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Numerical Modeling in Physical
Oceanography
Joanna Staneva
ICBM, Univerity of Oldenbirg
What do we study?
Geophysical fluid dynamics
• Oceanography
• Meteorology
• Climate dynamics
MOTIVATION
• Changes in the world ocean are going to be
important factors in any global change processes.
• It is difficult to build an „universal'' ocean model that
can treat accurately phenomena on all spatial and
temporal scales in the various ocean basins.
• This is due both to finite computer size and CPU
speed, and to an imperfect description of the
physical processes, such as turbulence.
• Ocean modeling efforts have diversified into different
classes,
Classification of Ocean Models
Free Surface and Rigid Lid Models
• In reality, the ocean surface is free to deform under
the influence of wind, heating and tidal forces. We find
wind-driven waves and surges up to several meters
high on its surface. These are typically short-lived,
have short spatial scales and fast wave speeds.
• In order to avoid the severe limitation on the time step
due to the fast gravity waves, one puts a rigid lid on
the ocean as this affects the large-scale motions only
slightly. The first such model was formulated by Bryan.
• This model has been recently reformulated by
Killworth et al. to retain the free surface by treating the
fast modes separately. Models that treat the fast
waves implicitly have been developed by Hurlburt et
al.
Fixed Level, Isopycnal, SigmaCoordinate and Semi-Spectral Models
• The models of Bryan and Killworth et al. have all used
fixed levels in the vertical -direction, with a variable
spacing of the depth levels.
• On the other hand, Blumberg and Mellor and Haidvogel
et al. have introduced a stretched coordinate - referred
to as sigma defined as =z/D, where D is the fluid
depth.
• In addition Haidvogel et al. have introduced a semispectral representation of the vertical dimension (sigma
layout) in terms of Chebyshev polynomials (collocation
method).
Barotropic vs. Baroclinic Models
• Ocean models describe the response of a variable
density ocean to atmospheric momentum and heat
forcing. This response can very simply be represented
in terms of eigenmodes of a linearized system of
equations.
• The 0-th mode is equivalent to the vertically-averaged
component of the motion, also known as the
barotropic mode.
• The higher modes are called baroclinic modes and are
associated with higher order components of the
vertical density profile.
Barotropic vs. Baroclinic Models
Barotropic vs. Baroclinic Models
• Many ocean models make the hydrostatic shallow
water approximation, in which the pressure depends
only on the depth , i.e. it's given by the classic
hydrostatic relation
dp/dz=z
• This relation holds if the horizontal dimensions of the
ocean volume under consideration are much larger
than the vertical dimension, hence the shallow water
designation.
• A particular form of the baroclinic models are the socalled reduced gravity models. These are essentially
isopycnal models of several deformable layers where
the lowest layer has infinite depth and zero velocity.
Barotropic Models
Why a barotropic model is interesting and important?
• The free surface elevation couples directly to the
barotropic mode. (using of satellite altimeter
measurements of the free surface elevation). Thus
information from altimeters may first enter the ocean
model through the barotropic mode, where it
represents a direct forcing
• The presence of the fast free surface gravity waves.
The simple explicit finite-difference schemes treating
such waves are subject to severe time step limitations,
so that solution of the barotropic mode may lead to
large CPU requirements.
Model Equations for a Barotropic Ocean
Navier-Stokes equations for incompressible flow on a rotating Earth
where:
Boundary conditions
closed boundaries in a rectangular geometry
Forcing for Barotropic Models
The circulation in a barotropic ocean is generally
the result of two kinds of ``forcings'':
• the wind stress at the ocean's surface
• the source-sink mass flows at the basin
boundaries. The source flows could be ocean
currents that enter the basin due to wind forcing
in an adjacent basin, or enter simply to replace
mass driven out by the wind or baroclinic
pressure gradients in the model basin. Sink
flows have similar origins.
Distribution of Temperature and Salinity
Annual mean sea surface temperature
Distribution of Temperature and Salinity
Annual mean sea surface salinity
Bottom Topography
• Bottom depth is one of the most important parameters
for a realistic ocean model.
• Bottom depth is derived from acoustic soundings from
a ship.
• Very high resolution (approaching 1 km) bottom
topography is generally not available in the public
domain.
• The best resolution public domain topography thus far
is the gridded ETOP5 at NCAR, which contains 5 min
resolution of Earth's topography.
• Another topographic data set is the DBDB-5 (Digital
Bathymetry Data Base at 5 min intervals), developed
by the U.S. Naval Oceanographic
On finite differencing
• Typical equation in Oceanography and
Geophysical Fluid Dynamics (GFD)
• On finite differences
• Finite difference form of equations
• Higher order schemes
• Time evolution
• More complex models and grid arrangements
• PROBLEMS!
On finite differencing
• Typical equation in Oceanography and
Geophysical Fluid Dynamics (GFD)
• On finite differences
• Finite difference form of equations
• Higher order schemes
• Time evolution
• More complex models and grid arrangements
• PROBLEMS!
Typical equations in Oceanography
• Equations

    
U
  D   F (t , x, )
t
x x  x 
• Boundary conditions
  b(t )
or
 (t )
 a( (t ))  b(t )
x x  0
On finite differencing
• Typical equation in Oceanography and
Geophysical Fluid Dynamics (GFD)
• On finite differences
• Finite difference form of equations
• Higher order schemes
• Time evolution
• More complex models and grid arrangements
• PROBLEMS!
Taylor expansion
1  ( x )
1  2 ( x ) 2 1  3 ( x ) 3
 ( x  x )   ( x ) 
x 
x 
x  ...
2
3
1! x
2! x
3! x
For the first order derivative
 ( x )   ( x )   ( x  x ) O( x 2 )


x
x
x
which is correct to
order:
O ( x 2 )
~ O ( x )
x
More accurate scheme
 ( x)
1  2 ( x) 2 1  3 ( x) 3
 ( x  x)   ( x) 
x 
x 
x  ...
2
3
x
2 x
6 x
 ( x)
1  2 ( x) 2 1  3 ( x) 3
 ( x  x)   ( x) 
x 
x 
x  ...
2
3
x
2 x
6 x
For the first order derivative
 ( x )   ( x  x )   ( x  x ) O( x 3 )


x
2x
x
which is correct to
order:
O( x 2 )
Second order derivative
 ( x)
1  2 ( x) 2 1  3 ( x) 3
 ( x  x)   ( x) 
x 
x 
x  ...
2
3
x
2 x
6 x
 ( x)
1  2 ( x) 2 1  3 ( x) 3
 ( x  x)   ( x) 
x 
x 
x  ...
2
3
x
2 x
6 x
(1)+(2) and rearranging:
 2 ( x )  ( x  x )  2 ( x )   ( x  x ) O( x 4 )


2
2
x
x
x 2
which is correct to
order:
O( x 2 )
Thin fluid layers
Large aspect ratio
Highly turbulent
Gulf stream: Re~1012
Large variety of scales
Parameterizations are important in geophysical fluid
dynamics
Timescales
• Atmospheric low pressures:
•
•
•
•
•
•
•
•
10 days
Seasonal/annual cycles:
0.1-1 years
Ocean eddies:
0.1-1 year
El Nino:
2-5 years.
North Atlantic Oscillation:
5-50 years.
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean:
4.000 years.
Glacial-interglacial timescales: 10.000-200.000 years.
Tropical hurricane Floyd
Timescales
• Atmospheric low pressures:
• Seasonal/annual cycles:
years
•
•
•
•
•
•
•
10 days
0.1-1
Ocean eddies:
0.1-1 year
El Nino:
2-5 years.
North Atlantic Oscillation:
5-50 years.
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean:
4.000 years.
Glacial-interglacial timescales: 10.000-200.000 years.
Plankton bloom
Plankton bloom
Plankton bloom
Plankton bloom
Timescales
• Atmospheric low pressures:
• Seasonal/annual cycles:
• Ocean eddies:
•
•
•
•
•
•
10 days
0.1-1 years
0.1-1 year
El Nino:
2-5 years.
North Atlantic Oscillation:
5-50 years.
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean:
4.000 years.
Glacial-interglacial timescales: 10.000-200.000 years.
Timescales
• Atmospheric low pressures:
• Seasonal/annual cycles:
• Ocean eddies:
10 days
0.1-1 years
0.1-1 year
• El Nino:
2-5 years.
•
•
•
•
•
North Atlantic Oscillation:
5-50 years.
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean:
4.000 years.
Glacial-interglacial timescales: 10.000-200.000 years.
Normal state
Initial ENSO state
The ENSO state
The ENSO state
Timescales
•
•
•
•
Atmospheric low pressures:
Seasonal/annual cycles:
Ocean eddies:
El Nino:
• North Atlantic Oscillation:
years.
•
•
•
•
10 days
0.1-1 years
0.1-1 year
2-5 years.
5-50
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean:
4.000 years.
Glacial-interglacial timescales: 10.000-200.000 years.
Positive NAO phase
Negative NAO phase
Timescales
•
•
•
•
Atmospheric low pressures:
Seasonal/annual cycles:
Ocean eddies:
El Nino:
• North Atlantic Oscillation:
• Turnovertime of atmophere:
10 days
0.1-1 years
0.1-1 year
2-5 years.
5-50 years.
10 years.
• Anthropogenic forced climate change: 100 years.
• Turnover time of the ocean:
4.000 years.
• Glacial-interglacial timescales: 10.000-200.000 years.
Ozon at Antartic
Timescales
•
•
•
•
•
•
•
Atmospheric low pressures:
10 days
Seasonal/annual cycles:
0.1-1 years
Ocean eddies:
0.1-1 year
El Nino:
2-5 years.
North Atlantic Oscillation:
5-50 years.
Turnovertime of atmophere:
10 years.
Anthropogenic forced climate change: 100 years.
• Turnover time of the ocean:
years.
4.000
• Glacial-interglacial timescales: 10.000-200.000 years.
Temperature in the North Atlantic
Ocean conveyer belt
Timescales
•
•
•
•
Atmospheric low pressures:
Seasonal/annual cycles:
Ocean eddies:
El Nino:
• North Atlantic Oscillation:
10 days
0.1-1 years
0.1-1 year
2-5 years.
5-50 years.
• Turnovertime of atmophere:
10 years.
• Anthropogenic forced climate change: 100 years.
• Turnover time of the ocean:
4.000 years.
• Glacial-interglacial timescales: 10.000-200.000 years.
Orbital forcing
Glacial-interglacial cycles
Ice coverage, sea level
What model will we use?
Mathematical models
• Development and evaluation of coupled modelling
capabilities
calibrated
against
data
from
observations.
• The numerical models aim to develop the necessary
tools for the integrated study of physical and
ecosystem dynamics.
• Great efforts have to be made to ensure the
interdisciplinary connections. The oil and
chemical spill modeling can be integrated in the
"real-time" simulations. The assimilation would
additionally enhance this mathematical tool.
Numerical models Some Words of Caution
Numerical models of ocean currents have many
advantages.
• They simulate flows in realistic ocean basins
with realistic bottom topography.
• They include the influence of viscosity and nonlinear dynamics.
• They are used to calculate possible future flows
in the ocean.
• They interpolate between sparse observations
of the ocean produced by ships, drifters, and
satellites.
Numerical models Some Words of Caution
Numerical models are not without problems.
"There is a world of difference between the
character of the fundamental laws, on the one
hand, and the nature of the computations required
to breathe life into them, on the other''-Berlinski
(1996).
• The models can never give complete descriptions
of the oceanic flows even if the equations are
integrated accurately.
• The problems arise from several sources.
Numerical models Some Words of Caution
Discrete equations are not the same as continuous
equations
• Discretization is essential for computer
implementation and cannot be dispensed with.
• The essence of the difficulty is that the dynamics
of discrete systems is only cloosely related to that
of continuous systems-indeed the dynamics of
discrete systems is far richer than that of their
continuous counterparts-and the approximations
involved can create spurious solutions.
Numerical models Some Words of Caution
Calculations of turbulence are difficult
• Numerical models provide information only at grid
points of the model. They provide no information about
the flow between the 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. Clearly, such a model can be used only
for flow in a small box.
• Practical ocean models have grid points spaced tens
to hundreds of kilometers in the horizontal, and tens to
hundreds of meters in the vertical.This means that
turbulence cannot be calculated directly, and the
influence of turbulence must be parameterized.
Numerical models Some Words of Caution
Practical models must be simpler than the real
ocean
• Models of the ocean must run on available
computers.
• This means oceanographers further simplify their
models, usually giving up resolution in the
horizontal or vertical.
• They cannot, for example, run the most detailed
models of oceanic circulation for thousands of
years to understand the role of the ocean in
climate.
Numerical models Some Words of Caution
Initial conditions are not well known.
How to initialize the model?
• We do not know accurately the present velocity and
density in the ocean.
• The best we can do is to start at rest using the best
estimates of the ocean's density field, such as that
contained in cöimatic data produced by Levitus
• .... or we can use the output from an earlier run of the
model or a similar model. Still, there are difficulties.
• ....and the oceans take hundreds of years to come to
equilibrium with the atmosphere, so models must run
for hundreds of years to get the right deep circulation.
Numerical models Some Words of Caution
Numerical code has errors
Do you know of any software without bugs (errors)?
• Numerical models use many subroutines each with many
lines of code which are converted into instructions
understood by the computer circuitry using other software
called a compiler. Eliminating all software errors is
impossible.
• With careful testing, the output may be correct, but the
accuracy cannot be guaranteed. Plus, numerical calculations
cannot be more accurate than the accuracy of the floatingpoint numbers and integers used by the computer. Round-off
errors cannot be ignored.
Numerical models Some Words of Caution
Summary
• Despite these many sources of errors, most are small
in practice.
• Numerical models of the ocean are giving the most
detailed and complete views of the circulation
available to oceanographers.
• Some of the simulations contain unprecedented
details of the flow.
• I included the words of warning not to lead you to
believe the models are wrong, but to lead you to
accept the output with a grain of salt.
Numerical Models in Oceanography
• 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.
(including models for describing planetary waves, the
interaction of the flow with sea-floor features, or the
response of the upper ocean to the wind ).
• Simulation models are used for calculating realistic
circulationof oceanic regions. The models are often
very complex because all important processes are
included, and the output is difficult to interpret.
Primitive - Equation Models
Geophysical Fluid Dynamics Laboratory Modular Ocean Model
MOM
• perhaps the most widely used model growing out of the
original Bryan-Cox code.
• It consists of a large set of modules that can be configured to
run on many different computers to model many different
aspects of the circulation.
• The model is widely use for climate studies and and for
studying the ocean's circulation over a wide range of space
and time scales
• The model uses the momentum equations, equation of state,
and the hydrostatic and Boussinesq approximations. Subgridscale motions are reduced by use of eddy viscosity. Versions
3+ of the model has a free surface, realistic bottom
topography, and it can be coupled to atmospheric models.
Primitive - Equation Models
Semtner and Chervin's Global Model
• was perhaps the first, global, eddy-resolving model based on
the Bryan-Cox models
• It has much in common with the MOM and it provided the first
high resolution view of ocean dynamics.
• It has a resolution of 0.5° x 0.5° with 20 levels in the vertical.
• It has simple eddy viscosity, which varies with scale; and it
does not allow static instability.
• In contrast with earlier models, it is global, it resolves the
largest turbulent eddies, and it has realistic bottom topography
and coastlines. Originally, it had a rigid lid to eliminate fastmoving waves such as tides.
• More recent versions of the model have a free surface,
eliminating the restrictions of the rigid lid.
Primitive - Equation Models
Parallel Ocean Program Model
• produced by Smith, Dukowicz, and Malone(1992).
The modifications included removing the rigid-lid at the surface,
improving the numerical algorithms, and adding realistic
coasts, islands, and unsmoothed bottom topography.
• The model has 1280 x 896 grid points on a Mercator grid
which extends from 78° S to 78° N, and 20 levels in the
vertical. The Mercator projection gives grid spacing that
decreases with latitude at the same rate that the diameter of
typical eddies decreases. Horizontal resolution varies from
6.5km at the highest latitudes to 31.25 km at the Equator.
• The model was initialized using temperature and salinity
calculated from Semtner's(1993) 0.25° model. The model was
then integrated for a 10-year period beginning in 1985 using
various surface-forcing functions.
Primitive - Equation Models
Miami Isopycnal Coordinate Ocean Model MICOM
• All the models just described use x, y, z coordinates. Such a
coordinate system has disadvantages. For example, mixing in
the ocean is easy along surfaces of constant density, and
difficult across such surfaces.
• A more natural coordinate system uses x, y, , where  is
density. A model with such coordinates is called an isopycnal
model.
• Essentially, (z) is replaced with z(). Furthermore, because
isopycnal surfaces are surfaces of constant density, horizontal
mixing is always on constant-density surfaces in this model.
Instantaneous, near-surface geostrophic currents in the
Atlantic for October 1, 1995 calculated from the Parallel
Ocean Program numerical model developed at the Los
Alamos National Laboratory.The length of the vector is
the mean speed in the upper 50 m of the ocean; the
direction is the mean direction of the current.
Output of Bleck's Miami Isopycnal Coordinate Ocean
Model MICOM. It is a high-resolution model of the Atlantic
showing the Gulf Stream, its variability, and the circulation
of the North Atlantic (FromBleck).
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 topography,
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.
Storm-Surge Models
• Storms coming ashore across wide, shallow, continental
shelves drive large changes of sea level at the coast called
storm surges. The surges can cause great damage to coasts
and coastal structures.
• Calculating storm suges is not easy. Here are some reasons, in
a rough order of importance.
1. The distribution of wind over the ocean is not well known.
2. The shoreward extent of the model's domain changes with
time.
3. The drag coefficient of wind on water is not well known for
hurricane force winds. The drag coefficient of water on the
seafloor is also not well known.
4. The models must include waves and tides which influence sea
level in shallow waters.
5. Storm surge models must include the currents generated in a
stratified, shallow sea by wind.
Topographic map of the Gulf of Maine showing
important features of the Gulf.
(From Lynch et al, 1993).
Triangular, finite-element grid used to compute flow in the Gulf
of Maine shown in the previous figure. Note that the size of the
triangles varies with depth and rate of change of depth. (From
Lynch et al, 1993).
Coupled Ocean and Atmosphere Models
• Coupled numerical models of the atmosphere and the
ocean are used to study the climate system, its natural
variability, and its response to external forcing.
• The most important use of the models has been to study
how Earth's climate might respond to a doubling of CO2
in the atmosphere.
• Other important uses of coupled models include studies
of El Niño and the meridional overturning circulation. The
former varies over periods of a few years, the latter
varies over a period of a few centuries.
Nested model
Allows a very fine model set-up for a limited area using results
from coarse grid simulations.
Involves two simulations running simultaneously:
The main run – the whole area (relatively coarse grid)
The secondary model - much smaller area (finer numerical grid)
Merging grids
BSH Model
Regions
MIKE 21 - NESTED
HYDRODYNAMIC MODULE
 Using a nested grid provides
a feedback mechanism
between the large and fine
grid boundaries,
 It is possible to resolve the
flow fields in narrow channels
or inlets within a coarse grid
nearshore model using the
enhanced resolution of finer
grids.
DATA ASSIMILATION
METHOD: We solve for the IC and forcing, which
provide the best model-data comparisons
Global Ocean example:
Variability of Sea Level Height and oceanic state
using 4D-VAR data assimilation
We assimilate: T/P data, SST, transports, T,S
Control parameters: NCEP data and Levitus
climatology
Sea level variability
obtained by Global Ocean
Assimilation
WHY A MODEL CAN DO BETTER THAN
CORRELATIONS BETWEEN EVENTS?
 The synergy between the different human forcings cannot be
assessed from simple correlations
observations and historical correlations.
between
ecological
 Mechanistic models, which describe the kinetics between
biological and chemical compartments as a function of
meteorological and human forcings provide a powerful tool which
encompass this complexity.
 When validated the model can be used for management
purpose.
Black Sea example
 Implementing such a model for the north-western
Black Sea is our objective during the DANUBS project.
 The ecological model ERSEM is used in order to
assess the response of the north-western Black Sea
ecosystem to human-induced changes and
meteorological forcing
MIT General circulation model
MIT General circulation model
•
•
•
•
•
•
•
•
General fluid dynamics solver
Atmospheric and ocean physics
Sophisticated mixing schemes
Biogeochemical modules
Efficient solvers
Sophisticated coordinate system
Automatic adjoint schemes
Data assimilation routines
• Finite difference scheme
• F77 code
• Portable
MIT General circulation model
Spherical coordinates
“Cubed sphere”
MIT General circulation model
Some computational aspects
Experiments with
60*60*20 grid points
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Run time
Specfp (swim)
700
600
500
400
300
200
100
0
Compiler
• Portland Group (PGF) ca
• GNU fortran compiler
• Intel Fortran Compiler (IFC)
8000 s
7750 s
6225 s
Important Concepts
1.
2.
3.
4.
5.
Numerical models solve discrete equations, which are not the
same as the equations of motion described in earlier
chapters.
Numerical models cannot reproduce all turbulence of the
coean because the grid points are tens to hundreds of
kilometers apart.
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.
Numerical models can be forced by real-time oceanographic
observations from ships and satellites to produce forecasts of
oceanic currents especially eddies.
Coupled ocean-atmosphere models have much coarser
spatial resolution so that that they can be integrated for
hundreds of years to simulate the natural variability of the
climate system and its response to increased CO2 in the
atmosphere.
Thanks for your attention