The surface roughness and planetary boundary layer

Applications of the entrainment process to layers at the boundary, which meet the self similarity requirements of the logarithmic profile, have been studied. By accepting that turbulence has dominating scales related in scale length to the height above the surface, a layer structure is postulated wherein exchange is rapid enough to keep the layers internally uniform. The diffusion rate is then controlled by entrainment between layers. It has been shown that theoretical relationships derived on the basis of using a single layer of this type give quantitatively correct factors relating the turbulence, wind and shear stress for very rough surface conditions. For less rough surfaces, the surface boundary layer can be divided into several layers interacting by entrainment across each interface. This analysis leads to the following quantitatively correct formula compared to published measurements. 1 $$\begin{gathered} \frac{{\sigma _w }}{{u^* }} = \left( {\frac{2}{{9Aa}}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \left( {1 - 3^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{a}{k}\frac{{d_n }}{z}\frac{{\sigma _w }}{{u^* }}\frac{z}{L}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \hfill \\ = 1.28(1 - 0.945({{\sigma _w } \mathord{\left/ {\vphantom {{\sigma _w } {u^* }}} \right. \kern-\nulldelimiterspace} {u^* }})({z \mathord{\left/ {\vphantom {z L}} \right. \kern-\nulldelimiterspace} L})^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \hfill \\ \end{gathered} $$ where \(u^* = \left( {{\tau \mathord{\left/ {\vphantom {\tau \rho }} \right. \kern-0em} \rho }} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) , σ _w is the standard deviation of the vertical velocity,z is the height andL is the Obukhov scale lenght. The constantsa, A, k andd _n are the entrainment constant, the turbulence decay constant, Von Karman's constant, and the layer depth derived from the theory. Of these,a andA, are universal constants and not empirically determined for the boundary layer. Thus the turbulence needed for the plume model of convection, which resides above these layers and reaches to the inversion, is determined by the shear stress and the heat flux in the surface layers. This model applies to convection in cool air over a warm sea. The whole field is now determined except for the temperature of the air relative to the water, and the wind, which need a further parameter describing sea surface roughness. As a first stop to describing a surface where roughness elements of widely varying sizes are combined this paper shows how the surface roughness parameter,z ₀, can be calculated for an ideal case of a random distribution of vertical cylinders of the same height. To treat a water surface, with various sized waves, such an approach modified to treat the surface by the superposition of various sized roughness elements, is likely to be helpful. Such a theory is particularly desirable when such a surface is changing, as the ocean does when the wind varies. The formula, 2 $$\frac{{0.118}}{{a_s C_D }}< z_0< \frac{{0.463}}{{a_s C_D (u^* )}}$$ is the result derived here. It applies to cylinders of radius,r, and number,m, per unit boundary area, wherea _s =2rm, is the area of the roughness elements, per unit area perpendicular to the wind, per unit distance downwind. The drag coefficient of the cylinders isC _D . The smaller value ofz _o is for large Reynolds numbers where the larger scale turbulence at the surface dominates, and the drag coefficient is about constant. Here the flow between the cylinders is intermittent. When the Reynolds number is small enough then the intermittent nature of the turbulence is reduced and this results in the average velocity at each level determining the drag. In this second case the larger limit forz ₀ is more appropriate.     

The boundary value problem of Poisson’s equation with Stokes’ boundary condition

The following Poisson’s equation with the Stokes’ boundary condition is dealt with $$\left\{ \begin{gathered} \nabla ^2 T = - 4\pi Gp outside S, \hfill \\ \left. {\frac{{\partial T}}{{\partial h}} = \frac{1}{\gamma }\frac{{\partial y}}{{\partial h}}T} \right|_s = - \Delta g, \hfill \\ T = O\left( {r^{ - 3} } \right) at infinity, \hfill \\ \end{gathered} \right.$$ whereS is reference ellipsord. Under spherical approximation transformation, the ellipsoidal correction terms about the boundary condition, the equation and the density in the above BVP are respectively given. Therefore, the disturbing potentialT can he obtained if the magnitudes aboveO(ε⁴) are neglected.     

Electromagnetic induction in the earth due to stationary and moving sources

A new approach to the theory of electromagnetic induction is developed that is applicable to moving as well as stationary sources. The source field is considered to be a standing wave generated by two waves travelling in opposite directions along the surface of the earth. For a stationary source the incident waves have velocities of the same magnitude, however for a moving source the velocities of the two incident waves are respectively increased and decreased by the velocity of the source. Electromagnetic induction in the earth is then considered as refraction of these waves and gives, for both stationary and moving sources, the magnetotelluric relation: $$\frac{{ - E_y }}{{H_x }} = \left( {\frac{{i\omega \mu }}{\sigma }} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - i\frac{{v^2 }}{{\omega \mu \sigma }}} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where ν is the wavenumber of the source, μ is the permeability (4π·10^?7) and σ is the conductivity of the earth. ω is the angular frequency of the variation observed on the earth. For a stationary source the observed frequency is the same as the source frequency, however the effect of moving a time-varying source is to make the observed frequency different from the frequency of the source. Failure to recognise this in previous studies led to some erroneous conclusions. This study shows that a moving source isnot “electromagnetically broader” than a stationary source as had been suggested.     

Reoccurrence formulas for the largest earthquake magnitudes based on the modified cumulative frequency-magnitude relationship

A modified formula of the cumulative frequency-magnitude relation has been formulated and tested in a previous paper by the authors of this study. Based on the modified relationship, the following reoccurrence formulas have been obtained.

1. For the ‘T-years period’ larger earthquake magnitude,M _T $$M_T = \frac{1}{{A_3 }}ln\frac{{A_2 }}{{(1/T) + A_1 }}.$$
2. For the value of the maximum earthquake magnitude, which is exceeded with probabilityP inT-years period,M _PT $$M_{PT} = \frac{{ln(A_2 .T)}}{{A_3 }} - \frac{{ln[A_1 .T - ln(1 - P)]}}{{A_3 }}.$$
3. For the probability of occurrence of an earthquake of magnitudeM in aT-years period,P _MT $$P_{MT} = 1 - \exp [ - T[ - A_1 + A_2 \exp ( - A_3 M)]].$$

The above formulas provide estimates of the probability of reoccurrence of the largest earthquake events which are significantly more realistic than those based on the Gutenberg-Richter relationships; at least for numerous tested earthquake samples from the major area of Greece.     

Characteristics of rain electricity in Nigeria II — Experimental and theoretical relations

Analysis of data, covering four rainy seasons, of rain current, point-discharge current and potential gradient reveal novel relations in the form (i) $$Q_{r + } /Q_{r - } = k_1 (T_{r + } /T_{r - } )^{1.1} $$ for rain charge and duration ratios; and (ii) $$Q_{p - } /Q_{p + } = k_2 (T_{p - } /T_{p + } )^{1.1} $$ for point charge and duration ratios, where thek's are constants; and (iii) $$i_r = - \alpha (i_p - c)$$ for rain and point-discharge current densities, where α has the same value for all types of rain andc is a constant controlled by the rainfall intensityR. For rain not associated with point discharge the relation takes the familiar form $$i_r = - AR(E - \bar E)$$ Theoretical values are obtained for \ga andA on the basis of the Wilson ion-capture theory as worked out in detail by Whipple and Chalmers.     

Application of the Pi theorem to the wear rate of gouge formation in frictional sliding of rocks

A simple law of wear rate is examined for the process of gouge generation during the frictional sliding of simulated faults in rocks, by use of the Pi theorem method (dimensional analysis) and existing experimental data. The relationship between wear rate (t/d) and the applied stress can be expressed by the power-law relations $$\frac{t}{d} = C_\sigma \sigma ^{m\sigma } ,\frac{t}{d} = C_\tau \tau ^{m\tau }$$ wheret is the thickness of the gouge generated on the frictional surfaces,d is the fault displacement, σ and τ are normal stress and shear stress, respectively, andC _σ,C _τ,m _σ andm _τ are constants. These results indicate that the exponent coefficientsm _σ andm _τ and the coefficientsC _σ andC _τ depend on the material hardness of the frictional surfaces. By using the wear rates of natural faults, these power-law relationships may prove to be an acceptable palaeopiezometer of natural faults and the lithosphere.     

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form $$\dot \varepsilon = A'\left[ {sinh (\alpha \sigma )} \right]^n \exp \left[ {{{ - (E * + PV * )} \mathord{\left/ {\vphantom {{ - (E * + PV * )} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right]$$ where \(\dot \varepsilon \) This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both.     

Die Berechnung der totalen Menge gelöster Stoffe in Seen

Total content is used mostly for balances. In order to calculate the areas of different depths, the contours are cut out of a topographical map and weighed. The formula is derived from the substance concentration (taken as a linear function of the depth), times the volume of a truncated cone. Content of one layer: $$\frac{{\Delta z}}{{l2}}\left[ {\left( {f_l + f_u } \right)^2 \cdot \left( {c_l + c_u } \right) + 2c_l f_l^2 + 2c_u f_u^2 } \right]$$ f_u Square root of upper area f₁ Square root of lower area c_u Concentration at upper depth c₁ Concentration at lower depth Δz Difference of depths (i.e. thickness of layer) Sum of contents of all layers gives total content of the lake.     

A theoretical value for von Karman's constant

This paper extends the theory of the entity and entrainment model of turbulence to obtain a numerical value of von Karman's constant,k=0.37. The formula is, $$k = (2a^3 /A)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \ln \beta $$ where,a=1/12 is the entrainment constant,A=1 is the turbulent decay constant, and β is the ratio in height of the successive self-similar layers of the theory, where β is evaluated as β=e ². These new values fork and β improve the surface roughness length estimates derived from this theory.     

Performance of the complete travel-time equation of state at simultaneous high pressure and temperature

The complete travel-time equation of state (CT-EOS) is presented by utilizing thermodynamics relations, such as; $$K_T = K_S (1 + \alpha \gamma T)^{ - 1} , \gamma = \frac{{\alpha K_S }}{{\rho C_P }}, \left. {\frac{{\partial C_P }}{{\partial P}}} \right)_T = - \frac{T}{\rho }\left[ {\alpha ^2 + \left. {\frac{{\partial \alpha }}{{\partial T}}} \right)_P } \right], etc.$$ The CT-EOS enables us to analyze ultrasonic experimental data under simultaneous high pressure and high temperature without introducing any assumption, as long as the density, or thermal expansivity, and heat capacity are also available as functions of temperature at zero pressure. The performance of the CT-EOS was examined by using synthesized travel-time data with random noise of 10^?5 and 10^?4 amplitude up to 4 GPa and 1500 K. Those test conditions are to be met with the newly developed GHz interferometry in a gas medium piston cylinder apparatus. The results suggest that the combination of the CT-EOS and accurate experimental data (10^?4 in travel time) can determine thermodynamic and elastic parameters, as well as their derivatives with unprecedented accuracy, yielding second-order pressure derivatives (?² M/?P ²) of the elastic moduli as well as the temperature derivatives of their first-order pressure derivatives ?² M/?P?T). The completeness of the CT-EOS provides an unambiguous criterion to evaluate the compatibility of empirical EOS with experimental data. Furthermore because of this completeness, it offers the possibility of a new and absolute pressure calibration when X-ray (i.e., volume) measurements are made simultaneously with the travel-time measurements.     

Long-term earthquake prediction in the Aegean area based on a time and magnitude predictable model

The Aegean and surrounding area (34°N–43°N, 18°E–30°E) is separated into 76 shallow and intermediate depth seismogenic sources. For 74 of these sources intervent times for strong mainshocks have been determined by the use of instrumental and historical data. These times have been used to determine the following empirical relations: $$\begin{gathered} \log T_t = 0.24M_{\min } + 0.25M_p - 0.36\log \dot M_0 + 7.36 \hfill \\ M_f = 1.04M_{\min } - 0.31M_p + 0.28\log \dot M_0 - 4.85 \hfill \\ \end{gathered} $$ whereT ₁ is the interevent time, measured in years,M _min the surface wave magnitude of the smallest mainshock considered,M _p the magnitude of the preceding mainshock,M _f the magnitude of the following mainshock, \(\dot M_0 \) the moment rate in each source per year. A multiple correlation coefficient equal to 0.74 and a standard deviation equal to 0.18 for the first of these relations were calculated. The corresponding quantities for the second of these relations are 0.91 and 0.22. On the basis of the first of these relations and taking into consideration the time of occurence and the magnitude of the last mainshock, the probabilities for the occurrence of mainshocks in each seismogenic source of this region during the decade 1993–2002 are determined. The second of these relations has been used to estimate the magnitude of the expected mainshock.     

The TKE dissipation rate in the northern South China Sea

The microstructure measurements taken during the summer seasons of 2009 and 2010 in the northern South China Sea (between 18°N and 22.5°N, and from the Luzon Strait to the eastern shelf of China) were used to estimate the averaged dissipation rate in the upper pycnocline 〈ε _p〉 of the deep basin and on the shelf. Linear correlation between 〈ε _p〉 and the estimates of available potential energy of internal waves, which was found for this data set, indicates an impact of energetic internal waves on spatial structure and temporal variability of 〈ε _p〉. On the shelf stations, the bottom boundary layer depth-integrated dissipation $ {\widehat{\varepsilon}}_{\mathrm{BBL}} $ reaches 17–19 mW/m², dominating the dissipation in the water column below the surface layer. In the pycnocline, the integrated dissipation $ {\widehat{\varepsilon}}_{\mathrm{p}} $ was mostly ～10–30 % of $ {\widehat{\varepsilon}}_{\mathrm{BBL}} $ . A weak dependence of bin-averaged dissipation $ \overline{\varepsilon} $ on the Richardson number was noted, according to $ \overline{\varepsilon}={\varepsilon}_0+\frac{\varepsilon_{\mathrm{m}}}{{\left(1+ Ri/R{i}_{\mathrm{cr}}\right)}^{1/2}} $ , where ε ₀ + ε _m is the background value of $ \overline{\varepsilon} $ for weak stratification and Ri _cr?=?0.25, pointing to the combined effects of shear instability of small-scale motions and the influence of larger-scale low frequency internal waves. The latter broadly agrees with the MacKinnon–Gregg scaling for internal-wave-induced turbulence dissipation.     

Partikelgrössenverteilung und Natürliche Koagulation im Zürichsee

The size distribution of suspended particles in Lake Zürich water shows always the same shape, irrespective of the total concentration of particles, depth or season. The particle size distribution can be described by a function of the form $$\frac{{\Delta {\rm N}(d_p )}}{{\Delta d_p }} = n(d_p ) = {\rm A}d_p^{ - m} $$ where N (d_p)=concentration of particles with diameters between d_p and d_p+Δd_p [cm^?3], d_p=particle diameter [μm], A=constant of the particle size distribution, n(d_p)=particle size distribution function. m was found to be about 3.5. Model calculations show that coagulation determines the particle size distribution. The lake model consists in three completely mixed parts: the epilimnion, the thermocline and the hypolimnion. The effect of outflow of particles by a river, sedimentation and coagulation on the particle size distribution were investigated.     

Estimation du débit de SO2, HCl et HF émis par le volcan Vulcano (Italie)

The flow rate of SO₂, HCl and HF was calculated from a transfer coefficient obtained by measuring the concentration of a tracer gas (SF₆) emitted at the plume and analysed down wind with a field gas chromatograph. The results obtained are: $$\begin{gathered} SO_2 = 2.3 \pm 0.4 t/day \hfill \\ HCl = 6.4 \pm 0.4 t/day \hfill \\ HF = 0.11 \pm 0.3 t/day \hfill \\ \end{gathered} $$      

Estimates of fracture parameters of earthquakes

A new estimate of the fracture parameters of earthquakes is provided in this paper. By theMuskhelishvili method (1953) a number of basic relations among fracture-mechanics parameters are derived. A scheme is proposed to evaluate the slip weakening parameters in terms of fault dimension, average slip, and rise time, and the new results are applied to 49 events compiled in the earthquake catalogue ofPurcaru andBerckhemer (1982). The following empirical relations are found in the paper: $$\begin{gathered} \frac{{\tau _B - \tau _f }}{{\tau _\infty - \tau _f }} = 2.339 \hfill \\ {{\omega _c } \mathord{\left/ {\vphantom {{\omega _c } {W = 0.113}}} \right. \kern-\nulldelimiterspace} {W = 0.113}} \hfill \\ \log G_c \left( {{{dyne} \mathord{\left/ {\vphantom {{dyne} {cm}}} \right. \kern-\nulldelimiterspace} {cm}}} \right) = 2 \log L (km) + 6.167 \hfill \\ \log \delta _c (cm) = 2 \log L (km) - 1.652 \hfill \\ \end{gathered} $$ whereG _c is the specific fracture energy,ω _c the size of the slip weakening zone,δ _c the slip weakening displacement,τ _B ?τ _f the drop in strength in the slip weakening zone,τ _∞?τ _f the stress drop,L the fault length, andW the fault width. The investigation of 49 shocks shows that the range of strength dropτ _B ?τ _f is from several doze to several hundred bars at depthh<400 km, but it can be more than 10³ bars ath>500 km; besides, the range of the sizeω _c of the strength degradation zone is from a few tenths of a kilometer to several dozen kilometers, and the range of the slip weakening displacementδ _c is from several to several hundred centimeters. The specific fracture energyG _c is of the order of 10⁸ to 10¹¹ erg cm^?2 when the momentM ₀ is of the order of 10²³ to 10²⁹ dyne cm.     

Flow velocity profiles in the Lower Scheldt estuary

Recent acoustic Doppler current profiler (ADCP)-measurements in the Scheldt estuary near Antwerp, Belgium, revealed anomalous, i.e. anti-clockwise circulations in a left bend during the major part of the flood period; these circulations were established shortly after the turn of the tide. During ebb, anti-clockwise circulations persisted, as predicted by classical theory. These data were analysed with a 3D and a 1DV-model. The 3D simulations reveal that the anomalous circulations are found when salinity is included in the computations—without salinity “normal” circulations were found. From analytical and 1DV simulations, it is concluded that a longitudinal salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ may induce a near-bed maximum in flow velocity reversing the direction of the secondary currents. The 1DV-model was then used to assess the contribution of various processes one by one. It was found that because of a reduction in vertical mixing, the vertical velocity profile is not at equilibrium during the first phase of accelerating tide, further enhancing the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ . A small vertical salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ appeared to have a very large effect as the crosscurrents of the secondary circulations induced by ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ became an order of larger magnitude. However, at the site under consideration, the effects of transverse salinity gradients, generated by differential advection in the river bend, were dominant: adverse directions of the secondary circulations were found even when the vertical velocity profile became more regular with a more or less logarithmic shape, i.e. when the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ and ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ did not play a dominant role anymore. It is argued that data on the secondary velocity structure, which can be measured easily owing to today’s developments in ADCP equipment, may serve as an indicator for the accuracy at which the salinity field is computed with 3D numerical models. Moreover, the large effect of the salinity structure on the velocity field must have a large impact on the morphological development of estuaries, which should therefore be accounted for in morphological modelling studies.     

On the dynamics of extensional basin

Geological and geophysical data from the North China-Bohai Basin and “Basin and Range” Province were examined and compared. They are similar to each other in many respects. Surficial geological structures are characterized by a series of half-grabens with their one flank constituted by normal fault. Those extensional structures usually extend to a depth of 6–8 km. Therefore, the stress condition in the upper 8 km can be written as $$\sigma _2 > \sigma _x > \sigma _y$$ wherex, y denote the directions of maximum compression and maximum tension on the horizontal plane, whilez signifies the vertical direction. Some people think that this kind of stress condition exists through the entire crust in the extensional basin. However, the focal mechanisms of the earthquakes in the extensional basins with focal depths usually at 12–20 km are dominated by strike-slip faults. The stress condition in the focal regions can be expressed by $$\sigma _x > \sigma _z > \sigma _y .$$ Geodetic measurements conducted before and after the Tangshan earthquake in 1976 and the Xingtai earthquake in 1966 showed that both horizontal and vertical surficial deformations with magnitudes of a similar order occurred during the earthquakes. The surficial deformations during the earthquakes can be explained by a summation of the motions produced by both stress fields in the upper crust and the middle crust. Dynamical processes other than the homogeneous horizontal regional tectonic field are required to explain the vertical variation of the stress condition in the upper and middle crusts. Evidence from the seismic refractions, reflections and the three-dimensional seismic tomography from both local earthquakes and teleseismic events provide convincing evidence that magmatic intrusions from the uppermost mantle to the middle crust occur near the hypocenters of both the Tangshan and Xingtai earthquakes. The variation from the extensional stress regime at the upper crust to the compressional stress regime in the middle and lower crusts is considered to be the common feature in extensional basins. And the magmatic intrusions from the upper mantle to the middle crust observed in the extensional basin is suggested to be its genetic cause. Numerical simulations of magmatic intrusion from the uppermost mantle to the middle crust were studied. Both the intruded compression and the thermal stress due to magmatic intrusion were considered, also the viscoelasticity of the middle and lower crusts were assumed. The results successfully explain the vertical variation of the stress condition in the crust and the process producing an extensional basin.     

Attenuation relations for strong seismic ground motion in the Himalayan region

Strong motion data from various regions of India have been used to study attenuation characteristics of horizontal peak acceleration and velocity. The strong ground motion data base considered in the present work consists of various earthquakes recorded in the northern part of India since 1986 with magnitudes 5.7 to 7.2. Using these data, relations for horizontal peak acceleration and velocity, which are $$\begin{gathered} log_{10} a = 1.14 + 0.31M + 0.65log_{10} R \hfill \\ log_{10} v = 0.571 + 0.41M + 0.768log_{10} R \hfill \\ \end{gathered} $$ have been proposed wherea is the peak horizontal acceleration in cm/sec²,v is the peak horizontal velocity in mm/sec,M is body wave magnitude, andR is the hypocentral distance in km. The proposed relations are in reasonable agreement with the small amount of strong ground motion data available for the northern part of India. The present results will be useful in estimating strong ground motion parameters and in the earthquake resistant design in the Himalayan region.