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- Author or Editor: Prashant D. Sardeshmukh x

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## Abstract

The baroclinic *χ* (chi) problem is the problem of diagnosing the three-dimensional distribution of large-scale vertical motion from the vorticity budget. A solution technique is developed in which a preliminary guess for the associated horizontal divergence field is refined iteratively until the budget is balanced. The advantage of diagnosing the vertical motion in this manner, especially in the tropics, is discussed.

An example of the process applied to six boreal winters of ECMWF data suggests several improvements over the diabatically initialized ECMWF analyses, such as much stronger ascent over the convective regions of Africa and South America and a more clearly defined ITCZ in the eastern Pacific. Diabatic heating fields, estimated as a balance requirement in the thermodynamic equation using these dynamically consistent vertical velocities, also seem more realistic. Some ideas on how to combine such heating estimates with rainfall or satellite information are presented.

The more general problem of adjusting both the vorticity and divergence fields minimally from a preliminary analysis so as to make them consistent with the vorticity budget is also considered. It is suggested from a scale argument that in most cases the adjustment to the vorticity field will be much smaller than that to the divergence field. The *χ* problem, in which one adjusts only the divergence field, thus provides a useful approximation to a rather more demanding general problem. Solutions to the general problem are nevertheless feasible, and have implications for the problem of four-dimensional data assimilation with dynamical constraints as well as the spinup problem in numerical weather prediction.

## Abstract

The baroclinic *χ* (chi) problem is the problem of diagnosing the three-dimensional distribution of large-scale vertical motion from the vorticity budget. A solution technique is developed in which a preliminary guess for the associated horizontal divergence field is refined iteratively until the budget is balanced. The advantage of diagnosing the vertical motion in this manner, especially in the tropics, is discussed.

An example of the process applied to six boreal winters of ECMWF data suggests several improvements over the diabatically initialized ECMWF analyses, such as much stronger ascent over the convective regions of Africa and South America and a more clearly defined ITCZ in the eastern Pacific. Diabatic heating fields, estimated as a balance requirement in the thermodynamic equation using these dynamically consistent vertical velocities, also seem more realistic. Some ideas on how to combine such heating estimates with rainfall or satellite information are presented.

The more general problem of adjusting both the vorticity and divergence fields minimally from a preliminary analysis so as to make them consistent with the vorticity budget is also considered. It is suggested from a scale argument that in most cases the adjustment to the vorticity field will be much smaller than that to the divergence field. The *χ* problem, in which one adjusts only the divergence field, thus provides a useful approximation to a rather more demanding general problem. Solutions to the general problem are nevertheless feasible, and have implications for the problem of four-dimensional data assimilation with dynamical constraints as well as the spinup problem in numerical weather prediction.

## 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

The momentum budget for January 1987 is evaluated with global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The dissipation term is diagnosed from the budget as a balance requirement, that is, as that required to balance the sum of the advection, Coriolis, pressure gradient, and local tendency terms. This is then compared with the parameterized subgrid-scale effects in the ECMWF model's momentum equation, with a view of identifying possible errors in those parameterizations.

The balance requirement does not support the high parameterized values of orographically induced gravity-wave drag in the lower stratosphere. A deeper analysis also does not suggest a major role for turbulent vertical transports above the boundary layer. On the other hand, our budget does indicate that more effort be spent on a better representation of the potential enstrophy cascade associated with Rossby wave breaking in the upper troposphere. These statements are qualified by the errors in the balance requirement itself. The extent to which this is a problem is discussed.

A distinctive feature of these calculations is their internal consistency., that is, all the terms in the budget are evaluated as in the version of the ECMWF model used for assimilating the observations. This offers several advantages, not the least of which is that it makes our budget residuals identical to the systematic initial tendency errors of the operational weather forecasts, thus facilitating their computation and routine monitoring. As such, our calculations explain a large fraction of the systematic short-range forecast errors and, because of their local character, provide clues as to the possible sources of those errors. Experiments with and without gravity-wave drag are described to illustrate its large contribution during this period to the southerly wind error of the operational weather forecasts at 70 mb over western North America.

## Abstract

The momentum budget for January 1987 is evaluated with global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The dissipation term is diagnosed from the budget as a balance requirement, that is, as that required to balance the sum of the advection, Coriolis, pressure gradient, and local tendency terms. This is then compared with the parameterized subgrid-scale effects in the ECMWF model's momentum equation, with a view of identifying possible errors in those parameterizations.

The balance requirement does not support the high parameterized values of orographically induced gravity-wave drag in the lower stratosphere. A deeper analysis also does not suggest a major role for turbulent vertical transports above the boundary layer. On the other hand, our budget does indicate that more effort be spent on a better representation of the potential enstrophy cascade associated with Rossby wave breaking in the upper troposphere. These statements are qualified by the errors in the balance requirement itself. The extent to which this is a problem is discussed.

A distinctive feature of these calculations is their internal consistency., that is, all the terms in the budget are evaluated as in the version of the ECMWF model used for assimilating the observations. This offers several advantages, not the least of which is that it makes our budget residuals identical to the systematic initial tendency errors of the operational weather forecasts, thus facilitating their computation and routine monitoring. As such, our calculations explain a large fraction of the systematic short-range forecast errors and, because of their local character, provide clues as to the possible sources of those errors. Experiments with and without gravity-wave drag are described to illustrate its large contribution during this period to the southerly wind error of the operational weather forecasts at 70 mb over western North America.

## Abstract

The impact of the climatological seasonally varying 300-mb flow on the North Pacific/North American response to remote anomalous forcing is considered in the context of a linear barotropic model. WKB theory suggests that the total wavenumber of stationary Rossby waves over the Pacific increases from about 7 in January to 8.5 by June, with the reverse occurring during fall. This change is accompanied by monthly changes in the location and shape of the Rossby waveguide itself. Using a diagnostic tool called the influence function, it is shown that the most sensitive area of forcing for producing a large response over the United States shifts from the east Pacific in late winter to the west Pacific by late spring. As spring progresses, there is also a marked increase in the sensitivity to smaller-scale forcing in both of these regions, particularly the west Pacific. The amplitude of the forced response can potentially be larger in June than any other month of the year. These results suggest that the evolution of extreme springtime weather events over North America may depend critically upon the precise timing and geographical structure of forcing anomalies over both the east and the west Pacific.

In this model, low-frequency variability within and downstream of the Rossby waveguide is sensitive to the annual cycle of the ambient flow. This suggests that the impact of the annual cycle must be taken into account in any complete theory of low-frequency variability. The impact is large enough to raise the possibility of significant interactions across timescales. In other words, it is possible for a steady forcing to produce an unsteady response and, equally, for an unsteady forcing to produce a seasonal-mean response. In such situations, particularly during the northern spring and fall seasons, investigating low-frequency anomalies as departures from three-month seasonal climatologies may lead to confusion and may not be useful.

## Abstract

The impact of the climatological seasonally varying 300-mb flow on the North Pacific/North American response to remote anomalous forcing is considered in the context of a linear barotropic model. WKB theory suggests that the total wavenumber of stationary Rossby waves over the Pacific increases from about 7 in January to 8.5 by June, with the reverse occurring during fall. This change is accompanied by monthly changes in the location and shape of the Rossby waveguide itself. Using a diagnostic tool called the influence function, it is shown that the most sensitive area of forcing for producing a large response over the United States shifts from the east Pacific in late winter to the west Pacific by late spring. As spring progresses, there is also a marked increase in the sensitivity to smaller-scale forcing in both of these regions, particularly the west Pacific. The amplitude of the forced response can potentially be larger in June than any other month of the year. These results suggest that the evolution of extreme springtime weather events over North America may depend critically upon the precise timing and geographical structure of forcing anomalies over both the east and the west Pacific.

In this model, low-frequency variability within and downstream of the Rossby waveguide is sensitive to the annual cycle of the ambient flow. This suggests that the impact of the annual cycle must be taken into account in any complete theory of low-frequency variability. The impact is large enough to raise the possibility of significant interactions across timescales. In other words, it is possible for a steady forcing to produce an unsteady response and, equally, for an unsteady forcing to produce a seasonal-mean response. In such situations, particularly during the northern spring and fall seasons, investigating low-frequency anomalies as departures from three-month seasonal climatologies may lead to confusion and may not be useful.

## Abstract

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.

## Abstract

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.

## Abstract

The steady linear response of a spherical baroclinic atmosphere to an equatorial diabatic heat source having a simple horizontal and vertical structure is examined. This source is imposed upon representative zonally symmetric as well as zonally varying flows during the boreal winter. Two climatologies are considered. One is a 6-year average of global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The other is a 30-year average, taken from a general circulation model (GCM) run at the Geophysical Fluid Dynamics Laboratory in Princeton.

The extratropical response is found to be very sensitive to the basic state around which the governing primitive equations are linearized, and in the case of the ECMWF climatology, to the longitudinal position of the source with respect to the climatological waves. There is also some sensitivity to the vertical level of maximum heating, although again this is more evident in the case of the ECMWF basic state.

These results are discussed in terms of simple theoretical ideas, and implications are drawn for the short-range climate prediction problem. The evidence presented here suggests that subtle differences in the ambient flow can give rise to very different low-frequency normal modes, and thence to drastically different responses to tropical perturbations imposed upon that flow.

## Abstract

The steady linear response of a spherical baroclinic atmosphere to an equatorial diabatic heat source having a simple horizontal and vertical structure is examined. This source is imposed upon representative zonally symmetric as well as zonally varying flows during the boreal winter. Two climatologies are considered. One is a 6-year average of global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The other is a 30-year average, taken from a general circulation model (GCM) run at the Geophysical Fluid Dynamics Laboratory in Princeton.

The extratropical response is found to be very sensitive to the basic state around which the governing primitive equations are linearized, and in the case of the ECMWF climatology, to the longitudinal position of the source with respect to the climatological waves. There is also some sensitivity to the vertical level of maximum heating, although again this is more evident in the case of the ECMWF basic state.

These results are discussed in terms of simple theoretical ideas, and implications are drawn for the short-range climate prediction problem. The evidence presented here suggests that subtle differences in the ambient flow can give rise to very different low-frequency normal modes, and thence to drastically different responses to tropical perturbations imposed upon that flow.

## 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

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 relative impacts of tropical diabatic heating and stratospheric circulation anomalies on wintertime extratropical tropospheric variability are investigated in a linear inverse model (LIM) derived from the observed zero lag and 5-day lag covariances of 7-day running mean departures from the annual cycle. The model predicts the covariances at all other lags. The predicted and observed lag covariances are generally found to be in excellent agreement, even at the much longer lag of 21 days. This validates the LIM’s basic premise that the dynamics of weekly averages are effectively linear and stochastically driven, which justifies further linear diagnosis of the system.

Analysis of interactions among the LIM’s variables shows that tropical diabatic heating greatly enhances persistent variability over most of the Northern Hemisphere, especially over the Pacific Ocean and North America. Stratospheric effects are largely confined to the polar region, where they ensure that the dominant pattern of sea level pressure variability is the annular Arctic Oscillation rather than the more localized North Atlantic Oscillation. Over the North Atlantic, both effects are important, although some of the stratospheric influence is ultimately traceable to tropical forcing. In general, the tropically forced anomalies extend through the depth of the troposphere and into the stratosphere, whereas stratospherically generated anomalies tend to be largest at the surface and relatively weak at midtropospheric levels. Some persistent variability is, however, found even in the absence of these “external” forcings, especially near the amplitude maxima of the leading eigenmodes of the internal extratropical tropospheric evolution operator. One of these eigenmodes has a circumglobal zonal wavenumber-5 structure with maxima over the Arabian Sea and the central Pacific, and two others are associated with north–south dipole variations across the North Atlantic jet. Overall, tropical influences are generally found to be larger than stratospheric influences on extratropical tropospheric variability and have a pronounced impact on the persistent, and therefore the potentially predictable, portion of that variability.

## Abstract

The relative impacts of tropical diabatic heating and stratospheric circulation anomalies on wintertime extratropical tropospheric variability are investigated in a linear inverse model (LIM) derived from the observed zero lag and 5-day lag covariances of 7-day running mean departures from the annual cycle. The model predicts the covariances at all other lags. The predicted and observed lag covariances are generally found to be in excellent agreement, even at the much longer lag of 21 days. This validates the LIM’s basic premise that the dynamics of weekly averages are effectively linear and stochastically driven, which justifies further linear diagnosis of the system.

Analysis of interactions among the LIM’s variables shows that tropical diabatic heating greatly enhances persistent variability over most of the Northern Hemisphere, especially over the Pacific Ocean and North America. Stratospheric effects are largely confined to the polar region, where they ensure that the dominant pattern of sea level pressure variability is the annular Arctic Oscillation rather than the more localized North Atlantic Oscillation. Over the North Atlantic, both effects are important, although some of the stratospheric influence is ultimately traceable to tropical forcing. In general, the tropically forced anomalies extend through the depth of the troposphere and into the stratosphere, whereas stratospherically generated anomalies tend to be largest at the surface and relatively weak at midtropospheric levels. Some persistent variability is, however, found even in the absence of these “external” forcings, especially near the amplitude maxima of the leading eigenmodes of the internal extratropical tropospheric evolution operator. One of these eigenmodes has a circumglobal zonal wavenumber-5 structure with maxima over the Arabian Sea and the central Pacific, and two others are associated with north–south dipole variations across the North Atlantic jet. Overall, tropical influences are generally found to be larger than stratospheric influences on extratropical tropospheric variability and have a pronounced impact on the persistent, and therefore the potentially predictable, portion of that variability.

## 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 δ**L** 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 δ**L** 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.