Raymond Hide at 80 • WHite: Meteorology

4.24 A&G•August2010•Vol.51

Raymond Hide at 80 • WHite: Meteorology

Raymond Hide’s scientific interests and achievements extend over wide areas of geophysical fluid dynamics and

magnetohydrodynamics, including studies of planetary atmospheres, the oceans, laboratory flow systems and liquid planetary cores. This article concentrates on the importance to ter-restrial meteorology, as practised by National Meteorological Services such as the Met. Office, of the rotating annulus laboratory experiments pioneered by Raymond from 1950 onwards.

During the late 1970s and the 1980s I was fortunate enough to work in the Met. Office’s Geophysical Fluid Dynamics Laboratory, recalled fondly as Met O 21, which Raymond Hide directed from 1967 to 1990. Raymond led research with Richard Barnes, Mike Bell and Clive Wilson on atmospheric angular momen-tum fluctuations and their relation to length-of-day changes and polar motion as determined by astronomical measurements (Hide et al. 1980, Barnes et al. 1983, Bell et al. 1991, Bell 1994). Rotating annulus studies involved numerical modelling as well as experimental and theoreti-cal work; during my time in Met O 21, those engaged in the work included Mike Bell, Roger Carter, Phil Hignett, Nick Jackson, Ian James, Doug Johnson, Peter Jonas, Peter Read and Bob Small as well as Raymond himself.

For the purposes of presentation, it is help-ful to distinguish three ways in which rotating annulus studies have contributed to terrestrial meteorology – and continue so to do: by providing a conceptual and demonstra-tional perspective for the Earth-bound subject; by providing a special test bed for numerical models of the type used in weather forecasting and climate simulation; by providing independent support for the sim-plified quasi-geostrophic model (on which much current understanding of atmospheric weather systems is based). These headings are considered successively below.

A perspective for terrestrial meteorology The relevance of rotating annulus flows to terrestrial meteorology tends to be under-estimated. I believe this is partly because the many fundamental ideas (and familiar terms) that have arisen from annulus studies have become so much part of the conceptual furni-ture of meteorology that their laboratory origin is easily overlooked. The relevance of annulus studies is also sometimes misinterpreted. Both aspects are addressed here.

The rotating annulus (here referred to simply as “the annulus”) and the Earth’s atmosphere (“the atmosphere”) are both rotating, thermally driven systems subject to gravity. Figures 1 and 2 show the set-up of the annulus via cross-section and plan views respectively. The inner cylinder of the annulus (identified with polar

regions of the atmosphere) is held at a lower temperature than the outer cylinder (identified with equatorial regions of the atmosphere). The geometry of the annulus is essentially cylindri-cal, in contrast to the quasi-spherical geometry of the atmosphere. Rotation occurs about the local vertical in the annulus, but in the atmo-sphere the rotation vector is aligned with (or against) the local vertical only at the poles, and is actually perpendicular to it at the equator. Neither the inner nor the outer cylinder of the annulus is present in the atmosphere, of course; furthermore, although the temperature differ-ence between equator and pole (in either hemi-sphere) is maintained by differential heating, it is not held constant as it is in many annulus experiments. The atmosphere consists of a com-pressible gas subject to phase changes of water substance, while annulus flow is that of a nearly incompressible fluid.

From these basic observations it is clear that the two systems share some key features, but the annulus is not an engineering model of the atmosphere. Instead, the annulus has the advan-tages of simplicity, controllability and the rich-ness of the flow phenomena that it reveals.

The simplicity of the annulus means that it poses a less formidable problem of understand-ing and simulation than does the atmosphere. Fundamental concepts and structures – such as Hadley cells, baroclinic waves, jet streams and Ekman layers – can be demonstrated on the laboratory scale, with a minimum of extraneous complication. The controllability of the annulus means that experiments can be conducted over wide ranges of externally imposed conditions (rotation rate, temperature difference, aspect ratio etc) to an extent that – fortunately! – is not possible in the atmosphere. The richness of the flow phenomena found in the annulus is reflected in the extensive range of regimes seen, and in the associated behaviour (in terms of transitions, hysteresis and non-uniqueness, for example).

The simplicity and controllability aspects of annulus experiments are transparent, but the flow

phenomena aspect deserves more discussion. The annulus demonstrates the ranges of phenomena that are possible in rotating fluids (Lorenz 1967, Hide and Mason 1975); in fluid dynamical terms, it gives a context to the atmosphere, a cultural background of possible behaviour.

Raymond Hide describes in his article in this issue (2010) how the annulus has stimulated the development of key notions of chaos and predictability. As another example of annu-lus contextualization, the very occurrence of the steady wave regime may be considered. Experiments show that the conditions under which steady wave flow occurs become more restricted as the radius of the inner cylinder is reduced (Hide 2010, Hide and Mason 1970); and numerical experiments suggest the same for spherical domains (Geisler 1983). The steady wave regime does not disappear, and its erosion by the irregular wave regime obeys an empirical rule that may be interpreted in terms of instabil-ity of the steady wave pattern (Hide 2010). In any case, because the annulus is not an engi-neering model of the atmosphere, its relevance to terrestrial meteorology does not hinge on the survival of steady waves as the inner cylin-der radius decreases. The steady wave regime in the annulus establishes the possibility of steady wave flow in rotating baroclinic fluids; meteorologists know well that quasi-steady flow occurs in the atmosphere in some regions, at some times (perhaps giving rise to weeks or months of anomalous weather). Annulus behav-iour thus provides a useful conceptual back-ground for notions of hemispheric or regional flow regimes in the atmosphere (Marshall and Molteni 1993), including the phenomenon known to synoptic meteorologists as “block-ing”. All may be related to the occurrence of finite amplitude, steady solutions of the govern-ing equations (White 1990) – particularly of the quasi-geostrophic model (see below).

Test bed for numerical models The use of numerical models in weather fore-

The atmosphere, the annulus and quasi-geostrophic theory

Andy White examines the contribution of Raymond Hide’s rotating annulus experiments to meteorological modelling and theory.

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casting is widespread in National Meteoro-logical Services such as the Met. Office, and is an activity known as Numerical Weather Prediction (NWP). Typically, a human fore-caster – well versed in the behaviour of both the atmosphere and the numerical models – bases a weather forecast on the available NWP results, making such adjustments and interpretations as appear appropriate in the light of experience.

The numerical models used in NWP are based on the equations of classical compressible fluid dynamics and thermodynamics expressed in discretized form (Holton 1992, Houghton 2002). For example, the Met. Office’s numeri-cal model (known locally as the Unified Model, and corporately as the MetUM) in its global manifestation delivers forecasts of atmos-pheric variables – such as wind components, temperature, pressure and humidity – at order 10 million points distributed over the horizontal extent and depth of the atmosphere (Davies et al. 2005, Staniforth and Wood 2008). The basic method is numerical time integration, the gov-erning equations (such as Newton’s second law of motion) being well suited to time-marching techniques. Similar models are used for climate simulation, and the same technique of time integration is used (outside the Met. Office) in numerous other contexts such as flow simula-tion for engineering.

The physical processes that force and dissi-pate motion in the atmosphere are complicated in their variety and in their spatial and tempo-ral variability. Many of the relevant processes (such as latent heating by condensation of water vapour in clouds) may occur on space scales too small to be resolved on a global grid of 10 million points, and are represented in terms of resolved fields by schemes known as parametri­zations, whose development is an area of active research (see the report of the Board on Atmo-spheric Sciences and Climate 2005 and Stensrud 2007). In contrast, the forcing and dissipative processes in the annulus may be represented to high accuracy using well-established expres-

sions involving molecular viscosity and ther-mal conductivity. The annulus thus provides an opportunity to test the core formulation of numerical models without the uncertainties and ambiguities that arise when comparing results against the atmosphere’s reality.

Model formulation may be tested in two ways: Qualitative tests: are the different flow types and the various phenomena that are seen in the lab system reproduced by the numerical model? Quantitative tests: how do the simulated results compare with careful measurements made in the lab system?

Both sorts of test have been conducted and reported (Hignett et al. 1985, White 1988, Read et al. 2000). Results have been encouraging – up to the limit of reasonable expectation, in my personal view – and have afforded strong sup-port to the discrete mathematical techniques used in NWP models. An important incidental consequence has been that scientists developing NWP models have gained confidence in their model building activity.

It may be noted that the arrangement of vari-ables on the computational grid used by the Met. Office’s model (the MetUM) is in essential respects the same as that used in the numeri-cal model of annulus flow that was tested by Hignett and co-workers (1985); and that both are based on non-hydrostatic equations of motion (in one case for incompressible flow, in the other case for compressible flow). The use of non-hydrostatic models in NWP is desirable for the smallest resolvable scales of motion, but is not obviously appropriate for the larger scales (in which hydrostatic balance is known to be a very good approximation). The feasibility of the technique is confirmed by its good performance in the annulus context (where both nonhydro-static boundary layers and hydrostatically bal-anced interior flows occur).

It is indicative of Raymond Hide’s scientific reach that his work on atmospheric angular momentum fluctuations (referred to briefly above) provides an entirely different way of

testing numerical weather prediction and cli-mate simulation models: by comparing forecast global angular momentum fluctuations with length-of-day changes and polar motion deter-mined by astronomical measurements (Hide et al. 1980, Barnes et al. 1983, Bell et al. 1991). The technique also provides a valuable global consistency test of wind and surface pressure observations in the atmosphere. This angular momentum work continued in the 1990s as a diagnostics sub-project of AMIP (Atmospheric Model Intercomparison Project) in which Ray-mond collaborated with two US groups under the auspices of the World Climate Research Pro-gramme and the Global Atmospheric Research Programme’s Working Group on Numerical Experimentation (Hide et al. 1997).

Finally, it should be emphasized that numeri-cal models of annulus flow have been exten-sively used to analyse and increase physical understanding of the relevant dynamics (see, for example, Williams 1972, Dietrich 1973, Quon 1976, James et al. 1981, Tucker and Long 1996, Read 2003, Pérez-Pérez et al. 2010). The present discussion, with its focus on direct relevance to weather forecasting and climate simulation, has perforce played down this key scientific aspect of annulus flow simulation.

Support for QG theory In addition to hydrostatic balance, a good approximation in the extratropical atmosphere outside the planetary boundary layer is geo­strophic balance. This term denotes an idealized equilibrium involving the horizontal pressure gradient force and the dominant horizontal components of the Coriolis force. It is closely associated with the well-known relationship between the pressure field and the wind that is expressed in Buys Ballot’s law, of which I know no more compact statement than the version often quoted by Raymond Hide (and ascribed by him to Hubert Kirk, sometime Chief Forecaster at the Met. Office): If you stand with your back to the wind, and high pressure is to your right, then you are in the northern hemisphere.

Geostrophic balance relates the horizontal flow (the wind) to the horizontal pressure gra-dient via an equation known as the geostrophic approximation. Hydrostatic balance relates the density field to the vertical pressure gradi-ent via an equation known as the hydrostatic approximation. By combining the hydrostatic and geostrophic approximations, various other relationships between meteorological fields may be deduced that are useful where the flow is close to geostrophic balance (Holton 1992, White 2002). Perhaps the principal example is the ther-mal wind equation, which Raymond might state whimsically (and concisely) as: If you stand with your back to the wind shear (with height), and high temperature is to your right, then you are in the northern hemisphere.

b

d T Tb bT Ta a

a

g

turntable

fluid fluid

1: Fluid fills an annulus of inner radius a, outer radius b and depth d. The inner and outer cylinders are held at temperatures Ta and Tb respectively; Ta < Tb when the inner cylinder is identified with polar regions of the atmosphere. The annulus stands on a turntable with which it is co-axial; the angular velocity of co-rotation about the vertical (defined by gravity g) is Ω.

2: Plan view of the annulus shown in vertical cross-section in figure 1. The turntable on which the annulus stands is not shown.

fluid

fluid

fluid

b

a

Tb

Ta

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Raymond Hide at 80 • WHite: Meteorology

These relationships are valuable in conceptual terms, but they are all diagnostic in character: they (approximately) relate meteorological fields at each instant of time, but give no information about how these fields evolve in time. In order to forecast, prognostic equations are required: equations containing time derivatives.

One way of re-introducing time derivatives is to return to the full equations of fluid motion, or close approximations to them; this is the rationale of the NWP models discussed above. Another way of re-introducing time deriva-tives (giving more insight but less accuracy) is to assume that the geostrophic and hydrostatic approximations apply to the real atmosphere in some zero-order sense and to seek the equations that describe the relationships between the vari-ous fields at the first order. The relevant order-ing parameter is a Rossby number Ro defined as the ratio of a typical wind or flow speed relative to the rotating system (be it Earth or annulus) to the product of the Coriolis parameter and the horizontal space scale of the motion concerned. Ro is simply a measure of horizontal flow accel-erations in the rotating frame as compared with the Coriolis force per unit mass; Ro is zero if geostrophic balance is exact.

The mathematical procedure, originated by J G Charney (1948), is described in textbooks on meteorological dynamics (Gill 1982, Ped-losky 1987, Holton 1992, James 1994, Dutton 1995, Vallis 2007) and is a matter of some com-plication (and variation in detail). As well as Ro << 1, it is assumed that a quantity known as the Burger number is of order unity; this corre-sponds to the Coriolis and buoyancy frequencies being in the same order-of-magnitude ratio as the vertical and horizontal scales of the motion. The result is a remarkably compact prognostic model known as the quasi-geostrophic model – the QG model – that describes the way in which the geostrophic flow slowly evolves when non-geostrophic effects are small (but not zero).

The core of the QG model is a single ellip-tic partial differential equation for the rate of change of a streamfunction of the geostrophic flow; it may be integrated by standard tech-niques (given appropriate boundary conditions). This prognostic equation – which embodies a conservation law for a quantity known as QG potential vorticity – is complemented by a diagnostic elliptic partial differential equa-tion for the vertical velocity, the source term of which depends only on the geostrophic streamfunction (and various constants). Other diagnostic relations enable the pressure, tem-perature and ageostrophic flow to be found if required. A summary presentation of the model is attempted in table 1.

To be sure, the QG model is significantly less complete than the full equations of motion. It applies only to motion that is close to geo-strophic balance, and many other approximate

models have been put forward that are formally more accurate (White 2002). Nevertheless, as a compromise amid the various demands of compactness, symmetry, tractability, accuracy and comprehensiveness, the QG model has no equal. It provides an invaluable simplified basis for describing and (in qualitative terms) under-standing the time evolution of the atmosphere on the appropriate space scales. It is also a natu-ral simplification from an applied mathemati-cian’s standpoint.

Prof. J A Dutton (1995) describes the QG model as “one of the great accomplishments of dynamic meteorology”; and it has aptly been said that the QG model provides “an arena for many important concepts of modern synoptic-scale meteorology”. Prof. G K Vallis (2007) com-bines these views: “Quasi-geostrophic theory, and the quasi-geostrophic potential vorticity equation, are quite beautiful, both for their aus-terity of description and richness of behaviour.” The QG model’s (many) advantages and (few) disadvantages are summarized in table 2.

Raymond Hide notes in his article (2010) that the simplest conceivable baroclinic insta-bility problem, that of Eady (1949), provides a

good account of the occurrence of the “upper transition” between axisymmetric and steady wave flow in the annulus; the level of agree-ment is improved if effects of Ekman layers are included (Barcilon 1964) or if more realistic azi-muthal flow structures are assumed (Williams 1974, Bell and White 1988). Raymond himself derived an extension of the Eady solution to the case in which Ekman layers are present, the lower bounding rigid surface may be sloping and the upper surface may be free or rigid and sloping (Hide 1968); a striking level of agree-ment with experimental determinations of the upper transition in sloping boundary systems was subsequently found (Mason 1975). The Eady solution, and the variants of it just noted, are all obtainable from the QG model. Their success in accounting for the upper transition in the annulus thus provides support for the QG model, and hence for a model that plays a central role in meteorology.

It may be noted also that the QG model (and in particular the Eady problem) is in some ways more appropriate to the annulus than to the atmosphere: the Coriolis parameter is spatially uniform in the annulus; the upper and lower

1: Summary of the quasi-geostrophic (QG) modelKey prognostic variable

ψ =ψ(x,y,z,t)(xandyarehorizontalcoordinates,zisheightandtistime.)ψisthestreamfunctionofthehorizontalgeostrophicflowvg:

vg=(ug,vg,0);ug=–∂ψ

 ___∂y ,vg=∂ψ

 ___∂y

Approximately, ψequalspressuredividedbytheproductofdensityandameanvaluef0oftheCoriolisparameterf.Contoursofψ atfixedzandtthusapproximateisobars.

Key prognostic equation

                                                    ∇32(∂ψ

 ___∂t )=–(∂ψ ___∂x      ∂     

 ___∂y –∂ψ ___∂y      ∂     

 ___∂x )(∇32ψ+f) (1)

Here,forastratifiedincompressiblefluidhavingbuoyancyfrequencyN =N (z),

                                                             ∇32≡     ∂

2

 ___∂x2 +    ∂2

 ___∂y2 +     ∂     

 ___∂z ( f02

 ___N2      ∂     

 ___∂z ) (2)

Thequantity∇32ψ +f thatappearsinequation(1)isknownastheQGpotentialvorticity.

Intheabsenceofheightvariations(i.e.when∂/∂z =0ineqn2)equation1reducestothebarotropicvorticityequation.

Conservation law form of equation 1

(     ∂      ___∂ t +ug

     ∂      ___∂ x +vg

     ∂      ___∂ y )(∇3

2ψ+f)=0

Elliptic p.d.e. for the vertical velocity (w)

(   ∂2

 ___∂ x2 +    ∂2

 ___∂ y2 + f0

2

 ___N 2   ∂2

 ___∂ z2 )w =G

Thisisthestratifiedincompressiblefluidformcorrespondingtoequations1and2.ThesourcetermGisaknownbutcomplicatedfunctionofψanditsspatialderivatives.

Notes. For simplicity, non-adiabatic and viscous effects have been neglected, but may be included if desired. See text for comments on range of validity. Boundary conditions are not considered and extension to stratified compressible fluids is straightforward. For detailed discussion see Charney 1948, Pedlosky 1987, Holton 1992, James 1994, Dutton 1995, White 2002, Vallis 2007.

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A&G•August2010•Vol.51 4.27

boundary conditions that are applied in the Eady problem can be more closely reproduced; and the lateral boundedness of the annulus (although attended by lateral boundary layers which the theory does not attempt to describe) is also more appropriate to the QG scenario than is the spherical atmosphere. Hence the annu-lus provides in some respects a more appropri-ate test bed for the Eady problem, and indeed for QG theory more generally (Williams et al. 2010), than does the atmosphere itself.

Concluding remarks This article has focused on the importance of rotating annulus experiments and associ-ated numerical and theoretical activity to the practical, everyday operations of a national meteorological service such as the Met. Office. Annulus experiments provide a fundamental cultural background by revealing the wide range of behaviour that is possible in thermally driven, rotating fluid systems; they also enable fundamental concepts and flow structures to be

appreciated on a laboratory scale, under con-trolled conditions. Given careful design and accurate measurement techniques, annulus flows provide a special test bed for numerical models of the type used in modern weather forecasting and climate simulation. The annu-lus is also important in supplying independent support for the QG model – an elegant sim-plified formulation of the governing equations that unifies many of the key concepts of mete-orological dynamics.

Emphasis here has been on a particular config-uration of the rotating annulus in which heat is input and output at the cylindrical boundaries. Other configurations play a similarly salient role in meteorology (and oceanography), and many of them were also pioneered by Raymond Hide during his long, productive and continu-ing career in geophysical fluid dynamics and magnetohydrodynamics. For discussion of these other configurations, the reader is referred to Raymond’s article (Hide 2010, this issue) and to the studies cited there.

Andy White works in the Dynamics Research branch of the Met. Office, Exeter, UK.Crown Copyright 2010.

References Barcilon V1964J. Atmos. Sci. 21291–299.Barnes R T Het al.1983Proc R Soc London A38731–73.Bell M J 1994Q. J. R. Meteorol Soc.120195–213.Bell M J and White A A1988J. Atmos. Sci.452571–2590.Bell M Jet al.1991Phil Trans R Soc London33455–92.Board on Atmospheric Sciences and Climate2005Improving the Scientific Foundation of Atmosphere-Land-Ocean Simulations: Report of a Workshop(NationalAcademiesPress).Charney J G1948Geophys. Publ.171–17.Davies T et al. 2005Q. J. R. Meteorol Soc.1331993–2010.Dietrich D E1973Pageoph1091826–1861.Dutton J A1995Dynamics of Atmospheric Motion(DoverPublications).Eady E T1949Tellus133–52.Geisler J E et al.1983J. Geophys Res.889706–9716.Gill A E 1982Atmosphere-Ocean Dynamics(AcademicPress).Hide R1968inThe Global Circulation of the Atmosphere ed.GACorby(RoyalMeteorologicalSociety,London)196–221.Hide R2010ApathofdiscoveryingeophysicalfluiddynamicsA&G514.16–4.23.Hide R and Mason P J 1975Advances in Physics2447–100.Hide R and Mason P J 1970Phil Trans R Soc London A268201–232.Hide R et al.1980Nature286114–117.Hide R et al. 1997J. Geophys Res.10216423–16438.Hignett P et al.1985Q. J. R. Meteorol Soc. 111131–154.Holton J R1992An Introduction to Dynamic Meteorol-ogy3rdEdition(AcademicPress).Houghton J2002The Physics of the Atmosphere(CambridgeUniversityPress).James I N1994Introduction to Circulating Atmospheres(CambridgeUniversityPress).James I Net al.1981Q. J. R. Meteorol Soc. 10751–78.Lorenz E N1967Laboratorymodelsoftheatmo­sphere,chapterVIinThe Nature and Theory of the General Circulation of the Atmosphere(WMO,Geneva).Marshall J C and Molteni F1993J Atmos Sci501792–1818.Mason P J1975Phil. Trans R. Soc. London A 278397–445.Pedlosky J1987Geophysical Fluid Dynamics(Springer­Verlag).Pérez-Pérez Eet al.2010Ocean Modelling32118–131.Quon C1976J. Comp. Phys20442–479.Read P L2003J. Fluid Mech.489301–323.Read P Let al. 2000Monthly Weather Review1282835–2852.Staniforth A and Wood N2008J. Comp. Phys2273445–3464.Stensrud D J2007Parameterization Schemes: Keys to Understanding Numerical Weather Prediction Models(CambridgeUniversityPress).Tucker P G and Long C A1996 Int. J. Num. Meth. Heat Fluid Flow639–50.Vallis G K2007Atmospheric and Oceanic Fluid Dynam-ics: Fundamentals and Large Scale Circulation(CUP).White A A1988Meteorological Magazine11754–63.White A A1990Meteorological Magazine1191–9.White A A2002inLarge-Scale Atmosphere-Ocean Dynamics I: Analytical Methods and Numerical ModelsedsJNorburyandIRoulstone(CUP)1–100.Williams G P1972Monthly Weather Review10029–41.Williams G P1974J. Fluid Mech.62643–655.Williams P Det al.2010 J. Fluid Mech. 649187–203.

2: Upside and downside of the QG model

advantages disadvantages

Representsanaturalfirststepbeyondthediagnosticgeostrophicapproximationintothetime­dependentrealm.Givesutilityandscientificcredibilitytothenotionofgeostrophy.

Quasi­geostrophyassumed(smallRossbynumberRo,Burgernumberoforderunity).

Conservationpropertiesareremarkablygood(incomparisonwiththoseofmostothermodelsofnearlygeostrophicflow).

Goodconservationpropertiesrequirerestrictionsonthespatialvariationsofmean­statebuoyancyfrequencyNandCoriolisparameterf.

Givesaunifyingarenaformanykeyconcepts,e.g.internal/externalRossbywaves,barotropicandbaroclinicinstability,omegaequation,geostrophicturbulence,quasi­steadyflowstructures,flowregimebehaviour,potentialvorticitydiagnostics(staticandprognostic).

Partlybecauseofthetwodisadvantagesoutlinedabove,QGisnoteasytoapplydirectlyindiagnosticstudiesofatmosphericdataoroutputofnon­QGnumericalmodels.

Retainsthenonlinearityofadvection(bythegeostrophicflow)butismorecompactandmathematicallytractablethanotherapproximatetime­dependentmodels;givesusefultheoreticalresultswithoutuseofadvancedmathematicaltechniques.

Manyapproximatemodelshavebeenproposedthatareformallymoreaccurate(undervariousconditions).

Viaitsdiagnosticellipticomegaequation,givesinsightintotheoccurrenceofverticalmotioninlarge­scaleatmosphericflow.

Revealsthepossibilityofquasi­steadyaswellastime­dependentflowsundernearly­geostrophicdynamics.

Manytechniquesthatareneededin(e.g.)dataassimilationandensembleforecastingfor/withnon­QGmodelscanbeexercisedandexaminedintheQGmodel(andbehaviourevenintheQGcontextmaybecontroversialornotwidelyappreciated).