Re: [asa] The Mathematics of Global Warming

I guess I'm not a mathematician either, but have developed and worked with
codes far more complicated that climate models.

I have in the past repeated here the aphorism that we used as a working
hypothesis: multiphysics nonlinear codes are not so much predictive models
as interpolation tools.

Turbulence modelling are interesting examples of this problem.

They are, in principal and by comparison, "simple" codes.
There are many successful "engineering"
models available, and have been available long before the advent of modern
computing. More sophisticated and detailed models are next to impossible
to
implement, even these are approximations (the problem of closure).

These complex and somewhat detailed models are used to hopefully develop
practical and implementable models. Such models would be implemented,
as reduced models, into more complex and multi-physics codes.

What these detailed models show, as I remember, is that chaos (turbulence)
does not merely work itself from larger to smaller scales (as was assumed
by much of the early turbulence work), but from smaller scales to larger
scales. This is MOST troubling for a computational model. For we might
hope and pray that what we miss in small scales can be averaged out and
hidden in cell size, treated as a kind of parametrized heating.

But if the small scales, that we don't and can't model because it is
sub-celluar, should have effect in large scales and on the celluar level,
that we do model, is seriously problematic, not only for physical
turbulence,
but also for numerical turbulence.

I don't know much of climate modeling. Nonlinear equations are usually
solved
explicitly, rather than implicitly. This results in a host of instability
issues. Time scales are everything, even the order in which the equations
are solved in a linear fashion is significant.

All of this is to say, and to highlight, the aphorism I began with, that
empirical results are absolutely essential. This is not to say that
numerical experiments, esp. on components and reduced physics aspects of
the
more complex physics are not valuable. They are.

But it is detailed (as detailed as we can get) empirical results that
provides
confidence in the code. It should be possible for complex climate models
to
have access to such empirical results. Remember however that the aphorism
councils interpolation; and this is the problem that climate models face.

We are being told that the climate models predict "unprecedented" effects
due
to unprecedented conditions. This suggests that computations are taking
on
the character of extrapolations, rather than interpolations. This is why
I have
for a very long time, since the early 90s, been very cautious of the
reported
climate model results.

I have spoken to others, those on this list and off, about these problems.
All the computational experts I speak to assure me that they are well
aware
of these problems. Nonetheless, they are still confident in the
computational
results.

bill

On Mon, 30 Nov 2009 13:14:04 +0000, Iain Strachan <igd.strachan@gmail.com>
wrote:
> OK I'm not an expert in the mathematical models used for climate
modelling,
> but I am familiar with the problems related to solving non-linear
> differential equations - a non-linear equation of order as low as three
can
> exhibit chaotic behaviour - which means that the solutions diverge with
> time
> due to arbitrarily small changes in initial conditions.
>
> I think this is the substantive point that is being made, and it is
correct
> in as far as it goes, were it not for the fact that other techniques can
be
> applied to counter this problem. For example a related problem is the
> modelling of turbulent fluid flow in Computational Fluid Dynamics (CFD)
> which arises as a result of the Navier-Stokes equations. Again, I am
not
> an
> expert in this, but my drinking partner is a world expert on this and
> author
> of some of the largest software packages in this field. This excerpt
from
> Wikipedia (on Navier-Stokes equations) gives some idea of the type of
> techniques employed:
>
> ----
> The numerical solution of the Navier-Stokes equations for turbulent flow
is
> extremely difficult, and due to the significantly different
mixing-length
> scales that are involved in turbulent flow, the stable solution of this
> requires such a fine mesh resolution that the computational time becomes
> significantly infeasible for calculation (see Direct numerical
> simulation<http://en.wikipedia.org/wiki/Direct_numerical_simulation>).
> Attempts to solve turbulent flow using a laminar solver typically result
in
> a time-unsteady solution, which fails to converge appropriately. To
counter
> this, time-averaged equations such as the Reynolds-averaged
Navier-Stokes
>
equations<http://en.wikipedia.org/wiki/Reynolds-averaged_Navier-Stokes_equations>
> (RANS),
> supplemented with turbulence models (such as the k-ε model), are used in
> practical computational fluid
> dynamics<http://en.wikipedia.org/wiki/Computational_fluid_dynamics>(CFD)
> applications when modeling turbulent flows. Another technique for
solving
> numerically the Navier-Stokes equation is the Large-eddy simulation
(LES).
> This approach is computationally more expensive than the RANS method (in
> time and computer memory), but produces better results since the larger
> turbulent scales are explicitly resolved.
> ----
>
> As far as I'm aware, this kind of technique is very successful and my
> friend's code was famously involved in reconstructing the events
following
> a
> famous UK railway station disaster where a front of flame shot up an
> escalator shaft (Kings Cross disaster). The results from the code were
> verified using a scale model.
>
> Hence I don't think it's as simple as saying that if there are
non-linear
> equations then you can't have any idea of the solution.
>
> I would also add that the writer refers to "wishful thinking" as a basis
> for
> economic policy.
>
> Would it not be more accurate to say that those who insist that climate
> change is NOT going to happen are indulging in wishful thinking?
>
> I for one have no great desire to see global warming come true. I would
> love it if the whole thing was found to be a complete hoax, and to wake
up
> one morning and find out it wasn't true, and that my grandchildren
weren't
> going to inherit a world ravaged by flood, famine, disease and war. To
> suggest that those who believe this will happen are indulging in
"wishful
> thinking" is absolutely preposterous.
>
> Iain
>
>
>
>
> On Mon, Nov 30, 2009 at 12:34 PM, John Walley <john_walley@yahoo.com>
> wrote:
>
>> I found this to be very interesting. I wonder if any of the
>> mathematicians
>> on the list have any comment?
>>
>> John
>>
>> November 30, 2009
>> The Mathematics of Global Warming
>> By Peter Landesman
>>
http://www.americanthinker.com/2009/11/the_mathematics_of_global_warm.html
>>
>> The forecasts of global warming are based on the mathematical solutions
>> of
>> equations in models of the weather. But all of these solutions are
>> inaccurate. Therefore no valid scientific conclusions can be made
>> concerning
>> global warming. The false claim for the effectiveness of mathematics is
>> an
>> unreported scandal at least as important as the recent climate data
>> fraud.
>> Why is the math important? And why don't the climatologists use it
>> correctly?
>>
>> Mathematics has a fundamental role in the development of all physical
>> sciences. First the researchers strive to understand the laws of nature
>> determining the behavior of what they are studying. Then they build a
>> model
>> and express these laws in the mathematics of differential and
difference
>> equations. Next the mathematicians analyze the solutions to these
>> equations
>> to improve the understanding of the scientist. Often the mathematicians
>> can
>> describe the evolution through time of the scientist's model.
>>
>> The most famous successful use of mathematics in this way was Isaac
>> Newton's demonstration that the planets travel in elliptical paths
around
>> the sun. He formulated the law of gravity (that the rate of change of
>> the
>> velocity between two masses is inversely proportional to the square of
>> the
>> distance between them) and then developed the mathematics of
differential
>> calculus to demonstrate his result.
>>
>> Every college physics student studies many of the simple models and
their
>> successful solutions that have been found over the 300 years after
>> Newton.
>> Engineers constantly use models and mathematics to gain insight into
the
>> physics of their field.
>>
>> However, for many situations of interest, the mathematics may become
too
>> difficult. The mathematicians are unable to answer the scientist's
>> important
>> questions because a complete understanding of the differential
equations
>> is
>> beyond human knowledge. A famous longstanding such unsolved problem is
>> the
>> n-body problem: if more than two planets are revolving around one
>> another,
>> according to the law of gravity, will the planets ram each other or
will
>> they drift out to infinity?
>>
>> Fortunately, in the last fifty years computers have been able to help
>> mathematicians solve complex models over short time periods. Numerical
>> analystshave developed techniques to graph solutions to differential
>> equations and thus to yield new information about the model under
>> consideration. All college calculus students use calculators to find
>> solutions to simple differential equations called integrals.
>> Space-travel
>> is possible because computers can solve the n-body problem for short
>> times
>> and small n. The design of the stealth jet fighter could not have been
>> accomplished without the computing speed of parallel processors. These
>> successes have unrealistically raised the expectations for the
>> application
>> of mathematics to scientific problems.
>>
>> Unfortunately, even assuming the model of the physics is correct,
>> computers
>> and mathematicians cannot solve more difficult problems such as the
>> weather
>> equations for several reasons. First, the solution may require more
>> computations than computers can make. Faster and faster computers push
>> back
>> the speed barrier every year. Second, it may be too difficult to
collect
>> enough data to accurately determine the initial conditions of the
model.
>> Third, the equations of the model may be non-linear. This means that no
>> simplification of the equations can accurately predict the properties
of
>> the
>> solutions of the differential equations. The solutions are often
>> unstable.
>> That is a small variation in initial conditions lead to large
variations
>> some time later. This property makes it impossible to compute solutions
>> over
>> long time periods.
>>
>> As an expertin the solutions of non-linear differential equations, I
can
>> attest to the fact that the more than two-dozen non-linear differential
>> equations in the models of the weather are too difficult for humans to
>> have
>> any idea how to solve accurately. No approximation over long time
>> periods
>> has any chance of accurately predicting global warming. Yet
>> approximation
>> is exactly what the global warming advocates are doing. Each of the
more
>> than 30 modelsbeing used around the world to predict the weather is
just
>> a
>> different inaccurate approximation of the weather equations. (Of
course
>> this is only an issue if the model of the weather is correct. It is
>> probably
>> not because the climatologists probably do not understand all of the
>> physical processes determining the weather.)
>>
>> Therefore, logically one cannot conclude that any of the predictions
are
>> correct. To base economic policy on the wishful thinking of these
>> so-called
>> scientists is just foolhardy from a mathematical point of view. The
>> leaders
>> of the mathematical community, ensconced in universities flush with
>> global
>> warming dollars, have not adequately explained to the public the above
>> facts.
>>
>> President Obama should appoint a Mathematics Czar to consult before he
>> goes
>> to Copenhagen.
>>
>> Peter Landesman mathmaze@yahoo.comis the author of the 3D-maze
>> bookSpacemazes for children to have fun while learning mathematics.
>>
>>
>>
>>
>>
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Received on Mon Nov 30 09:42:41 2009