Zonal Mean Fields Research Articles

Formation Mechanism of the Simulated SPCZ and Baiu Front Using a Regional Climate Model

Abstract The formation mechanisms of both the South Pacific convergence zone (SPCZ) and the baiu front are investigated using a regional climate model. Some idealistic numerical experiments are carried out assuming zonally uniform and temporally constant atmospheric fields obtained from ECMWF analysis data as initial and lateral boundary conditions. A rainfall zone similar to the SPCZ is reproduced using a zonal mean atmospheric field of the Southern Hemisphere (SH) summer. The simulated SPCZ in the idealized model framework is highly sensitive to the variation of SST during 1997–98 in a manner similar to observation. The SPCZ is extremely weak in an experiment under the zonal mean field of the Northern Hemisphere (NH) early summer. Experiments with a different intensity of zonal wind speed and baroclinicity suggest that a mild zonal wind (weak baroclinicity) weakens the precipitation of the SPCZ and even occasionally suppresses precipitation when it is too weak. The heat contrast between the Australian c...

Meridional Localization of Planetary Waves in Stochastic Zonal Flows

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.

Numerical Study on the Baiu Front Genesis by Heating Contrast between Land and Ocean.

The formation mechanism of the Baiu front, which appears in early summer and gives a typical rain band around Japan, was reporduced using a regional atmospheric model. The initial and boundary conditions of the model were derived from the European Center for Medium-Range Weather Forecasts (ECMWF) data. In the “realistic simulation”, temporary variable boundary conditions derived from twice-daily ECMWF data were utilized. In the “zonal mean simulation”, zonally uniform and temporally constant atmospheric fields obtained from ten-day averaged global zonal mean ECMWF data was utilized as the initial and boundary conditions. The simulated Baiu front in the zonal mean simulations, as well as that in the realistic simulations, has similar structures to the real Baiu front, i.e., the Low-Level Jet (LLJ) runs parallel to both the precipitation zone and the upper-level jet in the eastern part. Basically, the simulated Baiu front is formed by the deformation of the zonal mean field due to the Land/Sea contrast and topography. Although the distribution of the simulated rainfall over the Baiu front depends on the cumulus convective parameterizations, the fundamental structure of the Baiu front does not depend on them. A comparison between the zonal mean simulations of early and late June indicates that the Baiu front is formed at a higher latitudes in late June, when the upper-level jet is weak and located northward. The location of the Baiu front is quite sensitive to the zonal mean field. The Baiu front accompanied by the LLJ is also represented by numerical experiments without topography, which suggests that the Baiu front could be reproduced by two factors alone, the zonal mean field and the Land/Sea contrast. The orography, including the Tibetan plateau, intensifies the LLJ and the precipitation over the Baiu front. The LLJ also appears in the zonal mean simulation without a condensation process. However, the LLJ is formed along the eastern coast of the Eurasian continent and locates in the northern side of the upper-level jet eastward of Japan, which is a different feature from the zonal mean simulations with a condensation process. Accordingly, it is speculated that the condensation process acting on the atmospheric field is necessary to keep the LLJ in the southern side of the upper-level jet as in the real Baiu front.

The Influence of Wave– and Zonal Mean–Ozone Feedbacks on the Quasi-biennial Oscillation

The effects of wave and zonal mean ozone heating on the evolution of the quasi-biennial oscillation (QBO) are examined using a two-dimensional mechanistic model of the equatorial stratosphere. The model atmosphere is governed by coupled equations for the zonal mean and (linear) wave fields of ozone, temperature, and wind, and is driven by specifying the amplitudes of a Kelvin wave and a Rossby‐gravity wave at the lower boundary. Wave‐mean flow interactions are accounted for in the model, but not wave‐wave interactions. A reference simulation (RS) of the QBO, in which ozone feedbacks are neglected, is carried out and the results compared with Upper Atmosphere Research Satelliteobservations. The RS is then compared with three model experiments, which examine separately and in combination the effects of wave ozone and zonal mean ozone feedbacks. Wave‐ozone feedbacks alone increase the driving by the Kelvin and Rossby‐gravity waves by up to 10%, producing stronger zonal wind shear zones and a stronger meridional circulation. Zonal mean‐ ozone feedbacks (ozone QBO) alone decrease the magnitude of the temperature QBO by up to 15%, which in turn affects the momentum deposition by the wave fields. Overall, the zonal mean‐ozone feedbacks increase the magnitude of the meridional circulation by up to 30%. The combined effects of wave‐ozone and ozone QBO feedbacks generally produce a larger response then either process alone. Moreover, these combined ozone feedbacks produce a temperature QBO amplitude that is up to 30% larger than simulations without the feedbacks. Correspondingly, significant changes are also observed in the zonal wind and ozone QBOs. When ozone feedbacks are included in the model, the Kelvin and Rossby‐gravity wave amplitudes can be reduced by ;10% and still produce a QBO similar to simulations without ozone.

Cane-ZebiakモデルにおけるENSOの位相遷移と成長に果たす東西移流フィードバックの役割

The turnabout and growth mechanisms of the ENSO are diagnostically studied by analyzing the SST budget of the Cane-Zebiak model. The SST change rates, which are directly linked to the phase transition and growth of the ENSO, are attributed to two processes: the anomalous vertical advection of subsurface temperature by the mean upwelling (thermocline feedback), and the zonal advection of the climatological mean SST by the anomalous zonal current (zonal advection feedback). The contributions to the phase transition, and growth of the ENSO, can be systematically separated by decomposing the equatorial thermocline depth anomaly, and zonal current anomaly into their zonal mean and zonal contrast fields. It is found that the thermocline feedback, associated with the zonal mean thermocline depth anomaly, and the zonal advection feedback by the equatorial zonal mean zonal current anomaly are responsible for the phase transition of the ENSO. Those associated with zonal contrast fields are responsible for the growth of the ENSO. The two processes in the SST change contribute to the phase transition and growth of the ENSO in an almost equally significant manner. They are closely related, as a result of the geostrophic balance between the meridional gradient of the thermocline depth and the zonal current. These findings suggest that the conceptual understanding of the ENSO in the Cane-Zebiak model, should include both of these two processes.

夏のアジアモンスーン循環と熱源の季節変化

The Asian summer monsoon circulation, especially its climatological seasonal change, was studied as the model response to the prescribed zonal mean field and the prescribed diabatic heat source from the observation. The obtained results are summarized as follows. (1) During the Asian summer monsoon season, the prescribed deep heat sources in the southern part of Asia form the Tibetan High, the monsoon trough, the low-level circulation over South Asia, and furthermore, the downward motion in the western part of the Eurasian Continent. The heat sources near the surface over central Asia also induce downward motions aloft. (2) In early summer (June), the deep heat sources in the southern part of Asia tend to form southwesterly low-level flows and upward motion southeast of Japan. Those are considered to be the background for the Baiu formation in East Asia as well as heat lows produced in the southern part of Asia. The mid-latitude heat sources associated with the Baiu precipitation produce a low-level jet south of that. (3) Climatological seasonal change from early summer (June) to mid-summer (July) is characterized by an air temperature increase in the whole Northern Hemisphere and a northward shift of a weakened westerly jet. When in the model a zonal mean field in June is replaced by that in July, the major characteristics of the seasonal change are obtained qualitatively; low-level jets and upward motion areas in South Asia and East Asia shift from the ocean side of the coasts toward the land side. This change of vertical motion is consistent with the seasonal change of deep heat sources from June to July. (4) The climatological seasonal change from mid-summer (July) to late summer (August) is characterized by enhanced convective activity in the extended area of the subtropical western Pacific. When deep heat sources in July are replaced by those in August over the western Pacific only, the major characteristics of the seasonal change over the Pacific and the Indian Ocean are obtained. The expansion of the Tibetan High at the upper-level and the Pacific High at low-level over Japan is also simulated by the seasonal change of the western Pacific heat sources only. (5) The model simulation with the combination of the diabatic heat source for August and the zonal mean field for June is compared with the climatological August simulation. It is indicated that the zonal mean field delayed from its seasonal migration could be related to weak monsoon circulation and the associated precipitation anomalies in the mid-latitudes and the subtropics.

Distribution and seasonal variation of CFCS in the stratosphere: Comparison of satellite global data and a 2-D model

The Cryogenic Limb Array Etalon Spectrometer (CLAES) on the NASA Upper Atmosphere Research Satellite obtained global distributions of CF 2Cl 2 and CFCl 3 in the stratosphere between October 1991 and May 1993. This paper discusses the seasonal and interannual behavior of a number of prominent dynamical features observed in the zonal mean fields for both gases over the period January 1992 through February 1993 and compares these features with a two-dimensional model. The CFC fields are shown to exhibit many of the same seasonally variant features seen in other conserved tracer fields. These include a semiannual oscillation signature in the tropical upper-level mixing ratios; the development of a double-peaked tropical structure near the vernal equinox and its weakening, or absence, at the autumnal equinox; steep descent at winter high latitudes in both hemispheres with an associated midlatitude surf zone, which is stronger and more sustained in the southern hemisphere; and sharp subtropical gradients in the mixing ratio isopleths in the northern winter hemisphere. Distinct interannual variability is also observed in the CF 2Cl 2 morphology, particularly for the northern hemisphere for the periods January–February 1992 and 1993. Other than this latter feature, the two-dimensional model is generally found to reproduce the observed behavior, although usually with weaker manifestation than seen in the data, especially in regard to the double-peaked equinoctial structure.

Global atmospheric responses to Antarctic forcing

To evaluate the greatest impact that sea-ice anomalies around Antarctica could have on the global atmosphere, 15 year seasonal cycle simulations are conducted with the U.S. National Center for Atmospheric Research Community Climate Model version 2.1. Sensitivity simulations are performed with the following conditions: (1) all sea ice in the Southern Hemisphere is replaced by year-round open water, but the permanent ice shelves are retained (NSIS); and (2) all sea ice in the Southern Hemisphere and the major ice shelves are removed and replaced by open water (NISH). The results are compared to a standard run (CNT) with boundary conditions set for the present climate. The comparison shows that trains of positive and negative anomalies in zonal-mean fields extend into the tropical latitudes of the Northern Hemisphere. Anomalies are largest during April-October. The additional removal of the ice shelves in NISH enhances the response, as zonally averaged anomalies are similar in pattern to those in NSIS but are roughly twice as large poleward of 50° S, and only slightly larger farther north. Anomalies in the eddy fields are found in both hemispheres. in NISH, and to a lesser degree in NSIS. these anomalies appear to be related to a delayed northern advance over China during June of the rain front associated with the summer monsoon. Consequently, precipitation is enhanced in middle and southern China and decreased in northern China. Observational analyses have also found links between Antarctic sea-ice variations and modulations of the East Asian monsoon.

Dynamical Evolution of the Northern Stratosphere in Early Winter 1991/92, as Observed by the Improved Stratospheric and Mesospheric Sounder

Abstract Dynamical fields based on temperature measurements from the Improved Stratospheric and Mesospheric Sounder on the Upper Atmosphere Research Satellite are presented for the Northern Hemisphere stratosphere for the period 28 October 1991 through 18 January 1992. Interpretation of these fields gives a picture of the dynamical evolution of this period in terms of the zonal-mean fields and the synoptic structures. Among the features of interest are the movements of the zonal-mean jets and several periods of stratospheric warming, culminating in a near-major warming in January.

Acceleration, heating, and compositional mixing of the thermosphere due to upward propagating tides

The National Center for Atmospheric Research thermosphere‐ionosphere general circulation model (TIGCM) is utilized to evaluate the role of upward propagating diurnal and semidiurnal tides in determining the zonal mean states of the thermosphere and ionosphere above 100 km. Differences between various zonal mean fields from TIGCM simulations are examined with and without tidal forcing at the lower boundary. The following effects are observed: (1) A westward jet of the order of 10–30 m s−1 in the equatorial lower thermosphere and weaker eastward flanking mid‐latitude jets are induced by eddy and molecular dissipation of the tidal motions. (2) Salient characteristics of the equatorial jet are independent of tidal phase. (3) Increases in the zonal mean N2 and O2 number densities of the order of 10–30% are produced above 110 km and between ± 30° latitude, mostly by virtue of the net heating due to tidal dissipation; these are accompanied by electron density depletions of the order of 10–15% above about 160 km due to the enhanced chemical loss. (4) A low‐latitude (< 30°) depletion (∼ 30–50%) of atomic oxygen occurs due to the increase in effective recombination rate within a tidally displaced air parcel. The latter may explain the low‐latitude depression in the OI (5577 Å) airglow intensity from satellite‐based observations. In addition, variations in the zonal mean densities of NO, N(4S), and N(2D) of the order of 30–60% are induced by reactions with the tidally modified distributions of O, O2, and temperature. The occurrence frequency of these effects is influenced by the amplitude of diurnal propagating tide assumed at the lower boundary of the TIGCM, which according to radar observations is exceeded about 20–40% of the time, depending on season.

Baroclinic waves in a shallow rotating annulus with temperature gradients imposed on the horizontal boundaries

Abstract A numerical investigation is made of flow and heat transfer characteristics of baroclinic waves using a new rotating cylindrical annulus, referred to as the Miller-Fowlis (MF) model. Special features of this annulus are: (1) the aspect ratio (height to width) is small, (2) the thermal gradients are imposed on the horizontal boundaries. In an effort to expand upon the previous experimental measurements, comprehensive numerical data are acquired for the finite-amplitude wave-present flow regime. Full three-dimensional Navier-Stokes equations are solved by utilizing a well established pseudospectral numerical technique. Three specific case studies are performed by using the parameter values which are selected by consulting the experimentally-obtained regime diagrams. The three-dimensional computations successfully captured the existence of finite-amplitude waves. Salient features of three-dimensional flow and thermal fields are presented. Both the zonal mean and deviatoric fields of essential flow variables are displayed and pertinent physical interpretations are attempted. Comprehensive examinations of flow properties and energetics are undertaken; post processing analyses and discussions are rendered. In the present MF model, the sidewall layers are passive in nature; the dominant character of global flow is determined by the Ekman layers. The zonal mean flow field is characterized by a triple-cell structure which consists of two dominant cells and a weak central cell. The similarities and discrepancies between the results of the present model and of the conventional rotating annulus model are delineated.

Tide/gravity-wave/mean-flow interactions in the mesosphere and lower thermosphere

Interactions between the solar diurnal tide, semidiurnal tide, internal gravity waves and mean zonal winds in the mesosphere and lower thermosphere are investigated using a quasi-nonlinear time dependent numerical model under solstice conditions. In the model, a coupled system of zonally-averaged nonlinear equations and perturbation equations for tides, including a parameterization for gravity wave propagation, is integrated simultaneously making allowance for mutual interactions. It is shown that the propagation of internal gravity waves is affected not only by the mean zonal wind but also by the tidal waves, and the tidal wave fields as well as the mean zonal winds in the upper mesosphere and lower thermosphere weaken as a result of the momentum deposition due to gravity waves. It is also shown that the solar tides make a significant contribution to the generation of the zonal mean fields, especially an equatorial easterly jet and cellular meridional circulation structures in the upper mesosphere and lower thermosphere. Preliminary results of full nonlinear calculations of the solar tides using a general circulation model, in which the tidal wave breakdown is simulated by a convective adjustment, are also presented.

The Stationary Response to Large-Scale Orography in a General Circulation Model and a Linear Model

Abstract Stationary waves generated over orography in a linear model and a general circulation model (GCM) are compared to examine how the atmosphere's response is established for small mountains and how linear theory breaks down over large orographic features. Both models have nine vertical levels and are low-resolution (R15) spectral models. The linear model solves the stationary linear primitive equations. The GCM's control integration uses zonally uniform and hemispherically symmetric boundary conditions, with a global swamp surface. Five experiments are performed by perturbing the GCM with Gaussian mountains of various heights introduced in midlatitudes. The stationary wave model is linearized about zonal mean fields from the GCM climatology. The linear model's response to a Gaussian mountain at 45°N latitude is dominated by a single wave train radiating toward the southeast. For mountain heights between 0.7 and 2 km, the GCM's stationary waves are similar to the linear model response to orography, a...

Baroclinic waves propagating from a high-latitude source

A time-dependent primitive-equation model of the southern hemisphere winter troposphere with an idealized dome-shaped continent centred at 80°S has been used to assess the potential of Antarctica to force stationary waves and to determine the location of the mid-latitude baroclinic storm track. Several integrations of the model for two hundred days were made with the zonal-mean fields fixed at the climatological values. In the time mean of each run an anticyclonic wave response formed over the mountain and a planetary wave propagated into the mid latitudes with substantial amplitude. This suggests that much of the observed stationary asymmetry in the southern, mid and high latitudes may be due to Antarctica. The response to the Antarctic forcing was different in a steady, linear version of the model. However, interaction between the steady waves appears to be of minor importance to the time-mean pattern. Rather, E-vector fields point to a substantial effect of the transients on the pattern. In particular, the low-frequency transients appear to be important in the balance of the time-mean equations near the mountain.

The Atmosphere’s Response to the Ice Sheets of the Last Glacial Maximum

This paper discusses some modeling results that indicate how the atmospheric response to the topography of the continental ice of the Last Glacial Maximum (LGM) may be related to the cold North Atlantic Ocean of that time. Broccoli and Manabe (1987) used a three-dimensional general circulation model (GCM) of the atmosphere coupled with a fixed-depth, static ocean mixed-layer model with ice-age boundary conditions to investigate the individual influences of the CLIMAP ice sheets, snow-free land albedos, and reduced atmospheric CO2 concentrations. They found that the ice sheets are the most influential of the ice-age boundary conditions in modifying the northern hemisphere climate, and that the presence of continental ice sheets alone leads to cooling over the North Atlantic Ocean.One approach for extending these GCM results is to consider the stationary waves generated by the ice sheets. Cook and Held (1988) showed that a linearized, steady-state, primitive equation model can give a reasonable simulation of the GCM’s stationary waves forced by the Laurentide ice sheet. The linear model analysis suggests that the mechanical effect of the changed slope of the surface, and not changes in the diabatic heating (e.g. the high surface albedos) or time-dependent transports that necessarily accompany the ice sheet in the GCM, is largely responsible for the ice sheet’s influence. To obtain the ice-age stationary-wave simulation, the linear model must be linearized about the zonal mean fields from the GCM’s ice-age climate. This is the case because the proximity of the cold polar air to the region of adiabatic heating on the downslope of the Laurentide ice sheet is an important factor in determining the stationary waves. During the ice age, cold air can be transported southward to balance this downslope heating by small perturbations in the meridional wind, consistent with linear theory. Since the meridional temperature gradient is more closely related to the surface albedo (ice extent) than to the ice volume, this suggests a mechanism by which changes in the stationary waves and, therefore, their cooling influence at low levels over the North Atlantic Ocean, can occur on time scales faster than those associated with large changes in continental ice volume.

The Atmosphere’s Response to the Ice Sheets of the Last Glacial Maximum

This paper discusses some modeling results that indicate how the atmospheric response to the topography of the continental ice of the Last Glacial Maximum (LGM) may be related to the cold North Atlantic Ocean of that time. Broccoli and Manabe (1987) used a three-dimensional general circulation model (GCM) of the atmosphere coupled with a fixed-depth, static ocean mixed-layer model with ice-age boundary conditions to investigate the individual influences of the CLIMAP ice sheets, snow-free land albedos, and reduced atmospheric CO2 concentrations. They found that the ice sheets are the most influential of the ice-age boundary conditions in modifying the northern hemisphere climate, and that the presence of continental ice sheets alone leads to cooling over the North Atlantic Ocean. One approach for extending these GCM results is to consider the stationary waves generated by the ice sheets. Cook and Held (1988) showed that a linearized, steady-state, primitive equation model can give a reasonable simulation of the GCM’s stationary waves forced by the Laurentide ice sheet. The linear model analysis suggests that the mechanical effect of the changed slope of the surface, and not changes in the diabatic heating (e.g. the high surface albedos) or time-dependent transports that necessarily accompany the ice sheet in the GCM, is largely responsible for the ice sheet’s influence. To obtain the ice-age stationary-wave simulation, the linear model must be linearized about the zonal mean fields from the GCM’s ice-age climate. This is the case because the proximity of the cold polar air to the region of adiabatic heating on the downslope of the Laurentide ice sheet is an important factor in determining the stationary waves. During the ice age, cold air can be transported southward to balance this downslope heating by small perturbations in the meridional wind, consistent with linear theory. Since the meridional temperature gradient is more closely related to the surface albedo (ice extent) than to the ice volume, this suggests a mechanism by which changes in the stationary waves and, therefore, their cooling influence at low levels over the North Atlantic Ocean, can occur on time scales faster than those associated with large changes in continental ice volume.

Effects of solar tides on the zonal mean circulation in the lower thermosphere: solstice condition

Effects of momentum deposition due to solar diurnal and semi-diurnal tidal waves on the zonal mean circulation in the mesosphere and lower thermosphere for a solstice condition are discussed. In the present model, the system of zonally averaged equations and the system of perturbation equations are integrated simultaneously, so that the propagation of tidal waves is affected not only by the basic mean fields but also by the induced zonal mean fields due to the momentum deposition. Results for two different vertical eddy diffusion profiles are presented. It is shown that the solar tides make a significant contribution to the generation of the mean zonal winds in the upper mesosphere and the lower thermosphere. Below 120 km the main contribution is due to propagating diurnal tides, while above 120 km it is due to semidiurnal tides.

Chemistry and transport in a three‐dimensional stratospheric model: Chlorine species during a simulated stratospheric warming

Calculations of coupled chemistry and transport for the stratosphere are carried out for a 6‐day period during the February 1979 stratospheric major warming, using winds derived from a spectral forecast model. All major families of stratospheric chemistry (odd oxygen, odd nitrogen, odd hydrogen, and odd chlorine), as well as longitudinally varying reaction rate coefficients and photolysis rates, are included in the model. Eight constituents and/or families are transported in the model; additional ones are held fixed or inferred by photochemical equilibrium approximations. Results presented include zonal mean fields, latitude‐longitude distributions (and their changes with time), vertical profiles, and time series of the mixing ratios of transported constituents and families as well as of their total stratospheric column amounts. The results obtained show the relative importance of chemistry and transport for the chlorine‐containing species ClO, ClONO2, HCl, and HOCl. Dynamical effects dominate the variability of HCl, while diurnal ones dominate that of ClONO2 and ClO. Diurnal chemistry and dynamical variability are of similar magnitude for HOCl. The effects of strong planetary wave activity may be seen as large longitudinal variability of the total HCl and ClONO2 columns in the stratosphere; in middle and high northern latitudes it is sufficiently large that it exceeds the diurnal variability of the column.

1979年1月の非定常波の構造と振舞いについて

A numerical calculation of the atmospheric 16-day wave mode under a realistic zonal mean wind distribution and a linearr model simulation of transient waves during January 1979 were performed. The results were compared with the analysis of the FGGEIIIb data set for the same period.It was found that the 16-day wave mode solution using a zonal mean field during January exhibits barotropic features of the normal mode in the troposphere and lower stratosphere. In the upper atmosphere, it bears more resemblance to a forced baroclinic wave mode due to the significant influences of the zonal mean wind and damping.The linear model simulation was performed by using the time-dependent wavenumber 1 component of the geopotential height at the 500mb level of the FGGEIIIb data set as the lower boundary condition. The behavior of the transient wave was found to be similar to that of the 16-day wave mode solution.The results of the numerical simulation and the structure of the 16-day wave mode solution are in good agreement with the analysis of the FGGEIIIb data set. This implies that the transient waves of January 1979 were dominated by the atmospheric 16-day wave mode.

Energetics analysis of the observed and simulated general circulation using three-dimensional normal mode expansions

The energetics characteristics of the observed and simulated general circulation are analyzed using three-dimensional normal mode expansions. The data sets involved are the Goddard Laboratory for Atmospheres (GLA) analysis and simulation data and the Geophysical Fluid Dynamics Laboratory (GFDL) analysis data. The spectral energy properties of the Rossby and gravity modes and energy transformations are presented. Significant influences of model characteristics and the assimilation techniques are observed in the barotropic energy spectrum, particularly for the gravity mode. Energy transformations of the zonal mean field in the GLA analysis and simulation are similar, but distinctly different from that in the GFDL analysis. However, overall, the energy generation in the baroclinic mode is largely balanced by the sink in the barotropic mode. The present study may demonstrate utilities of the three-dimensional normal mode energetics in the analysis of the general circulation. DOI: 10.1111/j.1600-0870.1986.tb00474.x