Reduced models of atmospheric low-frequency variability: Parameter estimation and comparative performance

Low-frequency variability (LFV) of the atmosphere refers to its behavior on time scales of 10–100 days, longer than the life cycle of a mid-latitude cyclone but shorter than a season. This behavior is still poorly understood and hard to predict. The present study compares various model reduction strategies that help in deriving simplified models of LFV.Three distinct strategies are applied here to reduce a fairly realistic, high-dimensional, quasi-geostrophic, 3-level (QG3) atmospheric model to lower dimensions: (i) an empirical–dynamical method, which retains only a few components in the projection of the full QG3 model equations onto a specified basis, and finds the linear deterministic and the stochastic corrections empirically as in Selten (1995) [5]; (ii) a purely dynamics-based technique, employing the stochastic mode reduction strategy of Majda et al. (2001) [62]; and (iii) a purely empirical, multi-level regression procedure, which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in Kravtsov et al. (2005) [3]. The empirical–dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi-level regression models; the extended Kalman filter was used for the parameter estimation.Overall, the reduced models perform better when more statistical information is used in the model construction. Thus, the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full QG3 model’s LFV, i.e. its autocorrelations and spectra, as well as the nonlinear properties, i.e. the persistent flow regimes that induce non-Gaussian features in the model’s probability density function. The empirical–dynamical models capture the basic statistical properties of the full model’s LFV, such as the variance and integral correlation time scales of the leading LFV modes, as well as some of the regime behavior features, but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes. Dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical–dynamical models, but do much less well on the full QG3 model’s nonlinear dynamics. These results are discussed in terms of their implications for a better understanding and prediction of LFV.     

Crossover scaling behaviour of the quasi-one-dimensional n-vector models

Crossovers in the quasi-one-dimensional n-vector models are discussed by making new scaling hypotheses. Predictions regarding critical point shifts and amplitudes are made for one-to two- and three-dimensional crossovers for 1 ? n ? ∞. Many new scaling functions are obtained exactly and studied by series-expanding methods.     

New nonlinear mechanisms of midlatitude atmospheric low-frequency variability

This paper studies the dynamical mechanisms potentially involved in the so-called atmospheric low-frequency variability, occurring at midlatitudes in the Northern Hemisphere. This phenomenon is characterised by recurrent non-propagating and temporally persistent flow patterns, with typical spatial and temporal scales of 6000-10 000 km and 10-50 days, respectively.We study a low-order model derived from the 2-layer shallow-water equations on a β-plane channel. The main ingredients of the low-order model are a zonal flow, a planetary scale wave, orography, and a baroclinic-like forcing.A systematic analysis of the dynamics of the low-order model is performed using techniques and concepts from dynamical systems theory. Orography height (h₀) and magnitude of zonal wind forcing (U₀) are used as control parameters to study the bifurcations of equilibria and periodic orbits. Along two curves of Hopf bifurcations an equilibrium loses stability () and gives birth to two distinct families of periodic orbits. These periodic orbits bifurcate into strange attractors along three routes to chaos: period doubling cascades, breakdown of 2-tori by homo- and heteroclinic bifurcations, or intermittency ( and ).The observed attractors exhibit spatial and temporal low-frequency patterns comparing well with those observed in the atmosphere. For the periodic orbits have a period of about 10 days and patterns in the vorticity field propagate eastward. For , the period is longer (30-60 days) and patterns in the vorticity field are non-propagating. The dynamics on the strange attractors are associated with low-frequency variability: the vorticity fields show weakening and strengthening of non-propagating planetary waves on time scales of 10-200 days. The spatio-temporal characteristics are “inherited” (by intermittency) from the two families of periodic orbits and are detected in a relatively large region of the parameter plane. This scenario provides a characterisation of low-frequency variability in terms of intermittency due to bifurcations of waves.     

On the corrections to scaling in three-dimensional Ising models

The leading correction-to-scaling amplitudes for the spin-1/2, nearest-neighbor sc, bcc, and fee Ising models are considered with the particular aim of determining their signs. On the basis of previous two-variable series analyses by Chen, Fisher, and Nickel and renormalization group=4–d expansions, it is concluded that the correction amplitudes for the susceptibility, correlation length, specific heat, and spontaneous magnetization arenegative for all three lattices. Thus, for example, the effective exponent _eff(T) asymptotically approaches the true susceptibility exponent fromabove. Other earlier and more recent high-temperature series and field-theoretic analyses are seen to be consistent with this result. However, the usual nonasymptotic, perturbative field-theoretic approaches are essentially committed to positive correction amplitudes. The question of the signs therefore relates directly to the applicability of these non-asymptotic field-theoretic calculations to three-dimensional Ising models as well as to different experimental systems.     

On the variability of the transmission coefficient of the atmospheric optical window

Summary We use experimental data to show that the amplitudeA of the oscillations impressed upon upper atmospheric electrons by the electric field vectors of incident solar radiation increases with the solar energy wavelength λ over the visible spectrum. Calculations based upon experimental data show that the transmission coefficientT of the upper atmosphere for incident solar energy (say at intensityI ₀) also increases with λ over the visible spectrum, implying thatT increases withA over the latter range. Finally we show thatI ₀ ^1/2 λ² bears a positive linear relationship withT over the visible solar spectrum. The latter conclusion, which confirms theoretical results contained in a recent paper, is then used to show that the 100 000 year eccentricity cycle effectively causes a maximum change in the total solar energy reaching the lower atmosphere of approximately 4.5% which is enough to trigger an ice age.     

Degeneracy induced scaling of the correlation length for periodic models

The broken symmetric phase of scalar models exhibits an infrared fixed point which is induced by the degenerate effective potential. The definition of the correlation length in the infrared regime enables us to determine the type of the phase transition in the model. It is shown that the massive sine-Gordon model exhibits a continuous, while the layered sine-Gordon model has an infinite order Kosterlitz–Thouless type phase transition.     

Cascade-like and scaling behavior of wind velocity increments in the atmospheric surface layer

By using a large amount of data collected in the atmospheric surface layer, we analyze the probability density functions (PDFs), the probability of return and the moments of wind velocity increments. Results show that the PDFs change from the non-Gaussian long-tailed distributions to Gaussian with the increase of time scales. This is similar to what has been observed and interpreted as an indication of cascade in the fully developed homogeneous and isotropic turbulence. Besides, both the probability of return and the moments are found to be scaling with time scales. We then compare above results with the truncated Lévy flights and the log-normal PDF model. It is found that although both models show the cascade-like behavior in the PDFs and the scaling behavior in the probability of return and the moments under some conditions, they are not good enough for quantitatively describing the random process of wind velocity increments.     

Contribution of strings to the observed variability of extragalactic sources of radiation

The mechanism for the fast variability introduced in the observed luminosity of compact extragalactic sources of radiation by the diffraction of electromagnetic radiation by cosmic strings is studied. Cosmic strings at cosmological distances can produce not only the observed variability but also a large increase in the observed luminosity of quasars. Curves of the variation of the luminosity are obtained and the characteristic times of the oscillations of the luminosity are presented for a string with density 10¹⁶ g/cm. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 6, 385–390 (25 September 1996)     

Anomalous scaling exponents in nonlinear models of turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.     

A mechanistic model of mid-latitude decadal climate variability

A simple heuristic model of coupled decadal ocean-atmosphere modes in middle latitudes is developed. Previous studies have treated atmospheric intrinsic variability as a linear stochastic process modified by a deterministic coupling to the ocean. The present paper takes an alternative view: based on observational, as well as process modeling results, it represents this variability in terms of irregular transitions between two anomalously persistent, high-latitude and low-latitude jet-stream states. Atmospheric behavior is thus governed by an equation analogous to that describing the trajectory of a particle in a double-well potential, subject to stochastic forcing. Oceanic adjustment to a positional shift in the atmospheric jet involves persistent circulation anomalies maintained by the action of baroclinic eddies; this process is parameterized in the model as a delayed oceanic response. The associated sea-surface temperature anomalies provide heat fluxes that affect atmospheric circulation by modifying the shape of the double-well potential. If the latter coupling is strong enough, the model’s spectrum exhibits a peak at a periodicity related to the ocean’s eddy-driven adjustment time. A nearly analytical approximation of the coupled model is used to study the sensitivity of this behavior to key model parameters.     

Global analysis of solar and atmospheric neutrino data

A global analysis of solar (including the recent SNO result), atmospheric, and reactor neutrino data is presented in terms of three-and four-neutrino oscillations. We first present the allowed regions of solar and atmospheric oscillation parameters assuming three-neutrino families, showing that in this framework it is possible to reconcile the two anomalies and providing an unified fit of all the observables at a time. Then, we consider scenarios where a sterile neutrino is added to the three standard ones and the mass spectra present two separate doublets. We evaluate the allowed active-sterile admixture in both solar and atmospheric oscillations, showing that, although the Super-Kamiokande data disfavor both the pure ν_μ→ν _s atmospheric channel and, in combination with SNO, the pure ν _e →ν _s solar channel, the result from the combined analysis still favors close-to-pure active and sterile oscillations and disfavors oscillations into a near-maximal active-sterile admixture.     

Minimal modelling of atmospheric energy cycle and high-frequency variability

Summary In this paper we build a minimal model able to describe the essential features of the atmospheric energy cycle and high-frequency variability. The model is obtained by projecting the quasi-geostrophic, two-level system on a single Fourier mode in the longitudinal directions and two eigenfunctions of a Sturm-Liouville problem in the meridional direction. The dynamical system, consisting of 14 equations, shows statistical properties in agreement with the observations and other low-order models.     

Characteristics of atmospheric gravity waves observed using the MU (Middle and Upper atmosphere) radar and GPS (Global Positioning System) radio occultation

The wind velocity and temperature profiles observed in the middle atmosphere (altitude: 10–100 km) show perturbations resulting from superposition of various atmospheric waves, including atmospheric gravity waves. Atmospheric gravity waves are known to play an important role in determining the general circulation in the middle atmosphere by dynamical stresses caused by gravity wave breaking. In this paper, we summarize the characteristics of atmospheric gravity waves observed using the middle and upper atmosphere (MU) radar in Japan, as well as novel satellite data obtained from global positioning system radio occultation (GPS RO) measurements. In particular, we focus on the behavior of gravity waves in the mesosphere (50–90 km), where considerable gravity wave attenuation occurs. We also report on the global distribution of gravity wave activity in the stratosphere (10–50 km), highlighting various excitation mechanisms such as orographic effects, convection in the tropics, meteorological disturbances, the subtropical jet and the polar night jet.     

Breakdown of Kolmogorov scaling in models of cluster aggregation

We describe a model of cluster aggregation with a source which provides a rare example of an analytically tractable turbulent system. The steady state is characterized by a constant mass flux from small masses to large. Thus it can be studied using a phenomenological theory, inspired by Kolmogorov's 1941 theory, which assumes constant flux and self-similarity. We prove that such self-similarity is violated in dimensions less than or equal to two. We then use dynamical renormalization group techniques to show that the scaling of multipoint correlation functions implies nontrivial multifractality. The analytical results are supported by Monte Carlo simulations.     

Universality of entropy scaling in one dimensional gapless models

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.     

Local coupling of shell models leads to anomalous scaling

We consider cascade models of turbulence which are obtained by restricting the Navier-Stokes equation to local interactions. By combining the results of the method of extended self-similarity and a novel subgrid model, we investigate the inertial range fluctuations of the cascade. Significant corrections to the classical scaling exponents are found. The dynamics of our local Navier-Stokes models is described accurately by a simple set of Langevin equations proposed earlier as a model of turbulence [20]. This allows for a prediction of the intermittency exponents without adjustable parameters. Excellent agreement with numerical simulations is found.     

On scaling properties of cluster distributions in Ising models

Scaling relations of cluster distributions for the Wolff algorithm are derived. We found them to be well satisfied for the Ising model ind=3 dimensions. Using scaling and a parametrization of the cluster distribution, we determine the critical exponent/=0.516(6) with moderate effort in computing time.     

Casimir scaling as a test of QCD vacuum models

Recent accurate lattice measurements of static potentials between sources in various representations of the gauge group SU(3) performed by Bali, provide a crucial test of different QCD vacuum models. The Casimir scaling of the potential observed for all measured distances can be explained as being due to strong suppression of higher cumulants contribution.