13 13 1 Introduction The Amplitude As a ubiquitous source of motion both astrophysical and geophysical convection has attracted theoretical attention since the last century In the ocean many different scales are called convection from the deep circulation due to the seasonal production of Arctic bottom water see chapter 1 to micromixing of salt fingers see chapter 8 In the atmosphere convection dominates the flow from subcloud layers to Hadley cells It is proposed that convection in the earth s core powers the geomagnetic field The nonperiodic reversals of that field captured in the rock define the evolution of the ocean basin Recent recognition that this latter process is caused by convection in the mantle has produced a new geophysics In the past understanding the central features of convection has come from the isolation of simplest mechanistic examples Although large scale geophysical convection never coincides with the idealized simplest problem these examples e g Lord Rayleigh s study of the Btnard cells have generated much of the formal language of inquiry used in the field Students of dynamic oceanography have favored this formal language mixed in equal parts with more pragmatic engineering tongues when interpreting oceanic convective processes Speculations beyond these mathematically accessible problems take the form of hypotheses experiments and numerical experiments in which one seeks to isolate the central processes responsible for the qualitative and quantitative features of fully evolved flow fields The many facets of turbulent convection represent the frontier This chapter reviews only a narrow path toward that frontier This path is aimed at an understanding of the elementary processes responsible for the amplitude of convection in the belief that quantitative theories permit the theorist the least self deception Of course the heat flux due to a prescribed thermal contrast like the flow due to a given stress has been observed for a century The relation between force and flux has been rationalized with models emerging largely from linear theory and kinetic theory in particular with the use of observationally determined eddy conductivities estimated for the oceans in Sverdrup Johnson and Fleming 1942 Early theoretical interpretations of oceanic transport processes that go beyond these simple beginnings were explored by Stommel 1949 while current usage and extensions of mixing theories are discussed in chapter 8 Central to the most recent of such proposals is the idea that some large scale of the motion or density field is steady or statistically stable while turbulent transport due to smaller scales can be parameterized Changing the amplitude of the small scale transports is pre of Convection Willem V R Malkus 384 Willem V R Malkus I 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 model building and its quasi linear form is used on smallscale phenomena as well from inviscid marginal stability for the purpose of quantifying aspects of the wind mixed layer Pollard Rhines and Thompson 1973 to viscous marginal stability for the purpose of quantifying double diffusion Linden and Shirtcliffe 1978 It has not yet been possible to establish either the limits of validity or generalizability of this quasi linear use of marginal stability in the geophysical setting There can be little doubt that it is incorrect that fluids typically are destabilized by the extreme fluctuations yet it appears to be the only quantifying concept of sufficient generality to have been used in oceanic phenomena from the largest to the smallest scales Of course our idealizations in the realm of geophysics are all incorrect We turn to observation to establish in what sense and in what degree these idealizations are good first order descriptions of reality This chapter explores the hierarchy of quantifying idealizations in convection theory The quasi linear marginal stability problem is drawn from the full formal statement for stability of the flow A theory of turbulent convection based on marginal stability is presented incorporating both the qualitative features determined by inviscid processes and the quantitative aspects determined by dissipative processes Observations provide better support for both the quantitative and qualitative results from quasi linear marginal stability theory than might have been anticipated encouraging its continued application in the oceanic setting 13 2 Basic Boussinesq Description The primary simplification that permitted mathematical progress in the study of motion driven by buoyancy was the Boussinesq statement of the equations of motion In retrospect the central problem was to translate the correct energetic statement leading terms in an asonic asymptotic expansion away from a basic adiabatic hydrostatic temperature distribution This expansion is usually made in two small parameters one is the ratio of the height of the convecting region to the total adiabatic depth of the fluid while the second is the ratio of the superadiabatic temperature contrast across the convecting region to the mean temperature In suitably scaled variables the leading equations of the expansion are V u O 13 1 1 Du Dr VP V2u RaTk 13 2 DT V 2T 13 3 Dt where D a Dt at v v Ra yATd3 K KV k is the unit vector in the antidirection of gravitational acceleration d the depth of the convecting region AT the superadiabatic temperature contrast K the thermometric conductivity of the fluid and v is its kinematic viscosity Ra the Rayleigh number and a the Prandtl number Other symbols are defined above The Boussinesq equations retain the principal advective nonlinearity but have no sonic solutions Higher order equations are linear and inhomogeneous forced by the lower order solutions The most accessible problem in free convection has been the study of motion in a horizontal layer of fluid bounded by good thermal conductors at prescribed temperatures Such a layer is the thermal equivalent of the constant stress layer in shear flow This is seen by taking an average over the horizontal plane of each term in the heat equation One writes from 13 3 aT at az aT az 13 4 u VP where the overbar indicates the horizontal average For steady or statistically steady convection T at vanishes and one may integrate 13 4 twice to obtain into the approximate form Nu yWT 0 where u is the vector velocity of the fluid P
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