Time-dependent Version Research Articles

The seasonal steady circulation of the Eastern Mediterranean determined with the adjoint method

Abstract In this paper we use a rather unconventional approach to determine the steady seasonal circulation of the Eastern Mediterranean. Traditional calculations rely either on prognostic models spun-up with different forcing functions or on inverse methods having rather simple dynamics. In the present applications one of the most sophisticated inverse techniques, the adjoint method of control theory, is used to find the model state that is optimally consistent with the model dynamics, with a prescribed climatology and is steady in time. The model used is the GFDL primitive equation model in its fully time-dependent non-linear version forced by seasonal wind-stress fields that are kept steady for each calculation. The prescribed climatology consists of the seasonal hydrographies of the temperature and salinity fields. Steadiness upon the seasonal time scale is required as a term in the cost function of the adjoint that penalizes the tendencies of the prognostic variables. This use of the adjoint method reconstructs the steady seasonal wind-driven circulation in an ocean with a prescribed baroclinic structure. As such, it is equivalent to a prognostic spin-up calculation with steady winds and the robust diagnostic applied, i.e. adding a term that relaxes the temperature and salinity fields to the seasonal climatologies with a time constant of 3 months. To assess the “success’ of these calculations, the success of the inversion must be quantified. The examination of the final data misfits and steady state residuals shows that steady state has indeed been reached. The steady-state residuals are always much smaller than the data misfits and both of them are always small, well below the one standard deviation value for each field. Thus, we can assess that a meaningful solution has indeed been attained. To assess further if these solutions are reasonable, we have carried out for comparison robust diagnostic calculations with a time constant of 3 months. The circulations thus obtained are extremely similar to the adjoint solutions in reproducing the overall patterns as well as the individual sub-basin scale gyres and interconnecting currents and meandering jets. The circulations obtained with the two approaches are also equally strong. However, both the adjoint and the robust diagnostic results produce an overall barotropic transport that is one order of magnitude bigger than that observed. They also both show anomalously strong vortex structures in regions of sharp topographic breaks connecting the deep interior to the shelves, for which no observational evidence is available. These unrealistic features can be explained by taking into account that with the short time scale of 3 months used in both approaches biased solutions may be obtained. These biases are due to inconsistencies between the rough topography used and the smooth climatologies, that lead to a misrepresentation of the important JEBAR effect. This explanation is supported by a further robust diagnostic calculation in which the time constant is increased in the deep layers that gives a circulation intensity much more realistic. Overall, this application of the adjoint method to the GFDL model shows that it can be successfully used to find meaningful optimal solutions. These solutions also prove to be reasonable when compared with analogous robust diagnostics results.

The diabatic approximation in the classical frame

The time-dependent version of a Golden Rule treatment for studying vibrational predissociation (VP) of a model triatomic van der Waals (vdW) X ċ BC system shows the VP rate as the Fourier transform of an autocorrelation function. Analytical fits to this time dependent function allow the rewriting of such a rate in terms of the autocorrelation time. This suggests the possibility of using a diabatic vibrational approximation within the classical frame. In particular, it would enable the study of the VP of vdW clusters containing a fast vibrational diatomic subunit, a current bottleneck in quasiclassical trajectory (QCT) calculations.

Systematic measurements of K-shell ionization by 3He ions of energy 0.6–3.6 MeV

K-shell ionization cross sections for A1, P, Ar, Ti, Cr, Fe, Ni, Ge, Se, Nb and Pd were measured for incident 3He ions of energy 0.6–3.6 MeV. The experimental results were compared with the predictions of the existing theories of K-shell ionization by charged particles, i.e. the ECPSSR theory, the simplified SCA-model of Lægsgaard et al., the SCA calculations with approximate Andersen's recipe for the electron binding energy and, finally, the fully time-dependent SCA-FTD calculations of Trautmann and Kauer, which also use the Andersen's recipe for estimation of the binding energy, but in the time-dependent version. Good agreement is observed between the experimental data and the predictions of the ECPSSR theory. The fully time-dependent treatment of binding effect has essentially improved an agreement of the SCA calculations with experimental results for higher reduced velocities, but still there are discrepancies in the low-velocity region. On the other hand, the approximate Andersen's recipe simulates quite well the binding effect.

A time-dependent version of P�lya's urn

A process is defined that consists of drawing balls from an urn and replacing them along with extra balls. The process generalizes the well-knownPolya urn process. It is shown that the proportion of red balls in the urn converges to a random limit that may have a nonzero probability of being 0 or 1, but is nonatomic elsewhere.

Comments on a time-dependent version of the linear-quadratic model

The accuracy and interpretation of the “LQ + time” model ( E = D( α + βd) - γT) are discussed. Evidence is presented, based on data in the literature, that this model does not accurately describe the changes in isoeffect dose occurring with protraction of the overall treatment time during fractionated irradiation of the lung. This lack of fit of the model explains, in part, the surprisingly large values of γ/α that have been derived from experimental lung data. The large apparent time factors for lung suggested by the model are also partly explained by the fact that γT/ α, despite having units of dose, actually measures the influence of treatment time on the effect scale, not the dose scale, and is shown to consistently overestimate the change in total dose. The unusually high values of α/β that have been derived for lung using the model ( ≈5 Gy) are shown to be influenced by the method by which the model was fitted to data. Reanalyses of the data using a more statistically valid regression procedure produce estimates of α/β more typical of those usually cited for lung ( ≈ 3 Gy). Most importantly, published isoeffect data from lung indicate that the true deviation from the linear-quadratic (LQ) model is nonlinear in time, instead of linear, and also depends on other factors such as the effect level and the size of dose per fraction. Thus, we do not advocate the use of the “LQ + time” expression as a general isoeffect model.

Calculation of the VIB-rotational state density of polyatomic molecules by adiabatic switching

The problem of estimating the vibrational and vib-rotor state density (both local and cumulative) for polyatomic molecules is approached using the Hertz invariant of classical Hamiltonian dynamics. Such state densities are key input to statistical theoreis of reaction dynamics such as that of RRKM and transition-state theory (TST), and have recently found to be seriously under-estimated by standard Harmonic and direct count Morse oscillator methods. In the semi-classical limit (which is quite appropriate for the very high state densities of interest here) the cumulative state density at energy E is simply the volume of the energy shell, V( E), divided by h N , h being Planck's constant, and N the number of degrees of freedom. The local state density is then h −N d V( E)/d E. Use may thus be made of the fact, noted by Hertz, that the energy shell is an adiabatic invariant for ergodic Hamiltonian systems. This invariance, combined with the time-dependent version of Liouville's theorem allows rigorous estimation of the phase volume of ergodic Hamiltonian systems which are adiabatically connected to other ergodic systems of known phase volume. As such ergodic reference systems with known phase volume are not easily found, we show that, when taken as an averaging method, the Hertz invariant gives excellent results when the reference Hamiltonian is not only non-ergodic, but in fact is separable. Thus, using an N degree of freedom isotropic oscillator reference, classical adiabatic switching is shown to give cumulative and local state densities for a model system with numbers of degrees of vibrational freedom ranging from 4, 6, 8…20… and finally up to 1000. In the regime beyond 8 degrees of freedom primitive Monte-Carlo sampling is inapplicable in a practical sense.

Optimal Decision Making and the Value of Information in a Time-Dependent Version of the Cost-Loss Ratio Situation

Abstract A time-dependent version of the cost-loss ratio situation is described and the optimal use and economic value of meteorological information are investigated in this decision-making problem. The time-dependent situation is motivated by a decision maker who contemplates postponing the protect/do not protect decision in anticipation of obtaining more accurate forecasts at some later time (i.e., shorter lead time), but who also recognizes that the cost of protection will increase as lead time decreases. Imperfect categorical forecasts, calibrated according to past performance, constitute the information of primary interest. Optimal decisions are based on minimizing expected expense and the value of information is measured relative to the expected expense associated with climatological information. Accuracy and cost of protection are modeled as exponentially decreasing functions of lead time, and time-dependent expressions for expected expense and value of information are derived. An optimal lead time...

Time-resolved photodissociation dynamics studied by absorption and emission spectroscopies

Absorption and emission spectroscopies of photodissociating molecules are very refined tools for probing the short-lived transient species. The analogy between the energy or the time dependent versions of the quantum theory is discussed for the three successive steps of the dynamical process : photon absorption, radiative relaxation by Raman emission and final energy redistribution in the fragments. It is shown how informations concerning these different time scale processes can be related to the parameters of the potential energy surfaces and to the excitation laser wavelength. Two well-documented triatomic systems, namely ICN and 03 are considered to illustrate how absorption and emission spectra can be used to study reaction dynamics.

Transient electrodynamic response of thin-film superconductors to laser radiation.

We examine a theory of the transient electrodynamics of thin-film superconductors irradiated by both cw- and pulsed-laser light. The theory employs the Rothwarf-Taylor rate equations in conjunction with a time-dependent version of Parker's ${T}^{\mathrm{*}}$ model of nonequilibrium superconductivity, to determine the microwave and far-infrared electrodynamics. This formulation is sufficiently general that it can also be utilized for problems in which either static or dynamic gratings are written on the superconductor by two laser beams. We apply the theory to studies of time-dependent microwave transmission through thin superconducting films and microwave propagation on superconducting waveguides. A perturbative formulation of the electrodynamics, to first order in the quasiparticle response, is also presented.

An evaluation of the effect of reductions in ambient levels of primary pollutants on sulfate and nitrate wet deposition

The effect of reductions in the initial gas phase concentrations of SO 2, NO x and reactive hydrocarbons on sulfate and nitrate production and deposition under precipitating conditions is studied using a one-dimensional time-dependent version of the STEM-II model. Simulations corresponding to highly polluted, moderately polluted and rural conditions are evaluated for both daytime and night-time conditions. The simulation results indicate that complex relationships can exist between the initial concentrations of primary pollutants and the resultant sulfate and nitrate deposition. The results also demonstrate a strong coupling between the gas phase photochemical cycles and acid formation in clouds within storm events.

Chemical physics without the Born-Oppenheimer approximation: The molecular coupled-cluster method.

The Born-Oppenheimer (BO) and Born-Huang (BH) treatments of molecular eigenstates are reexamined. It is argued that in application of the BO approximation to nonrigid molecules and chemical dynamics involving single potential-energy surfaces (PES's), errors on the order of tens of percents can easily occur in many computed properties. Introduction of a BH expansion (in BO states) will always lead to poor convergence when the BO approximation fails; its diagonal (or adiabatic) approximation will not change this situation. The main problem in the above applications is the absence of well-developed, well-separated minima in the PES (or no minima at all). Inspired by a non-BO view of a molecule by Ess\'en [Int. J. Quantum Chem. 12, 721 (1977)], a molecular coupled-cluster (MCC) method is formulated. An Ess\'en molecule consists of neutral subunits (``atoms''), weakly interacting (``bonds'') in some spatial arrangement (``structure''). The quasiseparation in collective and individual motions within the molecule comes about by virtue of the virial theorem, not the smallness of the electron-to-nuclear mass ratio. The MCC method not only should converge well in the cluster sizes, but it also is capable of describing electronic shell and molecular geometric structures. It can be viewed as the workable formalism for Ess\'en's physical picture of a molecule. The time-independent and time-dependent versions are described. The latter one is useful for scattering, chemical dynamics, laser chemistry, half-collisions, and any other phenomena that can be described as the time evolution of many-particle wave packets. Close relationship to time-dependent Hartree-Fock theory exists. A few implementational aspects are discussed, such as symmetry, conservation laws, approximations, numerical techniques, as well as a possible relation with a non-BO PES. Appendixes contain mathematical details.

Time-dependent solution of pre-mixed laminar flames with a known temperature profile

A computer program designed for the evaluation of molecular flows interacting through chemical kinetics and molecular diffusion is described. Measured values of temperature profile and mass flow are used. The starting profiles and the hot boundary values are calculated by a kinetics approximation found by neglecting diffusion. A time-dependent method is used together with successive grid refinements. The successive grid refinements reduced the execution times by a factor of 5 for a H 2/air flame at a pressure of 1 atm. For a CH 4/O 2 flame at 0.05 atm the reduction due to grid refinements was a factor 50 or more according to the estimations. The execution times for the test flames were a factor 4 slower than a current implementation of the steady state method. Possible optimizations of the present time-dependent version can decrease that difference significantly. The computed concentration profiles agreed with published computed results within 1 %.

Correlation effects in ion neutralization scattering with the use of a time-dependent coupled-cluster approach.

We consider the problem of ion neutralization scattering from surfaces. For large kinetic energies, the motion of the ion can be treated classically. The electronic part is assumed to be described by a time-dependent Newns-Anderson Hamiltonian. The ion is supposed to have a closed-shell structure with one empty orbital outside the shell, which can take up, at the most, two electrons from the metal. One can obtain a time-dependent Hartree-Fock (TDHF) solution for this using a procedure suggested earlier. [T. B. Grimley, V. C. Jyothi Bhasu, and K. L. Sebastian, Surf. Sci. 124, 305 (1983)]. We show that this solution is defective in that it predicts that the probability that the ion comes back as a neutral species is always less than 0.5, thus illustrating that one has to include electron correlation in order to describe the process correctly.For this we make use of the time-dependent version of the coupled-cluster approach. In this, one assumes the wave function to have the form exp[${T}_{0}$(t)+${T}_{1}$(t)+${T}_{2}$(t)+. . .]\ensuremath{\Vert}${\ensuremath{\Phi}}_{0}$〉 where \ensuremath{\Vert}${\ensuremath{\Phi}}_{0}$〉 is a Slater determinant and ${T}_{n}$(t) can create n-particle hole excitations in it. We take ${T}_{1}$(t) as a linear combination of all possible single-particle hole-excitation operators while ${T}_{2}$(t) is taken as a linear combination of just those two-particle hole-excitation operators which transfer two electrons to the orbital of the ion from the solid, neglect ${T}_{n}$(t) for n>2, and derive differential equations for the matrix elements of the operators ${T}_{1}$(t) and ${T}_{2}$(t). These differential equations are solved numerically to obtain the wave function at any time t.New theorems which enable us to calculate all the expectation values that arise in our treatment of the problem are presented. Also, we have derived expressions for the excitation spectrum, produced as a result of the collision, by particles which come back as ions and also by those which have taken up one or two electrons from the solid. The method is applied to the scattering of lithium ions from the Ni(100) surface and also from a Ni surface contaminated with alkali atoms. The calculations show that TDHF theory is not a bad approximation if one is concerned just with the approach of the ion to the surface. But, in treating an ion which leaves the surface, TDHF fails. The predicted values of charge-transfer probabilities are considerably different in the two theories. Also, calculation of the excitation spectrum produced as a result of the collision show that the spectra are much broader for a contaminated surface having a lower work function than for the clean surface.

The atmospheric carbon dioxide response to oceanic primary productivity fluctuations

Data on the rate of primary production, the rate of sinking and oxidation of organic detritus, the rate of regeneration of calcium carbonate, the average rate of vertical mixing, and the vertical profile of dissolved carbon are consistent with a model of the ocean in which the downward transport of carbon by the sinking and oxidation of organic particulate matter is balanced by upward hydrodynamic mixing of dissolved inorganic carbon. A time dependent version of this model is used to make rough estimates of the effect of periodic and impulsive changes in the rate of primary production on the atmospheric CO2 concentration. The computations suggest that a 1% decrease in marine biospheric productivity could result in a steady state increase of the atmospheric CO2 concentration of from 0.5 to 2.5% depending on the rate of vertical mixing. Large but short term fluctuations in productivity, such as a die-off of the marine biosphere followed by an exponential recovery period, are estimated to produce smaller perturbations.

Unitary approximations to transition probabilities: Generalized impact parameter theory and the time dependent projection operator method

A general unitary approximation scheme is presented for transition probabilities based upon the application of the time dependent version of the Zwanzig–Feshbach projection operator method to the generalized impact parameter approximation. The overall approximation scheme is based upon the time dependent picture of the binary collision process. To begin with, the translational and internal motions are decoupled with the translational motion being treated in one of three ways, namely, as a straight line trajectory, as a single curved reference classical trajectory, or as a collection of such trajectories, each of which is associated with a pair of internal states. The internal dynamics is then parameterized by the set of given trajectories which are functions of the impact parameter. This motion is then formally solved using the Zwanzig–Feshbach projection operator method. A set of unitary approximations to these solutions are then presented, based upon a time disordered approximation. These latter approximations are applicable to each of the three treatments of the translational motion. As well, the basis used to describe the internal degrees of freedom may be either atomic or molecular (diabatic or adiabatic). Thus this combined approximation scheme provides a systematic theory for testing various effects associated with either the treatment of the translational motion or the internal motion.

Density-Functional Theory for Time-Dependent Systems

The response of an interacting many-particle system to a time-dependent external field can usually be treated within linear response theory. Due to rapid experimental progress in the field of laser physics, however, ultra-short laser pulses of very high intensity have become available in recent years. The electric field produced in such pulses can reach the strength of the electric field caused by atomic nuclei. If an atomic system is placed in the focus of such a laser pulse one observes a wealth of new phenomena [1] which cannot be explained by traditional perturbation theory. The non-perturbative quantum mechanical description of interacting particles moving in a very strong time-dependent external field therefore has become a prominent problem of theoretical physics. In principle, it requires a full solution of the time-dependent Schrodinger equation for the interacting many-body system, which is an exceedingly difficult task. In view of the success of density functional methods in the treatment of stationary many-body systems and in view of their numerical simplicity, a time-dependent version of density functional theory appears highly desirable, both within and beyond the regime of linear response.

Population of Excited States of Al10+ in a Plasma by a Time-Dependent Model

The present model is the time-dependent version of a previous model (Sureau et al. 1983) in which the population distribution was assumed in steady state. A finite set of levels is partitioned in four subsets: the Z-ion ground-level and, contingently, the first near-degenerated levels (subset 1); all the successive excited Z-ion levels up to n=5 (subset 2); a finite number of higher Rydberg levels (because of the limitation of the series in the plasmas) which are assumed in LTE with the Z+l−ion ground-level (subset 3, called the thermal band); and the Z+l ion ground-level (subset 4).The physical processes explicitly considered are the radiative cascades and the transitions between the Z-ion bound levels induced by electron-ion collisions. The radiative-transition probabilities are given by ab-initio calculations using a modified Hartree-Fock method including the spin-orbit interaction (Sureau et al., 1984). The collision rates are derived by the Van Regemorter formula multiplied by an adjustable parameter Fc.

On the Existence of Solutions to Time-Dependent Hartree-Fock Equations

where Xj=(x], xj, xj)ei2, Aj= ^ ( d / 3 x j ) 2 and Q(x V(x) are real functions such that V(x) = V(—x). If the system obeys the Fermi statistics, it is natural to treat (1.1) in the anti-symmetric subspace of L(R). Taking note of this anti-symmetry and using the variational principle, Dirac ([3], [4]) has derived the following time-dependent version of Hartree-Fock equation in order to obtain an approximate solution of (1.1) :

Transient magnetic field calculations with TRIDIF

TRIDIF is a time-dependent diffusion version of the well-known PANDIRA-POISSON-TRIM triangular mesh magnet code. Modifications which allow TRIDIF to treat field diffusion in materials with time-varying permeabilities are described. A heuristic dipole alignment model which described major and minor hysteresis loops and frequency dependence has been included to provide the requisite permeabilities. Good agreement between the measured and computed magnetic fields was found for a simple test experiment.

The strong convergence of Schrödinger propagators

Time dependent versions of the Trotter-Kato theorem are discussed using nonstandard analysis. Both standard and nonstandard results are obtained. In particular, it is shown that if a sequence of generators converges in the strong resolvent topology at each time to a limiting generator and if the sequence of generators and limiting generator uniformly satisfy Kisynski type hypotheses then the corresponding Schrodinger propagators converge strongly. The results are used to analyze time dependent, form bounded perturbations of the Laplacian.