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NASA NTRS1995

Intermittent turbulence in solar wind from the south polar hole

The magnetic fields measured by the Ulysses spacecraft are used to study solar wind turbulence in the fast solar wind from the south polar hole. The spacecraft was at about 46 deg south latitude and 3.9 AU. For a magnetic field with a Gaussian distribution the power spectrum (second-order structure function) is sufficient to completely characterize the turbulence. However, the actual distribution is non-Gaussian so that the effects of intermittency must be taken into account. The observed spectral exponents include effects of intermittency and cannot be directly compared with the standard second-order spectral theories such as the Kolmogorov and Kraichnan theories. To permit a better comparison of the observations with the theoretical models, we study the structure characteristics of the data. We find the exponents of the second-order structure functions (power spectra) and the higher-order normalized structure functions for the components of the magnetic fields. We show that these sets of exponents can be approximately described by two basic numbers: the spectral exponent and the intermittency exponent. The intermittency exponent characterizes correlation properties of the energy cascade from large to small scales. Before comparing the observations to the theoretically expected values, a reduction must be made to the observed spectral exponent. The amount of the reduction depends on both the intermittency exponent and the model of the energy cascade assumed in the turbulence theory. We reduce the measured spectral indices according to a simple model for Alfven turbulence that is described here. We then compare our reduced spectral indices with second-order spectral theory. The reduced spectral indices for the period range of 1 min to about a half hour are remarkably constant and in good agreement with the value of 3/2. Thus our treatment is self-consistent. Our tentative conclusion is that the high-frequency turbulence appears to agree with the model of random-phased Alfven waves. This tentative conclusion must be tested by further theoretical and observational work.

Ruzmaikin, A. A.↗

NASA NTRS1994

Multiparameter radar study of rainfall: Potential application to area-time integral studies

Multiparameter radars measure one or more additional parameters in addition to the coventional reflectivity factor. The combination of radar observations from a multiparameter radar is used to study the time evolution of rainstorms. A technique is presented to self-consistently compare the area-time integral (ATI) and rainfall volume estimates from convective storms, using two different measurements from a multiparameter radar. Rainfall volumes for the lifetime of individual storms are computed using the reflectivity at S band (10-cm wavelength) as well as one-way specific attenuation at X band (3-cm wavelength). Area-time integrals are computed by summing all areas in each radar snapshot having reflectivities (S band) in excess of a preselected threshold. The multiparameter radar data used in this study were acquired by the National Center for Atmospheric Research (NCAR) CP-2 radar during the Cooperative Huntsville Meteorological Experiment (COHMEX) and the Convection and Precipitation/Electrification Experiment (CaPE), respectively. ATI studies were accomplished in this work using multiparameter radar data acquired during the lifetime of six convective events that occurred in the COHMEX radar coverage area. A case study from the COMHEX field campaign (20 July 1986) was selected to depict the various stages in the evolution of a storm over which the ATI and rainfall volume computations were performed using multiparameter radar data. Another case study from the CaPE field campaign (12 August 1991) was used to demonstrate the evolution of a convective cell based on differential reflectivity observations.

Raghavan, R.↗

NASA NTRS1995

Mixed Carrier Conduction in Modulation-doped Field Effect Transistors

The contribution of more than one carrier to the conductivity in modulation-doped field effect transistors (MODFET) affects the resultant mobility and complicates the characterization of these devices. Mixed conduction arises from the population of several subbands in the two-dimensional electron gas (2DEG), as well as the presence of a parallel path outside the 2DEG. We characterized GaAs/AlGaAs MODFET structures with both delta and continuous doping in the barrier. Based on simultaneous Hall and conductivity analysis we conclude that the parallel conduction is taking place in the AlGaAs barrier, as indicated by the carrier freezeout and activation energy. Thus, simple Hall analysis of these structures may lead to erroneous conclusions, particularly for real-life device structures. The distribution of the 2D electrons between the various confined subbands depends on the doping profile. While for a continuously doped barrier the Shubnikov-de Haas analysis shows superposition of two frequencies for concentrations below 10(exp 12) cm(exp -2), for a delta doped structure the superposition is absent even at 50% larger concentrations. This result is confirmed by self-consistent analysis, which indicates that the concentration of the second subband hardly increases.

Schacham, S. E.↗

NASA NTRS1998

Theory of the Shape of the Heliospheric Termination Shock: Axial Symmetry

The heliospheric termination shock must exhibit asymmetry in its shape, due in part to internal latitude variations in the solar wind, and in part to the special directions defined by the external interstellar flow and/or the galactic magnetic field. This asymmetry shows up naturally in numerical simulations of the interaction between the heliosphere and local interstellar medium. However, to date only one analytical treatment of the theory has appeared, which discussed the modification of the shock shape due to solar wind latitude variations in the presence of spherically symmetric outer boundary conditions. In the present report, we discuss an extension and generalization of this theory to an axially symmetric gasdynamic system, in which departure from spherical symmetry may be due either to internal solar wind variations or to the directional properties of the external interstellar medium. It is shown that for steady flow the post-shock region is characterized by an infinite set of quantities that are conserved along streamlines; among these invariants are the stagnation pressure and a quantity closely related to vorticity. Moreover, for a given latitude profile of the (supersonic) solar wind, the geometry of the termination shock uniquely determines the valued of these invariants at the points where the streamlines emerge from the shock. A self-consistent solution of the problem thus requires a match between the shock geometry and distant boundary conditions. The theory is applied to give estimates of termination-shock asymmetry for various internal and external conditions.

Barnes, Aaron↗

NASA NTRS1994

The interstellar medium in the starburst regions of NGC 253 and NGC 3256

We discuss observations of the (C II) 158 micrometers, (O I) 63 micrometers, (Si II) 35 micrometers, (O III) 52,88 micrometers, and (S III) 33 micrometers fine-structure transitions toward the central 45 seconds of the starburst galaxies NGC 253 and NGC 3256. The (C II) and (O I) emission probably originates in photodissociated gas at the surfaces of molecular clouds, although a small (less than or approximately 30%) contribution to the (C II) flux from H II regions cannot be ruled out. The (O III) and (S III) lines originate in H II regions and the (Si II) flux is best explained as originating in H II regions with some contribution from photodissociation regions (PDRs). The gas phase silicon abundance is nearly solar in NGC 253, which we interpret as evidence for grain destruction in the starburst region. We find that the photodissociated atomic gas has densities approximately 10(exp 4)/cu cm and temperature 200-300 K. About 2% of the gas is in this phase. The thermal gas pressure in the PDRs, P(PDR)/k approximately 1-3 x 10(exp 6) K/cu cm, might represent the 'typical' interstellar gas pressure in starburst systems. The Far Ultraviolet (FUV) radiation fields illuminating the clouds are 10(exp 3)-10(exp 4) stronger than the local Galactic FUV field and come from the contribution of many closely packed O and B stars. For the central 250 pc of NGC 253, we find that the H II gas has an average density n(sub e) is approximately 400/cu cm. This corresponds to a thermal pressure P(H II)/k approximately 7 x 10(exp 6) K/cu cm which is approximately P(PDR)/k, suggesting that the ionized gas is in pressure equilibrium with the photodissociated gas at the surfaces of molecular clouds. The H II gas fills a significant fraction, approximately 0.01-0.3, of the volume between the clouds. The effective temperature of the ionizing stars in NGC 253 is greater than or approximately 34,500 K; 2 x 10(exp 5) O7.5 stars would produce the observed Lyman countinuum photon luminosity. The average separation between the stars is approximately 3 pc. Applying the simple model for the interstellar medium in galactic nuclei of Wolfire, Tielens, & Hollenbach (1990), we find the molecular gas in the central regions of NGC 253 and NGC 3256 to be distributed in a large number (5 x 10(exp 3) to 5 x 10(exp 5)) of small (0.5-2 pc), dense (approximately 10(exp 4)/cu cm) clouds (or alternatively 'thin-flattened' structures) with volume filling factors 10(exp -3) to 10(exp -2), very different from the local Interstellar Medium (ISM) of the Galaxy. We suggest a self-consistent scenario for the ISM in NGC 253 in which clouds and H II gas are in pressure balance with a supernova-shocked, hot 1-3 x 10(exp 6) K, low-density (approximately 10(exp 4)/cu cm), all pervasive medium. A feedback mechanism may be indicated in which the pressure generated by the supernovae compresses the molecular clouds and triggers further massive star formation. The similarity of ISM parameters deduced for NGC 253, NGC 3256, and M82 (Lord et al. 1993) suggests that the ISM properties are independent of the luminosity of the starburst or the triggering mechanism, but are rather endemic to starburst systems. The starburst in NGC 3256 appears to be a scaled-up version of the NGC 253 and M82 starbursts.

Carral, P.↗

NASA NTRS2003

Unified Models of Turbulence and Nonlinear Wave Evolution in the Extended Solar Corona and Solar Wind

The PI (Cranmer) and Co-I (A. van Ballegooijen) made significant progress toward the goal of building a "unified model" of the dominant physical processes responsible for the acceleration of the solar wind. The approach outlined in the original proposal comprised two complementary pieces: (1) to further investigate individual physical processes under realistic coronal and solar wind conditions, and (2) to extract the dominant physical effects from simulations and apply them to a one-dimensional and time-independent model of plasma heating and acceleration. The accomplishments in the report period are thus divided into these two categories: 1a. Focused Study of Kinetic MHD Turbulence. We have developed a model of magnetohydrodynamic (MHD) turbulence in the extended solar corona that contains the effects of collisionless dissipation and anisotropic particle heating. A turbulent cascade is one possible way of generating small-scale fluctuations (easy to dissipate/heat) from a pre-existing population of low-frequency Alfven waves (difficult to dissipate/heat). We modeled the cascade as a combination of advection and diffusion in wavenumber space. The dominant spectral transfer occurs in the direction perpendicular to the background magnetic field. As expected from earlier models, this leads to a highly anisotropic fluctuation spectrum with a rapidly decaying tail in the parallel wavenumber direction. The wave power that decays to high enough frequencies to become ion cyclotron resonant depends on the relative strengths of advection and diffusion in the cascade. For the most realistic values of these parameters, though, there is insufficient power to heat protons and heavy ions. The dominant oblique waves undergo Landau damping, which implies strong parallel electron heating. We thus investigated the nonlinear evolution of the electron velocity distributions (VDFs) into parallel beams and discrete phase-space holes (similar to those seen in the terrestrial magnetosphere) which are an alternate means of heating protons via stochastic interactions similar to particle-particle collisions. 1b. Focused Study of the Multi-Mode Detailed Balance Formalism. The PI began to explore the feasibility of using the "weak turbulence," or detailed-balance theory of Tsytovich, Melrose, and others to encompass the relevant physics of the solar wind. This study did not go far, however, because if the "strong" MHD turbulence discussed above is a dominant player in the wind's acceleration region, this formalism is inherently not applicable to the corona. We will continue to study the various published approaches to the weak turbulence formalism, especially with an eye on ways to parameterize nonlinear wave reflection rates. 2. Building the Unified Model Code Architecture. We have begun developing the computational model of a time-steady open flux tube in the extended corona. The model will be "unified" in the sense that it will include (simultaneously for the first time) as many of the various proposed physical processes as possible, all on equal footing. To retain this generality, we have formulated the problem in two interconnected parts: a completely kinetic model for the particles, using the Monte Carlo approach, and a finite-difference approach for the self-consistent fluctuation spectra. The two codes are run sequentially and iteratively until complete consistency is achieved. The current version of the Monte Carlo code incorporates gravity, the zero-current electric field, magnetic mirroring, and collisions. The fluctuation code incorporates WKJ3 wave action conservation and the cascade/dissipation processes discussed above. The codes are being run for various test problems with known solutions. Planned additions to the codes include prescriptions for nonlinear wave steepening, kinetic velocity-space diffusion, and multi-mode coupling (including reflection and refraction).

Cranmer, Steven R.↗

NASA NTRS1994

Airborne boundary layer flux measurements of trace species over Canadian boreal forest and northern wetland regions

Airborne heat, moisture, O3, CO, and CH4 flux measurements were obtained over the Hudson Bay lowlands (HBL) and northern boreal forest regions of Canada during July - August 1990. The airborne flux measurements were an integral part of the NASA/Arctic Boundary Layer Expedition (ABLE) 3B field experiment executed in collaboration with the Canadian Northern Wetlands Study (NOWES). Airborne CH4 flux measurements were taken over a large portion of the HBL. The surface level flux of CH4 was obtained from downward extrapolations of multiple-level CH4 flux measurements. Methane source strengths ranged from -1 to 31 mg m(exp -2)/d, with the higher values occurring in relatively small, isolated areas. Similar measurements of the CH4 source strength in the boreal forest region of Schefferville, Quebec, ranged from 6 to 27 mg m(exp -2)/d and exhibited a diurnal dependence. The CH4 source strengths found during the ABLE 3B expedition were much lower than the seasonally averaged source strength of 51 mg m(exp -2)/d found for the Yukon-Kuskokwim delta region of Alaska during the previous ABLE 3A study. Large positive CO fluxes (0.31 to 0.53 parts per billion by volume (ppbv) m/s) were observed over the inland, forested regions of the HBL study area, although the mechanism for the generation of these fluxes was not identified. Repetitive measurements along the same ground track at various times of day near the Schefferville site also suggested a diurnal dependence for CO emissions. Measurements of surface resistance to the uptake of O3 (1.91 to 0.80 s/cm) for the HBL areas investigated were comparable to those observed near the Schefferville site (3.40 to 1.10 s/cm). Surface resistance values for the ABLE 3B study area were somewhat less than those observed over the Yukon-Kuskokwim delta during the previous ABLE 3A study. The budgets for heat, moisture, O3, CO, and CH4 were evaluated. The residuals from these budget studies indicated, for the cases selected, a moderate net photochemical production of O3 present in the boundary layer over the HBL that coincided with an in situ destruction of CO, although the mechanism responsible for the destruction of CO was not identified. Results from the O3 budget analysis indicate the importance of in situ photochemical production and its possible dominance over surface deposition to the local O3 budget at the Schefferville site. Measurements of the in situ production of O3 indicated a direct relationship between the presence of biomass burning or large-scale pollution effects. Residuals from budget calculations for conserved quantities (heat, moisture, and CH4) were compared with their respective surface fluxes to provide a measure of the internal self-consistency of the flux measurements.

Ritter, John A.↗

NASA NTRS1976

Calculations of molecular ionization energies using a self-consistent-charge Hartree-Fock-Slater method

A numerical-variational method for performing self-consistent molecular calculations in the Hartree-Fock-Slater (HFS) model is presented. Molecular wavefunctions are expanded in terms of basis sets constructed from numerical HFS solutions of selected one-center atomlike problems. Binding energies and wavefunctions for the molecules are generated using a discrete variational method for a given molecular potential. In the self-consistent-charge (SCC) approximation to the complete self-consistent-field (SCF) method, results of a Mulliken population analysis of the molecular eigenfunctions are used in each iteration to produce 'atomic' occupation numbers. The simplest SCC potential is then obtained from overlapping spherical atomlike charge distributions. Molecular ionization energies are calculated using the transition-state procedure; results are given for CO, H2O, H2S, AlCl, InCl, and the Ni5O surface complex. Agreement between experimental and theoretical ionization energies for the free-molecule valence levels is generally within 1 eV. The simple SCC procedure gives a reasonably good approximation to the molecular potential, as shown by comparison with experiment, and with complete SCF calculations for CO, H2O, and H2S.

Rosen, A.↗

NASA NTRS1978

Structure of current sheets in magnetic holes at 1 AU

Current density profiles in several types of interplanetary magnetic holes were calculated assuming that the currents flow in planar sheets and that the magnetic field varies only in the direction normal to the sheet. The planarity was verified in four holes which were observed by two suitably spaced spacecraft. The structure of the current sheets ranges from very simple in some holes to very complex in others. The observed structures are found to be qualitatively consistent with models based on self-consistent solutions of Vlasov's equation and Maxwell's equations. Examples of complex, irregular magnetic holes are also presented, and they are shown to contain multiple, current sheets in which currents flow parallel to one another.

Fitzenreiter, R. J.↗

NASA NTRS1981

The self-consistent calculation of pseudo-molecule energy levels, construction of energy level correlation diagrams and an automated computation system for SCF-X(Alpha)-SW calculations

The self consistent calculation of the electronic energy levels of noble gas pseudomolecules formed when a metal surface is bombarded by noble gas ions is discussed along with the construction of energy level correlation diagrams as a function of interatomic spacing. The self consistent field x alpha scattered wave (SCF-Xalpha-SW) method is utilized. Preliminary results on the Ne-Mg system are given. An interactive x alpha programming system, implemented on the LeRC IBM 370 computer, is described in detail. This automated system makes use of special PROCDEFS (procedure definitions) to minimize the data to be entered manually at a remote terminal. Listings of the special PROCDEFS and of typical input data are given.

Schlosser, H.↗

NASA NTRS1986

Self-consistent modelling of the polar thermosphere and ionosphere to magnetospheric convection and precipitation (invited review)

It has recently been demonstrated that the dramatic effects of plasma precipitation and convection on the composition and dynamics of the polar thermosphere and ionosphere include a number of strong interactive, or feedback, processes. To aid the evaluation of these feedback processes, a joint three dimensional time dependent global model of the Earth's thermosphere and ionosphere was developed in a collaboration between University College London and Sheffield University. This model includes self consistent coupling between the thermosphere and the ionosphere in the polar regions. Some of the major features in the polar ionosphere, which the initial simulations indicate are due to the strong coupling of ions and neutrals in the presence of strong electric fields and energetic electron precipitation are reviewed. The model is also able to simulate seasonal and Universal time variations in the polar thermosphere and ionospheric regions which are due to the variations of solar photoionization in specific geomagnetic regions such as the cusp and polar cap.

Rees, D.↗

NASA NTRS1985

Alfvenic heating of the chromosphere and corona

The behavior in the chromosphere of Alfven waves propagating on closed magnetic field lines (e.g. coronal loops) is considered. It is found that the observed chromospheric nonthermal velocities are consistent with the predicted behavior of Alfven waves. If they are indeed Alfven waves, then the observed motions imply energy fluxes which are sufficient to heat the corona and chromosphere. It is further shown that the observed motions can reproduce the observed chromospheric heating, if the heating occurs via a Kolmogoroff cascade. But a definitive analysis will require a self-consistent treatment of nonlinear effects.

Hollweg, J. V.↗

NASA NTRS1990

Spiral structure of M51: Streaming motions across the spiral arms

The atomic hydrogen (HI) and the H alpha emission line in the grand-design spiral galaxy M51 have been observed with the Westerbork Synthesis Radio Telescope and the Taurus Fabry-Perot imaging spectrometer, respectively. Across the inner spiral arms significant tangential and radial velocity gradients are detected in the H alpha emission after subtraction of the axi-symmetric component of the velocity field. The shift is positive on the inside and negative on the outside of the northern arm. Across the southern arm this situation is reversed. The direction of the shifts is such that the material is moving inward and faster compared to circular rotation in both arms, consistent with the velocity perturbations predicted by spiral density wave models for gas downstream of a spiral shock. The observed shifts amount to 20 to 30 km (s-1), corresponding to streaming motions of 60 to 90 km (s-1) in the plane of the disk (inclination angle 20 degrees). Comparable velocity gradients have also been observed by Vogel et al. in the CO emission from the inner northern arm of M51. The streaming motions in M51 are about 2 to 3 times as large as the ones found in HI by Rots in M81, and successfully modelled by Visser with a self-consistent density wave model. Researchers have not been able to detect conclusively streaming motions in the HI emission from the arms, perhaps due to the relatively poor angular resolution (approx. 15 seconds) of the HI observations.

Tilanus, R. P. J.↗

NASA NTRS1991

Extension of convection modeling into the high-latitude ionosphere - Some theoretical difficulties

The Rice Convection Model (RCM) is extended and merged with empirical models so as to cover the entire high-latitude ionosphere with the aim of providing precipitation and electric field inputs for ionosphere and thermosphere modelers and producing a model in which the boundaries of the precipitation and electric field patterns maintain physically consistent relationships to each other. The computed auroral electron energy flux, plotted as a function of latitude, exhibited an exaggerated two-peak structure. When no floor was placed under the precipitation rate, the minimum between the two peaks was much too deep to be consistent with typical observations. The regions of excessively weak precipitation map to equatorial distances of 15-35 RE and thus to the regions of the plasma sheet that were not included in previous self-consistent convection calculations.

Wolf, R. A.↗

NASA NTRS1993

The velocity fields of elliptical galaxies: Steps toward a solution of the intrinsic shape problem

One of the few surviving signatures, at low redshift, of the process of galaxy formation should be the distribution of shapes of elliptical galaxies. Yet the problem of inferring this distribution from the observed ellipticals is still unsolved, because insufficient use has been made of kinematic information. The levels of 'sophistication' of the theoretical models and of the observations have up to now been poorly matched. The kinematic data available for most ellipticals consists of only major and minor axis spectra; and Franx et al. (1991) find, using simple geometric models, that the addition of only one kinematic parameter (the ratio of minor axis to major axis rotation velocity) to the photometry is just not enough to finely constrain the intrinsic shape distribution. On the other hand, the more elaborate self-consistent models (e.g., Levison and Richstone 1987, Statler 1987) have made only infrequent and model-dependent predictions of complicated velocity patterns, mostly at small radii, and have not discussed how they change with shape.

Statler, Thomas S.↗

NASA NTRS1995

A global 3-D MHD model of the solar wind with Alfven waves

A fully three-dimensional solar wind model that incorporates momentum and heat addition from Alfven waves is developed. The proposed model upgrades the previous one by considering self-consistently the total system consisting of Alfven waves propagating outward from the Sun and the mean polytropic solar wind flow. The simulation region extends from the coronal base (1 R(sub s) out to beyond 1 AU. The fully 3-D MHD equations written in spherical coordinates are solved in the frame of reference corotating with the Sun. At the inner boundary, the photospheric magnetic field observations are taken as boundary condition and wave energy influx is prescribed to be proportional to the magnetic field strength. The results of the model application for several time intervals are presented.

Usmanov, A. V.↗

NASA NTRS1996

Transient Creep of a Composite Lower Crust: Constitutive Theory - 1

A composite model is proposed to describe the time-dependent response of the Earth's lower crust. The motivation for such it model is twofold: First, new observations of widespread postseismic deformation indicate that the deep continental crust responds viscoelastically, having both long-and short-term decay times. Second, by any number of observationally based rationales, the lower crust is compositionally and structurally heterogeneous over many length scales. For heterogeneities that have much smaller characteristic lengths than the minimum deformation wavelength of interest, the aggregate rheology can be described by composite media theory. For wavelengths of the order of the thickness of the lower crust (approx. = 25-40 km) and larger, composite theory may be applied to heterogeneities that are smaller than about several hundred meters, or equivalent to the vertical extent of a thick lower crustal mylonitic shear zone. The composite media theory developed here is constructed using both Eshelhy-Mori-Tanaka theory for aligned generalized spheroidal inclusions and a generalized self-consistent method. The inclusions and matrix are considered to be Maxwellian viscoelastic: a rheology that is consistent with past homogeneous models of postseismic stress relaxation. The composite theory presented here introduces a transient response to a suddenly imposed stress field which does not appear in homogeneous Maxwell models. Analytic expressions for the amplitude and duration of the transient and for the effective long-and short-term viscosities of the composite are given which describe the sensitivity to inclusion concentration (phi), to shape, and to ratio of inclusion-to-matrix viscosity (R).

Ivins, Erik R.↗

NASA NTRS2001

A Hydrodynamic Theory for Spatially Inhomogeneous Semiconductor Lasers: Microscopic Approach

Starting from the microscopic semiconductor Bloch equations (SBEs) including the Boltzmann transport terms in the distribution function equations for electrons and holes, we derived a closed set of diffusion equations for carrier densities and temperatures with self-consistent coupling to Maxwell's equation and to an effective optical polarization equation. The coherent many-body effects are included within the screened Hartree-Fock approximation, while scatterings are treated within the second Born approximation including both the in- and out-scatterings. Microscopic expressions for electron-hole (e-h) and carrier-LO (c-LO) phonon scatterings are directly used to derive the momentum and energy relaxation rates. These rates expressed as functions of temperatures and densities lead to microscopic expressions for self- and mutual-diffusion coefficients in the coupled density-temperature diffusion equations. Approximations for reducing the general two-component description of the electron-hole plasma (EHP) to a single-component one are discussed. In particular, we show that a special single-component reduction is possible when e-h scattering dominates over c-LO phonon scattering. The ambipolar diffusion approximation is also discussed and we show that the ambipolar diffusion coefficients are independent of e-h scattering, even though the diffusion coefficients of individual components depend sensitively on the e-h scattering rates. Our discussions lead to new perspectives into the roles played in the single-component reduction by the electron-hole correlation in momentum space induced by scatterings and the electron-hole correlation in real space via internal static electrical field. Finally, the theory is completed by coupling the diffusion equations to the lattice temperature equation and to the effective optical polarization which in turn couples to the laser field.

Li, Jianzhong↗