Trace gas exchange at the air/water interface: Measurements of mass accommodation coefficients

A liquid jet of 90 m diameter and variable length has been utilized to determine absorption rates and, hence, mass accommodation coefficients , of atmospheric trace gases. The compounds investigated are HCl (0.01), HNO₃ (0.01), N₂O₅ (0.005), peroxyacetyl nitrate (>0.001), and HONO (0.005). It is concluded that the absorption of these trace gases by liquid atmospheric water is not significantly retarded by interfacial mass transport. The strengths and limitations of the liquid jet technique for measuring mass accommodation coefficients are explored.     

Propagation of the sea-surface temperature anomalies in the Tropical Atlantic

Summary The evolving modes of the sea-surface temperature (SST) in the Tropical Atlantic on the short interannual (IA) timescale were obtained by performing the extended empirical orthogonal function (EEOF) analyses on this variable separately for the 106-year (1871–1976) and 20-year (1881–1900; 1901–1920; 1921–1940; 1941–1960) periods. The equatorial and inter-hemispheric patterns manifest in the first EEOF mode of each analysis as part of the short IA evolution of the SST anomalies in the Tropical Atlantic. Another outstanding feature of the first EEOF mode of each analysis concerns the propagations of the SST anomalies in the meridional direction within the 20°N–20°S band and in the zonal direction in the sector 40°W–20°W. For all analyses, the SST anomalies propagate northward from the equator to 15°N and southward from 20°N to 15°N, with the same sign anomalies merging approximately at 15°N. On the other hand, the SST anomalies propagate westward in the sector 40°W–20°W with a propagation rate close to that of the phase speed of the fastest baroclinic Rossby wave in the ocean. So, the observed propagations of the SST anomalies in the 20°N–20°S band might result from the combined effect of the surface oceanic currents in this band and the baroclinic Rossby waves in the ocean.     

Effect of land management on ecosystem carbon fluxes at a subalpine grassland site in the Swiss Alps

Summary The influence of agricultural management on the CO₂ budget of a typical subalpine grassland was investigated at the Swiss CARBOMONT site at Rigi-Seebodenalp (1025m a.s.l.) in Central Switzerland. Eddy covariance flux measurements obtained during the first growing season from the mid of spring until the first snow fall (17 Mai to 25 September 2002) are reported. With respect to the 10-year average 1992–2001, we found that this growing season had started 10 days earlier than normal, but was close to average temperature with above-normal precipitation (100–255% depending on month). Using a footprint model we found that a simple approach using wind direction sectors was adequate to classify our CO₂ fluxes as being controlled by either meadow or pasture. Two significantly different light response curves could be determined: one for periods with external interventions (grass cutting, cattle grazing) and the other for periods without external interventions. Other than this, meadow and pasture were similar, with a net carbon gain of –128±17g Cm^–2 on the undisturbed meadow, and a net carbon loss of 79±17g Cm^–2 on the managed meadow, and 270±24g Cm^–2 on the pasture during 131 days of the growing season, respectively. The grass cut in June reduced the gross CO₂ uptake of the meadow by 50±2% until regrowth of the vegetation. Cattle grazing reduced gross uptake over the whole vegetation period (37±2%), but left respiration at a similar level as observed in the meadow.     

An observational aircraft-based study of sea-breeze frontogenesis

In the summer of 1988/89 flights were carried out in the Coorong coastal area of South Australia to investigate sea-breeze fronts. The flights yielded data sets of the structure of the fronts in the cross-frontal direction with a spatial resolution of approximately 3 m. The study is focused on the budgets of sensible and latent heat in the vicinity of the front and on frontogenesis/frontolysis processes which are closely related to budget considerations.The frontogenesis relationships and the budgets were established on a 2 km length scale by low-pass filtering of the space series. As the wind components were measured with high accuracy, all processes which determine frontogenesis could be evaluated and are displayed in x,z-cross-sections: these are the confluence, shear and diabatic effects, all of which play a role in q/x-, q/z-, /x- as well as /z-frontogenesis. A detailed analysis is given for two different states of frontal development. The presented results shed much light on the governing physical processes in the frontal region with strong emphasis on the effects of confluence-generated updrafts, on shear instabilities causing bulges and clefts in the frontal surface as well as producing the elevated frontal head, and on processes related to differential heating and moistening.     

On Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models

Recently Wilson and Flesch (Boundary-Layer Meteorology, 84, 411-426, 1997) suggested that the average increment d z to the orientation = arctan(w/u) of the Lagrangian velocity-fluctuation vector can be used to distinguish the better Lagrangian stochastic models within the well-mixed class. Here it is demonstrated that the specification of d z constitutes neither a sufficient or universally applicable criterion to distinguish the better Lagrangian stochastic models within the well-mixed class. The hypothesis made by Wilson and Flesch that Lagrangian stochastic models with /P_E irrotational are zero-spin models, having d z=0, is proven     

The carbon isotopic ratio of atmospheric carbon dioxide at Tsukuba,Japan

To find out the secular and seasonal trends of the ¹³C value and CO₂ concentration in the surface air and the determination of the ¹³C in the atmospheric CO₂ collected at Tsukuba Science City was carried out during the period from July 1981 to October 1983. The monthly average of the ¹³C value of CO₂ in the surface air collected at 1400 LMT ranged from -7.52 to \s-8.45 with an average of -7.96±0.25 and the CO₂ concentration in the air varied from 334.5 l 1^-1 to 359 l 1^-1 with an average of 347.2±6.3 l 1^-1. The ¹³C value is high in summer and low in winter and is negatively correlated with the CO₂ concentration. In general, the relationship between the ¹³C and the CO₂ concentration is explainable by a simple mixing model of two different constant carbon isotopic species but the relationship does not always follow the model. The correlation between the ¹³C value and the CO₂ concentration is low during the plant growth season and high at other times. The observed negative deviation of the ¹³C value from the simple mixing model in the plant growth season is partly due to the isotopic fractionation process which takes place in the land biota.     

Arctic hazes in summer over Greenland and the North American Arctic: II. Nature and concentrations of accumulation-mode and giant particles

Airborne measurements made during August 1985 over Greenland and its environs show that both accumulation-mode (0.1 m D2.0 m) and giant (D2 m) particles were present in relatively high concentrations in arctic haze layers and that the accumulation-mode particles dominated light scattering. Particles with diameters (D) between 1 and 4 m consisted predominately of mixed materials, small and dense inclusions, and probably organic compounds containing sulfur. Many of the particles from 0.1 to 1 m in diameter were also of mixed composition, with sulfuric acid, ammonium sulfate and organics probably the dominant constituents.     

Occurrence of roll circulations in a shallow boundary layer

Meteorological measurements taken at the Näsudden wind turbine site during slightly unstable conditions have been analyzed. The height of the convective boundary layer (CBL) was rather low, varying between 60 and 300 m. Turbulence statistics near the ground followed Monin-Obukhov similarity, whereas the remaining part of the boundary layer can be regarded as a near neutral upper layer. In 55% of the runs, horizontal roll vortices were found. Those were the most unstable runs, with -z _i/L > 5. Spectra and co-spectra are used to identify the structures. Three roll indicators were identified: (i) a low frequency peak in the spectrum of the lateral component at low level; (ii) a corresponding increase in the vertical component at mid-CBL; (iii) a positive covariance {ovvw} together with positive wind shear in the lateral direction (V/z) in the CBL. By applying these indicators, it is possible to show that horizontal roll circulations are likely to be a common phenomenon over the Baltic during late summer and early winter.     

PHYSICAL IMPACTS OF CLIMATIC CHANGE ON DELTAIC COASTAL SYSTEMS (II): DRIVING TERMS

The evaluation of the impact of climatic change on deltaic coastal systems requires establishing a reference situation. As deltaic coastal zones are highly dynamic areas, it is necessary to define the reference situation as a function of their present evolutive stage. This implies characterizing coastal processes as a complex system defined by the links between driving or forcing terms and the coastal response at different temporal and spatial scales (long/large, medium and episodic scales). Starting from classical previous works on deltaic systems and including all available field and conceptual knowledge to properly consider present conditions (largely regulated rivers, subsidence, etc.) a detailed methodology to analyse forcing or driving agents at the three mentioned scales is presented. This methodology which developed the general approach presented in a companion paper, will be followed by a future paper focusing on the analysis of the coastal response.     

Re-analysis of the nuclear winter phenomenon

Summary An analysis of the report of the (U.S.) National Academy of Sciences (NAS) on atmospheric effects of a nuclear exchange leads to conclusions that differ from those of the NAS and of the earlier TTAPS and AMBIO studies. Any cooling of the earth's surface is likely to beshort-lived because of rapid removal of the smoke clouds originating from nuclear burst-initiated fires, andminor because of appreciable green-house effects due to several distinct physical causes. (One of these, neglected in prior analyses, is the infrared absorption from cirrus clouds produced directly by the nuclear bursts.) Taken together, these effects may even induce slight surface warming (nuclear summer) instead of cooling (nuclear winter). The consequences to atmospheric ozone are similarly ambiguous; depending on the detailed nuclear scenario, the net ozone content may increase-rather than decrease as argued by TTAPS. Experiments could settle some uncertainties.With 2 Figures     

The Effect of Atmospheric Stability on the Surface-Layer Characteristics in a Low-Wind Area of Tropical West Africa

This study details the observed effects ofatmospheric stability on characteristics of thesurface layer in a low wind speed (U = 1.5 m s-1)regime of tropical West Africa. Theaerodynamic roughness length, z0, anddisplacement height, d, obtained from profilewind-speed data at our bush land site (height 2 m)have values of 0.24 ± 0.10 m and 1.54 ± 0.04 mrespectively. In the unstable range (-2.5 < Ri < -0.1; Riis gradient Richardson number), thestandard deviation in wind speed fluctuations, u, increased from 0.57 ± 0.19 m s-1 toa maximum of 0.7 ± 0.2 m s-1 in near-neutralconditions, and in the stable range, the parameterdecreased rapidly to 0.41 ± 0.15 m s-1 at Ri 0.2.In the same stability range, the horizontal winddispersion, , decreased withincreasing stability from 19 ± 8 deg. to 13 ± 5 deg.The surface-layer integral quantity, u/u*, when plottedas a function of stability, is in agreement with theempirical results. The ratio ofsensible heat flux (estimated) to the net radiationranged between 0.1 and 0.2 at nighttime,increasing to about 0.5 during the daytime, and showeda strong dependency on season.     

A new theory of the energy spectrum

In a recent paper, the author introduced a new viscous boundary layer, called the mesolayer, in turbulent shear flow. Its importance stems from its location between the inner and outer regions which are controlled by the law of the wall and Reynolds number similarity, respectively. This intrusion prevents the classical overlap assumption which appears to be fundamental in the derivation of the classical logarithmic behavior. The mesolayer has a thickness proportional to Taylor's microscale . This, and the analogy between the energy equation for the spectrum function of isotropic turbulence and the momentum equation for shear flow, suggest the existence of a similar region in wavenumber space with wavenumber k ~ ^-1. This mesoregion separates the inner region k ~ k _s(where k _s^-1 and is the Kolmogorov length) and the outer region k k _e(where k _e ^-1 and l is the energy-containing eddy size) and again invalidates the overlap assumption which appears to be fundamental in the derivation of the classical k ^-5/3-behavior of the inertial subrange.Incorporation of the mesoregion into the argument leads to a new theory with k ^-5/3-behavior in two regions (^-1 k k _s) and (k _e k ^-1) although with two different coefficients of proportionality (Kolmogorov constants). This leads to a wandering of the spectrum curve about the classical k ^-5/3 line similar to a wandering in turbulent shear flow about the logarithmic curve. This is clearly indicated by the data for the variation of the Kolmogorov constant.Other data support the new theory. In particular, the location of the point k _mwhere the curve of the nonlinear energy-transfer function goes through zero shows agreement with the theory, i.e., k _m^-1.     

The Askervein Hill Project: Mean wind variations at fixed heights above ground

This is one of a series of papers on the Askervein Hill Project. It presents results on the variations in mean wind speed at fixed heights (z) above the ground from linear arrays of anemometer posts and towers. Most of the data are for z = 10 m but some are for z = 3 m. Selected and directionally grouped data from the 55 Mean Flow runs are presented together with mean flow data from Askervein '83 Turbulence runs. Comparisons are made between the data and guideline estimates of fractional speed-up ratio at hilltop locations and between the data and MS3DJH/3 model predictions along the tower lines. There is good agreement in most cases.     

Measurements of solar radiation and estimation of optical depth in the high Andes of Peru

Summary During an expedition to the high Andes of Southern Peru in June–July 1977, measurements of direct solar radiation in four spectral bands (0.270–0.530–0.630–0.695–2.900 ) were conducted at six sites in elevations ranging from sea level to 5645 m. These measurements were evaluated in Langley plots to determine total optical depths () and irradiances at the top of the atmosphere. In addition, water vapor optical depths (_wv) were calculated from the mean radiosounding over Lima during the expedition, and Rayleigh (_ray) and ozone (_oz) optical depths were obtained from published tabulations. Subtracting _ray, _oz, and _wv from yielded estimates of aerosol optical depth _aer. The components _ray and _oz decrease from the shorter towards the longer wavelength bands and from the lower towards the higher elevation sites; _aer also decreases towards the higher elevations. Particularly pronounced is the decrease of _aer and from the lowlands of the Pacific coast to the highlands of the interior, reflecting the effect of a persistent lower-tropospheric inversion and the contrast from the marine boundary layer to the clear atmosphere of the high Andes.With 4 Figures     

Stable Isotope Ratios: Hurricane Olivia

The oxygen and hydrogen isotopic compositions of rains from HurricaneOlivia (1994) in the eastern Pacific were measured. The rains werecollected on 24 and 25 September during airplane flights conducted at anelevation of 3 km. Hurricane Olivia peaked in intensity to a category-4storm between the two dates. Isotope ratios of rains from HurricaneOlivia were markedly lower ( ¹⁸O = –13.9to –28.8) than that of rain collected from a thunderstormat an elevation of 2.3 km outside the influence of Olivia (¹⁸O = –3.8). A distinct decrease in isotoperatios from the first day to the next ( ¹⁸O =–18.4 to –21.9) in Hurricane Olivia wasattributed to decreased updraft velocities and outflow aloft. Thisshifted the isotopic water mass balance so that fewer hydrometeors werelifted and more ice descended to flight level. A decrease in the averagedeuterium excess from the first day to the next (d = 15.5 to 7.1)was attributed to an increase in the relative humidity of the watervapor `source' area. We hypothesize that the `source' region for therain was in the boundary layer near the storm center and that becausethe hurricane was at peak intensity prior to the second day the relative humidity was higher.     

The behaviour of aerosol optical depth during clear summer days in Athens,Greece

Summary The integral aerosol optical depths (k ) at the hour of 08:20 Local Standard Time (LST), are compared with those calculated previously at 11:20 and 14:20 LST, for clear days during summer in Athens over the period 1962–1988. The mean values at 08:20 LST were consistently lower than the values at 11:20 and 14:20 LST. The influence of the vertical wind profile on the values ofk was also investigated. A comparison was made of the wind profiles at 02:00 and 14:00 LST, for days in which the 11:20 and 14:20 LST values ofk were 0.200 andk 0.350, respectively. The corresponding bulk wind shear s was also found for the period 1980–1988. The most significant results occurred with the first category of days. The resultant wind velocities from the surface to the 900 hPa level, in each hour were higher by 2–4 m·s^–1 with respect to the corresponding values for the second category. At 02:00 LST the bulk wind shear showed a considerable difference (1.8) between the two categories of days in the surface to 700 hPa layer at 02:00 LST. Finally, the associated weather conditions that appear to initiate a period of low values ofk (k 0.200) at 11:20 and 14:20 LST were examined for the period 1980–1988. Fifteen such cases were identified and it was found that they all occurred after the passage of weak cold fronts.With 6 Figures     

Modification Of The Standard ∈-Equation For The Stable Abl Through Enforced Consistency With Monin–Obukhov Similarity Theory

A condition is derived for consistency of the standard-equation with Monin–Obukhov (MO) similarity theory of thestably-stratified surface layer. The condition is derivedby extending the procedure used to derive the analogous condition forneutral theory to stable stratification. It is shown that consistencywith MO theory requires a function of flux Richardson number, Ri_f, to be absorbed into either of two closure parameters, c ₁ or c ₂.Inconsistency, on the other hand, results if constant values of these are maintained for all Ri_f, as is done in standardapplication of the equation, and the large overpredictions ofturbulence found in such application to the one-dimensionalstable atmospheric boundary layer (1D-SBL) are traced to thisinconsistency. Guided by this, we formulate a MO-consistent-equation by absorbing the aforementioned function intoc ₁, and combine this with a Level-2.5 second-orderclosure model for vertical eddy viscosity and diffusivities.Numerical predictions of the 1D-SBL by the modified model converge to a quasi-steady state, rectifying the predictive failure of the standard -equation for the case.Quasi-steady predictions of non-dimensional variables agree stronglywith Nieuwstadt's theory. Qualitative accuracy of predictionsis inferred from comparisons to field data, large-eddy simulationresults and Rossby-number similarity relationships.     

On the similarity in atmospheric fluctuations of carbon dioxide,water vapor and temperature over vegetated fields

This paper describes the similarity between atmospheric fluctuations of carbon dioxide, water vapor and temperature using data which cover a wide range of instability (0.02 < < 10). The is the Monin-Obukhov stability parameter including the humidity effect.The spectral analysis shows that the coherency between fluctuations of carbon dioxide and water vapor or temperature is very close to unity, and the phase difference is basically out of phase for whole frequency ranges analyzed. The stability dependence of the normalized standard deviation of carbon dioxide is very similar to those of water vapor and temperature. The normalized standard deviation is about 2.5 under near neutral conditions, and it decreases with increasing instability following the -1/3; power law as (-)^-1/3. The skewness factors of carbon dioxide, water vapor and temperature show a systematic departure with increasing instabilities for 0.02 < s- < 1, and level off at high instabilities for 1 < -\s < 10. The stability dependence of the flatness factors is not so clear as that noted in the standrard deviation and skewness factors. Dissipation rates of carbon dioxide, water vapor and temperature variance are well related to the spectral peak wavelength. This seems to be real since the local production and local dissipation rates are the main terms, almost balancing one another in the variance budget equations for scalar entities.     

Thermal comfort of man in different urban environments

Summary On July 29, 1985, a hot summer day, biometeorological measurements were performed simultaneously in three different urban structures within the city of Munich and in the trunk space of a nearby tall spruce forest. Based on the results of these experiments the following thermophysiologically relevant biometeorological indices were calculated: Predicted mean vote, skin wettedness and physiologically equivalent temperature. These three indices are derived from different models for the human energy balance. They allow the assessment of the thermal components of the microclimates at the selected sites with regard to application in urban planning. The results quantitatively show the great heat stress in the urban structure street canyon, exposed to south, whereas in the trunk space of the tall spruce forest there is nearly an optimal climate even on hot summer days. Between these extremes the results for street canyon, exposed to north show a little higher heat load than for backyard with trees.

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Zusammenfassung An einem heißen Sommertag, dem 29. Juli 1985, wurden in drei Stadtstrukturen in München und im Stammraum eines nahegelegenen Fichtenhochwaldes zeitgleich biometeorologische Messungen durchgeführt. Mit den Meßergebnissen wurden folgende thermophysiologisch relevante biometeorologische Indizes berechnet: Predicted mean vote, Hautbenetzungsgrad und physiologisch äquivalente Temperatur. Diese drei Indizes beruhen auf verschiedenen Modellen zur menschlichen Energiebilanz. Mit den drei Indizes wurden die thermischen Komponenten der Mikroklimate an den ausgewählten Meßplätzen im Hinblick auf Stadtplanungsaufgaben bewertet. Die Ergebnisse zeigen quantitativ die relativ große Hitzebelastung bei der Stadtstruktur Straßenschlucht, nach Süd exponiert, während im Stammraum des Fichtenhochwaldes selbst an heißen Sommertagen nahezu optimale Bedingungen herrschen. Zwischen diesen Extremen liegen die Ergebnisse für die anderen Meßplätze, wobei für Straßenschlucht, nach Nord exponiert die Wärmebelastung etwas höher als für Innenhof mit Bäumen ist.

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With 6 Figures     

The atmospheric tide as a continuous spectrum: Lunar semidiurnal tide in surface pressure

Summary The standard equations for the theory of atmospheric tides are solved here by an integral representation on the continuous spectrum of free oscillations. The model profile of back-ground temperature is that of the U.S. Standard Atmosphere in the lower and middle atmosphere, and in the lower thermosphere, above which an isothermal top extends to arbitrarily great heights. The top is warm enough to bring both the Lamb and the Pekeris modes into the continuous spectrum.Computations are made for semidiurnal lunar tidal pressure at sea level at the equator, and the contributions are partitioned according to vertical as well as horizontal structure. Almost all the response is taken up by the Lamb and Pekeris modes of the slowest westward-propagating gravity wave. At sea level, the Lamb-mode response is direct and is relatively insensitive to details of the temperature profile. The Pekeris mode at sea level has an indirect response-in competition with the Lamb mode-and, as has been known since the time of its discovery, it is quite sensitive to the temperature profile, in particular to stratopause temperature. In the standard atmosphere the Lamb mode contributes about +0.078 mb to tidal surface pressure at the equator and the Pekeris mode about –0.048 mb.The aim of this investigation is to illustrate some consequences of representing the tide in terms of the structures of free oscillations. To simplify that task as much as possible, all modifying influences were omitted, such as background wind and ocean or earth tide. Perhaps the main defect of this paper's implementation of the free-oscillation spectrum is that, in contrast to the conventional expansion in the structures of forced oscillations, it does not include dissipation, either implicity or explicity, and thus does not satisfy causality. Dissipation could be added implicity by means of an impedance condition, for example, which would cause up-going energy flux to exceed downgoing flux at the base of the isothermal top layer. To achieve complete causality, however, the dissipation must be modeled explicity. Nevertheless, since the Lamb and Pekeris modes are strongly trapped in the lower and middle atmosphere, where dissipation is rather weak (except possibly in the surface boundary layer), more realistic modeling is not likely to change the broad features of the present results.Symbols a earth's mean radius; expansion coefficient in (5.3) - b recursion variable in (7.4); proximity to resonance in (9.2) - c sound speed in (2.2); specific heatc _p in (2.2) - f Coriolis parameter 2sin in (2.2) - g standard surface gravity - h equivalent depth - i ; discretization index in (7.3) - j index for horizontal structure - k index for horizontal structure; upward unit vectork in (2.2) - m wave number in longitude - n spherical-harmonic degree; number of grid layers in a model layer - p tidal pressure perturbation; background pressurep ₀ - q heating function (energy per mass per time) - r tidal state vector in (2.1) - s tidal entropy perturbation; background entropys ₀ - t time - u tidal horizontal velocityu - w tidal vertical component of velocity - x excitation vector defined in (2.3); vertical coordinate lnp _*/p ₀ [except in (3.8), where it is lnp /p ₀] - y vertical-structure function in (7.1) - z geopotential height - A constant defined in (6.2) - C spherical-harmonic expansion coefficient in (3.6) - D vertical cross section defined in (5.6) and (5.9) - E eigenstate vector - F vertical-structure function for eigenstate pressure in (3.2) [re-defined with WKB scaling in (7.2)] - G vertical-structure function for eigenstate vertical velocity in (3.2) [re-defined with WKB scaling in (7.2)] - H pressure-scale height - I mode intensity defined in (8.1) - K quadratic form defined in (4.4) - L quadratic form defined in (4.4); horizontal-structure magnification factor defined in (5.11) - M vertical-structure magnification factor defined in (4.6) - P eigenstate pressure in (3.2); tidal pressure in (6.2) - R tidal state vector in (5.1) - S eigenstate entropy in (3.2); spherical surface area, in differential dS - T background molecular-scale (NOAA, 1976) absolute temperatureT ₀ - U eigenstate horizontal velocityU in (3.2); coefficient in (7.3) - V horizontal-structure functionV for eigenstate horizontal velocity in (3.2); recursion variable in (7.3) - W eigenstate vertical velocity in (3.2) - X excitation vector in (5.1) - Y surface spherical harmonic in (3.7) - Z Hough function defined in (3.6) - +dH/dz - (1––)/2 - Kronecker delta; Dirac delta; correction operator in (7.6) - equilibrium tide elevation - (square-root of Hough-function eigenvalue) - ratio of specific gas constant to specific heat for air=2/7 - longitude - - - background density ₀ - eigenstate frequency in (3.1) - proxy for heating functionq =c _P/t - latitude - tide frequency - operator for the limitz - horizontal-structure function for eigenstate pressure in (3.2) - Hough function defined in (6.2) - earth's rotation speed - horizontal gradient operator - ()₀ background variable - ()_* surface value of background variable - () value at base of isothermal top layer - Õ state vector with zerow-component - , energy product defined in (2.4) - | | energy norm - ()^* complex conjugate With 10 Figures