Analysis of aeromagnetic anomalies over the central part of the Narmada-Son lineament

The Narmada-Son lineament (NSL) forms a major tectonic feature on the Indian subcontinent. The importance of this lineament lies in its evolution as well as its tectonic history. The lineament seems to have been active since Precambrian times. In order to understand the history of its evolution, it is necessary to know what igenous activity has been taking place along this lineament, and how the Deccan trap volcanics, which cover large areas along this lineament, have erupted.For the study of this problem an analysis of the aeromagnetic anomaly map lying between 76°15 to 77°30E and 21°45 to 22°50N has been carried out. Four different profiles (B ₁ B ₁,B ₂ B ₂,B ₃ B ₃ andB ₄ B ₄) have been drawn in N-S direction over this area and interpreted in terms of the intrusive bodies present within or below the surface of Deccan trap exposures. Inversion and forward modelling techniques have been adopted for interpretation purposes. An analysis of frequency spectra along the profiles has also been carried out to estimate the average depth of the different magnetic bodies. These results have been correlated with the available geological information. It has been found that most of the small wavelength anomalies are caused by dyke-like bodies within or below the Deccan trap at a depth of less than 0.5 km.     

Cyclic variations of the initial height of the sporadicE-layer at middle geographical latitudes

Summary The variations of the initialh E _s height are investigated in the solar cycle 1957–1968, deriving the regressive dependency:h E _s =121.4–6·10^–2 R referring to the median monthly values at a solar zenith angle =75°. The similar variations ofh E _s (R) during the day and night are interpreted as a domination of the sporadic layer formation from a redistribution of the day-time ionization and secondary participation of nightly ionizing sources. The analogous cyclich E _s andh E variations confirm this conclusion while the seasonal variations in the state of the sporadic layer show outlined dynamical effects. The comparatively not big cyclic variation in the spatial state of theE-region are considered to confirm the predominating ionizing action of the ultraviolet range (933–1038 Å) in the lower part of theE-region, while the soft X-radiation influences mainly the near maximum part of this region.     

The derivatives of the lunar disturbing potential harmonics with respect to euler's angles

Summary The derivatives of the harmonicsP _n ^(k) (sin _O)cos kT_O andP _n ^(k) (sin _O)sin kT_O, occurring in the development of the lunar disturbing potential, are derived upto n=4 and for k== 0, 1, ..., n. The equatorial co-ordinates _OT_O are referred to the Moon's mass centre; the procedure for the solar disturbing potential is formally identical.     

Deformation of a homogeneous gravitating sphere by internal dislocations

Summary An explicit solution is obtained for the system of equations describing the spheroidal motion in a homogeneous, isotropic, gravitating, elastic medium possessing spherical symmetry. This solution is used to derive the Green's dyad for a homogeneous gravitating sphere. The Green's dyad is then employed to obtain the displacement field induced by tangential and tensile dislocations of arbitrary orientation and depth within the sphere.Notation G Gravitational constant - a Radius of the earth - A _o =4/3 G - Perturbation of the gravitational potential - Circular frequency - V _p ,V _s Compressional and shear wave velocities - k _p =/V _p - k _s =/V _s - k _p [(2.8)] - , [(2.17)] - f _l ⁺ Spherical Bessel function of the first kind - f _l ^– Spherical Hankel function of the second kind - x =r - y =r - x _o =r _o - y _o =r_o - x =r k _s - y =r k _p - x _o =r _o k _s - y _o =r _o k _p - =a - =a - [(5.17)] - _{m, l}      

UV spectrometry for measuring particulate organic matter in natural water

Ultraviolet (UV) spectra were measured to characterize the dissolved and particulate organic matter in a bog and to investigate the seasonal fluctuation of this organic matter. The optical density spectra in the UV region of dissolved organic matter (DOM) were proportional to ^–4, where is the wavelength of incident light. There was also small absorption on the ^–4 spectra. The optical density at a wavelength of 280 nm was proportional to the concentration of dissolved organic carbon (DOC). The spectra and the dependency of optical density on DOC did not vary throughout the investigation period. The UV spectra of acid extractions of particulate organic matter (POM) showed two absorption peaks at the wavelengths of 220 nm and 335 nm. These peaks are well represented by the sum of two Lorentz functions. The ratio of the absorbances,A ₂₈₀/A ₃₃₅, was dependent on the nitrogen/carbon ratio of POM and the chlorophyll fraction in particulate organic carbon (POC), and not dependent on POC itself. The dependency of POC absorption at 280 nm varied with this ratio which fluctuated seasonally.     

Studies on precipitable water over tropical stations in relation to monsoon flow

Summary The mean monthly precipitable water at four tropical stations Madras (13°00N, 80°11E), Waltair (17°42N, 83°18E), Bombay (18°54N, 72°49E) and Nagpur (21°06N, 79°03E) are evaluated for the layer surface to 500 mb (0–5.4 km) of the atmosphere using radiosonde data available for seven years period (1959–1965). The mean monthly precipitable water for the above four stations is also estimated from dew point temperature.The precipitable water in the air column at any station is examined in relation to monsoon flow. The higher values of precipitable water are found to occur over the regions when there is good supply of moisture by the monsoon flow as well as low level convergence. These studies are believed to provide useful information in forecasting the monsoon circulation over the country.     

Die Abplattungsfunktion einparametriger sphäroidischer Gleichgewichtsfiguren

Zusammenfassung Aus der schon früher bewiesenen Eindeutigkeit des Dichtegesetzes der sphäroidischen Gleichgewichtsfiguren ergibt sich die Möglichkeit, aus einer gegebenen Gleichgewichtsfigur durch «Entblätterung» eine unendliche Reihe neuer Gleichgewichtsfiguren derselben Rotationsgeschwindigkeit mit ständig abnehmender Größe und Masse zu bilden. Auf diese Weise kann man jede beliebige innere Niveaufläche der ursprünglichen Figur bloßlegen und aus der von ihr umschlossenen TeilmasseE, der Rotationsgeschwindigkeit , der Äquatorachsea und der Äquatorachsea _h des homogenen Ausgangsellipsoides der zugehörigen Figurenreihe (,C) die Abplattung berechnen und gewinnt so die Abplattungsfunktion im Innern einer Gleichgewichtsfigur gänzlich unabhängig von derClairautschen Differentialgleichung. Die Methode der Entblätterung ist auch auf höherparametrige Gleichgewichtsfiguren mit unstetigem Dichtegesetz anwendbar; sie liefert für dasWiechertsche Modell die Kernoberfläche in einer Tiefe von 3864.75 km und einen Dichtesprung von 17.84.

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Summary In a previous paper I have shown, that the law of density of the spheroidal figures of equilibrium is strictly individual. This makes it possible from a given figure of equilibrium by the so called method of «stripping of leaves» to gain an infinite series of new figures of equilibrium with permanently decreasing size and mass. In this manner any inner level surface of the given figure can be laid bare and its flattening can be calculated from the enclosed part of massE, the rotation velocity , the equator axisa and from the equator axisa _h of theMacLaurin ellipsoid in the beginning of the series (,C) belonging to the seeked figure. So the function of flattening in the interior of any equilibrium figure is gained totally independant fromClairaut's differential equation. The method of «stripping of leaves» also can be used for figures of equilibrium with more than one form-parameter and with discontinual density distribution. It yields forWiechert's model the surface of the core in a depth of 3864.75 km and a discontinuity of 17.84 in density.

     

On reflection and diffraction of pressure waves from floating ice

Summary The diffraction of a pressure wave arising from the edge of the floating ice due to an incident wave of the formH[t–(xcos/)+ (ysin/)], whereH(t) is the Heaviside step function, has been studied. The ice sheet has been taken to be moving with a velocity much less than the velocity of sound in the liquid. The problem has been solved using an approximate method of solving Wiener Hopf problems. Finally, the pressure field at a large distance from the edge has been derived in a closed form evaluating the integrals by the saddle point method.     

Upper air mean dry-bulb and dew-point temperature profiles over Gauhati airport

Summary The best fit curves for upper air mean dry-bulb and dew-point temperatures over Gauhati airport (26°05N, 91°43E, 49 metres a.m.s.l.), for the month of April, have been calculated with the equation,x=A+By+Cy ²,y being the log value in mb of the isobaric level under consideration andx, the mean dry-bulb or dew-point temperature as the case may be, at the isobaric level under consideration. The values of constantsA, B andC for morning dry-bulb and dew-point curves come to be –29.54559, –93.65766 and +37.35048 and –118.84791, –31.15503 and +25.63585 respectively and values of these constants for evening curves come to be –35.86214, –94.15694 and +38.61870 and –127.55970, –29.97192 and +26.36538 respectively. These best fit curves help in finding out mean desired temperatures at any isobaric level in forecasting of thunderstorms and hailstorms, at a station, by dry-bulb and dew-point temperature anomaly technique propsed earlier by the authors.     

Structure and evolution of the terrestrial planets

Geo-scientific planetary research of the last 25 years has revealed the global structure and evolution of the terrestrial planets Moon, Mercury, Venus and Mars. The evolution of the terrestrial bodies involves a differentiation into heavy metallic cores, Fe-and Mg-rich silicate mantles and light Ca, Al-rich silicate crusts early in the history of the solar system. Magnetic measurements yield a weak dipole field for Mercury, a very weak field (and local anomalies) for the Moon and no measurable field for Venus and mars. Seismic studies of the Moon show a crust-mantle boundary at an average depth of 60 km for the front side, P- and S-wave velocities around 8 respectively 4.5 km s^–1 in the mantle and a considerable S-wave attenuation below a depth of 1000 km. Satellite gravity permits the study of lateral density variations in the lithosphere. Additional contributions come from photogeology, orbital particle, x-and -ray measurements, radar and petrology.The cratered surfaces of the smaller bodies Moon and Mercury have been mainly shaped by meteorite impacts followed by a period of volcanic flows into the impact basins until about 3×10⁹ yr before present. Mars in addition shows a more developed surface. Its northern half is dominated by subsidence and younger volcanic flows. It even shows a graben system (rift) in the equatorial region. Large channels and relics of permafrost attest the role of water for the erosional history. Venus, the most developed body except Earth, shows many indications of volcanism, grabens (rifts) and at least at northern latitudes collisional belts, i.e. mountain ranges, suggesting a limited plate tectonic process with a possible shallow subduction.List of Symbols and Abbreviations a=R _e mean equatorial radius (km) - A(r, t) heat production by radioactive elements (W m^–3) - A, B equatorial moments of inertia - b polar radius (km) - complex amplitude of bathymetry in the wave number (K) domain (m) - C polar moment of inertia - C _Fe moment of inertia of metallic core - C _Si moment of inertia of silicate mantle - C _p heat capacity at constant pressure (JK^–1 mole^–) - C _nm,J _nm,S _nm harmonic coefficients of degreen and orderm - C/(MR _e ² ) factor of moment of inertia - d distance (km) - d nondimensional radius of disc load of elastic bending model - D diameter of crater (km) - D flexural rigidity (dyn cm) - E Young modulus (dyn cm^–2) - E maximum strain energy - E energy loss during time interval t - f frequency (Hz) - f flattening - F magnetic field strength (Oe) (1 Oe=79.58A m^–1) - g acceleration or gravity (cms^–2) or (mGal) (1mGal=10^–3cms^–2) - mean acceleration - g _e equatorial surface gravity - complex amplitude of gravity anomaly in the wave number (K) domain - g free air gravity anomaly (FAA) - g Bouguer gravity anomaly - g _t gravity attraction of the topography - G gravitational constant,G=6.67×10^–11 m³kg^–1s^–2 - GM planetocentric gravitational constant - h relation of centrifugal acceleration (² R _e ) to surface acceleration (g _e ) at the equator - J magnetic flux density (magnetic field) (T) (1T=10⁹ nT=10⁹ =10⁴G (Gauss)) - J ₂ oblateness - J _nm seeC _nm - k ₍₀₎ (zero) pressure bulk modulus (Pa) (Pascal, 1 Pa=1 Nm^–2) - K wave number (km^–1) - K ^* thermal conductivity (Jm^–1s^–1K^–1) - L thickness of elastic lithosphere (km) - M mas of planet (kg) - M _Fe mass of metallic core - M _Si mass of silicate mantle - M(r) fractional mass of planet with fractional radiusr - m magnetic dipole moment (Am²) (1Am²=10³Gcm³) - m _b body wave magnitude - N crater frequency (km^–2) - N(D) cumulative number of cumulative frequency of craters with diameters D - P pressure (Pa) (1Pa=1Nm^–2=10^–5 bar) - P _z vertical (lithostatic) stress, see also _z (Pa) - P _n ^m (cos) Legendre polynomial - q surface load (dyn cm^–2) - Q seismic quality factor, 2E/E - Q _s ,Q _p seismic quality factor derived from seismic S-and P-waves - R=R ₀ mean radius of the planet (km) (2a+b)/3 - R _e =a mean equatorial radius of the planet - r distance from the center of the planet (fractional radius) - r _Fe radius of metallic core - S _nm seeC _nm - t time and age in a (years), d (days), h (hours), min (minutes), s (seconds) - T mean crustal thickness from Airy isostatic gravity models (km) - T temperature (°C or K) (0°C=273.15K) - T _m solidus temperature - T sideral period of rotation in d (days), h (hours), min (minutes), s (seconds), =2/T - U external potential field of gravity of a planet - V volume of planet - V _p ,V _s compressional (P), shear (S) wave velocity, respectively (kms^–1) - w deflection of lithosphere from elastic bending models (km) - z, Z depth (km) - z (K) admittance function (mGal m^–1) - thermal expansion (°C^–1) - viscosity (poise) (1 poise=1gcm^–1s^–1) - co-latitude (90°-) - longitude - Poisson ratio - density (g cm^–3) - mean density - ₀ zero pressure density - _m , _Si average density of silicate mantle (fluid interior) - average density of metallic core - _t , _top density of the topography - density difference between crustal and mantle material - electrical conductivity (^–1 m^–1) - _r , radial and azimuthal surface stress of axisymmetric load (Pa) - _z vertical (lithostatic) stress (seeP _z ) - _II second invariant of stress deviation tensor - latitude - angular velocity of a planet (=2/T) - ages in years (a), generally 0 years is present - B.P. before present - FAA Free Air Gravity Anomaly (see g - HFT High Frequency Teleseismic event - LTP Lunar Transient Phenomenon - LOS Line-Of-Sight - NRM Natural Remanent Magnetization Contribution No. 309, Institut für Geophysik der Universität, Kiel, F.R.G.     

Approximate Conditions for the Off-Axis Triplication in Transversely Isotropic Media

The presented approximate formulas yield a critical value of anisotropy parameter , for which an incipient off-axis SV-wave triplication occurs in transversely isotropic media. The formulas are simple but approximate the exact solution with a high accuracy. The best results are obtained using the third-order approximation, which yields accuracy at least 30 times higher than the formulas presented by Thomsen and Dellinger (2003). The formula works safely for parameters  = a ₃₃/a ₄₄ > 2 and 0.2 >  = (a ₁₁ – a ₃₃)/2a ₃₃ > –0.2, and yields critical values of from 0.1 to 0.7. Outside this interval, it is recommended to use an exact solution.     

The relations between the linear and bilinear atmospheric tidal fields in geometric height and in log-pressure coordinates

The relationships between the linearized meteorological variables as expressed in geometric height and in log-pressure coordinates are derived from the assumptions of classical atmospheric tidal theory. While the horizontal velocity components are the same to first-order in the two coordinate systems, a linearized vertical velocity differencew-H ₀ occurs because of the periodic vertical displacement of the constant pressure surfaces due to time-dependent, hydrostatic density perturbations; a linearized temperature differenceT- also results when these displacements occur in the presence of a zero-order vertical gradient of temperature. Both of these differences can be expressed in terms of the tidal geopotential field. For a given tidal mode, both differences are generally proportional to the square root of the ratio of the tidal mode's equivalent depth and the atmospheric scale height; the temperature difference is also proportional to the background temperature lapse rate. It is further shown that the two classical tidal vertical structure equations commonly derived in their respective geometric height and log-pressure coordinate systems are in fact identical to first-orderas long as the same thermotidal forcing function is used. Expressions for the zonal-mean components of the tidal bilinear fluxes, formed by zonally averaging the product of two longitudinally varying, linearized tidal fields, are also derived for the two coordinate systems. For the bilinear fields the largest relative differences (a few tens of percent) are for the tidal zonal-mean forcing per unit mass of the zonal wind. For Earth and Mars, differences between the tidal vertical velocity fields are generally less than 25% but may be significantly larger in the Martian atmosphere during one of its episodic planetary-scale dust storms. Tidal temperature differences are generally smaller.     

Variations of theD-region aeronomical parameters under non-equilibrium conditions

Summary In this paper an attempt is made to define the time variations of theD-region main aeronomical parameters under non-equilibrium conditions by using experimental data by the A3 method, phase-height method and satellite data for the short-wave X-radiation. The SID effect of day-time and time variations of 21 August 1969 electron concentrationN(t) and recombination coefficient (t) are investigated. It is shown that during SID effect (t) increases.

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Résumé Dans le présent travail on a procédé à une expérience pour déterminer les variations temporelles des principaux paramètres aéronomiques en utilisant des données expérimentales par la méthode A3 de mesure de l'absorption ionosphérique, la méthode phase-altitude et des données satellites aux rayons X durs. On a examiné les variations diurnes et les variations temporelles lors de SID effet durant le 21 Août 1969, de la densité électroniqueN(t) et du coefficient de récombinaison (t). On a montré que lors de SID effet (t) s'accroît.

     

Ionospheric and geomagnetic responses to changes in IMF B z : a superposed epoch study

Superposed epoch studies have been carried out in order to determine the ionospheric response at mid-latitudes to southward turnings of the interplanetary magnetic field (IMF). This is compared with the geomagnetic response, as seen in the indices K_p, AE and Dst. The solar wind, IMF and geomagnetic data used were hourly averages from the years 1967–1989 and thus cover a full 22-year cycle in the solar magnetic field. These data were divided into subsets, determined by the magnitudes of the southward turnings and the concomitant increase in solar wind pressure. The superposed epoch studies were carried out using the time of the southward turning as time zero. The response of the mid-latitude ionosphere is studied by looking at the F-layer critical frequencies, f_oF2, from hourly soundings by the Slough ionosonde and their deviation from the monthly median values, f_oF2. For the southward turnings with a change in B_z of B_z > 11.5 nT accompanied by a solar wind dynamic pressure P exceeding 5 nPa, the F region critical frequency, f_oF2, shows a marked decrease, reaching a minimum value about 20 h after the southward turning. This recovers to pre-event values over the subsequent 24 h, on average. The Dst index shows the classic storm-time decrease to about –60 nT. Four days later, the index has still to fully recover and is at about –25 nT. Both the K_p and AE indices show rises before the southward turnings, when the IMF is strongly northward but the solar wind dynamic pressure is enhanced. The average AE index does register a clear isolated pulse (averaging 650 nT for 2 h, compared with a background peak level of near 450 nT at these times) showing enhanced energy deposition at high latitudes in substorms but, like K_p, remains somewhat enhanced for several days, even after the average IMF has returned to zero after 1 day. This AE background decays away over several days as the Dst index recovers, indicating that there is some contamination of the currents observed at the AE stations by the continuing enhanced equatorial ring current. For data averaged over all seasons, the critical frequencies are depressed at Slough by 1.3 MHz, which is close to the lower decile of the overall distribution of f_oFl values. Taking 30-day periods around summer and winter solstice, the largest depression is 1.6 and 1.2 MHz, respectively. This seasonal dependence is confirmed by a similar study for a Southern Hemisphere station, Argentine Island, giving peak depressions of 1.8 MHz and 0.5 MHz for summer and winter. For the subset of turnings where B_z > 11.5 nT and P 5 nPa, the response of the geomagnetic indices is similar but smaller, while the change in f_oF2 has all but disappeared. This confirms that the energy deposited at high latitudes, which leads to the geomagnetic and ionospheric disturbances following a southward turning of the IMF, increases with the energy density (dynamic pressure) of the solar wind flow. The magnitude of all responses are shown to depend on B_z. At Slough, the peak depression always occurs when Slough rotates into the noon sector. The largest ionospheric response is for southward turnings seen between 15–21 UT.     

Determination of phase transition pressures of ZnTe under quasihydrostatic conditions

Pressure behavior of ZnTe at room temperature was studied using an X-ray energy dispersive method on a DIA type cubic anvil apparatus (SAM-85) at NSLS-X17B1. By using powdered polyethylene, the sample and NaCl for a pressure scale were held under quasihydrostatic conditions, which were confirmed by X-ray diffraction method. Two high-pressure phase transitions were confirmed using X-ray powder diffraction simultaneously with electrical resistance measurements. The phase transition pressures under quasihydrostatic conditions were determined to be 9.6 GPa, at which the resistance increased, and 12.0 GPa, which was the midpoint of a large resistance decrease. Errors in the pressure determinations were estimated to be less than 0.2 GPa. These pressure values may depend on grain size and anisotropic stress effects on the calibrant. From X-ray observation of ZnTe, the bulk modulus of the zinc blende structure was calculated to beK ₀=51(3) GPa andK ₀ =3.6(0.8), and the first transition at 9.6 GPa was found to have about 9% volume change. It was consistent with an anomaly in the pressure generating curves.     

High-level warmings and total ozone over a tropical region

The rocketsonde data obtained from the launchings made at Thumba (8°3215N, 76°5148E) during the winter period 1970–71, as already reported, have indicated that warmings of noticeable magnitude occurred at high levels (upper stratosphere and mesosphere) over this tropical station during the period mentioned. The mean monthly radiosonde temperatures of 50, 100 and 300 mb levels at Thumba (Trivandrum) and Delhi (28°35N, 77°12E) during the same period have also pointed out certain anomalies consistent with the warmings referred to above at Thumba. The radiosonde temperatures of the two stations, Thumba (Trivandrum) and Delhi, have now been examined, along with the values of total ozone, for the ten winter periods commencing from 1961–1962. The analysis has pointed out the possibility of high-level warmings also having occurred in the past over the Indian region during the winters of 1963–1964 and 1967–1968, which are also the periods when prominent warmings are definitely known to have occurred at higher latitudes. The behaviour of total ozone has been found to be different in the different years of the warmings. The features noticed have been presented and discussed.     

The magnetic observatory of Pendeli (Athens,Greece)

Summary A new magnetic observatory, named the Magnetic Observatory of Pendeli, was established and put in operation in 1958 near Athens (Greece). This Observatory was organized by and belongs to the Greek «Institute for Geology and Subsurface Research». The geographical position of the Pendeli Observatory is given by =38° 02.8, =23°51.8 andh=495 m (above sea level). The gemagnetic coordinates of the same are =36°.2, =102°.0. The Observatory is situated near Pendeli Mt. (18 km NEE of Athens). The site of the Observatory consists of marmor underlain by mica schists, both magnetically inactive.The building of the Observatory is constructed of stone and its roof made of tiles. The magnetograph room is in the underground of the building. The magnetic and thermal conditions in the variometer room are satisfactory enough.The variometers of the Observatory forH andZ are of the magnetic balance type. ForD a fibre suspension declinometer is used. The scale values of the variometers are _H =7.2 /mm, _D =1.0/mm (7.6 /mm), _Z =11.5 /mm. The speed of recording amounts to 12 mm/h and the width of the record is 9 cm.The Pendeli Observatory has been operating since April 1958. The record is changed every day. The scale and base-line values are determined every 10 days. The room for the absolute measurements is found in the ground floor of the building. A field magnetic theodolite is being used in the absolute measurements.D is measured with two magnets in a fibre suspension declinometer.H is measured by means of the deflection oscillation method and with a QHM as well. For the measurement ofI an earthinductor is available. The values of the magnetic elements are properly corrected in order to represent the external normal field.     

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.Nomenclature a _i inner sphere radius - a _d outer sphere radius - A ₁ defined by equation (5) - A ₂ defined by equation (6) - B ₁ defined by equation (7) - B ₂ defined by equation (8) - c dimensional factor for bispherical coordinates - C constant in equation (4) - d narrowest distance between the two eccentric spheres - E ² operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates - G modified vorticity, defined in equation (22) - G ^* non-dimensional modified vorticity, defined in equation (28) - h metric coefficient of bispherical coordinate system, defined in equation (18) - k _w thermal conductivity of water - K ₁ defined by equation (9) - K ₂ defined by equation (10) - N _Re Reynolds number=2a _dU/gn - N _Pe,h Peclet number=2a _dU/ - n integer counter - q heat flux - r radius - r ^* non-dimensional radius=r/a _d - S surface area - t time - t ^* non-dimensional time=t/a _d ² - T temperature - T _o temperature at inner sphere surface - T _a temperature at outer sphere surface - T ^* non-dimensional temperature;=(T–T _o)/(T_a–T_o) - u velocity - u _r radial velocity in spherical coordinates - u angular velocity in spherical coordinates - u radial velocity in bispherical coordinates - u angular velocity in bispherical coordinates - U free stream velocity - u _r ^* =u _r/U - u ^* =u /U - u ^* =u /U - u ^* =u /U Greek symbols a ₁ small displacement - vorticity, defined in equation (17) - ^* non-dimensional vorticity, defined in equation (27) - radial bispherical coordinates - _o bispherical coordinate of inner sphere - _a bispherical coordinate of outer sphere - angular coordinate in spherical coordinates - thermal diffusivity - _w thermal diffusivity of water - kinematic viscosity - angular bispherical coordinate - spherical coordinate - streamfunction - non-dimensional streamfunction for spherical coordinates, = /(U a _d ² ) - ^* non-dimensional streamfunction for bispherical coordinates, defined in equation (26)     

Nutrient-chlorophyll trajectories across trophic gradients

We estimate the response of chl-a (mg · m^–3) to changes in concentrations of total phosphorus (TP) by calculating the slopeS = chl-a/TP in chl-a =f(TP) graphs. Results show that in years where algae are P-limited oligotrophic lakes respond less (median slope 0.21) to changes in nutrient concentrations than eutrophic lakes, (median slope 0.31) and these again less than hypereutrophic lakes, (median slope 1.02). We find no saturation value for the slope within the TP range considered (6–480 mg · m^–3). Chl-a in eutrophic lakes responds more frequently to non-nutrient factors than oligotrophic and hypereutrophic lakes. Results obtained by replacing TP with a new nutrient parameter, TP = 0.056 · TP · IN^0.226, in which inorganic nitrogen, IN, is factored in, suggest that nitrogen has an influence on chl-a in oligotrophic lakes. Blue-green algae respond less to changes in TP than other algal species, e.g., diatoms.