A theory of convection: Modelling by two buoyant interacting fluids

Abstract

A new model of convection and mixing is presented. The fluid is envisioned as being composed of two buoyant interacting fluids, called thermals and anti-thermals. In the context of the Boussinesq approximation, pairs of governing equations are derived for thermals and anti-thermals. Each pair meets an Invariance Principle as a consequence of the reciprocity in the roles played by thermals and anti-thermals. Each pair is transformed into an average equation for which interaction terms cancel and another very simple equation linking the two fluid properties. An important parameter of the model is the fraction, f, of area occupied by thermals to the total area. A dynamic saturation equilibrium between thermals and antithermals is assumed. This implies a constant values of f throughout the system. The set of equations is written in terms of mean values and root-mean-square fluctuations, in keeping with equations of turbulence theories. The final set consists of four coupled non-linear differential equations. The model neglects dissipation and can be applied to any convective situations where molecular viscosity and diffusivity may be neglected. Applications of the model to mixed-layer deepening and penetrative convection are presented in subsequent papers.     

The motion of magnetic fronts in spiral galaxies

Abstract

We study the nonlinear asymptotic thin disc approximation to the mean field dynamo equations, as applicable to spiral galaxies. The circumstances in which sharp magnetic field structures (fronts) can propagate radially are investigated, and an expression for the speed of propagation derived. We find that the speed of an interior front is proportional to η//R _? (where η is the diffusivity and R_t the galactic radius), whereas an exterior front moves with speed of order , where γ is the local growth rate of the dynamo. Numerical simulations are presented, that agree well with our asymptotic results. Further, we perform numerical experiments using the 'no-z' approximation for thin disc dynamos, and show that the propagation of magnetic fronts in this approximation can also be understood in terms of our asymptotic results.     

Automated first and second order moment equations for a set of stochastic differential equations of type AŻ+BZ=C(t)

The generation of the second and higher order moment equations for a set of stochastic differential equations based on Ito's differential lemma is difficult, even for small system of equations. From the knowledge of the statistical properties of the Gaussian white noises associated with the parameters and input coefficients of a set of stochastic differential equations of typeA.+B.Z=C(t), a way to automatically generate the second order moment equations in a computer is presented in this paper. The resulting set of first and second order moment equations is also presented in the same state-space form of the original set of stochastic differential equations through a vectorization of the correlation matrix, which takes advantage of its symmetry. The procedure involved here avoids the inversion of matrixA to apply Ito's differential lemma. Therefore, the presented numerical implementation reduces the computational effort required in the formulation and solution of the moment equations. Moreover, other robust and efficient numerical deterministic integration schemes can be equally applied to the solution of the moment equations.     

Point-source inertial particle dispersion

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,?v,?t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x,?v,?t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x,?v)-variables decouple and the determination of p(x,?v,?t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity.     

3. Hydrometry

Abstract

The derived model achieves the rainfall/discharge conversion by means of two individual equations, namely, a production function and a modulation function.     

Incorporating velocity shear into the magneto-Boussinesq approximation

Motivated by consideration of the solar tachocline, we derive, via an asymptotic procedure, a new set of equations incorporating velocity shear and magnetic buoyancy into the Boussinesq approximation. We demonstrate, by increasing the magnetic field scale height, how these equations are linked to the magneto-Boussinesq equations of Spiegel and Weiss (Magnetic buoyancy and the Boussinesq approximation. Geophys. Astrophys. Fluid Dyn. 1982, 22, 219–234).     

Magnetic buoyancy and the Boussinesq approximation

Abstract

The full Boussinesq equations for hydromagnetic convection are derived and shown to include the effects of magnetic buoyancy. Instabilities caused by magnetic buoyancy are analyzed and their roles in double convection are brought out.     

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.     

Inferring groundwater system dynamics from hydrological time-series data

Abstract

The problem of identifying and reproducing the hydrological behaviour of groundwater systems can often be set in terms of ordinary differential equations relating the inputs and outputs of their physical components under simplifying assumptions. Conceptual linear and nonlinear models described as ordinary differential equations are widely used in hydrology and can be found in several studies. Groundwater systems can be described conceptually as an interlinked reservoir model structured as a series of nonlinear tanks, so that the groundwater table can be schematized as the water level in one of the interconnected tanks. In this work, we propose a methodology for inferring the dynamics of a groundwater system response to rainfall, based on recorded time series data. The use of evolutionary techniques to infer differential equations from data in order to obtain their intrinsic phenomenological dynamics has been investigated recently by a few authors and is referred to as evolutionary modelling. A strategy named Evolutionary Polynomial Regression (EPR) has been applied to a real hydrogeological system, the shallow unconfined aquifer of Brindisi, southern Italy, for which 528 recorded monthly data over a 44-year period are available. The EPR returns a set of non-dominated models, as ordinary differential equations, reproducing the system dynamics. The choice of the representative model can be made both on the basis of its performance against a test data set and based on its incorporation of terms that actually entail physical meaning with respect to the conceptualization of the system.

Citation Doglioni, A., Mancarella, D., Simeone, V. & Giustolisi, O. (2010) Inferring groundwater system dynamics from hydrological time-series data. Hydrol. Sci. J. 55(4), 593–608.     

Large-scale vortices in helical turbulence of incompressible fluid

Abstract

The interaction of a mean flow with a random fluctuation field is considered. This interaction is described by the averaged Navier-Stokes equation in which terms nonlinear in the fluctuation field are expressed in terms of the mean flow and the statistical properties of the fluctuation field, which is assumed to be homogeneous, isotropic, and helical. Averaged equations are derived using a functional technique. These equations are solved for a mean background flow that depends linearly on the position vector. The solutions show that large-scale vortices may arise in this system.     

A modal α2-dynamo in the limit of asymptotically small viscosity

Abstract

A spherical α²-dynamo is presented as an expansion in the free decay modes of the magnetic field. In the limit of vanishing viscosity the momentum equation yields various asymptotic expansions for the flow, depending on the precise form of the dissipation and boundary conditions applied. A new form for the dissipation is introduced that greatly simplifies this asymptotic expansion. When these expansions are substituted back into the induction equation, a set of modal amplitude equations is derived, and solved for various distributions of the α-effect. For all choices of α the solutions approach the Taylor state, but the manner in which this occurs can vary, as previously found by Soward and Jones (1983). Furthermore, as hypothesized by Malkus and Proctor (1975), but not previously demonstrated, the post-Taylor equilibration is indeed independent of the viscosity in the asymptotic limit, and depending on the choice of a may be either steady-state or oscillatory.     

PERSONALIA / PERSONALIA

Abstract

A stepwise multiple linear regression analysis was used to study factors that affect the mean annual rainfall in Israel. The factors investigated were latitude, elevation, and easterly distance from the sea. Separate equations, relating the 30-year mean rainfall to these factors were derived for the northern and southern parts of the country. The regression equations reduced most of the variance of the rainfall data for the 211 rain-gaging stations included in the study.     

A class of analytic solutions of the magnetic force-free field equations

Abstract

Formation of electric current sheets in the corona is thought to play an important role in solar flares, prominences and coronal heating. It is therefore of great interest to identify magnetic field geometries whose evolution leads to variations in B over small length-scales. This paper considers a uniform field B ₀[zcirc], line-tied to rigid plates z = ±l, which are then subject to in-plane displacements modeling the effect of photospheric motion. The force-free field equations are formulated in terms of field-line displacements, and when the imposed plate motion is a linear function of position, these reduce to a 4 × 4 system of nonlinear, second-order ordinary differential equations. Simple analytic solutions are derived for the cases of plate rotation and shear, which both tend to form singularities in certain parameter limits. In the case of plate shear there are two solution branches—a simple example of non-uniqueness.     

Stability of the subseismic wave equation for the Earth's fluid core

Abstract

The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.     

Homogeneous region delineation based on annual flood generation mechanisms

Abstract

Flood distributions can have unimodal or multimodal densities due to different flood generation mechanisms such as snowmelt and rainfall in the annual flood series. When applying nonparametric frequency analysis to annual flood data from the province of New Brunswick in Canada, unimodal, bimodal and heavy-tailed distribution shapes were found. By grouping basins with similarly-shaped densities on a geographical basis, homogeneous regions were delineated. Regional equations derived for a homogeneous region gave lower integral square errors than those of province-wide equations.     

Dynamic model of mesoscale eddies. Eddy parameterization for coarse resolution ocean circulation models

In the framework of the eddy dynamic model developed in two previous papers (Dubovikov, M.S., Dynamical model of mesoscale eddies, Geophys. Astophys. Fluid Dyn., 2003, 97, 311–358; Canuto, V.M. and Dubovikov, M.S., Modeling mesoscale eddies, Ocean Modelling, 2004, 8, 1–30 referred as I–II), we compute the contribution of unresolved mesoscale eddies to the large-scale dynamic equations of the ocean. In isopycnal coordinates, in addition to the bolus velocity discussed in I–II, the mesoscale contribution to the large scale momentum equation is derived. Its form is quite different from the traditional down-gradient parameterization. The model solutions in isopycnal coordinates are transformed to level coordinates to parameterize the eddy contributions to the corresponding large scale density and momentum equations. In the former, the contributions due to the eddy induced velocity and to the residual density flux across mean isopycnals (so called Σ-term) are derived, both contributions being shown to be of the same order. As for the large scale momentum equation, as well as in isopycnal coordinates, the eddy contribution has a form which is quite different from the down-gradient expression.     

On the wave field generated by a variable wind

Abstract

By using an empirical expression relating the rate of increase in wave energy to the local wind speed, an equation for the phase speed at the peak of the wave spectrum is derived. The solution of the equation is determined for some simple wind fields. In particular, the wave field caused by a localised storm moving steadily over an unbounded ocean is considered. It is also shown that only a small fraction of the momentum transferred from the wind into the water propagates away from a local storm area in the form of wave momentum.     

Radial expansion of the magnetohydrodynamic equations for axially symmetric configurations

Abstract

We introduce a general expansion approach to obtain a fully consistent closed set of magnetohydrodynamic equations in two independent variables, which is particularly useful to describe axially symmetric, time-dependent problems with weak variation of all quantities in the radial direction. This is done by considering the hierarchy of expanded magnetofluid equations in cylindrical coordinates and equating terms with equal powers in the radial coordinate r. From geometrical considerations it is shown that the radial expansions of the pertaining physical quantities are either even series or odd series in r; this introduces a significant reduction in the number of variables and equations. The closure of the system is provided by appropriate boundary conditions. Among other possible applications, the method is relevant for the analysis of structure and dynamics of magnetic field concentrations in stellar atmospheres.     

Effect of pressure gradients and line-tying on the magnetic stability of coronal loops

Abstract

In this paper we study the stability of an idealised magnetostatic coronal loop, incorporating both the effect of line-tying, due to the dense photosphere, and of pressure gradients. The stability equations may be solved analytically for our particular equilibrium. From the marginally stable case, the critical conditions separating instability from stability are derived. It is found that stretching or twisting a loop eventually makes it kink unstable.     

Improving rainfall–runoff modelling through the control of uncertainties under increasing climate variability in the Ouémé River basin (Benin,West Africa)

ABSTRACT

The objective of this paper is to understand how the natural dynamics of a time-varying catchment, i.e. the rainfall pattern, transforms the random component of rainfall and how this transformation influences the river discharge. To this end, this paper develops a rainfall–runoff modelling approach that aims to capture the multiple sources and types of uncertainty in a single framework. The main assumption is that hydrological systems are nonlinear dynamical systems which can be described by stochastic differential equations (SDE). The dynamics of the system is based on the least action principle (LAP) as derived from Noether’s theorem. The inflow process is considered as a sum of deterministic and random components. Using data from the Ouémé River basin (Benin, West Africa), the basic properties for the random component are considered and the triple relationship between the structure of the inflowing rainfall, the corresponding SDE that describes the river basin and the associated Fokker-Planck equations (FPE) is analysed.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR D. Gerten