Philip Sura

Multiplicative Noise and Non-Gaussianity: A Paradigm for Atmospheric Regimes?

Abstract Atmospheric circulation statistics are not strictly Gaussian. Small bumps and other deviations from Gaussian probability distributions are often interpreted as implying the existence of distinct and persistent nonlinear circulation regimes associated with higher-than-average levels of predictability. In this paper it is shown that such deviations from Gaussianity can, however, also result from linear stochastically perturbed dynamics with multiplicative noise statistics. Such systems can be associated with much lower levels of predictability. Multiplicative noise is often identified with state-dependent variations of stochastic feedbacks from unresolved system components, and may be treated as stochastic perturbations of system parameters. It is shown that including such perturbations in the damping of large-scale linear Rossby waves can lead to deviations from Gaussianity very similar to those observed in the joint probability distribution of the first two principal components (PCs) of weekly av...

Reconciling Non-Gaussian Climate Statistics with Linear Dynamics

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 predictionsbeyondaweek.Their success seems atoddswith the factthattheobserved statisticscan besignificantly 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 K‐S 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 wintermean diabatic forcing as a constant forcing. Despite the complete neglect of diabatic variations, the model reproduces the observed K‐S 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.

A Global View of Non-Gaussian SST Variability

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.

A note on estimating drift and diffusion parameters from timeseries

Abstract Estimating the deterministic drift and stochastic diffusion parameters from discretely sampled data is fraught with the potential for error. We derive a simple way of estimating the error due to the finite sampling rate in these parameters for a univariate system using a straightforward application of the Ito–Taylor expansion. The error is calculated up to first order in the finite sampling time increment Δt . We then compare the approximate results with the analysis of numerically generated timeseries where the answer is known. Furthermore, a meteorological real world example is discussed.

Stochastic Analysis of Southern and Pacific Ocean Sea Surface Winds

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 proces...

Regime Transitions in a Stochastically Forced Double-Gyre Model

A reduced-gravity double-gyre ocean model is used to study the influence of an additive stochastic wind stress component on the regime behavior of the wind-driven circulation. The variance of the stochastic component (spatially coherent white noise) representing the effect of atmospheric transient eddies is chosen to be spatially inhomogeneous. This is done to account for the observed concentration of eddy activity along the North Atlantic and North Pacific storm tracks. As a result the double-gyre model with a spatially inhomogeneous stochastic forcing shows a bimodal behavior. One regime shows a quasi-antisymmetric; the second regime a nonsymmetric flow pattern. It is suggested that the nonsymmetric regime corresponds to one member of a known nonsymmetric pair of stationary solutions. Actually no stationary solutions are explicitly calculated in this study. The bimodality does not appear without a spatially inhomogeneous stochastic forcing nor with spatially homogeneous stochastic forcing. Therefore, the regime transitions are induced by the inhomogeneity of the white noise variance. The study suggests that the stochastic forcing enables the system to reach the neighborhood of an unstable fixed point that is not reached without the spatially inhomogeneous stochastic wind field. The unstable fixed point then acts to steer the model in a temporarily persistent regime.

Climatology of Non-Gaussian Atmospheric Statistics

AbstractA 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–Fe...

An Evaluation of Tropical Cyclone Genesis Forecasts from Global Numerical Models

Tropical cyclone (TC) forecasts rely heavily on output from global numerical models. While considerable research has investigated the skill of various models with respect to track and intensity, few studies have considered how well global models forecast TC genesis in the North Atlanticbasin. This paper analyzes TC genesis forecasts from five global models [Environment Canada’s Global Environment Multiscale Model (CMC), the European Centre for Medium-Range Weather Forecasts (ECMWF) global model, the Global Forecast System (GFS), the Navy Operational Global Atmospheric Prediction System (NOGAPS), and the Met Office global model (UKMET)] over several seasons in the North Atlantic basin. Identifying TCs in the model is based on a combination of methods used previously in the literature and newly defined objective criteria. All modelindicated TCs are classified as a hit, false alarm, early genesis, or late genesis event. Missed events also are considered. Results show that the models’ ability to predict TC genesis varies in time and space. Conditional probabilities when a model predicts genesis and more traditional performance metrics (e.g., critical success index) are calculated. The models are ranked among each other, and results show that the best-performing model varies from year to year. A spatial analysis of each model identifies preferred regions for genesis, and a temporal analysis indicates that model performance expectedly decreases as forecast hour (lead time) increases. Consensus forecasts show that the probability of genesis noticeably increases when multiple models predict the same genesis event. Overall, this study provides a climatology of objectively identified TC genesis forecasts in global models. The resulting verification statistics can be used operationally to help refine deterministic and probabilistic TC genesis forecasts and potentially improve the models examined.

Daily to Decadal Sea Surface Temperature Variability Driven by State-Dependent Stochastic Heat Fluxes

Abstract The classic Frankignoul–Hasselmann hypothesis for sea surface temperature (SST) variability of an oceanic mixed layer assumes that the surface heat flux can be simply parameterized as noise induced by atmospheric variability plus a linear temperature relaxation rate. It is suggested here, however, that rapid fluctuations in this rate, as might be expected, for example, from gustiness of the sea surface winds, are large enough that they cannot be ignored. Such fluctuations cannot be fully modeled by noise that is independent of the state of the SST anomaly itself. Rather, they require the inclusion of a state-dependent (i.e., multiplicative) noise term, which can be expected to affect both persistence and the relative occurrence of high-amplitude anomalies. As a test of this hypothesis, daily observations at several Ocean Weather Stations (OWSs) are examined. Significant skewness and kurtosis of the distributions of SST anomalies is found, which is shown to be consistent with a multiplicative nois...

Interpreting wind-driven Southern Ocean variability in a stochastic framework

A stochastic model is derived from wind stress and bottom pressure gauge data to examine the response of the Antarctic Circumpolar Current (ACC) transport to wind stress forcing. A general method is used to estimate the drift and diffusion coefe cients of a continuous stationary Markovian system. As a e rst approximation, the response of the ACC to wind stress forcing can be described by a multivariate Ornstein-Uhlenbeck process: Gaussian red noise wind stress drives the evolution of the ACC transport, which is damped by a linear drag term. The model indicates that about 30( 610)% of ACC variability is directly driven by the wind stress. This stochastic model can serve as a null hypothesis for studies of wind driven ACC variability. A more accurate stochastic description of the wind stress over the Southern Ocean requires a multiplicative noise component. The variability of the wind stress increases approximately linearly with increasing wind stress values. A multiplicative stochastic process generates a power-law distribution rather than a Gaussian distribution. A simple stochastic model shows that non-Gaussian forcing could have a signie cant impact on the velocity (or transport) probability density functions (PDFs) of the wind-driven circulation. The net oceanic damping determines whether the distribution of the oceanic e ow is Gaussian (small damping) or resembles the distribution of the atmospheric forcing (large damping).