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1.
Abstract

Laboratory experiments are conducted on the instantaneous release of a constant volume of intermediate density fluid along the interface of a two-layer fluid system in rigid body rotation about a vertical axis. Such a system leads to the development of a thin lens of fluid with anticyclonic motion which grows in radial extent to reach an equilibrium radius as the radial motion is captured by Coriolis effects, becomes unstable to non-symmetric dusturbances, and finally decays due to the action of fluid viscosity. Scaling and dimensional analysis arguments are advanced for the growth, equilibrium and decay phases of the motion. The scaling analysis for the initial growth and equilibrium phases are shown to be in good agreement with the experimental observations. The lens decay data do not collapse under the assumption of a simple Ekman spindown model assuming immiscible fluids. An empirical fit of the data for the decay phase is presented.  相似文献   

2.
Abstract

Flow details inside the buoyant boundary layer in the heat-up process of a contained, stably stratified, fluid are presented. Numerical solutions were obtained for the heatup problem in a cylinder considered by Sakurai and Matsuda (1972). By plotting the scaled vertical velocity W versus the scaled temperature θ as functions of the normal distance from the sidewall, the precise shape of the buoyant layer spiral is constructed. The analogy between this spiral and the Ekman spiral in rotating fluids is apparent. As the Rayleigh number Ra increases, the magnitude of the scaled vertical velocity increases substantially, but the scaled temperature does not vary appreciably. The buoyant layer thickness is determined by measuring the zero-crossing normal distance for the vertical velocity. The buoyant layer suction increases significantly as Ra increases. The effects of vertical level and of time on the qualitative behavior of buoyant layer flows are found to be small. The buoyant layer flows decay over the heat-up time scale t n ; t h characterizes the time span over which the overall adjustment process in the inviscid interior region is accomplished. This work clarifies that the analogy between heat-up and spin-up, which has been known to exist in the main body of inviscid fluid, applies equally well to the boundary layer regions.  相似文献   

3.
Abstract

It is shown that magnetic fields generated by flows v r,(r,t)er+vT where vT is an arbitrary toroidal component (er˙vT≡V≡vT≡0), cannot be maintained indefinitely against ohmic dissipation. The poloidal field variable max |r 2 B r| is shown to decay strictly monotonically with an undetermined decay rate. A bound on the growth of the toroidal field norm ∥T∥1 is established solely dependent on the rate of conversion of poloidal to toroidal field, so that when the poloidal field is negligible then ∥T∥1 decays strictly monotonically. The main application of these results is to models of stellar evolution based on axisymmetric differential rotation and spherically symmetric contraction. This symmetric velocity theorem overlaps with two already known theorems, namely the toroidal velocity theorem where v r≡0 and the radial velocity theorem where vT≡0. The new theorem does not entirely include the already established ones, principal differences being in the rates of decay and the field variables for which the decay is proven (see Table 1).  相似文献   

4.
Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number Tm ? 105. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.  相似文献   

5.
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6.
Abstract

In a laboratory model ocean, fluid in a rotating tank of varying depth is subjected to “wind-stress”, For a certain range of the parameters, Ekman number E and Rossby number R, a homogeneous fluid displays steady, westward intensified flow. For the same range of E and R, a two-layer fluid can have baroclinic instabilities. The parameter range for the various kinds of instabilities is mapped in a regime diagram. The northward transport in the western boundary current is measured as it varies with Rossby number for both homogeneous and two-layer fluid.  相似文献   

7.
Abstract

The superconducting-gravimeter data of Melchior and Ducarme (1986) has been interpreted as internal motion in the Earth's core by Aldridge and Lumb (1987) using a Poincaré model. Several low-order modes with periods of 13–16 hours have been tentatively identified in the core which is taken to be an incompressible, homogeneous fluid within a rigid, rotating container. The identification is based on asymptotic values of the frequencies which change slowly with time while the modes decay with an e-folding time of about 280 days. The slow change in frequency with time implies a small temporal variation in the rotation rate of the core. This mean flow is a nonlinear effect often observed in laboratory experiments designed to excite Poincaré modes. Interaction among modes during free ringdown is also observed in those experiments and apparently in the data of Melchior et al. (1988) as well. Laboratory work thus provides the link to extend the Poincaré model to include viscous and nonlinear effects in order to interpret the gravimetric observations as core modes.  相似文献   

8.
Abstract

A method has recently been proposed for finding the radius rc of the electrically-conducting fluid core of a planet of outer radius rs from observations of the magnetic field B in the accessible region near or well above the surface of the planet (Hide, 1978). The method is based on the supposition that when the magnetic field is produced by hydromagnetic dynamo action in the core, implying that the magnetic Reynolds number R there is large, (a) fluctuations in B can occur everywhere on the comparatively short advective time-scale τ A associated with fluid motions in the core and so can fluctuations in the quantity N, defined for any closed surface S as the total number of intersection of magnetic lines of force with S, provided that S lies well outside the core, but (b) at the surface of the core, where lines of magnetic force emerge from their region of origin, concomitant fluctuations in N are negligibly small, of the order of τ A/τO where τ O (= RτA ) is the Ohmic decay time of the core.

A proof of this supposition follows directly from the general expression derived in the present paper showing that when S is a material surface the time rate of change of N is equal to minus twice the line integral of the current density divided by the electrical conductivity around all the lines on S where the magnetic field is tangential to S. This expression (which Palmer in an accompanying paper rederives and extends to the relativistic case using the mathematical formalism of Cartan’s exterior calculus) also provides a direct demonstration of the well-known result that although high electrical conductivity, sufficient to make R ? 1, is a necessary condition for hydromagnetic dynamo action, such action is impossible in a perfect conductor, when R→ ∞.  相似文献   

9.
Abstract

It is found that in a rotating stratified fluid bounded by a single rigid wall, edge waves may occur at all frequencies less than or equal to N sin a (a is the angle of the wall from the horizontal and N the Brunt‐Vaisala frequency). These decay exponentially away from the boundary, in a distance of O(S) wavelengths, for α = O(1), or O(S ‐1) wavelengths, for αS ≤ O(1), where S is the ratio of N to the Coriolis parameter f, taken for illustration to be large. The phase and energy both move with a component to the left, facing shallow water. The waves could, for example, appear as an internal tide at the continental rise or as baroclinic meandering of currents over a slope.

The low‐frequency limit, αS ? 1, is studied in detail. To allow for large scales of motion other rigid boundaries and variations in f are included. The edge (actually “bottom") waves then merge with topographic‐planetary waves as the wavelengths increase; the familiar depth‐independent mode is found to be possible in the sea for wavelengths exceeding about 450 km. The ß‐effect introduces modes complementary to that trapped at the bottom, which instead are isolated from it.  相似文献   

10.
Abstract

‘‘Helicity'’ density H ≡ u · ω and other pseudo-scalar fields such as P ≡ ω · Vlnρ (which is related to Ertel potential vorticity) are useful quantities in theoretical fluid dynamics and magneto-fluid dynamics. Here u denotes the Eulerian flow velocity relative to the chosen frame of reference, ω ≡ V × u is the corresponding relative vorticity and ρ the mass density of the fluid. A general expression is readily obtained for ?H/?t (where t denotes time) in terms of P and the ‘‘superhelicity'’ density S ≡ ω · V × ω which, in fluids of low viscosity, has its highest values in boundary layers. One need for such a relationship became evident during an attempt to interpret the findings of laboratory experiments on thermal convection in rotating fluids in containers of various geometrical shapes and topological characteristics.

In electrodynamics an analogous expression can be found relating the time rate of change of ‘‘magnetic helicity'’ A · B to ‘‘magnetic superhelicity'’ B · ? × B (where B · ? × A is the magnetic field) and a scalar quantity analogous to P which involves non-Ohmic contributions to the relationship between the electric current density and the electric field.  相似文献   

11.
Abstract

The variational lower bound v > 0.39π2 determined by Ivers (1984) for the infimum decay rate v of axisymmetric poloidal magnetic fields is corroborated numerically and an upper bound v < 0.66°2 established. This is achieved by correcting and extending results for certain flows considered by Chandrasekhar (1956).  相似文献   

12.
Abstract

The study of the mechanisms controlling the stratification in closed fluid regions is an important branch of geophysical fluid dynamics. Part of this subject can be handled with a simple linear model, consisting of a buoyancy layer at the non-horizontal boundaries of a container and an advective-diffusive interior coupled by volume continuity. The model is valid under the following conditions: firstly, the buoyancy-frequency characterizing the solution must be sufficiently large to give rise to a flow pattern of boundary layer type and, secondly, the non-horizontal walls must not have too large thermal conductivity.

The main purpose of the present paper is to summarise previous work done by the authors in this field and to present some consequences of their theory not previously discussed.

Three important cases are discussed; certain stationary solutions, the decay of a given stratification and the build up of a stratification in a homogeneous fluid. The experimental results concerning the afore-mentioned cases are presented.  相似文献   

13.
Abstract

Measurements have been made of the net horizontal force F acting on a sphere moving with horizontal velocity U (Reynolds numbers in the range 102-104) through a stratified fluid rotating about a vertical axis with uniform angular velocity Ω. In both homogeneous and stratified rotating fluids with small Rossby number R(R = U/Ωa ? 1 where a is the radius of the sphere) the force F is of magnitude 2ΩρUV (where ρ is the density of the fluid and V is the volume of the sphere). In a homogeneous fluid the relative directions of F and U were found to depend on the quantity F = 8Ωa 2/UD (where D is the depth of the fluid in which the object is placed (Mason, 1975)). In a rotating stratified fluid the relative directions of F and U are found to depend on the inverse Froude number k(k = Na/U where N 2 = (g/δ)?ρ/?z) provided D > 4aΩ/N. In a homogeneous fluid with F ? 1 the force F is mainly in the U direction (a drag force due to inertial wave radiation) and is ～ ?0.4 |MX 2ΩρUV For F ? 1 a “Taylor column” occurs and the force, in correspondence with theoretical expectations, is ～ - 2Ω |MX UρV In a rotating stratified fluid with N ～2Ω and k ? 1 the force F is mainly in the U direction but is roughly one half of that occurring in the homogeneous situation with F ? 1 (tentatively explained as due to the evanescence of inertia-gravity disturbances). In a rotating stratified fluid with k ? 1 the flow should have no vertical motion (as with F ? 1) and again in correspondence with theoretical expectations the drag is ～ ?2 Ω |MX UρV. In a non-rotating stratified fluid the drag coefficient C D(C D = F U/½?ρU 2) was measured in the range k = 0.1 to 10 and had a maximum value ～ 1.2 for k ～ 3.  相似文献   

14.
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15.
Abstract

The equations for a relativistic perfect fluid result from the requirement that the total mass-energy be stationary with respect to variations δxα(a, b, c, s) in the space-time location of the fluid particle identified by Lagrangian labels (a, b, c) at the point s on its world-line. By considering variations of the Lagrangian labels that leave the specific volume and entropy unchanged, we obtain a general covariant statement of vorticity conservation. The conservation laws for circulation, potential vorticity, and helicity are simple corollaries. This Noether-theorem derivation shows that the vorticity laws have no analogues in particle mechanics, where the corresponding particle labels cannot be continuously vaned.  相似文献   

16.
Abstract

Vorticity, although not the primary variable of fluid dynamics, is an important derived variable playing both mathematical and physical roles in the solution and understanding of problems. The following treatment discusses the generation of vorticity at rigid boundaries and its subsequent decay. It is intended to provide a consistent and very broadly applicable framework within which a wide range of questions can be answered explicitly. The rate of generation of vorticity is shown to be the relative tangential acceleration of fluid and boundary without taking viscosity into account and the generating mechanism therefore involves the tangential pressure gradient within the fluid and the external acceleration of the boundary only. The mechanism is inviscid in nature and independent of the no-slip condition at the boundary, although viscous diffusion acts immediately after generation to spread vorticity outward from boundaries. Vorticity diffuses neither out of boundaries nor into them, and the only means of decay is by cross-diffusive annihilation within the fluid.  相似文献   

17.
Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height zc. The height zc , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above zc. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z &Gt; zc is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ?-8/3 characterizes the marginal state of the wave at z = zc .  相似文献   

18.
Abstract

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large.

With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, Rm , based on U is small of order q and on exterior region in which Rm is of order unity.  相似文献   

19.
Abstract

The problem of the removal of the degeneracy of the patterns of convective motion in a spherically symmetric fluid shell by the effects of rotation is considered. It is shown that the axisymmetric solution is preferred in sufficiently thick shells where the minimum Rayleigh number corresponds to degree l = 1 of the spherical harmonics. In all cases with l > 1 the solution described by sectional spherical harmonics Yl l (θ,φ) is preferred.  相似文献   

20.
Abstract

The flow of a rotating homogeneous, incompressible fluid past a long ridge is investigated. An analysis is presented for flows in which E ? 1, Ro ～ E½, H/D ～ E0, h/D ～ E½ and cosα ～ E0 where E is the Ekman number, Ro the Rossby number, H/D the fluid depth to ridge width ratio, h/D the ridge height to ridge width ratio and α the angle between the free stream flow and a line perpendicular to the ridge axis. The analysis includes effects of the nonlinear inertial terms. Particular examples of a ridge of triangular cross section and a sinusoidal topography are investigated in some detail. Experiments are presented for a triangular ridge which are in good agreement with the theory.  相似文献   

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