About coricolus force

1. Introduction
Although the meteorological science enjoys a relative
wealth of colorful figures and events, from past
and present, the educators rarely make use of this historical
dimension, as was recently pointed out in this
journal by Knox and Croft (1997). A case where a historical
approach proves to be illuminating is in the
teaching of the Coriolis force, named after French
mathematician Gaspard Gustave Coriolis (1792–
1843). On a rotating earth the Coriolis force acts to
change the direction of a moving body to the right in
the Northern Hemisphere and to the left in the Southern
Hemisphere. This deflection is not only instrumental
in the large-scale atmospheric circulation, the development
of storms, and the sea-breeze circulation
(Atkinson 1981, 150, 164; Simpson 1985; Neumann
1984), it can even affect the outcome of baseball tournaments:
a ball thrown horizontally 100 m in 4 s in
the United States will, due to the Coriolis force, deviate
1.5 cm to the right.
2. The mathematical derivation
The “Coriolis acceleration,” as it is called when m
is omitted, is traditionally derived by a coordinate
transformation. The relation between the acceleration
of a vector, B, in a coordinate system fixed relative to
the stars (f) and a system, (r), rotating with an angular
velocity, w, is
d
dt
d
f dt r
B B
B æè
öø
=æè
öø
+w ´ . (1)
The procedure is to apply (1) first to the position
vector r, then to its velocity v to get the relative ve-
Corresponding author address: Anders Persson, ECMWF,
Shinfield Park, Reading RG2 9AX, United Kingdom.
E-mail: a.persson@ecmwf.int
In final form 20 March 1998.
©1998 American Meteorological Society
How Do We Understand
the Coriolis Force?
Anders Persson
European Centre for Medium-Range Weather Forecasts, Reading, Berkshire, United Kingdom
ABSTRACT
The Coriolis force, named after French mathematician Gaspard Gustave de Coriolis (1792–1843), has traditionally
been derived as a matter of coordinate transformation by an essentially kinematic technique. This has had the consequence
that its physical significance for processes in the atmosphere, as well for simple mechanical systems, has not
been fully comprehended. A study of Coriolis’s own scientific career and achievements shows how the discovery of the
Coriolis force was linked, not to any earth sciences, but to early nineteenth century mechanics and industrial developments.
His own approach, which followed from a general discussion of the energetics of a rotating mechanical system,
provides an alternative and more physical way to look at and understand, for example, its property as a complementary
centrifugal force. It also helps to clarify the relation between angular momentum and rotational kinetic energy and how
an inertial force can have a significant affect on the movement of a body and still without doing any work. Applying
Coriolis’s principles elucidates cause and effect aspects of the dynamics and energetics of the atmosphere, the geostrophic
adjustment process, the circulation around jet streams, the meridional extent of the Hadley cell, the strength and location
of the subtropical jet stream, and the phenomenon of “downstream development” in the zonal westerlies.
1374 Vol. 79, No. 7, July 1998
locity vr, combine the expressions, and arrive at an
expression for the absolute acceleration a,
a = ar + 2w ´ vr + w ´ (w ´ r), (2)
which for a rotating observer is composed of the observed
acceleration ar; the Coriolis acceleration
2w ´ vr, which only depends on the velocity; and the
centrifugal acceleration w ´ (w ´ r), which only depends
on the position (French 1971, 522; Pedlosky
1979, 17–20; Gill 1982, 73). For a given horizontal
motion the strongest horizontal deflection is at the
poles and there is no horizontal deflection at the equator;
for vertical motion the opposite is true (Fig. 1).
3. Frustration and confusion?
The mathematics involved in the derivation of the
Coriolis force is quite straightforward, at least in comparison
with other parts of meteorology, and cannot
explain the widespread confusion that obviously surrounds
it [for a recent example see an article by Kearns
(1998) in Weatherwise]. The late Henry Stommel
appreciated the sense of frustration that overcomes
students in meteorology and oceanography who encounter
the “mysterious” Coriolis force as a result of
a series of “formal manipulations”:
Clutching the teacher’s hand, they are carefully
guided across a narrow gangplank over the
yawning gap between the resting frame and the
uniformly rotating frame. Fearful of looking
down into the cold black water between the
dock and the ship, many are glad, once safely
aboard, to accept the idea of a Coriolis force,
more or less with blind faith, confident that it
has been derived rigorously. And some people
prefer never to look over the side again.
(Stommel and Moore 1989).
This article will suggest that the main problem
with the teaching of the Coriolis force does not lie so
FIG. 1. Since the Coriolis force is the cross product between the rotational vector of the earth (W) and the velocity vector, it will
take its maximal values for motions perpendicular to the earth’s axis (a) and vanish for motions parallel to the earth’s axis (b). Affected by
a maximal Coriolis force is, for example, air rising equatorward (or sinking poleward) in the midlatitudes. The Coriolis force vanishes
for air rising poleward (or sinking equatorward). Horizontal west–east winds on the equator are indeed affected by the Coriolis force,
although the deflection is completely in the vertical direction.
(a)
(b)
Bulletin of the American Meteorological Society 1375
much with the mathematics of the derivation, but the
purely kinematic nature of the derivation. It readily
provides the “approved” answer, but the price to pay
is a pedagogical difficulty to bridge the gap between
this formalistic approach and a genuine physical
understanding.
4. The mechanical interpretation of the
Coriolis force
Many textbooks are anxious to tell the student that
the Coriolis force is a “fictitious force,” “an apparent
force,” “a pseudoforce,” or “mental construct.” The
centrifugal force, however, although equally fictitious,
is almost always talked about as a force. This leaves
the impression that some fictitious forces are more fictitious
than others.
Some textbooks, with an ambition to avoid “formal
manipulations” and appeal to the student’s intuition,
derive the Coriolis force (in two dimensions) for
a moving body on a turntable, separately for tangential
and radial movements, making use of the variations
of the centrifugal force, w r2, and conservation
of angular momentum, w2 r, respectively. However,
problems arise when the physical conditions under
which this motion takes place are not taken into account
or addressed. In most rotating systems the dominating
force is not the Coriolis force but the centrifugal
force. A man walking at a pace of 1 m s-1 at a distance
of 3 m from the center of rotation on a turntable, making
one revolution in 2 s, will experience a centrifugal
acceleration that is five times stronger than the
Coriolis acceleration. Only at a distance of 0.6 m from
the center are the two forces of equal strength. The
centrifugal effect is eliminated if there is no interaction
between the rotating disk and the body, like a ball
rolling on the disk without friction. But then the problem
is brought back to the traditional kinematic approach
with transformations between coordinate
systems.
Conservation of angular momentum only applies
when there is no torque about the vertical axis. A person
moving on the rotating surface will exert a torque
on it through friction and the conservation will therefore
only apply to the man and the turntable together,
and angular momentum will not be conserved for either
of them. For quite natural reasons these problems
vanish if we consider movements on the rotating earth.
Due to its ellipsoid shape, the gravitational forces
balance the centrifugal force on any body, as long as
it does not move (Fig. 2). When it moves, the balance
is altered. For an eastward movement the centrifugal
force is increased and the body is deflected toward the
equator, to the right of the movement. For a westward
movement the centrifugal force is weakened and can
no longer balance the gravitational force, which is the
physical force that moves the body in the poleward
FIG. 2. The balance between the centrifugal force and the gravitation g*. In the early stages of the earth’s development (a) the
centrifugal force pushed the soft matter toward the equator. A balance was reached when the equatorially directed centrifugal force
balanced the poleward gravitational force and changed the spheroidal earth into an ellipsoid (b).
(a) (b)
1376 Vol. 79, No. 7, July 1998
direction to the right of the movement (Durran 1993;
Durran and Domonkos 1996).
The same result follows from a consideration of
conservation of angular momentum of a body moving
on the earth without being affected by torques. For
poleward (equatorward) movements a decrease (increase)
of the distance to the earth’s axis of rotation is
associated with an increase of the westerly (easterly)
zonal speed. These deflections (pointing to the left of
the motion on the Southern Hemisphere) are manifestations
of the Coriolis force. The application of angular
momentum conservation has, however, given rise
to some misunderstandings.
Some textbooks (and educational sites on the
World Wide Web) explain qualitatively the Coriolis
deflection of a meridional movement as a consequence
of the air’s origin at another latitude where its velocity
due to the earth’s rotation was different (e.g., Battan
1984, 117–118). But this does not relate to the principle
of conservation of angular momentum, but to
conservation of absolute velocity. This misunderstanding
is deceptive because it yields a deflection in the
right direction, but only explains half of the Coriolis
acceleration w × vr instead of 2 w × vr. The seriousness
of the mistake lies not primarily in the numerical
error, but in the confusion between two fundamental
mechanical principles: conservation of linear momentum
and conservation of angular momentum. This potential
misunderstanding is acknowledged by Eliassen
and Pedersen (1977, 98), who make it clear how two
kinematic effects each contribute half of the Coriolis
acceleration: relative velocity and the turning of the
frame of reference. This can also be understood from
simple kinematic considerations (Fig. 3).
5. Foucault, Ferrel, and Buys Ballot
It was only in the late eighteenth century that it was
realized that angular momentum conservation, which
so far had only been applied to celestial mechanics
(Kepler’s second law), could also be applied to mechanics.
So when George Hadley in 1735 realized that
the earth’s rotation deflects air currents, he discussed
it only in terms of conservation of absolute velocity
(Lorenz 1967, 2, 61; Lorenz 1969, 5; see facsimile of
Hadley’s paper in Shaw 1979). This needs to be emphasized
since Gill in his book on ocean and atmospheric
dynamics repeatedly claims that Hadley indeed
made use of conservation of angular momentum (Gill
1982, 23, 189, 369, 506, 549). Although Gill quotes
Hadley’s mathematics in detail (on p. 23) he does not
seem to realize that it relates to conservation of absolute
velocity. Similar misinterpretations of Hadley’s
work can be found in other authorative texts, most
recently in this journal (Lewis 1998, 39, 53).
When the effect of the earth’s rotation was debated
by scientists during the eighteenth century, it was always
in Hadley’s terms of absolute velocity. There was
no notion of any Coriolis force until the midnineteenth
century when Jean Bernard Léon Foucault
demonstrated that a simple pendulum would be deflected
by the earth’s rotation and at latitude j make a
full revolution in one pendulum day, which is
24 h/sinj, double the time for an inertia oscillation
(Fig. 4). This discovery attracted wide scientific and
popular attention and, together with reading Newton’s
and Laplace’s works, inspired William Ferrel in 1856
to conclude that the direction of the wind is parallel
to the isobars, its strength dependent on the latitude
and the horizontal pressure gradient (Khrgian 1970,
222; Kutzbach 1979, 36–38). Independently of Ferrel,
the Dutch meteorologist C. H. D. Buys Ballot in 1857
published his rule based on empirical data according
to which low pressure is to the left if you have the wind
at your back (Snelders and Schurrmans 1980).
Foucault was not an academic and there are good
reasons to assume that he had never heard about Co-
FIG. 3. A straight movement from O to P on a rotating disc/
turntable/merry-go-round follows a curved path. The track is along
the curved arc and the direction of the motion (relative to the fixed
axis) has turned through an angle 2 a while the system has rotated
only an angle, a. (From an idea by R. S. Scorer.)
Bulletin of the American Meteorological Society 1377
riolis when he designed his experiment. So where does
Coriolis enter? In a debate in this journal 30 years ago
(Burstyn 1966; Landsberg 1966; Haurwitz 1966;
Jordan 1966) the question was indeed raised if he has
any place in the meteorological science? One participant,
who was familiar with Coriolis’s original paper
through Dugas (1955, 374–383), noted that he derived
his force “not by transforming coordinates or conservation
of angular momentum, either of which might
suggest themselves to us, but by considering the energetics
of the system, which appears to us to be doing
it the hard way” (Burstyn 1966). This is the closest
any meteorologist appears to have come to find out
what Coriolis actually did. Fortunately his papers have
recently been reprinted (Coriolis 1990). They reveal
that his work is not only of historic interest, but also
opens up new ways to look at the deflective force and
its role in atmospheric dynamics.
6. Gaspar Gustave Coriolis
Gaspard Gustave Coriolis was born on 21 May
1792 in Paris to a small aristocratic family that was
FIG. 4. A graphical illustration, an extension of Scorer’s idea
in Fig. 3, of why it takes half the time for a body to make an inertial
oscillation (12 h/sinj where j is the latitude) than for the arch
of a Foucault pendulum to make a full rotation 24 h/sinj). One
body is moving under inertia (full arrows), the other is a pendulum
bob (open arrows), both seen from above and following the
earth’s rotation of a/t (degrees/time interval). During a time interval
4 t, under which the body following an inertia circle has
changed its direction by 8 a, the pendulum bob has only turned
half of that or 4 a. The heart of the matter is that the effect of the
Coriolis deflection is reversed when the swinging bob moves in the
opposite direction, while this is not the case for the inertia movement.
The explanation also makes it clear that it is irrelevant if
the bob is moving as a pendulum or in any other way, as long as
the direction is constant in space and is reversed periodically. (The
change of direction per swing in the figure is grossly exaggerated.) FIG. 5. Gaspard Gustave Coriolis (1792–1843).
impoverished by the French Revolution (Fig. 5). The
young Gaspard early showed remarkable mathematical
talents. At 16 he was admitted to the École
Polytechnique where he later became a teacher. As one
of his biographers noted, it was this teaching that inspired
his work (Costabel 1961; see also Tourneur
1961; Société Amicale des Anciens Élèves de l’Ecole
Polytechnique 1994, 7, 122; Lapparent 1895). The
education of mechanics in France at the time
was dominated by statics, which was suited only for
problems related to constructional work, not for machines
driven by water or wind. Lagrange’s mechanical
theories were criticized, not because they were
wrong, but because they were difficult to apply to
practical problems. A movement developed with the
chief goal to raise the education of workers, craftsmen,
and engineers in “méchanique rationelle.” Coriolis
was among the first to promote the reform and in 1829
he published a textbook, Calcul de l’Effet des Machines
(Calculation of the Effect of Machines), which
presented mechanics in a way that could be used by
industry. It was only now that the correct expression
for kinetic energy, m v2/2, and its relation to mechanical
work became established (Grattan-Guinness 1997,
330, 449; Kuhn 1977).
1378 Vol. 79, No. 7, July 1998
During the following years Coriolis worked to
extend the notion of kinetic energy and work to rotating
systems. The first of his papers, “Sur le principe
des forces vives dans les mouvements
relatifs des machines”
(“On the principle of kinetic
energy in the relative motion
in machines”), was read to the
Académie des Sciences (Coriolis
1832). Three years later
came the paper that would make
his name famous, “Sur les équations
du mouvement relatif des
systèmes de corps” (“On the
equations of relative motion of a
system of bodies”; Coriolis 1835).
Coriolis’s papers do not deal
with the atmosphere or even the
rotation of the earth, but with the
transfer of energy in rotating
systems like waterwheels. The
1832 paper established that
the relation between potential
and kinetic energy for a body,
m, moving with a velocity, v0,
affected by a force, P, which
makes it accelerate to a velocity,
v1, is the same in a rotating
system as in a nonrotational
(Fig. 6):
mv1
2/2 - mv0
2/2 = ò P cos Q ds, (3)
where Q is the angle between P and ds. Coriolis
applied this relation on problems in nineteenthcentury
technology; we can, for example, relate it to
the increase of speed and kinetic energy of a satellite
falling toward the earth. The work is done by the
gravitational force along the projection on the
satellite’s trajectory.
Three years later, in 1835, Coriolis went back to
analyze the relative motion associated with the system,
in particular the centrifugal force. It is directed perpendicular
to the moving body’s trajectory (seen from a
fixed frame of reference), which for a stationary body
is radially out from the center of rotation. For a moving
body this is not the case; it will point off from the
center of rotation. The centrifugal force can therefore
be decomposed into one radial centrifugal force,
m w2 r, and another, -2 m w v, the “Coriolis force.” It
is worth noting that Coriolis called the two components
“forces centrifuges composées” and was interested
in “his” force only in combination with the radial
centrifugal force to be able to compute the total centrifugal
force.
FIG. 6. Coriolis’s first theorem: a body, m, on a rotating turntable
moving with a speed, v0, is subject to a force, P, and displaced
along a trajectory, ds, and accelerates to v1. The change in
kinetic energy, corresponding to the change of potential energy, is
due to the work done by the driving force P along the projection
of the distance ds where Q is the angle between P and ds. (To make
the dynamic discussion complete, Coriolis also considered the
centripetal force Pe and the balancing centrifugal force Fe, both
acting to keep the body in a fixed position in the absence of a driving
force P. Both Fe and Pe cancel out in the energy equation.)
(b)
(a)
FIG. 7. Coriolis’s second theorem is most easily understood when the rotating system is
viewed first from a fixed frame of reference (a), then in the rotating frame of reference (b).
The total centrifugal force acting on the body m moving with a velocity V is directed perpendicular
to the tangent of the movement, along the radius of curvature (Fig. 6a). It can be
decomposed into two centrifugal forces; one m w2 r, directed from the center of rotation and
a second, -2mV, w, the Coriolis force, perpendicular to the relative motion Vr (Fig. 6b).
Bulletin of the American Meteorological Society 1379
It is significant that Coriolis did not make use of
angular momentum conservation. His two theorems
actually help us to understand that this important principle
has its explanatory limitations.
7. Kinetic energy and angular
momentum
It is a common misconception that kinetic energy
and angular momentum, if not the same, at least vary
synchronously, just like linear momentum and kinetic
energy. To show that this is not the case, let us make
use of one of the popular actors on the pedagogical
scene: the rotating ice skater. She increases her rotation
when she horizontally moves her arms inward.
With no friction against ice and air she conserves her
angular momentum. This is the standard argument.
More rarely addressed is the question of what happens
to her rotational kinetic energy. One might be tempted
to assume that it also remains conserved, but this is
wrong. It will increase significantly.
A short and elegant proof is found in one of
R. Feynman’s lectures on physics: with her arms
stretched out the ice skater has a moment of inertia, I;
an angular velocity, w; and an angular momentum, L
= Iw, which remains constant. The rotational kinetic
energy, K = Iw2/2 or L w /2, increases with increasing
rotation, when L remains unchanged (Feynman
et al. 1977, 19-8).
But from where does the extra kinetic energy
come? Contracting one’s arms while standing still
does not constitute any work in a mechanical sense;
while rotating one has to apply a force inward to
counter the centrifugal force; the extra energy comes
from the work done along this force. This follows exactly
from Coriolis’s first theorem: the increase in rotation
is achieved by the work done by her muscles
when she pulled in her arms.
It is not correct to say that the ice skater “makes
use of” the Coriolis force “in order to” increase her
rotation. Since the Coriolis force is always directed
perpendicular to the movement of a body, it can only
change its direction, not its speed and kinetic energy:
it does no work. Nor is it quite true that she increases
her rotation “in order to” conserve angular momentum.
Even if friction from the ice and the air will
slightly decrease her angular momentum, she will still
increase her rotation and kinetic energy. A satellite entering
the earth’s outer atmosphere will decrease its
angular momentum, due to friction, but, as mentioned
earlier, still increase its speed and kinetic energy.
Contrary to what might intuitively be expected, the
increase will be proportional to the strength of the resisting
frictional force (French 1965, 471–473).
The limitations of trying to interpret all kinds of
rotating motion solely from an angular momentum
perspectives become clear if we perform another experiment
with the (by now exhausted) ice skater. Let
her carry heavy weights in her hands. When she contracts
her arms, she will still conserve angular momentum,
but this will not prepare us for what happens next:
at some stage in the contraction she will be unable to
continue to move her arms inward. This occurs when the
outward centrifugal force has increased so much that
it balances the centripetal force from her arm muscles.
An interesting situation arises when she slackens
her arms. The centrifugal force drives them outward
and her rotational kinetic energy is converted into potential
energy. Work is done, not by the centrifugal
force, but by her muscles, doing negative work.
“Negative work” is a well-established concept in mechanics
and simply means the work done when a force
moves in a direction opposite to that in which it works.
When a ball passively rolls uphill, inertia might “do
the job,” but the conversion from kinetic to potential
energy is done by gravity, doing negative work. So far
the author has seen the notion of negative work explicitly
mentioned only once in the meteorological literature
(Ertel 1938, 48).
Coriolis’s first and second theorems put the deflective
force into a dynamical context and make clear
what it does—and does not do. All this is possible
because Coriolis avoided a kinematic approach and instead
derived “his” force through a dynamic analysis
of a mechanical system. This is a circumstance that is
more remarkable than it appears at first glance.
8. The development of kinematics
In 1834, at the same time as Coriolis worked on
his derivations, one of his colleagues at the École
Polytechnique, André-Marie Ampère, published a
major philosophical treatise, “Essai sur la Philosophie
des Sciences.” He noted that from the experience by
earlier scientists, from Kepler to Euler, it was found
to be possible to study motions without necessarily
considering the forces that create or result from this
motion. Ampère suggested that this approach could
form a new branch of mechanics, which he called
cinématique. Kinematics was soon accepted as a new
1380 Vol. 79, No. 7, July 1998
discipline. It was soon developed to higher levels,
among others by George Stokes, working on his equations
for fluids and solids (Smith and Wise 1989, 199,
360–372; Koetiser 1994). From fluid mechanics the
step was not far to dynamic meteorology.
The use of kinematics in dynamic and synoptic
meteorology includes not only parameters like divergence
and vorticity (absolute, relative, and potential),
but also more complicated relations like the geostrophic
and thermal wind relations, the omega equation,
Sutcliffe’s equation, the Q vector, etc. Many of
them are based on constraints like balance conditions
or conservation properties and are useful to analyze
and predict atmospheric movements: “When the causative
forces are disregarded, motion descriptions are
possible only for points having constrained motion,
that is, moving on determined paths. In unconstrained
or free motion the forces determine the shape of the
path” (Encyclopaedia Britannica, s.v. “kinematics”).
Although the use of constraints makes kinematics
potentially less powerful a tool than dynamics
(Hess 1979, 198), there is no contradiction between
the two. They complement each other: the former describes
“how,” the latter “why.”
As Knox and Croft (1997, 903) point out, thinking
in terms of divergence, vorticity, development
equations, etc., ties well with “the teddy bear of the
good ol’ equations of motion,” in particular when the
student gets confused with the former. One source of
such confusion is misinterpretations of kinematic
models, as pointed out by Holton (1993) and Persson
(1996, 1997). A common malpractice is to defy the
limitations of kinematics and try to deduce causal relations.
This leads easily to metaphysical reasonings
like, “In order to maintain geostrophic balance, the
wind has to . . .”. Since in a kinematic system any variable
can be appointed as a “cause,” it is not more true
to say that the divergence term in the vorticity equation
is the cause of changes in the vorticity term, than it
is to say that a change of vorticity is the cause of divergence
in the wind field. Further confusion comes from
the introduction of kinematical and statistical concepts
from turbulence theory, for example, “Reynolds
stresses,” to represent physical mechanism, in spite of
warnings from Pedlosky (1979, 172) among others.
It goes beyond the scope of this article to discuss
the misuse of kinematic conceptual models, except
when the Coriolis force is directly involved. This is
the case with the pedagogical use of angular momentum
conservation to explain certain features in the
general circulation.
9. Angular momentum in the
atmosphere
The principle of angular momentum conservation
is dependent on the condition that there is no net
torque in the direction of the rotation, like friction or
pressure gradient forces. The conservation therefore
applies only to the atmosphere as a whole, since the
global rate of working of the pressure gradient force
is zero. It does not apply to parts of it where local pressure
gradients exert torques. This was well known by
the meteorological generation before the Second
World War. As a consequence, they were critical to
angular momentum conservation as a model for the
large-scale circulation. Exner (1917) and Haurwitz
(1941) published tables to show how a parcel moving
meridionally just 10° of latitude under conservation
of angular momentum would experience a wind
increase of 100 m s-1 or more (Table 1).
The leading British meteorologist of the time,
David Brunt, regarded the whole idea of air moving
from one latitude to another while retaining its original
angular momentum, as “highly misleading” since air
TABLE 1. Increase in speed of a ring of air displaced 10° meridionally
under conservation of angular momentum (after Exner
1917, 24, 179–81; Haurwitz 1941, 121). Both Exner and Haurwitz
concluded that large-scale meridional displacements of air masses
under conservation of angular momentum hardly occurred in the
atmosphere, at least not at higher latitudes.
From To Velocity To Velocity
lat lat u m s-1 lat u m s-1
90° 80° -81.5
80° -70° -117.3 90° ¥
70° 60° -123.2 80° 234.7
60° 50° -118.3 70° 179.8
50° 40° -105.3 60° 151.1
40° 30° -87.7 50° 125.9
30° 20° -65.9 40° 98.9
20° 10° -41.0 30° 71.5
10° 0° -14.0 20° 43.0
0° 10° 14.2
Bulletin of the American Meteorological Society 1381
set in motion cannot travel to another latitude unless it
is guided by “a suitable arranged” pressure gradient.
If there were no pressure gradients, the air would be
constantly deviated due to the Coriolis force, describe an
inertia oscillation, and return to its latitude. Air moving
meridionally 20 m s-1 at 60° would turn back after
160 km (Brunt 1941, 404–405). The reason a hemispheric
Hadley circulation is not possible is not that it
would be baroclinicly unstable in the midlatitudes, as is
often stated, but that air conserving angular momentum
will be unable to move very far poleward. As Durran
(1993) recently pointed out, an inertial oscillation is
also a “constant angular momentum oscillation.”
Carl-Gustaf Rossby, although at first a proponent
of explaining the general circulation in terms of angular
momentum conservation (Rossby 1941), soon
became skeptical when he realized that the existence
of zonal pressure gradients violated the principle.
Those who tried to avoid the problems by considering
longitudinal averages of pressure and wind in a
hemispheric ring of air did not cut any ice with him.
They were compared with Ptolemaic astronomers who
added epicycles to their system to explain the motions
of the planets (Rossby 1949b, 18, 23, 26). This was
the start of the famous Rossby–Starr–Palmén debate,
which soon came to focus on other aspects of the general
circulation (Lewis 1998).
It remains a topic for historical research to find out
why and how angular momentum conservation, in
spite of the previous scientific objections, in the late
1940s became established as the key to understand the
general circulation. Although its basis was never theoretically
challenged, except by James (1953), its has
been accepted with mixed feelings. Whereas Hartmann
(1994, 150) regards angular momentum conservation
as a “heavy constraint” on the atmospheric movements,
others see it as “a clearly hopeless description”
(James 1994, 81–82) due to the unrealistically strong
zonal winds that would develop at middle and high
latitudes, a point made already by Helmholtz in the
1880s (Lorenz 1967, 66–71).
10. Coriolis’s approach applied to the
atmosphere
So far we have seen that the dynamics of an ice
skater can be understood to only a limited extent exclusively
from an angular momentum point of view,
while Coriolis’s two theorems cover alternative and
complementary aspects. This will prove valuable for
the understanding of atmospheric processes, in particular
for the wind and pressure relation.
a. Geostrophic adjustment
Lorenz (1967, 29, 65) has commented on a frequent
tendency among meteorologists to assume that the
wind field is somehow produced by the height field
in a simple one-way process and to overlook that they
determine one another through mutual effects. The
common textbook discussion relates to the acceleration
of a (subgeostrophic) wind into a confluent isobaric
field. Rarely discussed is what happens in a
diffluent isobaric field: the pressure gradient force
weakens and the (supergeostrophic) wind is deflected
toward higher pressure by the Coriolis force. The
mutual wind and pressure adjustment is a consequence
of the mass and energy transport across the isobars
(Uccellini and Johnson 1979; Uccellini 1990, 125).
b. Jet stream dynamics
The three-dimensional flow around a jet stream,
traditionally described in kinematic terms of vorticity
advection, can also be understood in Coriolis’s
terms of forces, energy, and in particular inertia. The
mass and kinetic energy released at the entrance of a
jet stream will be rapidly carried through the jet by the
flow itself. At the exit the velocity is reduced and the
kinetic energy is converted back to the “reservoir of
potential energy” (Kung 1977). The pressure pattern
is slowly driven downstream while the air is rapidly
passing through. Further energy discussions explain
the tendency for upward motion and cyclogenesis at
the right entrance and the left exit (Fig. 8). The tradi-
FIG. 8. A schematic illustration of a jet stream with geopotential
lines, wind vectors, and two frontal systems and the inferences
that can be made from it (see appendix).
1382 Vol. 79, No. 7, July 1998
tional kinematic vorticity explanation can neither account
for the vertical (indirect) circulation at the exit
as a response to the kinetic energy arriving under inertia
from the entrance, nor can it explain the speed
by which a development at the right entrance of the
jet rapidly may affect and even initiate a development
at the left exit. The mechanism by which the available
potential energy is rapidly transported downstream,
temporarily converted into kinetic energy, and then
at the jetstream exit converted back to available potential
energy is what forecasters since long ago have
recognized as one of the links in a “downstream development”
chain, which can spread over half of the
hemisphere in less than a week.
c. Energy balance
The process of conversion between potential and
kinetic energy in the free atmosphere is adiabatic and
reversible (van Mieghem 1973). There are widespread
misconceptions that it is irreversible (see, e.g., Rossby
1949a, 163; Carlson 1991, 16, 109, 114, 443; Lewis
1998) and can only go back to potential energy after
having been dissipated by friction to heat (Petterssen
and Smedbye 1971; Carlson 1991, 122). The wellknown
energy boxes (Fig. 9) only show the net conversion
between two almost equally strong fluxes of
potential and kinetic energy, and the long-term ultimate
and irreversible dissipation of kinetic energy into
heat (Lorenz 1967, 103; van Mieghem 1973, 157,
Uccellini 1990, 125; Kung 1971, 61; Peixoto and Oort
1992, 311). Turbulent and frictional dissipations are
important sinks for kinetic energy in the boundary
layer but not in the free atmosphere where most of the
kinetic energy is converted back to potential energy.
Work is done, but not by the Coriolis force, as stated,
for example, by Starr (1969, 198, 256), but by the pressure
gradient force, doing negative work. The fact that
large-scale motion can convert kinetic energy into potential
energy distinguishes it from three-dimensional
turbulence, where the total kinetic energy is constant
but redistributed among the scales. When L. F.
Richardson wrote his famous verse about “Big whirls
have little whirls/that feed upon their velocity/And
little whirls have lesser whirls/and so on to viscosity,”
he thought about normal turbulence, not the largescale
circulation as is implied by some textbooks
(e. g., Wallace and Hobbs 1977, 437).
d. The subtropical jet and the Hadley cell
One of the strongest and most persistent wind systems
is the subtropical jet on the poleward side of a
strong Hadley cell (Fig. 10). Fifty years after its discovery
it is still, according to a recent article in
Weatherwise, “the most frequently slighted” of all the
jet streams and is in need of “a new publicity agent”
(Grenci 1997). Curiously, there does not seem to be
any consensus among the experts about its mechanism
(Wiin-Nielsen and Chen 1993, 151) or why it is not
stronger (Hartmann 1994, 153). This might be due to
attempts to explain it along the line of angular momentum
conservation and transport. Coriolis’s approach
would lead us to seek the explanation in the
north–south pressure gradient in the Hadley circulation,
which drives the air northward and increases its
speed, while at the same time the Coriolis force deflects
this increased wind toward the east. It is the
pressure gradient force that increases the wind and
kinetic energy, not the Coriolis force, as stated, for
example, by Starr (1954, 271) and Wiin-Nielsen and
Chen (1993, 161). A rather weak pressure gradient of
5 hPa over 10 latitude degrees yields an acceleration
of 0.4 mm sec-2, which does not appear much, but after
one day has resulted in a wind speed of 40 m s-1
and transported air 2000 km, a distance equal from the
equator to 20° latitude.
When the supergeostrophic wind on the poleward
side of the Hadley cell adjusts geostrophically, zonal
pressure gradients are set up or enhanced that prevent
angular momentum from being conserved. With the
earth’s particular differential heating and rotation, this
FIG. 9. Two ways to depict the “energy boxes” of the atmosphere’s
general circulation: (a) the conventional, which shows
the net fluxes, but might easily be misunderstood to mean that
the flux can only go one way; (b) a pedagogically more secure
representation where it clearly comes out that the net flux is
the difference between two almost equally large fluxes in both
directions.
(b)
(a)
Bulletin of the American Meteorological Society 1383
deflection occurs at about 30° latitude
and this determines its meridional extent
of the Hadley cell. On another planet,
with weak differential heating and/or
faster rotation, the Hadley cell will be
confined to the equatorial band where
there will be a strong equatorial jet
stream, as shown in illuminating experiments
by Williams (1988). A simple
mechanic model that illustrates this is the
previously discussed ice skater, who was
unable to contract because of heavy
weights in her hands.
11. Coriolis’s legacy
In 1836 Coriolis was elected into the
Academie de Science and in 1838
became deputy director at the École
Polytechnique. In 1843 his health deteriorated
and he died while working on a
revision of his 1829 book. The importance
of his work was not realized outside
mechanics until 1859 when the
French Academy organized a discussion
concerning the effect of the earth’s rotation
on water currents like rivers
(Khrgian 1970, 222; Kutzbach 1971, 92;
Gill 1982, 210, 371). Coriolis’s name
began to appear in the meteorological literature
at the end of the nineteenth century,
although the term “Coriolis force”
was not used until the beginning of this
century.
All major discoveries about the general
circulation and the relation between
the pressure and wind fields were made
without any knowledge about Gaspard
Gustave Coriolis. Nothing in today’s
meteorology would have been different
if he and his work had remained forgotten.
The “deflective force” would just
have been named after Foucault or Ferrel.
(The letter f for the “Coriolis parameter”
may be an early attempt to honor one of
them.) However, had Coriolis’s work
been read and his dynamic approach understood,
today’s confusion about the
deflective force and the role of inertia in
the atmospheric processes might have
(a)
(b)
FIG. 10. The large-scale circulation at 500 (a) and 200 hPa (b) on 6 January
1997 at 1200 UTC. The jet streams are not only stronger at 200 hPa than at 500
hPa, but there is a strong subtropical jet at 200 hPa over northern Africa and
southern Asia, which is not found at 500 hPa. At around this time, two famous
European balloonists, R. Branson and P. Lindstrand, took off from Morocco in
an attempt to sail around the world by taking advantage of the subtropical jet
stream.
1384 Vol. 79, No. 7, July 1998
been avoided. So he is well qualified to lend his name
to the Coriolis force. Had he been with us today, he
might have been one of the few who had understood
and taught it properly!
Acknowledgments. This article has developed from inspiring
discussions with colleagues during the last years, in particular
with George Platzman and Adrian Simmons who restored my
appreciation of mathematics as a powerful tool. I am also grateful
to David Anderson, François Bouttier, and Jim Holton, who
patiently convinced me that some traditional truths actually are
not wrong, just because they are traditional. I am also indebted
to Professor Emeritus Enzo O. Macagno at the University of
Iowa, who provided an introduction to the history of kinematics
on the World Wide Web (enzo-macagno@uiowa.edu). One of
the anonymous reviewer’s comments helped me to further clarify
the content. Jean-Pierre Javelle, who forwarded copies and reprints
of Coriolis’s original papers, is also kindly acknowledged.
Appendix: A nonkinematic look at the
jet stream circulation
At the right entrance of the jet the wind is accelerated
by the pressure gradient force and is deflected to
the right by the Coriolis force; at the exit the pressure
gradient force weakens, and the Coriolis force drives
the wind to the right, up the pressure gradients and the
wind is decelerated by the pressure gradient force. As
potential energy converts into kinetic energy at the entrance,
the pressure gradient weakens; as kinetic energy
converts back to potential energy at the exit, the
pressure gradient strengthens—the whole pressure
pattern moves downstream.
The downgradient transport of mass at the entrance
of the jet stream becomes part of a full threedimensional
circulation where rising motion occurs for
warm air, sinking for cold. This means an overall lowering
of the point of gravity, which is consistent
with the loss of potential energy. The upgradient transport
at the jet exit forces for similar reasons a threedimensional
circulation where cold air is rising and
warm air sinking, which rises the point of gravity in
consistence with the increase in potential energy. The
tendency of rising motion at the right entrance and the
left exit of the jet streams are favorable locations for
synoptic developments.
The flow through the core of the jet is roughly geostrophic
and uneffected by any net force, since the
pressure gradient force and the Coriolis force balance
each other. Depending on the curvature of the trajectory
of the air parcels passing through the jet the velocity
is supergeostrophic (anticyclonic trajectory) or
subgeostrophic (cyclonic trajectory). The generation
of supergeostrophic winds results in wind oscillations
with a period of about one pendulum day and explains
the cycloid shape of most anticyclonic jet streams.
The flow through the jet stream carries rapidly both
mass and energy from one end of the jet to the other.
Baroclinic, kinetic energy-generating processes at the
right entrance, including latent heat releases from a developing
cyclone (or hurricane in extreme cases), rapidly
affect the conditions at the left exit and can even
initiate a development.
With several jet streams lined up after each other,
the impact can spread downstream like a “domino effect”
from one cyclone to the next. The well-known
notion of “downstream development” in the midlatitude
westerlies, sometimes described in kinematic or
graphical terms of “group velocity,” is essentially a
matter of energy transfer and transport.
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