On the spin-up of a stably stratified,electrically conducting fluid. Part 1: Spin-up controlled by the hartmann layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady Hartmann boundary layer control the fluid interior. It is shown how to obtain the known spin-up times for a homogeneous, nonconducting fluid (τ = E ^-½), a stably stratified, nonconducting fluid (τ = (σS/E, E ^?1) and a homogeneous conducting fluid (τ = α^?1 E ^-½) from the present formulation where τ = v/ωt, E = v/ωL ², σS = vN²/κω² and 2α² = σB²/pω. The problem is solved in the parameter range E?α²?1, α²/E?σS using the Laplace transform and two new spin-up times are obtained. Combined into one expression, they are τ = (1 + δ)α^?1E^-½ where δ = σμv. The spin-up mechanism is investigated and it is found that, in contrast to the homogeneous, conducting case, torsional Alfvén waves may be instrumental in the spin-up of a stratified conducting fluid. The effects of viscous and ohmic diffusion on the torsional Alfvén wave fronts are studied and the following regimes are identified: 0 < δ ?E/α², spin-up by meridional circulation of electric current with no Alfvén waves; E^-½/α ? δ ? 1, spin-up by meridional circulation of electric current with transient Alfvén waves; α/E^½ ? δ ? α²/E, spin-up by meridional circulation of current with weak Alfvén waves; 1 ? δ ? α/E^½, spin-up by strong Alfvén waves; α^½/E ? δ ? α²/E, spin-up by viscous diffusion with transient Alfvén waves; α/E ? δ < ∞, spin-up by viscous diffusion with no Alfvén waves.     

Linear spin-up in a sliced cylinder

Abstract

spin-up and spin-down in a circular tank with a uniformly sloping bottom are studied experimentally and numerically for small values of the relative change in the angular velocity of the tank. Generally, the initial single-cell flow evolves into a number of smaller vortices. The evolution is compared with an analytical model based on an expansion of the flow field in linear Rossby waves (Pedlosky and Greenspan, 1967). Although it is possible to tune the experimental parameters in such a way that agreement with the theory is found, in most cases the experiments show shedding of vortices in the initial stage of the spin-up or spin-down, a phenomenon not described by the analytical model. Nonetheless, in such cases the analytical model still accounts for other observations: the alternating generation of cyclonic and anticyclonic vortices in the eastern part of the tank and their subsequent westward motion.     

Does measuring azimuthal variations in sap flux lead to more reliable stand transpiration estimates?

Stand transpiration (E) estimated using the sap‐flux method includes uncertainty induced by variations in sap flux (F) within a tree (i.e. radial and azimuthal variations) and those between trees. Unlike radial variations, azimuthal variations are not particularly systematic (i.e. higher/lower F is not always recorded for a specific direction). Here, we present a theoretical framework to address the question on how to allocate a limited number of sensors to minimize uncertainty in E estimates. Specifically, we compare uncertainty in E estimates for two cases: (1) measuring F for two or more directions to cover azimuthal variations in F and (2) measuring F for one direction to cover between‐tree variations in F. The framework formulates the variation in the probability density function for E (σ_E) based on F recorded in m different azimuthal directions (e.g. north, east, south and west). This formula allows us to determine the m value that minimizes σ_E. This study applied the framework to F data recorded for a 55‐year‐old Cryptomeria japonica stand. σ_E for m = 1 was found to be less than the values for m = 2, 3 and 4. Our results suggest that measuring F for one azimuthal direction provides more reliable E estimates than measuring F for two or more azimuthal directions for this stand, given a limited number of sensors. Application of this framework to other datasets helps us decide how to allocate sensors most effectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Experiments in ocean circulation modelling

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.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 2: Spin-up controlled by the mac layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady MAC layer control the fluid interior, The MAC layer is a new boundary layer first studied by Loper (1976a) which controls the fluid in the parameter range E/α² ? σS ? α²/E, α² ? 1 Where E = vωL², 2α² = σB²/pω and σS = vN²/κω;². The problem is solved using the Laplace transform and four new spin-up times are obtained. Combined into one expression they are t = ω;^?1E-½[1+(σSE/α⁶)^½ + δα^-2] [1+(σSE/α^{6 1/4}]^?1 where δ = σμv. The internal spin-up mechanisms for this problem are shown to be very similar to those discussed in part 1 (Loper, 1976b). The ten known spin-up times are summarized and their inter-relationships are investigated. It is shown how to obtain the seven hydromagnetic spin-up times from a simple torsional Alfvén wave model involving a single parameter which measures the strength of the boundary layer dissipation. Finally, the present theory is applied to the solar spin-down problem and it is found that if the magnetic field in the solar interior is at least as strong as the interplanetary field of 10^-5 gauss, then the hydromagnetic spin-down time is much shorter than the Eddington-Sweet time and is comparable to the age of the sun.     

Wave hierarchies in alluvial river flows

Abstract

The long wave equations governing the flow in alluvial rivers and channels are considered. The linearized equations are re-cast in the form of a single equation of wave hierarchy type as discussed by Whitham (1974). The dynamic and kinematic waves are of third and second order respectively. Behaviour at the wave fronts is considered and a roll-wave type instability is revealed.

For stable flow, the theory is used to make both qualitative and quantitative predictions in the areas of short and long term floods, tidal waves and channel dredging.

The non-uniformity in the quasi-steady theory on bedform development [see, for example, Reynolds (1985)] as the Froude number, F, approaches unity is also discussed, and appropriate scalings are obtained to derive a theory which remains valid when F ～ 1.     

Flows within the Ekman layer during spin-up of a thermally stratified fluid

Abstract

Finite-difference numerical solutions were obtained to present the flow and temperature field details within the transient Ekman layer during spin-up of a thermally stratified fluid in a cylinder. This complements the earlier studies on stratified spin-up which examined the flows in the interior core region. As the stratification increases, the following changes in the flow field are noticeable. The radial velocity in the Ekman layer decreases in magnitude. The azimuthal flows adjust smoothly from the interior region to the endwall boundary, and the Ekman layer in the azimuthal flow field fades. Vertical motions are inhibited, resulting in a weakened Ekman pumping. The axial vorticity field behaves similarly to the azimuthal flows. The temperature deviation from the equilibrium profile decreases, and the heat transfer flux from the endwall to the fluid decreases. The thickness of the thermal layer is larger than the velocity layer thickness. Illustrative comparisons of the relative sizes of the terms in the governing equations are conducted in order to assess the stratification effect in the adjustment process of the fluid.     

Symmetric baroclinic instability for small ekman numbers

Abstract

The stability of a zonal shear flow to symmetric baroclinic perturbations is examined when the Ekman number, E, is asymptotically small. It is assumed, following Antar and Fowlis (1982), that the zonal Row is generated by imposing a constant horizontal temperature gradient γ* at the horizontal boundaries, and by maintaining a constant temperature difference δT* between them. The boundaries are at rest relative to a rotating frame.

Features of the neutral stability curve are determined for several ranges of values of δT/E ^1/3, where δT = δT*/Hγ* and H is the depth of the fluid layer, and all values of the Prandtl number, [sgrave]. In some cases it is possible to determine the whole curve analytically. The most important feature of the results is that the neutral stability curve is closed.

The results are compared to the numerical integrations of Antar and Fowlis (1982). The qualitative features of the solutions are in accord and the quantitative results are, in most cases, as good as can be expected for E only as small as ～ 10^?4. The implications of the results for experimental observations of symmetric baroclinic instability are explored.     

A laboratory study of baroclinic instability

Abstract

Experimental and theoretical results are presented for a simple system which exhibits baroclinic instability. We consider the motion of two immiscible fluids with densities ρ ₁ and ρ ₂ contained in a cylinder rotating with angular frequency ω. The motion is driven by a contact lid rotating with frequency ω + ω. In this paper ω, ω, 2(ρ ₂ – ρ ₁)/(ρ ₂ + ρ ₁), and the geometry are such that the interface does not intersect the “ground” (e.g. an almost horizontal boundary). The motions are described by two-layer quasi-geostrophic equations which are identical, except perhaps for the presence of interfacial friction and tension, with those used in meteorology and oceanography. For small enough internal Froude number F = 4ω² L ²/(g(Δρ/ρ)H) or small enough Rossby number ? = ω/2ω the flow is steady and axisymmetric, the velocity field in each layer being determined primarily by frictional effects in top, bottom, and interfacial Ekman layers. For certain (F, ?) the flow becomes non-axisymmetric. The transition points for the case where the basic potential vorticity gradient is due to interface slope alone have been carefully measured and are in very good agreement with a linear instability theory which neglects sidewall effects. Some preliminary observations of supercritical motion, which include repeatable amplitude and wavenumber vacillation, are reported.     

Streamflow sensitivity analysis to climate change for a large water‐limited basin

Partitioning precipitation (P) between streamflow (Q) and actual evapotranspiration (E_a) on a basin scale is controlled by climate change in combination with catchment characteristics. Fu's formulation of the Budyko framework was used to estimate Q as a function of two meteorological variables, P and potential evaporation (E_p), and one adjustable parameter reflecting characteristics of catchment conditions (ω). Results show that ω reflects the impacts of catchment characteristics on the partitioning of P between Q and E_a for the different water yielding regions. As predicted, Q was more sensitive to P than to comparable changes in E_p for the whole of the Yellow River Basin (YRB), a water‐limited basin, while it was shown to be highly sensitive to changes in P, E_p, and ω in the low water yielding region (LWYR) of the basin, followed by YRB and the high water yielding region of the basin. The high sensitivity of Q to P, E_p, and ω in LWYR indicates that the management of catchments within these zones is critical to the management of overall basin flow, mitigating impacts of climate change on Q. The Budyko framework, incorporating the adjustable parameter ω, outlines interactions between Q, climate, and characteristics specific to different water yielding regions. It also provides a new approach in understanding hydrological process response to climate change. Due to the obscure physical attributes of ω, an explanation of the parameter using soil or vegetation characteristics will aid in the understanding of the eco‐hydrological behaviour of catchments and help to provide more detailed catchment management options for which to mitigate climate change with respect to concerns regarding agricultural water usage. Copyright © 2013 John Wiley & Sons, Ltd.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

Inflow and outflow in a rotating circular container—A laboratory model for the Loop Current of the Gulf of Mexico

The Loop Current of the Gulf of Mexico is simulated in the laboratory. A circular tank is filled with water and is placed off-center on a rotating table and the flow field is generated by injecting and withdrawing water at two openings on the wall. The free surface becomes parabolic due to balance of gravitational and centrifugal forces, simulating the latitudinal change of the Coriolis parameter (-effect) in the ocean. The flow characteristics depend on the influx and the rate of rotation and can be classified according to non-dimensional parameters (Rossby, Ekman and Froude numbers denoted byR ₀,E andF, respectively). When the influx is small and the rotation rate is large (smallR ₀,E andF) the flow will be almost linear, and the fluid flows along the side-wall boundary layer under constraint of the -effect. For a very large influx (largeR ₀ andE) inertial forces become very large compared to the Coriolis force and the flow behaves like a potential flow. The flow studied had characteristics between these two extreme cases and hasR ₀ andF similar to the Gulf circulation, though similarity inE is ambiguous. Photographs of the flow indicate that the inflow penetrates further into the interior when the rotation rate is increased while the influx is kept constant. The numerical analysis of the non-linear vorticity equation confirms this for the parameters corresponding to the experiment. In addition, the photographs reveal eddies embedded on both sides of the main stream, particularly near the inflow region. These eddies are intensified and become uniform in size as the influx increases. It is pointed out that such eddies were actually observed near the Loop Current north of the Yucatan Straits.     

Nonlinear baroclinic equilibration at finite supercriticality

The nonlinear equilibration of finite amplitude baroclinic waves in Phillips two-layer model is investigated at finite supercriticality. The aims are to quantify the robustness and relevance of the nonlinear theory of Warn, Gauthier and Pedlosky (WGP) for the evolution of the developing baroclinic wave, and to assess the tightness of pseudomomentum and improved pseudoenergy bounds for disturbance amplitude and energy. A high-resolution numerical model is used to perform a parameter sweep in (β,?W)-space, where β is the inverse criticality of the initial flow, and W is the ratio of the channel width to the (internal) Rossby radius. At low supercriticalities, the main predictions of WGP are found to be accurate at short times, but at long times the fully nonlinear results are found to diverge from WGP's solution. The mechanism for equilibration involves the elimination of the lower layer potential vorticity (PV) gradient, but as the supercriticality increases this is achieved by the roll-up of a train of opposite-signed vortices, rather than by coarse-grain PV homogenization as in WGP. Peak wave amplitudes are typically ≈90% of the maximum attainable under the pseudomomentum bound. New formulae are given for the pseudoenergy bound on disturbance energy which, unlike the WGP solution and the pseudomomentum bound, have non-trivial dependence on W. A detailed assessment is made of the extent to which these bounds are attained.     

Gravity current down a steeply inclined slope in a rotating fluid

The sinking of dense water down a steep continental slope is studied using laboratory experiments, theoretical analysis and numerical simulation. The experiments were made in a rotating tank containing a solid cone mounted on the tank floor and originally filled with water of constant density. A bottom gravity current was produced by injecting more dense coloured water at the top of the cone. The dense water plume propagated from the source down the inclined cone wall and formed a bottom front separating the dense and light fluids. The location of the bottom front was measured as a function of time for various experimental parameters. In the majority of runs a stable axisymmetric flow was observed. In certain experiments, the bottom layer became unstable and was broken into a system of frontal waves which propagated down the slope. The fluid dynamics theory was developed for a strongly non-linear gravity current forming a near-bottom density front. The theory takes into account both bottom and interfacial friction as well as deviation of pressure from the hydrostatic formula in the case of noticeable vertical velocities. Analytical and numerical solutions were found for the initial (t < 1/f), intermediate (t 1/f), and main (t 1/f) stages, where f is the Coriolis parameter. The model results show that during the initial stage non-linear inertial oscillations are developed. During the main stage, the gravity current is concentrated in the bottom layer which has a thickness of the order of the Ekman scale. The numerical solutions are close to the same analytical one. Stability analysis shows that the instability threshold depends mainly on the Froude number and does not depend on the Ekman number. The results of laboratory experiments confirm the similarity properties of the bottom front propagation and agree well with the theoretical predictions.     

Anisotropic P-wave attenuation measured from a multi-azimuth surface seismic reflection survey

A system of aligned vertical fractures produces azimuthal variations in stacking velocity and amplitude variation with offset, characteristics often reported in seismic reflection data for hydrocarbon exploration. Studies of associated attenuation anisotropy have been mostly theoretical, laboratory or vertical seismic profiling based. We used an 11 common‐midpoint‐long portion of each of four marine surface‐seismic reflection profiles, intersecting each other at 45° within circa 100 m of a common location, to measure the azimuthal variation of effective attenuation, Q⁻¹_eff and stacking velocity, in a shallow interval, about 100 m thick, in which consistently orientated vertical fracturing was expected due to an underlying salt diapirism. We found qualitative and quantitative consistency between the azimuthal variation in the attenuation and stacking velocity, and published amplitude variation with offset results. The 135° azimuth line showed the least apparent attenuation (1000 Q⁻¹_eff= 16 ± 7) and the fastest stacking velocity, hence we infer it to be closest to the fracture trend: the orthogonal 45° line showed the most apparent attenuation (1000Q⁻¹_eff= 52 ± 15) and slowest stacking velocity. The variation of Q⁻¹_eff with azimuth φ is well fitted by 1000Q⁻¹_eff = 34 − 18cos[2(φ+40°)] giving a fracture direction of 140 ± 23° (±1SD, derived from ‘bootstrapping’ fits to all 11⁴ combinations of individual common‐midpoint/azimuth measurements), compared to 134 ± 47° from published amplitude variation with offset data. The effects of short‐window spectral estimation and choices of spectral ratio bandwidth and offset ranges used in attenuation analysis, individually give uncertainties of up to ±13° in fracture direction. This magnitude of azimuthal variation can be produced by credible crack geometries (e.g., dry cracks, radius 6.5 m, aspect ratio 3 × 10⁻⁵, crack density 0.2) but we do not claim these to be the actual properties of the interval studied, because of the lack of well control (and its consequences for the choice of theoretical model and host rock physical properties) and the small number of azimuths available here.     

Displacement and rectification of planetary fluids

Abstract

Laboratory experiments on the decay (spin-up) of fluid motion on the β-plane are compared with theory. Under weakly dissipative conditions, some particles conserve potential vorticity during the entire decay. We also study the rectified mean flow which is produced by the lateral Reynolds Stress when a low frequency force is applied to the planetary fluid. The possible connection of effects with oceanic phenomena is briefly discussed.     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r ₀, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

Rotating flow over long shallow ridges

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 ～ E^½, H/D ～ E⁰, h/D ～ E^½ 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.     

Comment on rotating hydraulics

Abstract

Numerical solutions of the axisymmetric flows during the relatively early phase of spin-up from rest of a stratified fluid in a cylinder are presented. Detailed results are given for a cylinder of aspect ratio of O(l) and for a minute Ekman number, showing axisymmetric spin-up for three values of the stratification parameter. As the stratification increases, the meridional circulation is confined to a region closer to the Ekman layers. An axisymmetric shear wave propagates radially inward from the sidewall, but, unlike the strictly vertical front for a homogeneous fluid, the interface which separates rotating from nonrotating fluid is bow-shaped. For a stratified fluid, the axial vorticity distribution is nonuniform both in the vertical and in the radial directions. With increasing stratification, diffusive vorticity production near the sidewall is more pronounced. Axisymmetric flows in the early phase of spin-up of a stratified fluid are controlled by both the inviscid dynamic effect and the viscous diffusion effect. At a location close to the Ekman layers, the inviscid effect outweighs the viscous effect, in much the same way as in a homogeneous fluid. However, at a location close to mid-depth, the viscous diffusion effect, enhanced by substantial flow gradients in that region, is dominant. This points to the necessity of including the direct effect of viscous diffusion in the interior in formulating an analytical model of stratified spin-up problems.     

Is rainfall intensity significant in the rainfall–runoff process within tropical rainforests of northeast Queensland? The Hewlett regression analyses revisited

Following the statistical analyses of long‐term rainfall‐runoff records from research basins in humid temperate latitudes, Hewlett and co‐workers extended the global challenge to disprove their findings that rainfall intensity was non‐significant. This paper responds to Hewlett's challenge as no preceding analyses have involved forested basins in a tropical cyclone‐prone area. Based on a 7 year rainfall‐runoff record, quickflow (QF), peak flow (QP) and quickflow response ratios (QRR) were regressed as dependent variables against rainfall parameters (intensity, P_i, amount, P), storm duration, D and antecedent flow, I. These data sets were categorised into total streamflow (Q) classes and stratified into three seasons, (monsoon, post‐monsoon and dry) for forested and cleared catchments. Where rainfall variable collinearity met acceptable levels, the addition of P_i to regression models including P, D, I contributed up to 9% and 66% of the respective variations in quickflow and peak flow. For the highest Q storm classes (monsoon), P_i alone accounted for up to 67% and 91% of the variation in Q_F and Q_P respectively and was the dominant influence on Q_P for all seasons. The very high rainfall intensities experienced in the monsoon season is a causal factor why these results differ from those of other research drainage basins. Surprisingly, P_i continued to have a significant influence on Q_F for dry season classes when less‐intense rainfall occurs. Further the results were similar for both catchments across all seasons. P was the dominant independent variable affecting Q_F above a threshold Q of 50 mm (monsoon), as rainfall contributes directly to saturation overland flow and return flow under saturated conditions. Further although QRR increased with increasing Q for each season, the regression results for that parameter were poor possibly due to the non‐linearity of the rainfall‐runoff relationship. Copyright © 2010 John Wiley & Sons, Ltd.