Low-wavenumber Modes Research Articles

Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties

Numerical simulations of forced isotropic turbulence are most often formulated in Fourier space, where forcing is applied to low-wavenumber modes. For applications in physical space, low-wavenumber forcing is difficult to implement. The linear forcing recently proposed by Lundgren [“Linearly forced isotropic turbulence,” in Annual Research Briefs (Center for Turbulence Research, Stanford, 2003), pp. 461–473], where a force proportional to velocity is applied, is an attractive alternative but not much is known about its properties. Using numerical experimentation, various properties of the linear forcing are explored: (i) it is shown that when implemented in physical space, linear forcing gives the same results as in spectral implementations; (ii) it is shown that the linearly forced system converges to a stationary state that depends on domain size and Reynolds number, but not on the spectral shape of the initial condition; (iii) it is also shown that the extent of Kolmogorov −5∕3 range is similar to that achieved using the standard band-limited forcing schemes but the integral length scale l=urms3∕ε is smaller, thus reducing the effective scaling range for a given resolution. It is concluded that linear forcing is a useful alternative method that does not require transformation to Fourier space and is easily integrated into physical-space numerical codes.

Low‐wavenumber dynamics of L‐alanine

AbstractL‐Alanine is one of the simplest amino acids, the building block for many polypeptides and proteins. A previous Raman scattering study showed an anomalous temperature dependence of the intensity of the two lowest Raman modes, attributed to dynamic localization of vibrational energy. The present Raman data, collected in the b(cc)b scattering geometry, were aimed at understanding the dynamics of the low‐wavenumber modes in connection with the reported anomalies in other physical properties. Our study shows that the energy and intensity of the observed external modes show unusual behaviour in the temperature range ∼150–250 K. The anomalous temperature dependence of the peak of the NH3+ torsional mode is confirmed. Its wavenumber has a singularity around ∼220 K and its full width at half‐maximum does not increase linearly with temperature. The CH3 torsion and CO2− rock undergo similar discontinuities below ∼250 K. The intensity of the two lowest wavenumbers librons at 42 and 48 cm−1 shows an anomalous temperature dependence and vary in an opposite way from that reported for the c(bb)a scattering geometry. The nature of the subtle symmetry breaking occuring below ∼250 K is discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Stability of pressure-driven creeping flows in channels lined with a nonlinear elastic solid

The effect of pressure gradients on the stability of creeping flows of Newtonian fluids in channels lined with an incompressible and impermeable neo-Hookean material is examined in this work. Three different configurations are considered: (i) pressure-driven flow between a rigid wall and a wall lined with a neo-Hookean material; (ii) pressure-driven flow between neo-Hookean-lined walls; and (iii) combined Couette–Poiseuille flow between a rigid wall and a neo-Hookean-lined wall. In each case, a first normal stress difference whose magnitude depends on depth arises in the base state for the solid, and linear stability analysis reveals that this leads to a short-wave instability which is removed by the presence of interfacial tension. For sufficiently thick solids, low-wavenumber modes become unstable first as the applied strain increases above a critical value, whereas for sufficiently thin solids, high-wavenumber modes becomes unstable first. Comparison of the dimensionless critical strains shows that configurations (i) and (ii) are more difficult to destabilize than Couette flow past a neo-Hookean solid. For configuration (iii), the nonlinear elasticity of the solid leads to two physically distinct critical conditions, in contrast to what happens when a linear elastic material is used. The mechanisms underlying the behaviour of the critical strains are explained through an analysis of the interfacial boundary conditions.

Micro‐Raman study of the fatigue fracture and tensile behaviour of polyamide (PA 66) fibres

AbstractThe micro/nano structural evolution of two different types of ultra‐high‐performance polyamide (PA) 66 fibres, before and after tensile fatigue loading and failure, was studied using Raman spectroscopy. The nanostructure of these PA fibres consists of amorphous and crystalline domains. Two probes were considered: (i) low‐wavenumber collective modes at ∼100 and ∼70 cm−1 as representative of the crystalline and amorphous domains, respectively; and (ii) NH stretching mode at ∼3300 cm−1, as representative of the inter‐chain behaviour. The analysis of the wavenumber distribution revealed that the skin/core heterogeneity is a maximum for the amorphous domains. The residual stress in the skin was evaluated at about 200 and 400 MPa from calibrations plots for crystalline and amorphous domains, for the two types of fibres. This stress disappears after thermal treatment around Tg. In situ analysis at different strain levels shows that the amorphous regions accommodate the stress up to ∼350 MPa, from which point elastic behaviour is observed. The analysis of a series of fibres failed in fatigue showed that amorphous domains are highly stressed during the failure and remnant compression can be measured. The extent of the stressed region extends to typically 100–400 µm from the fracture. Examination of the NH wavenumber shift shows that the fracture squeezes the polymer chains together. Fatigue failure is explained by the loss of compliance in the amorphous region. Copyright © 2004 John Wiley & Sons, Ltd.

A Raman spectroscopic study of Fe?Mg olivines

End-member synthetic fayalite and forsterite and a natural solid-solution crystal of composition (Mg1.80,Fe0.20)SiO4 were investigated using Raman spectroscopy. Polarized single-crystal spectra were measured as a function of temperature. In addition, polycrystalline forsterite and fayalite, isotopically enriched in 26Mg and 57Fe, respectively, were synthesized and their powder spectra measured. The high-wavenumber modes in olivine consist of internal SiO4 vibrations that show little variation upon isotopic substitution. This confirms conclusions from previous spectroscopic studies that showed that the internal SiO4 vibrations have minimal coupling with the lower-wavenumber lattice modes. The lowest wavenumber modes in both forsterite and fayalite shift in energy following isotopic substitution, but with energies less than that which would be associated with pure Mg and Fe translations. The low-wavenumber Raman modes in olivine are best described as lattice modes consisting to a large degree of mixed vibrations of M(2) cation translations and external vibrations of the SiO4 tetrahedra. The single-crystal spectra of forsterite and Fo90Fa10 were recorded at a number of temperatures from room temperature to about 1200 °C. From these data the microscopic Gruneisen parameters for three different A g modes for both compositions were calculated, and also the structural state of the solid solution crystal was investigated. Small discontinuities observed in the wavenumber behavior of a low-energy mixed Mg/T(SiO4) mode between 700 and 1000 °C may be related to minor variations in the Fe–Mg intracrystalline partitioning state in the Fo90Fa10 crystal, but further spectroscopic work is needed to clarify and quantify this issue. The mode wavenumber and intensity behavior of internal SiO4 vibrations as a function of temperature are discussed in terms of crystal field and dynamic splitting and also ν1 and ν3 coupling. Crystal-field splitting increases only very slightly with temperature, whereas dynamical-field splitting is temperature dependent. The degree of ν1–ν3 coupling decreases with increasing temperature.

(Nano)structure, skin/core and tension behaviour of polyamide fibres

AbstractThe micro/nanostructural evolution of ultrahigh‐performance and textile (containing a dispersion of TiO2 powder) polyamide 66 fibres during tensile loading and failure initiation was studied using Raman spectroscopy low‐wavenumber collective modes (∼100 cm−1) as a mechanical probe representative of the crystalline domains. The analysis of the wavenumber distribution revealed the skin/core heterogeneity. A compressive residual stress in the skin was evaluated at about 200 MPa from calibration plots. The homogeneity of the TiO2 grain dispersion was mapped on fibre sections and scanned along their lengths. In situ analysis at different strain levels shows that the amorphous region accommodates the stress up to ca 350 MPa, from which point elastic behaviour is observed. Particular attention was devoted to analysing crack formation. Copyright © 2004 John Wiley & Sons, Ltd.

Effects of rotation on turbulent mixing: Nonpremixed passive scalars

We study by direct numerical simulations the effects of uniform solid-body rotation on passive scalar mixing in turbulent flow, with a focus on the unsteady problem of nonpremixed scalars in forced rotating turbulence with isotropic initial conditions in the velocity field. The expectation of reduced mixing as a result of reduced spectral transfer is readily verified in several aspects, including slower decay rates for the scalar variance, increased scalar mixing times, and slower relaxation of the probability density function from its initially bimodal form to a near-Gaussian shape. Spectral transfer in the scalar field is shown to be dominated by very low-wavenumber velocity modes, and strongly suppressed at higher wavenumbers in the scalar field. Considerable departure from local isotropy is observed in the scalar gradient fluctuations, which are smaller in the direction along the axis of rotation where there is less mixing than in the orthogonal plane. A partial explanation is given in terms of the influence of a modified turbulence velocity structure on directional characteristics of spectral transfer, which leads to anisotropy in the scalar gradient spectra as well as one-dimensional spectra of the scalar field. The observed anisotropy is stronger than that for the velocity field, especially for high rotation rates, and is more pronounced at Schmidt number 1 than at 1/8. A reduction in intermittency compared with nonrotating turbulence is also observed.

Low‐temperature Raman spectra of Sr0.66Ba0.34Nb2O6 single‐crystal fibers

AbstractA possible structural change in Sr0.66Ba0.34Nb2O6 single‐crystal fibers when the temperature decreases from 295 to 10 K was investigated by dielectric constant measurements and Raman spectroscopy. An anomaly observed in the dielectric constant and the loss factor is associated with the variation in the intensity and wavenumber of certain Raman modes. We observe that the relative intensity of low wavenumber modes has a singular behavior for the y(xz)ȳ scattering geometry. A high‐wavenumber band disappears when the temperature is lowered to 10 K for the y(xz)ȳ geometry and shifts to higher wavenumber for the y(zz)ȳ geometry. These observations can be interpreted as evidence of a structural phase transition undergone by Sr0.66Ba0.34Nb2O6 single‐crystal fibers at temperatures below 100 K. Copyright © 2003 John Wiley & Sons, Ltd.

Cordierite IV: structural heterogeneity and energetics of Mg?Fe solid solutions

The local structural heterogeneity and energetic properties of 22 natural Mg-Fe cordierites, ideal formula (Mg,Fe)(2)Al4Si5O18.x(H2O,CO2), were investigated at length scales given by powder infrared spectroscopy (IR) and also by published electronic absorption spectra. The studied samples have iron mole fractions from X-Fe = 0.06 to 0.82 and cover most of the Mg-Fe cordierite binary. Variations in wavenumbers and line widths of the IR bands were determined as a function of composition. Most modes shift linearly to lower wavenumbers with increasing X-Fe, except those at high wavenumbers located between 900 and 1,200 cm(-1). They are vibrations that have a large internal (Si,Al)O-4 character and are not greatly affected by Mg-Fe exchange on the octahedral site. The lower wavenumber modes can be best characterized as lattice vibrations having mixed character. The systematics of the wavenumber shifts suggest small continuous variations in the 'average' cordierite structure with Mg-Fe exchange and are consistent with an ideal volume of mixing, DeltaV(mix)= 0, behavior (Boberski and Schreyer 1990). IR line broadening was measured using the autocorrelation function for three wavenumber regions in order to determine the range of structural heterogeneity between roughly 2 and 100 Angstrom (0.2-10.0 nm) in the solid solution. In order to do this, an empirical correction was first made to account for the effect that small amounts of channel Na have on the phonon systematics. The results show that between 1,200 and 540 cm(-1) the line widths of the IR bands broaden slightly and linearly with increasing X-Fe. Between 350 and 125 cm(-1) nonlinear behavior was observed and it may be related to dynamic effects. These results suggest minimal excess elastic enthalpies of mixing for Mg-Fe cordierite solid solutions. Channel Na should affect measurably the thermodynamic properties of natural cordierites as evidenced by variations in the IR spectra of Na-containing samples. Occluded H2O (Class I) and CO2 should have little interaction with the framework and can be considered nearly 'free' molecules. They should not give rise to measurable structural heterogeneity in the framework. The contribution of the crystal field stabilization energy (CFSE) of octahedral Fe2+ to the energetics of Mg-Fe cordierites was also investigated using published electronic absorption spectra (Khomenko et al. 2001). Two bands are observed between 8,000 and 10,500 cm(-1) and they represent electronic dd-excitations of octahedral Fe2+ derived from the T-5(2g) --> E-5(g) transition. They shift to higher wavenumbers with increasing X-Mg in cordierite. An analysis gives slightly asymmetric excess -DeltaCFSE across the Mg-Fe cordierite join with a maximum of about -550 J/mole towards iron-rich compositions.

Vibrational Analysis of trans-Stilbene in the Ground and Excited Singlet Electronic States Revisited

The Raman and infrared bands of trans-stilbene in the ground electronic state (S0) and an excited singlet state (S1) are assigned on the basis of density functional calculations at the 6-311+G** level for the S0 state and configuration interaction single calculations with the 6-311+G** basis set for the S1 state. Not only the wavenumbers of normal modes but also Raman activities, Raman depolarization ratios, and infrared intensities are calculated and used for band assignments. The vibrational patterns of some characteristic modes are discussed. It is found that, with respect to a few low-wavenumber modes in both the S0 and S1 states, the results of the present calculations are inconsistent with those derived from an analysis of the fluorescence excitation spectra and dispersed fluorescence spectra of trans-stilbene in a supersonic jet.

Structural Investigations on Octaethylporphyrin Using Density Functional Theory and Polarization-Sensitive Resonance Coherent Anti-Stokes Raman Scattering Spectroscopy

Probing the structure and dynamics of porphyrins by vibrational spectroscopy is of major interest because this cyclic tetrapyrrole system constitutes the chromophore in different very complex biological systems carrying out essential processes in nature such as oxygen transport and storage (hemoglobin and myoglobin, respectively), electron transfer (cytochrome c), and energy conversion (chlorophyll). Most of the biological porphyrins are β-substituted, i.e., substituted at the outer pyrrole carbon atoms. Investigations on the structure of octaethylporphyrin (OEP) using density functional theory (DFT) as well as linear and nonlinear Raman spectroscopic techniques have been performed. The optimized geometry of OEP reveals a centrosymmetric molecule with local D2h symmetry of the porpyhrin macrocycle showing an excellent agreement with the X-ray structure of OEP. The DFT-derived harmonic vibrational wavenumbers together with the corresponding eigenvectors of several low-wavenumber modes and prominent ag(D2h) and b1g(D2h) modes of OEP are presented. Resonance Raman (RR) and multiplex polarization-sensitive resonance coherent anti-Stokes Raman scattering (PS RCARS) spectroscopy have been applied to OEP in dichloromethane to obtain complementary vibrational spectroscopic information. The RR spectrum obtained by excitation within the B or Soret bands reveals mainly ag modes, whereas the corresponding Raman measurements in resonance with the Q bands have been foiled by excessive fluorescence. In the PS RCARS spectra acquired with Q-band excitation, only b1g modes are detected. The different enhancement pattern of this β-substituted free-base porphyrin in comparison with metalloporphyrins (MP) can reasonably be explained in terms of symmetry lowering (D4h → D2h) supporting the DFT-derived structure.

Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.

Maintenance of Tropical Intraseasonal Variability: Impact of Evaporation–Wind Feedback and Midlatitude Storms

An intraseasonal tropical oscillation with a period of 20–80 days is simulated in the Neelin–Zeng Quasi-Equilibrium Tropical Circulation Model. This model is an intermediate-level atmospheric model that includes primitive equation nonlinearity, radiative– convective feedbacks, a simple land model with soil moisture, and a Betts–Miller-type moist convective adjustment parameterization. Vertical temperature and moisture structures in the model are based on quasi-equilibrium profiles taken from deep convective regions. The tropical intraseasonal variability is reasonably broadband. The eastward propagating 20–80-day variability is dominated by zonal wavenumber 1, shows features similar to an irregular Madden–Julian oscillation (MJO), and exhibits amplitude and phase speeds that vary both seasonally and between events. At higher wavenumbers, the model has a distinction between the low-frequency MJO-like band and the moist Kelvin wave band, similar to that found in observations. In the model, it is conjectured that this arises by interaction of the wavenumber-1 moist Kelvin wave with the zonally asymmetric basic state. Experiments using climatological sea surface temperature forcing are conducted using this model to examine the effects of evaporation–wind feedback and extratropical excitation on the maintenance of intraseasonal variability, with particular attention paid to the low wavenumber mode in the 20–80-day band. These experiments indicate that evaporation–wind feedback partially organizes this intraseasonal variability by reducing damping, but it is not by itself sufficient to sustain this oscillation for the most realistic parameters. Excitation by extratropical variability is a major source of energy for the intraseasonal variability in this model. When midlatitude storms are suppressed, tropical intraseasonal variability is nearly eliminated. However, the eastward propagating intraseasonal signal appears most clearly when midlatitude excitation is aided by the evaporation–wind feedback.

High-pressure Raman study and pressure-induced phase transitions of sodium niobate NaNbO3

Raman spectroscopic studies have been carried out on antiferroelectric sodium niobate NaNbO3 under pressures up to 22 GPa. The Raman spectra show a rich variety of changes with pressure and evidence for several phase transitions were observed. A first-order phase transition was found at 7.0 GPa. A low wavenumber mode at 42 cm−1 appears at 6.0 GPa as a precursor, which softens at higher pressure. This band is obscured above 7.0 GPa after the first order phase transition. Another possible phase transition is proposed at 12 GPa, indicated by the sign change of the gradient of wavenumber-pressure plots for several modes and the disappearance of the Raman band for the strong ν1 mode. The spectral changes above 6.0 GPa closely resemble those of KNbO3, and the results are interpreted in comparison with LiNbO3 and KNbO3. The high-pressure phase after the first transition at 7.0 GPa may be ferroelectric and the phase above 12 GPa is probably the paraelectric phase. Copyright © 2000 John Wiley & Sons, Ltd.

Application of femtosecond coherence spectroscopy to the observation of nuclear motions in heme proteins and transparent solutions

The technique of femtosecond coherence spectroscopy (FCS) was applied to simple transparent organic fluids (chloroform and fenchone) and light-absorbing heme proteins (myoglobin derivatives). This study demonstrates that time-domain coherence measurements are in good agreement with standard frequency-domain Raman and resonance Raman scattering experiments. The unique capability of FCS to detect the low-wavenumber nuclear motions driven by the pumping fields and by the electronic state changes associated with photoinduced chemical reactions is also demonstrated. The latter response cannot be probed using traditional Raman spectroscopy. In all studies of myoglobin derivatives a low-wavenumber mode was observed near 40 cm−1. A sequence of low-wavenumber modes was also observed near 80, 120 and 160 cm−1 that could be an indication of an overtone sequence. Copyright © 2000 John Wiley & Sons, Ltd.

Vibrational spectra of transition metal phosphosilicides MSi4P4 (M = Fe, Ru, Os)

MSi4P4 compounds (M = Fe, Ru, Os) were studied at 300 K by Raman and IR spectroscopy. The low symmetry and the absence of a center of inversion were confirmed. The low-wavenumber modes are downshifted after substitution of Ru and Os for Fe. These modes were assigned to relative motions of octahedra and tetrahedra (SiP4 or PSi4) units. The upshift of high-energy modes was related to electronic transfer between the metal and the Si–P host lattice and to steric effects due to the atomic size increase from Fe to Os. Copyright © 1999 John Wiley & Sons, Ltd.

Calculation of relaxation rates from microscopic equations of motion.

For classical systems with anharmonic forces, Newton's equations for particle trajectories are nonlinear, while Liouville's equation for the evolution of functions of position and momentum is linear and is solved by constructing a basis of functions in which the Liouvillian is a tridiagonal matrix, which is then diagonalized. For systems that are chaotic in the sense that neighboring trajectories diverge exponentially, the initial conditions determine the solution to Liouville's equation for short times; but for long times, the solutions decay exponentially at rates independent of the initial conditions. These are the relaxation rates of irreversible processes, and they arise in these calculations as the imaginary parts of the frequencies where there are singularities in the analytic continuations of solutions to Liouville's equation. These rates are calculated for two examples: the inverted oscillator, which can be solved both analytically and numerically, and a charged particle in a periodic magnetic field, which can only be solved numerically. In these systems, dissipation arises from traveling-wave solutions to Liouville's equation that couple low and high wave-number modes allowing energy to flow from disturbances that are coherent over large scales to disturbances on ever smaller scales finally becoming incoherent over microscopic scales. These results suggest that dissipation in large scale motion of the system is a consequence of chaos in the small scale motion.

Cyanide binding and active site structure in heme-copper oxidases: normal coordinate analysis of iron-cyanide vibrations of a3(2+)CN- complexes of cytochromes ba3 and aa3.

The cyanide isotope-sensitive low-frequency vibrations of ferrous cyano complexes of cytochrome a3 are studied for cytochrome ba3 from Thermus thermophilus and cytochrome aa3 from bovine heart. Cyanide complexes of ba3 display three isotope sensitive frequencies at 512, 485, and 473 cm-1. The first is primarily an Fe-C stretching motion, whereas the lower wavenumber modes are bending motions. These iron-cyanide vibrations are independent of the redox levels of the other metal centers in the protein. On the other hand, the fully reduced bovine derivative complexed with cyanide gives rise to a bending vibration at 503 cm-1 and a stretching vibration at 469 cm-1. That is, the ordering of the stretching and bending frequencies is reversed from that of the bacterial protein. These results are analyzed by normal coordinate calculations to obtain comparative models for the binuclear O2 reducing site of the two proteins. We find that the observed frequencies are consistent with a linear Fe-C-N group and larger Fe-C stretching force constant (2.558 mdyn/A) for ba3 and a slightly bent Fe-C-N group (angle approximately 170 degrees) and a smaller Fe-C stretching force constant (2.335 mdyn/A) for aa3. Thus, there are significant differences in the interaction of cyanide with ferrous a3 in the two proteins that are most likely caused by a weaker proximal histidine interaction and stronger peripheral heme electron withdrawing effects in ba3. Possible sources of these protein-induced effects are discussed. Using the analysis developed here, comparison of the FeCN stretching and bending frequencies of the ferrous bovine a3-CN complex to those obtained from the ferric a3-CN complex suggests that upon conversion of the resting to the fully reduced protein, a conformational change occurs that constrains the ligand binding site.

Density fluctuations in JIPP T-IIU tokamak plasmas measured by a heavy ion beam probe

Multiple and small sample volume measurements of the density turbulence and potential profile measurements in tokamak plasmas were conducted using a heavy ion beam probe. The obtained wavenumber-frequency spectrum S(k, omega ) shows that the cross-sections of neutral beam injection (NBI) heated plasmas are divided into three regions of different turbulence characteristics outside the layer where the direction of the density turbulence propagation reverses, a low frequency and low wavenumber mode travelling in the ion diamagnetic drift direction dominates. The region enclosed by this reversal layer is divided into two parts during nearly perpendicular NBI heating: a region where the propagation velocity is near the Etau /Bt poloidal rotation velocity and a bad curvature region of very small wavenumber and high propagation velocity. The region of high propagation velocity, found in NBI plasmas, disappears in the ohmic plasmas. In addition, a small component that propagates in the ion diamagnetic drift direction is observed in NBI plasmas

Stability of a Two-Time-Level Semi-Implicit Integration Scheme for Gravity Wave Motion

Abstract A study is made of the computational stability of semi-implicit treatments of gravity wave motion suitable for use with two-time-level advection schemes. The analysis is for horizontally uniform reference values of temperature and surface pressure, and for hybrid pressure-based vertical coordinates. Stability requires the use of reference temperatures that are warmer than those that can be used safely with the corresponding three-time-level scheme. The reference surface pressure should also be higher. When stable, the two-time-level scheme is damping, although the largest scales are damped less than by the three-time-level scheme if the latter uses a typical time filtering. The first-order decentered averaging of gravity wave tendencies used in a number of semi-Lagrangian models reduces the need for a relatively warm reference temperature profile but causes a quite substantial damping of otherwise well-represented low-wavenumber modes. The low-wavenumber damping can be avoided by using an alterna...