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1.
Using linear and weakly nonlinear stability theory, the effects of Soret and Dufour parameters are investigated on thermohaline convection in a horizontal layer of rotating fluid, specifically the ocean. Thermohaline circulation is important in mixing processes and contributes to heat and mass transports and hence the earth’s climate. A general conception is that due to the smallness of the Soret and Dufour parameters their effect is negligible. However, it is shown here that the Soret parameter, salinity and rotation stabilise the system, whereas temperature destabilises it and the Dufour parameter has minimal effect on stationary convection. For oscillatory convection, the analysis is difficult as it shows that the Rayleigh number depends on six parameters, the Soret and Dufour parameters, the salinity Rayleigh number, the Lewis number, the Prandtl number, and the Taylor number. We demonstrate the interplay between these parameters and their effects on oscillatory convection in a graphical manner. Furthermore, we find that the Soret parameter enhances oscillatory convection whereas the Dufour parameter, salinity Rayleigh number, the Lewis number, and rotation delay instability. We believe that these results have not been elucidated in this way before for large-scale fluids. Furthermore, we investigate weakly nonlinear stability and the effect of cross diffusive terms on heat and mass transports. We show the existence of new solution bifurcations not previously identified in literature. 相似文献
2.
O. M. Podvigina 《Izvestiya Physics of the Solid Earth》2011,47(5):440-445
The onset of Boussinesq convection in a horizontal layer of an electrically conducting incompressible fluid is considered.
The layer rotating about a vertical axis is heated from below; a vertical magnetic field is imposed. Rigid electrically insulating
boundaries are assumed. The loss of stability of the trivial steady state, which occurs as the Rayleigh numbers increase,
can be accompanied by the development of a monotonic or an oscillatory instability, depending on the parameter values of the
problem at hand (the Taylor number, the Chandrasekhar number, the kinematic and the magnetic Prandtl numbers). When the instability
is monotonic, the emerging convective rolls themselves are also unstable if the Taylor number is sufficiently large (the so-called
Küppers-Lortz instability takes place). In the present work it is studied how the critical value of the Rayleigh number, the
type of the trivial steady state instability, and the critical value of the Taylor number for the Küppers-Lortz instability
depend on the kinematic and the magnetic Prandtl numbers. We consider the values of the Prandtl number not exceeding 1, which
is typical for the outer core of the Earth. 相似文献
3.
Abstract The modal expansion procedure has been used to analyze penetrative convection that arises when a thin unstable layer is embedded between two stable regions. The Boussinesq approximation is applied in which the effect of compressibility and stratification are neglected. Various calculations have been made, with one and two modes, for Rayleigh numbers ranging from the critical value to more than 105 times critical. The effect of decreasing the Prandtl number has also been investigated. It is found that in the nonlinear regime, the convective motions penetrate substantially into the stable regions. The flux of kinetic energy plays a crucial role in such penetration, and its existence puts some requirements on the motions: in the single-mode case, they need to be three-dimensional. The extent of penetration amounts to about half of the thickness of the unstable layer on each side of it when the degree of instability and that of stability are comparable in the two domains; it increases as the stability of the outer region is lowered. The penetration depth appears to be independent of all other parameters defining the problem. 相似文献
4.
B. I. Birger 《Izvestiya Physics of the Solid Earth》2011,47(6):541-554
Stationary solutions including wave solutions with constant amplitudes are found for nonlinear equations of thermal convection
in a layer with nonlinear rheology. The solution is based on the Fourier expansion of unknown velocities and temperatures
with only the first and first two terms retained in the velocity and temperature series, respectively. This method, which
can be regarded as the Lorenz method, yields the Lorenz equations that fairly well describe the thermal convection in a layer
with Newtonian rheology if the Rayleigh number is not very large. The obtained generalization of the Lorenz equations to the
case of an integral (having a memory) nonlinear rheology implies that only the first term is retained in the Fourier series
for the stress components, i.e., the nonlinear rheological equation is harmonically linearized. However, in the Fourier series
of temperature, it is essential to keep the second term: this term, which is independent of the horizontal coordinate, models
the thermal boundary layer that characterizes the developed convection. We constructed the bifurcation curves that describe
the stationary convection in the nonlinear integral medium simulating the rheology of the mantle, and analyzed the stability
of stationary convective flows. The Lorenz method is applied to study small-scale thermal convection in the lithosphere of
the Earth. 相似文献
5.
Results are presented from both linear stability analysis and numerical simulations of three-dimensional nonlinear convection in a Boussinesq fluid in an annular channel, under experimental boundary conditions, rotating about a vertical axis uniformly heated from below. The focus is placed on the Prandtl number Pr = 7.0, representing liquid water at room temperature. The linear analysis shows that, when the aspect ratio is sufficiently small, there exists only one stationary mode that occupies the whole fluid container. When the aspect ratio is moderate or large, however, there exist three different linear solutions: (i) the outer sidewall-localized traveling wave propagating against the sense of rotation; (ii) the inner sidewall-localized traveling wave propagating in the same sense as rotation; and (iii) both the counter-traveling waves occurring simultaneously. Guided by the result of the linear stability analysis, fully three-dimensional simulations are then performed for a channel with a moderate aspect ratio. It is found that neither the prograde nor the retrograde mode is physically realizable near threshold and beyond. The dynamics of nonlinear convection in a rotating channel are chiefly characterized by the interaction between the sidewall-localized waves and the interior convection cells/rolls, producing an interesting and unusual nonlinear phenomenon. In order to compare with the classical Rayleigh–Bénard problem without vertical sidewalls, we also study linear and nonlinear convection at exactly the same parameters but in an infinitely extended layer with periodic horizontal conditions. This reveals that both the linear instability and nonlinear convection in a rotating channel are characteristically different from those in a rotating layer with periodic horizontal conditions. 相似文献
6.
Abstract Finite amplitude solutions for convection in a rotating spherical fluid shell with a radius ratio of η=0.4 are obtained numerically by the Galerkin method. The case of the azimuthal wavenumber m=2 is emphasized, but solutions with m=4 are also considered. The pronounced distinction between different modes at low Prandtl numbers found in a preceding linear analysis (Zhang and Busse, 1987) is also found with respect to nonlinear properties. Only the positive-ω-mode exhibits subcritical finite amplitude convection. The stability of the stationary drifting solutions with respect to hydrodynamic disturbances is analyzed and regions of stability are presented. A major part of the paper is concerned with the growth of magnetic disturbances. The critical magnetic Prandtl number for the onset of dynamo action has been determined as function of the Rayleigh and Taylor numbers for the Prandtl numbers P=0.1 and P=1.0. Stationary and oscillatory dynamos with both, dipolar and quadrupolar, symmetries are close competitors in the parameter space of the problem. 相似文献
7.
In many natural environments, such as in underwater hot springs and hydrothermal vents, thermal gradients are accompanied with changes in the concentration of chemical compounds transported to the seawater, causing the so-called double-diffusive, mixed convection. To study the physical scenarios in such systems, a vertical channel filled with a porous medium saturated with saline water is considered. The motion in the sediment-filled channel is induced by two buoyancy forces and an external pressure gradient, similar to the situation in a vent with an upward flow direction. The fluid flow has been modeled by an extended Darcy model, and the flow instability mechanisms have been studied numerically. The linear stability analysis is performed considering a wide range of Darcy number (Da = 10−5 -10−8). The instability boundary curve showed three distinct dynamic regimes: (i) Rayleigh-Taylor (R-T), (ii) log-log non-linear variation, and (iii) log-log linear variation. The domain of different regimes were sensitive to external pressure gradient as well as permeability. Similar to cross-diffusive natural convection in pure viscous fluids, a linear relationship between logarithmic absolute values of critical thermal Rayleigh number (∣RaT∣) and solute Rayleigh number (RaC) is found in the third regime. Based on the permeability, for any solute Rayleigh number (RaC), there existed a minimum value of Reynolds number (Re), below which R-T type of instability appeared. Above this minimum value, the instability was due to two buoyancy forces, known as buoyant instability. Simulations of secondary flow via energy analysis demonstrated the development of complex dynamics at the critical state in all three regimes characterized by transition of multi to uni-cellular structures and vice verse. 相似文献
8.
Joe M. Straus 《地球物理与天体物理流体动力学》2013,107(1):261-281
AbstractTwo upper bounding problems for thermal convection in a layer of fluid contained between perfectly conducting stress-free boundaries are treated numerically. Since the Euler equations resulting from this variational approach are simpler than the Navier-Stokes equations, they allow numerical calculations to be carried out economically to fairly large values of the Rayleigh number. The upper bounding problem formulated by Howard (1963), which yields a Nusselt number independent of Prandtl number, diverges from the correct behavior as the Rayleigh number increases. In hopes of coming closer to results of previous investigations of the Boussinesq equations of motion, a more restrictive upper bounding problem is formulated. For large Prandtl numbers the momentum equation is linearized and is used as an explicit side constraint on the variational problem, thereby forcing the solutions to more closely resemble the solutions of the Boussinesq equations. Numerical calculations at values of the Rayleigh number up to 1.5 × 105 indicate that the additional constraint decreases the upper bound on the Nusselt number; it appears that this upper bound differs by only a multiplicative factor from that calculated from solutions of the full equations of motion and may be a reasonable approximation for large Rayleigh numbers. 相似文献
9.
An infinite horizontal layer, with vertically stratified temperature and solute concentration, is considered in the case where the viscosity is exponentially dependent on temperature, and the Prandtl number is infinite. Its linear stability is investigated when the destabilizing thermal gradient acts against a stabilizing solute gradient. The analysis is performed using horizontal Fourier and vertical Chebyshev polynomial expansions. For the constant viscosity case, the laws well established in the free boundary configuration are seen to be directly suitable for the rigid one. In the variable viscosity case, characterised by a given viscosity contrast c, the scaling laws with c are settled extrapolating to the double diffusive situation the approach initiated by Stengel et al. (1982). In contrast with the constant viscosity case, the critical wave number is found to be strongly dependent on the solutal Rayleigh number in the marginal oscillatory obtained at large contrast values. 相似文献
10.
Abstract The annulus model considers convection between concentric cylinders with sloping endwalls. It is used as a simplified model of convection in a rapidly rotating sphere. Large azimuthal wavenumbers are preferred in this problem, and this has been exploited to develop an asymptotic approach to nonlinear convection in the annulus. The problem is further reduced because the Taylor-Proudman constraint simplifies the dependence in the direction of the rotation vector, so that a nonlinear system dependent only on the radial variable and time results. As Rayleigh number is increased a sequence of bifurcations is found, from steady solutions to periodic solutions and 2-tori, typically ending in chaotic behaviour. Both the magnetic (MHD convection) and non-magnetic problem has been considered, and in the non-magnetic case our bifurcation sequence can be compared with those found by previous two-dimensional numerical simulations. 相似文献
11.
12.
--This work deals with computational modelling designed to understand the dynamical mechanism of low frequency monsoonal transients that results from nonlinear divergent-rotational (&gif1;) kinetic energy (KE) conversions due to the effects of Coriolis force, vorticity and divergence during the summer monsoon 1988 over the latitudinal belt 20°S-30°N at 850 hPa and 200 hPa. The results show two distinct spectral peaks spanning 30-45 days and 18-25 days in the energy conversions from the transient divergent motions to rotational motions. Due to the latitudinal variation of the earth's rotational effects, the conversion from the transient divergent to rotational motions, associated mainly with wavenumbers 1 and 2, tend to be more pronounced to the north of 15°N on the 30-45-day and 18-25-day time scales in the upper and lower tropospheres, respectively. The contribution of the stationary waves to maintenance of the low frequency rotational flow due to the effect of divergence through barotropic instability is significant at the upper troposphere. Divergent to rotational KE conversion by wave-wave interaction due to divergence is identified as an important mechanism for maintenance of low frequency oscillations in the lower troposphere. The upper tropospheric planetary scale divergent motions associated with 30-45-day oscillation gain substantial energy through nonlinear &gif1; interaction due to vorticity. 相似文献
13.
本文研究了双扩散效应对处于地球物理极限q→0条件下旋转分层导电流体的磁流体动力学（MHD）不稳定性的影响。研究发现,只要|RS|足够大,对流总能在流体静力稳定分层的导电流体中发展。当磁场作用较弱,即Λ～Ο（1）时,对所有可能的角向波数m优先发展的是双扩散不稳定性。当Rs>0时,流体中发生几乎不变的“指状”对流。当Rs<0时,流体中发生缓慢东向运行的“扩散”振荡。当磁场作用较强,即Λ≥ΛM时,与原始Soward模式类似,优先发展的仅是m=1磁力驱动不稳定性,并东向运行,但双扩散效应使得正RT不稳定性与负RT不稳定性间的对称性破缺。 相似文献
14.
J. Klostermeyer 《地球物理与天体物理流体动力学》2013,107(1-4):117-138
Abstract Theoretical studies predict a parametric instability of finite-amplitude internal gravity waves which hitherto has been observed only in laboratory experiments. The occurrence of this process in the atmosphere is of basic interest because finite-amplitude gravity waves, which are almost ubiquitous especially at upper atmospheric heights, would produce unstable flows even at large Richardson numbers. Maximum entropy power spectra of a strong internal gravity wave in the thermosphere, which was generated by a volcanic eruption and detected on records of the Doppler shift of high-frequency radio waves, in fact show good agreement with the spectra of synthetic Doppler records obtained from a calculated unstable gravity wave. The frequencies and wavenumbers observed in the gravity wave domain satisfy in particular the theoretically predicted resonance conditions. The observed Doppler records also show two significant lines in the acoustic domain which probably result from a nonlinear interaction with the basic gravity wave. It is suggested that acoustic double peaks, which are commonly observed in high-frequency Doppler spectra in the presence of nearby thunderstorms, represent parametric instabilities of internal gravity waves generated by penetrative cumulus convection. 相似文献
15.
Brian Straughan 《地球物理与天体物理流体动力学》2013,107(1-4):227-242
Abstract A nonlinear energy stability analysis is presented for the penetrative convection model of Veronis (1963). For top temperatures between 4°C and 8°C the nonlinear stability boundary obtained is very close to the linear one of Veronis and enables a region of possible sub-critical instabilities to be determined. 相似文献
16.
D. W. Hughes 《地球物理与天体物理流体动力学》2013,107(1-4):99-142
Abstract Recent calculations suggest that the bulk of the solar toroidal field may be stored in a thin, convectively stable region situated between the convection zone proper and the radiative zone. Determining the stability properties of such a field is therefore important with implications for both the generation and escape of magnetic flux. The plane layer, linear stability analysis of Hughes (1985) is extended to incorporate the effects of uniform rotation. Detailed studies are made of interchange, or “axisymmetric” modes and of undular, or wavelike, motions, considering modes of both low and high frequency. The force due to rotation acts to constrain the fluid motions, a feature which is strongly stabilizing for direct modes, but can, in certain circumstances, be destabilizing for oscillatory motions. For the interchange modes we show that the instability discussed at length by Hughes (1985), driven by fields increasing with height, is still present and indeed may be enhanced by rotational effects. We also study the more conventional instabilities, discussing the transformation between direct and oscillatory modes and considering in detail some peculiar properties of the oscillatory instabilities. The more relevant instabilities in an astrophysical context are likely to be undular modes. Previous studies of low frequency modes driven by top heavy field gradients are extended to consider modes of various frequencies for a wide range of parameter values. Of particular interest is the occurrence of two distinct modes of instability for bottom heavy field gradients. We also exhibit some of the peculiar stability boundaries which can result when none of the competing influences in the problem is dominant. 相似文献
17.
There are several important wavenumber sampling issues associated with 2.5D seismic modelling in the frequency domain, which need careful attention if accurate results are to be obtained. At certain critical wavenumbers there exist rapid disruptions in the mainly smooth oscillatory spectra. The amplitudes of these disruptions can be very large, and this affects the accuracy of the inverse Fourier transformed frequency-space domain solution. In anisotropic elastic media there are critical wavenumbers associated with each wave mode—the quasi-P (qP) wave, and the two quasi-shear (qS1 and qS2) waves. A small wavenumber sampling interval is desirable in order to capture the highly oscillatory nature of the wavenumber spectrum, especially at increasing distance from the source. Obviously a small wavenumber sampling interval adds greatly to the computational effort because a 2D problem must be solved for every wavenumber and every frequency. The discretisation should be carried out up to some maximum wavenumber, beyond which the field becomes evanescent (exponentially decaying or diffusive). For receivers close to the source, activity persists beyond the critical wavenumber associated with the minimum shear wave velocity in the model. Fortunately, for receivers well removed from the source, the contribution from the evanescent energy is negligible and so there is no need to sample beyond this critical wavenumber. Sampling at Gauss–Legendre spacings is a satisfactory approach for acoustic media, but it is not practical in elastic media due to the difficulty of partitioning the integration around the different critical wavenumbers. We found to our surprise that in transversely isotropic media, the critical wavenumbers are independent of wave direction, but always occur at those wavenumbers corresponding to the maximum phase velocities of the three wave modes (qP, qS1 and qS2), which depend only on the elastic constants and the density. Additionally, we have observed that intermediate layers between source and receiver can filter out to a large degree, the sharp irregularities around the critical wavenumbers in the ω–k y spectra. We have found that, using the spectral element method, the singularities (poles) at the critical wavenumbers which exist with analytic solutions, do not arise. However, the troublesome spike-like behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuation. 相似文献
18.
We have performed laboratory experiments using a Hele-Shaw cell to model a saturated, porous layer with various sinusoidal upper boundaries. Our intent was to determine the range of conditions over which boundary topography can control the pattern of thermal convection within a porous layer, and thereby take the first step toward understanding why heat flow seems correlated with hypsography in many areas of the ocean floor.These experiments indicate that above the critical Rayleigh number, topography does not control the convection pattern, except when the topographic wavelength is comparable to the depth of water penetration. Scaled to the depth of the layer, the convective wavenumbers are restricted to values between 2.5 and 4.8—a range which brackets π, the natural wavenumber for convection in a porous slab with planar, isothermal, impermeable boundaries. Topographies within this range control the circulation pattern perfectly, with downwelling under valleys and upwelling aligned with topographic highs. Other topographies do not force the pattern, although in some cases, the convection wavenumber may be a harmonic of the topographic wavenumber. Unforced circulation cells wander and vary in size, because they are not locked to the topography.For these experiments we employed eight different topographies with non-dimensional wavenumbers between 1.43 and 8.17, and we studied the flow at Rayleigh numbers between zero and five times the critical Rayleigh number. The amplitude of each topography tapered linearly (over a factor of three to six) from one end of the apparatus to the other, and the mean topographic amplitude was 0.05 times the depth of the layer. Under these conditions, amplitude has only a minor effect on the structural form and vigor of supercritical convection.Our results may apply to submarine geothermal systems, sealed by a thin layer of impermeable sediment draped over the basement topography. In this case, the convection wavelength—as measured perhaps by the spatial periodicity of conductive heat flow—may be a good measure of the depth to which water penetrates the crust. Where the circulation correlates with the bottom topography, it may be because the topographic wavelength is comparable to the depth to which water penetrates the porous crust. 相似文献
19.
Convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral rigid sidewalls, stress-free top and bottom, uniformly heated from below is investigated. The sidewalls are assumed to be either perfectly insulating or conducting. Three different types of convection are identified when the channel is rotating sufficiently fast: (i) global oscillatory convection preferred for small Prandtl numbers in channels with intermediate or large aspect ratios (width to height ratio), (ii) wall-localized oscillatory convection representing the most unstable mode for moderate or large Prandtl numbers in channels with intermediate or large aspect ratios and (iii) global stationary convection preferred in channels with sufficiently small aspect ratios regardless of the size of the Prandtl number. The corresponding weakly nonlinear problem describing differential rotation and meridional circulation is also examined, showing that geostrophic, multiple-peaked (two prograde and two retrograde) differential rotation can be maintained by the Reynolds stresses in wall-localized convective eddies in a rapidly rotating channel. 相似文献
20.
瑞利波具有能量大、信噪比高等特点,可以用来反演介质内部的力学信息,近年来在浅层地球物理勘探、深层地震学研究以及超声波无损检测等多个领域都有较广泛的应用。目前大多数瑞利波的应用中都假设介质是弹性的,然而实际中岩石、土壤和金属等介质都在一定程度上体现出了黏弹性。当介质的黏弹性较强时仍然采用弹性假设研究其中瑞利波的反演将增大误差,因此有必要考虑黏弹性介质中的瑞利波反演,但是目前这方面的研究仍不够深入。本文研究黏弹性介质中瑞利波频散曲线和衰减系数曲线的反演问题,给出其在半空间中联合反演的方法,并对该方法的误差进行分析。 相似文献