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1.
Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional
(l) stationary. Fermat's functional
(l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral
(l) =
(x
k,x
k
')du, taken along curve l, where (x
k,x
k
') is the relevant Lagrangian, x
k are coordinates, u is a parameter used to specify the position of points along l, and x
k
' = dx
k÷du. If Lagrangian (x
k,x
k
') is a homogeneous function of the first degree in x
k
', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian (1)(x
k,x
k
'). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian (1)(x
k,x
k
') to derive the relevant Hamiltonian (1)(x
k,p
k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians (1)(x
k,x
k
'). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians (2). Parameter u along the curves is taken to correspond to travel time , and the second-degree Lagrangian (2)(x
k,
k
) is then introduced by the relation (2)(x
k,
k
) = [(1)(x
k,
k
)]2, with
k
= dx
k÷d. The second-degree Lagrangian (2)(x
k,
k
) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian (1)(x
k,
k
). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian (2)(x
k,p
k) from Lagrangian (2)(x
k,
k
), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both (2)(x
k,
k
) and (2)(x
k,p
k) can be expressed in terms of the wave propagation metric tensor g
ij(x
k,
k
), which depends not only on position x
k, but also on the direction of vector
k
. It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian (2)(x
k,
k
), not to the first-degree Lagrangian (1)(x
k,
k
). It is also shown that relations (2)(x
k,
k
) =
; and (2)(x
k,p
k) =
are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x
i. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases. 相似文献
2.
Rick Salmon 《地球物理与天体物理流体动力学》2013,107(2):167-179
Abstract The equations for a relativistic perfect fluid result from the requirement that the total mass-energy be stationary with respect to variations δxα(a, b, c, s) in the space-time location of the fluid particle identified by Lagrangian labels (a, b, c) at the point s on its world-line. By considering variations of the Lagrangian labels that leave the specific volume and entropy unchanged, we obtain a general covariant statement of vorticity conservation. The conservation laws for circulation, potential vorticity, and helicity are simple corollaries. This Noether-theorem derivation shows that the vorticity laws have no analogues in particle mechanics, where the corresponding particle labels cannot be continuously vaned. 相似文献
3.
4.
5.
The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,?v,?t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x,?v,?t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x,?v)-variables decouple and the determination of p(x,?v,?t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity. 相似文献
6.
Susan Solomon Harold S. Johnston Marta Kowalczyk Ivan Wilson 《Pure and Applied Geophysics》1980,118(1):58-85
The catalytic destruction of stratospheric ozone by the oxides of nitrogen is believed to be an important part of the global ozone balance. The lack of sufficient measurements of NO
x
concentrations has impeded efforts to quantify this process. Recent measurements of stratospheric nitrogen dioxide from ground-based stations as well as aircraft and balloons have provided a first approximation to a global distribution of NO2 vertical columns at sunset. These observed vertical columns have been translated into time-dependent vertical NO2 profiles by means of a one-dimensional atmospheric photochemical model. Using recent observations of air temperature and ozone along with this information, the independent instantaneous (one second) rates of ozone production from oxygen photolysis P(O3), of ozone destruction from pure oxygen species (Chapman reactions) L(O
x
), and of ozone destruction by nitrogen oxides L(NO
x
) were estimated over the three-dimensional atmosphere. These quantities are displayed as zonal average contour maps, summed over various latitude zones, summed over various altitude bands, and integrated globally between 15 and 45 km. Although the global summation between 15 and 45 km by no means tells the complete story, these numbers are of some interest, and the relative values are: P(O3), 100; L(O
x
), 15; L(NO
x
), 45±15. It is to be emphasized that this relative NO
x
contribution to the integrated ozone balance is not a measure of the sensitivity of ozone to possible perturbations of stratospheric NO
x
; recent model results must be examined for current estimates of this sensitivity. 相似文献
7.
R. Krishna Rao Vupputuri 《Pure and Applied Geophysics》1978,117(3):448-485
two-dimensional time dependent model of the stratosphere incorporating the major interactions between radiative-photochemical and dynamical processes is described. The main prognostic equations considered are the thermodynamic equation and the general conservation equation for the minor chemical constituents representing the odd oxygen (O
x
=O+('D)+O3), odd hydrogen (HO
x
=HO+HO2), N2O, odd nitrogen (NO
x
=NO+NO2+HNO3), CF2Cl2, CFCl3 and odd chlorine (Cl
x
=Cl+ClO+HCl). The zonal wind and mean meridional circulations are determined diagnostically by the integration of the thermal wind equation and the stream function equation in the meridional plane espectively. The large scale eddy processes are parameterized in terms of zonal mean quantities using the generalized diffusion formulation on a sloping surface. The radiative heating and cooling and the hotochemical sources and sinks are incorporated in a form which allows for the major interactions among the minor trace constituents, temperature and mean circulation.Two integrations consisting of natural stratosphere and a stratosphere contaminated by the chlorofluoromethanes through lower boundary fluxes are carried out for 23 model years by changing the declination of the sun every day and using 6-hour time step. The model simulations of temperature, mean circulation, ozone, HO
x
, N2O and NO
x
in the meridional plane for the normal stratosphere, show satisfactory agreement with the available observations. Based on the results of second integration it is found that the injection of chlorofluoromethanes in the atmosphere at the estimated current production rates can lead to significant changes in the meridional distribution of ozone, temperature and NO
x
in the middle and upper stratosphere. The results also indicate that the percentage total ozone depletion increases from tropics to high latitudes and from summer to winter high latitudes. Also discussed are the results of additional experiments incorporating the reaction of HO2 with NO and the reactions involving ClNO3. 相似文献
8.
River water quality models usually apply the Fischer equation to determine the longitudinal dispersion coefficient (Dx) in solving the advection–dispersion equation (ADE). Recently, more accurate formulas have been introduced to determine Dx in rivers, which could strongly affect the accuracy of the ADE results. A numerical modelling-based approach is presented to evaluate the performance of various Dx formulas using the ADE. This approach consists of a finite difference approximation of the ADE, a MATLAB code and a MS Excel interface; it was tested against the analytical ADE solution and demonstrated using eight well-known Dx formulas and tracer study data for the Chattahoochee River (USA), the Severn (UK) and the Athabasca (Canada). The results show that Dx has an important effect on tracer concentrations simulated with the ADE. Comparison between the simulated and measured concentrations confirms the appropriate performance of Zeng and Huai’s formula for Dx estimation. Use of the newly proposed equations for Dx estimation could enhance the accuracy of solving the ADE. 相似文献
9.
Inversion of gravity and magnetic anomalies over some bodies of simple geometric shape 总被引：1，自引：0，他引：1
The general expression for gravity and magnetic anomalies over thin sheets and sloping contacts may be expressed as a polynomial of the formFx
2+C1Fx+C2F+C3x3+C4x2+C5x+C6. The initial parameters of the source are obtained from the coefficientsC
1, C2,..., C6 which may be solved by inverting a 6×6 matrix. The initial parameters are modified by successive iteration process using the difference formula until the root mean square error between the observed and calculated anomalies is a minimum. The regional background which may be in the form of a polynomial is estimated by the computer itself. This method is applied on a number of field anomalies and is found to yield reliable estimates of depth and other parameters of the source. 相似文献
10.
Alice Courvoisier 《地球物理与天体物理流体动力学》2013,107(2):217-243
We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t)?=?(? y ?ψ?(x, y, t), ?? x ?ψ?(x, y, t), ?ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R m . We calculate this effect for different motions and investigate how its limit as k?→?0 depends on R m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence. 相似文献
11.
Diurnal sea breeze effects on inner-shelf cross-shore exchange 总被引：1，自引：0，他引：1
Cross-shore exchange by strong (cross-shore wind stress, τsx>0.05 Pa) diurnal (7–25 h) sea breeze events are investigated using two years of continuous wind, wave, and ocean velocity profiles in 13 m water depth on the inner-shelf in Marina, Monterey bay, California. The diurnal surface wind stress, waves, and currents have spectral peaks at 1, 2, and 3 cpd and the diurnal variability represents about 50% of the total variability. During sea breeze relaxation (−0.05<τsx<0.05 Pa), a background wave-driven inner-shelf Eulerian undertow profile exists, which is equal and opposite to the Lagrangian Stokes drift profile, resulting in a net zero Lagrangian transport at depth. In the presence of a sea breeze (τsx>0.05 Pa), a uniform offshore profile develops that is different from the background undertow profile allowing cross-shore Lagrangian transport to develop, while including Lagrangian Stokes drift. The diurnal cross-shore current response is similar to subtidal (>25 h) cross-shore current response, as found by Fewings et al. (2008). The seasonality of waves and winds modify the diurnal sea breeze impact. It is suggested that material is not transported cross-shore except during sea breeze events owing to near zero transport during relaxation periods. During sea breeze events, cross-shore exchange of material appears to occur onshore near the surface and offshore near the sea bed. Since sea breeze events last for a few hours, the long-term cross-shore transport is incremental each day. 相似文献
12.
Craig S. Schwandt Randall T. Cygan Henry R. Westrich 《Pure and Applied Geophysics》1993,141(2-4):631-642
Few diffusion coefficient values have been measured for silicate minerals at pertinent geologic conditions because of experimental restrictions. Until recently, analysis of diffusion couples was conducted principally with electron microprobes which have rather poor spatial resolution (micrometer scale). Ion microprobe analyses, however, eliminate many of the previous experimental restrictions; in depth profile mode they have excellent spatial resolution (tens of angstroms) and diffusion couples can be analyzed normal to the interface. Diffusion couples analyzed by ion microprobe must be well-defined and uniform; previous methods using solution precipitates to form the diffusion couples were heterogeneous and had limited success. A new approach, the thermal evaporation of25MgO under high vacuum onto a crystalline substrate (oxide, silicate), produces a 1000 Å thick25MgO
x
(x<1) thin film. This method yields an excellent diffusion couple for low-temperature diffusion experiments. Diffusion anneal experiments using this approach for garnet provide a Mg self-diffusion coefficient ofD=0.60±0.09×10–21 m2/s at 1000°C (logFO2=–11.3,P=1 atm,X
Almandine=0.24). 相似文献
13.
S. Jeevananda Reddy 《Pure and Applied Geophysics》1976,114(1):29-38
Summary A modified method of computing theS
q
-range in terms ofR
x
inH, D andZ has been suggested. Mean quiet-day daily rangesR
x
, have been computed for five Indian stations usingH, D andZ data for selected years. The seasonal and latitudinal variation ofR
x
(H), R
x
(Z) andR
x
(D) are discusses. Thee-season maximum inR
x
(H) andR
x
(Z) is attributed to the decrease in the distance between the foci during equinoxes; the electrojet and theS
q
-foci movement with seasons have little influence onS
q
(D). It is inferred that the electrojet current is independent of the worldwideS
q
current system and stands with its own return currents.The variation of lunar semi-diurnal tide inH[L
2
(H)] with the dip latitude in Indian region shows a secondary peak at 9° N dip latitude. This secondary peak in theL
2
(H) is termed as Lunar secondary electrojet, and it is suggested that this is possibly produced by magneto-acoustic oscillations due to the drift motion of the charged particles that produce the primary jet in a direct transverse to itself. 相似文献
14.
Based on measured stream nitrogen concentrations at outlets of 12 small sub‐areas (1·3–54·7 km2) in a largely forested catchment during the base flow period, we investigated the influences of discharges and different catchment characteristics on stream nitrogen concentration. Our field surveys were carried out during the 11‐month's period from April 2001 to February 2002 and the correlations between nitrogen concentrations and catchment characteristics were studied. The results showed that the vegetation cover was strongly correlated to total nitrogen (TN) and nitrate + nitrite ? nitrogen (NOx‐N) concentrations. That is, the TN and NOx‐N concentrations had positive correlations with mean normalized difference vegetation cover index (NDVI) of each sub‐area during dormant seasons (mean NDVI < 0 · 70) and had negative correlations during the growing season (mean NDVI ≥ 0 . 70). The significance of catchment characteristics to TN and NOx‐N concentrations was ranked as vegetation cover > soil > topography > land use, and the best models can account for 55–64% of the variance of TN and NOx‐N concentrations. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
15.
This paper aims to develop an improved understanding of the critical response of structures to multicomponent seismic motion characterized by three uncorrelated components that are defined along its principal axes: two horizontal and the vertical component. An explicit formula, convenient for code applications, has been derived to calculate the critical value of structural response to the two principal horizontal components acting along any incident angle with respect to the structural axes, and the vertical component of ground motion. The critical response is defined as the largest value of response for all possible incident angles. The ratio rcr/rsrss between the critical value of response and the SRSS response—corresponding to the principal components of ground acceleration applied along the structure axes—is shown to depend on three dimensionless parameters: the spectrum intensity ratio γ between the two principal components of horizontal ground motion characterized by design spectra A(Tn) and γA(Tn); the correlation coefficient α of responses rx and ry due to design spectrum A(Tn) applied in the x‐ and y‐directions, respectively; and β = ry/rx. It is demonstrated that the ratio rcr/rsrss is bounded by 1 and . Thus the largest value of the ratio is , 1.26, 1.13 and 1.08 for γ = 0, 0.5, 0.75 and 0.85, respectively. This implies that the critical response never exceeds times the result of the SRSS analysis, and this ratio is about 1.13 for typical values of γ, say 0.75. The correlation coefficient α depends on the structural properties but is always bounded between −1 and 1. For a fixed value of γ, the ratio rcr/rsrss is largest if β = 1 and α = ±1. The parametric variations presented for one‐storey buildings indicate that this condition can be satisfied by axial forces in columns of symmetric‐plan buildings or can be approximated by lateral displacements in resisting elements of unsymmetrical‐plan buildings. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
16.
Nikolai Petersen 《Pure and Applied Geophysics》1976,114(2):177-193
Summary The magnetic properties of basaltic rocks are dominated by the contained primary Fe–Ti oxides. At solidus temperature (1000°C) the composition of these primary oxides is restricted to titanomagnetite (Fe3-xTixO4) and hemoilmenites (Fe2-yTiyO3). The examination of 269 chemical analyses of the primary Fe–Ti oxides in basalts (in sensu lato) gives an average ofx=0.61 (T
c=168°C) for the titanomagnetites andy=0.89 (T
c=–121°C) for the hemoilmenites. If distinction is made between tholeiites, alkali basalts and andesites, a clear difference for thex-values is observed: the average for tholeiitesx=0.64 (T
c=144°C), for alkali basaltsx=0.52 (T
c=253°C), for andesitesx=0.38 (T
c=341°C).Environment of crystallization and cooling rate are major interrelated factors influencing subsequent changes in the mineralogy of the primary Fe–Ti oxides and resulting magnetic properties. This has been tested by studying the variation of magnetization and some of its parameters in three different basalt rock units: a dike, 180 cm, and two lava flows, 3 m and 33 m thick, respectively. Grain size and oxidation state of the titanomagnetites control the variation of magnetization in these basalt units. 相似文献
17.
The gravitational attraction produced by a geological body of irregular shape can be easily determined by dividing it into cubes of small size. The exact expression of the effect of a cube is very complicated, but it can be calculated by using an electronic computer. 4851 values of the gravitational attraction were determined for different positions of a cube with the side of 2l and the center in M(x0, y0, z0), for x0∈[0;20], y0∈[0;20] and z0∈[0;10]. Using these values, templates in double logarithmic representation were drawn, with x0 and z0 as parameters and y0 as abscissa, and with x0 and y0 as parameters and z0 in abscissa; this double set of templates permits a good interpolation for all cubes in the considered domain of M. The use of templates was tested to approximate the effect produced by a theoretical model of spherical shape and in a real case of a three-dimensional salt body of known shape based on a large number of boreholes. In both cases very good results were obtained. 相似文献
18.
The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M 2 tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.
相似文献19.
T. L. Chelidze H. Spetzler I. C. Getting Z. A. Avaliani 《Pure and Applied Geophysics》1990,134(1):31-43
The elastic properties of a physical model representing a damaged rock matrix were studied using a square lattice deformed under tensile stress. The elastic modulusM of such a system varies in agreement with percolation theory as|x–x
c
|
f
, wherex is the damage parameter andx
c
the threshold value of the damage parameter,f3.6. Atxx
c
the scale dependence ofM can be expressed asML
–f/v
, whereL is the size of the sample andv the correlation exponent in percolation theory.The experimental results are of interest in assessing elastic properties in earthquake focal zones and fault zones in general. 相似文献