Raymond Hide

Atmospheric Angular Momentum Fluctuations, Length-of-Day Changes and Polar Motion

Variations in the distribution of mass within the atmosphere and changes in the pattern of winds, particularly the strength and location of the major mid-latitude jet-streams, produce fluctuations in all three components of the angular momentum of the atmosphere on timescales upwards of a few days. In a previous study (Hide et al. 1980) it has been shown that variations in the axial component of atmospheric angular momentum during the Special Observing Periods in 1979 of the First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Program) are well correlated with changes in length-of-day. This would be expected if the total angular momentum of the atmosphere and ‘solid’ Earth were conserved on short timescales (allowing for lunar and solar effects) but not if angular momentum transfer between the Earth’s liquid core and solid mantle, which is accepted to be substantial and even dominant on timescales upwards of several years, were significant on timescales of weeks or months. Fluctuations in the equatorial components of atmospheric angular momentum should contribute to the observed wobble of the instantaneous pole of the Earth’s rotation with respect to the Earth’s crust, but this has not been shown conclusively by previous studies. In this paper we re-examine some aspects of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. Currently available meteorological data are not adequate for evaluating the usual wobble excitation functions accurately, but we show that partial integration leads to an expression involving simpler functions ─ here called ‘equatorial angular momentum functions’ ─ which can be reliably evaluated from available meteorological data. The length-of-day problem is treated in terms of a similar ‘axial angular momentum function’ ; and ‘effective angular momentum functions’ are defined in order to allow for rotational and surface loading deformation of the Earth. Daily values of these atmospheric angular momentum functions have been calculated from the ‘initialized analysis global database’ of the European Centre for Medium-Range Weather Forecasts (ECMWF). They are presented for the period 1 January 1981─30 April 1982, along with the corresponding astronomically observed changes in length-of-day and polar motion, published by the Bureau International de l’Heure (BIH). Changes in length-of-day during this period can be accounted for almost entirely by angular momentum exchange between the atmosphere and solid Earth, and the existence of a persistent fluctuation in this exchange, with a timescale of about 7 weeks, is confirmed. We also demonstrate that meteorological phenomena provide an important contribution to the excitation of polar motion. Our work offers a theoretical basis for future routine determinations of atmospheric angular momentum fluctuations for the purposes of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.

Free Hydromagnetic Oscillations of the Earth's Core and the Theory of the Geomagnetic Secular Variation

Free hydromagnetic oscillations of a rotating spherical shell of an incompressible fluid are investigated by means of a simple theoretical model. For each spatial harmonic, rotation gives rise to two distinct modes of oscillation, ‘magnetic’ and ‘inertial’, which propagate with different velocities. As an application of the theory, it is shown that if the strength of the toroidal magnetic field in the Earth’s core is 100 Oe, then many of the properties of the observed secular changes, including the slow westward drift, of the main geomagnetic field at the Earth’s surface can be accounted for in terms of the interaction of magnetic modes in the core with the Earth’s poloidal magnetic field. Concomitant magnetic variations due to inertial modes in the core would, owing to their relatively short periods (several days), fail to penetrate to the surface of the Earth, although the eddy currents induced in the lower mantle by these modes might affect the mechanical coupling between the mantle and the core.

Sloping convection in a rotating fluid

Abstract Laboratory experiments on thermal convection in a fluid which rotates about a vertical axis and is subject to a horizontal temperature gradient show that when the rotation rate Ω exceeds a certain critical value ΩR (which depends on the acceleration of gravity, the shape and dimensions of the apparatus, the physical properties of the fluid and the distribution and intensity of the applied differential heating) Coriolis forces inhibit overturning motion in meridian planes and promote a completely different type of flow which has been termed ‘sloping convection’ or ‘baroclinic waves’. The motion is then non-axisymmetric and largely confined to meandering ‘jet streams’, with trajectories of individual fluid elements inclined at only very small (though essentially non-zero) angles to the horizontal. The kinetic energy of the waves derives from the interaction of slight vertical motions with the potential energy field maintained by differential heating, and it is dissipated by friction arising largely...

Earth's variable rotation

Recent improvements in geodetic data and practical meteorology have advanced research on fluctuations in the Earths rotation. Theinterpretation of these fluctuations is inextricably linked with studies of the dynamics of the Earth-moon system and dynamical processes in the liquid metallic core of the Earth (where the geomagnetic field originates), other parts of the Earths interior, and the hydrosphere and atmosphere. Fluctuations in the length of the day occurring on decadal time scales have implications for the topography of the core-mantle boundary and the electrical, magnetic, and other properties of the core and lower mantle. Investigations of more rapid fluctuations bear on meteorological studies of interannual, seasonal, and intraseasonal variations in the general circulation of the atmosphere and the response of the oceans to such variations.

An experimental study of thermal convection in a rotating liquid

An investigation is described of the hydrodynamical flow that ensues when a liquid which rotates uniformly at O about a vertical axis is subject to a horizontal temperature gradient. Although O was sufficiently large for primary effects due to Coriolis forces to arise, centripetal forces never exceeded a small fraction of those due to gravity. Laboratory investigations of this type are of some geophysical interest. They may have a direct bearing on the study of the general atmospheric circulation, and with suitable extensions they may eventually lead to a better understanding of the hydrodynamical flow which is supposed to occur in the earth’s liquid core, where the geomagnetic field originates. Water, the only liquid which was used, filled the annular space between two concentric cylinders of radii a and b (b > a)to a depth of d cm. depth of d cm. The cylinders were maintained at different temperatures MATHS FORMULA. The general properties of the flow depend on the value of a certain parameter Θ≡2gd[ρ(Ta)−ρ(Tb)]/Ω2(b−a)2[ρ(Tb)+ρ(Ta)] where g is the acceleration of gravity and p(T) is the density of water at temperature T. re T. When 0 exceeds a certain value,O the flow is essentially a meridional circulation, in which the motion perpendicular to the axis of rotation is deflected by Coriolis forces. When 0 is somewhat less than O. the flow is characterized by a regular quasi-horizontal wave-like pattern in which the motion is almost but not entirely confined to a thin meandering ‘jet’ stream. The transition between these two regimes of flow takes place quite sharply when 0 = 0 Crit. = 1*58 +- 0*05. The train of waves in the wave-flow regime drifts relative to the rotating system at a uniform angular rate, in the same general direction as that of the flow in the top surface ‘jet ’ stream. The wave number m increases as 0 decreases, until a certain point is reached, corresponding to an amplitude to wavelength ratio of about two-thirds, when no further increase in m can be produced by reducing 0. At this point a steady repeating fluctuation of the flow pattern occurs. This phenomenon has been termed ‘vacillation’. At even smaller values of 0 the flow is ‘turbulent’ in the sense that rapid and complicated fluctuations occur. These flow phenomena appear to have their counterparts in the general atmospheric circulation. Specific investigations are described, including heat transport measurements and a study of the thermal structure of a typical flow field. Theoretical considerations lead to an interpretation of the meaning of 0, which is tentatively identified with appropriate Rossby and Richardson numbers, and some of the results of theories due to Davies (1956) and Kuo (1953) are compared with the experimental measurements. A certain measure of agreement is found.

Thermal Convection in a Rotating Annulus of Liquid: Effect of Viscosity on the Transition Between Axisymmetric and Non-Axisymmetric Flow Regimes

Abstract When a vertical rotating annulus of liquid is subject to a horizontal temperature gradient, provided that the coefficient of kinematical viscosity, ν¯, is not too great and the angular velocity of rotation,Ω is sufficiently high, four distinct regimes of hydrodynamical flow are possible, as shown in previous work by Hide. Only one of these regimes is characterized by symmetry about the axis of rotation. The principal properties of the flow depend largely on the dimensionless parameters Π2≡d/(b−a), Π4gdΔρ/ρ¯Ω2(b−a)2, Π5≡4Ω2(b−a)5/ν¯2d and Π6≡ν¯/κ¯, where d is the depth of the fluid,b and a are the radi of curvature of the surfaces of the annulus, g is the acceleration of gravity, ρ¯ is the mean density of the fluid, Δρ is the density contrast associated with the impressed horizontal temperature gradient and κ¯ is the thermal diffusivity of the fluid. In a diagram with log10Π5 as abscissa and log10Π4 as ordinate, axisymmetric flow is found outside an anvil-shaped region whose upper boundary lies be...

Detached shear layers in a rotating fluid

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω 0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω 1 rad/s a disk of radius a cm and thickness b ’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω 0 , Ω 1 and a , ranges of Rossby number, the modulus of e ≡ (Ω 1 + Ω 0 )/½ (Ω 1 + Ω 0 ), from 0·01 to 0·3, and Ekman number, E ≡ 2 v / a 2 (Ω 1 + Ω 0 ), from 10 −5 to 5 × 10 −4 were attained. Although the apparatus was axisymmetric, only when |e| did not exceed a certain critical value, |e T |, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |e| > |e T |, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when e was positive, and of a blunt ellipse when e was negative. The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when e is positive, depend in relatively simple ways on e and E . The dependence of|e T | on E can be expressed by the empirical relationship |e T | = AE n , where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing e.

Hydromagnetics of rotating fluids

The author reviews work on the dynamics of a rapidly rotating electrically conducting fluid in the presence of a corotating magnetic field. While the separate action of either rotation or a magnetic field produces strong dynamical constraints, their simultaneous action can result in comparatively weak net constraints and novel phenomena then arise. A systematic account of these phenomena is given and certain applications to natural systems (with emphasis on the dynamics of the Earths liquid core) are outlined.

Motions of the Earth's Core and Mantle, and Variations of the Main Geomagnetic Field.

Theoretical work on the magnetohydrodynamics of the earths liquid core indicates (a) that horizontal variations in the properties of the core-mantle interface that would escape detection by modern seismological methods might nevertheless produce measurable geomagnetic effects; (b) that the rate of drift, relative to the earths surface, of nonaxisymmetric features of the main geomagnetic field might be much faster than the average zonal speed of hydrodynamic motion of core material relative to the surrounding mantle; and (c) why magnetic astronomical bodies usually rotate. Among the consequences of (a) and (b) are the possibilities that (i) the shortest interval of time that can be resolved in paleomagnetic studies of the geocentric axial dipole component of the earths magnetic field might be very much longer than the value often assumed by many paleomagnetic workers, (ii) reversals in sign of the geomagnetic dipole might be expected to show some degree of correlation with processes due to motions in the mantle (for example, tectonic activity, polar wandering), and (iii) variations in the length of the day that have hitherto been tentatively attributed to core motions may be due to some other cause.

On source-sink flows in a rotating fluid

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the ( x, y )-plane, the m rigid side walls and the two rigid end walls being in contact with the fluid. Here ( x, y, z ) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container at Q cm 3 /sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocity u = −∇ϕ − ∇ × A , is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects on u of uniform rotation of the whole system with angular velocity Ω about the z -axis when the normal component of u on the side walls is independent of z. In the simplest conceivable case, D ≡ z u − z l is infinite (but D/Q remains finite). End effects are then negligible and u is everywhere independent of z. The solenoidal component of u , − ∇ × A , corresponds to j gyres, one for each of the j irreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within the m -ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇ 2 ϕ = 0, depends on Q but not on Ω, j and v (the coefficient of kinematic viscosity), ∇ × A depends on all these quantities but vanishes identically when j Ω = 0. When j Ω ≠ 0 but v → 0, ∇ 2 A + 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form. End effects cannot be ignored when D is finite. When D is independent of x and y and equal to D 0 (say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3( v /Ω) ½ (δ [Lt ] D 0 ), the end effects that then arise are only confined to these boundary layers when j = 0. When j ≠ 0 boundary-layer suction influences the flow everywhere; thus ∇ 2 A and ∇ϕ (but not ∇ × A ) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇ 2 A being the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δ s (typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δ s . When D is not everywhere independent of x and y , non- viscous end effects arise which produce relative vorticity in the main body of the fluid even when j = 0. Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.