A theoretical investigation of bed-form shapes

The deformation of movable boundaries under the action of an applied turbulent shear stress is well known. The resulting bed forms often are highly organized and nearly two-dimensional, which makes them an intriguing focus of study considering that they are generated in both steady and oscillatory turbulent flows. Many past studies share a common approach in which an infinitesimal perturbation is prescribed and the resulting growth or decay patterns are examined. In this approach, the bed forms are usually sinusoidal and the perturbation analysis does not provide a theoretical prediction of equilibrium bed-form geometry. An alternative approach is suggested here in which the forcing terms (pressure and stress) are prescribed parametrically and the governing equations are solved for the flow velocity and the associated boundary deformation. Using a multilayered approach, in which the bottom boundary layer is divided into a discrete, yet, arbitrary number of finite layers, analytical solutions for the horizontal current and bed profile are derived. The derivations identify two nondimensional parameters, p₀/u₀² and ₀/kh₀u₀², which modulate the amplitude of the velocity fluctuations and boundary deformation. For the case of combined pressure and stress divergence anomalies, the magnitude of the front face and lee slopes exhibit an asymmetry that is consistent with observed bed forms in steady two-dimensional flows.Responsible Editor: Jens Kappenberg     

Migrating sand waves

A simple mathematical model is described, which reproduces the major features of sand waves' appearance and growth and in particular predicts their migration speed. The model is based on the linear stability analysis of the flat configuration of the sea bottom subject to tidal currents. Attention is focused on the prediction of the complex growth rate that bottom perturbations undergo because of both oscillatory fluid motions and residual currents. While the real part _r of controls the amplification or decay of the amplitude of the bedforms, the imaginary part _i is related to their migration speed. Previous works on the migration of the sand waves (Németh etal. 2002) consider a forcing tide made up by the M2 constituent (oscillatory period equal to 12 h) plus the residual current Z0 and predict always a downcurrent migration of the bedforms. However, field cases exist of upcurrent-migrating sand waves (downcurrent/upcurrent-migrating sand waves mean bedforms moving in the direction of the steady residual tidal current or in the opposite direction, respectively). The inclusion of a tide constituent characterized by a period of 6 h (M4) is the main novelty of the present work, which allows for the prediction of the migration of sand waves against the residual current Z0. Indeed, the M4 tide constituent, as does also the residual current Z0, breaks the symmetry of the problem forced only by the M2 tide constituent, and induces sand-wave migration. The model proposed by Besio etal. (2003a) forms the basis for the present analysis. Previous works on the subject (Gerkema 2000; Hulscher 1996a,b; Komarova and Hulscher 2000) are thus improved by using a new solution procedure (Besio etal. 2003a) which allows for a more accurate evaluation of the growth rate for arbitrary values of the parameter r, which is the ratio between the horizontal tidal excursion and the perturbation wavelength. Responsible Editor: Jens Kappenberg     

Implementation of configuration dependent stiffness proportional damping for the dynamics of rigid multi-block systems

The distinct element method (DEM) has been used successfully for the dynamic analysis of rigid block systems. One of many difficulties associated with DEM is modeling of damping. In this paper, new procedures are proposed for the damping modeling and its numerical implementation in distinct element analysis of rigid multi-block systems. The stiffness proportional damping is constructed for the prescribed damping ratio, based on the non-zero fundamental frequency effective during the time interval while the boundary conditions remain essentially constant. At this time interval, the fundamental frequency can be estimated without complete eigenvalue analysis. The damping coefficients will vary while the damping ratio remains the same throughout the entire analysis. A new numerical procedure is developed to prevent unnecessary energy loss that can occur during the separation phases. These procedures were implemented in the development of the distinct element method for the dynamic analyses of piled multi-block systems. The analysis results for the single-block and two-block systems were in a good agreement with the analytic predictions. Applications to the seismic analyses of piled fourblock systems revealed that the new procedures can make a significant difference and may lead to much-improved results.     

The Ensemble Kalman Filter: theoretical formulation and practical implementation

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.Responsible Editor: Jörg-Olaf Wolff     

A consistent derivation of the wave-energy equation from basic hydrodynamic principles

Based on a decomposition of the velocity into mean flow, turbulent and wave components, momentum and hereafter a wave-energy equation is derived. It contains a turbulent energy dissipation term which is closed by applying a wave-related mixing length model and linear wave theory solutions. This closure produces a non-linear turbulent wave-energy dissipation including the wave energy in a 5/2 power law. The theory is able to predict correctly the shape of deep-water wave spectra according to Phillips' similarity law.Responsible Editor: Hans Burchard     

Development and sensitivity analysis of a model for assessing stratification and safety of Lake Nyos during artificial degassing

To prevent the recurrence of a disastrous eruption of carbon dioxide (CO₂) from Lake Nyos, a degassing plan has been set up for the lake. Since there are concerns that the degassing of the lake may reduce the stability of the density stratification, there is an urgent need for a simulation tool to predict the evolution of the lake stratification in different scenarios. This paper describes the development of a numerical model to predict the CO₂ and dissolved solids concentrations, and the temperature structure as well as the stability of the water column of Lake Nyos. The model is tested with profiles of CO₂ concentrations and temperature taken in the years 1986 to 1996. It reproduces well the general mixing patterns observed in the lake. However, the intensity of the mixing tends to be overestimated in the epilimnion and underestimated in the monimolimnion. The overestimation of the mixing depth in the epilimnion is caused either by the parameterization of the k-epsilon model, or by the uncertainty in the calculation of the surface heat fluxes. The simulated mixing depth is highly sensitive to the surface heat fluxes, and errors in the mixing depth propagate from one year to the following. A precise simulation of the mixolimnion deepening therefore requires high accuracy in the meteorological forcing and the parameterization of the heat fluxes. Neither the meteorological data nor the formulae for the calculation of the heat fluxes are available with the necessary precision. Consequently, it will be indispensable to consider different forcing scenarios in the safety analysis in order to obtain robust boundary conditions for safe degassing. The input of temperature and CO₂ to the lake bottom can be adequately simulated for the years 1986 to 1996 with a constant sublacustrine source of 18 l s^–1 with a CO₂ concentration of 0.395 mol l^–1 and a temperature of 26 °C. The results of this study indicate that the model needs to be calibrated with more detailed field data before using it for its final purpose: the prediction of the stability and the safety of Lake Nyos during the degassing process.Responsible Editor: Hans Burchard     

Mineral-scale Sr-isotope constraints on magma evolution and chamber dynamics in the Rum layered intrusion,Scotland

Sr isotopic zoning within single plagioclase crystals from rocks from Unit 9 of the Rum layered intrusion is used to infer events during crystal growth in a magma undergoing contamination. The ⁸⁷Sr/⁸⁶Sr diversity among minerals and between cores and rims of plagioclase crystals increases as the boundary between unit 9 and the overlying Unit 10 peridotite is approached. Models of near-solidus interaction of the cumulate with a fluid or melt, or large scale textural re-equilibration, cannot easily account for the systematic differences in ⁸⁷Sr/⁸⁶Sr between small crystals and the rims of larger crystals.We propose a simple interpretation in which crystal growth is concentrated along the cool margins of the reservoir. Crystals are subsequently advected to a site of accumulation at the base of the reservoir, probably by episodic plume-like dense downwellings allowing mixing of isotopically zoned and unzoned crystals.If the core-rim isotope variations are inherited from primary magmatic growth, then the small distances over which they are now preserved (1–2 mm) place constraints on the minimum cooling rate of the intrusion. Although the length scale of diffusive equilibration is influenced by a number of poorly-constrained variables (starting temperature, feldspar composition, temperature-time path) cooling was clearly very rapid with cooling to effective closure (~1,000 °C) within a few thousand years.Editorial responsibility: I. Parsons     

Clinopyroxenes from mantle-related xenocrysts in alkaline basalts from Hannuoba (China): augite–pigeonite exsolutions and their thermal significance

Optically homogeneous augite xenocrysts, closely associated with spinel–peridotite nodules, occur in alkali basalts from Hannuoba (Hebei province, China). They were studied by electron and X-ray diffraction to define the occurrence and significance of pigeonite exsolution microtextures. Sub-calcic augite (Wo₃₄) exsolved into En_62–62Fs_25–21Wo_13–17 pigeonite and En_46–45Fs_14–14Wo_40–42 augite, as revealed by TEM through diffuse coarser (001) lamellae (100–300 Å) and only incipient (100) thinner ones (<70 Å). C2/c augite and P2₁/c pigeonite lattices, measured by CCD-XRD, relate through a(Aug)?a(Pgt), b(Aug)?b(Pgt), c(Aug)≠c(Pgt) [5.278(1) vs 5.189(1)Å] and β(Aug)≠β(Pgt) [106.55(1) vs 108.55(2)°]. Cell and site volumes strongly support the hypothesis that the augite xenocrysts crystallised at mantle depth from alkaline melts. After the augite xenocrysts entered the magma, (001) lamellae first formed by spinodal decomposition at a T_min of about 1,100 °C, and coarsened during very rapid transport to the surface; in a later phase, possibly on cooling, incipient (100) lamellae then formed.     

Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example, P–T petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (a–x) relations of minerals, melt and fluid in multicomponent systems. Although the symmetric formalism—a macroscopic regular model approach to a–x relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition. We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation. We term this the asymmetric formalism (ASF). In the symmetric formalism, the a–x relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase. In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry. In the case of fluid mixtures, for example, H₂O–CO₂, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the a–x relations. In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the a–x relations. In this way, we determine the a–x relations for feldspars in the systems KAlSi₃O₈–NaAlSi₃O₈ and KAlSi₃O₈–NaAlSi₃O₈–CaAl₂Si₂O₈, for carbonates in the system CaCO₃–MgCO₃, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg₂SiO₄–SiO₂, as well as for fluids in the system H₂O–CO₂. In each case the a–x relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully. The asymmetric formalism provides a powerful and flexible way of handling a–x relations of complex phases in multicomponent systems for petrological calculations.