Abstract

The theory of empirical orthogonal functions (EOFs) is generalized in the context of normal modes on unidirectional sheared flows, at first using small disturbance to a monotonic potential vorticity basic state. A wave function is introduced (so called because of a partial analogy with quantum mechanical wave function), via the pseudomomentum, and is used to define the covariance matrix needed in an EOF analysis. The resulting new formalism comprises a fundamental, simpler and more physically insightful, version of EOF theory: it allows empirical reconstruction of the normal modes excited in an atmospheric time series, their respective variances, and phase speed relationship. This new approach permits quantitative and qualitative discussions of the underlying wave mechanisms present in a sheared flow. The theory is given for a hierarchy of models starting with the linearized quasigeostrophic equations. The extent to which these concepts carry over to nonlinear finite-amplitude disturbance is investigated. Included in these considerations are the nonlinear primitive equations in the hydrostatic approximation. The method is applied to 24 winters of the NMC dataset. The empirical results show the presence in the upper troposphere of normal modes, which oscillate in a statistical sense with their theoretically predicted natural frequencies. The normal modes are observed too with divergence e-folding times ranging between 2.5 and 4.5 days. The empirical normal-mode spectrum splits into a continuous and a discrete spectrum with oscillations of less than and greater than two weeks, respectively. The discrete spectrum is divided in forced meridional monopoles and meridional dipoles. In particular, the dipole with zonal wavenumber 2 shows a strong amplitude in phase space of the generalized EOF description for the North Atlantic blocking weather regimes.

Abstract

A new time invariant of the shallow-water model on the f plane for an isolated distribution of vorticity is obtained. The physical interpretation of the new invariant and its relation to past studies is discussed. It corresponds to a global nonlinear geostrophic balance of the localized distribution of vorticity, and defines a manifold in spectral space where the inertial frequency is filtered out exactly for all time. The author argues that if the initial distribution of vorticity is not on this manifold, it is not in balance. Thus, an exact realization of a slow manifold generated by a number of finite constraints is obtained. The new time invariant is obtained with no assumption on the Froude number, the Rossby number, and the amplitude characterizing the distribution of vorticity.

Abstract

Previous studies have shown that the Madden–Julian oscillation (MJO) has a global impact that may provide an important source of skill for subseasonal predictions. The extratropical response was found to be the strongest when the tropical diabatic heating has a dipole structure with convection anomaly centers of opposite sign in the eastern Indian Ocean and the western Pacific. A positive (negative) MJO dipole heating refers to that with heating (cooling) in the eastern Indian Ocean and cooling (heating) in the western Pacific. In this study, two aspects of the extratropical response to the MJO are examined: 1) nonlinearity, which answers the question of whether the response to a positive MJO dipole heating is the mirror image of that to a negative MJO, and 2) sensitivity to the initial state, which explores the dependence of the extratropical response on the initial condition of the westerly jet.

Ensemble integrations using a primitive-equation global atmospheric circulation model are performed with anomalous tropical thermal forcings that resemble a positive MJO (+MJO) and a negative MJO (−MJO). The response in the first week is largely linear. After that, significant asymmetry is found between the response in the positive MJO and the negative MJO. The 500-hPa negative geopotential height response in the North Pacific of the −MJO run is located about 30° east of the positive height response of the +MJO run. There is also an eastward shift of the extratropical wave train in the Pacific–North American region. This simulated nonlinearity is in agreement with the observations. The two leading response patterns among the ensemble members are identified by an empirical orthogonal function (EOF) analysis. EOF1 represents an eastward shift of the wave train, which is positively correlated with strengthening of the East Asian subtropical upper-troposphere westerly jet in the initial condition. On the other hand, EOF2 represents an amplification of the response, which is associated with a southward shift of the westerly jet in the initial state.

Abstract

The theory of empirical normal modes (ENMs) for a shallow water fluid is developed. ENMs are basis functions that both have the statistical properties of empirical orthogonal functions (EOFs) and the dynamical properties of normal modes. In fact, ENMs are obtained in a similar manner as EOFs but with the use of a quadratic form instead of the Euclidean norm. This quadratic form is a global invariant, the wave activity, of the linearized equations about a basic state. A general formulation is proposed for calculating normal modes from a generalized hermitian problem, even when the basic state is not zonal.

The projection coefficients of the flow onto a few leading ENWs generally have a more monochromatic behavior than that obtained for EOFS, which give them an intrinsically more predictable character. This property is illustrated by numerical experiments using the shallow water model on the sphere. It is shown, in particular, that the ENM coefficients, when used as predictors in a statistical linear model, provide better predictions of the behavior of the shallow water atmosphere than EOF coefficients. It is also shown that the choice of the basic state itself is crucial.

Abstract

The theory of empirical normal modes (ENMs) is adapted to diagnose gravity waves generated by a relatively high-resolution numerical model solving the primitive equations. The ENM approach is based on the principal component analysis (which consists of finding the most efficient basis explaining the variance of a time series), except that it takes advantage of wave-activity conservation laws. In the present work, the small-amplitude version of the pseudoenergy is used to extract from data quasi-monochromatic three-dimensional empirical modes that describe atmospheric wave activity. The spatial distributions of these quasi-monochromatic modes are identical to the normal modes of the linearized primitive equations when the underlying dynamics can be described with a stochastic linear and forced model, thus establishing a bridge between statistics and dynamics. This diagnostic method is used to study inertia–gravity wave generation, propagation, transience, and breaking over the Rockies, the North Pacific, and Central America in the troposphere–stratosphere–mesosphere Geophysical Fluid Dynamics Laboratory SKYHI general circulation model at a resolution of 1° of latitude by 1.2° of longitude. Besides the action of mountains in exciting orographic waves, inertia–gravity wave activity has been found to be generated at the jet stream level as a possible consequence of a sustained nonlinear and ageostrophic flow. In the tropical region of the model (Central America), the inertia–gravity wave source mechanism produced mainly waves with a westward vertical tilt. A significant proportion of these inertia–gravity waves was able to reach the model mesosphere without much dissipation and absorption.

Abstract

A diagnostic algorithm, based on the empirical normal mode decomposition technique, is proposed as a diagnostic tool in studies of the atmospheric variability. It begins by analyzing the transient eddies in terms of empirical modes that are orthogonal with respect to wave activities. Time-dependent amplitudes together with wave activity spectra are used to classify the modes and compute their propagation properties.

The algorithm is applied to a sequence of four Northern Hemisphere winters taken from the National Centers for Environmental Prediction reanalyses, with a focus on the upper troposphere and lower stratosphere, giving a set of empirical modes of wind, pressure, specific volume, and potential vorticity. Results indicate that most of the wave activity is carried by large-scale, eastward-propagating modes centered at middle and high latitudes. Some properties of the leading modes, such as their average phase speeds, are in good agreement with the predictions of linear dynamics.

Characteristics of the leading wavenumber-5 mode, such as its dipolar pressure pattern near the summer hemisphere tropopause, its propagation speed of 12 m s⁻¹ and decay rate of 3 days, can be explained by the theory of quasi modes, defined as superpositions of singular modes sharply peaked in the phase speed domain. Other large-scale, midlatitude modes also show properties compatible with the quasi-modal description, suggesting that quasi modes play an important role in the upper-troposphere dynamics.

Abstract

A long integration of a primitive equation dry atmospheric model with time-independent forcing under boreal winter conditions is analyzed. A variety of techniques such as time filtering, space–time spectral analysis, and lag regressions are used to identify tropical waves. It is evident that oscillations with intraseasonal time scales and a Kelvin wave structure exist in the model tropical atmosphere. Coherent eastward propagations in the 250-hPa velocity potential and zonal wind are found, with a speed of about 15 m s⁻¹. The oscillation is stronger in the Eastern Hemisphere than in the Western Hemisphere.

Interactions between the tropical and extratropical flows are found to be responsible for the simulated intraseasonal variability. Wave activity flux analysis reveals that a tropical influence occurs in the North Pacific region where a northeastward wave activity flux is found associated with the tropical divergent flow in the western and central Pacific. In the North Atlantic sector, on the other hand, a strong extratropical influence is observed with a southward wave activity flux into the Tropics. The extratropical low-frequency variability develops by extracting kinetic energy from the basic mean flow and through interactions with synoptic-scale transient eddies. Linear experiments show that the tropical atmospheric response to the extratropical forcing in the North Atlantic leads to an eastward-propagating wave in the tropical easterly mean flow of the Eastern Hemisphere.

Abstract

An algorithm based on the empirical normal mode analysis is used in a comparative study of the climatology and variability in dynamical-core experiments of the Global Environmental Multiscale model. The algorithm is proposed as a means to assess properties of the model's dynamical core and to establish objective criteria for model intercomparison studies. In this paper, the analysis is restricted to the upper troposphere and lower stratosphere. Two dynamical-core experiments are considered: one with the forcing proposed by Held and Suarez, later modified by Williamson et al. (called HSW experiment), and the other with a forcing inspired by the prescriptions of Boer and Denis (BD). Results are also compared with those of an earlier diagnosis of NCEP reanalyses. Normal modes and wave-activity spectra are similar to those found in the reanalysis data, although details depend on the forcing. For instance, wave-energy amplitudes are higher with the BD forcing, and an approximate energy equipartition is observed in the spectrum of wavenumber-1 modes in the NCEP data and the BD experiment but not in the HSW experiment. The HSW forcing has a relatively strong relaxation acting on the complete temperature field, whereas the BD forcing only acts on the zonal-mean temperature, letting the internal dynamics alone drive the wave-activity spectral cascade. This difference may explain why the BD forcing is more successful in reproducing the observed wave activity in the upper troposphere and lower stratosphere.

Abstract

The theory of empirical normal modes (ENMs) was applied in a diagnostic study of the inner spiral bands formed in a simulated hurricane using the high-resolution Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) nonhydrostatic mesoscale model version 5 (MM5). The ENM method has the capability to decompose simultaneously wind and thermal fields into dynamically consistent and orthogonal modes with respect to wave activities.

For wavenumber 1 and 2 anomalies, it was found that the leading modes are vortex Rossby waves. These modes explain 70%–80% of the statistical variances in a 24-h period. Gravity waves have small contribution in terms of wave activities.

Analysis of the Eliassen–Palm (EP) flux and its time-mean divergence shows that vortex Rossby waves are generated in the eyewall region where the radial gradient of the basic-state potential vorticity is large. In general, these waves propagate outward in the lower troposphere and inward in the upper troposphere. Consequently, they transport eddy momentum radially inward and outward, respectively. The wave activities also propagate slowly upward inside the eyewall and downward outside. The associated eddy heat transport tends to warm the air in the eye region. The vortex Rossby waves lead to, through wave–mean flow interaction as indicated by the divergence of the EP flux, an acceleration of the mean tangential wind in the lower and middle troposphere inside and outside the eyewall and a deceleration aloft in the eyewall region.

Abstract

Motivated by Dunkerton et al., a climatological study of 54 developing easterly waves in 1998–2001 was performed. Time-lagged composites in a translating reference frame following the disturbances indicate a weak meridional potential vorticity (PV) gradient of the easterly jet and a cyclonic critical layer located slightly to the south of the weak PV gradient, consistent with previous findings in the marsupial paradigm. Using a closed PV contour as a criterion for the formation of the cat’s-eye, it was shown that on average it takes ~2.6 days for open PV contours to transform to a closed coherent structure. Bootstrap analysis was then applied to determine the reliability of the easterly wave–like pattern in the composite perturbation PV analysis. It is suggested that the coexistence of a nonlinear critical layer and a region of weak meridional PV gradient over several days, found to occur in only ~25% of the easterly waves, might be a major factor to distinguish developing and nondeveloping disturbances. This finding may explain why only a small fraction of easterly waves contribute to tropical cyclogenesis. Additionally, an analytic time scale of the form was obtained, where Q is the mass sink, ε is the amplitude of the initial disturbance, and τ is the cat’s-eye formation time that governs the onset of nonlinearity for forced disturbances on a parabolic jet critical layer. This time scale is consistent with that found in 54 cases of easterly waves that developed into named storms, highlighting the importance of nonlinear and diabatic processes in cat’s-eye formation.