Quasi-geostrophic shallow-water vortex–patch equilibria and their stability     

Hanna Płotka  David G. Dritschel 《地球物理与天体物理流体动力学》2013,107(6):574-595

We examine the equilibrium form, properties, stability and nonlinear evolution of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L _D called the “Rossby deformation length” relating the strength of the stratification to that of the background rotation. Here, we generate steadily-rotating vortex equilibria for a wide range of γ?=?L/L _D , where L is the typical horizontal length scale of the vortex. We vary both γ (over the range 0.02?≤?γ?≤?10) and the vortex aspect ratio λ (over the range 0?<?λ?<?1). We find two modes of instability arising at sufficiently small aspect ratio λ?<?λ _c (γ): an asymmetric (dominantly wave 3) mode at small γ (or large L _D ) and a symmetric (dominantly wave 4) mode at large γ (or small L _D ). At marginal stability, the asymmetric mode dominates for γ???3, while the symmetric mode dominates for γ???3. The nonlinear evolution of weakly-perturbed unstable equilibria results in major structural changes, in most cases producing two dominant vortex patches and thin, quasi-passive filaments. Overall, the nonlinear evolution can be classified into three principal types: (1) vacillations for a limited range of aspect ratios λ when 5?≤?γ?≤?6, (2) filamentation and a single-dominant vortex for γ???1, and (3) vortex splitting – asymmetric for 1???γ???4 and symmetric for γ???4.  相似文献   

Nonlocal modons on the beta-plane     

John P. Boyd 《地球物理与天体物理流体动力学》2013,107(2-4):163-182

Abstract

The vortex pair known as a modon is a classical solitary wave in the sense that it decays exponentially with distance from the center of the wave whenever the modon's phase speed of the wave is outside the linear range. In contrast, when ?1 < c < 0, the modon “far field” is oscillatory so that the modon is “nonlocal” in the sense that it has nonzero amplitude even at arbitrarily far distances from the vortex maximum. However, Tribbia and Verkley have independently noted that the oscillatory far field may be very weak for some parameter ranges.  相似文献   

The tripole: A new coherent vortex structure of incompressible two-dimensional flows     

Lorenzo M. Polvani  Xavier J. Carton 《地球物理与天体物理流体动力学》2013,107(1-4):87-102

Abstract

Using a contour dynamical algorithm, we have found rotating tripolar V-state solutions for the inviscid Euler equations in two-dimensions. We have studied their geometry as a function of their physical parameters. Their stability was investigated with the aid of contour surgery, and most of the states were found to be stable. Under finite-amplitude perturbations, tripoles are shown to either fission into two asymmetric dipoles or to evolve into a shielded axisymmetric vortex, demonstrating the existence of two new ‘‘reversible transitions'’ between topologically distinct coherent vortex structures. These dynamical results are confirmed by pseudo-spectral simulations, with which we also show how continuous tripolar long-lived coherent vortex structures can be generated in a variety of ways.  相似文献   

Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system     

F.A.L. Pacheco 《水文科学杂志》2013,58(4):930-944

Abstract

A numerical technique is presented whereby aquifer hydraulic diffusivities (D) and macrodispersivities (α) are calculated by linear equations rewritten from flow and solute transport differential equations. The approach requires a GIS to calculate spatial and temporal hydraulic head (h) and solute concentration gradients. The model is tested in Portugal, in a semi-confined aquifer periodically monitored for h and chloride/sulphate concentrations. Average D (0.46 m²/s) and α (1975 m) compare favourably with literature results. The relationship between α and scale (L) is also investigated. In this context, two aquifer groups could be identified: the first group is heterogeneous at the “macroscopic” scale (solute travelled distances <1 km), but homogeneous at the “megascopic” scale. The overall scale dependency in this case is given by an equation of logarithmic type. The second group is heterogeneous at the macroscopic and megascopic scales, with a scale dependency of linear type.

Citation Pacheco, F.A.L., 2013. Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system. Hydrological Sciences Journal, 58 (4), 930–944.  相似文献   

Three-dimensional convection in spherical shells     

Philippe Machetel  Michel Rabinowicz  Pierre Bernardet 《地球物理与天体物理流体动力学》2013,107(1-2):57-84

Abstract

In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as:

I) Axisymmetrical solutions, analogous to rolls in plane geometry.

II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry.

III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a closed polyhedral network of descending (or ascending) currents. This type of circulation is called ‘polygonal’ in analogy with hexagonal circulation in plane geometry.

The behaviour of each of the eight solutions has been studied by increasing the Rayleigh number up to 15 times the critical number. A trend towards transitions from type (I) and type (II) solutions to type (III) solutions is observed. It is inferred that only the “polygonal” solutions are stable for a Rayleigh number greater than 15 times the critical number.  相似文献   

Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves     

R. T. Pierrehumbert 《地球物理与天体物理流体动力学》2013,107(1-4):285-319

Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ^{? l} at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ^{? l} form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.  相似文献   

Discussion: ‘Macroscale surface roughness and frictional resistance in overland flow’ by D. S. L. Lawrence     

Athol D. Abrahams 《地球表面变化过程与地形》1998,23(9):857-859

Lawrence argued that the inundation ratio Λ, defined as the mean flow depth d divided by the roughness height k, is the dominant control of flow resistance f and should be used as the primary variable when evaluating the hydraulics of overland flow on rough surfaces. Lawrence defined three flow regimes on the basis of Λ and developed an expression for f in terms of Λ for each regime. Common sense, however, suggests that f is independent of Λ where Λ < 1 because when roughness elements protrude through the flow, the value of f for the flow is the same regardless of the height of the elements. The error appears to have crept in as a result of Lawrence's representation of roughness elements by hemispheres. Lawrence found that f ∝ d/k, which she interpreted to mean f ∝ Λ. However, in her model the length dimension denoted by k is in fact half the breadth b/2 of the roughness elements. The distinction between k and b/2 is important, especially for roughness elements where k ≠ b/2. Thus, contrary to Lawrence's claim, f is not generally a function of Λ. Instead, f is a function of Λ only where Λ > 1. Where Λ < 1, f is a function of d/(b/2) or d/b. © 1998 John Wiley & Sons, Ltd.  相似文献   

Some interactions between small numbers of baroclinic,geostrophic vortices     

W. R. Young 《地球物理与天体物理流体动力学》2013,107(1-4):35-61

Abstract

Various interactions between small numbers (two and four) of baroclinic, geostrophic point vortices in a two-layer system are studied with attention to the qualitative changes in behavior which occur as size of the deformation radius is varied.

A particularly interesting interaction, which illustrates the richness of baroclinic vortex dynamics, is a collision between two hetons. (A heton is a vortex pair in which the constituent vortices have opposite signs and are in opposite layers. The “breadth” of a heton is the distance between its constituent vortices. A translating heton transports heat.) When two hetons, which initially have different breadths, collide, the result is either an exchange of partners, or a “slip-through” collision in which the initial structures are preserved. It is shown here that the outcome is always an exchange, provided the deformation radius is sufficiently small. This strongly contrasts with a collision between pairs of classical, one-layer vortices in which no exchange occurs if the initial ratio of the breadths is sufficiently extreme.

Finally the transport of passive fluid by a translating baroclinic pair is investigated. A pair of vortices in the top layer transports no lower layer fluid if the distance between the vortices is less than 1.72 deformation radii. By contrast, the size of the region trapped by a heton increases without bound as the spacing between the vortices increases.  相似文献   

A study on the characteristics of the information dimension D ₁ of the temporal and spatial distributions of earthquakes in an active fault zone     

Yebang Xu 《地震学报(英文版)》1992,5(2):389-398

The research of the information dimension (D ₁) in an active fault zone considers the contribution of each seismic event to information and reflects the characteristics of the temporal and spatial distributions of earthquakes from a new point of view, avoiding some short-comings of the research about the capacity dimension (D ₀). The results of calculation show that the information dimension of the temporal distribution in Xianshuihe active fault zone before Luhuo large earthquake isD ₁=0.1051. It is a consult creterion of large earthquakes in future in the fault zone. The information dimensions of the temporal distribution of the earthquakes in Anninghe active fault zone are respectivelyD ₁(t _N)=0.1363 (for the north section) andD ₁(t _S)=0.06710 (for the south section). The information dimensions of the spatial distribution are respectivelyD ₁(K _N)=1.053 (for the north section) andD ₁(K _S)=0.7758 (for the south section). The north section and the south section belong to two self-similar systems with different information dimensions respectively. The extent of the self-organization of seismic activity in the south section is higher than that in the north section. This is helpful for us to judge the major dangerous section in the key region of the seismic monitoring. The research about the information dimension of the temporal and the spatial distributions of earthquakes is significant for the exploration of active fault zones and seismic prediction. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,13, 372–379, 1991. This paper is sponsored by the Chinese Joint Seismological Science Foundation. The English version is improved by Zhenwen An.  相似文献   

海原断层系的分形研究   总被引：17，自引：0，他引：17       下载免费PDF全文

周尽 《地震工程学报》1991,13(3):78-85

本文根据Okubo等人测量圣安德烈斯断层系所用的复盖维数法,对海原断层系进行了分形测量,求出海原断层系的整体维数D_0=1.137,其景泰段D_0=1.109,海原段D_0=1.182。计算中未得到邵家水段和李使堡段的分维数。此外,文中还着重探讨了断层几何与地震活动性及其力学环境的关系,进一步证明了自相似断层的几何复杂性与地震活动密切相关,剪切断裂带具较低分维数(1.1—1.3),而在张性环境中形成的断裂体系具较高的分维数(1.5—1.6)。最后本文讨论了断层迹线图等因素对分形测量精度的影响。  相似文献   

Interactions of isolated vortices I: Modons colliding with modons     

James C. McWilliams  Norman J. Zabusky 《地球物理与天体物理流体动力学》2013,107(3-4):207-227

Abstract

Collisions between isolated(i.e. localized) dipolar vortex states, called modons, are examined in various numerical solutions of the quasigeostrophic, equivalent barotropic equation. For a range of parameters, the collision interactions are soliton-like in that the vorticity maxima are displaced (phase-shifted) with only small speed changes and little excitation of internal degrees of freedom. For other parameters, new “inelastic” effects are observed, including speed changes due to vorticity rearrangement, vorticity filamentation, modon “capture” or “fusion” in an overtaking collision, and the “fission” of a modon into its component vorticity monopoles in a head-on collision.  相似文献   

Magnetic instabilities in rapidly rotating spherical geometries. III. The effect of differential rotation     

David R. Fearn  Werner S. Weiglhofer 《地球物理与天体物理流体动力学》2013,107(1-4):163-184

Abstract

We investigate the influence of differential rotation on magnetic instabilities for an electrically conducting fluid in the presence of a toroidal basic state of magnetic field B ₀ = B_MB₀(r, θ)1 _φ and flow U₀ = U_MU₀ (r, θ)1_φ, [(r, θ, φ) are spherical polar coordinates]. The fluid is confined in a rapidly rotating, electrically insulating, rigid spherical container. In the first instance the influence of differential rotation on established magnetic instabilities is studied. These can belong to either the ideal or the resistive class, both of which have been the subject of extensive research in parts I and II of this series. It was found there, that in the absence of differential rotation, ideal modes (driven by gradients of B ₀) become unstable for A_c ? 200 whereas resistive instabilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B₀) require A_c ? 50. Here, Λ is the Elsasser number, a measure of the magnetic field strength and Λ_c is its critical value at marginal stability. Both types of instability can be stabilised by adding differential rotation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes only a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated Λ_c increased rapidly and the modes disappeared when R_m ≈ O(Λ_C), where the magnetic Reynolds number R_m is a measure of the strength of differential rotation. The main emphasis, however, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid system is destabilised by a suitable differential rotation once the magnetic Reynolds number exceeds a certain critical value (R_m )_c. Earlier work in the cylindrical geometry has shown that the differential rotation can generate an instability if R_m ) ?O(Λ). Those results, obtained for a fixed value of Λ = 100 are extended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength Λ on these modes of instability. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good agreement with the local analysis and the predictions inferred from the cylindrical geometry. For Λ = O(100), the critical value of the magnetic Reynolds number (R_m )_c Λ 100, depending on the choice of flow U₀ . Modes corresponding to azimuthal wavenumber m = 1 are the most unstable ones. Although the magnetic field B ₀ is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical and spherical geometries, (R_m )_c reaches a minimum value for 50 ≈ Λ ≈ 100. If Λ is increased, (R_m )_c ∝ Λ, whereas a decrease of Λ leads to a rapid increase of (R_m )_c, i.e. a stabilisation of the system. No instability was found for Λ ≈ 10 — 30. Optimum conditions for instability driven by unstable gradients of the differential rotation are therefore achieved for ≈ Λ 50 — 100, R_m ? 100. These values lead to the conclusion that the instabilities can play an important role in the dynamics of the Earth's core.  相似文献   

Space discontinuous Galerkin method for shallow water flows—kinetic and HLLC flux,and potential vorticity generation     

《Advances in water resources》2007,30(4):998-1015

In this paper, a second order space discontinuous Galerkin (DG) method is presented for the numerical solution of inviscid shallow water flows over varying bottom topography. Novel in the implementation is the use of HLLC and kinetic numerical fluxes¹ in combination with a dissipation operator, applied only locally around discontinuities to limit spurious numerical oscillations. Numerical solutions over (non-)uniform meshes are verified against exact solutions; the numerical error in the L₂-norm and the convergence of the solution are computed. Bore–vortex interactions are studied analytically and numerically to validate the model; these include bores as “breaking waves” in a channel and a bore traveling over a conical and Gaussian hump. In these complex numerical test cases, we correctly predict the generation of potential vorticity by non-uniform bores. Finally, we successfully validate the numerical model against measurements of steady oblique hydraulic jumps in a channel with a contraction. In the latter case, the kinetic flux is shown to be more robust.  相似文献   

A framework for sourcing of evaporation between saturated and unsaturated zone in bare soil condition     

E. Balugani  M.W. Lubczynski  K. Metselaar 《水文科学杂志》2013,58(11):1981-1995

ABSTRACT

Sourcing subsurface evaporation (E_ss) into groundwater (E_g) and unsaturated zone (E_u) components has received little scientific attention so far, despite its importance in water management and agriculture. We propose a novel sourcing framework, with its implementation in dedicated post-processing software called SOURCE (used along with the HYDRUS1D model), to study evaporation sourcing dynamics, define quantitatively “shallow” and “deep” water table conditions and test the applicability of water table fluctuation (WTF) and “bucket” methods for estimation of E_g and E_u separately.

For the “shallow” and “deep” water table we propose E_g?>?0.95E_ss and E_g = 0 criteria, respectively. Assessment of the WTF method allowed sourcing of very small fluxes otherwise neglected by standard hydrological methods. Sourcing with SOURCE software was more accurate than the standard “bucket” method mainly because of greater flexibility in spatio-temporal discretization. This study emphasized the dry condition relevance of groundwater evaporation which should be analysed by applying coupled flow of heat, vapour and liquid water.