
∂y
∂σ  D ∂σ 
2
2
2K M   ∂U 
2g
2Dq3
 ∂V  
∂ ρ˜
 + 
  +
+

KH
+ Fq
 ∂σ  
D   ∂σ 
ρo
B1l
∂σ
(7)
∂ q2 lD
∂Uq2 lD
∂Vq 2 lD
∂ω q2 l
∂  K q ∂ q2 l 
+
+
+
=
∂t
∂x
∂y
∂σ
∂ s  D ∂σ 
2
 K  ∂ U 2
Dq3
 +  ∂ V   + E g K ∂ρ˜  W
˜
+ E1l  M 
+ Fl

3
 ∂σ  
ρo H ∂σ 
B1
 D  ∂σ 
(8)
The transformation to the Cartesian vertical velocity is
 ∂ D ∂η 
 ∂D ∂η 
∂D ∂η
+
+
+
W =ω +U σ
+σ
+V  σ
 ∂x ∂x 
 ∂y ∂y 
∂t ∂t
The so-called wall proximity function is prescribed according to
˜ = 1+ E (l / kL)
W
2
where
L−1 = (η − z)− 1 + (H − z)− 1.
Also,
∂ρ˜ / ∂σ ≡ ∂ρ / ∂σ − c−s 2∂p / ∂σ
where cs is the speed of sound.
The horizontal viscosity and diffusion terms are defined according to:
Fx ≡
∂
(Hτ xx ) + ∂ Hτ xy
∂x
∂y
(
)
(9a)
27
Fy ≡
∂
∂
Hτ xy +
Hτ yy
∂x
∂y
(
)
(
)
(9b)
where
τ xx = 2AM
 ∂U
∂U
∂V 
∂V
, τ xy = τ yx = AM 
+
 , τ yy = 2AM
 ∂y
∂x
∂x 
∂y
(10a,b,c)
Also,
Fφ ≡
∂
∂
(Hqx ) +
Hqy
∂x
∂y
( )
(11)
where
q x ≡ AH
∂φ
,
∂x
qy ≡ AH
∂φ
∂y
(12a,b)
and where φ represents T, S, q2 or q2 l . It should be noted that these horizontal
diffusion terms are not what one would obtain by transforming the conventional forms
to the sigma coordinate system. Justification for the present forms will be found in
Mellor and Blumberg (1985). In (9a, b) and (11), H is used in place of D for the small
algorithmic simplification it offers for terms whose physical significance is
questionable.
The Smagorinsky Diffusivity
The Smagorinsky diffusivity for horizontal diffusion is expressed by:
AM = C∆x ∆ y
1
T
∇V + (∇V)
2
where
∇V + (∇V)T /2=[(∂u / ∂x)2 + (∂v / ∂x + ∂u / ∂y)2 / 2 + (∂v / ∂y)2 ]1 /2 . Values of C
the range, 0.10 to 0.20 seem to work well.
Vertical Boundary Conditions.
The vertical boundary conditions for (2) are
in
28
ω (0) = ω (-1) = 0
(13a,b)
The boundary conditions for (3) and (4) are
KM  ∂ U ∂ V 
,
= − (< wu(0 ) >, < wv(0) >), σ → 0
D  ∂σ ∂σ 
(14a,b)
where the right hand side of (14a,b) is the input values of the surface turbulence
momentum flux (the stress components are opposite in sign), and
[
]
KM  ∂ U ∂ V 
2
2 1/2
,
(U,V ), σ → − 1
= Cz U + V
D  ∂σ ∂σ 
(14c,d)
where


k2


Cz = MAX
2 , 0.0025

 ln{(1+ σ kb−1)H / zo }

[
]
(14e)
k = 0.4 is the von Karman constant and zo is the roughness parameter. The boundary
conditions on (5) and (6) are
KH  ∂ T ∂S 
,
= − (< wθ (0) >) , σ → 0
D  ∂σ ∂σ 
KH  ∂ T ∂S 
,
= 0 ,
D  ∂σ ∂σ 
σ → −1
(15a,b)
(15c,d)
The boundary conditions for (7) and (8) are
(q 2 (0),q 2l(0))
(q 2 (−1), q 2l(−1))
(
)
= B12/3 uτ2 (0), 0
(
(16a,b)
)
= B12/ 3 u2τ (−1), 0
(16c,d)
29
where B1 is one of the turbulence closure constants and uτ is the friction velocity at the
top or bottom as denoted.
The Vertically Integrated Equations
The equations, governing the dynamics of coastal circulation, contain fast moving
external gravity waves and slow moving internal gravity waves. It is desirable in terms
of computer economy to separate the vertically integrated equations (external mode)
from the vertical structure equations (internal mode). This technique, known as mode
splitting (Simons, 1974; Madala and Piacsek, 1977) permits the calculation of the free
surface elevation with little sacrifice in computational time by solving the velocity
transport separately from the three-dimensional calculation of the velocity and the
thermodynamic properties.
The velocity transport, external mode equations are obtained by integrating the internal
mode equations over the depth, thereby eliminating all vertical structure. Thus, by
integrating Equation (2) from σ =-1 to σ = 0 and using the boundary conditions (13a,b),
an equation for the surface elevation can be written as
∂η
∂U D
∂V D
+
+
= 0
∂t
∂x
∂y
(17)
After integration, the momentum equations, (3) and (4), become
∂ U D ∂U 2 D ∂U V D ˜
∂η
+
+
− F x − fV D + gD
= − < wu(0) > + < wu(-1) >
∂t
∂x
∂y
∂x
+ Gx −
gD
ρo
o o
∂ ρ′
∫-1 ∫σ  D ∂x
−
∂D
∂ρ ′ 
σ′
dσ ′ dσ
∂x
∂σ 
(18)
∂ V D ∂U V D ∂ V 2 D ˜
∂η
+
+
− Fy + fU D + gD
= − < wv(0) > + < wv(-1) >
∂y
∂t
∂x
∂y
+ Gy −
∂ρ ′ ∂ D ∂ ρ ′ 
D
−
σ′
dσ ′ dσ

ρ o ∫-1 ∫σ  ∂ y ∂y
∂σ 
gD
o o
(19)
30
The over bars denote vertically integrated velocities such as
U ≡
o
∫-1U dσ .
(20)
The wind stress components are − < ω u(0) > and − < ω u(0) > , and the bottom stress
components are − < ω u(−1) > and − < ωu(−1) > . The quantities F˜ x and F˜ y are defined
according to
 ∂U
∂ 
∂U 
∂ 
∂V  
+
H2A
HA
+
F˜ x =



M
∂ x 
∂x  
∂x 
∂y  M  ∂y
(21a)
 ∂U
∂ 
∂V 
∂ 
∂V 
+
H 2AM
+
F˜ y =

 HAM 


 ∂y
∂y 
∂x 
∂y 
∂x 
(21b)
and
The so-called dispersion terms are defined according to
Gx =
∂U 2D
∂ U VD
∂U 2 D
∂ UVD
+
− F˜ x −
−
+ Fx
∂x
∂y
∂x
∂y
(22a)
Gy =
∂U V D
∂V 2D
∂ UVD
∂ V2 D
+
− F˜ y −
−
+ Fy
∂x
∂y
∂x
∂y
(22b)
Note that, if AM is constant in the vertical, then the "F" terms in (22a) and (22b) cancel.
However, we account for possible vertical variability in the horizontal diffusivity; such
is the case when a Smagorinsky type diffusivity is used. As detailed below, all of the
terms on the right side of (18) and (19) are evaluated at each internal time step and then
held constant throughout the many external time steps. If the external mode is executed
cum sole, then Gx = Gy = 0.
Figure 1. Number of POM users over the years.
Figure 2. A digital terrain map (DTM) of the Dead
Sea and its surrounding (from Hall, 1996).
Figure 3. Smoothed bathymetry and horizontal grid of the
Dead Sea used in the model. Z: depth in meters.
0.5 m
1.2 m
1.5 m
3.0 m
50 m
Figure 4. The vertical grid used in the model.
Figure 5. Flow pattern and surface elevation generated
within few days as a result of pressure gradient errors in σgrid POM. Eta: change in elevations in meters.
Figure 6. Comparison between z-grid (horizontal gridlines)
and σ−grid (gridlines that follow the bathymetry).
3,0E-04
3,0E-15
No profile
2,5E-04
2,5E-15
2,0E-15
V e lo c it y
2,0E-04
1,5E-04
1,5E-15
1,0E-04
1,0E-15
Single precision
Double precision
5,0E-05
5,0E-16
0,0E+00
0,0E+00
0
500
1000
1500
2000
2500
3000
3500
Day
Figure 7. Stability tests: maximal surface velocity as a function of time in
simulations with zero initial velocity and no forcing.
Initial
profile
After 3
years
After 10
years
Figure 8. Change in thermo- and halo- clines profiles using the POM default
molecular diffusion (D = 10-5) with no external forcing
Figure 9. Comparison between model simulation and analytical solution
for wind-driven surface elevation in a box filled with seawater.
Figure 10. Comparison between slopes of the wind-driven surface
elevation of a box filled with Dead Sea water and regular seawater.
Velocity, m/s
-0.08
-0.04
0
0.04
0
20
Depth, m
40
60
U model
V model
U analytical
80
V analytical
100
120
140
Figure 11. Comparison between model simulation and analytical solution
for horizontal velocities of the Ekman layer under constant wind of 15 m/s
applied on a 300 meter deep box filled with seawater.
0.08
b.
Figure 12. Surface elevation (in meters) and flow direction after a) 10 and b) 30 days. Simulation with seawater and
constant northern wind of 15m/s.
a.
Figure 13. Temperature distribution (above 25°C) along N-S cross-section after a) 10 b) 20 and c) 30 days. Simulation
with seawater and constant northern wind of 15m/s.
b.
c.
Figure 14. Surface elevation, in meters, after a) 1 b) 2 and c) 3 days. Simulation with seawater and real wind starting on 1/1/98.
a.
a.
Figure 15. Temperature distribution (above 25°C) along N-S cross-section after a) 1 b) 3 days.
Simulation with initial seawater stratification and real wind starting on 1/1/98
b.
a.
Figure 16. Initial and final vertical a) temperature and b) salinity profiles.
Simulation with initial seawater stratification and real wind starting on 1/1/98.
b.
a.
Figure 17. a) Temperature (above 25°C) and b) salinity (above 35 g/Kg) distribution along N-S crosssection after 1 year. Simulation with seawater, constant heat flux of 400 W/m2 and zero wind forcing.
b.
100
80
60
40
20
0
100
Initial
January
20
February
March
April
May
40
June
July
August
60
September
October
November
80
December
0
0.5
1.0
1.5
1.5
Salinity
2.0
2.5
3.0
b.
Figure 18. Monthly vertical change in a) temperature and b) salinity distribution during 1 year.
Simulation with seawater, constant heat flux of 400 W/m2 and zero wind forcing.
a.
Temperature
°C
Temperatuire
(oC)
2 4 6 8 10 12 14 16 18
Initial
January
February
March
April
May
June
July
August
September
October
November
December
100
80
60
40
20
0
0
0
100
Initial
January
February
20
March
April
May
40
June
July
August
60
September
October
November
80
December
0
0.1
0.2
Salinity
0.3
b.
0.4
Figure 19. Monthly vertical change in a) temperature and b) salinity distribution during 1 year.
Simulation with seawater and real 1998 meteorological data.
a.
Temperature (oC)
4
12
16
8
Initial
January
February
March
April
May
June
July
August
September
October
November
December
100
80
60
40
20
0
0
0.5
100
Initial
January
20February
March
April
40May
June
July
August
60
September
October
November
80
December
0
1.0
1.5
Salinity
2.0
Figure 20. Monthly vertical change in a) temperature and b) salinity distribution during 1 year.
Simulation with Dead Sea brine, Dead Sea equation of state and real 1998 meteorological data.
Temperature (oC)
4
12
16
8
2.5
Initial
January
February
March
April
May
June
July
August
September
October
November
December
Density calculated using Pitzer's equations
(kg/m3)
1.25
1.2
1.15
1.1
1.05
1
1
1.05
1.1
1.15
1.2
1.25
Density calculated using equation of state (kg/m3)
Figure 21. Calculated densities: Pitzer’s equation vs. the new equation of
state.
gypsum precipitation rate (mol/s)
-9
7 10
-9
6 10
-9
5 10
-9
4 10
-9
3 10
-9
-9
rate = 2.0 x 10 x gypsum mass
-9
R = 0.92
2 10
2
1 10
0
0
0.5
1
1.5
2
2.5
3
3.5
mass of gypsum at steady state (g)
Figure 22. Gypsum precipitation from a 1:1 Dead Sea – Red Sea mixture.
η
σ= 0
z=0
z=
H(
x, y
)
σ = −1
Figure 23. The σ coordinate system.