UNIVERSITY OF LJUBLJANA
FACULTY OF MATHEMATICS AND PHYSICS
DEPARTMENT OF PHYSICS
Princeton Ocean Model
Seminar
Abstract
Princeton Ocean Model, usually called POM was developed by George Mellor and Alan
Blumberg around 1977. The model was developed and applied to oceanographic problems
at Princeton University. The model is under GNU license and it can be downloaded from
http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/. POM was applied to many
ocean forecasting systems, which predict hurricanes, tsunamis, ocean currents and ocean
waves. POM uses Arakawa C-grid and central difference LEAP-FROG scheme.
Advisor:
dr. Vlado Malačič
Author:
Gašper Štifter
Ljubljana, May, 2006
Contents
1 Introduction
1.1 Global ocean models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Coastal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
4
2 The Princeton Ocean Model
5
3 Basic equations
3.1 Dynamic and Thermodynamic Equations . . . . . . . . . . . . . . . . . . . . . .
6
7
4 Finite difference scheme
4.1 One dimensional case - Forward difference . . . . . . . . . . . . . . . . . . . . .
4.2 One dimensional case - Central difference . . . . . . . . . . . . . . . . . . . . . .
4.3 LEAP - FROG scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
8
8
9
5 Boundary conditions
10
6 Sigma coordinate system
11
7 Seamount problem
12
8 Adriatic Tsunami simulation
16
9 Conclusion
17
List of Figures
1
2
3
4
5
6
7
8
9
Forward Difference(solid line) and central-difference (dashed line) . . . . . . . .
Arrangement of points for computation in leap-frog method . . . . . . . . . . .
The sigma coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Seamount bottom topography, legend shows depth in meters . . . . . . . . . . .
Surface flux and surface elevation, legend shows surface elevation in meters . . .
Locations of current profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity profiles in points A, B, C in figure 6 . . . . . . . . . . . . . . . . . . . .
Velocity u - X section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adriatic Tsunami[6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
9
10
11
12
13
14
14
15
17
1
Introduction
Numerical models[1] of ocean currents have simulate flows in realistic ocean basins with a realistic sea-floor. They include the influence of viscosity and non-linear dynamics and with them
one can calculate and also predict flows in the ocean. Perhaps the most important, they interpolate between sparse observations of the ocean collected by ships, drifters, and satellites.
Numerical models are not absolutely accurate and could never give complete descriptions of the
oceanic flows even if the equations are integrated accurately. Models use algebraic approximations of the differential equations. We assume the ocean basins are filled with a grid of points
and time moves forward in tiny steps. The values of the current, pressure, temperature, and
salinity are calculated from their values at neighbouring points and values from the previous
time. Another limitation of numerical models is that they do provide information only at grid
points, the results don’t give information about the flow between the grid points. Yet, the
ocean is turbulent and any oceanic model capable of resolving the turbulence needs grid points
spaced millimeters apart, with time steps of milliseconds. Practical ocean models have grid
points spaced tens to hundreds of kilometers apart in horizontal direction and tens to hundreds
of meters apart in the vertical. That means the turbulence cannot be calculated directly and
the influence of turbulence must be parametrized. Models of the ocean must run on current
available computers, which have limited resources, that means oceanographers have to simplify
their models further, if they want to finish calculation in decent time. Oceanographers use
the hydrostatic and Boussinesq approximations and often use vertically integrated equations,
called the shallow-water equations. It is necessary to do the simplification of the real problem,
because we currently unable to run the most detailed models of oceanic circulation for the
time period of thousands of years to understand the role of the ocean in climate[7]. Numerical
models are used for many purposes in oceanography in general we can divide models into two
groups:
• Mechanistic models are simplified models used for studying processes. Because the
models are simplified, the output is easier to interpret than output from more complex
models. Many different types of simplified models have been developed, including models
for describing planetary waves, the interaction of the flow with sea-floor or the response
of the upper ocean to the wind. These are perhaps the most useful models, because they
provide insight into the physical mechanisms influencing the ocean.
• Simulation models are used for calculating realistic circulation of oceanic regions. The
models are often very complex because all important processes are included and the
output is difficult to interpret. The first simulation models were developed by Kirk Bryan
and Michael Cox at the Geophysical Fluid Dynamics laboratory at Princeton.
1.1
Global ocean models
Several types of global models are widely used in oceanography[1]. Most have grid points
about one tenth of a degree apart, which is sufficient to resolve mesoscale eddies. Geophysical
Fluid Dynamics Laboratory Modular Ocean Model MOM is perhaps the most widely
3
used model growing out of the original Bryan-Cox code. The model uses the momentum
equations, equation of state, and the hydrostatic and Boussinesq approximations. Subgrid–
scale motions are reduced by use of eddy viscosity, which neglects small–scale vortices (or
eddies). Global ocean models have an implementation of improved numerical schemes, a free
surface, realistic bottom features and many types of mixing including horizontal mixing along
surfaces of constant density and can be coupled to atmospheric models. Hybrid Coordinate
Ocean Model HYCOM, which is a Global ocean model, have implementation of real x, y, z
coordinates. That type of Cartesian coordinate system has both advantages and disadvantages.
It has high resolution in the surface mixed layers and also in shallower regions, but it is less
useful in the interior of the ocean. Below the mixed layer, mixing in the ocean is easily calculated
along surfaces of constant density, but difficult across the same surfaces.
1.2
Coastal models
The great economic importance of the coastal zone has led to the development of many different
numerical models for describing coastal currents, tides, and storm surges. The models extend
from the beach to the continental slope and they can include a free surface, realistic coasts and
bottom features, river runoff and atmospheric forcing. Because the models don’t extend very
far into deep water, they need additional information about deep–water currents or conditions
at the shelf break. Princeton Ocean Model developed by Blumberg and Mellor (1987)[5]
is widely used for describing coastal currents. It is a direct descendant of the Bryan-Cox
model. It includes thermodynamic processes, turbulent mixing, Boussinesq and hydrostatic
approximations. The model has been used for calculation of results, presented in this seminar,
it has been used for calculation of the three–dimensional velocity distribution, salinity, sea level,
temperature and turbulence for up to 30 days, over a region roughly 100-1000km on a side with
grid spacing of 1-50km, some of those results are presented in this document, the basin in this
example was a rectangular basin with an underwater island in the middle of basin.
1.3
Important concepts
• Numerical models are used to simulate oceanic flows with realistic and useful results.
The most recent models include heat fluxes through the surface, wind forcing, mesoscale
eddies, realistic coasts and sea-floor features, and more than 20 levels in vertical.
• Recent models are nowadays quite good, with resolution near 0.1o and has showed previously unknown aspects of the ocean circulation.
• Numerical models are still not perfect. They solve discrete equations, which are not as
accurate as the analytical equations of motion.
• Numerical models cannot reproduce all turbulence of the ocean, because the grid points
are tens to hundreds of kilometers apart. The influence of turbulent motion over smaller
distances must be calculated from theory and this introduces errors.
4
• Numerical models can be forced by real–time oceanographic data, received from ships
and satellites, to produce forecasts of oceanic variables.
2
The Princeton Ocean Model
The Princeton Ocean Model[8] (POM) is a community general circulation numerical ocean
model, that can be used to simulate and predict oceanic currents, temperatures, salinity and
other water properties. The model code was originally developed in late 1970’s at Princeton
University[4] and Analysis of Princeton[7] by Alan Blumberg and George Mellor with later
contributions from other people. The model incorporates the Mellor-Yamada turbulence scheme
developed in the early 1970’s by George Mellor and Ted Yamada; this turbulence sub–model
is widely used by oceanic and atmospheric models. In the past, early computer ocean models
such as the Bryan-Cox model (developed in the late 1960’s at the Geophysical Fluid Dynamics
Laboratory, GFDL, later became the Modular Ocean Model1 , MOM)), which aimed mostly at
coarse–resolution simulations of the large–scale ocean circulation, so there was still a need for a
numerical model, that can handle high–resolution calculation of coastal ocean. The BlumbergMellor[8] model, which later became known as POM, thus included new features, such as free
surface to handle tides, sigma vertical coordinates (i.e., terrain-following) to handle complex
topographies and shallow regions, a curvilinear grid, to better handle coastlines and a turbulence
scheme to handle vertical mixing. At the early 1980’s, the model was used primarily to simulate
estuaries, such as the Hudson-Raritan Estuary (by Leo Oey) and the Delaware Bay (Boris
Galperin), but also first attempts to use a sigma coordinate model, for basin-scale problems
which were simulated with the coarse resolution model of the Gulf of Mexico (Blumberg and
Mellor) and models of the Arctic Ocean (with the inclusion of ice-ocean coupling by Lakshmi
Kantha and Sirpa Hakkinen). The principal attributes of the POM model are as follows:
• It contains an embedded second moment turbulence closure sub-model to provide vertical
mixing coefficients.
• It is a sigma coordinate model in that the vertical coordinate is scaled on the water
column depth.
• The horizontal curvilinear orthogonal coordinates and an Arakava C differencing scheme.
• The horizontal time differencing is explicit whereas the vertical differencing is implicit.
This later eliminates time constraints for the vertical coordinate and permits the use of
fine vertical resolution in the surface and bottom boundary layers.
• The model is able to use free surface and a split time step. The external model portion
of the model is two–dimensional and uses a short time step based on the CFL2 condition
and the external wave speed. The internal mode is three–dimensional and uses a long
time step based on the CFL condition and the internal wave speed.
1
http://www.gfdl.gov/~smg/MOM/MOM.html
CourantFriedrichsLewy condition (CFL condition) is a necessary condition for convergence while solving
certain partial differential equations
2
5
• Complete thermodynamics have been implemented.
The turbulence closure sub-model is one that was introduced my George Mellor, 1973 and was
significantly advanced in collaboration with Tetsuji Yamada (Mellor and Yamada, 1974; Mellor
and Yamada, 1982). It is often cited in the literature as the Mellor and Yamada turbulence
closure model. There are other versions of model in existence such as a non-Boussinesq3 version
and a more general vertical coordinate version of which the sigma coordinate is a special case.
3
Basic equations
Here we consider how Newton’s Second Law of Motion can be written in a form, which can be
applied in oceanography:
F~ = m~a
(1)
~
F
It is convenient to write ~a = m
and think of the Law as acceleration, due to the resultant force
acting per unit mass. Resultant force is sum of all forces and is dependent of pressure, gravity,
frictional and tidal force. In Cartesian coordinates, with the approximation of hydrostatics
and f-plane4 approximation, the system of equations for free surface liquid, notation has the
following vector form:
~
dV
~ ×V
~ + ~g + F~ ,
= −α · ∇p − 2Ω
(2)
dt
~ ×V
~ is Coriolis therm, ~g is gravity and F~ are other forces including
where p is pressure, 2Ω
viscosity. Capital letter V denotes macroscopic – verticaly integreated velocity. Small letters
u, v are x and y component of microscopic–local flow velocity, w denotes microscopic verital
velocity. Equations can be written also as three component equations:
x:
ρ
du
+ f∗ w − f v
dt
y:
ρ
dv
+ fu
dt
z:
ρ
dw
− fu
dt
∂p ∂τ xx ∂τ xy ∂τ xz
+
+
+
∂x
∂x
∂y
∂z
xy
yy
∂τ
∂τ yz
∂p ∂τ
+
+
+
=−
∂y
∂x
∂y
∂z
xz
yz
∂τ
∂τ
∂τ zz
∂p
+
+
,
= − − ρg +
∂z
∂x
∂y
∂z
=−
(3)
(4)
(5)
where f = 2Ω sin ϕ is Coriolis parameter, f∗ = 2Ω cos ϕ is reciprotial Coriolis parameter and
effects the equations only near the equator, in most cases is neglected. Ω is angular velocity
around Earth’s axis, ϕ defines position on the Erath’s surface in north-south direction. Equations are written as three component equation, with the coordinates x, y, z and their respective
velocity components u, v and w, being positive in the east, north and upward directions respectively. The origin of coordinates being at the sea surface, ρ is density of sea watter, τ is
friction tensor and g is gravity.
3
4
Density differences are neglected unless differences are multiplied by gravity.
The f-plane approximation is an approximation where the Coriolis parameter, f, is set to a constant value.
6
3.1
Dynamic and Thermodynamic Equations
For simplifying the problem, two approximations are used[8]. Firstly, it is assumed that the
weight of the fluid identically ballances the pressure (hydrostatic assumption) and secondly,
density differences are neglected unless the differences are multiplied by gravity (Boussinesque
approximation).
Consider a system woth orthogonal Cartesian coordinates, x increasing eastwards, y increasing
northward and z increasing vertically upwards. The free surface is located at z = η(x, y, t)
and the bottom is at z = −H(x, y). If V~ is horizontal macroscopic–global velocity vector with
components and ∇ the horizontal gradient operator, W is macroscopic vertical velocity, the
continuity equation follows[2]:
∂W
∇ · V~ +
= 0.
(6)
∂z
The Raynolds momentum equations are:
!
∂U ~
∂U
∂U
1 ∂P
∂
KM
+ Fx
+ V · ∇U + W
− fV = −
+
∂t
∂z
ρ0 ∂x
∂z
∂z
!
∂V
∂V
1 ∂P
∂
∂V
~
KM
+ Fy
+ V · ∇V + W
+ fU = −
+
∂t
∂z
ρ0 ∂y
∂z
∂z
∂P
,
ρg = −
∂z
(7)
(8)
(9)
with ρ0 reference density, ρ density, g gravitational acceleration, P the pressure, KM 5 vertical
eddy diffusivity of turbulent momentum mixing. The latitudinal variation of the Coriolis parameter f is introduced by use of β 6 plane approximation.
The pressure at depth z can be obtained as:
p=
Z η
z
ρgdz = ρ0 gη +
Z Q
z
ρgdz = ρ0 gη + ps ,
(10)
where η is deviation of free surface from its undisturbed position Q and the ps is:
ps =
Z Q
z
ρgdz.
(11)
The conservation equations for temperature Θ and salinity S may be written as:
!
∂Θ ~
∂Θ
∂Θ
∂
KH
+ FΘ
+ V · ∇Θ + W
=
∂t
∂z
∂z
∂z
!
∂S ~
∂S
∂S
∂
KH
+ FS ,
+ V · ∇S + W
=
∂t
∂z
∂z
∂z
5
(12)
(13)
The turbulent transfer of momentum by eddies giving rise to an internal fluid friction, in a manner analogous
to the action of molecular viscosity in laminar flow, but taking place on a much larger scale. The value of the
coefficient of eddy viscosity (an exchange coefficient) is of the order of 1m2 s1 , or one hundred thousand times
the molecular kinematic viscosity.
6
An approximation whereby the Coriolis parameter, f, is set to vary linearly in space is called a beta plane
approximation.
7
where Θ is temperature and S is the salinity. The vertical eddy–macroscopic diffusivity for turbolent mixing of heat and salt is denoted as KH . Because the microscopic processes responsible
for fluid mixing are too complex to model in detail, oceanographers generally treat fluid mixing
as a macroscopic ”eddy” diffusion process. Using the temperature and salinity, the density is
derived according to an equation of state and has the form:
ρ = ρ (Θ, S) .
(14)
The density is ρ, that is, the density evaluated as a function of temperature and salinity, but
at atmospheric pressure and it provides accurate density information.
4
Finite difference scheme
A rectangular grid[7] xi , yi , zi is specified and the whole domain is represented as a set of grid
boxes. Lateral boundaries coincide with vertical planes passing though the grid nodes. The
upper boundary of the grid boxes changes in time while the lower boundary is fixed, beeing
determined by the bottom relief. The vertical step varies from surface to the bottom and the
number of layers is determined by the overall depth and changes from one (when the upper
and lower layers coincide) up to n.
4.1
One dimensional case - Forward difference
The most direct method for numerical differentiation of a function starts by expanding it in
Taylor series. This series advances the function one small step forward:
h2 ′′
h3 ′′′
f (x) + f (x) + . . . ,
(15)
2
6
where h is the step size. We obtain the forward difference derivate algorithm by solving (5) for
f (x):
′
f (x + h) = f (x) + hf (x)
f (x + h) − f (x)
(16)
h
h ′′
′
(17)
≃ f (x) + f (x) + . . . ,
2
where the subscript c denotes a computed expression. One can think of this approximation as
using two points to represent the function by a straight line in te interval from x to x + h. The
approximation has an error proportional to h.
′
fc ≃
4.2
One dimensional case - Central difference
An improved approximation to the derivative starts with the basic definition (6). Rather than
making a single step of h forward, we form central difference by stepping forward by h/2 and
backward by h/2:
f (x + h/2) − f (x − h/2)
′
fc ≈
= Dc f (x, h).
(18)
h
8
Figure 1: Forward Difference(solid line) and central-difference (dashed line)
The important difference from (6) is that when f (x − h/2) is substracted from f (x + h/2), all
terms containing an odd power of h in Taylor series disappear. Therefore, the central-difference
algorithm becomes accurate to one order higher in h; that is, h2 . If function is well behaved;
that is, if (f 3 h2 )/24 ≪ (f 2 h)/2, then one can expect the error by central differrence method
to be smaller than error, produced by the forward difference (6).
4.3
LEAP - FROG scheme
The leap-frog scheme used in POM[9] model described in the present paper is a central difference
scheme with trunction error of second order. If we look at the figure 2, the time derivate of η
would be:
i
1 h k+1
∂η
k
ηi,j − ηi,j
,
(19)
=
∂t
∆t
where partial derivates of M, N are:
∂M
∂x
∂N
∂y
1
k+ 1
k+ 1
Mi+ 12,j − Mi− 12,j
=
2
2
∆x
1
k+ 21
k+ 21
Ni,j+ 1 − Ni,j− 1 .
=
2
2
∆y
(20)
(21)
Indices i,j are grid indices of XY plane on chosen depth, while index k is time step in this
particular case. An example shown in figure 2 is two–dimensional and the third parameter is
time. It is possible to solve shallow watter equations (3-5) in 3D, while the fourth parameter is
time, but it is harder to draw and understand the scheme. Because it is easier to understand
9
Figure 2: Arrangement of points for computation in leap-frog method
2D scheme, in this paper it will be only two dimensional grids will be dicoused. Assuming that
values at k and k + 1/2 time steps are known, the only unknown η(i, j, k + 1) is derived as:
k+1
k
ηi,j
= ηi,j
−
∆t
∆t
k+ 1
k+ 1
k+ 1
k+ 1
Mi+ 12,j − Mi− 12,j −
Ni,j+21 − Ni,j−21 .
2
2
2
2
∆x
∆y
(22)
As seen in figure 2, values of M, N are specified at the edge of the box, while the result for
the variable η is specified at the center of box. This scheme is known in literature as Arakawa
C-grid. All other shallow watter equations (3-5) are similary implemented in the ocean model.
5
Boundary conditions
Wind stress components, kinematics conditions and buoyancy flux were specified at the sea
surface as[9]:
Nz
∂u
∂v
∂η
∂η
∂η
∂ρ
= τOx , Nz
= τOy ,
+u
+v
= w, Kz
= 0,
∂z
∂z
∂t
∂x
∂y
∂z
(23)
where τOx and τOy are components of wind stress vector at the surface in x and y direction
respectively. Boundary conditions at the bottom are specified as:
Nz
∂v
∂H
∂H
∂ρ
∂u
= τbx , Nz
= τby , u
+v
= w, Kz =
= 0,
∂z
∂z
∂x
∂y
∂t
(24)
where τbx and τby are bottom fricition stress written as:
(τbx , τby ) = αub|U~b |, αvb |U~b | .
10
(25)
Ub is current speed near bottom, bottom friction coeficient is choosen as α = 2.5·10−3, α = ρ0 Cd ,
where ρ0 is densitiy and Cd is drag coefficient at barometric pressure, given by:
−2
1
.
Cd = ln (H + zb ) /z0
κ
(26)
The bottom stress coefficient is determined by matching velocities with logarithmic law of wall.
zb and u, v are grid points and corresponding velocities in the grid point nearest to the bottom
and κ is the von Karman constant. The parameter z0 depends on the local bottom roughness
and is typically set for the ocean to z0 = 1cm, as suggested by Weatherly and Martin [1978].
Near bottom at the boundary layer, for determining fluid profile kinematic viscosity is used, so
the profile is not turbulent.
6
Sigma coordinate system
To simplify solving basic equations, while bottom topography is included in calculation, it
is vise to transform our basic equations(3-5) (shallow watter equations) in sigma coordinate
system[4]. The new coordinates would be:
x = x∗ ,
y = y∗,
σ=
z − η(t)
,
H(x) − η(x, t)
(27)
where σ becomes a vertical coordinate in z direction, but it has a value within [0,1]. σ = 0 at
the surface and σ = −1 at the bottom. x, y, t are ordinary Cartesian coordinates. If the basic
equations are transformed in sigma coordinates, then the equations are always in same form,
which means it is independent of the bottom topography.
Figure 3: The sigma coordinate system
11
7
Seamount problem
In this section we would examine, fluxes, surface elevation on sea-mount case with model based
on Leap-frog scheme. Bottom topography of the sea-mount problem is shown in figure 4:
Figure 4: Seamount bottom topography, legend shows depth in meters
The basin in figure7 4 is 3500 km long and 2500 km wide. In the center bottom rises from
−4500 m to −400 m. From west it flows prescribed flux with velocity 0.2m/s. In figure 5 it
is shown profile of the flux at 15th of 49 sigma level and surface elevation. In this case basin
was closed in all directions. It is clearly seen, what effect would have bottom topography on
currents and surface elevation. Coriolis parameter was set to 1−4 /s, salinity was neglected,
3D calculation was performed with temperature and salinity held fixed, advection scheme was
centred, as originally provided with POM, grid size was 65 × 49 points, there were 49 sigma
7
All plots are made with John Hunter’s Pomviz package in Matlab in standard form used in oceanography,
the graphs have units which are use as a standard in oceanography, end are derived from the package used in
Matlab http://staff.acecrc.org.au/~johunter/ozpom/readme_pomviz
12
Figure 5: Surface flux and surface elevation, legend shows surface elevation in meters
levels, duration of simulation was 0.05 day, initial print interval - frame interval was 0.0015
day, reference density ρ0 = 1025kg/m3, acceleration due to gravity was taken as constant
g = 9.806m/s2 , von Karman constant κ = 0.4, bottom roughness z0b = 0.01m, Smagorinsky
diffusivity was set 0.2, inverse turbulent Prandtl number 0, temperature boundary condition
was prescribed temperature, surface salinity boundary condition prescribed salinity, which was
set to value 35. One must have in mind, that sigma layers are not at constant depth, but
go along the bottom relief, as shown in figure 3. In figure 6 are shown points in the i, j grid,
where the velocity profiles were taken. Letters A, B and C are cells, where the velocities are
being calculated, as seen in figure 7. In figure 8, it is shown the u velocity profile at 33th slice.
As expected, velocity u is increased, when the bottom rises. Those results, velocity profiles,
currents, etc..., are typical results the of the POM model, for more complex basin or even a
numerical topography of a bay, we have to include different topology in the model so the sigma
levels can be calculate. As a result we would receive similar graphs, which are standard output
of POM model and can be coupled with MATLAB pre–written package, to present the results.
13
Figure 6: Locations of current profiles
Figure 7: Velocity profiles in points A, B, C in figure 6
14
Figure 8: Velocity u - X section
The units of all graphs are left in scientific number format, because that is the standardization
in oceanography and all visualization tools used in oceanography use that units as standard in
oceanography.
15
8
Adriatic Tsunami simulation
Dr. V. Malačič and B. Petelin, M. Sci, have performed a simulation of Adriatic Tsunami[6].
They simulated the spread of Tsunami generated by a sub–sea quake near Bar situated in a
seismically active region of Montenegro. An earthquake of magnitude 6.8 on the Richter scale
with an epicentre at 13km depth which could generate a wave of 0.25m at the open sea. For
this simulation the sea level – surface exactly, near Bar was initially perturbed for 10 minutes
duration with an oscillation of 0.5 meter amplitude and period of 30 seconds. The simulation
was performed using the 3–dimensional Princeton Ocean Model (POM), which is routinely used
by the MBS (Marine Biology Station in Piran) and by the National Institute for Geophysics
and Volcanology in Bologna, to forecast the circulation of water masses within the Adriatic sea.
The POM model had had a spatial grid of 5km x 5km, similar simulations were carried out at
the Geophysics Institute in Zagreb twenty years ago and the results published in the popular
journal Priroda (Ima li Tsunamija u Jadranskom moru, Mirko Orlič, Priroda, May-june,1984,
310-311). The POM’s simulation of Tsunami confirmed the earlier results. The wave takes
more than 6 hours to arrive at the Gulf of Trieste with strongly attenuated amplitude of less
than 0.004m (4 mm). The tsunami attenuates principally due to destructive interference from
waves that are reflected by coast and by the pronounced bathymetric variations (rising sea
floor) in the southern Adriatic basin. The broken region of periodic wave reinforcement due
to constructive interference also experiences destructive interference, with stray reflected waves
and so is not stable. In addition the propagating tsunami loses energy due to wave friction
in the shallow waters of the northern Adriatic. The numerical simulation may overestimate
this friction and it need to be investigated further. However, even with a drastic reduction in
simulated friction, the anticipated wave amplitude in the Gulf of Trieste is expected to be less
than 0.4 m because it must be reduced from the wave amplitude in the generation region near
Bar.
16
Figure 9: Adriatic Tsunami[6]
9
Conclusion
Numerical models of ocean circulation are among the most demanding tasks performed by
computers today. Although some simple models used on small region of the ocean, over a short
time period, can be solved on a personal computer, realistic global models take thousands of
hours on the fastest super computers available. One way to get this kind of computational
power is to link many computers together to work on a problem, this strategy is known as
”parallel computing”[7]. Much of the numerical work done at COAS utilizes a massively parallel
Connection Machine CM5 which consists of 64 fast processors linked together by a fast network
and coordinated by special software. Other modeling work at COAS uses a Silicon Graphics
Power Challenge and an IBM SP2. The computer facilities at COAS are among the finest
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available at any oceanographic research institution anywhere in the world. Even when the
fastest computers are used, numerical models are only approximations of the full dynamics and
thermodynamics, the solutions are therefore only approximations. One way to correct errors,
caused by algorithm in the model, is to apply a procedure known as ”data assimilation”,
which blends the model approximations with observations of the real ocean in a least–square
error sense. This blending takes into account, the errors produces by the model, dynamics
and thermodynamics, as well as measurement errors made by observations. Princeton Ocean
model was used for simulating many problems in dynamical oceanography. There are also
many other ocean models in use, but most widespeaded is the POM, developed by Mellor
and Blumberg. All models use similar Arakawa C-grid [8], but there are many differences in
routines for calculating turbulent terms in shallow watter equations. Those routines were not
described in this seminar. With faster computers and large clusters it is possible to solve
larger problems on larger, more detailed grid. Most advancing technology uses multi processor
hardware and most promising are graphics processors and GPU scientific clusters like Nvidia
Tesla. Research in physical oceanography is basically divided in two parts, model developing,
which is described in this paper and in theoretical physical oceanography. Theoretical physical
oceanographers are trying to describe problems analytically, but unfortunately, nowadays it is
still hard to solve even most simple oceanographic problems analytically, with an exception
of few examples. That is one reason more, why is research in that field so important. Most
difficult is to understand the turbulent terms in shallow watter theory and even harder to
find a perfect model for calculation of turbulent term, that is the reason why are so many of
ocean models in use. There is no universal model for solving oceanographic problems, we have
to change the model for every particular problem. Ocean models are applied to automatic
ocean forecasting systems like: Princeton Regional Ocean Forecasting System (GOM), New
York Harbor Observing and Prediction System (NYHOPS), U.S. Coastal Observing System
(COOS) and many others. Most important in our region is Adriatic Sea Forecasts (INGV)
in Italy. All those models use as input combination of data measured with ocean stations
and data received from satellites. This field of physics is important for weather prediction and
prediction of hurricanes like Catrina in the USA or well known Tsunami, which killed thousands
of people in Indonesia and India few years ago. Physical oceanographers have predicted in every
particular case possible catastrophe, but the governments in that regions did not react as they
should have.
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References
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Annex III, Toulouse, France, 10-11 May 1999.
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