Raymond Hide at 80 • Hide: Geophysical fluid dynamics
A path of discovery in geo Raymond Hide looks back over
his career and the development
of the annulus approach, from
his early experiments in a
Cambridge lab to the very
latest ideas in geophysical,
meteorological and planetary
fluid theory.
T
his article outlines events that set the
course of my scientific career. It was submitted for inclusion in this festschrift at
the request of Prof. Peter Read and colleagues,
who were kind enough to organize a related
one-day meeting on “Rotating Fluids in Geophysics” held in July 2009 at the Clarendon
Laboratory of Oxford University to mark my
80th birthday. The meeting was sponsored by
the Royal Astronomical Society and the Royal
Meteorological Society, membership of which
societies I have enjoyed for nearly 60 years.
The article is based on my contribution (entitled “Geomagnetism, vacillation, atmospheric
predictability and deterministic chaos”, Hide
2006) to a wide-ranging symposium on “Paths
of Discovery” in scientific research organized
by the Pontifical Academy of Sciences (PAS), a
small self-governing international body of scientists elected to membership with no account
taken of religious persuasion (if any), gender,
race or nationality (Cartlidge 2009). For the
symposium, members were invited to describe
and explain events bearing on discoveries with
which they had in some way been associated. I
must thank the PAS for permission to reproduce
material from that contribution here.
As the editors of the proceedings of the symposium emphasized: “Discoveries are at the
basis of new knowledge. Some are made upon
verification or ‘falsification’ of a theory but in
many cases serendipity plays a key role. Then a
discovery is made while something else is being
sought but the scientific mind and intuition of
the researcher become directed towards the
unexpected” (Arber et al. 2006). Serendipity
certainly featured in some of the events marking
the chosen path of discovery described in Hide
(2006), but not in the earliest.
Geophysicists now accept that the main
geomagnetic field must be a manifestation of
ordinary electric currents flowing in the Earth’s
metallic core and that the currents are generated by “self-exciting dynamo” action involving convective motions in the liquid outer core.
But in the 1940s the then controversial subject
4.16
1: General view of the apparatus used in the original annulus experiments in cases when the outer
wall of the annulus was heated and the inner walled cooled (from Hide 1953b). Cooling water
tubes are on the right and the Variac and transformer arrangement used to vary the power to the
heating element immersed in the outer water jacket are on the left. The turntable mounted on a
carefully levelled cast-iron plate was driven into rotation by a belt and pulley arrangement. The
vertical post near the middle of the picture carries a “Dove prism” rotoscope at the top (not seen
in picture), part of the belt drive to which can be seen running parallel to the post. The scale can be
judged by noting that the transformer on the left rests on two standard building bricks.
of magneto­hydrodynamics (MHD) was in its
infancy and the “geodynamo” hypothesis had
yet to gain convincing theoretical and observational support. It was against this background
that in 1947 the cosmic-ray physicist P M S
Blackett – then the head of the lively Department of Physics of the University of Manchester
and due to receive a Nobel Prize the following
year – offered a different hypothesis to account
for the magnetism of the Earth and other massive spinning astronomical bodies (Blackett
1947). The subsequent “falsification” of the
hypothesis by the findings of two crucial tests
– one involving a laboratory experiment devised
and executed by Blackett himself (1952) and
the other involving determinations of the geomagnetic field in deep coal mines from which
the variation of the strength of the field with
depth field could be evaluated (Runcorn et al.
1951) – had the effect of focusing the attention
of geophysicists on the geodynamo hypothesis.
The final event on our chosen path of discovery
took place in 1963, when E N Lorenz – a dynamical meteorologist at the Massachusetts Institute
of Technology (MIT) working on the predictability of weather patterns – published an article
in a meteorological journal reporting research
on deterministic non-periodic fluid flow (Lorenz
1963a). This eventually attracted wide attention
once it had been recognized by mathematicians
as a major contribution to an area of research
that became known in 1973 as “chaos theory”.
This continues to influence ideas and methodologies in many branches of science and engineering
(Thompson and Stewart 2002, Tong 1995).
Intermediate events started with a series of
laboratory experiments on thermal convection
in spinning fluids carried out from 1950–1953
at the Department of Geodesy and Geophysics
of Cambridge University, where I had the good
fortune to discover with simple apparatus a rich
variety of nonlinear regimes of flow (see figures
1 and 2). The experiments were motivated in the
first instance by my interest in the origin of the
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Raymond Hide at 80 • Hide: Geophysical fluid dynamics
physical fluid dynamics
2: Thermal convection in a rotating liquid annulus. Streak photographs of upper-surface flowpatterns at six values of the angular speed of rotation of the turntable, Ω (rad. sec.–1), namely
(a) 0.05, (b) 0.85, (c) 2.10, (d) 3.64, (e) 3.98 and (f) 6.32. In all cases shown, the sense of the basic
rotation was counterclockwise and the outer cylinder warmer than the inner cylinder, giving
relative flow with generally positive azimuthal components in the upper reaches of the liquid.
This could be reversed by reversing the temperature difference between the bounding cylinders
(cf. figure 9 of Hide 1966). (a): An example of “upper (axi)symmetric” flow, when convective heat
transfer is effected by slow meridional overturning associated with flow in end-wall Ekman
boundary layers. (b,c,d): Examples of steady regular non-axisymmetric sloping convection.
(e,f): Examples of nonsteady irregular non-axisymmetric sloping convection.
main geomagnetic field, acquired through contact as an undergraduate with Blackett’s Manchester colleague S K Runcorn and his research
team working on the “mine experiment”. But
by attracting the attention of meteorologists
engaged in research on large-scale atmospheric
motions and their predictability, the laboratory
experiments were destined to influence Lorenz’s
seminal mathematical work on nonlinear
dynamical systems and chaos theory.
In what follows, geomagnetism and motions
in the Earth’s liquid core are discussed first.
Next are outlined the various regimes of thermal convection in a rotating liquid annulus,
including various manifestations of “sloping
convection” − notably steady flows, periodic
“vacillation” and non-periodic “geostrophic
turbulence” − and other findings of the Cambridge experiments, together with their implications for geophysical and astrophysical fluid
dynamics and for dynamical meteorology and
atmospheric predictability. Then the celebrated
A&G • August 2010 • Vol. 51 Lorenz equations and other low-dimensional
mathematical models of nonlinear dynamical
systems are discussed and the article ends with
an epilogue on modern developments.
Geomagnetism and motions in the
Earth’s liquid core
Speculations about the origin of the Earth’s magnetism go back several centuries, but geophysicists now agree that the phenomenon must be
due to ordinary electric currents flowing within
the Earth’s metallic core, where they experience
least resistance. While chemical and thermoelectric effects are unlikely to be strong enough
to provide the electromotive forces needed to
maintain the currents against ohmic dissipation,
motional induction involving hydrodynamical
flow in the liquid outer core cannot be ruled out
on quantitative grounds. This is why theoretical geophysicists – now equipped with powerful supercomputers – are prepared to wrestle
with the mathematical complexities of “self-
exciting dynamos” in electrically conducting
fluids (Backus et al. 1996, Dormy and Soward
2007, Ghil and Childress 1982, Glatzmaier and
Roberts 1997, Gubbins and Herrero-Bervera
2007, Hide 2008, Jacobs 1994, Moffatt 1978,
Roberts 2011, Weiss 2002).
By the process of motional induction, dynamos convert the kinetic energy of an electrical
conductor moving through a magnetic field into
the magnetic energy associated with the electric
currents thus generated in the conductor. In selfexciting dynamos an infinitesimally weak adventitious background magnetic field suffices for the
starting-up process. The self-excitation principle was discovered in the 1860s by engineers
concerned with public electricity supply, who
experimented with devices in which the rotating armature was connected by sliding electrical
contacts to a suitably oriented stationary field
coil. Such devices are topologically more complex in structure than a continuous body of fluid
such as the Earth’s liquid outer core. So it is by no
means obvious that self-exciting dynamo action
is possible in a fluid: but it is. Theoreticians seeking “existence theorems” from the equations of
electrodynamics have shown that most flows of
sufficient rapidity and complexity of form in
an electrically conducting fluid could generate
and maintain a magnetic field against ohmic dissipation. G E Backus (1958) and A Herzenberg
(1958) published the first existence theorems
nearly four decades after J Larmor (1919), in a
short paper on the magnetic fields of sunspots,
first suggested that self-exciting dynamo action
might be possible in a moving fluid (see Dormy
and Soward 2007, Gubbins and Herrero-Bervera
2007, Hide 2008, Weiss 2002).
Larmor’s important idea appeared less attractive when a mathematical study by T G Cowling (1934) cast doubt on the possibility that
motional induction was capable of maintaining
magnetic fields of the limited class that possess
an axis of symmetry. Larmor (but few others at
the time) correctly saw, on quantitative grounds,
that some kind of dynamo mechanism was
needed to explain solar magnetism. This was,
of course, some years before radio-astronomers
discovered that the giant rapidly rotating planet
Jupiter possesses a strong magnetic field of
“reversed” polarity with respect to its rotation
axis and rock-magnetism research produced
convincing evidence that the geomagnetic field
must have switched in polarity on many occasions over geological time (see below). So it
was against a background of uncertainty that
Blackett (1947) proposed his new hypothesis to
explain the origin of the main geomagnetic field.
4.17
Raymond Hide at 80 • Hide: Geophysical fluid dynamics
According to the hypothesis, which invoked an
earlier suggestion associated with the names of
H A Wilson and E Schrödinger (see Schröder
and Treder 1997), the main magnetic fields of
the Earth, Sun and the star 78 Virginis were all
manifestations of a new law of Nature, whereby
any rotating gravitating body, irrespective of its
chemical composition, would be magnetic in virtue of its rotation. Its magnetic moment would
be proportional to its spin angular momentum,
with the constant of proportionality equal to
the square root of the universal gravitational
constant divided by twice the speed of light.
Blackett’s hypothesis attracted wide attention for, if correct, it would provide a basis for
settling a fundamental issue in physics, that of
unifying the laws of electromagnetism and gravity. E C Bullard, an academic geophysicist with
wide-ranging research interests, quickly pointed
out that the hypothesis might be test­able from
determinations of the vertical variation of
the geomagnetic field in the upper reaches of
the Earth. Blackett responded by setting up a
research team under Runcorn charged with the
task of measuring the field in deep coal mines.
The hypothesis, which was shown theoretically
by S Chapman and Runcorn to imply that the
strength of the horizontal component of the geomagnetic field would exhibit a decrease with
depth, was successfully “falsified” by the findings of the “mine experiment” (Runcorn et al.
1951), which indicated an increase with depth.
It was also falsified by a delicate laboratory
experiment carried out by Blackett (1952) himself, for which he developed an astatic magnet­
ometer of high sensitivity (which was to prove
valuable in palaeomagnetic work).
These findings left geophysicists concerned
with the origin of the main geomagnetic field
with little choice but to confront the mathematical complexities of “geodynamo” theory and
the MHD of the Earth’s core (see Backus et
al. 1996, Dormy and Soward 2007, Gubbins
and Herrero-Bervera 2007, Glatzmaier and
Roberts 1997, Roberts 2011, Weiss 2002). The
complexities stem from the nonlinearity of the
equations of MHD that govern flows in electrically conducting fluids. MHD phenomena such
as self-exciting fluid dynamos must abound in
large-scale natural systems such as stars and
planets, where values of the “magnetic Reynolds number” R = UL µσ can be high. (Here U
is a characteristic flow speed, L a characteristic
length scale, µ the magnetic permeability of the
fluid and σ its electrical conductivity.) But the
scope for investigating MHD phenomena on
the small scale of the terrestrial laboratory is
limited, owing to the difficulty of attaining high
values of R with available conducting fluids (see
Dormy and Soward 2007, Gubbins and Herrero-Bervera 2007, Lowes 2011).
Convective motions in the Earth and other
planets and the Sun and other stars are driven
4.18
by buoyancy forces due to the action of gravity on density inhomogeneities associated with
differential heating and cooling. Pioneering geodynamo theorists W M Elsasser (1939, 1950)
and J Frenkel (1945) were the first to suggest
that the influence of gyroscopic (Coriolis)
forces on patterns of convection in the Earth’s
liquid outer core might somehow account for
the approximate alignment of the geomagnetic
field with the Earth’s rotation axis, which has
been exploited by navigators using the magnetic
compass for nearly a thousand years.
Regimes of thermal convection in a
rotating liquid ‘annulus’; ‘vacillation’
and ‘geostrophic turbulence’
In 1948, as an impecunious physics undergraduate at Manchester University needing part-time
paid employment, I joined the “mine-experiment” as an assistant. The experience of working with Runcorn and his team stimulated my
interest in geophysics and led to my enrolment
in 1950 as a PhD student in the small Department of Geodesy and Geophysics at Cambridge
University, where research in geodesy and
seismology was already well established and
new initiatives were being taken in marine geo­
physics, geo­magnetism and palaeomagnetism.
Aware of the possible geophysical relevance
of research on thermally driven flows in rapidly
rotating fluids, on reaching Cambridge I undertook some preliminary laboratory experiments
on thermal convection in a cylindrical annulus
of liquid (water) spinning about a vertical axis
and subjected to an impressed axisymmetric
horizontal temperature gradient. The necessary
apparatus (see figure 1) was quickly designed
and assembled using readily available equipment and other resources. These included a
war-surplus synchronous electric motor, a steel
turntable formerly used for grinding rocks, a
supply of brass and glass tubing up to about
10 cm in diameter, and a Dove prism borrowed
from the Cavendish Laboratory for the purpose
of constructing a “rotoscope” to view flow patterns in a rotating system.
The resources of the Cambridge department
also included, crucially, a small workshop where
research students were able to design and construct apparatus under the watchful eye of the
department’s helpful and talented senior technician, L Flavell. Also available was a recording
camera incorporating a set of gal­van­ometers,
which later enabled me to investigate fluctuating temperature fields using a small array of
tiny thermocouples immersed in the convecting liquid. The recording camera had been used
some years earlier in a programme of seismological field work by Bullard who later, from
1950–1955, served as the director of the UK
National Physical Laboratory in Teddington,
where access to rare computing facilities enabled him to pioneer numerical work on self-
exciting dynamos (Bullard and Gellman 1954,
see also Dormy and Soward 2007, Gubbins and
Herrero-Bervera 2007).
My motivation when starting the experiments
amounted to little more than the hope that lab­
oratory work on buoyancy-driven flows in a liquid of low viscosity that were strongly influenced
by Coriolis forces due to general rotation might
produce interesting results and possibly provide
insights into motions in the Earth’s liquid outer
core. Luckily, promising lines of investigation
emerged as soon as the apparatus was operated
for the first time, when a striking persistent
regular flow pattern of four waves marked out
by a meandering jet stream was seen at the top
surface of the convecting liquid (see figure 2c).
Raising the value of Ω (say), the steady angular
speed of rotation (typically a few radians per
second) of the turntable on which the annular
convection chamber was mounted, had the effect
of increasing the number of regular waves, m
(say, see figures 2c, d), but not beyond a point at
which what would now be termed “geostrophic
turbulence” set in, with increasingly distorted
flow patterns exhibiting irregular (“chaotic”)
fluctuations (see figures 2e, f). Reducing Ω had
the opposite effect of decreasing m, but not
beyond a point at which the non-axisymmetric
(N) flow disappeared, giving way to axisymmetric (A) flow (see figures 2b, a).
In the experiments illustrated by figure 2,
heat enters the annular convection chamber by
conduction via the warm sidewall and leaves
by conduction via the cool sidewall. Within the
liquid itself, however, heat transport is effected
mainly by convective motions, circulating in
such a way that warm liquid moves towards the
cool cylinder and cool liquid towards the warm
cylinder. When the flow is axisymmetric (as in
figure 2a), it is the meridional component of the
circulation that accounts for the convective heat
transfer, and it also accounts for the conversion
of potential energy associated with the action
of gravity on the density field maintained by the
impressed heating and cooling via the sidewalls.
Lighter-than-average fluid rises and heavier
fluid sinks, thus establishing within the fluid
a bottom-heavy fractional density contrast in
the vertical in magnitude not much less than the
impressed horizontal fractional density contrast
Δρ /ρ associated with the temperature difference
between the cylindrical sidewalls (Hide 1967,
Williams 1967, McIntyre 1968).
The cool horizontal branch of the meridional
circulation lies below the warm horizontal
branch. Coriolis forces weaken that circulation
in the main body of the fluid, leaving it largely
confined to boundary layers with a strength
controlled by the thin Ekman boundary layer(s)
of thickness ~3δ on the end wall(s) where
δ = (ν / Ω)1/2
(1)
ν being the coefficient of kinematic viscosity
of the convecting liquid, to the extent that if
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Raymond Hide at 80 • Hide: Geophysical fluid dynamics
the flow were to remain axisymmetric as Ω
increases, convective heat transfer would
decrease rapidly, as Ω–3/2 . But when Ω is such
that the dimensionless parameter Θ (see equation 2) falls below a certain critical value Θ*, the
axisymmetric flow gives way to non-axisymmetric flow (see figures 2a and b). Convective
heat transfer is then effected not by meridional
overturning (as in the axisymmetric regime)
but by “sloping convection” involving flow
patterns illustrated by figures 2b–f (Hide 2011,
Hide and Mason 1975). Such flows are characterized by a meandering jet stream with the
lighter fluid generally rising and heavier fluid
sinking in accordance with energy-conversion
requirements, albeit at speeds much slower than
horizontal components of the flow.
Within the regular non-axisymmetric (RN)
regime (figures 2b, c and d) the rate of heat
transfer is nearly independent of Ω and about
20% less than when Ω = 0 (the flow then being
axisymmetric with no azimuthal component).
In contrast with the RN regime, heat transfer
within the irregular non-axisymmetric (IN)
regime decreases markedly with increasing Ω,
notwithstanding the increasingly turbulent character of the flow (see figures 2e, f). Some experts
on turbulent flows in other fluid systems (where
turbulence usually enhances heat transfer) found
the last result surprising, but it is clear that the
jet stream in a regular non-axisymmetic flow
pattern (see figures 2b, c and d) provides an efficient mechanism for transporting heat across the
annulus and, by conduction, collecting heat from
one sidewall and delivering it to the other.
It was necessary at an early stage to determine
how regime transitions depended not only on
Ω but also on Δρ /ρ and other impressed experimental conditions, namely, the depth, d, of the
liquid within the convection chamber, and the
width, (b–a), of the gap between the sidewalls,
keeping the radius of curvature, b, of the outer
sidewall fixed at 4.85 cm in the first instance.
Thus were deduced empirical criteria for the
occurrence of transitions (i) between the A
regime and the regular non-axisymmetric RN
regime, and (ii) between the RN regime and the
irregular non-axisymmetric IN regime.
The first of these transitions was found to
occur at a critical value Θ* (=1.6) of the dimensionless parameter
Θ = (gdΔρ / ρ) / (Ω² [b – a]²)
(2)
(which ranged from circa 10 –2 to 10), where g
denotes the acceleration due to gravity (typically
much stronger than centripetal acceleration),
indicating that loss of stability of the A regime
at the transition to the N regime involves the
conversion of “available potential energy” into
kinetic energy, as in “baroclinic instability”
(Eddy 1949). Later experiments using glycerol/
water mixtures indicated how Θ* depends on
the viscosity of the working liquid (see Fowlis
and Hide 1965, Hide 2011, Hide and Mason
A&G • August 2010 • Vol. 51 1975). The dependence is weak when, by design,
as in the first experiments, Ω is so high that
viscous forces in the main body of the fluid (but
not in boundary layers) are very much weaker
than Coriolis forces. This corresponds to values
of T in excess of ~107, where
T = 4Ω2 (b – a)5 / ν 2 d
(3)
But at lower values of Ω, the criterion shows
dependence on the coefficient of kinematic viscosity ν , to the extent that axisymmetric flow
is found to occur at all values of Θ when T is
less than a critical value T* ~ 2 × 105 (Fowlis and
Hide 1965, Hide 2011).
The ratio a / b ranged from 0.22 to 0.73 (with
b fixed at 4.85 cm) in the experiments and d varied by a factor of three (Hide 1953b, 1958). The
transition between the regular RN regime and
the irregular IN regime was found to satisfy a
surprisingly simple criterion when expressed in
terms of the number of waves, m (say). Denote
by m* the maximum number of waves for a
given gap-width, (b – a), between the bounding
cylinders (where m* = 5 in the experiments illustrated by figure 2). The corresponding length
of the wave, π (b + a) / m*, was never less than
circa 1.5 (b – a), so that m* is the nearest integer
to 2/3Γ, where
Γ = (b – a) / π (b + a)
(4)
is the dimensionless gap-width; m* ranged from
3 to 14 as (b – a) ranged from 3.79 cm to 1.31 cm.
“Geostrophic turbulence” sets in beyond that
point, most likely because the waves are then
short enough for nonlinear transfer of kinetic
energy between Fourier modes to occur, rendering the regular non-axisymmetric regime of
flow “barotropically” unstable (Hide 2011).
The procedure followed in most investigations
of the RN-regime involved setting Ω and other
quantities required to specify the impressed
experimental conditions at predetermined
values, and then waiting until transients had
died away before measuring various properties of the flow that persisted. In some cases
the persistent pattern of waves turned out to
be virtually steady (see figures 2b, c, d, apart
from a steady drift of the whole pattern relative to the sidewalls of the convection chamber). But in other cases the pattern exhibited
large-amplitude persistent regular periodic
fluctuations of various kinds. The simplest of
these consisted of amplitude fluctuations which
at their most pronounced were accompanied by
alternations in the number of waves, m, from
one cycle to the next. In other time-varying
cases the shape of the whole pattern would
waver or a sizeable local distortion of the wave
pattern, sometimes amounting to the splitting
of a wave, would progress around the pattern.
Significantly – in a manner reminiscent of the
behaviour of pinball machines found in amusement arcades and indicative of sensitivity to initial conditions resulting in “multiple equilibria”
– when a number of experiments were carried
out under the same impressed conditions there
was a spread in values of m of the patterns that
persisted, rather than a unique value of m. Thus,
in a large number of “Monte Carlo”-type trials
under the conditions, say, of the picture in figure
2c (where m happens to be equal to 4) – with
each trial starting with the thorough stirring
of the working liquid and then waiting for the
resulting transient small-scale motions to die
away – the resulting value of m of the pattern
that eventually persisted would be equal to 3, 4
or 5, occurring with relative frequencies found
to depend mainly on the value of Θ.
Various terms were used for the periodic
fluctuations of RN-type flows, including “vacillation” to denote the most extreme form of
periodic “wavering” seen near the transition
to the IN-regime, during one phase of which
the meandering jet stream gave way to separate
eddies, which in turn decayed allowing the jet
stream to reform, and so on. But when other
workers took up annulus experiments (see
Hide and Mason 1975 and articles in Navon
et al. 1998) some used the term “vacillation”
to signify any flow exhibiting regular periodic
fluctuations. This made it necessary to introduce the terms “shape vacillation”, “amplitude
vacillation”, “wave-number vacillation”, etc,
leaving “vacillation” on its own as an alternative term for the whole RN regime. RN flows
that are virtually steady, which occur within the
same general region of the Θ / T regime diagram
as the pronounced vacillations, are special cases
when fluctuations in the shape of the pattern are
so slight as to be imperceptible.
Before leaving Cambridge in mid-1953 (when
my research studentship came to an end) I completed my experimental work there by making
further investigations of (a) flow patterns and
temperature fields (using arrays of thermo­
couples) and their temporal variations, and (b)
the non-unique dependence of m on Θ, and (c)
the inhibiting effect of rotation on the rate of
total heat transfer, to which convection typically contributed up to nearly 10 times that by
conduction and could be rendered insensitive
to Ω by inserting a thin rigid radial barrier connecting the bounding cylinders and extending
over the whole depth of the fluid (Hide 1997c,
Hide and Mason 1975).
My main findings as reported in a PhD dissertation (Hide 1953b) were summarized in two
short papers (Hide 1953a, 1956), but several
years (including a period of compulsory National
Service) elapsed before details of methods and
results reached the open literature (Hide 1958).
Geophysical and astrophysical
fluid dynamics and dynamical
meteorology
General theoretical considerations of the
dynamics of convective heat transfer in spinning
fluids show that “Coriolis forces usually pro4.19
Raymond Hide at 80 • Hide: Geophysical fluid dynamics
mote flow patterns with substantial departures
from axial symmetry in systems characterized
by axial symmetry in their boundary conditions” (Hide 1977). Annulus flows exemplify
this result, which has wide implications in “geophysical and astrophysical fluid dynamics”. And
in view of the effective need for departures from
axial symmetry that is implied by the existence
theorems for self-exciting dynamos, the result
indicates one possibly key role played by Coriolis forces in the geodynamo process and the
MHD of the Earth’s core, namely the promotion
of large-scale non-axisymmetric flow patterns.
Another significant phenomenon was observed
during a brief study of thermal convection in
a rotating spherical (rather than cylindrical)
annulus subjected to a radial temperature gradient (Hide 1953b). The study was intended
to elucidate possible effects on motions in the
Earth’s liquid outer core that the presence of
the underlying solid inner core might produce,
thereby influencing the pattern of the observed
geomagnetic field. The experiments confirmed
what general theoretical arguments predicted,
namely that owing to Coriolis forces the most
striking feature of the flow would be an extensive cylindrical “detached shear layer” aligned
parallel to the rotation axis and girdling the
inner spherical surface, touching it along the
equator. Non-axisymmetric waves developed on
the detached shear layer under some conditions,
not yet fully investigated.
But of more immediate significance during
the course of the main experiments was the
new dimension they acquired in 1951 thanks
to the eminent Cambridge geophysicist and
mathematician H Jeffreys. In 1946 Jeffreys had
succeeded Eddington as Plumian Professor of
Astronomy and Experimental Philosophy and
his early career included several years (starting
during the first world war) spent on research at
the UK Meteorological Office, where he made
seminal theoretical contributions to dynamical
meteorology (Lorenz 1967, Lindzen 1990). He
commented casually that some of my flow patterns reminded him of large-scale motions in the
Earth’s atmosphere. Thus prompted, I started
reading meteorological literature in the underused Napier Shaw collection in the Rayleigh
Library of the Cavendish Laboratory, handicapped at first by the absence in Cambridge of
individuals from whom I could obtain useful
guidance (Jeffreys having lost interest in meteor­
ology nearly two decades earlier). Expert fluid
dynamicists in various departments there apparently found my experimental results “interesting
but mysterious”, apparently preferring laboratory studies focused on the validation of mathematical analyses, at a time when many of the
ideas and mathematical techniques needed for
interpreting the essentially nonlinear behaviour
exemplified by the experiments had yet to be
developed. But there were active dynamical
4.20
meteorologists, including E T Eady, working
in the Department of Meteorology at Imperial
College London, where I was invited to give
a talk about my experiments and encouraged
to join in discussions about large-scale atmo­
spheric flows.
At that stage I first met D Fultz, an American visitor on sabbatical leave visiting European meteorologists, oceanographers and fluid
dynamicists (Frenzen 1993). He told me about
the work of the so-called “HydroLab” which
he directed in the Department of Meteorology
of Chicago University. The Chicago HydroLab
had been established some years earlier at the
suggestion of leading US dynamical meteorologists C-G Rossby of Chicago University and
V P Starr of MIT for the purpose of designing
laboratory experiments that might shed light
on the general circulation of the Earth’s atmo­
sphere. Fultz (1951) had already made a careful
literature search for relevant studies, in which
he unearthed reports describing qualitative
studies of convection in spinning fluids made
by meteorologists F Vettin (in 1857 in Berlin)
and F M Exner (in 1923 in Vienna). These had
already been reproduced in the HydroLab in
the so-called “dishpan” experiments (in which
the convection chamber was an open domestic
aluminium American dishpan). Understandably
interested in my results, especially the discovery and subsequent investigations of the various
flow regimes observed in the controllable, geometrically simple and well-defined Cambridge
apparatus (which he dubbed “annulus” to distinguish it from the Chicago “dishpan” apparatus),
Fultz instructed his deputy at the HydroLab,
R R Long, to start annulus work there as soon
as possible, to which end he visited me in Cambrdige to obtain technical details about my
apparatus design, experimental techniques and
scientific results (see Frenzen 1993).
The Cambridge annulus experiments were
duly repeated and their findings confirmed at
the HydroLab (Fultz et al. 1959). And in his
successful efforts to bring the experiments to
the attention of other meteorologists, especially
members (including Lorenz) of the long-term
and wide-ranging General (Atmospheric) Circulation Project being undertaken in the Meteorology Department of MIT under Starr’s direction,
Fultz promoted the use of the term “vacillation”
and introduced nomenclature of his own. Thus,
the critical dimensionless parameter Θ (see
equation 1) that I had found to be the principal
determinant of the characteristics of the annulus
flows he termed the “thermal Rossby number”,
my regular and irregular non-axisymmetric
regimes (RN and IN) he designated the “Rossby
regime”, and axisymmetric flows he termed the
“Hadley regime” – after G Hadley whose celebrated paper on the cause of the Trade Winds
was published as early as 1735 (Lorenz 1967).
In September 1953 I was able to discuss my
Cambridge experiments at a small scientific
meeting organized by R R Long on “the use of
(laboratory) models in geophysical fluid dynamics” held at the Johns Hopkins University in Baltimore (see Hide 1956). Among those present
were Lorenz of MIT and several other leading
meteorologists involved in theoretical research
on dynamical processes in the atmosphere and
the predictability of atmospheric motions (see
e.g. Lorenz 1956, 1963a, b). To paraphrase
Lorenz’s comments on the value of laboratory
studies made some years later in his celebrated
monograph on the general circulation of the
atmosphere (Lorenz 1967):
“So far as their meteorological significance
is concerned the experiments, by indicating
the flow patterns that can occur and the conditions favourable to each, have made possible the separation of essential from minor
and irrelevant considerations in the theory
of the global atmospheric circulation. They
show, for instance, that while considerations
of water vapour may yet play an essential
role in the Tropics, it appears to be no more
than a modifying influence in temperate latitudes, because the hydro­dynamical phenomena found in the atmo­sphere, including even
cyclones, jet streams and fronts, also occur
in the laboratory apparatus where there is no
analogue of the condensation process. The
same remarks apply to topographic features,
which were intentionally omitted in the
(first) experiments. The so-called ‘betaeffect’ associated with the sphericity of the
spinning Earth – which produces a tendency
for the relative vorticity to decrease in northward flow and increase in southward flow
because of the variation with latitude of the
Coriolis parameter – now appears to play a
lesser role than had once been assumed. Certainly a numerical weather forecast would
fail if the beta-effect were disregarded, but
the beta-effect does not seem to be required
for the production of typical atmospheric
systems. The experiments have emphasized
the necessity for truly quantitative considerations of planetary atmospheres. These
considerations must, at the very least, be
sufficient to place the Earth’s atmosphere in
one of the free non-axisymmetric regimes of
thermal convection discovered in the laboratory work.”
In natural systems, “sloping convection” characteristic of the non-axisymmetric flow regimes
occurs on scales up to thousands of kilometres
within the atmospheres of astronomical bodies such as the Earth and other rapidly rotating planets. It is remarkable (and fortunate)
that small-scale laboratory experiments can
elucidate the process of sloping convection
and related buoyancy-driven flows that occur
when Coriolis forces exert a dominant influence. In “internally heated” laboratory annulus
A&G • August 2010 • Vol. 51
Raymond Hide at 80 • Hide: Geophysical fluid dynamics
experiments (see Hide and Mason 1975, Hide
et al. 1994, Read et al. 1997), in which heating
is introduced throughout the body of the liquid
(rather than via one of the sidewalls), sloping
convection in the regular non-axisymmetric
regime can take the form of one or more persistent eddies (rather than waves). These have
dynamic and thermal characteristics similar to
those of the Great Red Spot in Jupiter’s atmo­
sphere, which has persisted for centuries (at
least), much longer than any dynamical feature
of the highly unpred­ictable terrestrial atmo­
sphere – a finding with an obvious bearing on
theories of atmospheric predictability. Significantly, the experiments covered the whole range
of parameters corresponding to Θ and T for the
atmospheres of the solar planets, which range
in size by a factor of 10 and in rotation period
from around 10 hours (Jupiter, Saturn, Uranus
and Neptune) to several months (Venus). Some
aspects of the dynamical influence of planetary
sphericity on large-scale atmospheric motions
can be studied in the annular laboratory system
by introducing sloping end-walls (Hide 1969,
Hide and Mason 1975).
Theoretical fluid dynamics and
atmospheric predictability
Theoretical work in fluid dynamics is based on
the nonlinear four-dimensional (space and time)
partial differential equations (PDEs) in terms
of which the laws of dynamics and thermo­
dynamics can be expressed mathematically. The
equations of electrodynamics are also needed
when dealing with MHD flows in electrically
conducting fluids, including those encountered
in dynamo theory. Being highly intractable, the
equations yield to traditional analytical methods only in simple special cases when nonlinear
terms can be neglected or treated as small perturbations.
Under certain impressed boundary conditions,
the governing mathematical equations can have
multiple solutions, indicating sensitivity to
small variations in initial conditions. The simplest persistent solution in the case of thermal
convection in a rotating fluid annulus subject
to steady mechanical and thermal boundary
conditions that are symmetric about the axis of
rotation will be characterized by steadiness and
axial symmetry, i.e. independent of time and
of the azimuthal spatial coordinate. But flows
corresponding to such solutions, as in cases of
axisymmetric flows seen in the annulus experiments (see figure 2a) only occur in practice
under impressed conditions when the mathematical solutions satisfy two criteria, namely
(a) that they could develop from realistic initial
conditions, and (b) that they remain stable in
the presence of weak adventitious non-axisymmetric disturbances.
Under conditions when the steady axisymmetric mathematical solutions to the governing
A&G • August 2010 • Vol. 51 equations are unstable as a consequence of the
process of “baroclinic instability”, the observed
persistent flow that develops would be nonaxisymmetric with varying degrees of spatial
and temporal irregularity, ranging from spatially regular (periodic) flows that are steady
or fluctuate periodically (as in figures 2b, c, d),
to spatially irregular flows that fluctuate nonperiod­ically (as in figures 2e, f). The regular
flows are simpler than the irregular flows and
they occur when they are “barotropically” stable to all weak adventitious disturbances. Irregular flows occur under impressed conditions
when regular flows would be baro­tropically
unstable (Hide 2011).
The experiments described above were sufficiently wide-ranging and quantitative to reveal
all three general types of flow – namely (i) spatially axisymmetric and temporally steady flow,
(ii) spatially regular non-axisymmetric and
temporally steady or periodically fluctuating
flow, and (iii) spatially irregular non-axisymmetric and non-periodically (i.e. chaotically)
fluctuating flow – and to provide criteria for
transitions between them in terms of dimensionless parameters conveniently specifying the
impressed experimental conditions.
When considering any fluid-dynamical system
it is useful to seek sets of dimensionless parameters from the governing mathematical equations
by expressing the dependent and independent
variables in dimensionless form, leaving the
coefficients in the equations as pure numbers.
When boundary conditions are also taken into
account about 15 dimensional parameters are
needed to specify fully the characteristics of
the laboratory annulus system, including the
rotation speed, the geometry of the convection
chamber, the impressed differential heating and
cooling and the thermal and mechanical properties of the working liquid (Fowlis and Hide
1965, Hide and Mason 1975). Experiments
show that Θ defined by equation 1 is by far the
most significant dimensionless parameter, followed by a “Taylor” number defined by equation 2 and a “Prandtl” number
Π = ν / κ (5)
(where κ denotes thermometric conductivity),
together with the aspect ratios Γ (see equation
4) and
Δ = d / (b – a)
(6)
The past half-century has witnessed impressive progress in the application of numerical
methods that now exploit the power of modern
supercomputers, with dynamical meteorologists
in centres for weather and climate forecasting
at the forefront of these developments (see Ghil
et al. 2010, Navon et al. 1998, Norbury and
Roulstone 2002, White 2010, Wiin-Nielsen
2000). But the idea of calculating how the
weather will evolve by solving the equations of
hydro­dynamics using the meteorological data
describing the present weather as the initial
conditions goes back to work by V Bjerknes
and L F Richardson in the early 20th century.
H Poincaré – whose mathematical work on the
“three-body problem” in planetary dynamics
included ideas and methods which are now used
widely in chaos theory – issued a note of caution
at the time when he wrote (see Lighthill 1986,
Gribbin 2004):
“Why have meteorologists such difficulty in
predicting the weather with any certainty?
Why is it that showers and even storms seem
to come by chance, so that many people
think it quite natural to pray for rain or
fine weather, though they would consider it
ridiculous to ask for an eclipse (of the Sun
or Moon) by prayer. We see that great disturbances are generally produced in regions
where the atmosphere is in unstable equilibrium. The meteorologists see very well that
the equilibrium is unstable, that a cyclone
will be formed somewhere, but exactly
where they are not in a position to say; a
tenth of a degree (in temperature) more or
less at a given point, and the cyclone will
burst here and not there, and extend its
ravages over districts it would otherwise
have spared. If they had been aware of this
tenth of a degree, they could have known of
it beforehand, but observations were neither
sufficiently comprehensive nor sufficiently
precise, and that is why it all seems due to
the intervention of chance.”
Low-dimensional models, the Lorenz
equations and deterministic chaos
When studying particular aspects of the behaviour of a fluid-dynamical system, the governing nonlinear PDEs can be rendered more
tractable, albeit less reliable, by simplifying the
spatial and/or temporal representation of processes of secondary interest, as in the so-called
“intermediate” theoretical models such as the
“kinematic” dynamos introduced by Bullard
and Gellman (1954, Gubbins and HerreroBervera 2007, Dormy and Soward 2007), in
which nonlinear effects are eliminated ab
initio by neglecting dynamical effects due to
Lorentz ponderomotive forces. And in extreme
cases such as the “low-dimensional” theoretical models (sometimes called “toy” models)
employed when interest focuses on the influence
of non­linearity on temporal behaviour, further
simplifications are effected when formulating the model by “parameterizing” all spatial
structure. The resulting system is governed by
ordinary, rather than partial, differential equations (ODEs) needing comparatively modest
computers for their analysis, with solutions
bearing qualitatively if not quantitatively on
the prototype.
Low-dimensional models bearing on the
nonlinear behaviour of self-exciting fluid dynamos are provided by systems of Faraday-disc
4.21
Raymond Hide at 80 • Hide: Geophysical fluid dynamics
dynamos (Dormy and Soward 2007, Gubbins
and Herrero-Bervera 2007, Hide 1997a, 2008,
2011, Jacobs 1994, Moffatt 1978), the simplest
versions of which are those introduced in the
1950s by Bullard and T Rikitake (Bullard 1955,
Rikitake 1958). The autonomous set of nonlinear ODEs in three time-dependent variables that
govern the Rikitake system of two coupled disc
dynamos was shown by D W Allan (1962) to possess persistent non-periodic (i.e. chaotic) solutions (a finding shown much later to be crucially
dependent on the neglect of mechanical friction
in the original Bullard system, see Hide 1995).
In concurrent research relating to the effects of
nonlinear processes on atmospheric predictability Lorenz (1963b,a) investigated low-dimensional models of vacillation and other laboratory
flows as well as the nonlinear amplification of
effects of tiny errors in meteorological data
and its likely consequences for weather forecasting. Using mathematical and computational
techniques he investigated in impressive detail
a low-dimensional “toy” model of convection
governed by what later became known as the
“Lorenz set” of three (dimensionless) autonomous nonlinear ODEs, namely:
dx / dt = α (y – x), dy / dt = β x – y – xz,
dz / dt = xy – γ z
(7)
which contains two simple nonlinear terms, –xz
and +xy. Here x(t), y(t) and z(t) are the three
time (t)-dependent variables and α, β, and γ are
positive “control parameters” (Lorenz 1963a).
In one of his solution regimes he found nonperiodic behaviour that would be termed “deterministic chaos” by others nearly a decade later
in mathematical work on nonlinear dynamical
systems (see e. g. Lorenz 1993, Palmer 2009,
Sparrow 1982, Thompson and Stewart 2002,
Tong 1995).
Through its impact on the development of
ideas in that area, Lorenz’s (1963a) report on
the solutions of equation 7 became one of the
most influential scientific papers of the past few
decades. The Lorenz equations and related sets
of autonomous nonlinear ODEs continue to provide fruitful lines of mathematical research. As
J D Barrow (1990) has eloquently explained:
“The mainstream of mathematics has begun
to move away from the high ground of
extreme formalism to the study of particular
problems, notably those involving chaotic
nonlinear phenomena, and to seek motivation from the natural world. This is a return
to a distinguished tradition for … there are
complementary examples where our study
of the physical world has motivated the
invention of new mathematics. The contemplation of continuous motion by Newton
and Leibniz … led to the creation of the
calculus … [and] Fourier series arose from
the study of heat flow and optics. In the
20th century, the consideration of impulsive
forces led to the invention of “general4.22
ized functions” … (which) were used most
powerfully by Paul Dirac in his formulation
of quantum mechanics … In recent years
this trend towards specific applications has
been perpetuated by the creation of a large
body of dynamical systems theory, and most
notably the concept of a ‘strange attractor’,
as a result of a quest to describe turbulent
fluid motions. The growing interest in the
description of chaotic change which is characterized by the very rapid escalation of any
error in its exact description as time passes
has led to a completely new philosophy with
regard to the mathematical description of
phenomena. Instead of seeking more and
more mathematical equations to describe a
given phenomenon, one searches for those
properties which are possessed by almost
every possible equation governing change.
Such ‘generic’ properties, as they are called,
can therefore be relied upon to manifest
themselves in phenomena that do not possess very special properties. It is this class of
probable phenomena that are most likely to
be found in practice.”
The disorder and associated lack of predictability of motions in the Earth’s atmosphere and
also of flows encountered in other nonlinear
fluid systems – such as Lorenz’s toy model in the
chaotic regime and the laboratory annulus in the
irregular non-axisymmetric regime – are due to
instabilities associated with feedback and coupling. But nonlinear processes can in some circumstances promote stability and order, rather
than instability and disorder. Such behaviour
has been investigated by modifying the feedback
and coupling terms in well-known autonomous
sets of nonlinear ODEs (Hide 1997b, 2011,
Hide et al. 2004). Non­linear quenching of the
chaotic behaviour of the geodynamo associated
with modest changes in boundary conditions at
the surface of the Earth’s liquid core has been
invoked to account for the intermittency seen in
the irregular time series of geomagnetic polarity
reversals over geological time, with intervals
between reversals varying from 0.25 My to
50 My (Hide 1997b, 2000). And there are other
examples of nonlinear processes promoting
stability rather than instability. Such processes
underlie the stability of annulus flows in the
regime of vacillation, the comparative regularity of large-scale motions in the atmosphere of
the planet Mars and the durability of the Jovian
Great Red Spot and other long-lived eddies in
the atmospheres of the major planets (see Hide
et al. 1994, Read et al. 1999).
Epilogue
The events on our path of discovery started
with Blackett’s testable ideas about the origin
of the Earth’s magnetism. Even though the
ideas turned out to be wrong, they also started
another (better known) path of discovery in
geophysics. The construction of several highly
sensitive astatic magnetometers of the type
designed by Blackett (1952) for the laboratory
test of his hypothesis enabled palaeomagnetic
workers, including a group led by Blackett
himself, to make accurate determinations of
the magnetic orientation of a wide variety of
rocks collected from various continents. These
provided new evidence in support of the “continental drift” hypothesis that A Wegener – a
meteorologist – had put forward nearly 40 years
earlier (see Blackett et al. 1965, Good 2010),
thereby advancing the general acceptance of the
hypothesis and the emergence in geology of the
unifying concepts of plate tectonics.
It was fortunate for science that during his
early career Blackett survived the 1916 naval
battle of Jutland and a clumsy attempt on his
life made 10 years later at Cambridge by a
mentally disturbed student (Bird and Sherwin
2005, Farmelo 2009). A brilliant and versatile
physicist, Blackett went on to influence basic
and applied research in many branches of his
subject and to inspire and encourage other
research workers. During a memorable lecture
on sunspots to a student society in Manchester in the late 1940s, he gave a clear and convincing explanation of the essential physics of
“magnetohydrodynamic (Alfvén)” waves, long
before H Alfvén’s new theoretical ideas became
widely accepted and eventually recognized by
the award of a Nobel Prize in 1970.
Research environments have changed significantly over the past half-century. New groundbased and spacecraft observations extending
over many wavelengths in the electromagnetic
spectrum had a major impact on meteorology,
geomagnetism and other geophysical sciences.
In solar physics, Larmor’s prescient views on
solar magnetism have been abundantly vindicated by subsequent research (Weiss 2002,
Tobias and Weiss 2004) and theoretical research
in geomagnetism has benefited from investigations of the magnetic fields of other planets
(Mercury, Jupiter, Saturn, Uranus and Neptune), for which there was no evidence before
the mid-1950s, when radio astronomers discovered that Jupiter is a strong emitter of nonthermal radiation on decimetre and decametre
wavelengths. Observations of the atmospheres
of Venus, Mars, Jupiter, Saturn, Titan, Uranus
and Neptune now challenge theoretical ideas
underpinning research in terrestrial meteorology, ocean­ography and climatology.
Few areas of science have been left untouched
by the astonishing growth in power and availability of computers, which now support most
research projects, including combined laboratory and numerical investigations of flows in
spinning fluids and other nonlinear phenomena.
These flourish in several places, including the
Department of Physics of Oxford University
where, thanks to Prof. Peter Read and his team
A&G • August 2010 • Vol. 51
Raymond Hide at 80 • Hide: Geophysical fluid dynamics
of organizers, the scientific meeting connected
with this festschrift was held in July 2009. The
long-term prospects for such work appear bright,
mainly thanks to the opportunities afforded by
new techniques, advances in theoretical geophysical fluid dynamics, and new observations
bearing on basic dynamical processes underlying a wide range of natural phenomena. In
my own research over several decades on the
hydrodynamics and magnetohydrodynamics of
spinning fluids with applications to problems in
geophysics and planetary physics, I have benefited from enjoyable and fruitful interactions
with many talented colleagues – some able to
participate in the Oxford meeting – to whom
I owe a debt of gratitude. I must also thank
Michael Ghil, Peter Read, Lenny Smith, Andy
White and other contributors to this festschrift
for helpful comments during the preparation
of this article. ●
Raymond Hide, a past President of the RAS and
the R. Met. Soc., Emeritus Professor of Physics
at Oxford University and Emeritus Gresham
Professor of Astronomy, is an Honorary Senior
Research Investigator at the Department of
Mathematics, Imperial College London.
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A note on the
bibliography
To keep the wide-ranging bibliography
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● Geomagnetism and dynamo theory:
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4.23