Maximum Buoyancy Frequency Research Articles

Reservoir Drawdown Highlights the Emergent Effects of Water Level Change on Reservoir Physics, Chemistry, and Biology

AbstractWater level drawdowns are increasingly common in lakes and reservoirs worldwide as a result of both climate change and water management. Drawdowns can have direct effects on physical properties of a waterbody (e.g., by altering stratification and light dynamics), which can interact to modify the waterbody's biology and chemistry. However, the ecosystem‐level effects of drawdown remain poorly characterized in small, thermally stratified reservoirs, which are common in many regions of the world. Here, we intensively monitored a small eutrophic reservoir for 2 years, including before, during, and after a month‐long drawdown that reduced total reservoir volume by 36%. During drawdown, stratification strength (maximum buoyancy frequency) and surface phosphate concentrations both increased, contributing to a substantial surface phytoplankton bloom. The peak in phytoplankton biomass was followed by cascading changes in surface water chemistry associated with bloom degradation, with sequential peaks in dissolved organic carbon, dissolved carbon dioxide, and ammonium concentrations that were up to an order of magnitude higher than the previous year. Dissolved oxygen concentrations substantially decreased in surface waters during drawdown (to 41% saturation), which was associated with increased total iron and manganese concentrations. Combined, our results illustrate how changes in water level can have cascading effects on coupled physical, chemical, and biological processes. As climate change and water management continue to increase the frequency of drawdowns in lakes worldwide, our results highlight the importance of characterizing how water level variability can alter complex in‐lake ecosystem processes, thereby affecting water quality.

The oceanic mixed layer changes along with the state transition of the Beaufort Gyre

Recent state transition of the Beaufort Gyre has drawn great interest in the Arctic research community, but how the upper ocean hydrographic structure varies with this transition remains poorly understood. The upper ocean mixed layer plays an important role in climatic and ecological processes. Therefore, we analyze the Ice-Tethered Profiler (ITP) observations over the last two decades (2004–2022) to investigate the long-term trend of the mixed layer in the Arctic Ocean’s Beaufort Gyre (BG) from an observational perspective. Results show that the linear trend of the BG surface mixed layer depth (MLD) before and after 2015 has changed significantly, characterized by the vanishing or even reversal of the significant deepening trend. This transition is most pronounced in winter. The BG winter mixed layer is significantly cooler, saltier and denser in the mid-transition period (2013–2017) compared to the pre-transition period (2004–2012), but becomes significantly warmer, fresher and lighter in the post-transition period (2018–2022). The transition feature of the depth of maximum buoyancy frequency in the upper BG is similar to that of MLD, while this maximum decreases significantly in both the mid- and post-transition period when compared to their previous period. The deepening signal of MLD is propagated eastward, which coincides with the recent transition of BG position and freshwater distribution. Mechanism analysis further reveals that the reversal of winter MLD trend before and after 2015 may be due to changes in surface wind stirring and Ekman pumping. This study extends the investigation of the recent state transition of BG considering the upper hydrographic structure.

Climate Trend Estimation of the Powell Basin Hydrophysical Characteristics

The trend assessment of the dynamic state of the Powell Basin waters in the Weddell Sea was made on the basis of the hydrological data of the 79th cruise of the R/V AkademikMstislavKeldysh (January 16–February 6, 2020) and the World Ocean Database-2018 data for January-February from 1975 to 2020. At each node of the quarter-degree grid, a linear trend was constructed for the calculated values of the maximum buoyancy frequency and the maximum amplitude of the vertical component of the internal wave velocity. It is shown that the southwestern and northwestern parts of the Powell Basin differ significantly in their hydrophysical characteristics. In the northwest of the basin, the linear trend of the maximum buoyancy frequency is negative, the trends of the depths of the maximum values of the Väisälä-Brent frequency and the amplitude of the vertical velocity component are positive. In the southwestern part of the basin, the opposite is true: the trend of the maximum buoyancy frequency is positive, the trends in the depths of the maximum values of the Väisälä-Brunt frequency and the amplitude of the vertical velocity component are negative.

Application of different internal solitary wave theories for SAR remote sensing inversion in the northern South China Sea

This study discusses the theoretical method of the inversion of internal solitary waves (ISWs) characteristic parameters using synthetic aperture radar (SAR) images and ocean reanalysis data. Based on the horizontal distribution characteristics of the pixel intensity of the internal wave fringe in the image, an imaging theoretical model based on the extended Korteweg-de Vries (eKdV) and Benjamin-Ono (BO) theories in SAR was constructed, and the peak-to-peak method and curve-fitting method for calculating the characteristic wavelength of internal solitary waves were determined. The results show that the eKdV theoretical model has the advantage of inverting solitary waves with large amplitudes under the same marine stratified structure, and it provides a more reasonable characteristic parameter estimation than the traditional Korteweg-de Vries (KdV) theory. Although the BO theory is theoretically applicable to deep-water internal waves whose wavelength is far less than the water depth, it can still provide good inversion results in practical applications. Compared to the peak-to-peak method, the curve fitting method fully considers the pixel intensity variation characteristics of the internal wave fringe profile of the image, and the calculated characteristic parameters are more reasonable. Using the maximum buoyancy frequency and internal wave vertical mode to construct the marine stratification model of the deep waters of the northeastern South China Sea may have great uncertainty, and it is better to calculate the interface depth using the relative error of the coefficients of two-layer and continuous internal wave equations. In addition, this study also proposes an internal wave inversion method using continuous SAR images, which provides more reference and support for the selection of internal wave inversion models and reliability test of inversion results in a single SAR image.

Disk wakes in nonlinear stratification

Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number ( ${\it Re} $ ) of $5000$ . Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency ( $N_{max}$ ) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.

Experimental Investigation on the Vertical Structure Characteristics of Internal Solitary Waves

An experimental investigation of the vertical structure characteristics of internal solitary waves (ISWs) was systematically carried out in a large gravitationally stratified fluid flume. Four different stratifications were established, and basic elements of ISWs were measured by a conductivity probe array. The vertical distributions of the amplitude, characteristic frequency and waveform of two types of ISWs under different stratifications were obtained, and the experimental results were compared with the theoretical model. The study shows that most vertical structures of the amplitude under different stratifications agree with those of the theoretical model, while there are some deviations for ISWs with large amplitudes. Neither the two-layer model nor the continuously stratified model can effectively describe the variation in the characteristic frequency at different depths with amplitude. For a single small-amplitude ISW, the characteristic frequency first increases and then decreases with increasing depth. The characteristic frequency is largest at the depth of the maximum buoyancy frequency. For an ISW with a relatively large amplitude, there is likely to be a local minimum of the characteristic frequency near the depth where the maximum buoyancy frequency lies. In different stratifications, the sech2 function of KdV theory can describe the waveforms of ISWs at different depths well.

Rossby numbers and stiffness values inferred from gravity-mode asteroseismology of rotating F- and B-type dwarfs

Context.Multi-dimensional (magneto-)hydrodynamical simulations of physical processes in stellar interiors depend on a multitude of uncalibrated free parameters, which set the spatial and time scales of their computations.Aims.We aim to provide an asteroseismic calibration of the wave and convective Rossby numbers, and of the stiffness at the interface between the convective core and radiative envelope of intermediate-mass stars. We deduce these quantities for rotating dwarfs from the observed properties of their identified gravity and gravito-inertial modes.Methods.We relied on near-core rotation rates and asteroseismic models of 26 B- and 37 F-type dwarf pulsators derived from 4-yearKeplerspace photometry, high-resolution spectroscopy, andGaiaastrometry in the literature to deduce their convective and wave Rossby numbers. We computed the stiffness at the interface of the convective core and the radiative envelope from the inferred maximum buoyancy frequency at the interface and the convective turnover frequency in the core. We use those asteroseismically inferred quantities to make predictions of convective penetration levels, local flux levels of gravito-inertial waves triggered by the convective core, and of the cores’ potential rotational and magnetic states.Results.Our sample of 63 gravito-inertial mode pulsators covers near-core rotation rates from almost zero up to the critical rate. The frequencies of their identified modes lead to models with stiffness values between 102.69and 103.60for the B-type pulsators, while those of F-type stars cover the range from 103.47to 104.52. The convective Rossby numbers derived from the maximum convective diffusion coefficient in the convective core, based on mixing length theory and a value of the mixing length coefficient relevant for these pulsators, vary between 10−2.3and 10−0.8for B-type stars and 10−3and 10−1.5for F-type stars. The 17 B-type dwarfs with an asteroseismic estimate of the penetration depth reveal it to be in good agreement with recent theory of convective penetration that takes rotation into account. Theoretical estimates based on the observationally inferred convective Rossby numbers and stiffness values lead to local stochastically-excited gravito-inertial wave fluxes which may exceed those predicted for non-rotating cores, in agreement with observations. Finally, the convective core of rapid rotators is expected to have cylindrical differential rotation causing a magnetic field of 20–400 kG for B-type stars and of 0.1–3 MG for F-type stars.Conclusions.Our results provide asteroseismic calibrations to guide realistic (magneto-)hydrodynamical simultations of rotating (magnetised) core convection in stellar interiors of dwarfs and future modelling of transport and mixing processes in their interiors.

Cross-shelf sediment transport in the Yangtze Delta frontal zone: Insights from field observations

Abstract The Yangtze River, which has the largest runoff in China, discharges massive amounts of sediment into the East China Sea every year, which has an important influence on the biogeochemical processes of estuaries and adjacent seas. However, how much Yangtze River sediment is transported to the East China Sea shelf and the dynamic mechanisms of cross-shelf sediment transport remain unclear, especially given the lack of sediment dynamics studies based on field observations. In this study, we performed 25 h field observations of cross-shelf dynamics in the frontal zone of the Yangtze River (122.74°E, 31.04°N, water depth: 28 m) in the summer and winter of 2011. The observations show that the strong tidal current with the maximum current speed of 0.8 m/s causes bottom shear stress up to 0.4 Pa, which is the main force for seabed sediment erosion and suspension, while wind waves play a negligible role. Significant stratification appears at the observation site with a maximum buoyancy frequency of 0.01 s−2 in winter and 0.2 s−2 in summer, which inhibits the high turbidity water within 10 m from the seabed. The cross-shelf sediment transport at the mooring site is mainly driven by the baroclinic force in the middle and bottom layers, and is influenced by wind in the surface layer. During the observations, the sediment from the Yangtze Delta was transported to the southeast in winter and to the northwest in summer, with sediment transport fluxes of 2.8 and 2.3 g/s/m2, respectively.

Performance of one-dimensional hydrodynamic lake models during short-term extreme weather events

Numerical lake models are useful tools to study hydrodynamics in lakes, and are increasingly applied to extreme weather events. However, little is known about the accuracy of such models during these short-term events. We used high-frequency data from three lakes to test the performance of three one-dimensional (1D) hydrodynamic models (Simstrat, GOTM, GLM) during storms and heatwaves. Models reproduced the overall direction and magnitude of changes during the extreme events, with accurate timing and little bias. Changes in volume-averaged and surface temperatures and Schmidt stability were simulated more accurately than changes in bottom temperature, maximum buoyancy frequency, or mixed layer depth. However, in most cases the model error was higher (30–100%) during extreme events compared to reference periods. As a consequence, while 1D lake models can be used to study effects of extreme weather events, the increased uncertainty in the simulations should be taken into account when interpreting results.

Observations of Shelf Exchange and High‐Frequency Variability in an Alaskan Fjord

AbstractHigh‐frequency observations collected over 3 years are used to describe upper‐ocean variability in Behm Canal, a nonglacial fjord in Southeast Alaska. The fjord is sheltered by surrounding topography and connected to the outer continental shelf by a 400‐m‐deep strait. Summer conditions are characterized by strong near‐surface stratification, a sea breeze wind regime, and tidally dominated flows. In nonsummer months, baroclinic subinertial flows exceeding 0.5 m/s dominate the velocity record. The flow events represent a wind‐driven response to low‐pressure systems that impact the coast as they propagate across the Gulf of Alaska. The observations suggest that the storm systems generate downwelling events that propagate into the fjord with a mode‐one‐like vertical structure in the upper 60 m. Following the initial up‐fjord current pulse lasting approximately a day, a down‐fjord flow occurs lasting several days, the duration of the downwelling anomaly. These subinertial downwelling events likely are the dominant mechanism of shelf‐fjord exchange, with estimated fjord‐flushing times of 50 days. The downwelling events are accompanied by enhanced near‐inertial shear, which exhibits downward energy propagation. Evidence for enhanced energy near the maximum buoyancy frequency in the thermocline suggests high‐frequency internal wave trapping, with the primary energy source in the semidiurnal band associated with tidal and wind forcing. The observations highlight the importance of shelf‐fjord coupling through meteorological forcing and the existence of internal wave energy at a range of frequencies, which have implications for mixing and transport within the fjord.

Estimating depth of polarity conversion of shoaling internal solitary waves in the northeastern South China Sea

Internal Solitary Wave (ISW) normally converts its polarity from depression wave to elevation wave during shoaling process. This phenomenon has been verified by model studies and field observations. However, through the comparison between theoretical studies and observations, we find that Extended Kortweg-de Vries (EKdV) theory, often used by numerical simulations, cannot well predict the polarity conversion area of shoaling ISWs. Here we trace seismic reflectors within the water column, captured by exploration seismic data, and systematically analyze the polarity conversion area of the shoaling ISWs in the northeastern South China Sea (SCS). The results show that the ISWs in the northeastern SCS generally begin to convert their polarities from depression to elevation waves at the seafloor depth of about 200m, and turn into elevation waves at the seafloor depth of about 100m. The ISWs between depression and elevation waves are newly defined as transition waves, which are distributed between the seafloor depth of 130m and 220m; while elevation waves are distributed between 80m and 120m. We compared the observation with the theoretical critical conversion depth estimation by the two-layer model based on EKDV equation. The result suggests that the depth of the upper layer in the two-layer model can be selected as the depth of the maximum buoyancy frequency not the base of the mixing layer. Such selection can get better estimation of critical conversion depth of shoaling ISWs. Our analysis also suggests that the polarity conversion process is amplitude-dependent. The ratio of amplitude to depth for elevation waves is more than two times larger than that of the transition waves. This implies that the seafloor plays a more important role in shallow water than in deep water in controlling the amplitudes of ISWs during shoaling process.

Mixing and phytoplankton dynamics in a submarine canyon in the West Antarctic Peninsula

AbstractBathymetric depressions (canyons) exist along the West Antarctic Peninsula shelf and have been linked with increased phytoplankton biomass and sustained penguin colonies. However, the physical mechanisms driving this enhanced biomass are not well understood. Using a Slocum glider data set with over 25,000 water column profiles, we evaluate the relationship between mixed layer depth (MLD, estimated using the depth of maximum buoyancy frequency) and phytoplankton vertical distribution. We use the glider deployments in the Palmer Deep region to examine seasonal and across canyon variability. Throughout the season, the ML becomes warmer and saltier, as a result of vertical mixing and advection. Shallow ML and increased stratification due to sea ice melt are linked to higher chlorophyll concentrations. Deeper mixed layers, resulting from increased wind forcing, show decreased chlorophyll, suggesting the importance of light in regulating phytoplankton productivity. Spatial variations were found in the canyon head region where local physical water column properties were associated with different biological responses, reinforcing the importance of local canyon circulation in regulating phytoplankton distribution in the region. While the mechanism initially hypothesized to produce the observed increases in phytoplankton over the canyons was the intrusion of warm, nutrient enriched modified Upper Circumpolar Deep Water (mUCDW), our analysis suggests that ML dynamics are key to increased primary production over submarine canyons in the WAP.

A numerical experiment on the formation of the tropopause inversion layer associated with an explosive cyclogenesis: possible role of gravity waves

The tropopause inversion layer (TIL) is a persistent layer with high static stability. Although some mechanisms for the formation of the TIL have been proposed, the time evolution of the TIL under realistic conditions especially when factoring in the contribution of small-scale processes such as gravity waves is not well understood. To gain an understanding of this factor, we conducted a numerical experiment on an explosive cyclogenesis in mid-latitudes using a nonhydrostatic regional atmospheric model. Although the TIL in the model is consistent with previous observations in the sense that it is stronger in the negative vorticity areas, the relationship is clear only in the development and mature stages of a cyclone, suggesting that the evolution of the cyclone plays an important role in the formation of the TIL. To ascertain the effects of gravity waves on the TIL, vertical convergence at the tropopause is analyzed. Histograms of maximum buoyancy frequency squared within the TIL show that regions of vertical convergence have higher , in addition to regions with high ∂2 w/∂ z2, implying that waves having downward phase propagation also play an important role in the dynamical formation of the TIL. This tendency is clearer in regions of negative relative vorticity at the tropopause. By taking account of the fact that the gravity wave activities associated with the cyclone and the jet streak are enhanced during the development and mature stages of the cyclone, vertical convergence due to gravity waves associated with synoptic weather systems can be seen to be a key process in the formation of the negative correlation between the strength of the TIL and the local relative vorticity at the tropopause.

Interface depth used in a two-layer model of nonlinear internal waves

A two-layer model includes three parameters: interface depth h 1, upper layer density $$\rho_{1}$$ , and lower layer density $$\rho_{2}$$ . Many theoretical and laboratorial studies of internal waves, as well as most numerical models, are based on the two-layer assumption. However, these three parameters cannot be directly measured because a pycnocline in the real ocean has finite thickness, and the densities in both the mixed layer and the deep ocean are not constant. In the present study, seven different methods are used to determine the interface depth of the two-layer model and compared with the depth of maximum vertical displacement: the depth of maximum buoyancy frequency (Ν max), the depth where the first mode eigenfunction has its maximum (Φmax), the depth where the lowest mode temperature empirical orthogonal function has its maximum, the depth where either the two-layer Korteweg–de Vries (KdV) or Benjamin–Ono equation has closest coefficients with their continuously stratified counterparts, and the same KdV approach with stratification replaced by two idealized distributions. The multi-ship measurement conducted near the Luzon Strait is used for deep ocean comparison, and two measurements conducted in the east of Dongsha Atoll are used for shallow water comparison. The results show that in the deep ocean, the KdV approach with idealized type I stratification gives the interface closest to the depth of maximum vertical displacement. In shallow waters, the KdV approach agrees with the measurement best.

Self‐regulation of deep‐ocean internal wave continuum: Observations on related near‐inertial shear and high‐frequency vertical motions

It is shown using detailed yearlong acoustic current observations between 850 and 1350 m that large‐scale and small‐scale waves in the ocean interior are coupled in a self‐regulating fashion. Vertical current shear, a key to vertical mixing, is dominated near the local inertial frequency. Its value follows details of maximum buoyancy frequency ‘Nmax’ in thin layers of different density. Its frequency however, is determined by the minimum Nmin in the near‐homogeneous layers between the Nmax‐layers. These weakly stratified layers are maintained by overturning of density surfaces due to high‐frequency internal waves guided by the shear‐layers. Rare vertical current observations show a spectral continuum between [Nmin, Nmax] of low‐mode interfacial waves rather than wave‐groups propagating through smooth large‐scale stratification.

Analyses of internal solitary waves generated at the Columbia River plume front using SAR imagery

Internal solitons generated at the Columbia River plume front during ebb tide are displayed on a synthetic aperture radar (SAR) image taken on 26 July 2004 at 14:24:22 UTC. Scale analyses suggest that these internal solitons belong to the finite‐depth category. A theoretical model, relating the radar backscatter cross section to dynamic parameters of internal solitons in the continuous stratification, is developed. Using the model and background stratification data collected by the River Influences on Shelf Ecosystem (RISE) project in July 2004, we extract the soliton dynamic parameters of the half‐width, amplitude, phase speed, and average energy flux per crest length along a typical section across the internal solitons. The observed stratification is simulated by using a theoretical model parameterized by pycnocline depth, thickness, and maximum buoyancy frequency. The half‐width, amplitude, phase speed, and energy flux of the leading soliton at the section are 119.3 m, 9.3 m, 0.69 m s−1, and 417.4 W m−1 with the uncertainty ranges of ±14.7 m, ±2.6 m, ±0.10 m s−1, and ±281.6 W m−1, respectively. Sensitivity analyses reveal that the extraction of the soliton half‐width from the SAR image is not affected by the stratification. The amplitude is less influenced by the pycnocline thickness than the depth and is independent of the maximum buoyancy frequency.

High‐frequency internal waves in large stratified lakes

Observations are presented from Lake Biwa and Lake Kinneret showing the ubiquitous and often periodic nature of high‐frequency internal waves in large stratified lakes. In both lakes, high‐frequency wave events were observed within two distinct categories: (1) Vertical mode 1 solitary waves near a steepened Kelvin wave front and vertical mode 2 solitary waves at the head of an intrusive thermocline jet were found to have wavelengths ~64–670 m and ~13–65 m, respectively, and were observed to excite a spectral energy peak near 10-3 Hz. (2) Sinusoidal vertical mode 1 waves on the crests of Kelvin waves (vertically coherent in both phase and frequency) and bordering the thermocline jets in the high shear region trailing the vertical mode 2 solitary waves (vertically incoherent in both phase and frequency) were found to have wavelengths between 28–37 and 9–35 m, respectively, and excited a spectral energy peak just below the local maximum buoyancy frequency near 10−2 Hz. The waves in wave event categories 1 and 2 were reasonably described by nonlinear wave and linear stability models, respectively. Analysis of the energetics of these waves suggests that the waves associated with shear instability will dissipate their energy rapidly within the lake interior and are thus responsible for patchy turbulent events that have been observed within the metalimnion. Conversely, the finite‐amplitude solitary waves, which each contain as much as 1% of the basinscale Kelvin wave energy, will propagate to the lake perimeter where they can shoal, thus contributing to the maintenance of the benthic boundary layer.

On internal waves near the high‐frequency limit in an enclosed basin

Observations of the internal wave field using thermistor chains in Lake Kinneret (Israel) showed a substantial energy peak at frequencies just below the maximum buoyancy frequency. The waves were of the first vertical mode, occurred in groups, and resulted in isotherm displacements of up to 4 m in 22 m of water. The appearance of this high‐frequency internal wave energy was closely related to the strength of the wind field. A weaker relationship was found with the phase of the basin‐scale internal waves, with more energy observed under the same wind conditions when the metalimnion was closer to the surface. The relationship of the high‐frequency waves to the wind and the basin‐scale internal waves implies a generation mechanism associated with shear in the surface layer (observed to be approximately 0.08 s−1), similar to that responsible for generating high‐frequency internal waves in the equatorial Pacific. A linear stability analysis of the mean state during periods of high‐frequency wave activity revealed the existence of unstable modes, whose period and frequency bandwidth closely matched field observations.

On the dispersion relation for trapped internal waves

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp= 1/CpO+k/ωmax+ ∈(k) wherecpis the wave phase speed for a particular mode,CpOis the phase speed atk= 0, ωmaxis the maximum possible wave angular frequency and ωmax≤NmaxwhereNmaxis the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) =o(k) forklarge, and is bounded for finitek.In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp≈ (∫∞0N2(y)ydy)-½+k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functionsN2(x) which is taken to be a class of realvalued functions of a real variablexwhere O ≤x∞ such thatN2(x) =O(e-βx) asx→ ∞ and 1/β is an arbitrary length scale. It is demonstrated thatN2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax=Nmax. More generally, it is demonstrated that whenk→ ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that ifN2(x) isO(e-βx) asx→ ∞ and has a maximum valueN2maxthen a sufficient condition for 1/cp∼k/Nmaxto hold for largekfor the lowest mode is thatN2(t)/tis convex for O ≤t≤ 1 wheret= e-βx.