Baroclinic instability with variable static stability—a design study for a spherical atmospheric model experiment

Abstract

Exact solutions are obtained for a quasi-geostrophic baroclinic stability problem in which the rotational Froude number (inverse Burger number) is a linear function of the height. The primary motivation for this work was to investigate the effect of a radially-variable, dielectric body force, analogous to gravity, on baroclinic instability for the design of a spherical, synoptic-scale, atmospheric model experiment for a Spacelab flight. Such an experiment cannot be realized in a laboratory on the Earth's surface because the body force cannot be made strong enough to dominate terrestrial gravity. Flow in a rotating, rectilinear channel with a vertically variable body force and with no horizontal shear of the basic state is considered. The horizontal and vertical temperature gradients of the basic and reference states are taken as constants. Consequences of the body force variation and the other assumptions of the model are that the static stability (Brunt-Väisälä frequency squared) and the vertical shear of the basic state flow have the same functional form and that the transverse gradient of the potential vorticity of the basic state vanishes. The solutions show that the stability characteristics of the model are qualitatively similar to those of Eady's model. A short wavelength cutoff and a wavenumber of maximum growth rate are present. Further, the stability characteristics are quantitatively similar to Eady's results for parameters based on the vertically averaged Brunt-Väisälä: frequency. The solutions also show that the temperature amplitude distribution is particularly sensitive to the vertical variation of the static stability. For the static stability and shear decreasing (increasing) with height a relative enhancement of the temperature amplitude occurs at the lower (upper) surface. The other amplitudes and phases are only slightly influenced by the variation. The implication for the Spacelab experiment is that the variable body force will not significantly alter the dynamics from the constant gravity case. The solutions can be relevant to other geophysical fluid flows, including the atmosphere, ocean and annulus system in which the static stability undergoes variation with height.     

Reflection of hydrostatic mountain waves from spatially nonuniform layers

Abstract

The steady, hydrostatic, inviscid, Boussinesq flow of a stably stratified fluid over a bell-shaped ridge is investigated within the framework of a linear model. The three layer model atmosphere introduced is such that the Brunt-Väisälä frequency is constant in each layer but the interfaces of the middle layer are allowed to vary gently in the cross-ridge direction. In essence, the model can be tuned in both vertical and horizontal directions. These cross-ridge variations can produce significant differences in both the cross-ridge surface wind and the surface drag compared to the response obtained by use of a horizontally uniform reflecting layer. These changes are sensitive to both the vertical location of the middle layer and to the slope of its lower interface at the ridge crest. Many of these features are explained by means of a conventional layered-model analysis.     

The water column distributions of thorium isotopes in the western North Pacific

Four vertical profiles of²³⁰Th and²²⁸Th were determined using large volume water samples in the western North Pacific. An almost linear increase of²³⁰Th with depth was observed for all of the profiles for which the unidirectional first order scavenging model was difficult to explain. We developed a model which included a dissolved-particulate transformation as well as parameters of the scavenging model. Application of the model to the vertical distributions of total and the GEOSECS particulate Th isotopes (²³⁰Th and²³⁴Th) yielded the residence time of dissolved Th with respect to adsorption to particles and the turnover time of particulate Th to be 235 days and 57 days, respectively. The Th isotopes appeared to be carried down the water column by fine particles with a mean settling velocity of 1 m/day which continually release Th into sea water as well as pick up Th from the water along their journey to the bottom.For²²⁸Th, a large excess over²³²Th was observed throughout the water column with pronounced high concentrations in surface and bottom waters, suggesting that the²²⁸Th was derived from²²⁸Ra diffused out of sediments. The vertical distributions of²²⁸Th seemed to be significantly influenced by lateral mixing along isopycnals.     

On the stability of stratified plane couette flow

Abstract

Unbounded stratified plane Couette flow is shown to be stable against small amplitude disturbances. The Brunt-Väisälä frequency is assumed to be constant. Both viscosity and thermal diffusion are included, and shown to be stabilizing.     

The scattering of Rossby waves from finite abrupt topography

The scattering of first mode linear baroclinic Rossby waves by a top-hat ridge in a continuously stratified ocean, with Brunt-Väisälä frequency that decays exponentially with depth below a surface mixed layer, is the subject of this study. A numerical mode matching technique is used to calculate the transmission coefficients for the propagating modes over the ridge. It is found that the scattered field depends crucially upon the stratification. For example, when the majority of the density variation is confined to a thin thermocline, corresponding to a small e-folding scale, gamma ^?1, for the Brunt-Väisälä frequency, a large amount of the incident wave energy is reflected by a small amplitude ridge. Appreciable energy conversion between the propagating barotropic and baroclinic modes takes place in this case. An asymptotic analysis for a small amplitude ridge is presented that confirms these numerical results. In the limit gamma ^?1→ 0, it is demonstrated that the scattered field in the continuously stratified ocean model differs markedly from the two-layer solution. The latter does not exhibit appreciable reflection of the incident wave energy for a small amplitude ridge. In conclusion, the application of a two-layer ocean model to describe Rossby wave scattering by ridges in place of a continuously stratified model cannot be recommended.     

Transitions to broad cells in a nonlinear thermal convection system

Abstract

This paper explores the properties of a two-dimensional, Boussinesq convection model with an ad hoc term in the buoyancy tendency equation that represents a positive external feedback process acting on the buoyancy fluctuations. Linear stability analyses and nonlinear integrations are presented for the case of constant heat flux boundary conditions. Although the large wavenumber modes grow the fastest from a state of rest, the nonlinear solutions progressively evolve to cells of small wavenumber. Applications to mesoscale cellular convection in the atmosphere are discussed.     

Asymptotic aspects of the Boussinesq approximation for gases and liquids

Abstract

The asymptotic formulation of the Boussinesq approximation relates the pressure of the fluid to a thermodynamical quantity involving the heat capacity c_Po . In this paper we examine the implications of such a scaling, in particular: (i) the singular degeneracy of the equation of state ρ = ρ* (1 ?α* (T?;T*)) of a liquid: this equation of state is valid only for small values of the coefficient α T*; (ii) in which manner the scaling introduces the Mach number of the flow as a small parameter e for a compressible fluid. The equations at order zero with respect to ? are the same equations for gases and for liquids only if the thermodynamics of the medium is described by using the Brunt-Väisälä frequency instead of the temperature.     

Land-cover impacts on streamflow: a change-detection modelling approach that incorporates parameter uncertainty

Abstract

The effect of land-use or land-cover change on stream runoff dynamics is not fully understood. In many parts of the world, forest management is the major land-cover change agent. While the paired catchment approach has been the primary methodology used to quantify such effects, it is only possible for small headwater catchments where there is uniformity in precipitation inputs and catchment characteristics between the treatment and control catchments. This paper presents a model-based change-detection approach that includes model and parameter uncertainty as an alternative to the traditional paired-catchment method for larger catchments. We use the HBV model and data from the HJ Andrews Experimental Forest in Oregon, USA, to develop and test the approach on two small (<1 km²) headwater catchments (a 100% clear-cut and a control) and then apply the technique to the larger 62 km² Lookout catchment. Three different approaches are used to detect changes in stream peak flows using: (a) calibration for a period before (or after) change and simulation of runoff that would have been observed without land-cover changes (reconstruction of runoff series); (b) comparison of calibrated parameter values for periods before and after a land-cover change; and (c) comparison of runoff predicted with parameter sets calibrated for periods before and after a land-cover change. Our proof-of-concept change detection modelling showed that peak flows increased in the clear-cut headwater catchment, relative to the headwater control catchment, and several parameter values in the model changed after the clear-cutting. Some minor changes were also detected in the control, illustrating the problem of false detections. For the larger Lookout catchment, moderately increased peak flows were detected. Monte Carlo techniques used to quantify parameter uncertainty and compute confidence intervals in model results and parameter ranges showed rather wide distributions of model simulations. While this makes change detection more difficult, it also demonstrated the need to explicitly consider parameter uncertainty in the modelling approach to obtain reliable results.

Citation Seibert, J. & McDonnell, J. J. (2010) Land-cover impacts on streamflow: a change-detection modelling approach that incorporates parameter uncertainty. Hydrol. Sci. J. 55(3), 316–332.     

Convection in large bodies of water open to the atmosphere

Abstract

Convection in large bodies of water open to the atmosphere is studied as a two-layer system, which allows for the accurate modeling of the interfacial boundary conditions. The Biot number associated with the heat transfer across the interface becomes a result of the analysis rather than a condition. The linear and nonlinear analyses yield essentially the same stability results, providing a sharp stability limit.     

Effect of modelling scale on the assessment of climate change impact on river runoff

Abstract

The effect of using two distributed hydrological models with different degrees of spatial aggregation on the assessment of climate change impact on river runoff was investigated. Analyses were conducted in the Narew River basin situated in northeast Poland using a global hydrological model (WaterGAP) and a catchment-scale hydrological model (SWAT). Climate change was represented in both models by projected changes in monthly temperature and precipitation between the period 2040–2069 and the baseline period, resulting from two general circulation models: IPSL-CM4 and MIROC3.2, both coupled with the SRES A2 emissions scenario. The degree of consistency between the global and the catchment model was very high for mean annual runoff, and medium for indicators of high and low runoff. It was observed that SWAT generally suggests changes of larger magnitude than WaterGAP for both climate models, but SWAT and WaterGAP were consistent as regards the direction of change in monthly runoff. The results indicate that a global model can be used in Central and Eastern European lowlands to identify hot-spots where a catchment-scale model should be applied to evaluate, e.g. the effectiveness of management options.

Editor D. Koutsoyiannis; Associate editor F.F. Hattermann

Citation Piniewski, M., Voss, F., Bärlund, I., Okruszko, T., and Kundzewicz. Z.W., 2013. Effect of modelling scale on the assessment of climate change impact on river runoff. Hydrological Sciences Journal, 58 (4), 737–754.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

Simple water balance modelling of surface reservoir systems in a large data-scarce semiarid region / Modélisation simple du bilan hydrologique de systèmes de réservoirs de surface dans une grande région semi-aride pauvre en données

Abstract

Abstract Water resources in dryland areas are often provided by numerous surface reservoirs. As a basis for securing future water supply, the dynamics of reservoir systems need to be simulated for large river basins, accounting for environmental change and an increasing water demand. For the State of Ceará in semiarid Northeast Brazil, with several thousands of reservoirs, a simple deterministic water balance model is presented. Within a cascade-type approach, the reservoirs are grouped into six classes according to storage capacity, rules for flow routing between reservoirs of different size are defined, and water withdrawal and return flow due to human water use is accounted for. While large uncertainties in model applications exist, particularly in terms of reservoir operation rules, model validation against observed reservoir storage volumes shows that the approach is a reasonable simplification to assess surface water availability in large river basins. The results demonstrate the large impact of reservoir storage on downstream flow and stress the need for a coupled simulation of runoff generation, network redistribution and water use.     

A numerical study of the string function using a primitive equation ocean model

Abstract

We use results from a primitive-equation ocean numerical model (SCRUM) to test a theoretical 'string function' formulation put forward by Tyler and Käse in another article in this issue. The string function acts as a stream function for the large-scale potential energy flow under the combined beta and topographic effects. The model results verify that large-scale anomalies propagate along the string function contours with a speed correctly given by the cross-string gradient. For anomalies having a scale similar to the Rossby radius, material rates of change in the layer mass following the string velocity are balanced by material rates of change in relative vorticity following the flow velocity. It is shown that large-amplitude anomalies can be generated when wind stress is resonant with the string function configuration.     

Finite amplitude instability thresholds in penetrative convection

Abstract

A nonlinear energy stability analysis is presented for the penetrative convection model of Veronis (1963). For top temperatures between 4°C and 8°C the nonlinear stability boundary obtained is very close to the linear one of Veronis and enables a region of possible sub-critical instabilities to be determined.     

Influence of climate on alpine stream chemistry and water sources

The resilience of alpine/subalpine watersheds may be viewed as the resistance of streamflow or stream chemistry to change under varying climatic conditions, which is governed by the relative size (volume) and transit time of surface and subsurface water sources. Here, we use end‐member mixing analysis in Andrews Creek, an alpine stream in Rocky Mountain National Park, Colorado, from water year 1994 to 2015, to explore how the partitioning of water sources and associated hydrologic resilience change in response to climate. Our results indicate that four water sources are significant contributors to Andrews Creek, including snow, rain, soil water, and talus groundwater. Seasonal patterns in source‐water contributions reflected the seasonal hydrologic cycle, which is driven by the accumulation and melting of seasonal snowpack. Flushing of soil water had a large effect on stream chemistry during spring snowmelt, despite making only a small contribution to streamflow volume. Snow had a large influence on stream chemistry as well, contributing large amounts of water with low concentrations of weathering products. Interannual patterns in end‐member contributions reflected responses to drought and wet periods. Moderate and significant correlations exist between annual end‐member contributions and regional‐scale climate indices (the Palmer Drought Severity Index, the Palmer Hydrologic Drought Index, and the Modified Palmer Drought Severity Index). From water year 1994 to 2015, the percent contribution from the talus‐groundwater end member to Andrews Creek increased an average of 0.5% per year (p < 0.0001), whereas the percent contributions from snow plus rain decreased by a similar amount (p = 0.001). Our results show how water and solute sources in alpine environments shift in response to climate variability and highlight the role of talus groundwater and soil water in providing hydrologic resilience to the system.     

MHD stability of a filament immersed in a solar bipolar magnetic region

Abstract

The stability properties of the filament model of Low (1981) are investigated. We use a two-dimensional formalism based on the magnetohydrodynamic energy principle of Bernstein et al. (1958). For the parameter range observed in quiescent prominences this model describes stable horizontal oscillations with periods of about 3–6 min. In other parameter ranges we find instability which is driven exclusively by compressional effects. The Lorentz force has a continuously stabilizing influence. In addition, it seems that gravity is practically unimportant for the stability state of the equilibrium.     

Time of concentration: a paradox in modern hydrology

Abstract

The time of concentration is a primary parameter for a variety of modern hydrological models adopted in professional and scientific communities. Nevertheless, a universally accepted working definition of this parameter is currently lacking and several definitions can be found in the technical literature along with related estimation procedures. This study brings to light the inherent variability of these definitions through the empirical analysis of four small basins. These case studies demonstrate that available approaches for the estimation of the time of concentration may yield numerical predictions that differ from each other by up to 500%.

Editor D. Koutsoyiannis

Citation Grimaldi, S., Petroselli, A., Tauro, F. and Porfiri, M., 2012. Time of concentration: a paradox in modern hydrology. Hydrological Sciences Journal, 57 (2), 217–228.     

A review of: “Mean-field magnetohydrodynamics and dynamo theory”

Abstract

By F. Krause and K.-H. Rädler. Pergamon, Oxford, 1981. 271 pp. (Cloth £15.00 ($36.00) (ISBN 0 08025 0416))     

Non linear waves in two dimensional keplerian flows

Abstract

Accretion discs in astrophysics are fundamental for converting gravitational binding energy into observed electromagnetic radiation. We study the behavior of waves in a two dimensional supersonic Keplerian flow inside a given gravitational potential. We present the effects of shearing and rotation on short waves, and the numerical study of the dynamical stability of such flows with respect to various perturbations. We show that a large class of dynamical effects, due to pressure and associated to short time scales, may be excited.     

Stability of model performance and parameter values on two catchments facing changes in climatic conditions

Abstract

Hydrological models are often used for studying the hydrological effects of climate change; however, the stability of model performance and parameter values under changing climate conditions has seldom been evaluated and compared. In this study, three widely-used rainfall–runoff models, namely the SimHYD model, the HBV model and the Xin’anjiang model, are evaluated on two catchments subject to changing climate conditions. Evaluation is carried out with respect to the stability in their performance and parameter values in different calibration periods. The results show that (a) stability of model performance and parameter values depends on model structure as well as the climate of catchments, and the models with higher performance scores are more stable in changing conditions; (b) all the tested models perform better on a humid catchment than on an arid catchment; (c) parameter values are also more stable on a humid catchment than on an arid catchment; and (d) the differences in stability among models are somewhat larger in terms of model efficiency than in model parameter values.