Potential Vorticity and Potential Magnetic Field

PROUDMAN-TAYLOR THEOREM*

Raymond Hide

Department of Mathematics, Imperial College London.

* Encyclopaedia of Geomagnetism and Palaeomagnetism (eds. D. Gubbins & E. Herrero- Bevera), pages 852-853, Springer Geosciences, New York: Springer Verlag(2007)

Geophysicists accept a suggestion traceable back to a paper by Elsasser (1939) that the near-alignment of the main geomagnetic field (qv) with the Earth’s geographic polar axis is a manifestation of the influence on convective motions in the liquid, metallic outer core (qv) that is exerted by gyroscopic (Coriolis) forces associated with the Earth’s diurnal rotation. Such forces render core motions anisotropic and highly sensitive to the mechanical and other (thermal and electrical) boundary conditions imposed on the motions by the underlying solid inner core (qv) and the overlying mantle (qv) (for further references see Hide, 2000 (hereafter H00)). The influence of Coriolis forces on motions in spinning fluids is most pronounced in mechanically-driven flows satisfying the so- called Proudman-Taylor (PT) theorem. Stated in words, the theorem (see equation (5) below) shows that ‘slow and steady hydrodynamical motion of an inviscid (and electrically-insulating) fluid of constant density that is otherwise in steady ‘rigid-body’ rotation will be the same in all planes perpendicular to the axis of rotation’. Given in a paper by J. Proudman in 1916, the PT theorem was successfully tested by means of a laboratory experiment in which G. I. Taylor investigated the relative flow produced in a tank of water in rapid rotation about a vertical axis by the slow and steady horizontal motion of a solid body through the water (see Greenspan, 1968 (hereafter G68)). In accordance with the theorem, the flow was highly two-dimensional nearly everywhere and the

1 moving solid object carried with it a ‘Taylor column’ of water extending axially throughout the whole depth of the tank. We note here in passing (a) that the theorem had appeared much earlier in the literature, in a paper on tidal theory by S. S. Hough (Gill, 1982 (hereafter AG82)), and (b) that some writers find it convenient to refer to it as the ‘Proudman theorem’ (Hide, 1977 (hereafter H77)), to avoid confusion with another theorem, due to G. I. Taylor, concerning effects of Coriolis forces on fluid motions (see G68; Hide, 1997). Convective motions in the core (qv) cannot satisfy the PT theorem exactly because they are driven by buoyancy forces due to the action of gravity on density inhomogeneities, which give rise to axial variations of the horizontal components of the relative flow velocity (see equation (6) below). Core motions are also influenced by Lorentz forces associated with electric currents and magnetic fields in the core, which in regions well below the core-mantle boundary (qv) may be comparable in strength with Coriolis forces (see equations (1) and (2) below). But – as in theoretical work in dynamical meteorology and oceanography, see e.g. AG82; Pedlosky, 1987 (hereafter P87) – certain basic dynamical processes and phenomena can be elucidated by means of simplified theoretical models in which axial variations of the non-axial components of the relative flow velocity are neglected in the first instance. Characteristic shear waves that occur in fluids subject to Coriolis and/or Lorentz forces – such as Rossby waves, Alfvén (magnetohydrodynamic (qv)) waves and related hybrid waves (one important type of which may underlie the geomagnetic secular variation (qv)) – have been investigated in this way. And so have other phenomena, such as torsional oscillations of the core-mantle system associated with angular momentum transfer within the Earth’s core (Hide et al., 2000) and differential rotation produced in a rapidly-rotating spherical fluid annulus by potential vorticity (qv) mixing (Hide & James, 1983)

2 Geostrophic and magnetostrophic flows

Flows satisfying the PT theorem belong to a wider class of ‘geostrophic’ flows (see AG82; P87), for which the horizontal component of the local pressure gradient, grad p, is closely balanced by the horizontal component of the Coriolis force (per unit volume), 2ρΩxu (see equation (1)), but torques produced by buoyancy forces associated with horizontal density gradients give rise to systematic axial variations in the horizontal components of u (see equations (5) & (6) below). Consider a moving element of fluid of density ρ which at time t is located at a general point P with vector position r in a frame of reference that rotates with angular velocity Ω relative to an inertial frame. Newton’s second law is conveniently expressed as follows (see Hide, 1971 (hereafter H71)):-

2ρΩxu + gradp + ρ gradV + A = 0 (1) where the Eulerian flow velocity at P and the corresponding pressure and the potential due to gravitational plus centripetal effects are denoted by u, p, and V respectively. The ‘ageostrophic’ term, A, in equation (1) is defined as follows:

A = ρ[∂u/∂t + (u.grad)u – rxdΩ/dt] + curl(νρω) – jxB (2) if ω = curlu, the relative vorticity vector at P, and ν denotes the coefficient of kinematic viscosity. The term jxB in equation (2) is needed when dealing with electrically-conducting fluids; it represents the Lorentz force per unit volume acting on the fluid element if j is the electric current density and B the magnetic field at P. The flow is said to be geostrophic in regions where A is negligible in comparison with other terms in equation (1). Within the Earth’s core (qv), such regions probably arise in its upper reaches (LeMouël, 1984; Hide, 1995) where the toroidal part of the

3 geomagnetic field (qv) is no stronger than the poloidal part (qv) and the corresponding Lorentz term is typically no more than a few parts percent of the Coriolis term in magnitude. On the other hand, the flow is said to be in ‘magnetostrophic’ (rather than geostrophic) balance in any regions where the Lorentz force provides the main contribution to A and is comparable in magnitude with 2ρΩxu. Such regions probably arise at depth within the core, where the toroidal part of the magnetic field may be an order of magnitude stronger than the poloidal part. Following H71, now consider the balance of torques acting on the moving fluid element by taking the curl of equation (1), thereby obtaining the so-called ‘vorticity’ equation in the useful form

(2Ω.grad)(ρu) + gradV x gradρ + curlA + 2ΩC = 0. (3)

Here C = ∂ρ/∂t which, by the mass continuity equation, satisfies

div(ρu) + C = 0. (4)

The PT theorem follows immediately from equation (3), for when the conditions A = 0, grad ρ = 0 and C = 0 are satisfied, the equation reduces to

(2Ω.grad)u = 0, (5) implying that all three components of u are then independent of the axial coordinate. When the restriction to cases of fluids of constant density is relaxed we have

(2Ω.grad)(ρu) + gradV x gradρ = 0 (6) in place of equation (5). Flow patterns satisfying equations (5) or (6) are constrained by Coriolis forces to have no circulation in (meridian) planes containing the axis of rotation. But by equation (6) the horizontal

4 component of u is not independent of the axial direction, except in regions where grad ρ has no horizontal component. We note here in passing (a) that the specific helicity, u.ω, –a quantity of importance in dynamo theory (qv) – of flows satisfying equation (6) is generally non-zero (Hide, 1976), and (b) that with certain geometrical simplifications equation (6) leads to the meteorologist’s ‘thermal wind’ equation (qv) relating the rate of increase of the speed and direction of the geostrophic wind with height in the atmosphere to the horizontal gradient of temperature.

Ageostrophic effects and detached shear layers.

The full expression of the laws of mechanics given by equation (1) is, of course, prognostic, in the sense that when solved simultaneously with the full equations of thermodynamics and electrodynamics under all the relevant boundary conditions the equation gives the fields of all the dependent variables, p, u, ρ, j, B etc. But the expression for geostrophic flow to which equation (1) reduces in regions where the ageostrophic term, A, is negligible is diagnostic (rather than prognostic), in the sense that it provides useful relationships between u, p and ρ, but it cannot give full solutions satisfying all the relevant boundary conditions. It follows that flows in real systems cannot therefore be geostrophic everywhere. Regions of ageostrophic flow involving length scales so short that A is comparable in magnitude with 2ρΩxu must be present not only on bounding surfaces (in Ekman- Hartmann and Stewartson boundary layers (qv)) but also within the main body of the fluid, in ‘detached shear layers’. It is within such regions, where effects due to Coriolis forces are countered by strong ageostrophic effects, that meridional circulation can occur. Detached shear layers formed the ‘walls’ of the so-called ‘Taylor column’ of fluid that remained attached to the moving solid body in G. I. Taylor’s experiment. Geophysical examples of ageostrophic detached shear layers embedded in geostrophic flows are the jet

5 streams and their associated frontal systems seen in the Earth’s atmosphere and ‘western boundary’ currents such as the Gulf Stream in the Atlantic Ocean and the Kuroshio Current in the Pacific Ocean (see AG82; P87). Amongst the various laboratory investigations that were stimulated directly by the original ‘Taylor column’ experiment were several studies of the effect of the inner spherical boundary on patterns of mechanically-driven (G68; H77) and buoyancy-driven flows in a rotating spherical annulus of fluid (qv). These studies clearly demonstrated what general arguments based on equation (5) or equation (6) predict, namely that near the imaginary ‘tangent cylindrical surface’ (qv) in contact with the equator of the inner spherical surface and extending axially throughout the fluid, a detached shear layer would form inhibiting mixing of fluid in the ‘polar’ regions inside the cylinder with fluid in the ‘equatorial’ region outside the cylinder. Such behaviour is thought to bear on core dynamics (qv) and the influence of the solid inner core (qv) on structure of the geomagnetic field (qv) (for further references see H00 and Hide, 1966a,b).

Bibliography

Elsasser, W. M. (1939) On the origin of the Earth’s magnetic field. Phys. Rev., 55, 489-497. Gill, A. E. (1982) Atmosphere-Ocean Dynamics. New York: Academic Press; cited as AG82. Greenspan, H. P. (1968) The Theory of Rotating Fluids. Cambridge University Press; cited as G68. Hide, R. (1966a) The dynamics of rotating fluids and related topics in geophysical fluid dynamics. Bull. Amer. Meteorol. Soc., 47, 873-885. –––––– (1966b) Free hydromagnetic oscillations of the Earth’s core and the theory of the geomagnetic secular variation. Philos. Trans. Roy. Soc. London, A259, 615-647.

6 –––––– (1971) On geostrophic motion of a non-homogeneous fluid. J. Fluid Mech., 49, 745-751; cited as H71. –––––– (1976) A note on helicity. Geophys. (Astrophys.) Fluid Dyn., 7, 157-161. –––––– (1977) Experiments with rotating fluids; Presidential address. Quart. J. Roy. Meteorol. Soc., 103, 1-28; cited as H77. –––––– (1995) The topographic torque on the rigid bounding surface of a rotating gravitating fluid and the excitation of decadal fluctuations in the Earth’s rotation. Geophys. Res. Letters, 22, 1057-1059. –––––– (1997) On the effects of rotation on fluid motions in containers of various shapes and topological characteristics. Dyn. Atmos. Oceans, 27, 243-256. –––––– (2000) Generic nonlinear processes in self-exciting dynamos and the long-term behaviour of the main geomagnetic field, including superchrons. Philos. Trans. Roy. Soc., A358, 943-955; cited as H00. Hide, R., Boggs, D. H. & Dickey, J. O. (2000) Angular momentum fluctuations within the Earth’s core and torsional oscillations of the core-mantle system. Geophys. J. International, 143, 777- 786. Hide, R. & James, I. N. (1983) Differential rotation produced by large-scale potential vorticity mixing in a rapidly-rotating fluid. Geophys. J. Roy. Astron. Soc., 74, 301-312. LeMouël, J-L. (1984) Outer core geostrophic flow and the secular variation of the Earth’s magnetic field. Nature, 311, 734-735. Pedlosky, J. (1987) Geophysical Fluid Dynamics (second edition), New York: Springer Verlag; cited as P87.