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Philip Sura

Abstract

This paper shows that the synoptic variability of zonal and meridional midlatitude Pacific and Southern Ocean sea surface winds can be well described by a univariate stochastic dynamical system directly derived from data. The method used to analyze blended Quick Scatterometer (QuikSCAT)–NCEP winds is a general method to estimate drift and diffusion coefficients of a continuous stationary Markovian system. Almost trivially, the deterministic part consists of a simple, nearly linear damping term. More importantly, the stochastic part appears to be a state-dependent white noise term, that is, multiplicative noise. The need for a multiplicative noise term to describe the variability of midlatitude winds can be interpreted by the fact that the variability of midlatitude winds increases with increasing wind speed. The results indicate that a complete stochastic description of midlatitude winds requires a state-dependent white noise term, that is, multiplicative noise. A simple Ornstein–Uhlenbeck process is not sufficient to describe the wind data within a stochastic framework. The method used fails for tropical regions, suggesting that tropical variability might be non-Markovian.

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Philip Sura

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The concepts of multiplicative stochastic perturbations and noise-induced transitions are applied to a quasigeostrophic β-plane model of barotropic flow over topography. The spectral three-component low-order representation of this configuration yields the Charney–DeVore (CDV) model. The externally prescribed damping of the system is allowed to scatter around a mean value. The stochastic representation of the damping term leads to a multiplicative stochastic forcing. The Fokker–Planck equation and the stochastic differential equation of the low-order CDV model are solved numerically. It is found that the qualitative behavior of the system is a function of the multiplicative noise level. In particular, the effect of multiplicative noise is not simply a smoothing of the probability density function, as it would be for pure additive noise. Rather, multiplicative noise leads to the high-index state being favored over the low-index state. The concept of noise-induced transitions explains this behavior. The noise-induced transition of the stochastic low-order model is confirmed by numerical integrations of a corresponding gridpoint model with many more degrees of freedom than the spectral model. It is suggested that the statistics of the unresolved physical processes could be an important factor in understanding the behavior of midlatitude large-scale atmospheric dynamics.

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Philip Sura and Abdel Hannachi

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Understanding non-Gaussian statistics of atmospheric synoptic and low-frequency variability has important consequences in the atmospheric sciences, not least because weather and climate risk assessment depends on knowing and understanding the exact shape of the system’s probability density function. While there is no doubt that many atmospheric variables exhibit non-Gaussian statistics on many time (and spatial) scales, a full and complete understanding of this phenomenon remains a challenge. Various mechanisms behind the observed atmospheric non-Gaussian statistics have been proposed but remain, however, multifaceted and scattered in the literature: nonlinear dynamics, multiplicative noise, cross-frequency coupling, nonlinear boundary layer drag, and others. Given the importance of this subject for weather and climate research, and in an attempt to contribute to this topic, a thorough review and discussion of the different mechanisms that lead to non-Gaussian weather and climate variability are presented in this paper and an outlook is given.

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Maxime Perron and Philip Sura

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A common assumption in the earth sciences is the Gaussianity of data over time. However, several independent studies in the past few decades have shown this assumption to be mostly false. To be able to study non-Gaussian climate statistics, one must first compile a systematic climatology of the higher statistical moments (skewness and kurtosis; the third and fourth central statistical moments, respectively). Sixty-two years of daily data from the NCEP–NCAR Reanalysis I project are analyzed. The skewness and kurtosis of the data are found at each spatial grid point for the entire time domain. Nine atmospheric variables were chosen for their physical and dynamical relevance in the climate system: geopotential height, relative vorticity, quasigeostrophic potential vorticity, zonal wind, meridional wind, horizontal wind speed, vertical velocity in pressure coordinates, air temperature, and specific humidity. For each variable, plots of significant global skewness and kurtosis are shown for December–February and June–August at a specified pressure level. Additionally, the statistical moments are then zonally averaged to show the vertical dependence of the non-Gaussian statistics. This is a more comprehensive look at non-Gaussian atmospheric statistics than has been taken in previous studies on this topic.

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Philip Sura and Matthew Newman

Abstract

The basic effect of extratropical atmosphere–ocean thermal coupling is to enhance the variance of both anomalous sea surface temperatures (SSTs) and air temperatures (AIRT) due to a decreased energy flux between the atmosphere and ocean, called reduced thermal damping. In this paper it is shown that rapidly varying surface winds, through their influence upon the turbulent surface heat fluxes that drive this coupling, act to effectively weaken the coupling and thus partially counteract the reduced thermal damping. In effect, rapid fluctuations in wind speed somewhat insulate the atmosphere and ocean from each other.

The nonlinear relationship between the rapidly varying wind speed anomalies and SST and AIRT anomalies results in a rapidly varying component of the surface heat fluxes. The clear separation between the dynamical time scales of the ocean and atmosphere allows this rapidly varying flux to be simply approximated by a stochastic process in which rapidly varying wind speed is represented as Gaussian white noise whose amplitude is modulated by the more slowly evolving thermal anomalies. Such state-dependent (multiplicative) noise can alter the dynamics of atmosphere–ocean coupling because it induces an additional heat flux term, the noise-induced drift, that effectively acts to weaken both coupling and dissipation. Another key implication of the outlined theory is that air–sea coupling includes both deterministic and stochastic components.

The theory is tested by examining daily observations during extended winter (November–April) at several ocean weather stations (OWSs). Two important results are found. First, multiplicative noise at OWS P effectively decreases the coupling by about one-third, with about a 10% (20%) decrease in the damping of SST (AIRT). This suggests that multiplicative noise may be responsible for roughly half of the AIRT variability at OWS P on subseasonal time scales. Second, OWS observations reveal that joint probability distribution functions of daily averaged SST and AIRT anomalies are significantly non-Gaussian. It is shown that treating the rapidly varying boundary layer heat fluxes as state-dependent noise can reproduce this observed non-Gaussianity. It is concluded that the effect of state-dependent noise is crucial to understand and model atmosphere–ocean coupling.

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Philip Sura and Maxime Perron

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This study explores the dynamical role of non-Gaussian potential vorticity variability (extreme events) in the zonally averaged circulation of the atmosphere within a stochastic framework. First the zonally averaged skewness and kurtosis patterns of relative and potential vorticity anomalies from NCEP–NCAR reanalysis data are presented. In the troposphere, midlatitude regions of near-zero skewness coincide with regions of maximum variability. Equatorward of the Northern Hemisphere storm track positive relative/potential vorticity skewness is observed. Poleward of the same storm track the vorticity skewness is negative. In the Southern Hemisphere the relation is reversed, resulting in negative relative/potential vorticity skewness equatorward, and positive skewness poleward of the storm track. The dynamical role of extreme events in the zonally averaged general circulation is then explored in terms of the potential enstrophy budget by linking eddy enstrophy fluxes to a stochastic representation of non-Gaussian potential vorticity anomalies. The stochastic model assumes that potential vorticity anomalies are advected by a random velocity field. The assumption of stochastic advection allows for a closed expression of the meridional enstrophy flux: the potential enstrophy flux is proportional to the potential vorticity skewness. There is some evidence of this relationship in the observations. That is, potential enstrophy fluxes might be linked to non-Gaussian potential vorticity variability. Thus, extreme events may presumably play an important role in the potential enstrophy budget and the related general circulation of the atmosphere.

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Prashant D. Sardeshmukh and Philip Sura

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While it is obvious that the mean diabatic forcing of the atmosphere is crucial for maintaining the mean climate, the importance of diabatic forcing fluctuations is less evident in this regard. Such fluctuations do not appear directly in the equations of the mean climate but affect the mean indirectly through their effects on the time-mean transient-eddy fluxes of heat, momentum, and moisture. How large are these effects? What are the effects of tropical phenomena associated with substantial heating variations such as ENSO and the MJO? To what extent do variations of the extratropical surface heat fluxes and precipitation affect the mean climate? What are the effects of the rapid “stochastic” components of the heating fluctuations? Most current climate models misrepresent ENSO and the MJO and ignore stochastic forcing; they therefore also misrepresent their mean effects. To what extent does this contribute to climate model biases and to projections of climate change?

This paper provides an assessment of such impacts by comparing with observations a long simulation of the northern winter climate by a dry adiabatic general circulation model forced only with the observed time-mean diabatic forcing as a constant forcing. Remarkably, despite the total neglect of all forcing variations, the model reproduces most features of the observed circulation variability and the mean climate, with biases similar to those of some state-of-the-art general circulation models. In particular, the spatial structures of the circulation variability are remarkably well reproduced. Their amplitudes, however, are progressively underestimated from the synoptic to the subseasonal to interannual and longer time scales. This underestimation is attributed to the neglect of the variable forcing. The model also excites significant tropical variability from the extratropics on interannual scales, which is overwhelmed in reality by the response to tropical heating variability. It is argued that the results of this study suggest a role for the stochastic, and not only the coherent, components of transient diabatic forcing in the dynamics of climate variability and the mean climate.

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Philip Sura and Prashant D. Sardeshmukh

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The skewness and kurtosis of daily sea surface temperature (SST) variations are found to be strongly linked at most locations around the globe in a new high-resolution observational dataset, and are analyzed in terms of a simple stochastically forced mixed layer ocean model. The predictions of the analytic theory are in remarkably good agreement with observations, strongly suggesting that a univariate linear model of daily SST variations with a mixture of SST-independent (additive) and SST-dependent (multiplicative) noise forcing is sufficient to account for the skewness–kurtosis link. Such a model of non-Gaussian SST dynamics should be useful in predicting the likelihood of extreme events in climate, as many important weather and climate phenomena, such as hurricanes, ENSO, and the North Atlantic Oscillation (NAO), depend on a detailed knowledge of the underlying local SSTs.

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Prashant D. Sardeshmukh and Philip Sura

Abstract

Linear stochastically forced models have been found to be competitive with comprehensive nonlinear weather and climate models at representing many features of the observed covariance statistics and at predictions beyond a week. Their success seems at odds with the fact that the observed statistics can be significantly non-Gaussian, which is often attributed to nonlinear dynamics. The stochastic noise in the linear models can be a mixture of state-independent (“additive”) and linearly state-dependent (“multiplicative”) Gaussian white noises. It is shown here that such mixtures can produce not only symmetric but also skewed non-Gaussian probability distributions if the additive and multiplicative noises are correlated. Such correlations are readily anticipated from first principles. A generic stochastically generated skewed (SGS) distribution can be analytically derived from the Fokker–Planck equation for a single-component system. In addition to skew, all such SGS distributions have power-law tails, as well as a striking property that the (excess) kurtosis K is always greater than 1.5 times the square of the skew S. Remarkably, this KS inequality is found to be satisfied by circulation variables even in the observed multicomponent climate system. A principle of “diagonal dominance” in the multicomponent moment equations is introduced to understand this behavior.

To clarify the nature of the stochastic noises (turbulent adiabatic versus diabatic fluctuations) responsible for the observed non-Gaussian statistics, a long 1200-winter simulation of the northern winter climate is generated using a dry adiabatic atmospheric general circulation model forced only with the observed long-term winter-mean diabatic forcing as a constant forcing. Despite the complete neglect of diabatic variations, the model reproduces the observed KS relationships and also the spatial patterns of the skew and kurtosis of the daily tropospheric circulation anomalies. This suggests that the stochastic generators of these higher moments are mostly associated with local adiabatic turbulent fluxes. The model also simulates fifth moments that are approximately 10 times the skew, and probability densities with power-law tails, as predicted by the linear theory.

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Philip Sura and Sarah T. Gille

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Sea surface height anomalies measured by the Ocean Topography Experiment (TOPEX)/Poseidon satellite altimeter indicate high values of skewness and kurtosis. Except in a few regions, including the Gulf Stream, the Kuroshio Extension, and the Agulhas Retroflection, that display bimodal patterns of sea surface height variability, kurtosis is uniformly greater than 1.5 times the squared skewness minus an adjustment constant. This relationship differs substantially from what standard Gaussian or double-exponential noise would produce. However, it can be explained by a simple theory in which the noise is assumed to be multiplicative, meaning that a larger background state implies larger random noise elements. The existence of multiplicative noise can be anticipated from the equations of motion, if ocean dynamics are split into a slowly decorrelating deterministic component and a rapidly decorrelating contribution that is approximated as noise. Such a model raises the possibility of predicting the probabilities of extreme sea surface height anomalies from first physical principles and may provide a useful null hypothesis for non-Gaussian sea surface height variability.

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