The boundary layer contribution to intertropical convergence zones in the quasi-equilibrium tropical circulation model framework

Theories for the position and intensity of precipitation over tropical oceans on climate time scales have a perplexing disagreement between those that focus on the momentum budget of the atmospheric boundary layer (ABL) and those that focus on thermodynamic factors. In the case of narrow intertropical convergence zones (ITCZs), there is some evidence for both classes of theories, and there are large open questions on the interpretation of the moist static energy (MSE) and momentum budgets of these regions. We develop a model in which both types of mechanisms can operate and the interaction between them can be analyzed. The model includes a mixed-layer ABL, coupled to a free troposphere whose vertical structure follows the quasi-equilibrium tropical circulation model (QTCM) of Neelin and Zeng. The case analyzed here is axisymmetric, using a fixed sea surface temperature (SST) lower boundary condition with an idealized off-equatorial SST maximum. We examine a regime with small values of the gross moist stability associated with tropospheric motions, which is realistic but poses theoretical challenges. In both rotating (equatorial β-plane) and nonrotating cases, the model ITCZ width and intensity are substantially controlled by the horizontal diffusion of moisture, which is hypothesized to be standing in for nonaxisymmetric transients. The inclusion of the ABL increases the amplitude and sharpness of the ITCZ, contributing to the importance of diffusion. Analytical solutions under simplifying assumptions show that the ABL contribution is not singular in the nondiffusive limit; it just features an ITCZ more intense than observed. A negative gross moist stability contribution associated with the flow component driven by ABL momentum dynamics plays a large role in this. Because of the ABL contribution, the flow imports, rather than exports, MSE in the ITCZ, but we show that this can be understood rather simply. The ABL contribution can be approximately viewed as a forcing to the tropospheric thermodynamics. The ABL forcing term is in addition to thermodynamic forcing by net flux terms in the MSE budget, which otherwise is much as in the standard QTCM. The ABL momentum budget suggests that divergent flow in the ABL is controlled to a significant extent by the pressure gradient imprinted on the ABL by the SST gradient—termed the Lindzen–Nigam contribution—although we also find that the thermodynamics mediating this is nontrivial, especially in the rotating case. Nonetheless, when this component of the pressure gradient is artificially removed, the peak ITCZ precipitation is reduced by a fraction on the order of 15 to 25%, less than might have been expected based on the diagnosis of the ABL momentum budget.     

大气运动不稳定的变分原理(Ⅱ)

<正> 7 分层流模式 假设有J薄层均匀流体,其上边界面、密度和速度分别由Z_k;ρ_k和V_k表示,k=1,2,…,J(图3)。我们有如下的基本方程组 (曾庆存,1979):      

A non-oscillatory balanced scheme for an idealized tropical climate model

We propose a non-oscillatory balanced numerical scheme for a simplified tropical climate model with a crude vertical resolution, reduced to the barotropic and the first baroclinic modes. The two modes exchange energy through highly nonlinear interaction terms. We consider a periodic channel domain, oriented zonally and centered around the equator and adopt a fractional stepping–splitting strategy, for the governing system of equations, dividing it into three natural pieces which independently preserve energy. We obtain a scheme which preserves geostrophic steady states with minimal ad hoc dissipation by using state of the art numerical methods for each piece: The f-wave algorithm for conservation laws with varying flux functions and source terms of Bale et al. (2002) for the advected baroclinic waves and the Riemann solver-free non-oscillatory central scheme of Levy and Tadmor (1997) for the barotropic-dispersive waves. Unlike the traditional use of a time splitting procedure for conservation laws with source terms (here, the Coriolis forces), the class of balanced schemes to which the f-wave algorithm belongs are able to preserve exactly, to the machine precision, hydrostatic (geostrophic) numerical-steady states. The interaction terms are gathered into a single second order accurate predictor-corrector scheme to minimize energy leakage. Validation tests utilizing known exact solutions consisting of baroclinic Kelvin, Yanai, and equatorial Rossby waves and barotropic Rossby wave packets are given.     

Structure of tropical variability from a vertical mode perspective

A composite mesoscale precipitation event and a convectively coupled Kelvin wave produced by a diabatically accelerated cloud resolving model are compared. Special emphasis is placed on the vertical structure of density and moisture perturbations and the interaction of these perturbations with the composited dynamical fields. Both composites share the same general features, a gradual deepening and strengthening of convection followed by deep convection and a stratiform region, quite similar in character to observations and some recent idealized models. Composited frozen moist static energy (FMSE) perturbations are several times larger than virtual temperature perturbations, suggesting moisture is a dominant regulator of convection. An empirically derived two vertical mode decomposition of the dynamical and moisture fields is found to reproduce both composites quite well. The leading vertical modes of FMSE and virtual temperature variability are strongly correlated with the modes of vertical velocity variability; these correlations are strongest at near-zero time lags. Deep convection is associated with moistening in the lower and middle troposphere, while shallow convection is associated with a moist lower troposphere and dry middle and upper troposphere. To the extent that our numerical model is realistic, the empirical modal decomposition provides support for the use of two-mode idealized models for convective interaction with large-scale circulations and guidance for formulating feedbacks between convection and the thermodynamic profile in such models. The FMSE budget leads to an interpretation of the convective life-cycle as a recharge–discharge mechanism in column-integrated FMSE. The budget analysis places diabatic forcing, surface and radiative fluxes into the moist energetic framework. In particular, these fluxes are seen to prolong active convection, but play a passive role in its initiation. The modally decomposed FMSE budget highlights the dynamical importance of the second baroclinic mode in moistening the lower and middle troposphere before convective onset (recharging), and then discharging stored FMSE in the stratiform region.     

Multicloud Convective Parametrizations with Crude Vertical Structure

Recent observational analysis reveals the central role of three multi-cloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating two-day waves, and the Madden–Julian oscillation. The authors have recently developed a systematic model convective parametrization highlighting the dynamic role of the three cloud types through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low level heating and cooling corresponding respectively to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation and a nonlinear switch which favors either deep or congestus convection depending on whether the troposphere is moist or dry. Here several new facets of these multi-cloud models are discussed including all the relevant time scales in the models and the links with simpler parametrizations involving only a single baroclinic mode in various limiting regimes. One of the new phenomena in the multi-cloud models is the existence of suitable unstable radiative convective equilibria (RCE) involving a larger fraction of congestus clouds and a smaller fraction of deep convective clouds. Novel aspects of the linear and nonlinear stability of such unstable RCE’s are studied here. They include new modes of linear instability including mesoscale second baroclinic moist gravity waves, slow moving mesoscale modes resembling squall lines, and large scale standing modes. The nonlinear instability of unstable RCE’s to homogeneous perturbations is studied with three different types of nonlinear dynamics occurring which involve adjustment to a steady deep convective RCE, periodic oscillation, and even heteroclinic chaos in suitable parameter regimes.     

A non-oscillatory balanced scheme for an idealized tropical climate model

We use the non-oscillatory balanced numerical scheme developed in Part I to track the dynamics of a dry highly nonlinear barotropic/baroclinic coupled solitary wave, as introduced by Biello and Majda (2004), and of the moisture fronts of Frierson et al. (2004) in the presence of dry gravity waves, a barotropic trade wind, and the beta effect. It is demonstrated that, for the barotropic/baroclinic solitary wave, except for a little numerical dissipation, the scheme utilized here preserves total energy despite the strong interactions and exchange of energy between the baroclinic and barotropic components of the flow. After a short transient period where the numerical solution stays close to the asymptotic predictions, the flow develops small scale eddies and ultimately becomes highly turbulent. It is found here that the interaction of a dry gravity wave with a moisture front can either result in a reflection of a fast moistening front or the pure extinction of the precipitation. The barotropic trade wind stretches the precipitation patches and increases the lifetime of the moisture fronts which decay naturally by the effects of dissipation through precipitation while the Coriolis effect makes the moving precipitation patches disappear and appear at other times and places.     

Interpretation of simple and cloud-resolving simulations of moist convection–radiation interaction with a mock-Walker circulation

An idealized two-dimensional mock-Walker circulation in the tropical atmosphere forced by prescribed horizontal gradients in sea-surface temperature (SST) is discussed. This model problem includes feedbacks between cumulus convection and tropical large-scale circulations that have proved challenging for global climate models to predict accurately. Three-dimensional cloud-resolving model (CRM) simulations that explicitly simulate turbulent circulations within individual cloud systems across 4,096 and 1,024 km-wide Walker circulations are compared with a simple theoretical model, the Simplified Quasiequilibrium Tropical Circulation Model (SQTCM). This theoretical model combines the weak-temperature-gradient approximation with a unimodal truncation of tropospheric vertical structure coupled to highly simplified formulations of moist precipitating cumulus convection and its cloud-radiative feedbacks. The rainfall, cloud and humidity distribution, circulation strength, energy fluxes and scaling properties are compared between the models. The CRM-simulated horizontal distribution of rainfall and energy fluxes are adequately predicted by the SQTCM. However, the humidity distribution (drier subsidence regions and high-humidity boundary layers in the CRM), vertical structure and domain-size scaling of the circulation differ significantly between the models. For the SQTCM, the concept of gross moist stability – related to advection of moist static energy (MSE) out of tropospheric columns by the mean divergent circulation – is used to explain the width and intensity of the rainy region. Column MSE budgets averaged across the ascent branch of the simulated Walker circulation provide similar insight into the cloud-resolving simulations after consideration of the more complex horizontal and vertical circulation structure and the role of transient eddies. A nondimensional ascent-region moist stability ratio α, analogous to the SQTCM gross moist stability, is developed. One term of α is related to the vertical profiles of ascent-region mean vertical motion and ascent-region edge MSE; a second term is proportional to eddy export from the ascent region. Smaller α induces a narrower, rainier ascent region. The sensitivity of the SQTCM and CRM to a uniform 2 K increase in SST is compared, and the rainy upward branch of the circulation narrows in both models. MSE budget arguments are used to explain this behavior. In the simple model, the gross moist stability is a decreasing function of tropospheric temperature. Hence gross moist stability reduces and the ascent region narrows as the SST increases. In the CRM, increased atmospheric radiative cooling due to the warmer and moister troposphere destabilizes the MSE profile and decreases α, inducing a narrower ascent region. In the CRM, and to a lesser extent in the SQTCM, intensified shortwave cloud forcing in the warmer climate causes a negative radiative feedback on the SST change.Funding for this work from NSF grant DMS-0139794 is gratefully appreciated     

Experiments to study interactions between baroclinic lower flows and a stably stratified upper layer

Experiments were conducted on a rotating fluid annulus to study the basic interactions between baroclinic lower flows and a stably stratified upper layer. Sufficiently stable stratification is necessary for steady flows to emerge in the lower layer. Upward fluid motions make the baroclinic flows permeate into the upper layer. The stable stratification, however, suppresses upward motions so that zonal fluid velocities decrease with height. In fact, their maximum appears at the top level of the baroclinic lower layer and the sign of the radial temperature gradient changes there; namely, it is warmer on the inner side of the annulus in the upper layer. This temperature profile is reflected in a meridional fluid circulation mixing both layers. In the upper layer of the wave flow, there exists a critical level below and above which the zonal fluid velocities have opposite directions for the wave to have a phase shift of half a wavelength in appearance. The experimental results correspond to real atmospheric phenomena.     

Equilibrium Statistical Predictions for Baroclinic Vortices: The Role of Angular Momentum

We develop a point-vortex equilibrium statistical model for baroclinic quasigeostrophic vortices within the context of a two-layer quasigeostrophic fluid that evolves in all of space. Angular momentum, which follows from the rotational symmetry of the unbounded domain, is the key conserved quantity, introducing a length scale that confines the most probable states of the statistical theory. We apply the theory as a model of localized convection in a preconditioned gyre. To illustrate this application, the preconditioned cyclonic, largely barotropic gyres are modeled as „zero inverse temperature” states, which are explicit solutions to the mean-field equations with a Gaussian probability distribution of vortices. Convection is modeled by a cloud of point-vortex hetons – purely baroclinic arrangements of point vortices, cyclonic above and anticyclonic below – which capture the short-term, geostrophically balanced response to strong surface cooling. Numerical heton studies (Legg and Marshall, 1993, 1998) have shown that a preexisting barotropic rim current can suppress baroclinic instability and confine anomalies of potential vorticity and temperature introduced by the cold-air outbreak. Here, we demonstrate that the lateral extent of the most probable states of the statistical theory are constrained by the angular momentum. Without resolution of the detailed dynamics, the equilibrium statistical theory predicts that baroclinic instability is suppressed for preconditioned flows with potential vorticity of the same sign in each layer provided that the strength of convective overturning does not change the sign of potential vorticity in one of the layers. This result agrees with detailed simulations (Legg and Marshall, 1998) and supports the potential use of these statistical theories as parametrizations for crude closure. Received 12 April 2000 and accepted 21 September 2000     

Semi-implicit two-level predictor–corrector methods for non-linearly coupled,hydrostatic, barotropic/baroclinic flows

The unsteady shallow-water equations for barotropic/baroclinic (free-surface/density-stratified) flows with non-linear coupling of density transport and momentum are solved using a family of two-time-level, semi-implicit predictor–corrector methods (PC2). The PC2 methods are a general family that includes the popular TRIM method for hydrostatic flows. PC2 is characterised by four ‘θ’ parameters controlling the time ‘n’ and ‘n + 1’ weighting of (1) free surface gradient, (2) predictor step, (3) baroclinic gradient and (4) density temporal interpolation. Stability of the non-linear coupling between momentum and density transport for PC2 is examined in the inviscid limit. Central difference and quadratic (QUICK) spatial interpolation for density are compared. Second-order temporal accuracy for both barotropic and baroclinic flows is simultaneously obtained with appropriate θ parameters, which has previously been shown to be impractical for TRIM. The 2nd-order PC2 method has near-neutral non-linear stability (slightly positive amplification factor) where linear theory predicts exactly neutral stability. QUICK is shown to be preferable to central difference spatial discretisation to reduce the amplification factor. Adjusting the baroclinic weighting or adding small artificial viscosities can stabilise the model for non-linear internal wave simulations.     

Stability of zonal regimes in a truncated model of forced atmospheric flow

Summary The stability properties of zonal circulations induced by external forcing in a rotating atmosphere are investigated making use of a truncated model of the barotropic vorticity equation for forced, dissipative non-divergent flow in spherical geometry. Sufficient conditions for global and local asymptotic stability are found as a function of the dissipation time-scale, the coefficients of non-linear interaction between zonal flow and wave components, and the absolute rotation speed of the atmosphere. For weak, axisymmetric forcing fields the corresponding forced zonal circulation is a global attractor for states belonging to the configuration space of the model, while for larger forcing intensities it is only locally attracting, the extension of the basin of attraction being an increasing function of the absolute angular velocity of the atmosphere.

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Sommario Si analizzano le proprietà di stabilità di circolazioni zonali forzate in una atmosfera rotante facendo uso di un modello troncato a pochi modi dell'equazione di vorticità barotropica per flussi dissipativi in geometria sferica. Si determinano condizioni sufficienti per la stabilità asintotica sia locale che globale in funzione delle scale di tempo dissipative e di interazione nonlineare.

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This research was partly supported by C.N.R. through G.N.F.M.     

A MULTI-LEVEL NUMERICAL MODEL OF COASTAL UPWELLING: A DIAGNOSTIC STUDY

A two-dimensional baroclinic model is described for coastal upwelling in a vertical plane perpendicular to the coast. The model consists of equations of motion, continuity and turbulence energy along with equations for salinity and thermal energy and an equation of state. The role of density gradient in the baroclinic pressure gradient is investigated to understand the dynamics during the upwelling process. To represent the surface and bottom boundaries corresponding to a fixed computational level in the discretized equations, a set of non-dimensional co-ordinates is used. These co-ordinates are then transformed onto logarithmic co-ordinate axes to resolve effectively the boundary layers. The first experiment is carried out with a flat bottom to understand the dynamics of the upwelling and the structural features of the process by diagnostic analysis of the balance between various terms of the momentum equation. Starting from a state of rest, a spatially uniform alongshore wind stress corresponding to the mean monthly wind stress for the month of May is applied and held constant thereafter. The fluid is assumed to be incompressible and stratified, with the initial temperature and salinity having no horizontal variations but a uniform vertical gradient. As the upwelling phenomenon is transient in nature and keeping in mind the additional computational overheads, the response of the model is studied day-wise up to 4 days. In the second experiment the model is applied to study the upwelling off the east coast of India in a plane normal to the coast of Visakhapatnam. The analysis area extends to 100 km offshore with real topography. The results are presented day-wise for 4 days, comparing the balance between various terms in the upwelling region and in the open sea, and the dynamics of the baroclinic coastal jet is explained. © 1997 John Wiley & Sons, Ltd.     

A multi‐level adaptation model of circulation for the western Indian Ocean

A three‐dimensional, fully non‐linear semi‐diagnostic (adaptation) model is described. This model is used to compute the climatological mean circulation and to understand the role of local, steady forcing of the wind and thermohaline forcing on the observed circulation in the western tropical Indian Ocean. The model consists of equations of motion and continuity, sea surface topography, equations of state and temperature, and salinity diffusion equations. While the sea surface topography equation is solved by a successive overrelaxation technique, the other model equations are solved by a leap‐frog numerical scheme. Two versions of the model, having 18 and 33 levels in the vertical direction, were prepared to study climatological mean circulation in the western tropical Indian Ocean. The first numerical experiment is carried out with the 18‐level adaptation model to study the sensitivity of the solution to different values of eddy coefficients. The main scientific rationale behind these numerical experiments was to obtain the most appropriate values of the eddy coefficients for the realistic computation of climatological circulation in the western tropical Indian Ocean. Three numerical experiments were conducted for the month of February to understand the sensitivity of the model solution to different eddy coefficients. The model reproduced the circulation features during February, even with low values of horizontal and vertical eddy coefficients. In the second experiment, the adaptation model, with 33 levels in the vertical direction, is applied to study the seasonal mean climatological circulation at selected depths during Spring in the western tropical Indian Ocean. Adapted (steady state) results of currents, sea surface topography, temperature and salinity anomaly fields are presented. Reasonable agreement is obtained between the model results on currents and the observational data. The computed anomaly fields for temperature and salinity at selected depths during Spring show that the observed temperature and salinity data were adapted with surface wind, flow field and bottom relief of the ocean and that the observed data were found to be fully smoothed during the adaptation stage. Copyright © 1999 John Wiley & Sons, Ltd.     

Normal and typhoon wind loadings on a large cooling tower: A comparative study

Large cooling towers are sensitive to wind effects with their increasing heights and flexibilities. Unlike traditional approaches, which employed Code-defined normal winds to check the loading characteristics, this paper developed a framework for checking the typhoon-induced wind loading on a large cooling tower using Monte Carlo simulations and multi-fan wind tunnel tests. Some distinct characteristics of typhoon winds were compared with those of Code-defined normal winds. Furthermore, wind characteristics of incoming normal and typhoon winds in terms of vertical profile of mean wind speed, turbulence intensity, turbulence integral length scale and power spectrum density of fluctuating winds were well reproduced by a feedback control process of a multi-fan actively controlled wind tunnel. The surface wind pressure distributions of a large cooling tower under these conditions were then investigated by testing a 1:600 reduced scale model. Mean and fluctuating external wind pressures along the circumferential direction under various incoming winds were discussed and quantitatively formulated with eight-term trigonometric equations. Moreover, the cross correlations of wind pressures in the circumferential and meridian directions and correlations with structure base forces, i.e. integral drag and lift forces, were investigated. Non-Gaussian characteristics in terms of skewness and kurtosis of fluctuating wind pressures were also analyzed under two wind climates. Peak factors for modeling extreme wind pressures were examined and compared with those of various models. Finally, the extreme wind loads on a large cooling tower obtained from different wind pressure combinations were compared with peak-factor-theory-based results to identify an appropriate combination for structural design.     

Background current concept and chaotic advection in an oceanic vortex flow

The concept of background currents is offered in the examples of barotropic and baroclinic quasigeostrophic models. The background currents are characterized by constant value of potential vorticity, which minimize the energy of a system. Hamiltonian character of motion equations of fluid particles allows to apply such models to study chaotic advection.     

Modeling vortex-shedding effects for the stochastic response of tall buildings in non-synoptic winds

This study derives a model for the vortex-induced vibration and the stochastic response of a tall building in strong non-synoptic wind regimes. The vortex-induced stochastic dynamics is obtained by combining turbulent-induced buffeting force, aeroelastic force and vortex-induced force. The governing equations of motion in non-synoptic winds account for the coupled motion with nonlinear aerodynamic damping and non-stationary wind loading. An engineering model, replicating the features of thunderstorm downbursts, is employed to simulate strong non-synoptic winds and non-stationary wind loading. This study also aims to examine the effectiveness of the wavelet-Galerkin (WG) approximation method to numerically solve the vortex-induced stochastic dynamics of a tall building with complex wind loading and coupled equations of motions. In the WG approximation method, the compactly supported Daubechies wavelets are used as orthonormal basis functions for the Galerkin projection, which transforms the time-dependent coupled, nonlinear, non-stationary stochastic dynamic equations into random algebraic equations in the wavelet space. An equivalent single-degree-of-freedom building model and a multi-degree-of-freedom model of the benchmark Commonwealth Advisory Aeronautical Research Council (CAARC) tall building are employed for the formulation and numerical analyses. Preliminary parametric investigations on the vortex-shedding effects and the stochastic dynamics of the two building models in non-synoptic downburst winds are discussed. The proposed WG approximation method proves to be very powerful and promising to approximately solve various cases of stochastic dynamics and the associated equations of motion accounting for vortex shedding effects, complex wind loads, coupling, nonlinearity and non-stationarity.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

The interaction of equatorial waves with a barotropic shear: a potential test case for climate model dynamical cores

We compare the performances of two different numerical methods to solve the equatorial shallow water equations in a background meridional shear: A coarse-resolution Galerkin-truncation-based method and a finite-volume method, for the case of an equatorial Rossby wave. In the presence of the barotropic shear, the Rossby wave quickly loses its energy through shear interaction and excites other equatorially trapped waves despite the fact that the PDE system is linear. The two methods handle this sudden energy exchange across scales very differently. While the finite volume converges statistically to a coherent large-scale solution, the Galerkin truncation method results in large-scale and slow oscillations where energy is bumped back and forth within the resolved-scale spectrum, involving higher-order equatorial wave modes heavily depending on numerical resolution, that is, the number of Galerkin basis functions. The addition of an artificial viscosity for the coarse-resolution Galerkin method heavily damps the energy oscillations without significantly changing its transient dynamics. This provides an example of interaction across scales where the resolution of scales that are apparently not representative of the statistical solution are important for the transient dynamics. Therefore, non-linear interactions of equatorially trapped waves may provide an interesting test bed for the validation of climate model dynamical cores.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.