
Written
by V. Bjerknes, University of Stockholm
Translated
from the original German by Allen R. Greenberg, NOS.
Published
in the Magazine of Meteorology, January 1904
(Meteorologische Zeitschrift)
If,
as every scientifically inclined individual believes, atmospheric
conditions develop according to natural laws from their precursors,
it follows that the necessary and sufficient conditions for
a rational solution of the problems of meteorological prediction
are the following:
1: The condition of the atmosphere must be known at a specific
time with sufficient accuracy
2: The laws must be known, with sufficient accuracy, which determine
the development of one weather condition from another.
I.
The job of observational meteorology is to produce knowledge
of the condition of the atmosphere at a desired future time.
This problem has not been solved for the scientific weather
forecaster in even the most felicitous circumstances. Two gaps
are especially noticeable. Firstly, the only weather stations
reporting daily are on the land. On the sea, which constitutes
four fifths of the earth’s surface and therefore exercises
an overwhelming influence, no daily weather observations are
made. In addition, the regular weather service observations
are terrestrial in origin, and lack any information about the
condition of the upper atmosphere.
We
already possess the technical tools which will make it possible
to fill in these two gaps. Steamships which travel fixed routes
in the neighborhood of stations and send daily weather reports
can contribute with the help of wireless telegraphy. In addition,
given the great progress that has been made in aeronautical
meteorology during the last few years, it will no longer be
impossible to receive daily observations of the upper atmosphere
at fixed terrestrial stations or airborne platforms.
Hopefully, the time will also soon come, when a complete statement
of atmospheric conditions can be made either daily or on specified
dates. At that point, the first condition for scientific weather
forecasting will be met.
II.
The second question then presents itself, which is to what extent
we have sufficiently detailed knowledge of the laws according
to which one atmospheric condition develops into another.
The
atmospheric processes are mixtures of a physical and mechanical
nature. We can describe each of these processes with one or
more mathematical equations. We will have sufficient knowledge
of the laws governing the development of atmospheric processes
when we can write enough independent equations to calculate
all the unknown quantities. The state of the atmosphere at a
particular time will be determined, in a meteorological sense,
when we can determine the air speed, density, pressure, temperature,
and humidity at every point. The wind velocity will be represented
by three scalars, the three wind speed components, so that,
as a result, we need to be concerned with the calculation of
seven unknown quantities.
To calculate these quantities, we set up the following equations:
1. The three hydrodynamic equations of motion. These are differential
equations representing relationships between the three wind
speed components, the density, and the pressure.
2. The equation of continuity, which states the principle conservation
of mass during motion. This relationship is also expressed by
a differential equation, more precisely a relationship between
the wind speed components and air density.
3. The equation of state for the atmosphere, which is an infinite
series involving the density, pressure, temperature, and the
humidity of a given air mass.
4. The two major principles of the mechanical theory of heat,
which state, in two differential equations, how, as a result
of ongoing condition changes, the energy and entropy of a chosen
airmass are altered. In addition, these equations introduce
no new unknown quantities into the problem, because the energy
and entropy express themselves through the same transformations
that are found in the equations of state, and tie the changes
in these variables with the changes of other known quantities.
The other inputs are: firstly, the work done by the air mass,
which is determined by the same transformations which are found
in the dynamic equations; secondly, the externally determined
heat quantities, which will be obtained from physical data concerning
radiant energy transfer and the heating of the air caused by
the motion of the earth.
It
is given that a significant simplification of the problem arises
when there are no humidity changes, so that the amount of water
present in the air mass remains constant. We then have one fewer
variable, and one equation, namely that of the second principle,
which can be eliminated. On the other hand, if we have several
changes in the condition of the atmosphere, the application
of the second principle for each new condition produces a new
equation.
In
order to calculate the usual seven variables, we need seven
independent equations. Insofar as it is possible now to get
an overlook of the problem, we must also conclude that we possess
a sufficient knowledge of the atmospheric processes upon which
a scientific weather forecasting is based. It must also be acknowledged
that we may have overlooked important considerations because
of the incompleteness of our understanding. The intervention
of global processes of an unknown type is conceivable. Further,
overall atmospheric changes are accompanied by a long series
of accompanying phenomena of an optical and electrical nature
and the question remains, how significantly these influence
the atmospheric processes. The interconnections are self evident.
Rainbows are, for example, a modified refraction of solar energy,
and electrical charges have known influences on the condensation
processes. Up to this point, however, there are no indications
that these ancillary events significantly affect overall atmospheric
processes. At any rate, it’s the scientific method to
start with the simplest problem which can be stated, which is
precisely that with seven equations involving seven unknowns.
III.
Of the seven equations, only one is expressed as an infinite
series. The other six are partial differential equations. Of
the seven unknowns, one can be eliminated with the help of the
equation of state, and the solution then becomes one of the
integration of a system of six partial differential equations
with six unknowns and initial values obtained from observations
of the beginning atmospheric conditions.
It’s
not possible to obtain a rigorous mathematical integration of
this system of equations. Even the calculation of the motion
of three objects, which are mutually influenced according to
simple Newtonian law, goes substantially beyond today’s
mathematical tools. It’s self evident that calculation
of the substantially more complicated interactions of atmospheric
molecules is hopeless. The exact analytic solution would not
be what we need, even if we were able to obtain it. In order
for such a solution to be useful, it would of necessity include
all conceivable conditions, something which would introduce
an infinite number of singularities into the analytic solution.
The predictions need only concern themselves with average values
over large areas and long periods of time, for example hourly
reports by degrees of longitude as opposed to every second by
millimeters.
Accordingly, we abandon all thoughts of an analytic solution,
and restate the problem of the weather forecaster in the following
practical form:
Because of independent observations, the initial state of the
atmosphere is represented by a number of tables, which specify
the division of the seven variables from layer to layer in the
atmosphere. With these tables as the initial values, one can
specify similar new tables which represent the new values from
hour to hour.
Graphical or mixed graphical and numerical methods are needed
in order to solve the problem in this form, whether from the
partial differential equations, or from the physical/ dynamic
principles which underlie the equations. The effectiveness of
such methods can’t be questioned on an a priori basis.
Everything depends on successfully separating a single overwhelmingly
difficult problem into a series of sub problems, none of which
present impossible difficulties.
IV.
In order to implement this partition into sub problems, we must
call on the general principle, which underlies the Calculus
of Several Variables. For computational purposes, we can replace
the simultaneous changes of several variables with sequential
changes of individual variables or groups of variables. If we
proceed to very small intervals, we approach the exact methods
of differential calculus. If we retain a finite interval, then
we’ll make use of the numerical analysis approximation
techniques of finite differences and numerical integration.
This principle must not be blindly applied, because the practical
utility of the methods will, above all, depend on the natural
grouping of the variables which are contained in the mathematically
and physically well defined sub problems. Most importantly,
the primary division is fundamental. It must follow the major
problem in a natural order.
Such a natural order must also be specified. It lies on the
border between specific dynamic and physical processes, out
of which the atmospheric processes are synthesized. A segmentation
along these boundaries provides a decomposition of the main
problem into purely hydrodynamic and thermodynamic sub problems.
The link between the hydrodynamic and thermodynamic problems
is very easy to separate, so simple, in fact, that theoretical
Hydrodynamicists have used it to avoid all serious contact with
Meteorology, because the connection is made by the equations
of state. If one assumes that the temperature and humidity are
not involved in these equations, then we arrive at the conventionally
applied “supplementary equations” of hydrodynamics,
which are only relations between density and pressure. In that
manner, we are led to the study of fluid flow under the circumstances
that every explicit accounting of thermodynamic processes falls
away by itself.
Instead of allowing the temperature and humidity of the equations
of state to disappear entirely, we can regard them as fixed
values for small periods of time, with values given either by
observations or from previous computations. When the dynamic
problem is solved for this time interval, further calculations
can be made according to purely thermodynamic methods to obtain
temperature and humidity. These values can be used as known
quantities in the solution of the hydrodynamic problem in the
next time interval, etc.
V.
The general principle of the first decomposition of the main
problem has been stated. In the practical follow through, we
have the choice of several different paths, each according to
technical considerations, which introduce hypotheses about temperature
and humidity. To go into these considerations in greater detail
would be meaningless in such a general discussion.
The
next major question will be to what extent the hydrodynamic
and thermodynamic partial problems can be solved in a sufficiently
simple manner.
We
first consider the hydrodynamic problem, which is the real major
problem, as the dynamic equations provide the primary predictions.
Only in this manner can time be introduced into the problem
as an independent variable, as the thermodynamic equations don’t
involve time.
The
hydrodynamic problem is an excellent candidate for a graphical
solution. In order to solve the three dynamic equations, one
has to construct simple parallelograms for a suitable number
of chosen points, and solve graphically for intervening points.
The principal difficulty will be found in accounting for the
limits of the freedom of motion, which come from the equation
of continuity and the boundary conditions. The test of whether
the equation of continuity is satisfied is also left to graphical
methodology and, in this manner, no consideration can made of
the earth’s topography, because the constructions are
carried out on charts, that represent this topography in a conventional
manner.
There are also no substantial mathematical difficulties to be
found in the solution of the hydrodynamic partial problems.
A perceptible gap in the knowledge of the factors , with which
we are required to calculate, is also present insofar as we
have an incomplete knowledge of the viscous friction in the
motion of the air, because the friction depends on the difference
in the speed of small molecules, while meteorologists are compelled
to compute the average movement of extended masses of air. None
the less, one can’t utilize the laboratory obtained coefficients
of friction for the frictional elements in the hydrodynamic
equations. On the contrary, empirical results must be obtained
of the effective resistance to motion of large masses of air.
We already have sufficient data of this type to make the first
attempts at predictions of the motion of air, which will be
supplemented and corrected over time.
The thermodynamic partial problem is significantly
simpler to look at than the hydrodynamic one. One only takes out
of the solved hydrodynamic problem the work which the air masses
have performed during the displacements. With the knowledge of
this work, and the additional knowledge of the thermal changes
caused by radiant energy flux during the given time period, one
can obtain a new distribution of temperature and humidity from
known thermodynamic principles. From a mathematical standpoint,
the calculations are no more difficult than similar computations
made in laboratory experiments, which are made with air masses
at rest in enclosed spaces. Extensive preparatory work was done
in the studies of Hertz, V. Bezold, and others.
As in the hydrodynamic problem, the biggest difficulty in carrying
out the calculations is the incomplete nature of our understanding
of several of the factors. Initially, there will be uncertainty
in the estimation of the quantities of heat which the air mass
receives from radiant energy flux, and in the mass of water which
evaporates from the earth’s surface or which condenses from
clouds and falls as rain. We have sufficient knowledge for the
experimental initiation of the first calculations. With further
work, we will obtain increasingly exact values of the constants
which are associated with different continents and seas, different
atmospheric heights, differing weather conditions, and varying
degrees of cloud cover.
VI.
It’s certain that, if we proceed in the indicated manner,
that no intractable mathematical difficulties will be encountered.
When the graphical methods have been worked out and the necessary
tools implemented, the individual operations will be easy to
carry out. In addition, the number of single operations do not
need to become excessive. The number will depend on the length
of the time interval which is being used for the solution of
the dynamic partial problem. The smaller the time interval chosen,
the more complex the work, but the results are also more accurate.
Larger time intervals yield faster, less accurate results. Initial
attempts will provide experience. Only when substantial accuracy
is required might intervals of at most an hour be useful. It’s
highly unusual that air masses will traverse more than a degree
of longitude in an hour’s time and that, during this time,
their paths will substantially change. Therefore, the conditions
are fulfilled under which one can construct a simple parallelogram
with straight sides. When enough experience has been obtained,
it will also be easier to work with larger time intervals, say
of approximately six hours. For a 24 hour weather prediction,
it would be necessary to carry out four hydrodynamic constructions
and to calculate the thermodynamic corrections of temperature
and humidity four times.
It
might then be possible that, sometime in the future, a technique
of this sort could be put into practical, daily weather service
use. No matter how this procedure worked, sooner or later it
will be necessary to undertake a deeper scientific study of
atmospheric processes that is founded on the laws of Mechanics
and Physics. In this manner, one will necessarily arrive at
a method that is sketched here.
While
this is understood, there is also a general plan for dynamic
meteorological research.
The
principal task of observational meteorology is to work out the
most comprehensive picture of the physical and dynamic conditions
of the atmosphere from observations. This picture must certainly
have such a form as to be useful as a starting point for weather
predictions based on rational dynamicphysical methods.
Clearly,
this first introductory exercise is not of limited generality.
It is self evident that it is certainly more involved to represent
all heights of the atmosphere that what is now done at sea level.
In addition, we realize that our opportunity for direct observation
of the upper atmosphere will always remain very constrained.
It is therefore of greatest importance that data from the upper
atmosphere be most thoroughly utilized. From the directly measured
quantities we must calculate all associated elements in the
widest area. For that purpose, we must utilize the mathematical
relationships between the different data elements. Also, if
we want to use these sporadic observations to construct a thorough
picture of the state of the atmosphere, we must use dynamicphysical
methods in large areas.
The
second and most important task of theoretical meteorology will
ultimately be to take this picture of the condition of the atmosphere
as a starting point and construct future states, whether with
the methods outlined here or with comparable methodology. The
comparison of the constructed states with the observed ones
will in part yield verification of the validity of the methodology,
and in another part provide indications for better values of
constants and improvements of technique.
I will return at future opportunities to the major points of
this program.

Top of Page 

