Spectral Budget Research Articles

Revisiting the formulations for the longitudinal velocity variance in the unstable atmospheric surface layer

Because of its non‐conformity to Monin–Obukhov Similarity Theory (MOST), the effects of thermal stratification on scaling laws describing the streamwise turbulent intensity σu normalized by the turbulent friction velocity (u∗) continue to draw research attention. A spectral budget method has been developed to assess the variability of σu/u∗ under unstable atmospheric stratification. At least three different length‐scales—the distance from the ground (z), the height of the atmospheric boundary layer (δ) and the Obukhov length (L)–are all found to be controlling parameters in the variation of σu/u∗. Analytical models have been developed and supported by experiments for two limiting conditions: z/δ < 0.02, −z/L < 0.5 and 0.02 ≪ z/δ < 0.1, −z/L > 0.5. Under the first constraint, the turbulent kinetic energy spectrum is predicted to follow three regimes: k0, k−1 and k−5/3, divided in the last two regimes by a break‐point at kz = 1, where k denotes the wave number. The quantity σu/u∗ is shown to follow the much discussed logarithmic scaling, reconciled to Townsend's attached eddy hypothesis , where the coefficients B1 and A1 are modified by MOST for mildly unstable stratification. Under the second constraint, the turbulent energy spectrum tends to become quasi‐inertial, displaying k0 and k−5/3 with a break‐point predicted to occur at 0.3 < kz < 1. The work here brings together well‐established but seemingly unrelated theories of turbulence such as Kolmogorov's hypothesis, Townsend's attached eddy hypothesis, MOST and Heisenberg's eddy viscosity under a common framework.

Mesoscale Energy Spectra of the Mei-Yu Front System. Part I: Kinetic Energy Spectra

Abstract The mesoscale kinetic energy (KE) spectra of the mei-yu front system are investigated through idealized numerical simulations. In the mature stage, the upper-tropospheric KE spectrum resembles a −3 power law for wavelengths between 1000 and 400 km and shallows to a slope of approximately − at smaller wavelengths. A similar behavior can be observed in the lower stratosphere. At both levels, the rotational KE spectrum shallows nearly to the same extent as the divergent KE spectrum at smaller wavelengths, accounting for the transition in the total KE spectrum. About 12 h after the latent heating is turned off, the mesoscale KE spectra hardly show the distinct spectral transition, especially in the upper troposphere. The spectral KE budget for various height ranges is analyzed and compared. In the upper troposphere, the mesoscale KE is deposited through the buoyancy flux and removed by the advective nonlinearity and vertical pressure flux divergence. The buoyancy flux spectrum in the mature phase has a peak at scales of around 300 km and a plateau throughout the mesoscale, which suggests a significant injection of KE in the mesoscale. The negative contribution of the advective nonlinearity demonstrates that to some extent the mesoscale KE derives from a nonlinear upscale cascade, with the buoyancy-produced energy source located at the lower end of mesoscale spectrum. In the lower stratosphere, the mesoscale KE is deposited through the advective nonlinearity and vertical pressure flux divergence and removed by the buoyancy flux. This suggests that the lower-stratospheric KE spectrum is influenced by both the downscale energy cascade and vertically propagating IGWs.

Logarithmic scaling in the longitudinal velocity variance explained by a spectral budget

A logarithmic scaling for the streamwise turbulent intensity \documentclass[12pt]{minimal}\begin{document}$\sigma _u^2/{{u_*}^2}=B_1-A_1\break\ln \left({z}/{\delta }\right)$\end{document}σu2/u*2=B1−A1lnz/δ was reported across several high Reynolds number laboratory experiments as predicted from Townsend's attached eddy hypothesis, where u* is the friction velocity and z is the height normalized by the boundary layer thickness δ. A phenomenological explanation for the origin of this log-law in the intermediate region is provided here based on a solution to a spectral budget where the production and energy transfer terms are modeled. The solution to this spectral budget predicts A1 = (18/55)Co/κ2/3 and B1 = (2.5)A1, where Co and κ are the Kolmogorov and von Kármán constants. These predictions hold when very large scale motions do not disturb the k−1 scaling existing across all wavenumbers 1/δ < k < 1/z in the streamwise turbulent velocity spectrum Eu(k). Deviations from a k−1 scaling along with their effects on A1 and B1 are discussed using published data and field experiments.

Coherent structures and the k−1 spectral behaviour

Here we present unique evidence of a k−1 scaling behaviour in the atmospheric boundary layer and its connection to large scale coherent structures within the boundary layer. Wind lidar measurements were conducted above a lake under cold atmospheric conditions. The large coherent structures could be visually observed over Lake Geneva in Switzerland when cold air met the relatively warm water. Proper orthogonal decomposition of the experimental data acquired with the wind lidar clearly reveals coherent oscillations of both the fluctuating velocity field and the water aerosol field over the surface of the lake. Precise identification of the large coherent structures propagating in the flow allows for detailed analysis of their contribution to the total spectral budget. Additionally, it is shown that the experimental data agree well with recent theoretical predictions.

Indications of Stratified Turbulence in a Mechanistic GCM

Abstract The horizontal kinetic energy spectrum and its budget are analyzed on the basis of a general circulation model with simplistic parameterizations of radiative and latent heating. A spectral truncation at total wavenumber 330 is combined with a level spacing of either ~200 m or ~1.5 km from the midtroposphere to the lower stratosphere. The subgrid-scale parameterization consists of a Smagorinsky-type anisotropic diffusion scheme that is scaled by a Richardson criterion for dynamic instability and combined with a stress-tensor-based hyperdiffusion that acts only on the very smallest resolved scales. Simulations with both vertical resolutions show a transition from the synoptic −3 to the mesoscale slope in the upper-tropospheric kinetic energy spectrum. Analysis of the spectral budget indicates that the mesoscale slope can be interpreted as stratified turbulence, as has been proposed in recent works of Lindborg and others, only when a high vertical resolution is applied. In this case, the mesoscale kinetic energy around 300–150 hPa is dominated by the nonrotational flow, and the forward horizontal energy cascade is accompanied by an equally strong forward spectral flux due to adiabatic conversion. This adiabatic conversion mainly results from a vertical potential energy flux that originates in the midtroposphere, where the mesoscale adiabatic conversion is negative. For a conventionally coarse vertical resolution, however, the mesoscale slope in the troposphere is dominated by the rotational flow, and the spectral flux due to adiabatic conversion is not comparable to that associated with horizontal advection.

G050021 主流方向に系回転するMHD一様勇断乱流の直接数値計算([G05002]流体工学部門一般セッション(2):複雑流)

The direct numerical simulation for MHD homogeneous shear turbulence under the streamwise system-rotation isperformed. There are small differences for the kinetic and magnetic energy, Reynolds and Maxwell stresses in theseveral streamwise rotation cases by comparison with the results of the spanwise rotation MHD field. Immediately afterthe calculation start, however, the contribution of the energy transformation becomes large as the rotation increases.From the kinetic energy spectral budget it is found that the sum of the transfer and transformation spectral functions isbalanced with the dissipative one in the small scale.

The Role of Wake Production on the Scaling Laws of Scalar Concentration Fluctuation Spectra Inside Dense Canopies

The scalar concentration fluctuations within a plane parallel-to-the-ground surface were measured inside a model canopy composed of densely arrayed rods using the laser-induced fluorescence technique. Two-dimensional scalar concentration spectra were computed and were shown to exhibit an approximate −3 power-law scaling at wavenumbers larger than those associated with wake production during quiescent instances when von Karman vortex streets dominated the flow. However, during instances when sweeps disrupted the flow, the spectral exponents increased above −3. The −3 power-law for these concentration fluctuation spectra measurements was shown to be consistent with a simplified spectral budget for locally homogeneous and isotropic turbulence augmented with a relaxation time scale similarity argument that assumed a constant enstrophy injection rate and wake generation mechanism. Hence, the origin of this −3 power-law scaling here differs from the well-known −3 power-law result for the so-called inertial diffusive range derived for the scalar concentration spectrum at small Prandtl numbers.

Uncertainty analysis of photometer quality factor

The applicability of biased and random error models for determining the uncertainty of photometer quality factor using Monte Carlo simulations is investigated. In the biased error model the contribution to the measured value has the same sign and almost the same magnitude at neighbouring wavelengths, whereas in the random error model contributions of varying signs appear at the neighbouring wavelengths. Both error models are used with real spectral responsivity data and uncertainty budgets of two photometers. Analytical considerations supported by simulation results show that the random error model alone may easily underestimate the uncertainty of . The biased error model is recommended as the basis of the uncertainty evaluation of . It is also demonstrated by real photometer data that the uncertainty of can be highly sensitive to fine details of the photometer spectral responsivity when the biased error model is used.

3D modeling of satellite spectral images, radiation budget and energy budget of urban landscapes

DART EB is a model that is being developed for simulating the 3D (3 dimensional) energy budget of urban and natural scenes, possibly with topography and atmosphere. It simulates all non radiative energy mechanisms (heat conduction, turbulent momentum and heat fluxes, water reservoir evolution, etc.). It uses DART model (Discrete Anisotropic Radiative Transfer) for simulating radiative mechanisms: 3D radiative budget of 3D scenes and their remote sensing images expressed in terms of reflectance or brightness temperature values, for any atmosphere, wavelength, sun/view direction, altitude and spatial resolution. It uses an innovative multispectral approach (ray tracing, exact kernel, discrete ordinate techniques) over the whole optical domain.

Surface kinetic energy transfer in surface quasi-geostrophic flows

The relevance of surface quasi-geostrophic dynamics (SQG) to the upper ocean and the atmospheric tropopause has been recently demonstrated in a wide range of conditions. Within this context, the properties of SQG in terms of kinetic energy (KE) transfers at the surface are revisited and further explored. Two well-known and important properties of SQG characterize the surface dynamics: (i) the identity between surface velocity and density spectra (when appropriately scaled) and (ii) the existence of a forward cascade for surface density variance. Here we show numerically and analytically that (i) and (ii) do not imply a forward cascade of surface KE (through the advection term in the KE budget). On the contrary, advection by the geostrophic flow primarily induces an inverse cascade of surface KE on a large range of scales. This spectral flux is locally compensated by a KE source that is related to surface frontogenesis. The subsequent spectral budget resembles those exhibited by more complex systems (primitive equations or Boussinesq models) and observations, which strengthens the relevance of SQG for the description of ocean/atmosphere dynamics near vertical boundaries. The main weakness of SQG however is in the small-scale range (scales smaller than 20–30 km in the ocean) where it poorly represents the forward KE cascade observed in non-QG numerical simulations.

Resource Allocation and Quality of Service Evaluation for Wireless Communication Systems Using Fluid Models

Wireless systems offer a unique mixture of connectivity, flexibility, and freedom. It is therefore not surprising that wireless technology is being embraced with increasing vigor. For real-time applications, user satisfaction is closely linked to quantities such as queue length, packet loss probability, and delay. System performance is therefore related to, not only Shannon capacity, but also quality of service (QoS) requirements. This work studies the problem of resource allocation in the context of stringent QoS constraints. The joint impact of spectral bandwidth, power, and code rate is considered. Analytical expressions for the probability of buffer overflow, its associated exponential decay rate, and the effective capacity are obtained. Fundamental performance limits for Markov wireless channel models are identified. It is found that, even with an unlimited power and spectral bandwidth budget, only a finite arrival rate can be supported for a QoS constraint defined in terms of exponential decay rate

The Structure and Maintenance of Stationary Waves in the Winter Northern Hemisphere

Abstract Previous studies of extratropical stationary waves in the winter Northern Hemisphere (NH) often focused on effects of orography and land–ocean thermal contrast on the formation, structure, and maintenance of these waves. In contrast, research attention to tropical stationary waves was attracted by the summer monsoon circulations and the ENSO-related climate variability. Consequently, the structure and basic dynamics of tropical stationary waves and the relationship of these waves with those in mid–high latitudes have long been neglected. Thus, the following several distinct features of observed winter NH stationary waves have not been explained: 1) an abrupt change in the longitudinal phase across 30°N; 2) a transition from the vertical phase reversal of tropical stationary waves to the vertically westward tilt of extratropical stationary waves; and 3) a longitudinally quarter-phase relationship between stationary waves and east–west circulations, and a reversal of this relationship across 30°N. It is inferred from a spectral streamfunction budget analysis with the NCEP–NCAR reanalyses that these wave features are caused by the transition of wave dynamics from the Sverdrup regime in the Tropics to the Rossby regime in the mid–high latitudes. Based on the simplified vorticity equations of these two dynamic regimes, analytic solutions obtained with observed velocity potential fields (which were used to portray the global divergent circulation) confirm that the aforementioned distinct features of stationary waves are attributed to the dynamics transition across 30°N. Since east–west circulations are part of the global divergent circulation, it is revealed from a diagnosis of the velocity potential maintenance equation that this circulation component is maintained in the Tropics primarily by diabatic heating and in the mid–high latitudes by both horizontal heat advection and diabatic heating. Evidently, stationary waves are maintained by diabatic heating through the divergent circulation and the dynamics transition of these waves from the Sverdrup regime to the Rossby regime is attributed to strong midlatitude westerlies.

Barry Saltzman and the Theory of Climate

Abstract Barry Saltzman was a giant in the fields of meteorology and climate science. A leading figure in the study of weather and climate for over 40 yr, he has frequently been referred to as the “father of modern climate theory.” Ahead of his time in many ways, Saltzman made significant contributions to our understanding of the general circulation and spectral energetics budget of the atmosphere, as well as climate change across a wide spectrum of time scales. In his endeavor to develop a unified theory of how the climate system works, he played a role in the development of energy balance models, statistical dynamical models, and paleoclimate dynamical models. He was a pioneer in developing meteorologically motivated dynamical systems, including the progenitor of Lorenz’s famous chaos model. In applying his own dynamical-systems approach to long-term climate change, he recognized the potential for using atmospheric general circulation models in a complimentary way. In 1998, he was awarded the Carl-Gustaf Rossby medal, the highest honor of the American Meteorological Society “for his life-long contributions to the study of the global circulation and the evolution of the earth’s climate.” In this paper, the authors summarize and place into perspective some of the most significant contributions that Barry Saltzman made during his long and distinguished career. This short review also serves as an introduction to the papers in this special issue of the Journal of Climate dedicated to Barry’s memory.

Simple parameterizations of the radiation budget of uniform broadleaved and coniferous canopies

Simulations of the different components of the spectral radiation budget of structurally simple leaf and shoot canopies with varying canopy leaf area index (LAI) were performed. The aims were (1) to test a proposed parameterization of the budget using two spectrally invariant canopy structural parameters ( p and p t) governing canopy absorption and transmittance, respectively, and (2) to incorporate the effect of within-shoot scattering in the parameterization for shoot canopies. Results showed that canopy spectral absorption and scattering were well described by a single parameter, the canopy p value or ‘recollision probability’, which was closely related to LAI. The relationship between p and LAI was however different in leaf and shoot canopy: e.g., at the same LAI the recollision probability was larger in the shoot canopy. It was shown that the p value of the shoot canopy could be decomposed into the p value of an individual shoot ( p sh) and the p value of the leaf canopy with the same effective LAI (LAI e). The canopy p value allows calculation of canopy absorption and scattering at any given wavelength from the leaf (or needle) scattering coefficient at the same wavelength. To calculate canopy reflectance, separation of the downward and upward scattered parts is needed in addition. The proposed parameter p t worked rather well in the leaf canopy at moderate values of LAI, but not in the coniferous shoot canopy nor at high values of LAI. However, the simulated fraction of upward scattered radiation increased in a straightforward manner with LAI, and was not particularly sensitive to the leaf (or needle) scattering coefficient. Judged by this ‘smooth’ behavior, the existence of another kind of simple parameterization for this separation remains an interesting possibility.

The Horizontal Kinetic Energy Spectrum and Spectral Budget Simulated by a High-Resolution Troposphere–Stratosphere–Mesosphere GCM

Abstract Horizontal kinetic energy spectra simulated by high-resolution versions of the Geophysical Fluid Dynamics Laboratory SKYHI middle-atmosphere general circulation model are examined. The model versions considered resolve heights between the ground and ∼80 km, and the horizontal grid spacing of the highest-resolution version is about 35 km. Tropospheric kinetic energy spectra show the familiar ∼−3 power-law dependence on horizontal wavenumber for wavelengths between about 5000 and 500 km and have a slope of ∼−5/3 at smaller wavelengths. Qualitatively similar behavior is seen in the stratosphere and mesosphere, but the wavelength marking the transition to the shallow regime increases with height, taking a value of ∼2000 km in the stratosphere and ∼4000 km in the mesosphere. The global spectral kinetic energy budget for various height ranges is computed as a function of total horizontal wavenumber. Contributions to the kinetic energy tendency from nonlinear advective processes, from conversion of avai...

Radiative Effects on Temperature in the Stable Surface Layer

The interaction between longwave radiation and temperature fluctuations plays a role in the dissipation of temperature variance. This interaction is most easily described by spectral models of atmospheric turbulence and a spectral radiative dissipation function which gives the intensity of the damping at each radiative wavelength λ and wavenumber k. We have used a Corrsin–Pao closure for the spectral budgets of turbulent kinetic energy and temperature to study the coupling of radiation to turbulence. The spectral radiative dissipation function and a related integral have been fitted by analytical approximations with the correct asymptotic behavior. This resulted in a simple analytical formula for the dimensionless temperature spectrum as a function of Monin-Obukhov stability, and a new dimensionless parameter describing the relative importance of radiation in the temperature spectral budget. The radiative effects both on the temperature spectrum and on the dimensionless temperature variance can then be calculated. Based on typical values of the radiative dimensionless parameters for the surface layer, we conclude that radiative dissipation is probably negligible there.

A Theoretical and Experimental Investigation of Energy-Containing Scales in the Dynamic Sublayer of Boundary-Layer Flows

The existence of universal power laws at low wavenumbers (K) in the energy spectrum (Eu) of the turbulent longitudinal velocity (u) is examined theoretically and experimentally for the near-neutral atmospheric surface layer. Newly derived power-law solutions to Tchen's approximate integral spectral budget equation are tested for strong- and weak-interaction cases between the mean flow and turbulent vorticity fields. To verify whether these solutions reproduce the measured Eu at low wavenumbers, velocity measurements were collected in the dynamic sublayer of the atmosphere at three sites and in the inner region of a laboratory open channel. The atmospheric surface layer measurements were carried out using triaxial sonic anemometers over tall corn, short grass, and smooth desert-like sandy soil. The open channel measurements were performed using a two-dimensional boundary-layer probe above a smooth stainless steel bed. Comparisons between the proposed analytical solution for Eu, the dimensional analysis by Kader and Yaglom, and the measured Haar wavelet Eu spectra are presented. It is shown that when strong interaction between the mean flow and turbulent vorticity field occurs, wavelet spectra measurements, predictions by the analytical solution, and predictions by the dimensional analysis of Kader-Yaglom (KY) are all in good agreement and confirm the existence of a -1 power law in Eu(= Cuuu2* K-1, where Cuu is a constant and u* is the friction velocity). The normalized upper wavenumber limit of the -1 power law (Kz = 1, where z is the height above the zero-plane displacement) is estimated using two separate approaches and compared to the open channel and atmospheric surface-layer measurements. It is demonstrated that the measured upper wavenumber limit is consistent with Tchen's budget but not with the KY assumptions. The constraints as to whether the mean flow and turbulent vorticity strongly interact are considered using a proposed analysis by Panchev. It is demonstrated that the arguments by Panchev cannot be consistent with surface-layer turbulence. Using dimensional analysis and Heisenberg's turbulent viscosity model, new constraints are proposed. The new constraints agree with the open channel and atmospheric surface-layer measurements, Townsend's ‘inactive eddy motion hypothesis’, and the Perry et al. analysis.

Spectral Budget Analysis of the Short-Range Forecast Error of the NMC Medium-Range Forecast Model

The budget of the systematic component of the short-range forecast error in the National Meteorological Center's Medium-Range Forecast Model (NMC MRF) is examined. The budget is computed for the spectral coefficients and the variances of vorticity, divergence, virtual temperature, and specific humility at every time step during the 24-h model integration. Two months in winter and three months in summer, totaling 150 cases, were integrated with the budget diagnostics. The results of the budget of the spectral coefficients—that is, the budget of mean error—showed compensation among large terms except near the model boundary; therefore, it is difficult to point to a significant source of the systematic error in the free atmosphere. Near the model lower boundaries, dynamics cannot fully compensate physical forcing, and estimation of some physical processes responsible for the mean errors is possible. In contrast, the budget of the variance of the coefficients—that is, the energy budget—is more interesting and informative. The most apparent problem found in the model is a loss of rotational kinetic energy in the medium (total wavenumber n = 11–40) and small (n = 41–80) scales in the free atmosphere. About 50% of the loss is explained by the excessive horizontal and vertical diffusion. There is a strong indication that the rest of the loss of kinetic energy is related to the insufficient generation of available potential energy in the medium scale. To isolate further the cause of the error in the energetics, several forecasts with budget diagnostics were performed. The experiments showed complex interactions between the physics and dynamics and among the different physical processes. Particularly noteworthy are (a) the compensation between horizontal and vertical diffusion and (b) the balance among horizontal/vertical diffusion, the barotropic scale interaction, and the baroclinic conversion terms in the rotational kinetic energy equation. The results of this study guided the design and implementation of changes in the NMC model in the horizontal diffusion and the cumulus parameterization.

Mean and Transient Spectral Energy and Enstrophy Budgets

Abstract The scale-dependent behavior of atmospheric flow on the sphere is investigated in terms of the spectra and spectral budgets of enstrophy and kinetic and available potential energy. The decomposition into both the α = (n, m) and the more usual n-spectral forms is considered. Several novel spectral results are obtained. First, a direct test of the extent to which large-scale atmospheric flow satisfies the necessary and sufficient conditions for homogeneous and isotropic two-dimensional turbulent behavior is carried out by calculating “spectral teleconnections” from data. The conditions are satisfied for the higher wavenumber transient component of the flow. Second, the spectral budget equations are extended to include the decomposition into time mean and transient components, a method that, although common and straightforward in real-space calculations, is novel in the spectral domain. The terms in these budgets are evaluated from data. Finally, the flow of energy and enstrophy through α-spectral s...

A comparison of the spectral energy and enstrophy budgets of blocking versus nonblocking periods

The time mean spectral energy and enstrophy budgets for blocking versus nonblocking periods from the winters of 1978–79 and 1976–77 are calculated. The major differences in the 1978–79 cases were in the nonlinear interaction terms. A pronounced upscale cascade of kinetic energy and enstrophy from intermediate to planetary-scale wavenumbers during blocking was found. During December 1976, the energy and enstrophy budgets of a case of Atlantic blocking were very similar to the cases in 1978–79, but quite different from the persistent, greatly amplified, planetary-wave pattern that followed in January and February 1977 (as identified by Charney et al., 1981). Planetary-scale baroclinic processes were greatly increased during the 1977 event along with reduced intermediate-scale baroclinic activity and the absence of the upscale enstrophy cascade noted in the other blocking cases. The predictability time (Lilly, 1970) based on the enstrophy flux function showed an increased predictability for the January-February 1977 event but no significant differences between the other blocking cases compared to the nonblocking sample. However, the reversal of the low wavenumber enstrophy cascade during blocking does suggest that blocking may be more persistent (due to reduced dissipation of the large-scale circulation) and therefore more predictable. DOI: 10.1111/j.1600-0870.1984.tb00222.x