The destabilising nature of differential rotation

Abstract

In a rapidly rotating, electrically conducting fluid we investigate the thermal stability of the fluid in the presence of an imposed toroidal magnetic field and an imposed toroidal differential rotation. We choose a magnetic field profile that is stable. The familiar role of differential rotation is a stabilising one. We wish to examine the less well known destabilising effect that it can have. In a plane layer model (for which we are restricted to Roberts number q = 0) with differential rotation, U = sΩ(z)1 _?, no choice of Ω(z) led to a destabilising effect. However, in a cylindrical geometry (for which our model permits all values of q) we found that differential rotations U = sΩ(s)1 _? which include a substantial proportion of negative gradient (dΩ/ds ≤ 0) give a destabilising effect which is largest when the magnetic Reynolds number R _m = O(10); the critical Rayleigh number, Ra _c, is about 7% smaller at minimum than at R_m = 0 for q = 10⁶. We also find that as q is reduced, the destabilising effect is diminished and at q = 10^?6, which may be more appropriate to the Earth's core, the effect causes a dip in the critical Rayleigh number of only about 0.001%. This suggests that we see no dip in the plane layer results because of the q = 0 condition. In the above results, the Elsasser number A = 1 but the effect of differential rotation is also dependent on A. Earlier work has shown a smooth transition from thermal to differential rotation driven instability at high A [A = O(100)]. We find, at intermediate A [A = O(10)], a dip in the Ra_c vs. R_m curve similar to the A = 1 case. However, it has Ra_c ≤ 0 at its minimum and unlike the results for high A, larger values of R_m result in a restabilisation.     

The use of MRS in the determination of hydraulic transmissivity: The case of alluvial aquifers

Magnetic Resonance Sounding (MRS) is nowadays accepted as a new geophysical method that can be used for a reliable determination of the ground water content distribution in the top 150 m. A great effort has also been made in MRS development to deduce the hydraulic transmissivity, based on empiric relationships of the permeability with a factor F which is calculated with NMR parameters measured at laboratory scale. To use this relationship under field conditions a calibration coefficient C_T = T_pt / F has to be previously established, which demands the knowledge of the transmissivity T_pt evaluated in the pumping test. The transmissivity can then be calculated at any other site of the same aquifer using the relation T_mrs = C_TF. The C_T values reported suggest a certain relationship with the lithology, but with a great dispersion and contradictory results. MRS surveys carried out in alluvial aquifers in Spain have shown that the value of C_T evaluated at one site may not be valid at another place of the same aquifer, because of the great heterogeneity of this kind of geological environment. The demand of a pumping test at each site where a MRS is measured invalidates the method actually used for MRS transmissivity evaluation. More than 50 MRS have been used to propose a new methodology. The aquifers visited cover a great range of transmissivities (from 2 × 10^− 6 to 9 × 10^− 3 m²/s). The MRS signal amplitude varies between 20 and 1400 nV, the signal/noise ratio is in the range from 0.6 to 42, and the value of the decay time constant varies from 200 to 800 ms. It has been demonstrated that when the transmissivity increases, the value of F decreases, and C_T increases, except for certain groups of MRS taken at the same aquifer or part of one aquifer, for which F increases with T_pt, keeping C_T constant. A function C_T(F) of the type C_T = mF^− n has been obtained that allows the transmissivity evaluation without the need of T_pt. Considering that both values of transmissivity, T_pt and T_mrs, are subjected to deviations due to the experimental errors as well as due to evaluation errors, the prediction achieved by the proposed equation is rather good. To perform a better evaluation of the values of the coefficients m and n it is necessary to have a greater number of MR soundings of good quality and with a trustworthy inversion at locations where a really comparable and good performed pumping test is available, covering a sufficient range of transmissivities. Though the data we have used do not always fulfil these conditions, the result is promising. Once a trustable function is available, the forecast of the transmissivity using MRS will not need the existence of any pumping test in the area. The general extension of this methodology demands the availability of MRS taken at all kinds of geological and hydrogeological environments, which is impossible without the existence of a universal MRS data base.     

The null distribution of sample serial correlation coefficient

 The null distribution of the lag-k sample serial correlation coefficient (r _k , k=1,2,3) was investigated by Monte Carlo simulation. For a time series with normal, exponential, Pearson 3, EV1 (Gumbel), or generalized Pareto (GP) distribution type, the null distribution of its r _k can be approximated by the normal distribution with mean −1/(n−k) and variance 1/(n−1). But for a time series with the lognormal, EV2 or EV3 (Weibull) distribution type, the null distribution of r _k is skewed distributed. In such cases, a simulation technique is suggested to construct percentile confidence intervals at a given significance level.     

The coalescence of water drops II. The coalescence problem and coalescence efficiency

By the use of the model of approaching drops (Arbel and Levin, 1977) the coalescence efficiencies of drops are computed. It is found that for interactions of drops at their terminal velocities the coalescence depends both on the size of the large drop and on the size ratio of the interacting drops in agreement with the experimental results of Whelpdale and List (1971) and Levin and Machnes (1977).The results were found to be sensitive to the assumption of the drops deformation and to the critical separation distance. This distance is defined as the distance at which the drops begin to merge. The variations of the coalescence efficiency with these parameters is discussed.Appendix: List of symbols D distance between the deformed surfaces of the drops - D _o initial value ofD - D _s stop distance, the distance at which the impact velocity vanishes - D _c critical coalescence distance - E collection efficiency - E ₁ collision efficiency - E ₂ coalescence efficiency - E _2R coalescence efficiency for collisions with stationary targets - F _c centrifugal force - p ratio of the radii of the interacting drops - r _o initial distance between drops' centers - R _L radius of larger drop - R _s radius of smaller drop - R _D radius of deformation - v approach velocity of two deformed surfaces - v _o initial value ofv - V _i impact velocity (given negative sign when drops approach each other) - V _c critical impact velocity - W _i velocity of the smaller drop at infinity for it to reachD _o with velocityv _o - x _i impact distance, the distance between the trajectories of the two drops - x _c critical impact distance for coalescence -  average critical impact distance for coalescence - X _c critical impact distance for collisions - coefficient of deformation given in equation 1 - _i impact angle defined byWhelpdale andList (1971) given also inArbel andLevin (1977) - coefficient of deformation given in equation 2 - viscosity of air - _i impact angle used inArbel andLevin (1977) and here - _c critical angle for coalescence - average critical angle for coalescence On sabbatical leave (1976–77) from the Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The coupling of quasi-linear pitch angle and energy diffusion

Wave–particle interactions are described in the quasi-linear formulation by bounce-averaged diffusion coefficients for equatorial pitch angle α₀ and for momentum p, along with a mixed or cross diffusion coefficient D_α0p. Because the cross terms complicate the associated time-dependent diffusion equation, and because D_α0p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, D_α0p becomes increasingly important for small values of f_pe/f_ce, as does D_pp, and for small widths of the wave frequency and wavenormal angle distributions. Here we use very simple models of diffusion to investigate the effect of the cross terms, and to demonstrate numerical problems associated with them.     

The ionosphere and solar activity

Summary The correlation between monthly median critical frequencies and solar activity was determined for Washington, D.C. Results were compared with those from a similar study made for six Arctic stations. For noon data (E, F1 andF2 layers), a greater value off _c (at zero sunspot number) and a slightly greater slope were obtained for Washington than for the Arctic locations. The influence of increased solar activity on the behavior of the midnight ionosphere is discussed.     

The effect of bed mobility on resistance to overland flow

When sediment grains are transported as bed load in overland flow, there is a net transfer of momentum from the flow to the grains. When these grains collide with other grains, whether on the bed or in the flow, streamwise flow velocity decreases and resistance to flow increases. Resistance to flow generated in this manner is termed bed‐load transport resistance. Resistance to flow f over a plane bed may be partitioned into grain resistance f_g and bed‐load transport resistance f_bt. We use the symbols f_btf and f_btm to denote f_bt for flows over fixed beds and over mobile beds, respectively, and we compute the effect of bed mobility on flow resistance f_mob by subtracting f_btf from f_btm. The data for this study come from 54 flume experiments with fixed beds and 38 with mobile beds. On average f_mob is approximately equal to half of f_btm, which is about one‐quarter of f. Hence, f_mob is about one‐tenth of f. Predictive equations are developed for f_btf, f_btm and f_mob using dimensional analysis to identify the relevant independent variables and regression analysis to evaluate the coefficients associated with these variables. Values of f_mob are always positive which implies that mobile beds offer greater resistance to flow than do fixed beds. Evidently bed‐load grains colliding with mobile beds lose more momentum to the bed than do grains colliding with fixed beds. In other words, grain collisions with mobile beds are less elastic than those with fixed beds. Copyright © 2005 John Wiley & Sons, Ltd.     

The vertical structure of tidal currents

Abstract

The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = ε and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition ε>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth.

The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity ε or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is shown to depend upon accurate specification of both eddy viscosity and the bed stress condition.     

The mean energetic level. theory

Summary One of the important atmospheric levels, the mean energetic level (MEL), which in a sense reflects the energetics of the whole atmosphere, is defined. Its fundamental properties are shown. In order to describe the MEL correctly a new vertical coordinate is introduced and discussed. The new coordinate, , is defined as the ratio of height and temperature. The MEL is shown to be a level with constant value of . Some incorrect conclusions concerning the MEL, derived in the past, have been corrected.List of symbols used c _p specific heat of air at constant pressure - c _v specific heat of air at constant volume - e base of natural logarithms - E total potential energy - f Coriolis parameter - g acceleration of gravity - i specific internal energy - I internal energy - J enthalpy - k unit vector pointing upwards - p pressure - Q diabatic heating rate - R gas constant of the air - t time - T temperature - v horizontal velocity - v ⁽³⁾ three-dimensional velocity - w vertical velocity in thez-system - z height - temperature growth rate (T/z) - Pechala's vertical coordinate (z/T) - generalized vertical velocity in the -system (d/dt) - specific potential energy - potential energy - density of the air - Ruppert function - T(1–)^–1 - ( ) _S quantity at the sea level - ( )* quantity at the MEL     

The role of the helicity spectrum function in turbulent dynamo theory

Abstract

It is shown that, for general homogeneous turbulence, the anti-symmetric part of the spectrum tensor can be expressed in terms of a single scalar function H(k,ω) (the helicity spectrum function). Under the first-order smoothing approximation, the coefficients α _ij β _ijk in the expansion of the mean electromotive force in terms of the mean magnetic field are determined; α _ij is a weighted integral of H(k,ω), and β _ijk contains a part β(a)ijk which is likewise a weighted integral of H(k, ω). When the turbulence is axisymmetric, β(a)ijk contains Rädler's (1969a) “Ω ∧ J-effect”. It is shown that when the turbulence is statistically symmetric about a plane perpendicular to the axis of symmetry, then βij = O but the Rädler effect is non-zero. Explicit expressions for αij and βijk are given when the velocity field is generated by random forcing in a rotating medium. Finally, it is shown by means of a local analysis that the Rädler effect, in conjunction with uniform mean shear, can give rise to non-oscillatory dynamo action, and it is argued that this effect may be significant in the well-mixed interior of a stellar convection zone, where by symmetry the α-effect may be weak.     

The anomalous surface wave in uniaxially-stressed elastic material

Summary For a semi-infinite incompressible elastic medium which is subjected to a large primary extension (or compression) in a directionOX ₁ parallel to the free surfaceOX ₁ X ₂, the dispersion equation is derived for surface waves of small amplitude. It is shown in particular that in some cases for directions of propagation close toOX ₂, the surface waves exhibit novel features in that the displacement at the surface is almost parallel toOX ₁, that is, nearly transverse to the direction of wave-propagation.     

The Displacement and Strain Field of Three-dimensional Rheologic Model of Earthquake Preparation

Based on the three-dimensional elastic inclusion model proposed by Dobrovolskii, we developed a rheological inclusion model to study earthquake preparation processes. By using the Corresponding Principle in the theory of rheologic mechanics, we derived the analytic expressions of viscoelastic displacement U(r, t) , V(r, t) and W(r, t), normal strains ε_xx (r, t), ε_yy (r, t) and ε_zz (r, t) and the bulk strain θ (r, t) at an arbitrary point (x, y, z) in three directions of X axis, Y axis and Z axis produced by a three-dimensional inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model. Subsequent to the spatial-temporal variation of bulk strain being computed on the ground produced by such a spherical rheologic inclusion, interesting results are obtained, suggesting that the bulk strain produced by a hard inclusion change with time according to three stages (α, β, γ) with different characteristics, similar to that of geodetic deformation observations, but different with the results of a soft inclusion. These theoretical results can be used to explain the characteristics of spatial-temporal evolution, patterns, quadrant-distribution of earthquake precursors, the changeability, spontaneity and complexity of short-term and imminent-term precursors. It offers a theoretical base to build physical models for earthquake precursors and to predict the earthquakes.     

The spatial and temporal changes inV p /V s before strong earthquakes in Kamchatka

A method was developed to obtain from a signal station the spatial and temporal distribution ofV _p /V _s ratios before earthquakes of magnitude>6. It was shown thatV _p /V _s values strongly depend upon the relative positions of the stations, the future large earthquake and the foci of the smaller earthquakes used forV _p /V _s determination. The appearance of a zone of anomalousV _p /V _s values with linear dimensions of the order of 100 km was noted at least 4 years before a deep earthquake of magnitude 7. Similar size anomalous zones were detected one year before some magnitude 6 earthquakes. V _p /V _s values decreased by a small but distinct amount during this time. Additionally, local short term minima inV _p /V _s ratios were observed some months before the major event. The epicenters of the large earthquakes were located within the 100 km size zone where the gradients of theV _p /V _s field were largest.     

The nonlinear stabilization of a zonal shear flow instability

Abstract

The development of initially small perturbations in a weakly supercritical zonal shear flow on a β-plane is studied. Two different scenarios of evolution are possible. If the supercriticality is sufficiently small, the growth of a perturbation is stopped in the viscous critical layer regime; for this case the evolution equation (corrected by the inclusion of a quintic nonlinearity) is derived. At greater supercriticality the nonlinearity cannot stop the growth of the perturbation in a linear (viscous or unsteady) critical layer regime, and the evolution is more complicated. Transition to a nonlinear critical layer regime leads to a reduction in the growth rate and to a slowing (but not a stopping) of the increase in amplitude, A. These are connected to the formation of a plateau (S=constant) of width L=O(A ^½) in the profile of absolute vorticity, S. Careful analysis reveals that the growth in amplitude ceases only when the whole instability domain (where the slope of unperturbed S-profile is positive) becomes covered again by the plateau.     

The role of wavegroup velocity in the blocking problem

Considering the blocking problem as a baroclinic instability problem in a dispersive wave system with diabatic heating effects, it is of great interest to investigate the role of wavegroup velocityv _gr in blocking processes, becausev _gr controls the energy transfer in the wave field. Using a Newtonian Cooling —type of forcing with a phase differencek to the main field and taking the linearized version of a two-level model, the phase speedc _r, the group velocityv _gr and the growth ratekc _i have been obtained as analytical functions of the mean zonal windU, the thermal windU _T, the coefficient of diabatic heating x, the phase differencek and the wavelengthL. Now the hypothesis is introduced, that a blocking should be expected, ifv _gr has a maximum value in the vicinity ofL _o, for whichc _r vanishes and thee-folding timet=1/kc _i (kc _i>0) is smaller than 6 days (see condition (20) in the text). One finds, that for special parameter combinations (U _T, U, ), where 15 m/secU _T25m/sec,U=10m/sec, 0.8·10^–51.5·10^–5 [sec^–1], certain valuesL _o with an appropriate phase differencek exist, which satisfy the conditions mentioned above (for values see Table 2). TherebyL _o varies within the range 8500 km <L _o<11000 km corresponding to the preferred planetary blocking wavenumber 2 in middle latitudes 50°<<70° N.     

The size of the auroral belt during magnetic storms

Using the auroral boundary index derived from DMSP electron precipitation data and the Dst index, changes in the size of the auroral belt during magnetic storms are studied. It is found that the equatorward boundary of the belt at midnight expands equatorward, reaching its lowest latitude about one hour before Dst peaks. This time lag depends very little on storm intensity. It is also shown that during magnetic storms, the energy of the ring current quantified with Dst increases in proportion to L_e^–3, where L_e is the L-value corresponding to the equatorward boundary of the auroral belt designated by the auroral boundary index. This means that the ring current energy is proportional to the ion energy obtained from the earthward shift of the plasma sheet under the conservation of the first adiabatic invariant. The ring current energy is also pronortional to E_mag, the total magnetic field energy contained in the spherical shell bounded by L_e and L_eq, where L_eq corresponds to the quiet-time location of the auroral precipitation boundary. The ratio of the ring current energy E_R to the dipole energy E_mag is typically 10%. The ring current leads to magnetosphere inflation as a result of an increase in the equivalent dipole moment.     

The deepening of the wind-Mixed layer

Abstract

A simple model is given that describes the response of the upper ocean to an imposed wind stress. The stress drives both mean and turbulent flow near the surface, which is taken to mix thoroughly a layer of depth h, and to erode the stably stratified fluid below. A marginal stability criterion based on a Froude number is used to close the problem, and it is suggested that the mean momentum has a strong role in the mixing process. The initial deepening is predicted to obey

where u. is the friction velocity of the imposed stress, N the ambient buoyancy frequency, and t the time.

After one-half inertial period the deepening is arrested by rotadeon at a depth h = 2^2/4 u.{(Nf)⁺

where f is the Coriolis frequency. The flow is then a “mixed Ekman” layer, with strong inertial oscillations superimposed on it. Three quarters of the mean energy of the deepening layer is found to be kinetic, and only one-quarter potential.

Heating and cooling are included in the model, but stress dominates for time-scales of a day or less. Non-uniform stratification and currents existing prior to the onset of the wind are easily included.

Agreement between the first formula above and laboratory experiments of Kato and Phillips is very satisfactory; the second formula is consistent with observations of Francis and Stommel, though a more thorough test is needed. Oceanic observations in general support the assumption of slab-like mean profiles and direct response of the fluid to local winds.     

The effect of long-term aerial exposure on intertidal mudflat erodibility

Intertidal zones by definition are exposed to air at low tide, and the exposure duration can be weeks (e.g. during neap tides) depending on water level and bed elevation. Here we investigated the effect of varying exposure duration (6 h to 10 days) on intertidal mudflat erosion (measured using the EROMES device), where the effects of water content and biofilm biomass (using chlorophyll-a content as a proxy, Chl-a μg g⁻¹) were taken into account. Sediments were collected between spring and summer (in October 2018, January 2019 and February 2019) from an intertidal site in the Firth of Thames, New Zealand. Longer exposure duration resulted in more stable sediments [higher erosion threshold (Ƭ_cr, N m⁻²) and lower erosion rate (ER, g m⁻² s⁻¹)]. After 10 days, exposure increased Ƭ_cr by 1.7 to 4.4 times and decreased ER by 11.6 to 21.5 times compared with 6 h of exposure. Chl-a and water content changed with exposure duration and were significantly correlated with changes in Ƭ_cr and ER. The stability of sediments after two re-submersion periods following exposure was also examined and showed that the stabilizing effect of exposure persisted even though water content had increased to non-exposure levels. Re-submersion was associated with an increase in Chl-a content, which likely counteracted the destabilizing influence of increased water content. A site-specific model, which included the interplay between evaporation and biofilm biomass, was developed to predict water content as a function of exposure duration. The modelled water content (W_Mod.) explained 98% of the observed variation in water content (W_Obs.). These results highlight how the exposure period can cause subtle changes to erosion regimes of sediments. An understanding of these effects (e.g. in sediment transport modelling) is critical to predicting the resilience of intertidal zones into the future, when sea-level rise is believed to exacerbate erosion in low-lying areas. © 2020 John Wiley & Sons, Ltd.     

The application of uranium series disequilibrium concepts to sediment yield determination

The determination of uranium series disequilibria in fluvial environments is proposed as a method of calculating catchment mass balances. The technique is based on two main principles. Firstly, ²³⁴U is more mobile than ²³⁸U, especially during the early stages of weathering. Secondly, uranium is far more mobile than either thorium or protactinium. Consequently, teaching during weathering results in the loss of the uranium found in the fresh rock, leaving the two immobile daughters behind. The ratio of uranium carried by sediment to that dissolved, U_S/U_W can, therefore, be determined from river water and sediment isotopic activity ratios. Fluxes of uranium can then be calculated from average concentrations in the water and the associated sediment, from which a sediment yield can be inferred. The Witham catchment in Lincolnshire has been used to test the proposed method. A U_S/U_W ratio of between 5 and 7 is determined and a sediment yield of 2.51 ± 2.12 tonnes yr^?1 km^?2 is proposed. Although some problems concerning environmental chemistry have arisen, the validity of the approach is confirmed by the close correspondence between the results obtained and those inferred by earlier workers using more conventional methods.