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## Abstract

A strong linear relationship between the mean and skewness of global sea surface winds (both zonal and meridional) is shown to exist, such that where the wind component is on average positive, it is negatively skewed (and vice versa). This relationship is observed in reanalysis, satellite, and buoy data. This relationship between the mean and skewness fields of sea surface winds follows from the nonlinear surface drag predicted for a turbulent boundary layer by Monin–Obukhov similarity theory since forcing perturbations speeding the wind up are subject to a stronger drag force than perturbations slowing it down. Furthermore, it is demonstrated that the results of an empirical fit of observed surface winds to a stochastic differential equation presented in a recent study by Sura are consistent with the white-noise limit of the momentum equations for a turbulent boundary layer subject to fluctuating forcing, albeit with a somewhat different physical interpretation.

## Abstract

A strong linear relationship between the mean and skewness of global sea surface winds (both zonal and meridional) is shown to exist, such that where the wind component is on average positive, it is negatively skewed (and vice versa). This relationship is observed in reanalysis, satellite, and buoy data. This relationship between the mean and skewness fields of sea surface winds follows from the nonlinear surface drag predicted for a turbulent boundary layer by Monin–Obukhov similarity theory since forcing perturbations speeding the wind up are subject to a stronger drag force than perturbations slowing it down. Furthermore, it is demonstrated that the results of an empirical fit of observed surface winds to a stochastic differential equation presented in a recent study by Sura are consistent with the white-noise limit of the momentum equations for a turbulent boundary layer subject to fluctuating forcing, albeit with a somewhat different physical interpretation.

## Abstract

This study considers the probability distribution of sea surface wind speeds, which have historically been modeled using the Weibull distribution. First, non-Weibull structure in the observed sea surface wind speeds (from SeaWinds observations) is characterized using relative entropy, a natural information theoretic measure of the difference between probability distributions. Second, empirical models of the probability distribution of sea surface wind speeds, parameterized in terms of the parameters of the vector wind probability distribution, are developed. It is shown that Gaussian fluctuations in the vector wind cannot account for the observed features of the sea surface wind speed distribution, even if anisotropy in the fluctuations is accounted for. Four different non-Gaussian models of the vector wind distribution are then considered: the bi-Gaussian, the centered gamma, the Gram–Charlier, and the constrained maximum entropy. It is shown that so long as the relationship between the skewness and kurtosis of the along-mean sea surface wind component characteristic of observations is accounted for in the modeled probability distribution, then all four vector wind distributions are able to simulate the observed mean, standard deviation, and skewness of the sea surface wind speeds with an accuracy much higher than is possible if non-Gaussian structure in the vector winds is neglected. The constrained maximum entropy distribution is found to lead to the best simulation of the wind speed probability distribution. The significance of these results for the parameterization of air/sea fluxes in general circulation models is discussed.

## Abstract

This study considers the probability distribution of sea surface wind speeds, which have historically been modeled using the Weibull distribution. First, non-Weibull structure in the observed sea surface wind speeds (from SeaWinds observations) is characterized using relative entropy, a natural information theoretic measure of the difference between probability distributions. Second, empirical models of the probability distribution of sea surface wind speeds, parameterized in terms of the parameters of the vector wind probability distribution, are developed. It is shown that Gaussian fluctuations in the vector wind cannot account for the observed features of the sea surface wind speed distribution, even if anisotropy in the fluctuations is accounted for. Four different non-Gaussian models of the vector wind distribution are then considered: the bi-Gaussian, the centered gamma, the Gram–Charlier, and the constrained maximum entropy. It is shown that so long as the relationship between the skewness and kurtosis of the along-mean sea surface wind component characteristic of observations is accounted for in the modeled probability distribution, then all four vector wind distributions are able to simulate the observed mean, standard deviation, and skewness of the sea surface wind speeds with an accuracy much higher than is possible if non-Gaussian structure in the vector winds is neglected. The constrained maximum entropy distribution is found to lead to the best simulation of the wind speed probability distribution. The significance of these results for the parameterization of air/sea fluxes in general circulation models is discussed.

## Abstract

The probability distribution of sea surface wind speeds, *w*, is considered. Daily SeaWinds scatterometer observations are used for the characterization of the moments of sea surface winds on a global scale. These observations confirm the results of earlier studies, which found that the two-parameter Weibull distribution provides a good (but not perfect) approximation to the probability density function of *w*. In particular, the observed and Weibull probability distributions share the feature that the skewness of *w* is a concave upward function of the ratio of the mean of *w* to its standard deviation. The skewness of *w* is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). An analytic expression for the probability density function of *w*, derived from a simple stochastic model of the atmospheric boundary layer, is shown to be in good qualitative agreement with the observed relationships between the moments of *w*. Empirical expressions for the probability distribution of *w* in terms of the mean and standard deviation of the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surface wind vector components. The significance of these distributions for improvements to calculations of averaged air–sea fluxes in diagnostic and modeling studies is discussed.

## Abstract

The probability distribution of sea surface wind speeds, *w*, is considered. Daily SeaWinds scatterometer observations are used for the characterization of the moments of sea surface winds on a global scale. These observations confirm the results of earlier studies, which found that the two-parameter Weibull distribution provides a good (but not perfect) approximation to the probability density function of *w*. In particular, the observed and Weibull probability distributions share the feature that the skewness of *w* is a concave upward function of the ratio of the mean of *w* to its standard deviation. The skewness of *w* is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). An analytic expression for the probability density function of *w*, derived from a simple stochastic model of the atmospheric boundary layer, is shown to be in good qualitative agreement with the observed relationships between the moments of *w*. Empirical expressions for the probability distribution of *w* in terms of the mean and standard deviation of the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surface wind vector components. The significance of these distributions for improvements to calculations of averaged air–sea fluxes in diagnostic and modeling studies is discussed.

## Abstract

The statistical structure of sea surface wind speeds is considered, both in terms of the leading-order moments (mean, standard deviation, and skewness) and in terms of the parameters of a best-fit Weibull distribution. An intercomparison is made of the statistical structure of sea surface wind speed data from four different datasets: SeaWinds scatterometer observations, a blend of Special Sensor Microwave Imager (SSM/I) satellite observations with ECMWF analyses, and two reanalysis products [NCEP–NCAR and 40-yr ECMWF Re-Analysis (ERA-40)]. It is found that while the details of the statistical structure of sea surface wind speeds differs between the datasets, the leading-order features of the distributions are consistent. In particular, it is found in all datasets that the skewness of the wind speed is a concave upward function of the ratio of the mean wind speed to its standard deviation, such that the skewness is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). This relationship between moments is also found in buoy observations of sea surface winds. In addition, the seasonal evolution of the probability distribution of sea surface wind speeds is characterized. It is found that the statistical structure on seasonal time scales shares the relationships between moments characteristic of the year-round data. Furthermore, the seasonal data are shown to depart from Weibull behavior in the same fashion as the year-round data, indicating that non-Weibull structure in the year-round data does not arise due to seasonal nonstationarity in the parameters of a strictly Weibull time series.

## Abstract

The statistical structure of sea surface wind speeds is considered, both in terms of the leading-order moments (mean, standard deviation, and skewness) and in terms of the parameters of a best-fit Weibull distribution. An intercomparison is made of the statistical structure of sea surface wind speed data from four different datasets: SeaWinds scatterometer observations, a blend of Special Sensor Microwave Imager (SSM/I) satellite observations with ECMWF analyses, and two reanalysis products [NCEP–NCAR and 40-yr ECMWF Re-Analysis (ERA-40)]. It is found that while the details of the statistical structure of sea surface wind speeds differs between the datasets, the leading-order features of the distributions are consistent. In particular, it is found in all datasets that the skewness of the wind speed is a concave upward function of the ratio of the mean wind speed to its standard deviation, such that the skewness is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). This relationship between moments is also found in buoy observations of sea surface winds. In addition, the seasonal evolution of the probability distribution of sea surface wind speeds is characterized. It is found that the statistical structure on seasonal time scales shares the relationships between moments characteristic of the year-round data. Furthermore, the seasonal data are shown to depart from Weibull behavior in the same fashion as the year-round data, indicating that non-Weibull structure in the year-round data does not arise due to seasonal nonstationarity in the parameters of a strictly Weibull time series.

## Abstract

Air–sea exchanges of momentum, energy, and material substances of fundamental importance to the variability of the climate system are mediated by the character of the turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. The probability density function (pdf) of sea surface wind speeds *p*(*w*) is a mathematical object describing the variability of surface winds that arises from the physics of the turbulent atmospheric planetary boundary layer. Previous mechanistic models of the pdf of sea surface wind speeds have considered the momentum budget of an atmospheric layer of fixed thickness and neutral stratification. The present study extends this analysis, using an idealized model to consider the influence of boundary layer thickness variations and nonneutral surface stratification on *p*(*w*). It is found that surface stratification has little direct influence on *p*(*w*), while variations in boundary layer thickness bring the predictions of the model into closer agreement with the observations. Boundary layer thickness variability influences the shape of *p*(*w*) in two ways: through episodic downward mixing of momentum into the boundary layer from the free atmosphere and through modulation of the importance (relative to other tendencies) of turbulent momentum fluxes at the surface and the boundary layer top. It is shown that the second of these influences dominates over the first.

## Abstract

Air–sea exchanges of momentum, energy, and material substances of fundamental importance to the variability of the climate system are mediated by the character of the turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. The probability density function (pdf) of sea surface wind speeds *p*(*w*) is a mathematical object describing the variability of surface winds that arises from the physics of the turbulent atmospheric planetary boundary layer. Previous mechanistic models of the pdf of sea surface wind speeds have considered the momentum budget of an atmospheric layer of fixed thickness and neutral stratification. The present study extends this analysis, using an idealized model to consider the influence of boundary layer thickness variations and nonneutral surface stratification on *p*(*w*). It is found that surface stratification has little direct influence on *p*(*w*), while variations in boundary layer thickness bring the predictions of the model into closer agreement with the observations. Boundary layer thickness variability influences the shape of *p*(*w*) in two ways: through episodic downward mixing of momentum into the boundary layer from the free atmosphere and through modulation of the importance (relative to other tendencies) of turbulent momentum fluxes at the surface and the boundary layer top. It is shown that the second of these influences dominates over the first.

## Abstract

Nonlinear principal component analysis (NLPCA) is a generalization of traditional principal component analysis (PCA) that allows for the detection and characterization of low-dimensional nonlinear structure in multivariate datasets. The authors consider the application of NLPCA to two datasets: tropical Pacific sea surface temperature (SST) and tropical Indo–Pacific sea level pressure (SLP). It is found that for the SST data, the low-dimensional NLPCA approximations characterize the data better than do PCA approximations of the same dimensionality. In particular, the one-dimensional NLPCA approximation characterizes the asymmetry between spatial patterns characteristic of average El Niño and La Niña events, which the 1D PCA approximation cannot. The differences between NLPCA and PCA results are more modest for the SLP data, indicating that the lower-dimensional structures of this dataset are nearly linear.

## Abstract

Nonlinear principal component analysis (NLPCA) is a generalization of traditional principal component analysis (PCA) that allows for the detection and characterization of low-dimensional nonlinear structure in multivariate datasets. The authors consider the application of NLPCA to two datasets: tropical Pacific sea surface temperature (SST) and tropical Indo–Pacific sea level pressure (SLP). It is found that for the SST data, the low-dimensional NLPCA approximations characterize the data better than do PCA approximations of the same dimensionality. In particular, the one-dimensional NLPCA approximation characterizes the asymmetry between spatial patterns characteristic of average El Niño and La Niña events, which the 1D PCA approximation cannot. The differences between NLPCA and PCA results are more modest for the SLP data, indicating that the lower-dimensional structures of this dataset are nearly linear.

## Abstract

Salinity dynamics in a simple two-box model of the thermohaline circulation (THC) is considered. The model parameterizes fluctuating eddy transport and buoyancy forcing by two independent stochastic processes. The associated stationary probability density function is calculated analytically, and its structure is analyzed in the space of the three parameters of the model. It is found that over a broad range of model parameters in which the stationary density is technically bimodal, the population of one regime is very much larger than that of the other, so the system behaves effectively unimodally. This preferential population of one regime is denoted stabilization. This phenomenon is only relevant if the timescale of THC variability is less than the mean residence times of the destabilized regime, so that the system may be described by its stationary probability density. These average residence times are calculated, and it is found that stabilization occurs over a broad range of parameter values. The stabilization phenomenon has important consequences for the stability of the THC. It is shown that the inclusion of stochastic processes in the model results in random hysteresis responses to steady changes in freshwater forcing, such that the transitions between regimes generally occur some distance away from the bifurcation points at which transitions occur in the deterministic model.

## Abstract

Salinity dynamics in a simple two-box model of the thermohaline circulation (THC) is considered. The model parameterizes fluctuating eddy transport and buoyancy forcing by two independent stochastic processes. The associated stationary probability density function is calculated analytically, and its structure is analyzed in the space of the three parameters of the model. It is found that over a broad range of model parameters in which the stationary density is technically bimodal, the population of one regime is very much larger than that of the other, so the system behaves effectively unimodally. This preferential population of one regime is denoted stabilization. This phenomenon is only relevant if the timescale of THC variability is less than the mean residence times of the destabilized regime, so that the system may be described by its stationary probability density. These average residence times are calculated, and it is found that stabilization occurs over a broad range of parameter values. The stabilization phenomenon has important consequences for the stability of the THC. It is shown that the inclusion of stochastic processes in the model results in random hysteresis responses to steady changes in freshwater forcing, such that the transitions between regimes generally occur some distance away from the bifurcation points at which transitions occur in the deterministic model.

## Abstract

The effect of stochastic fluctuations in the zonal-mean velocity field on the energy dispersion of planetary stationary waves is considered, using the nondivergent, barotropic vorticity equation. It is found that for small noise levels, the oscillatory structure of the solutions is not altered. However, for noise levels comparable to or larger than those observed in the circulation at 500 mb, the marginal density functions of the solution process in the subpolar region cluster near zero. This indicates that fluctuations in the velocity field inhibit the poleward dispersion of stationary wave energy. This localization phenomenon appears whether the ensemble average of the mean zonal flow is superrotation or has a simple two-jet structure.

## Abstract

The effect of stochastic fluctuations in the zonal-mean velocity field on the energy dispersion of planetary stationary waves is considered, using the nondivergent, barotropic vorticity equation. It is found that for small noise levels, the oscillatory structure of the solutions is not altered. However, for noise levels comparable to or larger than those observed in the circulation at 500 mb, the marginal density functions of the solution process in the subpolar region cluster near zero. This indicates that fluctuations in the velocity field inhibit the poleward dispersion of stationary wave energy. This localization phenomenon appears whether the ensemble average of the mean zonal flow is superrotation or has a simple two-jet structure.

## Abstract

The spatial structure of asymmetries in sea surface temperature (SST) and surface air temperature (SAT) between average El Niño and La Niña events is considered. It is demonstrated that in historical SST and SAT reconstructions, the anomaly spatial pattern that changes sign between El Niño and La Niña events (the “linear” signal) strongly resembles that of principal component analysis (PCA) mode 1, while that which does not change sign (the “nonlinear” signal) resembles the pattern of PCA mode 2. The linear and nonlinear patterns also strongly resemble the standard deviation and skewness fields, respectively. Furthermore, temporal subsampling of long (130 yr) SST reconstructions suggests that the magnitude of the nonlinear signal and its similarity to PCA mode 2 are functions of the strength of ENSO, as measured by the standard deviation of the PCA mode-1 time series. Finally, it is found that of several coupled general circulation models (GCMs) considered, the spatial and temporal structure of the El Niño–La Niña asymmetry is captured only by the GFDL R30 model, despite large biases in its covariance structure.

## Abstract

The spatial structure of asymmetries in sea surface temperature (SST) and surface air temperature (SAT) between average El Niño and La Niña events is considered. It is demonstrated that in historical SST and SAT reconstructions, the anomaly spatial pattern that changes sign between El Niño and La Niña events (the “linear” signal) strongly resembles that of principal component analysis (PCA) mode 1, while that which does not change sign (the “nonlinear” signal) resembles the pattern of PCA mode 2. The linear and nonlinear patterns also strongly resemble the standard deviation and skewness fields, respectively. Furthermore, temporal subsampling of long (130 yr) SST reconstructions suggests that the magnitude of the nonlinear signal and its similarity to PCA mode 2 are functions of the strength of ENSO, as measured by the standard deviation of the PCA mode-1 time series. Finally, it is found that of several coupled general circulation models (GCMs) considered, the spatial and temporal structure of the El Niño–La Niña asymmetry is captured only by the GFDL R30 model, despite large biases in its covariance structure.

## Abstract

Open-ocean deep convection is a highly variable and strongly nonlinear process that plays an essential role in the global ocean circulation. A new view of its stability is presented here, in which variability, as parameterized by stochastic forcing, is central. The use of an idealized deep convection box model allows analytical solutions and straightforward conceptual understanding while retaining the main features of deep convection dynamics. In contrast to the generally abrupt stability changes in deterministic systems, measures of stochastic stability change smoothly in response to varying forcing parameters. These stochastic stability measures depend chiefly on the residence times of the system in different regions of phase space, which need not contain a stable steady state in the deterministic sense. Deep convection can occur frequently even for parameter ranges in which it is deterministically unstable; this effect is denoted wandering unimodality. The stochastic stability concepts are readily applied to other components of the climate system. The results highlight the need to take climate variability into account when analyzing the stability of a climate state.

## Abstract

Open-ocean deep convection is a highly variable and strongly nonlinear process that plays an essential role in the global ocean circulation. A new view of its stability is presented here, in which variability, as parameterized by stochastic forcing, is central. The use of an idealized deep convection box model allows analytical solutions and straightforward conceptual understanding while retaining the main features of deep convection dynamics. In contrast to the generally abrupt stability changes in deterministic systems, measures of stochastic stability change smoothly in response to varying forcing parameters. These stochastic stability measures depend chiefly on the residence times of the system in different regions of phase space, which need not contain a stable steady state in the deterministic sense. Deep convection can occur frequently even for parameter ranges in which it is deterministically unstable; this effect is denoted wandering unimodality. The stochastic stability concepts are readily applied to other components of the climate system. The results highlight the need to take climate variability into account when analyzing the stability of a climate state.