1. The History of the Weather Prediction Equations and
Calculating Sea Breeze
Matthew Smith
1 Introduction
Definition 1. Weather is the state of the atmosphere at a particular place and time,
regarding cloudiness, heat, precipitation, wind, etc.
Weather is important to everybody. It can determine what you are going to do on
any particular day, and in extreme cases, has the power to destroy large buildings and
take peoples lives. Without being able to predict the weather accurately, thousands if
not millions of people every year would suffer the consequences of not being prepared
for hurricanes, tornadoes and violent thunderstorms, which can bring strong winds and
floods. Furthermore, it allows farmers to efficiently grow crops as they can estimate
precisely how much water is going to rain onto their fields.
As technology became more and more advanced during the twentieth century, it be-
came apparent to Mathematicians and Scientists that it may be possible to model the
atmosphere using the equations of fluid dynamics and thermodynamics. With millions
of people living in areas that rely on accurate forecasts of their weather, it is paramount
for us that we are able to predict weather of all types all over the planet. How can this
be done? The atmosphere is, or can be considered to be, a fluid. As such, the idea of
Weather Prediction is to sample the state of a fluid (the atmosphere) at a given time to
estimate the state of the fluid at some time in the future.
This project will give an outline of the development of meteorological study through
human history, focusing on the work of Vilheim Bjerknes and giving examples where
mathematical theory and the general equations of fluid mechanics can be used to deter-
mine the conditions of the weather at some time in the future.
2 History
2.1 Aristotle of Greece
The Greek philospher Aristotle wrote Meteorologica, which means ”a study of things
that fall from the sky”, at some time close to 340 BC. This was the first book in history
that defined meteorology and it was a source of innovation and inspiration that led to
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2. a development of thinking in terms of how meteorology was thought about by scientists
thereafter. His theories were based on the elements of earth, fire, water and air combined
with rationalilty, rather than scientific experimentation where everything is measured ex-
actly. However, this book was considered the most important contribution to meterology
for around 2,000 years, even though it was riddled with errors. But, because it was filled
with very accurate descriptions of winds and the types of weather that wind can bring,
the book was seen as a very legitimate attempt of the practical forecasting that we use
today.
2.2 Archimedes
Archimedes of Syracuse was a scholar who discovered fundamental laws of physics, as
well as at the time state of the art mathematics. In contrast to Aristotle before him,
Archimedes’ work can be considered to be ”exact”, as his theories and ideas were based
on a far greater insight into equations and laws based of mathematics and physics. He
also had a respect for experimental testing of each law.
Arcihmedes is famous for discovering the principle of buouancy whilst he was taking
a bath, which in essence meant he had discovered the basic principles of hydrostatics, or
fluids at rest. These principles enabled scientists nearly 2,000 years later to answer some
of the basic questions as to why air circulates around the Earth. Buoyancy is a key factor
involving the patterns of clouds and mists we see on a daily basis.
2.3 Leonhard Euler - Parcelling Up Fluids
Up until the mid 1700s, Newtonian’s methods were only applied to solid objects, and
scientists of these times attempted to adapt them for the problem of a fluid. Their
problem was trying to apply Newton’s method to a substance which had billions upon
billions of tiny particles, yet, by definition was not solid. Leonhard Euler was the first
person to overcome this problem by introducing the idea of a ”fluid parcel”.
Definition 2. [5] A fluid parcel is an extremely small amount of fluid, but is large enough
to possess billions of molecules so that the properties of mass, density and temperature of
the liquid are still represented, however, it is small enough so that these properties do not
differ between any two points within the parcel.
Archimedes before him had found that pressure is responsible for initiating and main-
taining the flow of a fluid, and Euler went on to formulate precise equations that tell us
how pressure can act upon the fluid to change the properties of motion of a fluid parcel.
2.4 Cleveland Abbe
By the late nineteenth century, scientists were using Newton’s laws to calculate the times
of sunrise and sunset, the phases of the moon and the tides. Cleveland Abbe, in 1901, who
was the founder of the United States Weather Bureau, was the first to propose that the
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3. atmosphere is governed by the same principles of thermodynamics and hydrodynamics
that had been studied in the previous century.
2.5 Vilheim Bjerknes
In 1904, a scientist from Norway called Vilheim Bjerknes (1862-1951) published a paper
suggesting that weather forecasting could be formulated as a problem in Mathematics
and Physics. He stated that there were two key issues when trying to successfully model
the weather; these were:
• a sufficient knowledge of the state atmosphere at a specific point in time
• an understanding of the laws according to which one state of the atmosphere de-
velops from another [1]
Bjerknes labelled these steps as the ”diagnostic” and ”prognostic” steps. The first step,
diagnostic, requires adequate obervational data to define the structure of the atmosphere
at a particular time. The second step, prognostic, requires a set of equations for each
variable describing the atmosphere, which can be solved in order to describe future condi-
tions. These two points are still valid to this day, and are what forms the basis of modern
day Weather Prediction.
2.6 The First Numerical Forecast
On 20th May 1910, a British Mathematician called Lewis Fry Richardson became the
first person to attempt a weather forecast through direct computation [2]. Previously,
meteorologists had only attempted forecasts using past data of the weather, finding pat-
terns in their records, and then extrapolating this date forward. He attempted to use
a mathematical model of the principle features of the atmosphere, using data recorded
at 7am that day, to predict the atmospheric conditions six hours later [1]. His forecast
was almost a complete failure, however, further analysis into his forecast showed that
with a few adjustments, he may not have been too far off. Even though this forecast was
incredibly unrealistic, the thought and execution of predicting the weather numerically
had begun.
Richardson’s first forecast took months of work, and as a result, to track the weather
all over the globe, he envisioned a ”forecast factory” where he calculated that 64,000 hu-
mans would be needed [1], each responsible for their own small area of the globe, in order
to keep ”pace with the weather” and to predict it. They would be housed in a circular
theatre type building, with a map on the walls and cieling, with a conductor coordinating
the calculations using coloured lights, as seen below. As a result of Richardson’s forecasts
being filled with errors, his monumental work was ingored for several decades [2].
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4. Figure 1: Richardson’s vision of a forecast factory [10]
2.7 The Invention of Computers
In 1946, the first electronic computer, named ENIAC (short for Electronic Numerical
Integrator and Computer) was installed in Philadelphia at Pennsylvania University [1]. At
the same time, a U.S. mathematician called John von Neumann was working on building
even better machines at the Institute for Advanced studies in Princeton. It would only
take four more years for Jule Charnet, Ragnar Fj¨ortoft and John von Neumann to make
the first numerical weather prediction. They used the absolute vorticity conservation
equation to compute their prediction and they used the ENIAC at Aberdeen [1]. This
forecast marks the start of modern day numerical weather prediction.
This was the start of what we now see on our televisions everyday, with meteorologists
being paid to monitor weather all over the globe, using Physics, Mathematics and com-
puters. Computing a weather forecast numerically is a topic that only began to take shape
in the second half of the twentieth century, as a result of the continuous advancements
in computing [3]. From meteorogical observations, we are able to numerically compute
future values of the atmosphere’s characteristic parameters in order to solve equations
describing its behaviour.
The equations used are the general equations of fluid mechanics, where we apply
certain simplifications which are justified by the orders of magnitude of the various terms
in the specific instance of the Earth’s atmosphere and the scales to be described.
3 Bjerknes’s work on Circulation
3.1 The Circulation Theorem
To begin with, we give a definition of circulation, in this context.
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5. Definition 3. [12] [13] Circulation is the amount of force that pushes along a closed
boundary or path.
Definition 4. [12] A barotropic fluid is a fluid whose density is only a function of pres-
sure.
Vilheim Bjerknes had a vision that using measurements of pressure and density (or
temperature), he would be able to predict average winds even when local conditions were
complicated as a result of differing strength of gusts and eddies.
Kelvin’s circulation theorem states:
Theorem 1. [11] In a barotropic ideal fluid with conservative body forces, the circulation
around a closed curve moving with the fluid remains constant with time, where:
DΓ
Dt
= 0 (1)
with Γ being the circulation around a material contour C(t) defined by:
Γ(t) =
C
u · ds (2)
Proof. [8] [11]
D
Dt
( u · ds) =
Du
Dt
· ds + u ·
D(ds)
Dt
(3)
Substituting Euler’s equation into the first term of the right hand side gives:
Du
Dt
· ds = −
1
ρ
P · ds = −
1
ρ
dP = 0 (4)
Then, the second term on the right hand side is:
D(ds)
Dt
= du (5)
which implies that:
u·
D(ds)
Dt
= u·du = (udu+vdv+wdw) =
1
2
d(u2
+v2
+w2
) =
1
2
d(q2
) = 0 (6)
Thus both terms on the right hand side are equal to zero.
Kelvin’s version of the circulation theorem states that the rate of change of circulation
as measured by an observer moving with the flow is zero. To add to this, Bjerknes
contributed that DC/Dt = 0, i.e. circulation can be generated or destroyed when there
is nonalignment of surfaces of constant pressure and density in the fluid caused by varying
temperatures or pressures.
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6. 3.2 Circulation and Sea Breeze
Bjerknes adapted this Circulation theorem and applied it to consider the creation of a sea
breeze near a coast warmed by the sun. The circulation is defined by a countour integral
C= v · dl where v is the wind vector and dl is a small piece of the path around which
the integral is performed. Typically, a path begins over the ocean and moves over the
shore and onto the land for approximately thirty kilometres, from which it then returns
into the upper atmosphere. This integral adds up the total component of the wind that
blows in the direction of the chosen path, where on each piece of path dl, v · dl is the
path component of the wind times by the length of path. [5]
In the atmosphere, there are different layers which have different pressures, and since
these layers are generally horizontal, so are the pressure surfaces. Layers of air of differing
temperature and density press down on layers below due to the Earth’s gravity, and
Bjerknes looked for applications where heating in the atmosphere changes the density
of the air more significantly than the pressure. He studied a situation where the Sun
has heated the air above a coastal plain more so than the ocean next to it, causing the
surfaces of constant density to dip over the land [5]. This happens as a result of the air
above the coastal plain being heated more and becomes less dense as it expands more.
Figure 2: Here we see an ideallized sea breeze forming at the coastline, shown with the
sea to the left and land to the right. Dashed lines represent surfaces of constant pressure,
solid lines represent the situation we have highlighted above. [5]
The arrows G and B are notations for vectors. The vector B is perpendicular to the
density surface, whereas the vector G is perpendicular to the pressure surface. In this
situation, we have that
DC
Dt
= (G × B) · ndA (7)
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7. where n is the unit vector perpendicular to the plane containing the area element dA and
A is the area enclosed by the contour. This new integral is the net sum of misalignment
of
G × B (8)
inside the contour path. When (8) is equal to 0, Kelvin’s theorem applies. However, when
G and B are no longer aligned, circulation is created and then a sea breeze is blown onto
the land [5].
4 Equations for Fluid Motion
As we have mentioned before, until the mid 1700’s, Newtonian mechanics had only been
applied to solid objects, and so the challenge that scientists faced at the time was to
extend these laws to fluids. Leonhard Euler, who introduced the concept of fluid parcels,
working alongside his compatriot Daniel Bernoulli in St. Petersburg, constructed the
basis of what nowadays we call hydrodynamics. [5]
Archimedes had discovered that pressure forces are responsible for initiating and main-
taining the steady flow of a fluid, and Euler would go on to formulate precise equations
that tell us how pressure acts to change the motion of a parcel of fluid. He discovered
that if there was a difference in pressure on either side of the fluid parcel then this would
result in acceleration in the same direction of largest pressure. Once Euler had worked
this out, he was able to apply Newton’s laws of motion to his idea of fluid parcels. The
result was three equations governing the motion of a fluid parcel through the three di-
mensions of space; one equation for vertical motion, one equation for horizontal motion
in the northward and eastward directions. The principle that as a fluid flows through a
region, such as a length of pipe, fluid cannot magically disappear, so fluid matter cannot
be created or destroyed in motion, creates a fourth equation, called the conservation of
mass [5].
4.1 Euler’s Equations
Here, we will define Df/Dt to be the rate of change with time of a quantity f, where f
is defined to be on fluid parcels [5]. The wind is simply the movement of multiple air
parcels each with velocity v = (u, v, w), density ρ, and mass δm = ρδV where δV is
the volume of the air parcel [5]. For convenience, we will say that the wind has speed u
in the east direction, v in the northern direction, and w in the veritcally upwards direction.
We have that the mass of the air parce, by definition, must be constant, and hence:
D(δm)
Dt
= 0 (9)
Therefore, the density ρ of the air parcel must be inversely proportional to its volume
and the rate of change in volume is related to the divergence divv.
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8. Thus, since δm = ρδV , we have that:
Dρ
Dt
= −ρdivv (10)
Let F be the sum of the gravitational, pressure gradient and frictional forces. Then, the
rate of change of momentum, δmv can we written as:
D(δmv)
Dt
= FδV (11)
The major complication for the Earth’s atmosphere is that we must study the equation
Dv/Dt = F/ρ [5] when the fluid rotates around with the planet, when it is subject to
gravitational forces, and when it is effected by heat and moisture.
4.2 The Hydrostatic Equation
The key formula that describes how the pressure of a gas varies as the volume in which it
is contained is known as Boyle’s law. This law was discovered independently by a French
physicist Edme Mariotte, and by Robert Hooke, Boyle’s assistant. [5]
Remark. [7] Boyle’s Law states that:
P ∝
1
V
(12)
where P is the pressure and V is the volume of a gas
The hydrostatic equation involves describing the way pressure varies with height in
the atmosphere [14]. It was solved by a man called Halley in 1685, where he used Boyle’s
Law to derive the first expression for calculating height from pressure.
Constant temperature in the air column is assumed, then, the change in density ∆ρn
causes a corresponding change in the pressure ∆pn, where the constant value of the
difference between conditions for the layers of air is R∗
. Then, we have:
∆pn = R∗
∆ρn (13)
We can use (12) to eliminate ∆pn, which gives us:
R∗
∆ρn = −ρng∆zn (14)
which suggests that the change in height is directly related to the change in density. Here,
g is the denotion for gravity, and zn is the denotion for height in the air column. Solving
this equation to find a formula for the height we get:
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9. gH = R∗
ln(
ρ
ρ0
) = R∗
ln(
p
p0
) (15)
This gives the height, H, above the reference level at pressure p0 and density ρ0.
4.3 Equations describing Weather Evolution
Combining the equation for the conservation of energy, the equation of state for the gas
that makes up the atmosphere, and Euler’s four equations of fluid mechanics provides
us with six of the seven equations that form the basis of our physcial model of the
atmosphere and oceans. The final equation we have yet to discuss will allow us to
describe how moisture is carried around and how it affects temperature [5]. The addition
of this seventh equation will allow us to describe a mathematical model that relates to
the life-giving properties that our atmosphere has.
So, as we have said, in total, there are seven equations that form the basis of modern
weather prediction. Four of them comprise the wind equations:
Du
Dt
=
F1
ρ
,
Dv
Dt
=
F2
ρ
,
Dw
Dt
=
F3
ρ
(16)
as well as the density equation,
Dρ
Dt
= −ρdivv (17)
where F is the vector F = (F1, F2, F3) that includes all of the forces acting upon the
fluid parcel, and v = (u, v, w). The equation of state says that:
p = ρRT (18)
because the atmosphere behaves very much like an ideal gas, so that the pressure is
directly proportional to the product of temperature T and the density ρ.
We also have the equation relating to energy conservation to describe, which can also
be referred to as the first law of thermodynamics [5]. The heating of the atmosphere
from the sun changes the internal energy of the gas and through pressure, compresses
the gas. The internal energy of the gas is proportional to the temperature T, and we will
write the proportional constant to be cv, which is known as the specific heat at constant
volume. The compression of the gas, or the change of density as we follow a fluid parcel,
can be written as the amount −(p/ρ2
)Dρ/Dt [5]. Therefore, the heating rate, Q, on the
fluid parcel can we written as:
Q = cv
DT
Dt
− (
p
ρ2
)
Dρ
Dt
(19)
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10. The heat from the sun causes moisture from the ground to evaporate, and causes heated
air near the ground to rise. As the air rises, it can often reach a level of the atmosphere
at which the temperature is low enough for some of the water vapour to condense, and
thus become visible as a cloud. So hence, the final quantity we need to consider involves
the amount of water vapour, q, within the air parcel. We have that:
Dq
Dt
= S (20)
where S, the net supply of water to an air parcel through processes such as evaporation
and condensation, is itself involved in the processes that affect the heating rate Q.
Thus, we have seven equations and seven variables: ρ, v = (u, v, w), p, T, and q [5].
It is quite compelling to think that the endless and seemingly random cloud patterns we
see moving across our skies every day can be understood in terms of the seven equations
given above.
5 Conclusion
Meteorology is a subject that puzzled Mathematicians and scientists for centuries. From
Aristotle of Greece to the trio of Jule Charnet, Ragnar Fj¨ortoft and John Von Neumann,
it was the collective effort of hundreds of years of work which enabled the latter three
scientists to finally numerically compute mans first ever weather forecast. Through the
adaptation of Newton’s methods for determining the state of a solid at some point in the
future, Euler thought of the idea of ”fluid parcels”, which were small yet very significant,
that enabled him to discover the equations for fluid motion, that would be used by
numerous scientists after him in order to predict the state of the atmosphere at a certain
time in the future.
Without the exceptional work of the scientists like Archimedes, Euler and Bjerknes,
humans would still be very much in the dark about the weather conditions they would
have to experience and endure on a day to day basis. Thus, many people owe their
safety and security of being able to live where and how they do to the scientists and
meteorologists who throughout the centuries have ensured that weather prediction is
as precise and as accurate as possible, with improvements continuously being made by
companies such as The Met Office in the United Kingdom and NOAA in the United
States.
In this project, I hoped to expand upon the readers knowledge of how humans in the
past few hundreds of years have been able to create a mathematical model that allows
us nowadays to be able to make accurate and reliable forecasts on an hourly basis and I
do hope that this has been the case.
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11. References
[1] Coiffier, J. Fundamentals of Numverical Weather Prediction. Cambridge: Cambridge
University Press, 2011
[2] Ehrendorfer, M. Spectral numerical weather prediction models. Philadelphia: Society
for Industrial and Applied Mathematics 2012
[3] Warner, T. T. Numerical Weather and Climate Prediction. Cambridge: Cambridge
University Press 2011
[4] www.wikipedia.com/vorticity
[5] Roulstone, I. and Norbury, J. Invisible in the Storm: The Role of Mathematics in
Understanding Weather. Princeton and Oxford: Princeton University Press 2013
[6] Haltiner, G. J. Numerical Weather Prediction. Norfolk, Virginia: Navy Weather
Research Facility. p1-5. 1968
[7] http://science.howstuffworks.com/boyles-law-info.htm
[8] www.see.ed.ac.uk/johnc/teaching/fluidmechanics4/2003-04/fluids2/node29.html
[9] en.wikipedia.org/wiki/LewisFryRichardson
[10] celebrating200years.noaa.gov/foundations/numericalwxpred/theater.html
[11] http://en.wikipedia.org/wiki/Kelvinscirculationtheorem
[12] http://www.ess.uci.edu/ yu/class/ess228/lecture.4.vorticity.all.pdf
[13] http://betterexplained.com/articles/vector-calculus-understanding-circulation-and-
curl/
[14] http://maths.ucd.ie/met/msc/fezzik/Phys-Met/Ch03-Slides-2.pdf
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