1. Numerical Modeling in Physical Oceanography
Joanna Staneva
ICBM, Univerity of Oldenbirg

2. What do we study?

3. Geophysical fluid dynamics
Oceanography
Meteorology
Climate dynamics

4. 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,

5. Classification of Ocean Models

6. 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.

7. Fixed Level, Isopycnal, Sigma-Coordinate 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 semi-spectral representation of the vertical dimension (sigma layout) in terms of Chebyshev polynomials (collocation method).

10. 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.

11. Barotropic vs. Baroclinic Models

12. 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 so-called reduced gravity models. These are essentially isopycnal models of several deformable layers where the lowest layer has infinite depth and zero velocity.

13. 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.

15. Model Equations for a Barotropic Ocean
Navier-Stokes equations for incompressible flow on a rotating Earth
where:

16. Boundary conditions
closed boundaries in a rectangular geometry

17. 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.

22. Distribution of Temperature and Salinity
Annual mean sea surface temperature

23. Distribution of Temperature and Salinity
Annual mean sea surface salinity

26. 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

29. 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!

30. 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!

31. Typical equations in Oceanography
Boundary conditions or

32. 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!

33. Taylor expansion For the first order derivative
which is correct to order:

34. More accurate scheme For the first order derivative
which is correct to order:

35. Second order derivative
(1)+(2) and rearranging:
which is correct to order:

36. Thin fluid layers Large aspect ratio

37. Highly turbulent Gulf stream: Re~1012

38. Large variety of scales
Parameterizations are important in geophysical fluid dynamics

39. Timescales Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

40. Tropical hurricane Floyd

41. Timescales Seasonal/annual cycles: 0.1-1 years
Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

42. Plankton bloom

43. Plankton bloom

44. Plankton bloom

45. Plankton bloom

46. Timescales Ocean eddies: 0.1-1 year Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

48. Timescales El Nino: 2-5 years. Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

49. Normal state

50. Initial ENSO state

51. The ENSO state

52. The ENSO state

53. Timescales North Atlantic Oscillation: 5-50 years.
Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

54. Positive NAO phase
Negative NAO phase

56. Timescales North Atlantic Oscillation: 5-50 years.
Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

57. Ozon at Antartic

58. Timescales Turnover time of the ocean: 4.000 years.
Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

59. Temperature in the North Atlantic

60. Ocean conveyer belt

61. Timescales North Atlantic Oscillation: 5-50 years.
Atmospheric low pressures: 10 days
Seasonal/annual cycles: years
Ocean eddies: year
El Nino: years.
North Atlantic Oscillation: years.
Turnovertime of atmophere: 10 years.
Anthropogenic forced climate change: 100 years.
Turnover time of the ocean: years.
Glacial-interglacial timescales: years.

62. Orbital forcing

63. Glacial-interglacial cycles

64. Ice coverage, sea level

65. What model will we use?

66. 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.

67. 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 non-linear 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.

68. 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.

69. 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.

70. 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.

71. 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.

72. 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.

73. 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 floating-point numbers and integers used by the computer. Round-off errors cannot be ignored.

74. 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.

75. 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.

76. 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. Subgrid-scale 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.

77. 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 fast-moving waves such as tides.
More recent versions of the model have a free surface, eliminating the restrictions of the rigid lid.

78. 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 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.

79. 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, r, where r is density. A model with such coordinates is called an isopycnal model.
Essentially, r(z) is replaced with z(r). Furthermore, because isopycnal surfaces are surfaces of constant density, horizontal mixing is always on constant-density surfaces in this model.

80.                                      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.

81. 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).

82. 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.

83. 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.
The distribution of wind over the ocean is not well known.
The shoreward extent of the model's domain changes with time.
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.
The models must include waves and tides which influence sea level in shallow waters.
Storm surge models must include the currents generated in a stratified, shallow sea by wind.

84.       Topographic map of the Gulf of Maine showing important features of the Gulf.
(From Lynch et al, 1993).

85. 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).

86. 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.

87. 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

88. BSH Model Regions

89. 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.

90. 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

91. Sea level variability obtained by Global Ocean Assimilation

92. WHY A MODEL CAN DO BETTER THAN CORRELATIONS BETWEEN EVENTS?
 The synergy between the different human forcings cannot be assessed from simple correlations between ecological observations and historical correlations.
 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.

93. 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

94. MIT General circulation model

95. 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

96. MIT General circulation model
Spherical coordinates
“Cubed sphere”

97. MIT General circulation model

98. Some computational aspects

99. Experiments with 60*60*20 grid points

100. Specfp (swim)

101. Compiler Portland Group (PGF) ca 8000 s GNU fortran compiler 7750 s
Intel Fortran Compiler (IFC) s

102. Important Concepts
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.

103. Thanks for your attention