An efficient flux difference splitting algorithm for unsteady duct flows of a real gas

A numerical scheme is presented for the solution of the Euler equations of compressible flow of a real gas in a single spatial coordinate. This includes flow in a duct of variable cross-section, as well as flow with slab, cylindrical or spherical symmetry, as well as the case of an ideal gas, and can be useful when testing codes for the two-dimensional equations governing compressible flow of a real gas. The resulting scheme requires an average of the flow variables across the interface between cells, and this average is chosen to be the arithmetic mean for computational efficiency, which is in contrast to the usual “square root” averages found in this type of scheme. The scheme is applied with success to five problems with either slab or cylindrical symmetry and for a number of equations of state. The results compare favourably with the results from other schemes.

An efficient numerical method for compressible flows of a real gas using arithmetic averaging

An efficient numerical method is presented for the solution of the Euler equations governing the compressible flow of a real gas. The scheme is based on the approximate solution of a specially constructed set of linearised Riemann problems. An average of the flow variables across the interface between cells is required, and this is chosen to be the arithmetic mean for computational efficiency, which is in contrast to the usual square root averaging. The scheme is applied to a test problem for five different equations of state.

A Riemann solver for compressible flow of a real gas in a Lagrangian frame

An approximate Riemann solver is presented for the compressible flow equations with a general (convex) equation of state in a Lagrangian frame of reference.

Real gas flows in a duct

A numerical scheme is presented for the solution of the Euler equations of compressible flow of a real gas in a single spatial coordinate. This include flow in a duct of variable cross-section as well as flow with cylindrical or spherical symmetry, and can prove useful when testing codes for the two-dimensional equations governing compressible flow of a real gas. The scheme is applied with success to a problem involving the interaction of converging and diverging cylindrical shocks for four equations of state and to a problem involving the reflection of a converging shock.

The full-spectrum correlated-k method for longwave atmospheric radiative transfer using an effective Planck function

The correlated k-distribution (CKD) method is widely used in the radiative transfer schemes of atmospheric models and involves dividing the spectrum into a number of bands and then reordering the gaseous absorption coefficients within each one. The fluxes and heating rates for each band may then be computed by discretizing the reordered spectrum into of order 10 quadrature points per major gas and performing a monochromatic radiation calculation for each point. In this presentation it is shown that for clear-sky longwave calculations, sufficient accuracy for most applications can be achieved without the need for bands: reordering may be performed on the entire longwave spectrum. The resulting full-spectrum correlated k (FSCK) method requires significantly fewer monochromatic calculations than standard CKD to achieve a given accuracy. The concept is first demonstrated by comparing with line-by-line calculations for an atmosphere containing only water vapor, in which it is shown that the accuracy of heating-rate calculations improves approximately in proportion to the square of the number of quadrature points. For more than around 20 points, the root-mean-squared error flattens out at around 0.015 K/day due to the imperfect rank correlation of absorption spectra at different pressures in the profile. The spectral overlap of m different gases is treated by considering an m-dimensional hypercube where each axis corresponds to the reordered spectrum of one of the gases. This hypercube is then divided up into a number of volumes, each approximated by a single quadrature point, such that the total number of quadrature points is slightly fewer than the sum of the number that would be required to treat each of the gases separately. The gaseous absorptions for each quadrature point are optimized such that they minimize a cost function expressing the deviation of the heating rates and fluxes calculated by the FSCK method from line-by-line calculations for a number of training profiles. This approach is validated for atmospheres containing water vapor, carbon dioxide, and ozone, in which it is found that in the troposphere and most of the stratosphere, heating-rate errors of less than 0.2 K/day can be achieved using a total of 23 quadrature points, decreasing to less than 0.1 K/day for 32 quadrature points. It would be relatively straightforward to extend the method to include other gases.

The full-spectrum correlated k method for longwave atmospheric radiative transfer: treatment of gaseous overlap

The correlated k-distribution (CKD) method is widely used in the radiative transfer schemes of atmospheric models, and involves dividing the spectrum into a number of bands and then reordering the gaseous absorption coefficients within each one. The fluxes and heating rates for each band may then be computed by discretizing the reordered spectrum into of order 10 quadrature points per major gas, and performing a pseudo-monochromatic radiation calculation for each point. In this paper it is first argued that for clear-sky longwave calculations, sufficient accuracy for most applications can be achieved without the need for bands: reordering may be performed on the entire longwave spectrum. The resulting full-spectrum correlated k (FSCK) method requires significantly fewer pseudo-monochromatic calculations than standard CKD to achieve a given accuracy. The concept is first demonstrated by comparing with line-by-line calculations for an atmosphere containing only water vapor, in which it is shown that the accuracy of heating-rate calculations improves approximately in proportion to the square of the number of quadrature points. For more than around 20 points, the root-mean-squared error flattens out at around 0.015 K d−1 due to the imperfect rank correlation of absorption spectra at different pressures in the profile. The spectral overlap of m different gases is treated by considering an m-dimensional hypercube where each axis corresponds to the reordered spectrum of one of the gases. This hypercube is then divided up into a number of volumes, each approximated by a single quadrature point, such that the total number of quadrature points is slightly fewer than the sum of the number that would be required to treat each of the gases separately. The gaseous absorptions for each quadrature point are optimized such they minimize a cost function expressing the deviation of the heating rates and fluxes calculated by the FSCK method from line-by-line calculations for a number of training profiles. This approach is validated for atmospheres containing water vapor, carbon dioxide and ozone, in which it is found that in the troposphere and most of the stratosphere, heating-rate errors of less than 0.2 K d−1 can be achieved using a total of 23 quadrature points, decreasing to less than 0.1 K d−1 for 32 quadrature points. It would be relatively straightforward to extend the method to include other gases.

Riemann solvers with primitive parameter vectors for two-dimensional compressible flows of a real gas

An analysis of averaging procedures is presented for an approximate Riemann solver for the equations governing the compressible flow of a real gas. This study extends earlier work for the Euler equations with ideal gases.

Precipitation and latent heating distributions from satellite passive microwave radiometry. Part I: improved method and uncertainties

A revised Bayesian algorithm for estimating surface rain rate, convective rain proportion, and latent heating profiles from satellite-borne passive microwave radiometer observations over ocean backgrounds is described. The algorithm searches a large database of cloud-radiative model simulations to find cloud profiles that are radiatively consistent with a given set of microwave radiance measurements. The properties of these radiatively consistent profiles are then composited to obtain best estimates of the observed properties. The revised algorithm is supported by an expanded and more physically consistent database of cloud-radiative model simulations. The algorithm also features a better quantification of the convective and nonconvective contributions to total rainfall, a new geographic database, and an improved representation of background radiances in rain-free regions. Bias and random error estimates are derived from applications of the algorithm to synthetic radiance data, based upon a subset of cloud-resolving model simulations, and from the Bayesian formulation itself. Synthetic rain-rate and latent heating estimates exhibit a trend of high (low) bias for low (high) retrieved values. The Bayesian estimates of random error are propagated to represent errors at coarser time and space resolutions, based upon applications of the algorithm to TRMM Microwave Imager (TMI) data. Errors in TMI instantaneous rain-rate estimates at 0.5°-resolution range from approximately 50% at 1 mm h−1 to 20% at 14 mm h−1. Errors in collocated spaceborne radar rain-rate estimates are roughly 50%–80% of the TMI errors at this resolution. The estimated algorithm random error in TMI rain rates at monthly, 2.5° resolution is relatively small (less than 6% at 5 mm day−1) in comparison with the random error resulting from infrequent satellite temporal sampling (8%–35% at the same rain rate). Percentage errors resulting from sampling decrease with increasing rain rate, and sampling errors in latent heating rates follow the same trend. Averaging over 3 months reduces sampling errors in rain rates to 6%–15% at 5 mm day−1, with proportionate reductions in latent heating sampling errors.

Gas-phase reactions of NO3 and N2O5 with (Z)-hex-4-en-1-ol, (Z)-hex-3-en-1-ol ('leaf alcohol'), (E)-hex-3-en-1-ol, (Z)-hex-2-en-1-ol and (E)-hex-2-en-1-ol

The night-time atmospheric chemistry of the biogenic volatile organic compounds (Z)-hex-4-en-1-ol, (Z)-hex-3-en-1-ol ('leaf alcohol'), (E)-hex-3-en-1-ol, (Z)-hex-2-en-1-ol and (E)-hex-2-en-1-ol, has been studied at room temperature. Rate coefficients for reactions of the nitrate radical (NO3) with these stress-induced plant emissions were measured using the discharge-flow technique. We employed off-axis continuous-wave cavity-enhanced absorption spectroscopy (CEAS) for the detection of NO3, which enabled us to work in excess of the hexenol compounds over NO3. The rate coefficients determined were (2.93 +/- 0.58) x 10(-13) cm(3) molecule(-1) s(-1), (2.67 +/- 0.42) x 10(-13) cm(3) molecule(-1) s(-1), (4.43 +/- 0.91) x 10(-13) cm(3) molecule(-1) s(-1), (1.56 +/- 0.24) x 10(-13) cm(3) molecule(-1) s(-1), and (1.30 +/- 0.24) x 10(-13) cm(3) molecule(-1) s(-1) for (Z)-hex-4-en-1-ol, (Z)-hex-3en-1-ol, (E)-hex-3-en-1-ol, (Z)-hex-2-en-1-ol and (E)-hex-2-en-1-ol. The rate coefficient for the reaction of NO3 with (Z)-hex-3-en-1-ol agrees with the single published determination of the rate coefficient using a relative method. The other rate coefficients have not been measured before and are compared to estimated values. Relative-rate studies were also performed, but required modification of the standard technique because N2O5 (used as the source of NO3) itself reacts with the hexenols. We used varying excesses of NO2 to determine simultaneously rate coefficients for reactions of NO3 and N2O5 with (E)-hex-3-en-1-ol of (5.2 +/- 1.8) x 10(-13) cm(3) molecule(-1) s(-1) and (3.1 +/- 2.3) x 10(-18) cm(3) molecule(-1) s(-1). Our new determinations suggest atmospheric lifetimes with respect to NO3-initiated oxidation of roughly 1-4 h for the hexenols, comparable with lifetimes estimated for the atmospheric degradation by OH and shorter lifetimes than for attack by O-3. Recent measurements of [N2O5] suggest that the gas-phase reactions of N2O5 with unsaturated alcohols will not be of importance under usual atmospheric conditions, but they certainly can be in laboratory systems when determining rate coefficients.

Structure–activity relations (SARs) for gas-phase reactions of NO3, OH and O3 with alkenes: an update

We present updated structure-activity relations (SARs) for the prediction of rate coefficients for gas-phase reactions with alkenes of the major atmospheric oxidants NO3, OH and O-3. Such SARs provide one way of incorporating essential information about reactivity into atmospheric models. Rate coefficients obtained from correlations relating the logarithms of the rate coefficients to the energies of the highest occupied molecular orbitals (HOMOs) of the alkenes were used to refine the SARs. SARs have an advantage for the user over the direct application of the correlations in that knowledge of the structure of the alkene of interest is sufficient to estimate rate coefficients, and no quantum-mechanical calculations need to be performed. A comparison of the values predicted by the SARs with experimental data where they exist allowed us to assess the reliability of our method.

Correlations for gas-phase reactions of NO3, OH and O3 with alkenes: an update

Methods are developed for predicting rate coefficients for reactions of initiators of tropospheric oxidation with unsaturated compounds that are abundant in the atmosphere; prognostic tools of this kind are essential for atmospheric chemists and modellers. To pursue the aim of exploring such tools, the kinetics of reactions of NO3, OH and O-3 with a series of alkenes are examined for correlations relating the logarithms of the rate coefficients to the energies of the highest occupied molecular orbitals (HOMOs) of the alkenes. A comparison of the values predicted by the correlations with experimental data (where the latter exist) allowed us to assess the reliability of our method. We used a series of theoretical methods to calculate the HOMO energies, and found that higher computational effort improves the agreement of the predicted rate coefficients with experimental values, especially for reactions of NO3 with alkenes that possess vinyllic halogen substituents. As a consequence, it is expedient to suggest new correlations to replace those presented by us and others that were based on the lower level of theory. We propose the following correlations for the reactions of NO3, OH and O-3 with alkenes: ln(k(NO3)/cm(3) molecule(-1) s(-1)) = 6.40(E-HOMO/eV) + 31.69, ln(k(OH)/cm(3) molecule(-1) s(-1)) = 1.21 (E-HOMO/eV)-12.34 and ln(k(O3)/cm(3) molecule(-1) s(-1)) = 3.28(E-HOMO/eV)-6.78. These new correlations have been developed using the larger experimental data sets now available, and the impact of the extended data on the quality of the correlations is examined in the paper. Atmospheric lifetimes have been calculated from both experimental and estimated rate coefficients to provide an overview of removal efficiencies for different classes of alkenes with respect to oxidative processes initiated by NO3, OH and O-3. A figure is presented to show the spatial scales over which alkenes may survive transport in competition with attack by NO3, OH and O-3. Removal by NO3 or OH is always more important than removal by O-3, and reactions with NO3 dominate for scales up to a few hundred metres.

Gas-phase rate coefficients for the reactions of O(3P), S(3P), Se(3P), and Te(3P) with alkenes: application of perturbation frontier molecular orbital theory, correlations, and structure-activity relations (SARs)

The kinetics of the reactions of the atoms O(P-3), S(P-3), Se(P-3), and Te((3)p) with a series of alkenes are examined for correlations relating the logarithms of the rate coefficients to the energies of the highest occupied molecular orbitals (HOMOs) of the alkenes. These correlations may be employed to predict rate coefficients from the calculated HOMO energy of any other alkene of interest. The rate coefficients obtained from the correlations were used to formulate structure-activity relations (SARs) for reactions of O((3)p), S(P-3), Se (P-3), and Te((3)p) with alkenes. A comparison of the values predicted by both the correlations and the SARs with experimental data where they exist allowed us to assess the reliability of our method. We demonstrate the applicability of perturbation frontier molecular orbital theory to gas-phase reactions of these atoms with alkenes. The correlations are apparently not applicable to reactions of C(P-3), Si(P-3), N(S-4), and Al(P-2) atoms with alkenes, a conclusion that could be explained in terms of a different mechanism for reaction of these atoms.

Gas-phase rate coefficients for the reactions of nitrate radicals with (Z)-pent-2-ene, (E)-pent-2-ene, (Z)-hex-2-ene, (E)-hex-2-ene, (Z)-hex-3-ene, (E)-hex-3-ene and (E)-3-methylpent-2-ene at room temperature

Rate coefficients for reactions of nitrate radicals (NO3) with (Z)-pent-2-ene, (E)-pent-2-ene, (Z)-hex-2-ene, (E)-hex-2-ene, (Z)-hex-3-ene, (E)-hex-3-ene and (E)-3-methylpent-2-ene were determined to be (6.55 +/- 0.78) x 10(-13) cm(3) molecule(-1) s(-1), (3.78 +/- 0.45) x 10(-13) cm(3) molecule(-1) s(-1), (5.30 +/- 0.73) x 10(-13) cm(3) molecule(-1) s(-1), (3.83 +/- 0.47) x 10(-13) cm(3) molecule(-1) s(-1), (4.37 +/- 0.49) x 10(-13) cm(3) molecule(-1) s(-1), (3.61 +/- 0.40) x 10(-13) cm(3) molecule(-1) s(-1) and (8.9 +/- 1.5) x 10(-12) cm(3) molecule(-1) s(-1), respectively. We performed kinetic experiments at room temperature and atmospheric pressure using a relative-rate technique with GC-FID analysis. The experimental results demonstrate a surprisingly large cis-trans (Z-E) effect, particularly in the case of the pent-2-enes, where the ratio of rate coefficients is ca. 1.7. Rate coefficients are discussed in terms of electronic and steric influences, and our results give some insight into the effects of chain length and position of the double bond on the reaction of NO3 with unsaturated hydrocarbons. Atmospheric lifetimes were calculated with respect to important oxidants in the troposphere for the alkenes studied, and NO3-initiated oxidation is found to be the dominant degradation route for (Z)-pent-2-ene, (Z)-hex-3-ene and (E)-3-methylpent-2-ene.

Time-resolved gas-phase kinetic study of the germylene addition reaction, GeH2 + C2D2

Time-resolved studies of germylene, GeH2, generated by laser. ash photolysis of 3,4-dimethyl-1-germacyclopent-3-ene, have been carried out to obtain rate coefficients for its bimolecular reaction with C2D2. The reaction was studied in the gas phase, mainly at a total pressure of 1.3 kPa (in SF6 bath gas) at five temperatures in the range 298-558 K. Pressure variation measurements over the range 0.13-13 kPa ( SF6) at 298, 397 and 558 K revealed a small pressure dependence but only at 558 K. After correction for this, the second-order rate coefficients gave the Arrhenius equation: log(k(infinity)/cm(3) molecule(-1) s(-1)) = (-10.96 +/- 0.05) + ( 6.16 +/- 0.37 kJ mol(-1))/RT ln 10 Comparison with the reaction of GeH2 + C2H2 (studied earlier) showed a similar behaviour with almost identical rate coefficients. The lack of a significant isotope effect is consistent with a rate-determining addition process and is explained by irreversible decomposition of the reaction intermediate to give Ge(P-3) + C2H4. This result contrasts with that for GeH2 + C2H4/C2D4 and those for the analogous silylene reactions. The underlying reasons for this are discussed.