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PerspectiveCite this article Vallis GK 2016 Geophysicalfluid dynamics whence whither and whyProc R Soc A 472 20160140httpdxdoiorg101098rspa20160140

Received 24 February 2016Accepted 22 July 2016

Subject Areasmeteorology oceanography astrophysics

Keywordsgeophysical fluid dynamics meteorologyoceanography fluid dynamics modellingsimulation

Author for correspondenceGeoffrey K Vallise-mail gvallisexeteracuk

A contribution to the special featurelsquoPerspectives in astrophysical and geophysicalfluidsrsquo

Electronic supplementary material is availableat httpdxdoiorg101098rspa20160140 orvia httprsparoyalsocietypublishingorg

Geophysical fluid dynamicswhence whither and whyGeoffrey K Vallis

College of Engineering Mathematics and Physical SciencesUniversity of Exeter Exeter EX4 4QF UK

GKV 0000-0002-5971-8995

This article discusses the role of geophysical fluiddynamics (GFD) in understanding the naturalenvironment and in particular the dynamics ofatmospheres and oceans on Earth and elsewhereGFD as usually understood is a branch of thegeosciences that deals with fluid dynamics and thatby tradition seeks to extract the bare essence of aphenomenon omitting detail where possible Thegeosciences in general deal with complex interactingsystems and in some ways resemble condensedmatter physics or aspects of biology where we seekexplanations of phenomena at a higher level thansimply directly calculating the interactions of allthe constituent parts That is we try to developtheories or make simple models of the behaviourof the system as a whole However these days inmany geophysical systems of interest we can alsoobtain information for how the system behavesby almost direct numerical simulation from thegoverning equations The numerical model itselfthen explicitly predicts the emergent phenomenamdashthe Gulf Stream for examplemdashsomething that isstill usually impossible in biology or condensedmatter physics Such simulations as manifested forexample in complicated general circulation modelshave in some ways been extremely successful andone may reasonably now ask whether understandinga complex geophysical system is necessary forpredicting it In what follows we discuss such issuesand the roles that GFD has played in the past and willplay in the future

1 PreliminariesWhat is geophysical fluid dynamics (GFD) Broadlyspeaking it is that branch of fluid dynamics concernedwith any and all things geophysical It thus deals withsuch fluid phenomena as the Earthrsquos interior volcanoes

2016 The Author(s) Published by the Royal Society All rights reserved

on November 7 2016httprsparoyalsocietypublishingorgDownloaded from

2

rsparoyalsocietypublishingorgProcRSocA47220160140

lava flows ocean circulation and planetary atmospheres In this article I will mainly discussmatters associated with atmospheres and oceans for that is my area of expertise and I will takeGFD to include the fields of dynamical meteorology and oceanography The moniker lsquoGFDrsquo hasalso come to imply a methodology in which one makes the maximum possible simplificationsto a problem perhaps a seemingly very complex problem seeking to reduce it to some bareessence It suggests an austere approach devoid of extraneous detail or superfluous descriptionso providing the fundamental principles and language for understanding geophysical flowswithout being overwhelmed by any inessentials that may surround the core problem In thissense GFD describes a method as well as an object of study

Although one might think that such a method is of course entirely appropriate in all scientificareas in some branches of science there is a tendency to embrace the complexity of reality by usingcomplicated models to which we add processes whenever possible rather than taking them awayThis approach certainly has its place especially if our concern is in making detailed predictionsof the behaviour of a complex system such as we might if we are engaged in weather forecasts orclimate predictions and climate models have grown enormously in complexity over the past fewdecades (figure 1) The GFD approach and the modelling approach have at times diverged whenboth approaches (and a range in between) may be needed to gain a proper understanding of aproblem Still the rift between the two fields is sometimes exaggeratedmdashthere are many signs ofa realization that we (the scientific community) really do need both

At the same time as we might be adding complexity in much of science it is usually takenas a given that we should be endlessly seeking the most fundamental level of understandingand this search has mostly served us wellmdashprovided we have a sensible notion of what lsquomostfundamentalrsquo means for any given problem In physics for example it is taken for grantedthat we should seek to unify the fundamental forces as much as possible a unified descriptionof the weak and electromagnetic forces is universally regarded as more satisfying and morefoundational than a description of the two separately There is no discussion but that we shouldtry to go further and physicists are now looking apparently without irony for a single theoryof everything

The simplifications sought by GFD are not quite like thatmdashthey are more akin to those soughtby biologists or condensed matter physicists or anyone dealing with a complex subject thatcontains emergent phenomena An emergent phenomenon is one that emerges from the collectivebehaviour of the constituents of a system and is not a property of its individual componentsmdashits equivalent atoms or its primitive building blocks emergence is a manifestation of a groupbehaviour Perhaps the most familiar example is temperature which is a collective property ofthe molecules of a system and is proportional to the mean kinetic energy of molecules in a gasthe phenomenon of phase transitions is another example in physics Biology itself may be anemergent phenomenon or at least aspects of it such as life speciation and evolution

Now although there may be a sense in which biology can be reduced to chemistry chemistryto physics and so on it is preposterous to seek an explanation for the emergence of biologicalsystems in the laws of physics One simply cannot understand biology based solely on the laws ofphysics as has been widely if not universally accepted for some time (eg [1]) At each new levelof complexity new properties emerge that do not depend on the details of the underlying laws andqualitatively different behaviour takes place Darwinian principles care nothing for string theoryand an evolutionary system can be built on the computer with no regard to the particular laws ofphysics The construction and function of a gene may be constrained by the laws of chemistry butat a higher level evolutionary principles are not it would not make sense to seek an explanationof macroscopic evolutionary laws in terms of chemistry or physics any more than it would makesense to try to understand Uncle Vanya using the rules of grammar even with a dictionary at handIndeed those rules are different in Russian and English but Uncle Vanya transcends that

Still there are some emergent properties that we can now simulate using the microscopiclaws applied to the atoms of the systemmdashthe Maxwellian distribution of velocities in a gas forexamplemdashbut we still would not dream of performing a numerical simulation of the individualmolecules to calculate the change in temperature of an ideal gas as it is adiabatically compressed


3


1960s

atmosphereland surface

atmosphereland surfacevegetation


atmosphere atmosphere

ocean ocean

sea ice sea ice

sulfateaerosols

solar forcingvolcanicaerosols


carboncycle

dustsea spraymineral

aerosols

vegetation interactivevegetation

biogeochemicalcycles

ice sheet


aerosols

carbonnitrogen

cycle


sulfateaerosols

sulfateaerosols

sea ice sea ice

ocean ocean ocean

1970sndash1980s 1990s 2000s 2010s

Figure 1 Growth of complexity of climate models over the past few decades Each component interacts with each othercomponent in models that are over 1 million lines of code (Online version in colour)

rather we seek a direct or macroscopic explanation By contrast in some other problemsit may be easiest to perform a straightforward computation of their atoms to see how thesystem evolvesmdashthe orbits of planets around their sun for example Where does GFD fit inthis spectrum

(a) Regarding geophysical fluid dynamicsSome of the main goals (and past triumphs) of GFD lie in explaining lsquofluid-dynamical emergentphenomenarsquo for example the Gulf Stream in the Atlantic or a hurricane for these are notproperties of a fluid parcel However compared with the complexity of biological systems oreven in some ways phase transitions these phenomena occupy something of a half-way housewe can seek high-level explanations (theories) of the phenomena but we can also simulate some


4


of these phenomena quite well using the basic laws of physics as expressed by the NavierndashStokesequations and associated thermodynamical and radiative equations These days we do a far betterjob of describing ocean currents using a numerical simulation than using any theory analyticalor otherwise that seeks to directly predict them using some more holistic method Similarlythe climate contains a turbulent fluid (the atmosphere) but our most accurate descriptions ofthe future climate are made by attempting to simulate the individual eddies over the course ofdecades and centuries somewhat akin to following molecules in a simulation of a gas ratherthan trying to construct a macroscopic theory of climate Given all this is there any need to seeka high-level explanation In other words do we still need GFD The answer it turns out is yesbut at the same time GFD needs to continue to evolve and to draw from and give to those largenumerical simulations else it will become irrelevant

There are two reasons why we might seek to understand a phenomenon The first is thatunderstanding is an end in itselfmdashwe gain something by virtue of a better understandingof the natural world In this sense science resembles the arts and humanities and scientificunderstanding lies alongside the pleasure one might find in listening to a Chopin nocturne (orcomposing a nocturne if one has the talent) in gazing upon a Turner or understanding the riseand fall of great civilizations A famous quote by Poincareacute comes to mind lsquoMathematics is notimportant because it enables us to build machines Machines are important because they giveus more time for mathematicsrsquo In our context GFD gives us an austere understanding of thebehaviour of the natural environment and that understanding increases our wonderment at itsbeauty to the benefit of all The beauty of a sunset or a cloud pattern and our respect for ahurricane are all increased by our knowledge of themmdashas is in a sibling field our wonder of aspiral galaxy

The second reason for seeking understanding is more prosaic it is that by understanding suchphenomena we are able to better predict them and thereby bring a practical benefit to societyfor example in the form of better weather forecasts or better climate projections Sometimes suchpredictions are made through the use of massive computer simulations solving the equations ofmotion ab initio and letting the phenomena emerge from the simulation It might therefore seemas if there is no longer a need (apart from the aesthetic one) to understand those phenomenathat emerge from the NavierndashStokes equations but this is plainly untrue If we see a hurricane inthe tropical mid-Atlantic we know it will usually move westward and not because a simulationtells us that We know that global warming is happening and will continue not because of theresults of complicated models but because of the basic laws of physics Even if our goals aresolely to improve numerical models then knowledge of the fundamentals is required as we willsee explicitly in the next few sections At the same time a purely analyticalndashtheoretical approachalone is insufficient and decrying the use of large numerical models is tilting at windmills for itis in the use of such models in conjunction with GFD that the future lies If we pursue GFD withouttaking account of developments in comprehensive numerical models we will pursue a dry jejunefield that will eventually become irrelevant

I should also emphasize that GFD is not or should not be a purely analyticalndashtheoreticalendeavour Rather and without seeking a definition a GFD approach means seeking the mostfundamental explanation of a phenomenon specifically in the geosciences and often of complexphenomena Using an idealized numerical model with simple equations (but perhaps complexoutput) certainly falls under the rubric of GFD and modern GFD relies as much on suchsimulations as it does on conventional lsquopaper and pencilrsquo theory

The rest of this article expands on these notions discussing some specific examples andreflecting upon where and how GFD might evolve in the future focusing on atmospheric andoceanic dynamics In all matters I express my own view and opinions and some of the historicaldiscussion can also be found in the endnotes of Vallis [2] In one or two places the description istechnical but readers from other fields may skim these parts without undue loss We divide therest of the article up into the past (whence) and the future (whither) with the lsquowhyrsquo permeatingall sections We skate through the early history at speed for the intent is not to give a historylesson (or a GFD lesson) but a sense of how the field developed and what it is


5


2 The early historyIt is hard to say when the subject of GFD began for that depends on the definition used Onemight call Archimedes (287ndash212 BC) the first GFDer for he rather famously investigated buoyancyas well as showing that a liquid will acquire a spherical form around a gravitationally attractingpoint Blithely skipping a couple of millennia and focusing more specifically on oceans andatmospheres we come to George Hadley (1685ndash1768) who put forth a theory for the trade windsin Hadley [3] He realized that the rotation of the Earth was of key importance and that inorder for the trade winds to exist there must be a meridional overturning circulation and thatcirculation now bears his namemdashthe Hadley cell His paper was perhaps not GFD in the modernsense however since he did not use the fluid dynamical equations of motionmdashhe could not havedone so since the Euler equations did not appear for another 20 years ([4] English translationin [5]) and the NavierndashStokes equations not until 1822 It may have been Laplace who in about1776 (English translation is in [6]) was the first to use the fluid equations in a GFD contextmdashhewrote down the linear shallow water equations on a sphere in the rotating frame of reference (andthus with the Coriolis terms) and forced by an external potential His goal was to understand thetides and he gave some partial solutions which were greatly extended by Hough as noted below

Moving forward to the mid- and late nineteenth century the notion of linearizing the (toocomplex) NavierndashStokes equations emerged as a way of understanding various geophysicalphenomenamdashKelvin waves being a notable example [7] Meteorology itself advanced too wefind in the work of William Ferrel and James Thomson papers on atmospheric circulation witha recognizably modern flavour [8ndash10] Using equations of motion Ferrel tried to account for themulti-celled structure of the Earthrsquos circulation and although his explanations were wrong (he didnot properly understand the role of zonal asymmetries in the flow and he envisioned a shallowcell existing beneath Hadleyrsquos Equator to Pole cell) he did have an essentially correct view of theCoriolis effect and the geostrophic wind Thompsonrsquos Bakerian Lecture in 1892 describes his ownwork along similar lines and also provides a review on the atmospheric circulation as seen at thattime showing a number of figures similar to figures 2 and 3 Next to one of them he remarks lsquoIt[is] suggestive of the most remarkable features that would probably present themselves in thewinds if the surface of the world were all oceanrsquo Thus did GFD emerge for it is this style ofreasoningmdasheliminating the continents as a kind of detail for the problem at handmdashthat reallyepitomizes the subject In any given GFD problem one must always face the question lsquoWhat is adetailrsquo

In the early twentieth century GFD as we recognize it today really began to take hold inthe use of the fluid dynamical equations of motion simplified if needs be to try to get at theheart of the problem This was enabled by some relevant advances in fluid dynamics itself theextension of Kelvinrsquos circulation theorem to rotating and baroclinic flow by Silberstein [12] andBjerknes [13] and Poincareacutersquos results on the effect of rotation on the shallow water equations [14]Thus for example Hough [1516] revisited Laplacersquos tidal problem and as part of his solutiondiscovered a form of the Rossby wave on the sphere a wave re-discovered half a centurylater in a simpler form by Rossby and collaborators [17]mdashand it was Rossbyrsquos simpler formthat enabled the wave to be properly understood Houghrsquos papers were perhaps the first toinvestigate mathematically the importance of Earthrsquos rotation on large-scale ocean currents (orat least Hough thought so) and he discussed whether ocean currents could be maintained byevaporation and precipitation in the absence of continents A few years later and following asuggestion by F Nansen Ekman [18] elucidated the nature of the wind-driven boundary layer atthe top of the oceanmdashor indeed in any rotating fluidmdashand this helped pave the way for the morecomplete understanding of ocean currents that came a half-century later in the work of Stommeland Munk Other areas of GFD were having similar advances and in dynamical meteorologyDefant [19] and Jeffreys [20] realized that non-axisymmetric aspects of the atmosphere wereessential for the meridional transport of heat and angular momentummdashJeffreys noted that lsquonogeneral circulation of the atmosphere without cyclones is dynamically possible when friction istaken into accountrsquo


6


(b)(a)

Figure 2 (ab) Schematics of the circulation of the atmosphere taken from the Bakerian Lecture of Thomson [10] (b) shows aPole to Equator Hadley cell underneath which is a shallow indirect cell the precursor of the Ferrel cell illustrated in figure 3

North Pole

South Pole Ferrel cell

Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream

Figure 3 Schematic of themodern view of the zonally averaged atmospheric circulation taken fromWallace amp Hobbs [11] Themain differences from figure 2 are the explicit recognition of the importance of zonal irregularities in the flow and the differentnature of the Ferrel cell (Online version in colour)

It is interesting that this periodmdashcall it the pre-modern eramdashis also marked by the firstdescriptions of possible numerical approaches to solving the fluid equations by Abbe [21]Bjerknes [22] and Richardson [23] Their methodologies were prescient but at the time impracticaland Richardsonrsquos actual attempt was wildly unsuccessful but it led to the revolution of numericalmodelling in the second half of the twentieth century

3 The modern eraThe modern era of GFD began around the middle of the twentieth century when Benny Goodmanwas in his prime and Miles Davis was emerging but well before the Beatlesrsquo first LP It began inatmospheric dynamics in the work of Rossby Charney Eady and others and in oceanography


7


with Stommel Munk and others By modern I mean work that has a direct influence on the workdone todaymdashit is no more than one or two intellectual generations from current research In whatfollows I would not give a comprehensive or historical view of this work Rather I will choose afew specific examples to show how this work both has led to a better understanding of our naturalworld (which might be regarded as its aesthetic justification) and has led (or not) to practicalbenefits sometimes through the development (or not) of better numerical models The exampleschosen are those I am familiar with and sometimes have been involved in but many others wouldserve the same purpose and may be equally or more important I have a large-scale bias and afew of the more egregious omissions are frontal theory (indeed all of mesoscale meteorology)convection gravity waves tides hydraulics and El Nintildeo

(a) Quasi-geostrophic theory and baroclinic instabilityOne of the first triumphs of modern GFD was the discovery of baroclinic instability and thetheoretical framework that enabled it namely quasi-geostrophic theory Charney [24] may becredited for the first systematic development of quasi-geostrophy although the term lsquoquasi-geostrophicrsquo seems to have been introduced by Durst amp Sutcliffe [25] and the concept used inSutcliffersquos development theory of baroclinic systems [2627] The full NavierndashStokes equationsare too complicated for most meteorological or oceanographic purposes and Charney himselfcomments in his 1948 paper lsquoThis extreme generality whereby the equations of motion apply tothe entire spectrum of possible motionsmdashto sound waves as well as cyclone wavesmdashconstitutesa serious defect of the equations from the meteorological point of viewrsquo Quasi-geostrophy is aparticular solution to this problem and a remarkable achievement for it reduces the complexityof the NavierndashStokes equations (three velocity equations a thermodynamic equation a masscontinuity equation an equation of state) to a single prognostic equation for one dependentvariable the potential vorticity with the other variables (velocity temperature etc) obtaineddiagnostically from itmdasha process known as lsquopotential vorticity inversionrsquo

Thus for a Boussinesq system the evolution of the entire system is given by

DqDt

= F q = nabla2ψ + f + f 20part

partz

(1

N2partψ

partz

) (31)

along with boundary and initial conditions Here q is the quasi-geostrophic potential vorticity Frepresents forcing and dissipation terms ψ is the streamfunction for the horizontal flow f is theCoriolis parameter f0 is a constant representative value of f and N is the buoyancy frequencyTemperature is related to the vertical derivative of streamfunction and velocity to the horizontalderivative and hydrostatic and geostrophic balances are built-in This equation is now standardfodder in textbooks such as Pedlosky [28] and Vallis [2] The beauty of this equation lies in itselimination of all extraneous phenomena that may exist in the original (primitive) equationsthereby making the flow comprehensible It is a line drawing of a rich and detailed landscape

The development of quasi-geostrophic theory was absolutely crucial to the development ofnumerical weather forecasts We noted earlier that L F Richardsonrsquos attempt in 1922 failedrather miserably and one reason for this would have been the presence of high-frequency wavesin his equations The development of quasi-geostrophic theory enabled the development ofnumerical models in the 1950s that filtered sound and gravity waves thereby avoiding the needfor complicated initialization procedures and allowing a much longer time step [29] The era ofusing quasi-geostrophic models as forecast tools passed fairly quickly and use of the primitiveequations (essentially the NavierndashStokes equations with hydrostatic balance and one or two othermild approximations) began in the 1960s but the early importance of quasi-geostrophy can hardlybe overstated for without quasi-geostrophy numerical weather forecasts simply might neverhave got off the ground Contrary to its reputation as a difficult subject GFD makes things easier

The development of quasi-geostrophic theory also allowed Charney [30] and Eady [31] toindependently develop the theory of baroclinic instabilitymdashwhich put simply is a mathematicaltheory of weather development and is thus one of the most important scientific theories


8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)

Figure 4 (a) Baroclinic growth rate in non-dimensional units with 2 4 and 8 vertical levels (solid lines as labelled) and acontinuous calculation (dashed line) The two-level result is the analytical result of (34) and other layer results are numericalThe continuous result is that of the Eadymodel (b) A similar calculation withβ = 0 with the two-level calculation (solid line)and a continuous stratification (dashed red line calculated numerically) (Online version in colour)

developed in the twentieth century The theory was further simplified by Phillips [32] whoderived the two-layer equations that when linearized about a constant shear may be writtenas (

part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2

(ψ prime2 minus ψ prime

1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)

where U is the sheared mean zonal flow kd is the inverse of the radius of deformation and β

is the latitudinal variation of the Coriolis term In Earthrsquos atmosphere U sim 10 m sminus1 and kd sim11000 kmminus1 whereas in the ocean U sim 01 m sminus1 and kd sim 1100ndash110 kmminus1 The simplicity ofthese equations allows one to analytically obtain a dispersion relation andor growth rate forperturbations and if (for illustrative purposes) we set β to zero we find

c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)

where c is the wave speed σ is the growth rate and K2 = k2x + k2

y is the total wavenumbersquared For a complex problem this is a remarkably simple result illustrated in figure 4 alongwith the corresponding result for a more complete model when β = 0 and the stratification iscontinuous and the results are quite similar There are two important findings that transcend thesimplifications of the two-level model namely

mdash The horizontal scale of instability is similar to or a little larger than the Rossby radiusof deformation Ld = 2πkd sim NHf which is a characteristic scale in GFD where bothrotation and stratification are important given a height H More elaborate calculationsbring up constant factors and the presence of β and continuous stratification (as in theCharney problem) further complicate the matter but nonetheless this is a transcendentresult

mdash The maximum growth rate is approximately σ sim Ukd or UfNH In the Earthrsquosatmosphere this is measured in days and in the ocean weeks On Mars it is weeks tomonths and the baroclinic waves are much steadier


9


The result already tells us something important for numerical models the grid size needs to besufficiently small to resolve the instability This is easily done for the atmosphere but is still notdone routinely for the ocean where the deformation radius is down to 10 km at high latitudesThe inability to resolve baroclinic eddies in the ocean suggests that we should parametrize theireffects which is easier said than done However imperfect as it is the parametrization of Gent ampMcWilliams [33] led to marked improvements in models of the ocean circulation to the degreethat arbitrary lsquoflux correctionsrsquo could be eliminated (Flux corrections are empirical fluxes betweenatmosphere and ocean that were added to coupled models in order that they do not drift too farfrom reality) The lesson here is twofold First theory is necessary to make numerical modelsperform better Second there are many cases where we cannot and should not expect theoryto substitute for numerical models for as model resolution increases with faster computers wecan expect to drop the GentndashMcWilliams parametrization even its redoubtable inventors wouldadmit that a computer model can do a better job than their theory given sufficient resolutionBaroclinic instability in the atmosphere is well resolved by modern general circulation models(GCMs) and to predict the atmosphere with anything less than a full-fledged numerical modelwith equations close to the full NavierndashStokes equations would be foolish

A numerical modeller of Earthrsquos atmosphere may these days have little knowledge of theearly work of Charney or Eady but this work played a significant role in the early developmentof the numerical modelling Even today a forecaster wishing to get a sense of the vertical velocityin a developing cyclone may look to the omega equation which is the quasi-geostrophic way ofdiagnosing vertical velocity

(b) Jets and surface westerliesBoth a triumph and an ongoing problem in GFD is to obtain a better understanding of zonaljets These jets are manifested in the surface westerlies on Earth and in the magnificent grandeurof the jets on Jupiter The ideas that we base todayrsquos theories on took form in the 1950s witha search for the cause of the surface wind pattern on Earthmdasheasterlies (westward winds) inthe tropics and westerlies in mid-latitudes and sometimes weak polar easterlies dependingon season There is no single paper that can be pointed to as a breakthrough heremdashStarr [34]Rossby [35] and Eady [36] all realized the importance of large-scale eddy motions and althoughEady realized that the large-scale eddies were the result of baroclinic instability (and not lsquojustturbulencersquo) even he could not properly crystallize its surface-wind and jet-producing essenceKuo [37] addressed the problem in a rather different way by considering the maintenance of zonalflows by the mechanism of vorticity transfer in a state with a meridional background gradient butthe situation remained opaque As late as 1967 Lorenz [38] noted that the cause of the polewardeddy momentum transport across mid-latitudes and hence the cause of the surface eastwardwinds was not at that time properly explained

A relatively simple explanation in terms of the momentum transport due to Rossby wavesemerged shortly after in papers by Dickinson [39] and Thompson [4041] although ironicallyThompsonrsquos initial motivation was oceanographic The essence of the explanation begins withthe dispersion relation for a barotropic Rossby wave

ω= ck = uk minus βkk2 + l2

equivωR (35)

implying a meridional group velocity

cyg = partω

partl= 2βkl

(k2 + l2)2 (36)

Now the velocity variations associated with the Rossby waves are

uprime = minusRe C il ei(kx+lyminusωt) and vprime = Re C ik ei(kx+lyminusωt) (37)


10


(a) (b)

Figure 5 A modern numerical simulation of a rapidly rotating terrestrial planet (a) Zonal velocity and (b) relative vorticity(Courtesy of Junyi Chai) (Online version in colour)

where C is a constant that sets the amplitude so that the associated momentum flux is

uprimevprime = minus 12 C2kl (38)

This is of opposite sign to the group velocity Thus a source of Rossby waves is associated witha convergence of eddy momentum flux so that a Rossby wave source in mid-latitudes associatedwith (for example) baroclinic instability will give rise to surface eastward flow

GCMs were of course already simulating the surface wind pattern reasonably well at thattime so one cannot say that the theory led to better simulations and the nonlinear transferof momentum by Rossby waves continues to be better simulated than theorized about forthe purposes of the general circulation But the theory does provide an underpinning for thesimulation and also provides needed guidance as we go out of the comfort zone of GCMs intomultiple jet regimes or superrotation on other planets or possibly a different wind regime in afuture or past climate on the Earth

Our understanding of multiple jetsmdashpossibly such as those seen on Jupitermdashcame througha different route that of geostrophic turbulence and wavendashwave interaction [42ndash44] with manyvariations The basic idea is that nonlinear interactions will because of the β-effect preferentiallyproduce zonal motion at large scale The concept is extraordinarily robust as might be suggestedfrom the vorticity equation on a β-plane namely

time derivative and nonlinear terms + βv= forcing and dissipation (39)

If β is large then the meridional velocity v must be small because otherwise the equationcannot balance Depending on the parameter regime the zonal flow is produced by wavendashwave interactions or wavendashmean flow interactions of various forms (eg [4546]) Explicationsin physical space as a potential vorticity staircase are also very revealing and may touchthe truth better than spectral arguments [4748] These various theoretical notions form thebasis for our theories of the production of jets not only on the Earth but also on giantplanets in accretion discs and possibly stellar interiors well beyond the capabilities of explicitdetailed numerical simulations Figure 5 shows a high-resolution (T512) primitive-equationsimulation of a rapidly rotating planet The numerical planet is terrestrial with a well-definedsurface but has other parameters similar to Jupiter It is a beautiful simulation full of jetsand eddies But it is nowhere close to the resolution needed to resolve the question as to thenature of Jupiterrsquos jets especially in the vertical especially given the contrasting paradigmsfor jet maintenance that require very different types of numerical model to properly simulatethem [49]


11


(c) Ocean circulationLet us turn our attention to the ocean and consider two related problems the thermocline andthe meridional overturning circulation

(i) The thermocline

The thermocline is that region of the upper ocean over which temperature varies quite rapidlyfrom its high surface values to its low abyssal values It is about 1 km in depth and over much ofthe ocean it coincides with the pycnocline the region of high-density variation If we take it as agiven that we wish to understand the oceanrsquos structuremdashafter all the ocean covers two-thirds ofthe globemdashthen we must understand the thermocline and this is a GFD problem

Many of our ideas about the thermocline stem from two papers published in 1950Welander [50] posited an adiabatic model based on the planetaryndashgeostrophic equations withno diffusion terms whereas Robinson amp Stommel [51] proposed a model that is intrinsicallydiffusive Over the years both models had their adherents and both models were developedtheoretically and somewhat independently [52ndash54] with some bridges between the two [5556]Building on all this work Samelson amp Vallis [57] proposed a model with an adiabatic andadvectively dominated upper regime (essentially the ventilated thermocline of [54]) and anunavoidably diffusive base (essentially the internal boundary layer of [52]) and two points ofview became one

In its most basic form the theory of the thermocline proceeds by way of deriving the so-calledM-equation which is a single partial differential equation that encapsulates the motion on verylarge scales in rotating stratified systems The planetaryndashgeostrophic equations of motion are

minus fv = minuspartφpartx

fu = minuspartφparty

b = partφ

partz(310)

and

nabla middot v = 0partbpartt

+ v middot nablab = κnabla2b (311)

In these equations f is the Coriolis parameter v is the three-dimensional velocity φ is the pressuredivided by the density b is the buoyancy and κ is a diffusion coefficient The vertical direction zhas much smaller scales than the horizontal (x and y) so that we can approximate κnabla2b by part2bpartz2If we now define a variable M such that φ = Mz the thermodynamic equation (311b) becomes

partMzz

partt+ 1

fJ(Mz Mzz) + β

f 2 MxMzzz = κMzzzz (312)

where J is the horizontal Jacobian and the subscripts on M denote derivatives and we take f = βyThe other variables are obtained using

u = minusMzy

f v= Mzx

f b = Mzz and w = β

f 2 Mx (313)

Equation (312) is amenable to attack by various meansmdashasymptotics similarity solutions andsimple numerical solutions among them [525859] We can simplify this model further if we seeksolutions of the form M(x z) = (x minus 1)M(z) and then in the steady case (312) reduces to the one-dimensional problem

β

f 2 WWzzz = κWzzzz (314)

Although this equationmdashstill nonlinear and of high ordermdashis at least approachable it looks a littlelike Burgerrsquos equation and the solution has shock-like features If we were to add an advectivemotion above the internal boundary layer then the entire upper ocean would become stratifiedeven at low diffusivity and a schematic of this is given in figure 6 The point is that algebraaside with a series of rational simplifications we can build a picture of the structure of thethermoclinemdasha picture that both makes testable predictions and gives us some understanding


12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy

Figure 6 Schematic of the overturning circulation and thermocline in a single basin in a single hemisphere (Online version incolour)

So who cares Where is the practical benefit of a better understanding of the thermocline orput another way why should a numerical modeller or an observer care about a somewhat esoterictheory At least one answer is that we now have a model of the upper ocean that can producea stratification that does not solely rely on a large diapycnal eddy diffusivity The magnitudeof that diffusivity is a matter for observation (and a different theory) but there is no need forheroic observational measures to find large amounts of mixing If observations show that mixingis small in the thermocline then modellers must use numerical methods that can adiabaticallysubduct water from the mixed layer and that do not introduce spurious mixing

(d) The meridional overturning circulationThe quasi-horizontal circulation of the oceanmdashthe gyres and western boundary currentsmdashobtained its conceptual model half a century ago in the work of Stommel [60] and Munk [61] butthe meridional overturning circulation (MOC) had to wait more than another half-century for itsown and still our ideas are evolving The MOC was first called the lsquothermohalinersquo circulationin the belief that it was largely driven by surface gradients of temperature and salinity andthe theory of the MOC was closely related to the diffusive theories (or internal boundary layertheories) of the thermocline The balance in the thermodynamic equation is then approximatelythe advectivendashdiffusive relation

wpartTpartz

= κpart2Tpartz2 (315)

which is closely related to (314) In order to produce a deep circulation of the magnitudeobserved this balance requires a diffusivity of about 1 times 10minus4 m2 sminus1 [62] which is about an orderof magnitude larger than that commonly observed in the thermocline [63] and although valuesof κ in the abyss and in boundary layers may be larger that alone cannot support an MOC asobserved The conundrum was overcome when over the years it became realized that the deepwater that sank in the North Atlantic upwelled in the Southern Ocean (figure 7) and not uniformlyin the subtropics and that this could take place with little vertical mixing [6465] This observationled to a picture of a wind-driven near-adiabatic pole-to-pole circulation as crystallized usingidealized models [6667] and now simulated with full GCMs Rather interestingly theoreticalmodels based on the equations of motion (eg [68ndash70]) really only properly emerged after thefirst numerical models and observations albeit with some simple conceptual-model precursors(eg [71]) and it was the use of idealized numerical models that led to the development of the


13


PacificndashIndonesianthroughflow133 plusmn 18

Tasman leakage136 plusmn 62

CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind

buoyancy loss buoyancy lossbuoyancy gain

AAIW

mixing

mixing

Equator

ice

South Pole North Pole

mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell

lower cell lower cell

(b)

(a)

Figure7 Twoviewsof themeridional overturning circulation not to scale (a) TheMOC ina singlebasin resembling theAtlanticand which is now reasonably well understood with a combination of idealized and comprehensive models and observationsGFD has yet to fully come to grips with the more complex multi-basin reality (b) (courtesy of K Speer) although it may yetbe a minor extension of the single-basin picture AABW Antarctic Bottom Water AAIW Antarctic Intermediate Water CDWCircumpolar Deep Water NADW North Atlantic Deep Water (Online version in colour)

theory So we cannot claim that analytical theory preceded the numerical models still the use ofidealized numerical models is part of GFD and the numerical models of Vallis [66] and Wolfe ampCessi [67] are close to being the simplest possible ones that capture the phenomenon even if theiroutput is complex

Regardless of the history theory and understanding can still help improve complex modelsand guide observations One example is the realization that an overly diffusive model maygive an incorrect picture of the circulation so that use of either an isopycnal model or carefully


14


designed advection schemes in conjunction with high vertical resolution in a height-coordinatemodel is needed [72] All told this is a story of how numerical models observations and simplerconceptual models (or lsquotheoryrsquo) can work in unison to give a fairly complete understandingalthough in this case it took a rather long time

4 Whither Do we still need geophysical fluid dynamics if we havecomprehensive numerical models

(a) GeneralitiesDo we still need understanding of the natural environment if we have comprehensive numericalmodels that simulate the emergent phenomena for us It is a reasonable question for certainlysome branches of science come to an endmdashthey die of natural causes [73] However the truthof the matter is that the presence of complicated numerical models of complex phenomena onlyincreases the need for understanding at a more basic level with such understanding coming fromboth analytical manipulations and ever more from simplified numerical models GFD has asubstantial role to play in this activity for many years to come but it must continue to evolveor it will wither and there will be no whither

Two types of contributions of GFD are immediately evident

(i) Improving the performance of comprehensive numerical models via parametrizations ofprocesses not treated properly by the models such as subgridscale processes

(ii) Understanding the behaviour of a system as a whole

Let us focus on the practical benefits of these activities rather than on understanding as an end ititself (which some readers may take as a given and some as irrelevant)

(i) The subgridscale

The subgridscale has with justification provided nourishment for basic fluid dynamics for manyyears and this will continue Large-eddy simulations rely on having some understanding ofturbulence and the aforementioned GentndashMcWilliams parametrization is a notable GFD exampleAs we enter an age of mesoscale-eddy-resolving ocean simulations it may lose its importancebut the parametrization of the submesoscale (eg [74]) will come to the fore and so on In theatmosphere convection occurs on the kilometre scale and it is impossible to properly resolve itin global climate models and will be for many years to comemdashalthough with heroic efforts wecan now resolve the atmospheric mesoscale (figure 8) The unresolved breaking of gravity wavesis a third example important in both atmosphere and ocean in the former the breaking bothprovides a drag on the flow and produces the quasi-biennial oscillation (QBO) and in the latter itprovides a finite diffusivity that partially drives the overturning circulation

These problems or their successors will be with us for years to come ([76] provide aquantification) and GFD and the style it brings is the field that addresses them Developing aparametrization sometimes seems to be regarded as dirty work whereas it should be regarded asa high calling for it is nothing more or less than developing a theory for a phenomenon andexpressing that theory in a very practical form

Still subgridscale improvements alone are unlikely to be enough and in any case will be hardto achieve since at any given model resolution it will be the interaction of the subgridscale withthe resolved scale that limits the model performance and we are unlikely to be able to improvethe model without some understanding of thatmdashin figure 8 we see small convective clustersorganized by the large-scale cyclone for example Doubling or quadrupling resolution is neverguaranteed to reveal problems that might involve the feedback of the small scale on the large afeedback that might act differently in a different climate or planetary regime Thus in conjunctionwith the inclusion of ever more detail a more holistic approach is needed as we now discuss


15


Figure 8 A snapshot of a global numerical simulation at about 12 km resolution beautifully resolving a mid-latitude cyclonesystem in the Pacific (From Orlanski [75]) (Online version in colour)

(ii) The system as a whole

The world is a complicated placemdashfar too complicated as von Neumann noted lsquoto allowanything but approximationsrsquomdashso how do we understand it The meaning of lsquounderstandingrsquois open to interpretationmdashin some branches of physics it seems to be taken simply to meanthe ability to predict the results of an experiment or observation But for a complex systemwe might mean the ability to simultaneously grasp the behaviour of the system at multiplelevels of granularity Thus for example to fully understand the climate system requires anunderstanding of a simplified version system as a whole (say a simple energy-balance model)through an intermediate model to a complicated GCMmdashthis is the renowned lsquohierarchy ofmodelsrsquo perhaps first adumbrated by Schneider amp Dickinson [77] and further discussed byColin de Verdiegravere [78] Hoskins [79] Held [80] and others and which is perhaps morecommonly discussed than implemented Adding complexity to a climate model is no guaranteeof improvement since we may simply be adding unconstrained feedbacks that cause differentmodels to diverge further Indeed todayrsquos climate models have not succeeded well in narrowingthe possible range of the global and regional warming we will experience in the yearsto come

The understanding that GFD and simpler models provides is of key practical importance forsuppose we put a lot of effort into producing a large numerical model and then that numericalmodel produces the wrong answer or it produces an answer that differs from another modelWhat then We should try to improve the subgridscale representation but in a complex systemlike the atmosphere with many feedbacks this is extremely difficult and virtually impossibleunless we have some level of intuitive understanding of the system as a whole as discussedby Held [80] If a GCM does not produce a good QBO we expect to be able to fix it byincreasing the resolution and lowering the diffusivity and by ensuring that tropical convectionproduces gravity waves of the correct magnitude But we only know this because we have anunderstanding of the nature of the QBO and we cannot apply the fix if we do not understandtropical convection and gravity waves And so it goes with other poorly simulated phenomenamdashthe intertropical convergence zone or biases in the Southern Ocean sea-surface temperature Thefix will ultimately come in the form of an improved parametrization in a new piece of code butthat code cannot be written without some prior understanding Many of these problems willinvolve the interaction of GFD with other aspects of the climate system and GFD must embrace


16


that This will be a practical challenge as well as a scientific challenge but modern computationaltools like scripting andor object-oriented languages (eg Python Julia) do make model buildingat different levels of complexity a viable task Let us be a little more specific about ripe scientificproblems in which GFDers or at least dynamicists might productively engage There are a hostof interesting pure fluids problems remaining but in what follows I have emphasized impureproblems

(b) Interaction with physics and chemistryFor historical reasons in the atmospheric science aspects associated with fluid mechanics havebecome known as lsquodynamicsrsquo and aspects associated with such things as moisture radiation andeven convection (a fluid dynamical problem) are known as lsquophysicsrsquo The demarcation may bethat the latter are not treated by the dynamical core of a model that solves the fluid equationsIn any case ripe problems for the atmospheric sciences lie in the interaction of GFD with thesephysics problems and in the ocean comparable problems lie in the interaction of dynamicswith biogeochemistry such as the carbon and nitrogen cycles and with plankton as well as thefiendish problems associated with the idiosyncrasies of the seawater equation of state

(i) Moisture and radiation

Almost the entire general circulation of the atmosphere is affected by the presence of moistureand needless to say life on Earth depends on water From a dynamical perspective watervapour is a decidedly non-passive tracer releasing significant energy into the atmosphere uponcondensation The condensation itself is a rapid process occurring much faster than a typicaladvective time scale even for convective scale flow and the criterion for condensationmdashthatthe level of water vapour exceed the saturated valuemdashis essentially an exponential functionof temperature as determined by the ClausiusndashClapeyron equation Evidently nonlinearitiesand feedback abound as well as a mathematical stiffness and understanding moist effects is achallenge both for numerical modellers and for theoretical dynamicists It is hardly original tosay that folding moisture into a GFD framework is a challenging and important problem formuch work has already taken placemdashsee [81ndash86] for miscellaneous examples and reviewsmdashbut itis a true statement nevertheless

A particular example lies in the area of convectively coupled equatorial waves and theMaddenndashJulian oscillation (MJO) [87] The MJO is an eastward travelling pattern in the tropicsmoving at a few metres per second and thus much more slowly than a Kelvin wave with anequivalent depth appropriate to a tropical atmosphere and it is sometimes called the 30ndash60 dayoscillation because it seems to recur on that time scale but it certainly should not be regardedas a linear wave with such a period It is the phenomenon that is most akin to mid-latitudeweather systems in the languid tropics and so obviously rather important to understand andsimulate Progress has been rather slow and we are still unable to give a crisp blackboard-styleexplanation nor are we able to unambiguously simulate it with a simple numerical model andalthough minimal models exist [88] our understanding of it is not at the same level as it is of drybaroclinic instability Our ability to simulate it has not moved much more quickly some cloud-resolving models do seem able to capture the phenomenon [89] but the progress of GCMs hasbeen slow [90] perhaps just outpacing our theoretical ideas The success of a cloud-resolvingmodel can be regarded as an existence proof that models will get there in the end but withouta better understanding of the phenomena our progress will continue to be slow and GCMs willimprove only incrementally It is a difficult problem but it is the kind of problem that GFD is wellplaced to attack (in addition to pure fluid problems) and that is obviously relevant to climateThe interaction of fluid dynamics with radiation is another such problemmdashit is this interactionthat gives us the height of the tropopause on Earth and other planets [91]mdashbut let us move onto oceanography


17


(ii) Ocean dynamics palaeoclimate and biogeochemistry

There are many important remaining problems in oceanography of a purely fluid dynamicalnature The meridional overturning circulation is one for although we now have variousconceptual and theoretical models of the MOC in a single basin we do not have a comparablemodel derived from a rational simplification of the equations of motion in multiple basinsThis is rather a basic gap in our field and such a model would be a considerable advanceOur understanding of the Southern Ocean is also at a slightly cruder level than ourunderstanding of the basins for one of the foundational dynamical balancesmdashSverdrup balancemdashcannot apply over large swaths of the Southern Ocean because of the lack of meridionalboundaries Mesoscale eddies then play a leading-order role with all the attendant difficultiesof geostrophic turbulence and it may be hard to make qualitative leaps beyond what isalready known

Another class of problem that is open to GFDers lies in the interaction of the circulationwith ocean chemistry and biology This interaction has not been neglectedmdashsee the books byWilliams amp Follows [92] and Mann amp Lazier [93]mdashbut its importance may be growing Theinherent complexity of the subject naturally suggests the use of complicated numerical modelsbut a GFD-like approach can also have value just as it does with climate modelling The carboncycle is one such areamdashthe uptake and sequestration of carbon by the ocean is an extraordinarilyimportant problem for it will in part determine the level of global warming that we suffer in thedecades and centuries ahead and it is a key aspect of the glacial cycles that Earth goes throughon the 100 000 year time scale [9495] The carbon cycle is complex as is the ocean circulationyet neither are so complex that a more holistic approach is not tenable At the simpler end of thescale carbon models have been studied to good effect using box models of the ocean circulationbut there is a very large jump to full GCMs [96] Box models are usefully transparent but mosthave virtually no proper fluid dynamics and problems such as the subtle differences betweenheat uptake and carbon dioxide uptake will require a much better treatment of the physicaloceanography That problem is particularly important in the long term for it determines howthe temperature of the planet will evolve as carbon is added to the atmosphere and whether thetemperature will continue to increase or will fall when carbon emissions cease Full GCMs witha carbon cycle (Earth system models) are part of that picture but are less able to help isolatemechanisms and models and arguments such as that of Burke et al [97] or Watson et al [95] thatat least try to incorporate the fluid dynamics in a simple and rational way can have an importantrole to play and their growing presence is a hopeful sign

Palaeoclimate is a particular area where the interaction of GFD (ocean dynamics in particular)with the rest of the climate system is key Observations are sparse and come largely by proxyand are insufficient to constrain the many semi-empirical parameters of ocean GCMs The resultis that different models often produce different answers when forced with the same boundaryconditions Theory is not a substitutemdashit alone cannot decide how the ocean might operate witha different configuration of continents or when totally covered by ice in a snowball Earth Rathertheory and idealized numerical simulations can provide rigorous and understandable results incertain configurations andor span a parameter space that is unattainable to a comprehensivemodel As with simulations of todayrsquos climate simulations with comprehensive models mayultimately provide the most accurate descriptions of the palaeo world but the absence of a largeset of observations with which to tune a model will make that task well-nigh impossible if it isregarded purely as a modelling problem

(c) Planetary atmospheresFor our last example we consider GFD on other planets in our Solar System and beyond Thediscovery of planets beyond our Solar Systemmdashover 3000 so far and countingmdashis surely oneof the most exciting scientific developments to have occurred in the last 20 years and it nowseems almost certain that there are millions and probably billions of planets in our galaxy alone


18


These exoplanets are a shot of adrenaline for GFD for they bring a host of new problems to thetable and since they lie beyond the Solar System they are within the remit of astronomers andastrophysicists and so bring a new cadre of scientists to the field

Exoplanets have atmospheres (and some probably have oceans) that come in a variety ofshapes and sizes and there can be no single theory of their circulation There is no planetaryequivalent of astronomyrsquos main sequence of stars that shows the relationship between stellarluminosity and effective temperature and on which most stars in the Universe lie Planetaryatmospheres differ from each other in their mass and composition their emitting temperaturethe size and rotation rate of the host planet whether they are terrestrial planets or gas giantsor something else and in a host of other parameters Some of these atmospheres will be asmultifaceted as Earthrsquos but in quite different ways and the combination of complexity andvariety provides a considerable challenge The complexity of any given atmosphere suggeststhat complicated numerical models are a natural tool but this is inappropriate as a generalmethodology for two reasons First it is simply impossible as a practical matter to build aGCM for each planetary atmosphere and ocean Second and more fundamentally the amountof data we have about these planets diminishes rapidly with distance from Earth We have ordersof magnitude less data about Venus and Mars than we do for Earth and we have orders ofmagnitude less data for Gliese 581d (a recently discovered potentially habitable exoplanet) thanwe do for Venus and Mars Any Earth-like GCM that is applied to an exoplanet will be vastlyunconstrained and the danger is that we will be able to get almost any answer we wish for aspecific planet The difficulty lies in reining in our ideas and modelsmdashexoplanets are such fertileground that anything will grow on them weeds as well as roses Thus we have to use simplermodels and by doing so we must engage fully in GFD However the information we obtain fromother planets comes entirely electromagnetically and so GFD is forced to interact with lsquophysicsrsquoas discussed above

Understanding and classifying the range of circulation types is a place for GFD to start (eg[9899]) This immediately gives rise to a number of poorly understood dynamical phenomenafor example superrotation which essentially is the prograde motion of an atmosphere at theEquator This motion implies that angular momentum has a local maximum in the interior ofthe fluid which in turn implies that the motion is eddying Rossby waves are the obvious wayto generate an upgradient transfer and in a slowly rotating regime (such as Titan) they seem tobe excited in conjunction with Kelvin waves with a Doppler shift allowing phase locking but byno means is the mechanism transparent [100101] Detailed GCMs still have trouble reproducingthe zonal wind on Venus [102] and the mechanism of superrotation on the gas giants almostcertainly differs again (eg [103104])mdasheven if the jets are shallow More generally the mechanismof the jet formation at all latitudes on Jupiter remains rather mysterious with various thoughtsabout the mechanism of coupling of the weather layer to the gaseous interior The original Joviandichotomy was that there is either a shallow weather layer rather like on a terrestrial planetwith β-plane turbulence giving rise to jets [105] or deep Taylor columns of which the jets aresurface manifestations [106] These remain useful end-members but other variations are possiblein which a weather layer is to a greater or lesser extent coupled to deeper jets giving rise to surfacevortices and jets that can equilibrate with little or no bottom friction and that are quite zonallyuniform (eg [107108]) whereas the baroclinic jets of figure 5 wander a little in the longitudinaldirection

Finally understanding the possible role of circulation in determining habitability is a GFDproblem of a grand nature The habitable zone is often taken to be the zone around a star inwhich a planet can support liquid water and if we know the atmospheric composition we can getan answer just using a one-dimensional radiativendashconvective modelmdashgenerally with complicatedradiation and simple dynamics But we know from our experience on the Earth how circulationmodifies temperatures so using a three-dimensional dynamical model with simple radiation is acomplementary approach and both will be needed if we are to gain an understanding of three-dimensional calculations with full radiation and the problem itself


19


5 A final note on educationWe cannot prescribe how the field will evolve but we can educate the next generation to beprepared for whatever comes and perhaps influence what they bring to it Numerical modelsare now such a part of the field that there is not so much a danger that students will not beexposed to them rather that they will not be exposed to the basic ideas that will enable themto understand them or infer whether the models are behaving physically Thus the very basicconcepts in GFD are as important perhaps more so than they ever were but can sometimesseem irrelevant relatedly there is a danger that analytical skills if not lost become divorcedfrom modelling skills Scientists will always have personal preferences and differing expertisebut combining analytical ideas with simple numerical models can be a very powerful tool inboth research and education and modern tools can be used to enable this at an early stage in theclassroom A numerical model transparently coded in 100 lines and run on a laptop can then playa similar role to that of a rotating tank in illustrating phenomena and explaining what equationsmean and the rift between theory models and phenomena then never opens

Although conventional books will remain important for years to come the next textbook ormonograph in GFD or really in any similar field could to great effect be written using a JupyterNotebook (formerly IPython Notebook) or similar which can combine numerical models withconventional text and equations (eg LATEX markup) figures and even symbolic manipulationin a single document enabling interactive exploration of both analytical and numerical GFDconcepts Such an effort would be a major undertaking so a collaborative effort may be neededperhaps like the development of open source software and the end product would hopefully befree like both beer and speech Research papers are a separate challenge but a first step mightbe to allow electronic supplementary material to be in the form of such a Notebook Thereare obvious difficulties with such a program but if one specifies that the text and figures beself-contained the material can at the least be read as a conventional article or book Anotherdifficulty is that software standards evolve faster than hardcopy printing methods it was 500years after the Gutenberg press before digital typography arrived Thus a Jupyter Notebookmight be unreadable in 20 years and Python might be passeacute but a lsquorealrsquo book will live on Thisproblem simply might not matter (immortality is not the goal) but if it does it might be wise notto couple the text and code too tightly In any case these are all simply obstacles to discuss andovercome rather than insurmountable barriers

I hope that it is by now clear that GFD has played an enormous role in the development ofour understanding of the natural world With the emergence of complicated models that roleis more important than ever it may be hidden like the foundations of a building but withoutthat foundation the edifice will come tumbling down I will continue to do GFD because it isinteresting important and fundamental Other peoplersquos motivation may differ but whatever thefuture holds GFD and the approach it brings has (or should have) an expansive role to play

Competing interests The author declares no competing interestsFunding The work was funded by the Royal Society (Wolfson Foundation) NERC NSF and the Newton FundAcknowledgements I am very grateful to two reviewers for their careful readings and thoughtful comments

References1 Anderson PW 1972 More is different broken symmetry and the nature of the hierarchical

structure of science Science 177 393ndash396 (doi101126science1774047393)2 Vallis GK 2006 Atmospheric and oceanic fluid dynamics Cambridge UK Cambridge University

Press3 Hadley G 1735 Concerning the cause of the general trade-winds Phil Trans 29 58ndash62

(doi101098rstl17350014)4 Euler L 1757 Principes geacuteneacuteraux du mouvement des fluides (General principles of the

motion of fluids) Acadeacutemie Royale des Sciences et des Belles-Lettres de Berlin Meacutemoires 11274ndash315


20


5 Frisch U 2008 Translation of Leonhard Eulerrsquos General principles of the motion of fluids(httparxivorgabs08022383)

6 Laplace P 1832 Elementary illustrations of the celestial mechanics of Laplace London UK JohnMurray

7 Thomson (Lord Kelvin) W 1879 On gravitational oscillations of rotating water Proc R SocEdinb 10 92ndash100 (doi101017S0370164600043467)

8 Ferrel W 1858 The influence of the Earthrsquos rotation upon the relative motion of bodies nearits surface Astron J V 97ndash100 (doi101086100631)

9 Ferrel W 1859 The motion of fluids and solids relative to the Earthrsquos surface Math Mon 1140ndash148 210ndash216 300ndash307 366ndash373 397ndash406

10 Thomson J 1892 Bakerian Lecture On the grand currents of atmospheric circulation PhilTrans R Soc Lond A 183 653ndash684 (doi101098rsta18920017)

11 Wallace JM Hobbs PV 2006 Atmospheric science an introductory survey 2nd edn AmsterdamThe Netherlands Elsevier

12 Silberstein L 1896 O tworzeniu sie wirow w plynie doskonalym (On the creation of eddiesin an ideal fluid) W Krakaowie Nakladem Akademii Umiejetnosci (Proc Cracow Acad Sci) 31325ndash335

13 Bjerknes V 1898 Uumlber einen hydrodynamischen Fundamentalsatz und seine Anwendungbesonders auf die Mechanik der Atmosphaumlre und des Weltmeeres (On a fundamentalprinciple of hydrodynamics and its application particularly to the mechanics of theatmosphere and the worldrsquos oceans) Kongl Sven Vetensk Akad Handlingar 31 1ndash35

14 Poincareacute H 1893 Theacuteorie des tourbillons (Theory of vortices [literally swirls]) Paris FranceG Carreacute Reprinted by Eacuteditions Jacques Gabay 1990

15 Hough SS 1897 On the application of harmonic analysis to the dynamical theory of the tidesPart I On Laplacersquos lsquoOscillations of the first speciesrsquo and on the dynamics of ocean currentsPhil Trans R Soc Lond A 189 201ndash258 (doi101098rsta18970009)

16 Hough SS 1898 On the application of harmonic analysis to the dynamical theory of the tidesPart II On the general integration of Laplacersquos dynamical equations Phil Trans R Soc LondA 191 139ndash185 (doi101098rsta18980005)

17 Rossby C-G Collaborators 1939 Relation between variations in the intensity of the zonalcirculation of the atmosphere and the displacements of the semi-permanent centers of actionJ Mar Res 2 38ndash55

18 Ekman VW 1905 On the influence of the Earthrsquos rotation on ocean currents Arch MathAstron Phys 2 1ndash52

19 Defant A 1921 Die Zirkulation der Atmosphaumlre in den gemaumlszligigten Breiten der ErdeGrundzuumlge einer Theorie der Klimaschwankungen (The circulation of the atmosphere inthe Earthrsquos midlatitudes Basic features of a theory of climate fluctuations) Geograf Ann 3209ndash266 (doi102307519434)

20 Jeffreys H 1926 On the dynamics of geostrophic winds Q J R Meteorol Soc 51 85ndash10421 Abbe C 1901 The physical basis of long-range weather forecasts Mon Weather Rev 29 551ndash

561 (doi1011751520-0493(1901)29[551cTPBOLW]20CO2)22 Bjerknes V 1904 Das Problem der Wettervorhersage betrachtet vom Standpunkte der

Mechanik und der Physic (The problem of weather forecasting as a problem in mathematicsand physics) Meteorol Z 1ndash7 Engl transl Y Mintz Reprinted in Shapiro MA Groslashnarings S1999 The life cycles of extratropical cyclones Boston MA American Meteorological Societypp 1ndash7

23 Richardson LF 1922 Weather prediction by numerical process Cambridge UK CambridgeUniversity Press (Reprinted by Dover Publications)

24 Charney JG 1948 On the scale of atmospheric motion Geofys Publ Oslo 17 1ndash1725 Durst CS Sutcliffe RC 1938 The importance of vertical motion in the development of tropical

revolving storms Q J R Meteorol Soc 64 75ndash84 (doi101002qj49706427309)26 Sutcliffe RC 1939 Cyclonic and anticylonic development Q J R Meteorol Soc 65 518ndash524

(doi101002qj49706528208)27 Sutcliffe RC 1947 A contribution to the problem of development Q J R Meteorol Soc 73

370ndash383 (doi101002qj49707331710)28 Pedlosky J 1987 Geophysical fluid dynamics 2nd edn Berlin Germany Springer29 Lynch P 2006 The emergence of numerical weather prediction Richardsonrsquos dream Cambridge

UK Cambridge University Press


21


30 Charney JG 1947 The dynamics of long waves in a baroclinic westerly current J Meteorol 4135ndash162 (doi1011751520-0469(1947)004lt0136TDOLWIgt20CO2)

31 Eady ET 1949 Long waves and cyclone waves Tellus 1 33ndash52 (doi101111j2153-34901949tb01265x)

32 Phillips NA 1954 Energy transformations and meridional circulations associated withsimple baroclinic waves in a two-level quasi-geostrophic model Tellus 6 273ndash286(doi101111j2153-34901954tb01123x)

33 Gent PR McWilliams JC 1990 Isopycnal mixing in ocean circulation models J PhysOceanogr 20 150ndash155 (doi1011751520-0485(1990)020lt0150IMIOCMgt20CO2)

34 Starr VP 1948 An essay on the general circulation of the Earthrsquos atmosphere J Meteorol 7839ndash43 (doi1011751520-0469(1948)005lt0039AEOTGCgt20CO2)

35 Rossby C-G 1949 On the nature of the general circulation of the lower atmosphere In Theatmospheres of the Earth and planets (ed GP Kuiper) pp 16ndash48 Chicago IL University ofChicago Press

36 Eady ET 1950 The cause of the general circulation of the atmosphere Cent Proc RoyMeteorol Soc 156ndash172

37 Kuo H-I 1951 Vorticity transfer as related to the development of the general circulationJ Meteorol 8 307ndash315 (doi1011751520-0469(1951)008lt0307VTARTTgt20CO2)

38 Lorenz EN 1967 The nature and the theory of the general circulation of the atmosphere WMOPublications vol 218 Geneva Switzerland World Meteorological Organization

39 Dickinson RE 1969 Theory of planetary wavendashzonal flow interaction J Atmos Sci 26 73ndash81(doi1011751520-0469(1969)026lt0073TOPWZFgt20CO2)

40 Thompson RORY 1971 Why there is an intense eastward current in the North Atlanticbut not in the South Atlantic J Phys Oceanogr 1 235ndash237 (doi1011751520-0485(1971)001lt0235WTIAIEgt20CO2)

41 Thompson RORY 1980 A prograde jet driven by Rossby waves J Atmos Sci 37 1216ndash1226(doi1011751520-0469(1980)037lt1216APJDBRgt20CO2)

42 Newell AC 1969 Rossby wave packet interactions J Fluid Mech 35 255ndash271 (doi101017S0022112069001108)

43 Rhines PB 1975 Waves and turbulence on a β-plane J Fluid Mech 69 417ndash443 (doi101017S0022112075001504)

44 Vallis GK Maltrud ME 1993 Generation of mean flows and jets on a beta plane andover topography J Phys Oceanogr 23 1346ndash1362 (doi1011751520-0485(1993)023lt1346GOMFAJgt20CO2)

45 Manfroi AJ Young WR 1999 Slow evolution of zonal jets on the beta plane J Atmos Sci 56784ndash800 (doi1011751520-0469(1999)056lt0784SEOZJOgt20CO2)

46 Tobias S Marston J 2013 Direct statistical simulation of out-of-equilibrium jets Phys RevLett 110 104502 (doi101103PhysRevLett110104502)

47 Marcus PS 1993 Jupiterrsquos great Red Spot and other vortices Ann Rev Astron Astrophys 31523ndash573 (doi101146annurevaa31090193002515)

48 Dritschel D McIntyre M 2008 Multiple jets as PV staircases the Phillips effect and theresilience of eddy-transport barriers J Atmos Sci 65 855ndash874 (doi1011752007JAS22271)

49 Vasavada AR Showman AP 2005 Jovian atmospheric dynamics an update after Galileo andCassini Rep Progress Phys 68 1935 (doi1010880034-4885688R06)

50 Welander P 1959 An advective model of the ocean thermocline Tellus 11 309ndash318(doi101111j2153-34901959tb00036x)

51 Robinson AR Stommel H 1959 The oceanic thermocline and the associated thermohalinecirculation Tellus 11 295ndash308 (doi101111j2153-34901959tb00035x)

52 Salmon R 1990 The thermocline as an internal boundary layer J Mar Res 48 437ndash469(doi101357002224090784984650)

53 Veronis G 1969 On theoretical models of the thermocline circulation Deep-Sea Res 31301ndash323

54 Luyten JR Pedlosky J Stommel H 1983 The ventilated thermocline J Phys Oceanogr 13292ndash309 (doi1011751520-0485(1983)013lt0292TVTgt20CO2)

55 Welander P 1971 The thermocline problem Phil Trans R Soc Lond A 270 415ndash421 (doi101098rsta19710081)

56 Colin de Verdiegravere A 1989 On the interaction of wind and buoyancy driven gyres J Mar Res47 595ndash633 (doi101357002224089785076172)


22


57 Samelson RM Vallis GK 1997 Large-scale circulation with small diapycnal diffusion thetwo-thermocline limit J Mar Res 55 223ndash275 (doi1013570022240973224382)

58 Hood S Williams RG 1996 On frontal and ventilated models of the main thermocline J MarRes 54 211ndash238 (doi1013570022240963213376)

59 Samelson RM 1999 Internal boundary layer scaling in lsquotwo-layerrsquo solutions of thethermocline equations J Phys Oceanogr 29 2099ndash2102 (doi1011751520-0485(1999)029lt2099IBLSITgt20CO2)

60 Stommel H 1948 The westward intensification of wind-driven ocean currents Trans AmerGeophys Union 29 202ndash206 (doi101029TR029i002p00202)

61 Munk WH 1950 On the wind-driven ocean circulation J Meteorol 7 79ndash93 (doi1011751520-0469(1950)007lt0080OTWDOCgt20CO2)

62 Munk WH 1966 Abyssal recipes Deep-Sea Res 13 707ndash730 (doi1010160011-7471(66)90602-4)

63 Ledwell JR Watson AJ Law CS 1993 Evidence for slow mixing across the pycnocline froman open-ocean tracer-release experiment Nature 363 701ndash703 (doi101038364701a0)

64 Toggweiler JR Samuels B 1998 On the oceanrsquos large-scale circulation in the limit ofno vertical mixing J Phys Oceanogr 28 1832ndash1852 (doi1011751520-0485(1998)028lt1832OTOSLSgt20CO2)

65 Webb DJ Suginohara N 2001 Vertical mixing in the ocean Nature 409 37 (doi10103835051171)

66 Vallis GK 2000 Large-scale circulation and production of stratification effects of windgeometry and diffusion J Phys Oceanogr 30 933ndash954 (doi1011751520-0485(2000)030lt0933LSCAPOgt20CO2)

67 Wolfe CL Cessi P 2011 The adiabatic pole-to-pole overturning circulation J Phys Oceanogr41 1795ndash1810 (doi1011752011JPO45701)

68 Samelson RM 2004 Simple mechanistic models of middepth meridional overturning J PhysOceanogr 34 2096ndash2103 (doi1011751520-0485(2004)034lt2096SMMOMMgt20CO2)

69 Nikurashin M Vallis GK 2011 A theory of deep stratification and overturning circulation inthe ocean J Phys Oceanogr 41 485ndash502 (doi1011752010JPO45291)

70 Nikurashin M Vallis GK 2012 A theory of the interhemispheric meridional overturningcirculation and associated stratification J Phys Oceanogr 42 1652ndash1667 (doi101175JPO-D-11-01891)

71 Gnanadesikan A 1999 A simple predictive model for the structure of the oceanic pycnoclineScience 283 2077ndash2079 (doi101126science28354102077)

72 Ilıcak M Adcroft AJ Griffies SM Hallberg RW 2012 Spurious dianeutral mixing and the roleof momentum closure Ocean Model 45 37ndash58 (doi101016jocemod201110003)

73 Horgan J 1996 The end of science facing the limits of science in the twilight of the scientific ageNew York NY Broadway Books

74 Fox-Kemper B Ferrari R Hallberg R 2008 Parameterization of mixed layer eddies Part ITheory and diagnosis J Phys Oceanogr 38 1145ndash1165 (doi1011752007JPO37921)

75 Orlanski I 2008 The rationale for why climate models should adequately resolve themesoscale In High resolution numerical modelling of the atmosphere and ocean pp 29ndash44 BerlinGermany Springer

76 Fox-Kemper B Backman SD Pearson B Reckinger S 2014 Principles and advances insubgrid modelling for eddy-rich simulations Clivar Exchanges 19 42ndash46

77 Schneider SH Dickinson RE 1974 Climate modeling Rev Geophys Space Phys 12 447ndash493(doi101029RG012i003p00447)

78 Colin de Verdiegravere A 2009 Keeping the freedom to build idealized climate models EOS 90224 (doi1010292009EO260005)

79 Hoskins BJ 1983 Dynamical processes in the atmosphere and the use of models Q J RMeteorol Soc 109 1ndash21 (doi101002qj49710945902)

80 Held IM 2005 The gap between simulation and understanding in climate modeling BullAm Meteorol Soc 86 1609ndash1614 (doi101175BAMS-86-11-1609)

81 Emanuel KA Fantini M Thorpe AJ 1987 Baroclinic instability in an environment of smallstability to slantwise moist convection Part I two-dimensional models J Atmos Sci 441559ndash1573 (doi1011751520-0469(1987)044lt1559BIIAEOgt20CO2)

82 Emanuel KA 1994 Atmospheric convection Oxford UK Oxford University Press83 Lapeyre G Held I 2004 The role of moisture in the dynamics and energetics of

turbulent baroclinic eddies J Atmos Sci 61 1693ndash1705 (doi1011751520-0469(2004)061lt1693TROMITgt20CO2)


23


84 Pauluis O Czaja A Korty R 2008 The global atmospheric circulation on moist isentropesScience 321 1075ndash1078 (doi101126science1159649)

85 Pauluis O Schumacher J 2010 Idealized moist RayleighndashBeacutenard convection with piecewiselinear equation of state Commun Math Sci 8 295ndash319 (doi104310CMS2010v8n1a15)

86 Lambaerts J Lapeyre G Zeitlin V 2011 Moist versus dry barotropic instability in ashallow-water model of the atmosphere with moist convection J Atmos Sci 68 1234ndash1252(doi1011752011JAS35401)

87 Kiladis GN Wheeler MC Haertel PT Straub KH Roundy PE 2009 Convectively coupledequatorial waves Rev Geophys 47 RG2003 (doi1010292008RG000266)

88 Majda AJ Stechmann SN 2009 The skeleton of tropical intraseasonal oscillations Proc NatlAcad Sci USA 106 8417ndash8422 (doi101073pnas0903367106)

89 Miura H Satoh M Nasuno T Noda AT Oouchi K 2007 A MaddenndashJulian oscillationevent realistically simulated by a global cloud-resolving model Science 318 1763ndash1765(doi101126science1148443)

90 Kim D et al 2009 Application of MJO simulation diagnostics to climate models J Clim 226413ndash6436 (doi1011752009JCLI30631)

91 Vallis GK Zurita-Gotor P Cairns C Kidston J 2015 The response of the large-scale structureof the atmosphere to global warming Q J R Meteorol Soc 141 1479ndash1501 (doi101002qj2456)

92 Williams RG Follows MJ 2011 Ocean dynamics and the carbon cycle principles and mechanismsCambridge UK Cambridge University Press

93 Mann K Lazier J 2013 Dynamics of marine ecosystems biological-physical interactions in theoceans New York NY John Wiley amp Sons

94 Archer D 2010 The long thaw Princeton NJ Princeton University Press95 Watson A Vallis GK Nikurashin M 2015 Southern ocean buoyancy forcing of ocean

ventilation and glacial atmospheric CO2 Nat Geosci 8 861ndash864 (doi101038ngeo2538)96 Toggweiler J Gnanadesikan A Carson S Murnane R Sarmiento J 2003 Representation of

the carbon cycle in box models and GCMs 1 Solubility pump Glob Biogeochem Cycles 171026 (doi1010292001GB001401)

97 Burke A Stewart AL Adkins JF Ferrari R Jansen MF Thompson AF 2015 Theglacial mid-depth radiocarbon bulge and its implications for the overturning circulationPaleoceanography 30 1021ndash1039 (doi1010022015PA002778)

98 Read P 2011 Dynamics and circulation regimes of terrestrial planets Plan Space Sci 59900ndash914 (doi101016jpss201004024)

99 Kaspi Y Showman AP 2015 Atmospheric dynamics of terrestrial exoplanets over a widerange of orbital and planetary parameters Astrophys J 804 60 (doi1010880004-637X804160)

100 Mitchell J Vallis GK 2010 The transition to superrotation in terrestrial atmospheresJ Geophys Res (Planets) 115 E12008 (doi1010292010JE003587)

101 Potter SF Vallis GK Mitchell JL 2014 Spontaneous superrotation and the role of Kelvinwaves in an idealized dry GCM J Atmos Sci 71 596ndash614 (doi101175JAS-D-13-01501)

102 Lebonnois S et al 2013 Models of Venus atmosphere In Towards understanding the climate ofVenus (ed L Bengtsson) pp 129ndash156 Berlin Germany Springer

103 Lian Y Showman AP 2010 Generation of equatorial jets by large-scale latent heating on thegiant planets Icarus 207 373ndash393 (doi101016jicarus200910006)

104 Schneider T Liu J 2009 Formation of jets and equatorial superrotation on Jupiter J AtmosSci 66 579ndash601 (doi1011752008JAS27981)

105 Williams GP 1978 Planetary circulations 1 Barotropic representation of Jovian andterrestrial turbulence J Atmos Sci 35 1399ndash1426 (doi1011751520-0469(1978)035lt1399PCBROJgt20CO2)

106 Busse F 1976 A simple model of convection in the Jovian atmosphere Icarus 29 255ndash260(doi1010160019-1035(76)90053-1)

107 Dowling TE Ingersoll AP 1989 Jupiterrsquos Great Red Spot as a shallow water system J AtmosSci 46 3256ndash3278 (doi1011751520-0469(1989)046lt3256JGRSAAgt20CO2)

108 Thomson S McIntyre ME 2016 Jupiterrsquos unearthly jets a new turbulent modelexhibiting statistical steadiness without large-scale dissipation J Atmos Sci 73 1119ndash1141(doi101175JAS-D-14-03701)


Preliminaries

Regarding geophysical fluid dynamics

The early history

The modern era

Quasi-geostrophic theory and baroclinic instability

Jets and surface westerlies

Ocean circulation

The meridional overturning circulation

Whither Do we still need geophysical fluid dynamics if we havecomprehensive numerical models

Generalities

Interaction with physics and chemistry

Planetary atmospheres

A final note on education

References

2


lava flows ocean circulation and planetary atmospheres In this article I will mainly discussmatters associated with atmospheres and oceans for that is my area of expertise and I will takeGFD to include the fields of dynamical meteorology and oceanography The moniker lsquoGFDrsquo hasalso come to imply a methodology in which one makes the maximum possible simplificationsto a problem perhaps a seemingly very complex problem seeking to reduce it to some bareessence It suggests an austere approach devoid of extraneous detail or superfluous descriptionso providing the fundamental principles and language for understanding geophysical flowswithout being overwhelmed by any inessentials that may surround the core problem In thissense GFD describes a method as well as an object of study

Although one might think that such a method is of course entirely appropriate in all scientificareas in some branches of science there is a tendency to embrace the complexity of reality by usingcomplicated models to which we add processes whenever possible rather than taking them awayThis approach certainly has its place especially if our concern is in making detailed predictionsof the behaviour of a complex system such as we might if we are engaged in weather forecasts orclimate predictions and climate models have grown enormously in complexity over the past fewdecades (figure 1) The GFD approach and the modelling approach have at times diverged whenboth approaches (and a range in between) may be needed to gain a proper understanding of aproblem Still the rift between the two fields is sometimes exaggeratedmdashthere are many signs ofa realization that we (the scientific community) really do need both

At the same time as we might be adding complexity in much of science it is usually takenas a given that we should be endlessly seeking the most fundamental level of understandingand this search has mostly served us wellmdashprovided we have a sensible notion of what lsquomostfundamentalrsquo means for any given problem In physics for example it is taken for grantedthat we should seek to unify the fundamental forces as much as possible a unified descriptionof the weak and electromagnetic forces is universally regarded as more satisfying and morefoundational than a description of the two separately There is no discussion but that we shouldtry to go further and physicists are now looking apparently without irony for a single theoryof everything

The simplifications sought by GFD are not quite like thatmdashthey are more akin to those soughtby biologists or condensed matter physicists or anyone dealing with a complex subject thatcontains emergent phenomena An emergent phenomenon is one that emerges from the collectivebehaviour of the constituents of a system and is not a property of its individual componentsmdashits equivalent atoms or its primitive building blocks emergence is a manifestation of a groupbehaviour Perhaps the most familiar example is temperature which is a collective property ofthe molecules of a system and is proportional to the mean kinetic energy of molecules in a gasthe phenomenon of phase transitions is another example in physics Biology itself may be anemergent phenomenon or at least aspects of it such as life speciation and evolution

Now although there may be a sense in which biology can be reduced to chemistry chemistryto physics and so on it is preposterous to seek an explanation for the emergence of biologicalsystems in the laws of physics One simply cannot understand biology based solely on the laws ofphysics as has been widely if not universally accepted for some time (eg [1]) At each new levelof complexity new properties emerge that do not depend on the details of the underlying laws andqualitatively different behaviour takes place Darwinian principles care nothing for string theoryand an evolutionary system can be built on the computer with no regard to the particular laws ofphysics The construction and function of a gene may be constrained by the laws of chemistry butat a higher level evolutionary principles are not it would not make sense to seek an explanationof macroscopic evolutionary laws in terms of chemistry or physics any more than it would makesense to try to understand Uncle Vanya using the rules of grammar even with a dictionary at handIndeed those rules are different in Russian and English but Uncle Vanya transcends that

Still there are some emergent properties that we can now simulate using the microscopiclaws applied to the atoms of the systemmdashthe Maxwellian distribution of velocities in a gas forexamplemdashbut we still would not dream of performing a numerical simulation of the individualmolecules to calculate the change in temperature of an ideal gas as it is adiabatically compressed


3


1960s





ocean ocean

sea ice sea ice

sulfateaerosols



carboncycle


aerosols



ice sheet


aerosols

carbonnitrogen

cycle


sulfateaerosols

sulfateaerosols

sea ice sea ice

ocean ocean ocean

1970sndash1980s 1990s 2000s 2010s





4








5






6


(b)(a)


North Pole


Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream





7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

3


1960s





ocean ocean

sea ice sea ice

sulfateaerosols



carboncycle


aerosols



ice sheet


aerosols

carbonnitrogen

cycle


sulfateaerosols

sulfateaerosols

sea ice sea ice

ocean ocean ocean

1970sndash1980s 1990s 2000s 2010s





4








5






6


(b)(a)


North Pole


Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream





7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

4








5






6


(b)(a)


North Pole


Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream





7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

5






6


(b)(a)


North Pole


Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream





7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

6


(b)(a)


North Pole


Ferrel cellL

L

L L

H H H

H H H

J

Hadleycells

Jtropospheric

jet stream





7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

7





DqDt


partz

(1

N2partψ

partz

) (31)





8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














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23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

8


1 2 3 40

01

02

03

04

0

01

02

03

04

grow

th r

ate

wavenumber

b = 0 b π 0

1 2 3 4wavenumber

2

4

8

(b)(a)



part

partt+ U

part

partx

)[nabla2ψ prime

1 + k2d2


1)

]+ partψ prime

1partx

(β + k2dU) = 0 (32)

and (part

parttminus U

part

partx

)[nabla2ψ prime

2 + k2d2


2)

]+ partψ prime

2partx

(β minus k2dU) = 0 (33)



c = U

(K2 minus k2

d

K2 + k2d

)12

or σ = Uk

(k2

d minus K2

K2 + k2d

)12

(34)






9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














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23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

9







equivωR (35)


cyg = partω

partl= 2βkl

(k2 + l2)2 (36)




10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












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21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

10


(a) (b)










11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







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17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

11



(i) The thermocline






b = partφ

partz(310)

and




partMzz

partt+ 1

fJ(Mz Mzz) + β



u = minusMzy

f v= Mzx


f 2 Mx (313)


β




12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

12


IT

Equator Pole

PDa

z

d

T (y1 z)

y1 y0

VT

DTSP DTSTy




wpartTpartz




13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

13




CDW 110plusmn 51

32deg S

64deg S

ATLANTIC

INDIAN

PACIFIC

wind


AAIW

mixing

mixing

Equator

ice


mixing

gyres

NADW

AABW

eddies

gyres

NADW 1

76plusmn 31

AABW 56plusmn 30

92 plusmn 27

CDW

upper c

ell up

per cell


(b)

(a)





14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

14














15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

15







16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

16








17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

17








18







19












20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

18







19












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21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

19












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21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

20



























21






























22






























23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

21






























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Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

22






























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Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

23




























Preliminaries


The early history

The modern era



Ocean circulation



Generalities




References

GeoffreyK.Vallisempslocal.ex.ac.uk › people › staff › gv219 › papers ›...

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