Chapter 15 - Numerical Models
We saw earlier that analytic
solutions to the equations of motion are difficult or impossible to obtain
for typical oceanic flows. The problem is due to the non-linear terms, friction,
and the need for realistic shapes for the sea-floor and coastlines. We have
also seen how difficult it is to describe the ocean from measurements. Satellites
can observe some processes almost everywhere every few days. But they observe
only some processes, and only near or at the surface. Ships can measure more
variables, and deeper into the water, but the measurements are sparse. Hence,
numerical models provide the only useful, global view of ocean currents. Let's
look at the accuracy and validity of the models, keeping in mind that although
they are only models, they provide a remarkably detailed and realistic view
of the ocean.
Words of Caution
Numerical models of ocean currents have many advantages. They simulate flows
in realistic ocean basins with a realistic sea-floor. They include the influence
of viscosity and non-linear dynamics. And they can calculate possible future
flows in the ocean. Perhaps, most important, they interpolate between sparse
observations of the ocean produced by ships, drifters, and satellites.
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.
The models can never give complete descriptions of the
oceanic flows even
if the equations are integrated accurately. The problems arise from several
Discrete equations are not the same as continuous equations
Chapter 7 we wrote down the differential equations describing the motion of
a continuous fluid. Numerical models use algebraic approximations to the differential
equations. We assume that the ocean basins are filled with a grid of points,
and time moves forward in tiny steps. The value of the current, pressure,
temperature, and salinity are
calculated from their values at nearby points and previous times. Ian Stewart
(1992), a noted mathematician, points out that
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 loosely
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.
Calculations of turbulence are
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.
Practical ocean models have grid points spaced tens to hundreds of kilometers apart
in the horizontal, and tens to hundreds of meters apart in the vertical. This means that
turbulence cannot be calculated directly, and the influence of
turbulence must be parameterized. Holloway (1994) states the problem succinctly:
Ocean models retain fewer degrees of freedom than the
actual ocean (by about 20 orders of magnitude). We compensate by applying
goo' to squash motion at all but the smallest retained scales. (We also use
non-conservative numerics.) This is analogous to placing a partition
in a box to prevent gas molecules from invading another region
of the box. Our oceanic models cannot invade most of the real oceanic
degrees of freedom simply because the models do not include them.
Given that we cannot do things 'right', is it better to do nothing?
That is not an option. 'Nothing' means applying viscous goo and wishing
for the ever bigger computer. Can we do better? For example, can we
guess a higher entropy configuration toward which the eddies tend to
drive the ocean (that tendency to compete with the imposed forcing and
By "degrees of freedom" Holloway means all possible motions from
the small-est waves and turbulence to the largest currents. Let s do a simple
We know that the ocean is turbulent with eddies as small as a few millimeters.
To completely describe the ocean we need a model with grid points spaced 1mm
apart and time steps of about 1 ms. The model must therefore have 360° × 180°
× (111 km/degree)2 × 1012 (mm/km)2 × 3 km × 106 (mm/km)
2.4 × 1027 data points for a 3km deep ocean covering the globe. The
global Parallel Ocean
Program Model described in the next section has 2.2 × 107 points.
So we need
1020 times more points to describe the real ocean. These are the missing
degrees of freedom that are missing.
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. We use the hydrostatic and Boussinesq approximations,
and we often use equations integrated in the vertical, the shallow-water equations
(Haidvogel and Beckmann, 1999: 37). We do this because we cannot yet run the
most detailed models of oceanic circulation for thousands of years to understand
the role of the ocean in climate.
Numerical code has errors
Do you know of
any software without bugs? Numerical models use many subroutines each with
many lines of code which are converted into instructions understood by processors
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. Lawrence et al.
(1999), examining the output of an atmospheric numerical model found an error
in the code produced by the FORTRAN-90 compiler used on the CRAY Research supercomputer
used to run the code. They also found round-off errors in the concentration
of tracers calculated from the model. Both errors produced important errors
in the output of the model.
Most models are not well verified or validated (Post and Votta, 2005).
Yet, without adequate verification and validation, output from numerical models
is not credible.
Despite these many sources of error, 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. Langer (1999), writing about the use of
computers in physics wrote:
All of who are involved in the sciences know that
the computer has become an essential tool for research... Scientific computation
has reached the
point where it is on a par with laboratory experiment and mathematical
theory as a tool for research in science and engineering,
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.