Upper-boundary Condition Research Articles

Influence of seawater loading on the thermal history of underlying strata in an oceanic basin—an example from the eastern Gulf of Papua

The eastern Gulf of Papua, at the easternmost end of the Papuan Basin, has been subjected to extended seawater loading during the Cenozoic, allowing an investigation into the influence of seawater loading on the thermal development of the underlying rock layers in the oceanic basin. Utilising paleontological, lithologic and paleobathymetric data, along with present-day temperature fields, this study reconstructed the burial history, thermal history and maturity of the eastern Papuan Basin using PetroMod2012.2 1D software. Burial history curves for the virtual wells were employed to trace the evolutionary stages in different regions, revealing a positive relationship between predicted vitrinite reflectance (R o) and modelled thermal history. R o values displayed a gradual decrease from northeast to southwest, with higher R o values observed in areas unaffected by seawater loading. The presence of seawater loading had a notable impact on thermal history, particularly influencing the upper boundary condition and the sediment–water interface temperature of the temperature field. This research enhances our understanding of the Meso-Cenozoic thermal history of the eastern Papuan Basin and reveals the factors influencing stratigraphic thermal evolution.

Ecological regulation of chemical weathering recorded in rivers

We provide a new generalized framework accounting for the effects of water and nutrient uptake by roots in the weathering zone on broad patterns of solute concentration - discharge relationships in rivers. As plants withdraw water from the subsurface, they decrease the area-normalized fluid drainage rate. This results in a discharge flux to springs and streams which is a function of water use by plants, leading to ecosystem regulation of the characteristic timescales of chemical weathering reactions and fluid flow rate. Our predictive framework quantifies the pore water and river solute chemistry generated from water-rock interactions, regulated by the effects of vegetation on water flow rates. To this we add the capacity for vegetation to enrich, passively admit, or discriminate against a given dissolved solute based on ecophysiological needs, thus further modifying the local concentration of a given chemical species in solution across a weathering profile. This uptake flux is not lost to the atmosphere in the same manner as water is removed from the subsurface via evapotranspiration. Rather, these extracted chemical species become incorporated into the topsoil as litter which may be partially or fully resolubilized, creating an upper boundary condition which depends on the uptake and recycling characteristics of the ecosystem. These processes are united through a new solution to the advection-reaction equation which allows vegetation to regulate broad patterns of stream concentration - discharge relationships. We show that ecosystems are capable of both accelerating and impeding water-rock reaction rates across the weathering profile, ultimately leading to both thermodynamic and ecological controls on the baseflow concentration of rock-derived solutes in rivers.

Internally heated and fully compressible convection: Flow morphology and scaling laws

In stars and planets natural processes heat convective flows in the bulk of a convective region rather than at hard boundaries. By characterizing how convective dynamics are determined by the strength of an internal heating source we can gain insight into the processes driving astrophysical convection. Internally heated convection has been studied extensively in incompressible fluids, but the effects of stratification and compressibility have not been examined in detail. In this work, we study fully compressible convection driven by a spatially uniform heating source in two-dimensional (2D) and 3D Cartesian, hydrodynamic simulations. We use a fixed temperature upper boundary condition which results in a system that is internally heated in the bulk and cooled at the top. We find that the flow speed, as measured by the Mach number, and turbulence, as measured by the Reynolds number, can be independently controlled by separately varying the characteristic temperature gradient from internal heating and the diffusivities. Two-dimensional simulations at a fixed Mach number (flow speed) demonstrate consistent power at low wave number as diffusivities are decreased. We observe convection where the velocity distribution is skewed towards cold, fast downflows and that the flow speed is related to the length scale and entropy gradient of the upper boundary where the downflows are driven. We additionally find a heat transport scaling law which is consistent with prior incompressible work. Published by the American Physical Society 2024

Rayleigh waves in a reservoir rocks lying on the bed of sandy rocks

An analysis of the propagation behaviour of Rayleigh waves has been carried out in anisotropic layer like reservoir rocks overlying a sandy medium. The most general case of anisotropic material i.e., of triclinic crystal has been taken. An analytical approach is incorporated in a state to obtain the solutions for layer and half-space. A closed relation of phase velocity and wave number is obtained as determinant form with weld contact boundary at the interface of lower half and the adjacent layer with the upper boundary condition as a rigid called dispersion relation. For the numerical interpretation of results, two different reservoir rocks of triclinic type have been taken. The outcomes of phase velocity and wave number have been illustrated by graphs obtained with the aid of Mathematica-7.

Stirring the base of the solar wind: On heat transfer and vortex formation

Context.Current models of the solar wind must approximate (or ignore) the small-scale dynamics within the solar atmosphere; however, these are likely important in shaping the emerging wave-turbulence spectrum that ultimately heats and accelerates the coronal plasma.Aims.This study strives to make connections between small-scale vortex motions at the base of the solar wind and the resulting heating and acceleration of the coronal plasma.Methods.TheBifrostcode produces realistic simulations of the solar atmosphere which facilitate the analysis of spatial and temporal scales which are currently at, or beyond, the limit of modern solar telescopes. For this study, theBifrostsimulation is configured to represent the solar atmosphere in a coronal hole region, from which the fast solar wind emerges. The simulation extends from the upper-convection zone (2.5 Mm below the photosphere) to the low corona (14.5 Mm above the photosphere), with a horizontal extent of 24 Mm × 24 Mm. The network of magnetic funnels in the computational domain influence the movement of plasma, as well as the propagation of magnetohydrodynamic waves into the low corona.Results.The twisting of the coronal magnetic field by photospheric flows efficiently injects energy into the low corona. Poynting fluxes of up to 2 − 4 kWm−2are commonly observed inside twisted magnetic structures with diameters in the low corona of 1–5 Mm. Torsional Alfvén waves are favourably transmitted along these structures, and subsequently escape into the solar wind. However, reflections of these waves from the upper boundary condition make it difficult to unambiguously quantify the emerging Alfvén wave-energy flux.Conclusions.This study represents a first step in quantifying the conditions at the base of the solar wind usingBifrostsimulations. It is shown that the coronal magnetic field is readily braided and twisted by photospheric flows. Temperature and density contrasts form between regions with active stirring motions and those without. Stronger whirlpool-like flows in the convection, concurrent with magnetic concentrations, launch torsional Alfvén waves up through the magnetic funnel network, which are expected to enhance the turbulent generation of magnetic switchbacks in the solar wind.

Effect of anisotropy and boundary conditions on Darcy and Brinkman porous penetrative convection

We investigate the effects of anisotropic permeability and changing boundary conditions upon the onset of penetrative convection in a porous medium of Darcy type and of Brinkman type. Attention is focussed on the critical eigenfunctions which show how many convection cells will be found in the porous layer. The number of cells is shown to depend critically upon the ratio of vertical to horizontal permeability, upon the Brinkman coefficient, and upon the upper boundary condition for the velocity which may be of Dirichlet type or constant pressure. The critical Rayleigh numbers and wave numbers are determined, and it is shown how an unconditional threshold for nonlinear stability may be derived.HighlightsShows how number of convection cells depends upon the temperature of the upperlayer and the anisotropy of the permeabilityShows how number of convection ceels depends upon the temperature of the upperlayer and the Brinkman coefficientShows how number of convection cells patters depends upon the upper boundarycondition on the velocity or the ambient pressure

NARX Neural Network Derivations of the Outer Boundary Radiation Belt Electron Flux

AbstractWe present two new empirical models of radiation belt electron flux at geostationary orbit. GOES‐15 measurements of 0.8 MeV electrons were used to train a Nonlinear Autoregressive with Exogenous input (NARX) neural network for both modeling GOES‐15 flux values and an upper boundary condition scaling factor (BF). The GOES‐15 flux model utilizes an input and feedback delay of 2 and 2 time steps (i.e., 5 min time steps) with the most efficient number of hidden layers set to 10. Magnetic local time, Dst, Kp, solar wind dynamic pressure, AE, and solar wind velocity were found to perform as predicative indicators of GOES‐15 flux and therefore were used as the exogenous inputs. The NARX‐derived upper boundary condition scaling factor was used in conjunction with the Versatile Electron Radiation Belt (VERB) code to produce reconstructions of the radiation belts during the period of July–November 1990, independent of in‐situ observations. Here, Kp was chosen as the sole exogenous input to be more compatible with the VERB code. This Combined Release and Radiation Effects Satellite‐era reconstruction showcases the potential to use these neural network‐derived boundary conditions as a method of hindcasting the historical radiation belts. This study serves as a companion paper to another recently published study on reconstructing the radiation belts during Solar Cycles 17–24 (Saikin et al., 2021, https://doi.org/10.1029/2020sw002524), for which the results featured in this paper were used.

Stable channel flow with spanwise heterogeneous surface temperature

Recent studies have demonstrated that large secondary motions are excited by surface roughness with dominant spanwise length scales of the order of the flow's outer length scale. Inspired by this, we explore the effect of spanwise heterogeneous surface temperature in weakly to strongly stratified closed channel flow (at$Ri_\tau =120$, 960;$Re_\tau = 180$, 550) with direct numerical simulations. The configuration consists of equally sized strips of high and low temperature at the lower and upper boundaries, while an overall stable stratification is induced by imposing an average temperature difference between the top and bottom. We consider the influence of the width of the strips (${\rm \pi} /8 \leq \lambda /h \leq 4{\rm \pi} $), Reynolds number, stability and upper boundary condition on the mean flow structure, skin friction and heat transfer. Results indicate that secondary flows are excited, with alternating high- and low-momentum pathways and vortices, similar to the patterns induced by spanwise heterogeneous surface roughness. We find that the impact of the surface heterogeneity on the outer layer depends strongly on the spanwise heterogeneity length scale of the surface temperature. Comparison to stable channel flow with uniform temperature reveals that the heterogeneous surface temperature increases the global friction coefficient and reduces the global Nusselt number in most cases. However, for the high-Reynolds cases with$\lambda /h \geq {\rm \pi} /2$, we find a reduction of the friction coefficient. At stronger stability, the vertical extent of the vortices is reduced and the impact of the heterogeneous temperature on momentum and heat transfer is smaller.

Numerical modeling of groundwater flow based on explicit and fully implicit schemes of finite volume method

This paper documents a novel numerical model for calculating the behavior of unsteady, one-dimensional groundwater flow by using the finite volume method. The developed model determined water table fluctuations for different scenarios as follows: Drainage and recession from an unconfined aquifer, and water table fluctuations above an inclined leaky layer due to ditch recharge with a constant and variable upper boundary condition. The Boussinesq equation, which is the governing equation in this domain, is linearized and solved numerically in both of the explicit and fully implicit conditions. Meanwhile, the upwind scheme is employed to discretize the governing equation. The computed outcomes of both the explicit and implicit approaches agreed well with the results of analytical solution and laboratory experiments. The main reason is that in the first half of simulation process explicit scheme obtains slightly better results and in the second half of the simulation process fully implicit scheme predicts more reliable outcomes that are hidden in the neighbor node points. As a final point, the numerical outcomes confirm that the developed model is capable of calculating satisfactory outcomes in engineering and science applications.

Numerical Modeling of Soil Water–Heat Transport under Oxo-Biodegradable Film Mulch and the Optimal Mulching Period

Timely monitoring and regulation of the transport and distribution of soil water and heat are appropriate methods to ensure and improve crop yields. With the degradation of the oxo-biodegradable film, the upper boundary condition is a time-variable boundary, and soil water-heat transport is more complicated, comparing with conventional plastic film. Based on HYDRUS-2D version 2.02 and the characteristics of mulching films, a soil water-heat coupled transport mathematical model of drip irrigation was established, and the simulated soil water content and temperature were tested against three-year field observations for calibration and validation purposes. The model prediction demonstrated that the induction periods of oxo-biodegradable film mulching differed in different hydrological years. In normal flow years, an induction period of 50–80 days had higher water use efficiency. While the window would increase to 50–100 days in low flow years. As for temperature, comparing with conventional plastic film mulching, the decreasing trend of temperature started to slow down when the induction period increased to 70 days. Therefore, the optimal induction period of the oxo-biodegradable film is 50–70 days.

Determination and assessment of phosphorus assimilation capacity applying Vollenweider approach for Hazar Lake

Eutrophication is a natural process for natural lakes and dams and the hydraulic retention time of this process is directly related to whether the assimilation capacity is exceeded depending on the current water quality and pressures consists of point and diffuse sources. Exceeding assimilation capacity is accepted good for biological productivity however this situation cannot be desired for water resources any time and preventive actions need to be taken to sustain good water quality. The Vollenweider OECD Method is a widely used and accepted approach in order to calculate current phosphorus loads for phosphorus limiting lakes and dams. In addition to OECD method, rational method is used frequently in calculating the flowrates via surface run-off after precipitation. In this study, the calculation of the assimilation capacity for phosphorus limiting lakes and dams using the Vollenweider Method and rational method for calculation of flowrates were applied for Hazar Lake’s Basin example. When the apply proposed approach Hazar Lake’s TP loading might be increased 3.7 times a year for desired oligotrophic upper boundary condition and 13.7 times a year for mesotrophic state boundary condition.

When time pressure counters the zero price effect

Purpose The freemium pricing model is dominant in digital products such as mobile applications. While limited evaluation of a product such as when a consumer is under time pressure, has been found to increase consumer preference for the free version (“the zero price effect”), this paper aims to explore moderators that attenuate or reverse that effect. Design/methodology/approach Three experiments test the role of anchoring effects induced by time pressure in moderating the zero price effect. Findings The studies offer evidence that anchoring effects induced by time pressure can be directed to reduce preference for free versions of products. In addition, these effects are mediated by the perceived performance risk of a product and an upper boundary condition for monetary price level is found. Research limitations/implications This research demonstrates exceptions to time pressure’s role in intensifying the zero price effect. Future research could focus on additional moderators of the effect such as the need for certainty and examine time pressure’s effect on in-app purchases. Practical implications These findings can be directly applied by marketers of digital products using a freemium pricing model who wants to use time pressure to create urgency with customers without pushing them toward the free version of a product. Originality/value This paper finds exceptions to the zero price effect where consumers exhibit a stronger preference for the paid (vs free) version of a product when under time pressure.

Reconstruction of the Radiation Belts for Solar Cycles 17–24 (1933–2017)

AbstractWe present a reconstruction of the dynamics of the radiation belts from solar cycles 17 to 24 which allows us to study how radiation belt activity has varied between the different solar cycles. The radiation belt simulations are produced using the Versatile Electron Radiation Belt (VERB)‐3D code. The VERB‐3D code simulations incorporate radial, energy, and pitch angle diffusion to reproduce the radiation belts. Our simulations use the historical measurements of Kp (available since solar cycle 17, i.e., 1933) to model the evolution radiation belt dynamics between L* = 1–6.6. A nonlinear auto regressive network with exogenous inputs (NARX) neural network was trained off GOES 15 measurements (January 2011–March 2014) and used to supply the upper boundary condition (L* = 6.6) over the course of solar cycles 17–24 (i.e., 1933–2017). Comparison of the model with long term observations of the Van Allen Probes and CRRES demonstrates that our model, driven by the NARX boundary, can reconstruct the general evolution of the radiation belt fluxes. Solar cycle 24 (January 2008–2017) has been the least active of the considered solar cycles which resulted in unusually low electron fluxes. Our results show that solar cycle 24 should not be used as a representative solar cycle for developing long term environment models. The developed reconstruction of fluxes can be used to develop or improve empirical models of the radiation belts.

The challenges of an in situ validation of a nonequilibrium model of soil heat and moisture dynamics during fires

Abstract. With the increasing frequency and severity of fire, there is an increasing desire to better manage fuels and minimize, as much as possible, the impacts of fire on soils and other natural resources. Piling and/or burning slash is one method of managing fuels and reducing the risk and consequences of wildfire, but the repercussions to the soil, although very localized, can be significant and often irreversible. In an effort to provide a tool to better understand the impact of fire on soils, this study outlines the improvements to and the in situ validation of a nonequilibrium model for simulating the coupled interactions and transport of heat, moisture and water vapor during fires. Improvements to the model eliminate the following two important (but heretofore universally overlooked) inconsistencies: one that describes the relationship between evaporation and condensation in the parameterization of the nonequilibrium vapor source term, and the other that is the incorrect use of the apparent thermal conductivity in the soil heat flow equation. The first of these made a small enhancement in the stability and performance of the model. The second is an important improvement in the physics underpinning the model but had less of an impact on the model's performance and stability than the first. This study also (a) develops a general heating function that describes the energy input to the soil surface by the fire and (b) discusses the complexities and difficulties of formulating the upper boundary condition from a surface energy balance approach. The model validation uses (in situ temperature, soil moisture and heat flux) data obtained in a 2004 experimental slash pile burn. Important temperature-dependent corrections to the instruments used for measuring soil heat flux and moisture are also discussed and assessed. Despite any possible ambiguities in the calibration of the sensors or the simplicity of the parameterization of the surface heating function, the difficulties and complexities of formulating the upper boundary condition and the obvious complexities of the dynamic response of the soil's temperature and heat flux, the model produced at least a very credible, if not surprisingly good, simulation of the observed data. This study then continues with a discussion and sensitivity analysis of some important feedbacks (some of which are well known and others that are more hypothetical) that are not included in the present (or any extant) model, but that undoubtedly are dynamically influencing the physical properties of the soil in situ during the fire and, thereby, modulating the behavior of the soil temperature and moisture. This paper concludes with a list of possible future observational and modeling studies and how they would advance the research and findings discussed here.

Mapping of Ice Storage Processes on the Moon with Time-dependent Temperatures

Abstract Lunar cold traps are defined by extremely low time-integrated sublimation loss, so that water ice is expected to accumulate in them. Due to the strong dependence of the sublimation rate on temperature, they have heretofore been delineated by the peak rather than the time-averaged sublimation rate. Here, time-averaged sublimation rates are calculated for the south polar region of the Moon based on 11 yr of Diviner surface temperature measurements. Data for each spatial pixel are binned according to subsolar (diurnal) and ecliptic (seasonal) longitude. Frequency-domain filtering of temperature time series is applied to interpolate and smooth the data set. The cold trap area is 17,000 km2 from 80°S to the south pole, about 24% larger than defined by peak temperature. In addition, subsurface ice stability is mapped based on solutions of the heat equation with Diviner surface temperatures as the upper boundary condition. Even a thin layer of dust reduces the sublimation loss dramatically. A third potential mechanism for ice storage, vapor pumping by temperature cycles, is also mapped, based on a model for the time-variable population of adsorbed water molecules on the lunar surface. The area of significant pumping is estimated to be 96,000 km2 poleward of 80°S. In summary, a new processing technique has been developed that exploits information about the time variability of lunar surface temperatures measured by Diviner, and it results in new maps for the geographic distribution of potential water-ice deposits in the south polar region of the Moon.

Effects of soil heterogeneity on subsurface water movement in agricultural fields: A numerical study

This study aims to numerically examine the effects of soil heterogeneity on subsurface water movement in agricultural fields under wheat (Triticum aestivum L.) and rice (Oryza sativa L.) crop covers. The 3-D Richards equation that accounts for root water uptake was coupled with a random field generation method to produce realizations of soil hydraulic properties, and the effects of soil heterogeneity were analyzed under realistic atmospheric boundary conditions. Four soil types and three different soil heterogeneity levels were considered to investigate the response of soil heterogeneity on soil water dynamics. It was observed that the effect of soil heterogeneity varies across soil types and its effect was more pronounced in silty clay loam soils. Further, an increase in soil heterogeneity level can increase or decrease the mean soil moisture, hydraulic conductivity, and flow velocity based on land cover, upper boundary condition, and atmospheric demand, and its effect was higher during irrigation periods. A maximum difference of 30% in soil moisture values was observed near the surface soil layers due to an increase in soil heterogeneity on irrigation days. The mean soil moisture content under rice crop cover was always lower than that of wheat crop cover due to higher atmospheric demand in the former. Soil layering effect on soil moisture dynamics, which is related to capillary and hydraulic barrier formation, was found to depend on the layering configuration, available soil moisture, and land cover. The numerical experiments provided useful insights into the complex soil moisture dynamics in agricultural fields.

Uncertainty and Risk Evaluation of Deep Geothermal Energy Source for Heat Production and Electricity Generation in Remote Northern Regions

The Canadian off-grid communities heavily rely on fossil fuels. This unsustainable energetic framework needs to change, and deep geothermal energy can play an important role. However, limited data availability is one of the challenges to face when evaluating such resources in remote areas. Thus, a first-order assessment of the geothermal energy source is, therefore, needed to trigger interest for further development in northern communities. This is the scope of the present work. Shallow subsurface data and outcrop samples treated as subsurface analogs were used to infer the deep geothermal potential beneath the community of Kuujjuaq (Nunavik, Canada). 2D heat conduction models with time-varying upper boundary condition reproducing climate events were used to simulate the subsurface temperature distribution. The available thermal energy was inferred with the volume method. Monte Carlo-based sensitivity analyses were carried out to determine the main geological and technical uncertainties on the deep geothermal potential and risk analysis to forecast future energy production. The results obtained, although speculative, suggest that the old Canadian Shield beneath Kuujjuaq host potential to fulfill the community’s annual average heating demand of 37 GWh. Hence, deep geothermal energy can be a promising solution to support the energy transition of remote northern communities.

A modeling study of the impact of stratospheric intrusion on ozone enhancement in the lower troposphere over the Hong Kong regions, China

Stratospheric intrusion is of great importance to transporting ozone (O3)-enriched air from stratosphere into the middle and lower troposphere. The Weather Research and Forecasting model coupled with Chemistry (WRF-Chem) was employed to simulate a typical stratospheric intrusion episode occurring on March 24, 2010 and examine the impact of stratospheric intrusion on the O3 enhancement in the lower troposphere. An upper boundary condition scheme was applied to the simulations to overcome the limitation of WRF-Chem in which O3 chemistry in the stratosphere is not included. The simulations were evaluated with various observational data including O3 sounding, surface O3, as well as the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim reanalysis data. The WRF-Chem showed reasonable performance on simulating the meteorological processes that control the stratospheric intrusion. Several findings were identified from the detailed analyses of the modeling results. First, the subtropical high-level jet was responsible for this stratospheric intrusion event. When Hong Kong was located at the south side of the subtropical jet where subsidence was dominated, the O3-enriched air was transported from stratosphere to troposphere. Second, the stratospheric intrusion caused substantial enhancement of O3 with the maximum O3 concentration higher than 80.0 ppbv in the lower above the atmospheric boundary layer in the Hong Kong region. Finally, the process analyses (PA) method is performed to quantify the impact of stratosphere intrusion on the O3 enhancement in the lower troposphere. The PA result illustrates that a substantial increase in the lower troposphere O3 is highly associated with vertical transport from the upper troposphere. These results indicated important role of stratospheric intrusion on O3 enhancement events and more observations are advised to better constrain and quantify its effects.

Analytical study of Love wave propagation in functionally graded piezo-poroelastic media with electroded boundary and abruptly thickened imperfect interface

An analytical study of propagation characteristic of Love wave in a layered structure comprising of a functionally graded piezo-poroelastic stratum with electroded upper boundary condition and a functionally graded piezo-poroelastic substrate having abrupt thickening at the common imperfect interface has been encapsulated in the present work which was not attempted before. Two distinct shapes of abrupt thickening are considered, viz. rectangular and parabolic. Dispersion equations are obtained using Fourier transformation and perturbation method and the results have been validated with classical cases. Numerical computations have been carried out for functionally graded piezo-poroelastic PZT-5H stratum and BaTiO3 substrate. The effect of various influencing parameters, viz. abrupt thickness ratio, functional gradient parameters, mechanical imperfect bonding, and electrical imperfect bonding parameter on the phase velocity of Love wave has been illustrated graphically. The study may find its application in the field of Love wave sensors, actuators, health monitoring system, and other SAW devices.

Barotropic modes, baroclinic modes and equivalent depths in the atmosphere.

The equivalent depth of a rotating fluid layer determines the phase speed of all free (i.e. unforced) linear waves that propagate in it. The equivalent depth can be estimated from the eigenvalues of a second‐order differential equation for the vertical dynamics where the mean temperature profile is a coefficient. The eigenfunctions dictate the vertical structure of the modes. This work combines analytic and numerical solutions of the vertical structure equation for various mean temperature profiles and for various combinations of top and bottom boundary conditions relevant to two atmospheric configurations: troposphere and combined troposphere‐stratosphere. Our formulation provides a clear definition of the barotropic mode and the countable baroclinic modes (including the special n = 0 mode). The barotropic mode exists only when the lower boundary condition is the vanishing of the vertical velocity (i.e. when a solid boundary bounds the layer from below) and the equivalent depth of this mode is about 10 km. The n = 0 baroclinic mode does not exist in a layer whose thickness exceeds a threshold value of about 20 km and therefore this mode does not exist in the combined troposphere‐stratosphere layer. The upper boundary condition affects the eigenvalues much more so than the details of the temperature profile, as the details of the temperature profile only affect the equivalent depth in the troposphere. Increasing the height of the upper boundary has little effect on the barotropic mode, but strongly influences the phase speed and vertical structure of the baroclinic modes; a general circulation model with a lid at or below the stratopause where, for example, planetary Rossby waves are present will therefore be incapable of correctly simulating the interaction of these waves with the mean flow. The values of the equivalent depth of baroclinic modes are approximately 10% of the actual layer's depth in all realistic cases.