Radial Derivative Research Articles

Uniform spectral representation of the Earth's inner density structures and their gravitational field

We introduce the generic expressions for computing the gravitational field(potential and its radial derivative) generated by an arbitrary density (contrast) layerwith a variable depth and thickness having a laterally-distributed radial density variation.The information on the geometry and density distribution of a volumetric mass layer isdescribed by means of spherical harmonics. These generic expressions can uniformly beapplied to model all major known density structures within the Earth’s interior usingmethods for a spherical harmonic analysis and synthesis of the gravitational field. Thisis demonstrated on specific examples given for various density models commonly adoptedfor the approximation of crust density structures.

Application of Möbius coordinate transformation in evaluating Newton's integral

Application of Möbius coordinate transformation in evaluating Newton's integralWe propose a numerical scheme which efficiently combines various existing methods of solving the Newton's volume integral. It utilises the analytical solution of Newton's integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky's truncation coefficients to a spectral representation of Newton's integral. The weak singularity of Newton's integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the Möbius transformation with an enhanced integration grid resolution at vicinity of the observation point.

The Radial Volume Derivative and the Critical Boundary Displacement for Cavitation

We study the displacement boundary value problem of minimizing the total energy $E(\mathbf{u})$ stored in a nonlinearly elastic body occupying a spherical domain B in its reference configuration over (possibly discontinuous) radial deformations $\mathbf{u}$ of the body subject to affine boundary data $\mathbf{u}(\mathbf{x})=\lambda\mathbf{x}$ for $\mathbf{x}\in \partial B$. For a given value of $\lambda$, we define what we call the radial volume derivative at $\lambda$, denoted by $G(\lambda )$, which measures the stability or instability of the underlying homogeneous deformation $\mathbf{u}^h_\lambda(\mathbf{x})\equiv \lambda\mathbf{x}$ to the formation of holes. We give conditions under which the critical boundary displacement $\lambda_{\mathrm{crit}}$ for radial cavitation is the unique solution of $G(\lambda )=0$. Moreover, we prove that the radial volume derivative $G(\lambda )$ can be approximated by the corresponding volume derivative for a punctured ball $B_\epsilon$, containing a pre-existing cavity of radius $\epsilon >0$ in its reference state, in the limit $\epsilon\rightarrow 0$ and we use this to propose a numerical scheme to determine $\lambda_{\mathrm{crit}}$. We illustrate these general results with analytical and numerical examples.

On conformal, harmonic mappings and Dirichlet’s integral

This paper has an expository character, however we present as well some new results and new proofs. We prove a complex version of Dirichlet?s principle in the plane and give some applications of it as well as estimates of Dirichlet?s integral from below. Some of the results in the plane are generalized to higher dimensions. Roughly speaking, under the appropriate conditions we estimate the n-Dirichlet integral of a mapping u defined on a domain ? ? Rn , n ? 2, by the measure of u(?) and show that equality holds if and only if it is injective conformal. Also some sharp inequalities related to the L2 norms of the radial derivatives of vector harmonic mappings from the unit ball in Rn, n ? 2, are given. As an application, we estimate the 2-Dirichlet integrals of mappings in the Sobolev space Wi2.

Improved Hardy-Sobolev inequalities for radial derivative

We prove some Hardy-Sobolev inequalities for radial derivative and obtain the corresponding sharp constant. Mathematics subject classification (2010): Primary 26D10, 46E35.

Viscous flow in domains with corners: Numerical artifacts, their origin and removal

Viscous flows in domains with boundaries forming two-dimensional corners are considered. We examine the case where on each side of the corner the boundary condition for the tangential velocity is formulated in terms of stress. It is shown that computing such flows numerically by straightforwardly applying well-tested algorithms (and numerical codes based on their use, such as COMSOL Multiphysics) can lead to spurious multivaluedness and mesh-dependence in the distribution of the fluid’s pressure. The origin of this difficulty is that, near a corner formed by smooth parts of the boundary, in addition to the solution of the formulated inhomogeneous problem, there also exists an eigensolution. For obtuse corner angles this eigensolution (a) becomes dominant and (b) has a singular radial derivative of velocity at the corner. Despite the bulk pressure in the eigensolution being constant, when the derivatives of the velocity are singular, numerical errors in the velocities calculation near the corner give rise to pressure spikes, whose magnitude increases as the mesh is refined. A method is developed that uses the knowledge about the eigensolution to remove the artifacts in the pressure distribution. The method is first explained in the simple case of a Stokes flow in a corner region and then generalized for the Navier–Stokes equations applied to describe steady and unsteady free-surface flows encountered in problems of dynamic wetting.

Mixing of passive tracers in the decay Batchelor regime of a channel flow

We report detailed quantitative studies of passive scalar mixing in a curvilinear channel flow, where elastic turbulence in a dilute polymer solution of high molecular weight polyacrylamide in a high viscosity water-sugar solvent was achieved. For quantitative investigation of mixing, a detailed study of the profiles of mean longitudinal and radial components of the velocity in the channel as a function of Wi was carried out. Besides, a maximum of the average value as well as a rms of the longitudinal velocity was used to determine the threshold of the elastic instability in the channel flow. The rms of the radial derivatives of the longitudinal and radial velocity components was utilized to define the control parameters of the problem, the Weissenberg Wiloc and the Péclet Pe numbers. The main result of these studies is the quantitative test of the theoretical prediction about the value of the mixing length in the decay Batchelor regime. The experiment shows large quantitative discrepancy, more than 200 times in the value of the coefficient C, which appears in the theoretical expression for the mixing length, but with the predicted scaling relation. There are two possible reasons to this discrepancy. First is the assumption made in the theory about the δ-correlated velocity field, which is in odds with the experimental observations. Second, and probably a more relevant suggestion for the significantly increased mixing length and thus reduced mixing efficiency, is the observed jets, the rare, localized, and vigorous ejection of the scalar trapped near the wall, which protrudes into the peripheral region as well as the bulk. They are first found in the recent numerical calculations and then observed in the experiment reported. The jets definitely strongly reduce the mixing efficiency in particular in the peripheral region and so can lead to considerable increase of the mixing length. We hope that this result will initiate further numerical calculations of the mixing length. Finally, we analyze statistical properties of the mixing in the decay Batchelor regime by studying the power spectra, the decay exponents scaling, the structure functions of a tracer and moments of PDF of passive scalar increments, and the temporal and spatial correlation functions and find rather satisfactory agreement with theory.

Semi-stochastic modification of second-order radial derivative of Abel–Poisson’s formula for validating satellite gravity gradiometry data

The geoid can be used to validate the satellite gravity gradiometry data. Validation of such data is important prior to their downward continuation because of amplification of the data errors through this process. In this paper, the second-order radial derivative of Abel–Poisson’s formula is modified stochastically to reduce the effect of the far-zone geoid and generate the second-order radial derivative of geopotential at 250 km level. The numerical studies over Fennoscandia show that this method yields the gradients with an error of 10 mE and when the long wavelength of geoid is removed from the estimator and restored after the computations (remove–compute–restore) the error will be in 1 mE level. We name this method semi-stochastic modification. The best case scenario is found when the degree of modification of the integral formula is 200 and the long wavelength geoid to degree 100 is removed and restored. In this case the geoid should have a resolution of 15′ × 15′ and the integration should be performed over a cap size of 3°.

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential ( T rr ) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T rr . The numerical investigations show satisfactory results when the area of T rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T rr measured on the orbit.

ChemInform Abstract: Selective Bond Fission in Methyl Mercaptan at 193 nm via Radial Derivative Coupling Between the 2 1A′′ and 1 1A′′ Adiabatic Electronic States

We investigate the origin of the observed fission of the stronger S–H bond over the weaker C–S bond in CH3SH excited at 193 nm using the complementary techniques of mass‐resolved photofragment time‐of‐flight spectroscopy and emission spectroscopy. The velocities and angular distributions of the CH3S and SH photofragments show that both C–S and S–H bond fission occur on a subpicosecond timescale and impart considerable energy to relative product translation. The dispersed emission from photoexcited CH3SH molecules in a flow cell evidences a progression in the CH3 umbrella mode and combination bands with one quantum in the C–S stretch, but no progression with S–H stretch. Examination of the results with reference to previous ab initio calculations of the excited state surfaces reveals the importance of nonadiabatic coupling in the dissociation dynamics. This is a clear example of selective bond fission upon excitation of an electronic state that is not repulsive in the bond that breaks. We discuss the impli...

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.

Evolution of discontinuous solutions to transport equations in stellarators

It was shown earlier that, in the range of rare collisions, transport equations for stellarators allow steady discontinuous solutions for the ambipolar electric field and for the plasma density and temperature gradients. Moreover, such solutions are non-single-valued; that is, their explicit form depends on the initial values of the ambipolar electric field. The time-independent transport equations are derived under the conventional quasineutrality condition; i.e., it is assumed that the electron and ion densities, N e and N i , are related by the relationship N e = ZN i (where Z is the ion charge number). In other words, the plasma charge density is assumed to be much less than the product e i N i . Under typical conditions, the corresponding inequality is satisfied by a large margin. However, if the electric field E has discontinuities, then it can be seen from the equation ▿·E = 4πρ that, at the discontinuity points, the charge density becomes infinite and the relationship N e = ZN i fails to hold, so it is necessary to replace it with N e = ZN i + ρ/e e . In the transport equations, this latter replacement produces additional terms, proportional to the second radial derivative of the field E. With these additional terms, the steady solutions are modified substantially. First, the ambipolar field and the derivatives of the density and temperatures all become continuous functions of the coordinates, a result that seems to be quite obvious. The second, not-so-obvious result is that the steady solutions become single-valued, i.e., independent of the initial values of the ambipolar electric field. It turns out that, in this case, two regimes are possible, depending on the values of the plasma parameters. In the first regime, the solution is unique and is independent of the initial conditions. In the second regime, two steady solutions can exist, depending on the initial conditions. One of the solution is similar to that obtained in the first regime, and the other differs from the first one both in the ambipolar field profile and in the dependence of the density and temperatures on the minor plasma radius. It cannot be excluded that different plasma confinement modes revealed in experiments are associated with the existence of such solutions.

Candidates for multiple impact craters?: Popigai and Chicxulub as seen by the global high resolution gravitational field model EGM2008

Abstract. In 2008 the new Earth Gravitational Model (EGM2008) was released. It contains a complete set of spherical harmonic coefficients of the Earth's gravitational potential (Stokes parameters) to degree 2190 and order 2159 and selected orders to degree 2190, that can be used for evaluation of various potential quantities with both the unprecedented accuracy and high spatial resolution. Two such quantities, the gravity anomaly and second-order radial derivative of the disturbing potential, were computed over selected areas with known impact craters. The displays of these derivatives for two such sites clearly show not only the strong circular-like features known to be associated with them but also other symmetrical structures which appear to make them multiple impact sites. At Popigai, Siberia, the series of circular features fall in a line from the "primary crater" in the southeast (SE) direction. At Chicxulub, Yucatán, there appears to be one more crater close to the "primary" in the northeast (NE) direction, as well as possibly others in the vicinity of the main crater (SW). Gravity information alone is not, however, proof of impact craters but it is useful in identifying candidate sites for further study, for examination by geologists and geophysicists. In the case of Chicxulub, a very recent single seismic profile suggests that a more likely explanation for the observed circular like gravity signal from EGM2008 NE of the "primary" is a pre-impact basin.

Integral-type operators between α-Bloch spaces and Besov spaces on the unit ball

Let H ( B ) denote the space of all holomorphic functions on the unit ball B ⊂ C n . The boundedness and compactness of the following integral-type operators T g ( f ) ( z ) = ∫ 0 1 f ( tz ) R g ( tz ) dt t and L g ( f ) ( z ) = ∫ 0 1 R f ( tz ) g ( tz ) dt t , z ∈ B , where g ∈ H ( B ) and R h ( z ) is the radial derivative of h, between α-Bloch spaces and Besov spaces on the unit ball are characterized. The results regarding the case when these operators map α-Bloch spaces to Besov spaces are nontrivial generalizations of recent one-dimensional results by Li and the present author.

Advective mixing in a nondivergent barotropic hurricane model

Abstract. This paper studies Lagrangian mixing in a two-dimensional barotropic model for hurricane-like vortices. Since such flows show high shearing in the radial direction, particle separation across shear-lines is diagnosed through a Lagrangian field, referred to as R-field, that measures trajectory separation orthogonal to the Lagrangian velocity. The shear-lines are identified with the level-contours of another Lagrangian field, referred to as S-field, that measures the average shear-strength along a trajectory. Other fields used for model diagnostics are the Lagrangian field of finite-time Lyapunov exponents (FTLE-field), the Eulerian Q-field, and the angular velocity field. Because of the high shearing, the FTLE-field is not a suitable indicator for advective mixing, and in particular does not exhibit ridges marking the location of finite-time stable and unstable manifolds. The FTLE-field is similar in structure to the radial derivative of the angular velocity. In contrast, persisting ridges and valleys can be clearly recognized in the R-field, and their propagation speed indicates that transport across shear-lines is caused by Rossby waves. A radial mixing rate derived from the R-field gives a time-dependent measure of flux across the shear-lines. On the other hand, a measured mixing rate across the shear-lines, which counts trajectory crossings, confirms the results from the R-field mixing rate, and shows high mixing in the eyewall region after the formation of a polygonal eyewall, which continues until the vortex breaks down. The location of the R-field ridges elucidates the role of radial mixing for the interaction and breakdown of the mesovortices shown by the model.

A mathematical model of the bathymetry-generated external gravitational field

A mathematical model of the bathymetry-generated external gravitational field The currently available global geopotential models and the global elevation and bathymetry data allow modelling the topography-corrected and bathymetry stripped reference gravity field to a very high spectral resolution (up to degree 2160 of spherical harmonics) using methods for a spherical harmonic analysis and synthesis of the gravity field. When modelling the topography-corrected and crust-density-contrast stripped reference gravity field, additional stripping corrections are applied due to the ice, sediment and other major known density contrasts within the Earth's crust. The currently available data of global crustal density structures have, however, a very low resolution and accuracy. The compilation of the global crust density contrast stripped gravity field is thus limited to a low spectral resolution, typically up to degree 180 of spherical harmonics. In this study we derive the expressions used in forward modelling of the bathymetry-generated gravitational field quantities and the corresponding bathymetric stripping corrections to gravity field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The expressions for the potential and its radial derivative are formulated for the adopted constant (average) ocean saltwater density contrast and for the spherical approximation of the geoid surface. These newly derived expressions are utilized in numerical examples to compute the gravitational potential and attraction generated by the ocean density contrast. The computation is realized globally on a 1 x 1 arc-deg geographical grid at the Earth's surface.

On an Integral‐Type Operator Acting between Bloch‐Type Spaces on the Unit Ball

Let 𝔹 denote the open unit ball of ℂn. For a holomorphic self‐map φ of 𝔹 and a holomorphic function g in 𝔹 with g(0) = 0, we define the following integral‐type operator: , z ∈ 𝔹. Here ℜf denotes the radial derivative of a holomorphic function f in 𝔹. We study the boundedness and compactness of the operator between Bloch‐type spaces ℬω and ℬμ, where ω is a normal weight function and μ is a weight function. Also we consider the operator between the little Bloch‐type spaces ℬω,0 and ℬμ,0.

Integral-Type Operators from Spaces to Zygmund-Type Spaces on the Unit Ball

Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol , , , , where is the radial derivative of . We characterize the boundedness and compactness of the integral-type operators from general function spaces to Zygmund-type spaces , where is normal function on .

Least-squares modification of extended Stokes’ formula and its second-order radial derivative for validation of satellite gravity gradiometry data

The gravity anomalies at sea level can be used to validate the satellite gravity gradiometry data. Validation of such a data is important prior to downward continuation because of amplification of the data errors through this process. In this paper the second-order radial derivative of the extended Stokes’ formula is employed and the emphasis is on least-squares modification of this formula to generate the second-order radial gradient at satellite level. Two methods in this respect are proposed: (a) modifying the second-order radial derivative of extended Stokes’ formula directly, and (b) modifying extended Stokes’ formula prior to taking the second-order radial derivative. Numerical studies show that the former method works well but the latter is very sensitive to the proper choice of the cap size of integration and degree of modification.

On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball

Let H ( B ) denote the space of all holomorphic functions on the open unit ball B of C n and g ∈ H ( B ) . We characterize the boundedness and compactness of the following integral-type operator L g f ( z ) = ∫ 0 1 R f ( tz ) g ( tz ) dt t , z ∈ B , where R f is the radial derivative of function f, from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball.