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On Some Global Problems of Geophysical Fluid Dynamics (A Review)T (Gravitational Differentiation/Rotational Pole/Mantle Convection) A

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On Some Global Problems of Geophysical Fluid Dynamics (A Review)T (Gravitational Differentiation/Rotational Pole/Mantle Convection) A Proc. Nati. Acad. Sci. USA Vol. 75, No. 1, pp. 34-39, January 1978 Geophysics On some global problems of geophysical fluid dynamics (A Review)t (gravitational differentiation/rotational pole/mantle convection) A. S. MONIN P. P. Shirshov Institute of Oceanology, Academy of Sciences of the Union of Soviet Socialist Republics, Moscow, Union of Soviet Socialist Republics Contributed by A. S. Monin, September 28,1977 The development of the earth sciences during recent years has of the moon, the sun, and other planets are not taken into con- been associated with a wide use of geophysical fluid dynamics sideration. These simplifications do not distort too much the (i.e., the dynamics of natural flows of rotating baroclinic class of motions inside the earth that we are interested in; if stratified fluids), a discipline whose scope includes turbulence necessary, the simplifications can be removed by the intro- and vertical microstructure in stably stratified fluids, gravity duction of appropriate corrections. Equations of motion are waves at the interface, internal gravity waves and convection, written for each of the layers, with the right-hand parts con- tides and other long-period waves, gyroscopic waves, the hy- taining volume forces associated with the presence of the drodynamic theory of weather forecasting, circulations of the gravity and magnetic fields and stresses of the viscous and planetary atmospheres and oceans, the dynamo theory of the magnetic origin. A magnetohydrodynamic equation of the generation of planetary and stellar magnetic fields, etc. magnetic field evolution (induction equation) should be added to the equations of motion. The energetics of the processes in the layers under consideration can be represented by mutual POTENTIAL VORTICITY transformations of kinetic, magnetic, and internal energies. The One of the basic notions in geophysical fluid dynamics, com- equation for kinetic energy density of each of the layers contains bining fluid rotation and stratification nonlinearly, is the po- the terms describing mutual transformations of kinetic and tential vorticity potential energy in the gravity field and viscous dissipation of kinetic energy. The equation for energy density of the magnetic Q = p-1 rot u grad n, field contains in its right-hand side the terms describing ohmic in which u is the absolute velocity of fluid motion, p the density, dissipation of the magnetic field energy (i.e., joule heat gen- v the specific entropy; Q is the invariant of fluid particles in their eration) and mutual transformations of kinetic and magnetic adiabatic motions (see, for instance, ref. 1). The evolution of the energy owing to the action of Maxwell field stresses (pondero- Q and q fields is the main object of study in the hydrodynamic motive forces). The thermodynamic equation for the density theory of weather forecasting [2, 3]. of internal energy contains the terms describing the dissipation The similar object in magnetic field dynamics is the "mag- of viscous and magnetic stresses, heat flux generated by con- netic field charge" ductivity of the medium, external heat sources inside the earth (including radioactivity), and energy dissipation of the pre- X = p-1H grad I, cession and tidal motions. Integrating these equations over the in which H is the magnetic field intensity; the quantity X is also volumes of the mantle, the liquid layer, and the solid core (the the adiabatic invariant of moving particles. The evolution of indices introduced further correspond to this succession) results the If field should be the principal object in the dynamo theory in nine expressions for kinetic K, magnetic M, and internal E of the generation of magnetic fields of celestial bodies (1, energy of each of the three layers. A diagram of energy trans- 4-6). formations is shown in Fig. 1 (potential energy of the liquid More detailed equations of the hydrodynamic theory of the layer P2 and energy of the geomagnetic field outside the earth geomagnetic field generation are considered in refs. 7 and 8, MO are also introduced there). The diagram shows the estimates in which the motions inside the Earth are described as made of energy components (in joules/100 years) obtained in ref. 8. up of the mantle rotation (observed by the astronomical One of the most interesting elements of the diagram is the rate methods and usually called the earth's rotation), magnetohy- of transformation K2 -* M2 describing the geomagnetic field drodynamic flows in the liquid layer ("liquid core"), and the generation by the dynamo mechanism in the liquid layer. displacement and rotation of the internal solid core. The mo- tions of these three layers interact due to hydromagnetic stresses GRAVITATIONAL DIFFERENTIATION at the internal and the external boundaries of the liquid layer As the second example of the use of the geophysical fluid dy- and the deformations of the gravity field with the displacements namics let us consider, following refs. 9-13, the problem of of the internal core. The model suitable for the study of the gravitational differentiation of the interior of a spherically dynamics of these three layers was considered in refs. 7 and 8. symmetric and initially homogeneous planet consisting of two It contains three simplifications: (i) rheological-the mantle components: light "mantle" and heavy "core" ones. The and the internal core are assumed to be absolutely solid, which question in essence concerns the Earth's evolution-the for- excludes from consideration some types of internal motions in mation of its core and mantle, the process giving the basic them (elastic oscillations, tidal deformations, and convective to the balance of the The in- motions in the mantle and the continental drift they produce); contribution energy planet (14). (ii) geometric-the mantle and the internal core are assumed Abbreviation: TME, tectono-magmatic epoch. to be spherically symmetric bodies, so that, in particular, their t By invitation. From time to time, reviews on scientific and techno- flattening due to rotation (and hence the ability for precession) logical matters of broad interest are published in the PROCEED- is not taken into account; (iii) dynamic-gravitational effects INGS. 34 Downloaded by guest on September 27, 2021 Geophysics: Monin Proc. Natl. Acad. Sci. USA 75 (1978) 35 FIG. 1. Diagram of the Earth's energy transformation. ternal structure of such a planet at different stages of its evo- with the parameters pi, al, and L chosen from the available lution can be estimated with the aid of the hydrostatic equation, information on the Earth's contemporary structure. The local which, with neglect of rotation, has the form equation of a planetary state is reduced to the form dp= Gm 2i7rGL m = 22Ck1= _p; g=2= 4 pr2dr. p = (p2 a F dr= Jo 272Ca a2ps2); k=ikark Here and further, p is the pressure, p the density, g the accel- The solutions of the hydrostatic equation in every chemically eration r con- of gravity, the radial coordinate, G the gravity homogeneous layer are given by the general integral stant. Under barotropic approximation (assumed for solid planets) the knowledge of the equation of state p = p(p) of 1 ar planetary material is sufficient for integrating the hydrostatic p=-(Asin.++Bcost); = L equation. Taking into account that the density variations in the planets of the solar system lie within a small interval of values The coefficients A and B can be found from the condition of 2 < p < 15 g/cm3, let us approximate the equation of state of continuity of pressure and the acceleration of gravity at the = each of the planetary composing materials by parabolas interfaces. Without loss of generality it can be put a2 1 and ail > 1. Let the planetary "core" material concentration be cl= c; then the "mantle" material concentration will be P = P1,2(P) = a,2Po(P); Po(P) = (P2 + 27rL2) C2= 1 - c. At some stage of the differentiation process char- Downloaded by guest on September 27, 2021 36 Geophysics: Monin Proc. Natl. Acad. Sci. USA 75 (1978) Table 1. Calculated values for various parameters of Earth during its evolution ri, r2, Pc, P+, P, Pc, P 1, II, x km km g/cm3 g/cm3 g/cm3 TPa TPa I/I. 1031 J 0 0 6393 11.34 0.230 - 1.12 0.89 0 0.2 2091 6386 12.38 10.86 7.43 0.289 0.204 1.09 0.92 0.32 0.4 2652 6381 12.93 10.45 6.96 0.323 0.183 1.06 0.94 0.73 0.6 3043 6376 13.38 10.05 6.48 0.351 0.163 1.03 0.97 1.11 0.8 3361 6372 13.75 9.65 6.00 0.375 0.144 1.01 0.99 1.49 0.863 3451 6371 13.86 9.52 5.84 0.382 0.138 1 1 1.61 1 3635 6368 14.08 9.25 5.50 -0.397 0.125 0.98 1.02 1.86 acterized by the fraction x of the already differentiated material proportional to the average concentration of core material in (the core mass is cxM) the average concentrations of the mantle the mantle: - and core material in the mantle are (1- c)/(1 - cx) and (c - cx)/(1 - cx), respectively, and the equation of state for the cM-= 47rkr12 C - CX mantle material has the-parabolic form with-the parameter dt 1 -Cx' in which k ifthe coefficient of proportionality, assumed to be A p ~~1-Cx constant. Integrating this equation simultaneously with the a(3ai; ~=ai(1 - C) + c(1 - X remaining equations of the model gives x(t) for all the planets The radial distributions of density and pressure for an arbitrary of interest here, and the assumption-of their simultaneous for- planet of mass M at the stage x contain five unknown param- mation makes it possible to determine the coefficient k from eters aip,, L/a1, fi, and the radii of the core and the entire the earth's age (4.6 billion years at x = 04863), which turns out planet, ri and r2.
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