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- Author or Editor: Prashant D. Sardeshmukh x
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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 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 winter-mean 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.
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 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 winter-mean 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.
Abstract
The first-order perturbation technique is reviewed as a tool for investigating the error and sensitivity of results obtained from the eigenanalysis of geophysical systems. Expressions are provided for the change in a system's eigenfunctions (e.g., normal modes) and their periods and growth rates associated with a small change δL in the system matrix L. In the context of data analysis, these expressions can be used to estimate changes or uncertainties in the eigenstructure of matrices involving the system's covariance statistics. Their application is illustrated in the problems of 1) updating a subset of the empirical orthogonal functions and their eigenvalues when more data become available, 2) estimating uncertainties in the growth rate and spatial structure of the singular vectors of a linear dynamical system, and 3) estimating uncertainties in the period, growth rate, and spatial structure of the normal modes of a linear dynamical system. The linear system considered in examples 2 and 3 is an empirical stochastic-dynamic model of tropical sea surface temperature (SST) evolution derived from 35 years of SST observations in the tropical Indo-Pacific basin. Thus, the system matrix L is empirically derived. Estimates of the uncertainty in L, required for estimating the uncertainties in the singular vectors and normal modes, are obtained from a long Monte Carlo simulation. The analysis suggests that the singular vectors, which represent optimal initial structures for SST anomaly growth, am more reliably determined from the 35 years of observed data than are the individual normal modes of the system.
Abstract
The first-order perturbation technique is reviewed as a tool for investigating the error and sensitivity of results obtained from the eigenanalysis of geophysical systems. Expressions are provided for the change in a system's eigenfunctions (e.g., normal modes) and their periods and growth rates associated with a small change δL in the system matrix L. In the context of data analysis, these expressions can be used to estimate changes or uncertainties in the eigenstructure of matrices involving the system's covariance statistics. Their application is illustrated in the problems of 1) updating a subset of the empirical orthogonal functions and their eigenvalues when more data become available, 2) estimating uncertainties in the growth rate and spatial structure of the singular vectors of a linear dynamical system, and 3) estimating uncertainties in the period, growth rate, and spatial structure of the normal modes of a linear dynamical system. The linear system considered in examples 2 and 3 is an empirical stochastic-dynamic model of tropical sea surface temperature (SST) evolution derived from 35 years of SST observations in the tropical Indo-Pacific basin. Thus, the system matrix L is empirically derived. Estimates of the uncertainty in L, required for estimating the uncertainties in the singular vectors and normal modes, are obtained from a long Monte Carlo simulation. The analysis suggests that the singular vectors, which represent optimal initial structures for SST anomaly growth, am more reliably determined from the 35 years of observed data than are the individual normal modes of the system.
Abstract
It is argued from SST observations for the period 1950–90 that the tropical Indo-Pacific ocean-atmosphere system may be described as a stable linear dynamical system driven by spatially coherent Gaussian white noise. Evidence is presented that the predictable component of SST anomaly growth is associated with the constructive interference of several damped normal modes after an optimal initial structure is set up by the white noise forcing. In particular, El Niño–Southern Oscillation (ENSO) growth is associated with an interplay of at least three damped normal modes, with periods longer than two years and decay times of 4 to 8 months, rather than the manifestation of a single unstable mode whose growth is arrested by nonlinearities. Interestingly, the relevant modes are not the three least damped modes of the system. Rather, mode selection, and the establishment of the optimal initial structure from which growth occurs, are controlled by the stochastic forcing. Experiments conducted with an empirical stochastic-dynamical model show that stochastic forcing not only adds energy to the system, but also plays a role in setting up the optimal structure.
It is shown that growth from modal interference can occur for as long as 18 months, which within the confines of this model defines a predictability limit for growth events. Growth associated with the stochastic forcing is also possible, but is unpredictable. The timescale on which the predictable and unpredictable components of SST growth become comparable to each other gives a more conservative predictability limit of 15 months.
The above scenario is based on empirical evidence obtained from SST anomalies alone. From the results of several tests based on statistical properties of linear and nonlinear dynamical systems, one may conclude that much of the ENSO cycle in nature is dominated by stable, forced dynamics.
Abstract
It is argued from SST observations for the period 1950–90 that the tropical Indo-Pacific ocean-atmosphere system may be described as a stable linear dynamical system driven by spatially coherent Gaussian white noise. Evidence is presented that the predictable component of SST anomaly growth is associated with the constructive interference of several damped normal modes after an optimal initial structure is set up by the white noise forcing. In particular, El Niño–Southern Oscillation (ENSO) growth is associated with an interplay of at least three damped normal modes, with periods longer than two years and decay times of 4 to 8 months, rather than the manifestation of a single unstable mode whose growth is arrested by nonlinearities. Interestingly, the relevant modes are not the three least damped modes of the system. Rather, mode selection, and the establishment of the optimal initial structure from which growth occurs, are controlled by the stochastic forcing. Experiments conducted with an empirical stochastic-dynamical model show that stochastic forcing not only adds energy to the system, but also plays a role in setting up the optimal structure.
It is shown that growth from modal interference can occur for as long as 18 months, which within the confines of this model defines a predictability limit for growth events. Growth associated with the stochastic forcing is also possible, but is unpredictable. The timescale on which the predictable and unpredictable components of SST growth become comparable to each other gives a more conservative predictability limit of 15 months.
The above scenario is based on empirical evidence obtained from SST anomalies alone. From the results of several tests based on statistical properties of linear and nonlinear dynamical systems, one may conclude that much of the ENSO cycle in nature is dominated by stable, forced dynamics.
Abstract
An assessment is made of the ability of the singular value decomposition (SYD) technique to recover the relationship between two variables x and y from a time series of their observations. It is shown that SVD is rigorously successful only in the special cases when either (i) the transformation linking x and y is orthogonal or (ii) the covariance matrix of either x or y is the identity matrix. The behavior of the method when theSE conditions are not met is also studied in a simple two-dimensional case.
That this caveat can be relevant in a meteorological context is demonstrated by performing an SVD analysis of a time series of global upper-tropospheric streamfunction and vorticity fields. Although these fields are linked by the two-dimensional Laplacian operator on the sphere, it is shown that the pairs of singular patterns resulting from the SVD analysis are not so related. The problem is apparent even for the first SVD pair and generally becomes worse for succeeding pairs These results suggest that any physical interpretation of SVD pairs may be unjustified.
Abstract
An assessment is made of the ability of the singular value decomposition (SYD) technique to recover the relationship between two variables x and y from a time series of their observations. It is shown that SVD is rigorously successful only in the special cases when either (i) the transformation linking x and y is orthogonal or (ii) the covariance matrix of either x or y is the identity matrix. The behavior of the method when theSE conditions are not met is also studied in a simple two-dimensional case.
That this caveat can be relevant in a meteorological context is demonstrated by performing an SVD analysis of a time series of global upper-tropospheric streamfunction and vorticity fields. Although these fields are linked by the two-dimensional Laplacian operator on the sphere, it is shown that the pairs of singular patterns resulting from the SVD analysis are not so related. The problem is apparent even for the first SVD pair and generally becomes worse for succeeding pairs These results suggest that any physical interpretation of SVD pairs may be unjustified.
Abstract
Any discussion of intraseasonal and interannual variability in the atmosphere must presume a reliable assessment of the observed variability. In spite of continued improvements in observing systems, quality control techniques, and data analysis schemes, however, and also because of them, this assessment remains difficult in the tropics.
In this paper the authors examine the mean tropical circulation during two Januarys, 1988 and 1989, as described by the circulation analyses produced at two weather prediction centers, the National Meteorological Center (NMC) in Washington, D.C., and the European Center for Medium-Range Weather Forecast (ECMWF) in Reading, England. In particular, the authors’ focus is on the change in the circulation between 1988 and 1989 as estimated by these two sets of analyses, especially the change in the 200-mb wind divergence associated with organized deep convection. The authors find that in many regions the discrepancy between thew estimates is of the order of the change itself. A comparison with maps of the outgoing longwave radiation (OLR) is not quantitatively useful in this regard.
One way out of this dilemma is to derive divergence fields that are consistent with the 200-mb vorticity balance. The authors do so by solving the “chi problem” of Sardeshmukh and Hoskins. Because the large-scale vorticity fields generated by NMC and ECMWF are highly correlated (∼98%), the divergence fields derived in this manner are also better correlated than the analyzed fields and enable a more reliable assessment of the observed change between these two periods.
Abstract
Any discussion of intraseasonal and interannual variability in the atmosphere must presume a reliable assessment of the observed variability. In spite of continued improvements in observing systems, quality control techniques, and data analysis schemes, however, and also because of them, this assessment remains difficult in the tropics.
In this paper the authors examine the mean tropical circulation during two Januarys, 1988 and 1989, as described by the circulation analyses produced at two weather prediction centers, the National Meteorological Center (NMC) in Washington, D.C., and the European Center for Medium-Range Weather Forecast (ECMWF) in Reading, England. In particular, the authors’ focus is on the change in the circulation between 1988 and 1989 as estimated by these two sets of analyses, especially the change in the 200-mb wind divergence associated with organized deep convection. The authors find that in many regions the discrepancy between thew estimates is of the order of the change itself. A comparison with maps of the outgoing longwave radiation (OLR) is not quantitatively useful in this regard.
One way out of this dilemma is to derive divergence fields that are consistent with the 200-mb vorticity balance. The authors do so by solving the “chi problem” of Sardeshmukh and Hoskins. Because the large-scale vorticity fields generated by NMC and ECMWF are highly correlated (∼98%), the divergence fields derived in this manner are also better correlated than the analyzed fields and enable a more reliable assessment of the observed change between these two periods.
Abstract
Single column models (SCMs) provide an economical framework for assessing the sensitivity of atmospheric temperature and humidity to natural and imposed perturbations, and also for developing improved representations of diabatic processes in weather and climate models. Their economy is achieved at the expense of ignoring interactions with the circulation dynamics; thus, advection by the large-scale flow is either prescribed or neglected. This artificial decoupling of the diabatic and adiabatic tendencies can often cause rapid error growth in SCM integrations, especially in the Tropics where large-scale vertical advection is important. As a result, SCMs can quickly develop highly unrealistic thermodynamic structures, making it pointless to study their subsequent evolution.
This paper suggests one way around this fundamental difficulty through a simple coupling of the diabatic and adiabatic tendencies. In essence, the local vertical velocity at any instant is specified by a formula that links the local vertical temperature advection to the evolution of SCM-generated diabatic heating rates up to that instant. This vertical velocity is then used to determine vertical humidity advection, and also horizontal temperature and humidity advection under an additional assumption that the column is embedded in a uniform environment. The parameters in the formula are estimated in a separate set of calculations, from the approach to equilibrium of a linearized global primitive equation model forced by steady heat sources. As a test, the parameterized dynamics are used to predict the linear model's local response to oscillating heat sources, and found to perform remarkably well over a wide range of space and time scales. In a second test, the parameterization is found to capture important aspects of a general circulation model's vertical advection and temperature tendencies and their lead–lag relationships with diabatic heating fluctuations at convectively active locations in the Tropics.
When implemented in the NCAR SCM, the dynamically coupled SCM shows a clear improvement over its uncoupled counterpart for tropical conditions observed during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). Coupling effectively stabilizes the SCM. As a result, short-term prediction errors are substantially reduced, the ensemble spread is reduced in ensemble runs, and the SCM is able to maintain realistic thermodynamic structures in extended runs. Such a dynamically coupled SCM should therefore be more useful not only for isolating physical parameterization errors in weather and climate models, but also for economical simulations of regional climate variability.
Abstract
Single column models (SCMs) provide an economical framework for assessing the sensitivity of atmospheric temperature and humidity to natural and imposed perturbations, and also for developing improved representations of diabatic processes in weather and climate models. Their economy is achieved at the expense of ignoring interactions with the circulation dynamics; thus, advection by the large-scale flow is either prescribed or neglected. This artificial decoupling of the diabatic and adiabatic tendencies can often cause rapid error growth in SCM integrations, especially in the Tropics where large-scale vertical advection is important. As a result, SCMs can quickly develop highly unrealistic thermodynamic structures, making it pointless to study their subsequent evolution.
This paper suggests one way around this fundamental difficulty through a simple coupling of the diabatic and adiabatic tendencies. In essence, the local vertical velocity at any instant is specified by a formula that links the local vertical temperature advection to the evolution of SCM-generated diabatic heating rates up to that instant. This vertical velocity is then used to determine vertical humidity advection, and also horizontal temperature and humidity advection under an additional assumption that the column is embedded in a uniform environment. The parameters in the formula are estimated in a separate set of calculations, from the approach to equilibrium of a linearized global primitive equation model forced by steady heat sources. As a test, the parameterized dynamics are used to predict the linear model's local response to oscillating heat sources, and found to perform remarkably well over a wide range of space and time scales. In a second test, the parameterization is found to capture important aspects of a general circulation model's vertical advection and temperature tendencies and their lead–lag relationships with diabatic heating fluctuations at convectively active locations in the Tropics.
When implemented in the NCAR SCM, the dynamically coupled SCM shows a clear improvement over its uncoupled counterpart for tropical conditions observed during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). Coupling effectively stabilizes the SCM. As a result, short-term prediction errors are substantially reduced, the ensemble spread is reduced in ensemble runs, and the SCM is able to maintain realistic thermodynamic structures in extended runs. Such a dynamically coupled SCM should therefore be more useful not only for isolating physical parameterization errors in weather and climate models, but also for economical simulations of regional climate variability.
Abstract
The equivalence of taking an isotropic, moving, spatial average of a two-dimensional field on the sphere to multiplying the coefficients in its spherical harmonics representation with factors that depend only on the total wavenumber n is discussed. Equivalent spatial averaging operators for several such spectral filters are displayed.
Abstract
The equivalence of taking an isotropic, moving, spatial average of a two-dimensional field on the sphere to multiplying the coefficients in its spherical harmonics representation with factors that depend only on the total wavenumber n is discussed. Equivalent spatial averaging operators for several such spectral filters are displayed.
Abstract
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.
Abstract
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.
Abstract
The effect of air–sea coupling on tropical climate variability is investigated in a coupled linear inverse model (LIM) derived from the simultaneous and 6-day lag covariances of observed 7-day running mean departures from the annual cycle. The model predicts the covariances at all other lags. The predicted and observed lag covariances, as well as the associated power spectra, are generally found to agree within sampling uncertainty. This validates the LIM’s basic premise that beyond daily time scales, the evolution of tropical atmospheric and oceanic anomalies is effectively linear and stochastically driven. It also justifies a linear diagnosis of air–sea coupling in the system.
The results show that air–sea coupling has a very small effect on subseasonal atmospheric variability. It has much larger effects on longer-term variability, in both the atmosphere and the ocean, including greatly increasing the amplitude of ENSO and lengthening its dominant period from 2 to 4 years. Consistent with these results, the eigenvectors of the system’s dynamical evolution operator also separate into two distinct, but nonorthogonal, subspaces: one governing the nearly uncoupled subseasonal dynamics and the other governing the strongly coupled longer-term dynamics. These subspaces arise naturally from the LIM analysis; no bandpass frequency filtering need be applied. One implication of this remarkably clean separation of the uncoupled and coupled dynamics is that GCM errors in anomalous tropical air–sea coupling may cause substantial errors on interannual and longer time scales but probably not on the subseasonal scales associated with the MJO.
Abstract
The effect of air–sea coupling on tropical climate variability is investigated in a coupled linear inverse model (LIM) derived from the simultaneous and 6-day lag covariances of observed 7-day running mean departures from the annual cycle. The model predicts the covariances at all other lags. The predicted and observed lag covariances, as well as the associated power spectra, are generally found to agree within sampling uncertainty. This validates the LIM’s basic premise that beyond daily time scales, the evolution of tropical atmospheric and oceanic anomalies is effectively linear and stochastically driven. It also justifies a linear diagnosis of air–sea coupling in the system.
The results show that air–sea coupling has a very small effect on subseasonal atmospheric variability. It has much larger effects on longer-term variability, in both the atmosphere and the ocean, including greatly increasing the amplitude of ENSO and lengthening its dominant period from 2 to 4 years. Consistent with these results, the eigenvectors of the system’s dynamical evolution operator also separate into two distinct, but nonorthogonal, subspaces: one governing the nearly uncoupled subseasonal dynamics and the other governing the strongly coupled longer-term dynamics. These subspaces arise naturally from the LIM analysis; no bandpass frequency filtering need be applied. One implication of this remarkably clean separation of the uncoupled and coupled dynamics is that GCM errors in anomalous tropical air–sea coupling may cause substantial errors on interannual and longer time scales but probably not on the subseasonal scales associated with the MJO.
Abstract
El Niño–Southern Oscillation (ENSO) is commonly viewed as a low-frequency tropical mode of coupled atmosphere–ocean variability energized by stochastic wind forcing. Despite many studies, however, the nature of this broadband stochastic forcing and the relative roles of its high- and low-frequency components in ENSO development remain unclear. In one view, the high-frequency forcing associated with the subseasonal Madden–Julian oscillation (MJO) and westerly wind events (WWEs) excites oceanic Kelvin waves leading to ENSO. An alternative view emphasizes the role of the low-frequency stochastic wind components in directly forcing the low-frequency ENSO modes. These apparently distinct roles of the wind forcing are clarified here using a recently released high-resolution wind dataset for 1990–2015. A spectral analysis shows that although the high-frequency winds do excite high-frequency Kelvin waves, they are much weaker than their interannual counterparts and are a minor contributor to ENSO development. The analysis also suggests that WWEs should be viewed more as short-correlation events with a flat spectrum at low frequencies that can efficiently excite ENSO modes than as strictly high-frequency events that would be highly inefficient in this regard. Interestingly, the low-frequency power of the rapid wind forcing is found to be higher during El Niño than La Niña events, suggesting a role also for state-dependent (i.e., multiplicative) noise forcing in ENSO dynamics.
Abstract
El Niño–Southern Oscillation (ENSO) is commonly viewed as a low-frequency tropical mode of coupled atmosphere–ocean variability energized by stochastic wind forcing. Despite many studies, however, the nature of this broadband stochastic forcing and the relative roles of its high- and low-frequency components in ENSO development remain unclear. In one view, the high-frequency forcing associated with the subseasonal Madden–Julian oscillation (MJO) and westerly wind events (WWEs) excites oceanic Kelvin waves leading to ENSO. An alternative view emphasizes the role of the low-frequency stochastic wind components in directly forcing the low-frequency ENSO modes. These apparently distinct roles of the wind forcing are clarified here using a recently released high-resolution wind dataset for 1990–2015. A spectral analysis shows that although the high-frequency winds do excite high-frequency Kelvin waves, they are much weaker than their interannual counterparts and are a minor contributor to ENSO development. The analysis also suggests that WWEs should be viewed more as short-correlation events with a flat spectrum at low frequencies that can efficiently excite ENSO modes than as strictly high-frequency events that would be highly inefficient in this regard. Interestingly, the low-frequency power of the rapid wind forcing is found to be higher during El Niño than La Niña events, suggesting a role also for state-dependent (i.e., multiplicative) noise forcing in ENSO dynamics.
