Form-drag instability in a barotropic and a baroclinic atmosphere

We discuss the form-drag instability for a quasi-geostrophic channel flow. We first study the characteristics of this instability in a barotropic flow, considering in detail the influence of the meridional scale and discussing which structure of the perturbation zonal flow must be chosen in order to describe properly this instability.We then consider a continuous quasi-geostrophic channel model in which the topography enters only through the bottom boundary condition, and we discuss how in this case the effects of the form-drag are felt by the mean zonal flow through the ageostrophic mean meridional circulation. Because the meridional structure of the perturbation zonal flow cannot simply be extended from the barotropic to the continuous case, we show how to modify it properly.We then study the baroclinic model in the particular case of constant (in the vertical) basic-state zonal flow and show how this case closely resembles the barotropic, demonstrating the barotropic nature of the form-drag instability.Symbols _t is the partial derivative with respect tot. - _x is the partial derivative with respect tox. - _y is the partial derivative with respect toy. - represents the geostrophic stream function. - u is the eastward component of the geostrophic wind. - v is the northward component of the geostrophic wind. - u _a is the eastward component of the ageostrophic wind. - v _a is the northward component of the ageostrophic wind. - w is the vertical component of the wind. - f is the Coriolis parameter=2 sin f _o+y. - f _o is the Coriolis parameter evaluated at mid-latitude. - N is the Brunt-Vaisala frequency. - [A] is the zonal (x) average ofA at constantp andy. - <A> is the horizontal (x andy) average ofA at constantp     

A scaling law of internal run-up duration

Wave tank experiments with long internal waves of elevation, of different initial length l, moving in a two-fluid system, interacting with a weak slope of 0.045 rad, show an onshore flow of the dense water, at the undisturbed pycnocline-slope intersection, of duration $11.3\sqrt{l/g'}Wave tank experiments with long internal waves of elevation, of different initial length l, moving in a two-fluid system, interacting with a weak slope of 0.045 rad, show an onshore flow of the dense water, at the undisturbed pycnocline-slope intersection, of duration 11.3?{l/g￠}11.3\sqrt{l/g'} (g′ reduced gravity). This period corresponds to that of a strong bottom current event measured in the stratified ocean at the Ormen Lange gas field, at 850 m depth, lasting for 24 hrs, corresponding to 11.2?{l/g￠}11.2\sqrt{l/g'}, using the width l = 300 km of the Norwegian Atlantic Current (NAC) at the site as length scale, suggesting a lateral sloshing motion of the NAC causing the event. The onshore velocity of the dense fluid has a maximal velocity of 0.4?{g￠h₂}0.4\sqrt{g'h_2} in laboratory and 0.5 ms^-1=0.3?{g￠h₂}^{-1}=0.3\sqrt{g'h_2} in the field (h ₂ mixed upper layer thickness). Run-up of the dense fluid, beyond the undisturbed pycnocline-slope intersection, has initially a front velocity of 0.35?{g￠h₂}0.35\sqrt{g'h_2}, corresponding to the velocity of the head of a density current on a flat bottom. Due to disintegration, an initially depressed pycnocline results in comparatively smaller run-up and velocity. While moving past the turning point, a dispersive wave train is formed in the back part of the depression wave, developing by breaking into a sequence of up to eight boluses moving by the undisturbed pycnocline-slope intersection.     

Approximate expression for the hilbert transform of a certain class of functions and their application to the ray theory of seismic waves

Summary Approximate expressions for the Hilbert transform of the functionf(t)=exp(- ₀ ² t ²/²) cos( ₀ t+v) are determined. This function, given a suitable choice of the three parameters ₀, and v, approximates a wide class of seismic signals very well. The approximate expressions for the Hilbert transform enable very simple formulae to be given for the elementary seismograms of the individual seismic body waves (in the zero approximation of the ray theory). This accelerates the computation of ray theoretical seismograms considerably.     

Mass transport in rotatory tidal currents

Summary A perturbation method is used to calculate steady non-linear effects in rotatory tidal currents. The inclusion of the Coriolis acceleration leads to a uniquely determined second order current system in the tidal stream. This is shown to be absent in high frequency gravity waves. It is found that particle drift, or mass transport, is critically dependent on this mean current, and consequently on the value off/,f being the Coriolis parameter, and the angular frequency of the oscillation.     

Effects of the bottom boundary condition in numerical investigations of dense water cascading on a slope

The flow of dense water along continental slopes is considered. There is a large literature on the topic based on observations and laboratory experiments. In addition, there are many analytical and numerical studies of dense water flows. In particular, there is a sequence of numerical investigations using the dynamics of overflow mixing and entrainment (DOME) setup. In these papers, the sensitivity of the solutions to numerical parameters such as grid size and numerical viscosity coefficients and to the choices of methods and models is investigated. In earlier DOME studies, three different bottom boundary conditions and a range of vertical grid sizes are applied. In other parts of the literature on numerical studies of oceanic gravity currents, there are statements that appear to contradict choices made on bottom boundary conditions in some of the DOME papers. In the present study, we therefore address the effects of the bottom boundary condition and vertical resolution in numerical investigations of dense water cascading on a slope. The main finding of the present paper is that it is feasible to capture the bottom Ekman layer dynamics adequately and cost efficiently by using a terrain-following model system using a quadratic drag law with a drag coefficient computed to give near-bottom velocity profiles in agreement with the logarithmic law of the wall. Many studies of dense water flows are performed with a quadratic bottom drag law and a constant drag coefficient. It is shown that when using this bottom boundary condition, Ekman drainage will not be adequately represented. In other studies of gravity flow, a no-slip bottom boundary condition is applied. With no-slip and a very fine resolution near the seabed, the solutions are essentially equal to the solutions obtained with a quadratic drag law and a drag coefficient computed to produce velocity profiles matching the logarithmic law of the wall. However, with coarser resolution near the seabed, there may be a substantial artificial blocking effect when using no-slip.     

Predicting earthquakes by analyzing accelerating precursory seismic activity

During 11 sequences of earthquakes that in retrospect can be classed as foreshocks, the accelerating rate at which seismic moment is released follows, at least in part, a simple equation. This equation (1) is ,where is the cumulative sum until time,t, of the square roots of seismic moments of individual foreshocks computed from reported magnitudes;C andn are constants; andt _fis a limiting time at which the rate of seismic moment accumulation becomes infinite. The possible time of a major foreshock or main shock,t _f,is found by the best fit of equation (1), or its integral, to step-like plots of versus time using successive estimates oft _fin linearized regressions until the maximum coefficient of determination,r ²,is obtained. Analyzed examples include sequences preceding earthquakes at Cremasta, Greece, 2/5/66; Haicheng, China 2/4/75; Oaxaca, Mexico, 11/29/78; Petatlan, Mexico, 3/14/79; and Central Chile, 3/3/85. In 29 estimates of main-shock time, made as the sequences developed, the errors in 20 were less than one-half and in 9 less than one tenth the time remaining between the time of the last data used and the main shock. Some precursory sequences, or parts of them, yield no solution. Two sequences appear to include in their first parts the aftershocks of a previous event; plots using the integral of equation (1) show that the sequences are easily separable into aftershock and foreshock segments. Synthetic seismic sequences of shocks at equal time intervals were constructed to follow equation (1), using four values ofn. In each series the resulting distributions of magnitudes closely follow the linear Gutenberg-Richter relation logN=a–bM, and the productn timesb for each series is the same constant. In various forms and for decades, equation (1) has been used successfully to predict failure times of stressed metals and ceramics, landslides in soil and rock slopes, and volcanic eruptions. Results of more recent experiments and theoretical studies on crack propagation, fault mechanics, and acoustic emission can be closely reproduced by equation (1). Rate-process theory and continuum damage mechanics offer leads toward understanding the physical processes.     

Wave boundary layers in rotating stratified fluid and near-inertial oscillations

Theory of wave boundary layers (WBLs) developed by Reznik (J Mar Res 71: 253–288, 2013, J Fluid Mech 747: 605–634, 2014, J Fluid Mech 833: 512–537, 2017) is extended to a rotating stratified fluid. In this case, the WBLs arise in the field of near-inertial oscillations (NIOs) driven by a tangential wind stress of finite duration. Near-surface Ekman layer is specified in the most general form; tangential stresses are zero at the lower boundary of Ekman layer and viscosity is neglected below the boundary. After the wind ceases, the Ekman pumping at the boundary becomes a linear superposition of inertial oscillations with coefficients dependent on the horizontal coordinates. The solution under the Ekman layer is obtained in the form of expansions in the vertical wave modes. We separate from the solution a part representing NIO and demonstrate development of a WBL near the Ekman layer boundary. With increasing time t, the WBL width decays inversely proportional to \( \sqrt{t} \) and gradients of fields in the WBL grow proportionally to \( \sqrt{t} \); the most part of NIO is concentrated in the WBL. Structure of the WBL depends strongly on its horizontal scale L determined by scale of the wind stress. The shorter the NIO is, the thinner and sharper the WBL is; the short-wave NIO with L smaller than the baroclinic Rossby scale L_R does not penetrate deep into the ocean. On the contrary, for L?≥?L_R, the WBL has a smoother vertical structure; a significant long-wave NIO signal is able to reach the oceanic bottom. An asymptotic theory of the WBL in rotating stratified fluid is suggested.     

THE ESTIMATION OF SIGNAL SPECTRA AND RELATED QUANTITIES BY MEANS OF THE MULTIPLE COHERENCE FUNCTION *

A seismic trace recorded with suitable gain control can be treated as a stationary time series. Each trace, χ_j(t), from a set of traces, can be broken down into two stationary components: a signal sequence, α_j(t) *s(t—τ_j), which correlates from trace to trace, and an incoherent noise sequence, n_j(t), which does not correlate from trace to trace. The model for a seismic trace used in this paper is thus χ_j(t) =α_j(t) * s(t—τ_j) +n_j(t) where the signal wavelet α_j(t), the lag (moveout) of the signal τ_j, and the noise sequence n_j(t) can vary in any manner from trace to trace. Given this model, a method for estimating the power spectra of the signal and incoherent noise components on each trace is presented. The method requires the calculation of the multiple coherence function γ_j(f) of each trace. γ_j(f) is the fraction of the power on traced at frequency f that can be predicted in a least-square error sense from all other traces. It is related to the signal-to-noise power ratio ρ_j(f) by where K_j(f) can be computed and is in general close to 1.0. The theory leading to this relation is given in an Appendix. Particular attention is paid to the statistical distributions of all estimated quantities. The statistical behaviour of cross-spectral and coherence estimates is complicated by the presence of bias as well as random deviations. Straightforward methods for removing this bias and setting up confidence limits, based on the principle of maximum likelihood and the Goodman distribution for the sample multiple coherence, are described. Actual field records differ from the assumed model mainly in having more than one correctable component, components other than the required sequence of reflections being lumped together as correlated noise. When more than one correlatable component is present, the estimate for the signal power spectrum obtained by the multiple coherence method is approximately the sum of the power spectra of the correlatable components. A further practical drawback to estimating spectra from seismic data is the limited number of degrees of freedom available. Usually at least one second of stationary data on each trace is needed to estimate the signal spectrum with an accuracy of about 10%. Examples using synthetic data are presented to illustrate the method.     

Frontal upwelling in a rotating two-layer fluid

Abstract

This paper describes the source-sink driven flow in a two-layer fluid confined in a rotating annulus. Light fluid is injected at the inner wall, while denser fluid is withdrawn at the outer wall. The interface between the immiscible fluids intersects the bottom and thus produces a front. The net transport from the source to the sink is carried by Ekman layers at the bottom and at the interface, and by Stewartson layers at the side walls. A detached Stewartson layer arises at the front, leading to a pronounced upwelling circulation.     

Grain-growth kinetics in wadsleyite: Effects of chemical environment

Grain-growth kinetics in wadsleyite was investigated using a multianvil high-pressure apparatus. Fine-grained wadsleyite aggregates were synthesized by isostatic hot-pressing and were subsequently annealed under high pressure and temperature in a controlled chemical environment. Wadsleyite samples show normal grain-growth characterized by a log-normal grain-size distribution following the relation, where n is a constant, L the grain-size at time t, L₀ the grain-size at time t = 0 and k is a rate constant that depends on temperature T and chemical environments (fO₂: oxygen fugacity in Pa, C_OH: water content in H/10⁶Si) as:

     

A multivariate gamma distribution arising from a Markov model

A Markov chain{X _t }, which has been useful for modelling in hydrology, can be specified by the Laplace transform (LT) of the conditional p.d.f. ofX _t+1 givenX _t =x _t , which is assumed to be of the exponential formH()exp{-G()x _t }. For appropriate choice ofH andG the marginal distribution ofX _t is the (univariate) gamma distribution. In this case, the joint p.d.f. ofX _t +1,...,X _t+n and its LT, are obtained, and this is extended to a seasonal version of the chain. A simple method of generating observations from these multivariate gamma distributions is noted, and the joint LT is applied to the problem of determining moments of weighted sums of such variables.     

Falling sphere observations of anisotropic gravity wave motions in the upper stratosphere over Australia

A study of inertial scale gravity wave motions in the region of the atmosphere between 30 and 60 km has been undertaken, using wind and temperature data derived from rocket-borne falling sphere density experiments performed over Woomera, Australia between 1962 nad 1976. The gross features of the wave field compare favorably with those found in similar northern hemispheric studies. Wave propagation is found to be both vertically and horizontally anisotropic. A rotary spectral analysis indicates predominately upgoing wave energy, suggesting that the majority of sources of these waves lie below 30 km. A detailed statistical investigation of the waves, made using the Stokes parameters technique, reveals that phase progression is also highly directional in the horizontal, with a significant zonal component in summer, but with a strong meridional component in winter. Propagation towards the southeast is inferred in summer, with the waves possibly emanating from tropospheric sources in equatorial regions to the north of Australia. The technique also shows that, on average, the waves appear to have mean ellipse eccentricities (=f/) around 0.4–0.45. Indirect estimates of a number of important wave parameters are made. In particular,v andw flux estimates are made over several height intervals. The vertical gradient of density weighted flux implies wave-induced mean flow accelerations of the order 0.1–1 ms^–1day^–1. This suggests that dissipating gravity waves are a significant source of the momentum residuals that are encountered in studies of satellite data from this region.     

The motion of magnetic fronts in spiral galaxies

Abstract

We study the nonlinear asymptotic thin disc approximation to the mean field dynamo equations, as applicable to spiral galaxies. The circumstances in which sharp magnetic field structures (fronts) can propagate radially are investigated, and an expression for the speed of propagation derived. We find that the speed of an interior front is proportional to η//R _? (where η is the diffusivity and R_t the galactic radius), whereas an exterior front moves with speed of order , where γ is the local growth rate of the dynamo. Numerical simulations are presented, that agree well with our asymptotic results. Further, we perform numerical experiments using the 'no-z' approximation for thin disc dynamos, and show that the propagation of magnetic fronts in this approximation can also be understood in terms of our asymptotic results.     

A study of the upward continuation of gravity data from a plane surface

Summary A summation method of upward continuation of gravity data has been considered under the assumption that observations are available at regular intervals. The upward continued value has been obtained as the sum of products of individual gravity values and corresponding theoretical coefficients. Besides the usual parameter involving horizontal and vertical distances, the theoretical coefficients have been generalized to be dependent also on i) the order of a low order polynomial assumed to represent the gravity variation around a grid point and ii) the weights assigned to the gravity values at the nearest four grid points used for least-squares determination of the polynomial. Since the observations in practical cases are available over a finite area only, the effect of truncation of the area of the integration has also been discussed separately. The method has been programmed and tested on a three-dimensional model, whose true gravity effects were computed at various levels over a finite area. Upward continued values have been computed under various assumptions about the gravity field in the outside region. Comparisons of these results with the true values indicate that the truncation effect becomes increasingly important for larger values of the ratio of elevation to grid separation and/or when the gravity field is not a local one. It has also been found that the greater is the above ratio, the less important is the effect of weights on the theoretical coefficients and practically vanishes(<10 ^–4 ) when the ratio is greater than5.0.     

The equatorial dynamics of a shallow,homogeneous ocean

Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ^? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ^? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ^5/3), then inertial effects become important at a distance ?^2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.     

Travel times for a linear variation of the velocity of bodily waves

Summary The expression of the travel timet of bodily waves propagating in a layered earth, as a function of the emerging angle is obtained, to complement the one given in a previous paper [1]²) of the epicentral distance . The waves velocityV is again assumed to vary linearly with the depth. Having both expressions it becomes possible to construct a theoretical functiont=t() and to explore the true values ofV by adjusting it to the experimental travel time tables.     

Vertical motion and diabatic heating over the Indian monsoon region during the Bay of Bengal depression of 5–8 July, 1979

The vertical velocity, , and the diabatic heating were computed at 800, 600, 400 and 200 mb surfaces using the Omega equation. The highest contribution to is from the diabatic heating produced by condensation associated with the precipitations appearing to be the main source of diabatic heating. The net radiative cooling and the thermal advection in the upper troposphere over the warm anticyclone result in diabatic cooling over the eastern part of the Bay of Bengal and adjoining northern and eastern regions.List of Symbols Used C _p Heat capacity at constant pressure - f Coriolis parameter - g Acceleration due to gravity - P Atmospheric pressure - Q Diabatic heating rate per unit mass - R Gas constant of air - S Static stability parameter - t Time - U, V Zonal and meridional wind components - Specific volume - Relative vorticity - Absolute vorticity - Potential temperature - Geopotential - Vertical velocity (dP/dt) - ₁ Adiabatic vertical velocity - ₂ Vertical velocity due to certain forcing - ₃ Diabatic vertical velocity - Isobaric gradient operator - ² Laplacian operator - J(A, B) Jacobian operator     

An application of the time- and magnitude-predictable model to long-term earthquake prediction in eastern Anatolia

In order to estimate the recurrence intervals for large earthquakes occurring in eastern Anatolia, this region enclosed within the coordinates of 36^∘–42^∘N, 35^∘–45^∘E has been separated into nine seismogenic sources on the basis of certain seismological and geomorphological criteria, and a regional time- and magnitude-predictable model has been applied for these sources. This model implies that the magnitude of the preceding main shock which is the largest earthquake during a seismic excitation in a seismogenic source governs the time of occurrence and the magnitude of the expected main shock in this source. The data belonging to both the instrumental period (M_S≥ 5.5) until 2003 and the historical period (I₀≥ 9.0 corresponding to M_S≥ 7.0) before 1900 have been used in the analysis. The interevent time between successive main shocks with magnitude equal to or larger than a certain minimum magnitude threshold were considered in each of the nine source regions within the study area. These interevent times as well as the magnitudes of the main shocks have been used to determine the following relations: