Unstable Density Stratification Research Articles

Internal waves in the earth's core

Summary. This paper investigates possible long-period oscillations of the earth’s fluid outer core. Equations describing free oscillations in a stratified, self-gravitating, rotating fluid sphere are developed using a regular perturbation on the equations of hydrodynamics. The resulting system is reduced to a finite set of ordinary differential equations by ignoring the local horizontal component of the earth’s angular velocity vector, S2, and retaining only the vertical component. The angular dependence of the eigensolutions is described by Hough functions, which are solutions to Laplace’s tidal equation. The model considered here consists of a uniform solid elastic mantle and inner core surrounding a stratified, rotating, inviscid fluid outer core. The quantity which describes the core’s stratification is the Brunt-Vaigla frequency N, and for particular distributions of this parameter, analytical solutions are presented. The interaction of buoyancy, and rotation results in two types of wave motion, the amplitudes of which are confined predominantly to the outer core: (1) internal gravity waves which exist when N2 > 0, and (2) inertial oscillations which exist when N2 < 4Q2. For a model with a stable density stratification similar to that proposed by Higgins & Kennedy (1971), the resulting internal gravity wave eigenperiods are all at least 8 hr, and the fundamental modes have periods of at least 13 hr. A model with an unstable density stratification admits no internal gravity waves but does admit inertial oscillations whose eigenperiods have a lower bound of 12 hr.

Finite strain and progressive deformation in models of diapiric structures

The distribution of strain within and around gravitationally produced diapiric structures was studied through the use of experimental models which were deformed in a large-capacity centrifuge. A new method of model construction was developed which is equivalent to building the model of initially square 1-mm elements. After deformation the elements assume shapes which approximate parallelograms and their finite strains can easily be calculated. If several initially identical models are deformed to different extents, the finite strain states of an element in each of the models define points on the deformation path of that element. The deformation path can be used to make estimates of the nature of the internal fabric which would be expected in the equivalent element of the natural structure. This method was applied to the study of the finite strain in diapiric ridges. The models demonstrate that the highest strain is always in the region above the diapir. Within the diapir initial vertical stretching is followed by vertical flattening. Large portions of the structure can be seen to suffer what would in natural examples be called polyphase deformation, even though all of the deformation was due to a single buoyant overturn of unstable density stratification. The strain patterns within the models support the contention that in salt diapirs the buoyant salt has a lower viscosity than the overlying sediments, but that in mantled gneiss domes the reverse is true.

On finite amplitude convection in a rotating magnetic system

The instabilities that can arise in a stratified, rapidly rotating, magnetohydrodynamic system such as the Earth’s core are often thought to play a key role in dynamo theory — that is, in the study of how the magnetic field in the system is maintained in the face of ohmic dissipation. An account of such instabilities is to be found in the M.A.C.- wave theory of Braginsky (1967), who, however, laid his greatest emphasis on the dissipationless modes, an idealization which leads to difficulties described below, ohmic and thermal diffusion is therefore restored, and three key dimensionless parameters are isolated: q , the ratio of thermal to ohmic diffusivities; A, a measure of the relative importance of Coriolis and magnetic forces; and R , a Rayleigh number, which is here the ratio of buoyancy to Coriolis forces. This study concentrates on a particular M.A.C.-wave model originally proposed by Braginsky. It consists of a horizontal layer containing a uniform horizontal magnetic field, B0, and rotated about the vertial, an adverse temperature gradient being maintained on the horizontal boundaries to provide the unstable density stratification. In the rotationally dominant case of large A, the principle of the exchange of stabilities holds, and the motions that arise in the marginal state are steady. The planform of the convection is in rolls orthogonal to B0. If q and A are sufficiently small the principle of the exchange of stabilities remains valid, but the planform consists of one or other of two families of rolls oblique to B0, or a combination of each. If q is large but A is small, the modes are again oblique, but overstability occurs, a type of oscillation which also arises when q is large and A takes intermediate values, although the motion is then in rolls transverse to B0. A theory is developed for the weakly nonlinear convection that arises when R exceeds only slightly the critical value Rc{q, A) at which marginal convection occurs. A critical curve q = ^D(A) is located which roughly divides the (qA) plane into regions of small q and of large q , although when q is large it separates the largeAq from the small. On the one side of the curve, where q or Xq are sufficiently small, it is concluded that, starting from an arbitrary initial perturbation, the convection that arises when R exceeds Rc will ultimately become a completely regular tesselated pattern filling the horizontal plane. On the other side of the curve the situation is considerably more complicated but it is argued that, for sufficiently large q and Aq, subcritical instabilities can occur and that supercritical bursting is likely; that is, the instability that arises from the general initial perturbation will focus into a small spot in a finite time. The relevance of the theory to sunspot formation is discussed. In an appendix, the form of the weakly nonlinear convection that arises when q differs only slightly from qB, and R only slightly from R c, is considered in situations in which the exchange of stabilities holds.

Thermal instability of nonhomogeneous fluids

The present paper is an attempt to investigate theoretically the stability of a continuously stratified layer of an incompressible, inviscid fluid, statically confined between two horizontal boundaries and heated uniformly from above. The basic density stratification is assumed to be unstable and of the exponential type, namely, p = p o e δz ( δ > 0 being a constant) depending only on the vertical coordinate ( z) while the fluid extends indefinitely in the horizontal directions. The principal results established in the paper are as follows: 1. (a) Neutral modes do not exist. 2. (b) The oscillatory modes of the system are damped irrespective of the unstable basic density stratification while the nonoscillatory modes are amplified irrespective of the stable nonadverse 1 1 The heating is qualified as nonadverse since, on account of thermal expansion, the fluid at the bottom will be heavier than the fluid at the top. heating. 3. (c) An upper bound for the modulus of the dimensionless growth rate of an arbitrary unstable mode is (R 2) 1 2 , where R 2 represents a dimensionless numbers corresponding to the gravitational effects of the basic nonhomogeneity of the fluid. 4. (d) The results are compared with the classical Bénard and Rayleigh-Taylor problems for inviscid fluids.

Experimental geodynamical models relating to continental drift and orogenesis

After introductory comments on scale-model theory, descriptions and discussions of experimental models of continental drift and orogenesis are presented. The driving mechanism of the models is the spontaneous overturn in layered systems with gravitationally unstable density stratification. The pattern of the overturn movement is similar to thermal convection currents. A model Pangaea breaks in tension above the upwelling branches of the overturn cells in the model mantle, and the fragments are set adrift by the horizontal limbs of the current. Though buckling occurs along the leading edges of the wandering continental fragments, this fold structure is not of the kind encountered in natural orogens. The latter kind of structures — including basement upheaval, batholith intrusion, nappe formation etc. — is generated in models with no net horizontal shortening across the orogen.

Instability of the Ekman Spiral with Applications to the Planetary Boundary Layers

The Ekman spiral boundary layer occurs in rotating systems near boundary surfaces perpendicular to the rotation axis when the flow is slow compared to the speed of rotation. New results are presented for the instability of the Ekman layer with a free boundary and for shear‐flow instability in the presence of density stratification: (1) For the Ekman boundary layer caused by wind stress on a free water surface the critical Reynolds number is much less than for the case of flow over a smooth rigid boundary. An example of this instability from a laboratory experiment is presented. (2) For the free‐surface case the distortion of the basic Ekman flow by finite‐amplitude motion is very large. (3) With unstable density stratification either shear‐flow instability or thermal convection may dominate according to the relative values of Rayleigh and Reynolds numbers. (4) With stable density stratification the shear‐flow instability may be resonant with internal gravity‐wave oscillations so that damping of the cellular circulations is minimized. Some of the implications of these results for the planetary boundary layers of the atmosphere and ocean are considered.