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MIT 12 000 - The Amplitude of Convection

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13The Amplitudeof ConvectionWillem V. R. Malkus13.1 IntroductionAs a ubiquitous source of motion, both astrophysicaland geophysical, convection has attracted theoreticalattention since the last century. In the ocean, manydifferent scales are called convection; from the deepcirculation due to the seasonal production of Arcticbottom water (see chapter 1) to micromixing of saltfingers (see chapter 8). In the atmosphere, convectiondominates the flow from subcloud layers to Hadley"cells." It is proposed that convection in the earth'score powers the geomagnetic field. The nonperiodicreversals of that field, captured in the rock, define theevolution of the ocean basin. Recent recognition thatthis latter process is caused by convection in the man-tle has produced a new geophysics.In the past, understanding the central features ofconvection has come from the isolation of "simplest"mechanistic examples. Although large-scale geophysi-cal convection never coincides with the idealized sim-plest problem, these examples (e.g., Lord Rayleigh'sstudy of the Btnard cells) have generated much of theformal language of inquiry used in the field. Studentsof dynamic oceanography have favored this formal lan-guage mixed in equal parts with more pragmatic en-gineering tongues when interpreting oceanic convec-tive processes.Speculations beyond these mathematically access-ible problems take the form of hypotheses, experiments,and numerical experiments in which one seeks to iso-late the central processes responsible for the qualitativeand quantitative features of fully evolved flow fields.The many facets of turbulent convection represent thefrontier. This chapter reviews only a narrow path to-ward that frontier. This path is aimed at an understand-ing of the elementary processes responsible for the am-plitude of convection, in the belief that quantitativetheories permit the theorist the least self-deception.Of course the heat flux due to a prescribed thermalcontrast, like the flow due to a given stress, has beenobserved for a century. The relation between force andflux has been rationalized with models emerginglargely from linear theory and kinetic theory-in par-ticular, with the use of observationally determined"eddy conductivities" (estimated for the oceans inSverdrup, Johnson, and Fleming, 1942). Early theoreti-cal interpretations of oceanic transport processes thatgo beyond these simple beginnings were explored byStommel (1949), while current usage and extensions of"mixing" theories are discussed in chapter 8.Central to the most recent of such proposals is theidea that some large scale of the motion or density fieldis steady or statistically stable, while turbulent trans-port due to smaller scales can be parameterized. Chang-ing the amplitude of the small-scale transports is pre-384Willem V. R. MalkusI __ _ __ _·_ _ _____sumed to lead to a new equilibrium for the large scale,so that the statistical equilibrium is marginally stable.This view lurks behind most traditional oceanic modelbuilding and its quasi-linear form is used on small-scale phenomena as well-from inviscid marginal sta-bility for the purpose of quantifying aspects of the windmixed layer (Pollard, Rhines, and Thompson 1973) toviscous marginal stability for the purpose of quantify-ing double diffusion (Linden and Shirtcliffe, 1978).It has not yet been possible to establish either thelimits of validity or generalizability of this quasi-linearuse of marginal stability in the geophysical setting.There can be little doubt that it is "incorrect"-thatfluids typically are destabilized by the extreme fluc-tuations-yet it appears to be the only quantifying con-cept of sufficient generality to have been used inoceanic phenomena from the largest to the smallestscales. Of course, our idealizations in the realm ofgeophysics are all "incorrect." We turn to observationto establish in what sense and in what degree theseidealizations are good "first-order" descriptions of real-ity.This chapter explores the hierarchy of quantifyingidealizations in convection theory. The quasi-linearmarginal-stability problem is drawn from the full for-mal statement for stability of the flow. A theory ofturbulent convection based on marginal stability is pre-sented, incorporating both the qualitative features de-termined by inviscid processes and the quantitativeaspects determined by dissipative processes.Observations provide better support for both thequantitative and qualitative results from quasi-linearmarginal-stability theory than might have been antic-ipated, encouraging its continued application in theoceanic setting.13.2 Basic Boussinesq DescriptionThe primary simplification that permitted mathemat-ical progress in the study of motion driven by buoyancywas the Boussinesq statement of the equations of mo-tion. In retrospect, the central problem was to translatethe correct energetic statement(u.VP) = ,into the approximate form(yWT)= 0,where u is the vector velocity of the fluid, P the pres-sure, y the coefficient of thermal expansion times theacceleration of gravity, W the vertical component ofvelocity, T the temperature field, the total dissipa-tion by viscous processes in the fluid, and the bracketsa spatial average over the entire fluid. This has beenachieved (e.g., Spiegel and Veronis, 1960; Malkus 1964)by recognizing that the Boussinesq equations are theleading terms in an asonic asymptotic expansion awayfrom a basic adiabatic hydrostatic temperature distri-bution. This expansion is usually made in two smallparameters; one is the ratio of the height of the con-vecting region to the total "adiabatic depth" of thefluid, while the second is the ratio of the superadiabatictemperature contrast across the convecting region tothe mean temperature.In suitably scaled variables, the leading equations ofthe expansion areV'u = O,1 DuDr- = -VP + V2u + RaTk,DT = V2T,Dt(13.1)(13.2)(13.3)whereD aDt atv yATd3v- Ra =K KVk is the unit vector in the antidirection of gravitationalacceleration, d the depth of the convecting region, ATthe superadiabatic temperature contrast, K the ther-mometric conductivity of the


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