Nontraditional Terms Research Articles

Balanced Convective Circulations in a Stratified Atmosphere. Part I: A Framework for Assessing Radiation, the Coriolis Force, and Drag

Abstract The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on an f plane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes.

Nontraditional hypsometric equation

AbstractThis study corrects the hypsometric equation by restoring the nontraditional terms to relax the hydrostatic approximation. The nontraditional terms include one Coriolis term and two metric terms in the vertical momentum equation. The hypsometric equations with and without the nontraditional correction are used to calculate the geopotential height of pressure levels using more than 300,000 selected tropical rawinsonde profiles. With westerlies between two pressure levels, the thickness of the layer increases, which reduces the upward pressure gradient forces to balance the upward nontraditional Coriolis forces; the opposite is true for easterlies. Hence, zonal winds are negatively correlated with traditional geopotential height biases aloft. At 500 hPa, for example, traditional geopotential height error in the tropical troposphere is on the order of at least 0.5 m, which is considerable with respect to geopotential height variability in tropical large‐scale flow, on the order of 10 to 15 m.

РЕЗУЛЬТАТИ ЕКСПЕРИМЕНТІВ ЗІ ШТУЧНОГО ОТРИМАННЯ ОВУЛЬОВАНОЇ ІКРИ ВІД ПЛІДНИКІВ СТЕРЛЯДІ ACIPENSER RUTHENUS (LINNAEUS) У НЕТРАДИЦІЙНІ СТРОКИ

РЕЗУЛЬТАТИ ЕКСПЕРИМЕНТІВ ЗІ ШТУЧНОГО ОТРИМАННЯ ОВУЛЬОВАНОЇ ІКРИ ВІД ПЛІДНИКІВ СТЕРЛЯДІ ACIPENSER RUTHENUS (LINNAEUS) У НЕТРАДИЦІЙНІ СТРОКИ

Geostrophic adjustment with gyroscopic waves: barotropic fluid without the traditional approximation

AbstractWe study geostrophic adjustment in rotating barotropic fluid when the angular speed of rotation $\boldsymbol{\Omega}$ does not coincide in direction with the acceleration due to gravity; the traditional and hydrostatic approximations are not used. Linear adjustment results in a tendency of any localized initial state towards a geostrophically balanced steady columnar motion with columns parallel to $\boldsymbol{\Omega}$. Nonlinear adjustment is examined for small Rossby numbers Ro and aspect ratio $H/L$ ($H$ and $L$ are the layer depth and the horizontal scale of motion), using multiple-time-scale perturbation theory. It is shown that an arbitrary perturbation is split in a unique way into slow and fast components evolving with characteristic time scales $(\mathit{Ro}f)^{-1}$ and $f^{-1}$, respectively, where $f $ is the Coriolis parameter. The slow component does not depend on depth and is close to geostrophic balance. On times $O(1/f\, \mathit{Ro})$ the slow component is not influenced by the fast one and is described by the two-dimensional fluid dynamics equation for the geostrophic streamfunction. The fast component consists of long gyroscopic waves and is a packet of inertial oscillations modulated by an amplitude depending on coordinates and slow time. On times $O(1/f\, \mathit{Ro})$ the fast component conserves its energy, but it is coupled to the slow component: its amplitude obeys an equation with coefficients depending on the geostrophic streamfunction. Under the traditional approximation, the inertial oscillations are trapped by the quasi-geostrophic component; ‘non-traditional’ terms in the amplitude equation provide a meridional dispersion of the packet on times $O(1/f\, \mathit{Ro})$, and, therefore, an effective radiation of energy from the initial perturbation domain. Another important effect of the non-traditional terms is that on longer times $O(1/f\, \mathit{Ro}^{2})$ a transfer of energy between the fast and the slow components becomes possible.

Equatorially Bounded Zonally Propagating Linear Waves on a Generalized β Plane

Abstract Meridionally confined zonally propagating wave solutions to the linear hydrostatic Boussinesq equations on a generalized equatorial β plane that includes the “nontraditional” Coriolis force terms associated with the poleward component of planetary rotation are calculated. Kelvin, Rossby, inertia–gravity, and mixed Rossby–gravity modes generalize from the traditional model with the dispersion relation unchanged. The effects of the nontraditional terms on all waves are the curving upward with latitude of the surfaces of constant phase and the equatorial trapping width of the solutions (the equatorial radius of deformation) increasing by order (Ω/N)2 compared to the traditional case, where Ω is the planetary rotation rate and N the buoyancy frequency. In addition, for the Rossby, inertia–gravity, and mixed Rossby–gravity modes, there is a phase shift of O(Ω/N) in the zonal and vertical velocity components relative to the meridional component, and their spatial structures are further modified by differences of O(Ω/N)2. For the Rossby and inertia–gravity waves, the modifications depend also on the phase speed of the wave. In the limit N ≫ Ω, the traditional approximation is justified and lines of constant phase in the y–z plane become horizontal, whereas for N ≪ Ω phase lines become everywhere almost parallel to the planetary rotation vector. In both limits, the phase lines are perpendicular to the dominant restoring force—respectively, gravity and the centrifugal force associated with the solid-body rotation of the atmosphere at rest in the rotating frame.

Women’s Employment and its Effects on Bengali Households of Shillong, India

This paper examines the effect of the employment of women on the structure and functions of households within the Bengali community in a part of northeastern India. Despite its demographic significance, this cultural grouping remains largely unstudied from a sociological point of view. The results of the present study show that the entrance of women into the paid work force, the rate of which has increased in recent years, has had a partially transformative effect on the household, in the sense that responsibility for domestic tasks and for domestic decision-making is increasingly being shared by both spouses. In addition, unmarried women are coming to view the marriage bargain in distinctly non-traditional terms. The results suggest that paid employment has the potential to alter deeply-embedded and well-established cultural norms.

Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients

In order to capture the complexities of two-phase flow in heterogeneous porous media, we have used the method of large-scale averaging and spatially periodic models of the local heterogeneities. The analysis leads to the large-scale form of the momentum equations for the two immiscible fluids, a theoretical representation for the large-scale permeability tensor, and a dynamic, large-scale capillary pressure. The prediction of the permeability tensor and the dynamic capillary pressure requires the solution of a large-scale closure problem. In our initial study (Quintard and Whitaker, 1988), the solution to the closure problem was restricted to the quasi-steady condition and small spatial gradients. In this work, we have relaxed the constraint of small spatial gradients and developed a dynamic solution to the closure problem that takes into account some, but not all, of the transient effects that occur at the closure level. The analysis leads to continuity and momentum equations for theβ-phase that are given by $$\begin{gathered} \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}} + \nabla \cdot \left\{ {\langle V_\beta \rangle } \right\} = 0, \hfill \\ \left\{ {\langle V_\beta \rangle } \right\} = - \frac{1}{{\mu _\beta }}K_\beta ^* \cdot \left( {\nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \rho _\beta g} \right) - u_\beta \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}}^* - U_\beta \cdot \nabla \frac{{\partial \left\{ {\varepsilon _\beta } \right\}^* }}{{\partial t}} - \hfill \\ - \frac{1}{{\mu _\beta }}\mathcal{M}_\beta :\nabla \nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \frac{1}{{\mu _\beta }}\mathcal{R}_\beta :\nabla \Phi _\beta - \frac{1}{{\mu _\beta }}\Phi _\beta \hfill \\ \end{gathered} $$ . Here {〈vβ〉} represents the large-scale averaged velocity for theβ-phase, {eβ}* represents the largescale volume fraction for theβ-phase andKβ* represents the large-scale permeability tensor for theβ-phase. We have considered only the case of the flow of two immiscible fluids, thus the large-scale equations for theγ-phase are identical in form to those for theβ-phase. The terms in the momentum equation involving\({{\partial \left\{ {\varepsilon _\beta } \right\}^* } \mathord{\left/ {\vphantom {{\partial \left\{ {\varepsilon _\beta } \right\}^* } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}}\) and\({{\nabla \partial \left\{ {\varepsilon _\beta } \right\}^* } \mathord{\left/ {\vphantom {{\nabla \partial \left\{ {\varepsilon _\beta } \right\}^* } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}}\) result from the transient nature of the closure problem, while the terms containing\(\nabla \nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta ,\nabla \Phi _\beta \) andΦβ are the results of nonlinear variations in the large-scale field. All of the latter three terms are associated with second derivatives of the pressure and thus present certain unresolved mathematical problems. The situation concerning the large-scale capillary pressure is equally complex, and we indicate the functional dependence of {pc}c by $$\left\{ {p_c } \right\}^c = \mathcal{F}\left( {\partial \left\{ {\varepsilon _\beta } \right\}^* ,\left( {\rho _\gamma - \rho _\beta } \right)g,\nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta ,\frac{{\partial \left\{ {\varepsilon _\beta } \right\}^* }}{{\partial t}},etc.} \right)$$ . Because of the highly nonlinear nature of the capillary pressure-saturation relation, small causes can have significant effects, and the treatment of the large-scale capillary pressure is a matter of considerable concern. On the basis of the derived closure problems, estimates ofuβ, etc., are available and they clearly indicate that the nontraditional terms in the momentum equation can be discarded whenlH ≪ℒ. HerelH is the characteristic length scale for the heterogeneities and ℒ is the characteristic length scale for the large-scale averaged quantities. WhenlH is not small relative to ℒ, the nontraditional terms must be considered and nonperiodic boundary conditions must be developed for the closure problem. Detailed numerical studies presented in Part II (Quintard and Whitaker, 1990) and carefully documented experimental studies described in Part III (Berlin et al., 1990) provide further insight into the effects of large spatial and temporal gradients.

Flow in porous media II: The governing equations for immiscible, two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important whenμβ/μγ is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.