Numerical studies of wind forced internal waves with a nonhydrostatic model

The nonhydrostatic pressure effects on the generation and propagation of wind-forced internal waves are studied with a two-dimensional numerical ocean model. A one-way directed wind pulse over a stratified ocean initiates surface and internal waves in a closed basin. The studies are performed with horizontal grid sizes in the range from 1 km to 62.5 m. The experiments are performed with both a hydrostatic and a nonhydrostatic model, facilitating systematic studies of the sensitivity of the numerical model results to the grid size and to the nonhydrostatic pressure adjustments. The results show that the nonhydrostatic pressure effects are highly dependent on the grid size and grow with increased resolution. In the internal depression wave, the horizontal nonhydrostatic pressure gradients reach the same order of magnitude as the hydrostatic gradients in the high-resolution nonhydrostatic studies. In these studies, the nonhydrostatic pressure gradients approximately balance the corresponding hydrostatic pressure gradients in the internal depression wave, and the wave degenerates into a train of soliton waves. The time for the soliton form to develop agrees with the steepening timescale calculated from Korteweg-de Vries theory. In the high-resolution hydrostatic model, the internal depression wave takes the form of a single wave front. When the internal waves are generated in the boundary layers, the nonhydrostatic pressure gradients are much smaller than the hydrostatic gradients and the generation processes are not effected by the nonhydrostatic pressure with the present range of grid sizes.     

The instability of internal gravity waves to localised disturbances

The instability of an internal gravity wave due to nonlinear wave-wave interaction is studied theoretically and numerically. Three different aspects of this phenomenon are examined. 1. The influence of dissipation on both the resonant and the nonresonant interactions is analysed using a normal mode expansion of the basic equations. In particular, the modifications induced in the interaction domain are calculated and as a result some modes are shown to be destabilised by dissipation. 2. The evolution of an initial unstable disturbance of finite vertical extent is described as the growth of two secondary wave packets travelling at the same group velocity. A quasi-linear correction to the basic primary wave is calculated, corresponding to a localised amplitude decrease due to the disturbance growth. 3. Numerical experiments are carried out to study the effect of a basic shear on wave instability. It appears that the growing secondary waves can have a frequency larger than that of the primary wave, provided that the shear is sufficient. The instability of waves with large amplitude and long period, such as tides or planetary waves, could therefore be invoked as a possible mechanism for the generation of gravity waves with shorter period in the middle atmosphere.     

A 3D unstructured non-hydrostatic ocean model for internal waves

A 3D non-hydrostatic model is developed to compute internal waves. A novel grid arrangement is incorporated in the model. This not only ensures the homogenous Dirichlet boundary condition for the non-hydrostatic pressure can be precisely and easily imposed but also renders the model relatively simple in its discretized form. The Perot scheme is employed to discretize horizontal advection terms in the horizontal momentum equations, which is based on staggered grids and has the conservative property. Based on previous water wave models, the main works of the present paper are to (1) utilize a semi-implicit, fractional step algorithm to solve the Navier-Stokes equations (NSE); (2) develop a second-order flux-limiter method satisfying the max–min property; (3) incorporate a density equation, which is solved by a high-resolution finite volume method ensuring mass conservation and max–min property based on a vertical boundary-fitted coordinate system; and (4) validate the developed model by using four tests including two internal seiche waves, lock-exchange flow, and internal solitary wave breaking. Comparisons of numerical results with analytical solutions or experimental data or other model results show reasonably good agreement, demonstrating the model’s capability to resolve internal waves relating to complex non-hydrostatic phenomena.     

重力波包在中层大气温度波导中传播的数值模拟

给出了重力波在中层大气温度波导中的反对称形式导制传播的线性理论模型,并采用二维非线性的数值模型对重力波波包在中层大气温度波导中的传播和演变过程进行了模拟研究.模拟的结果表明,下层大气激发的重力波能量进入波导区域后被俘获形成导制传播.重力波在波导内不停地来回反射,垂直方向的自由传播受到限制,能量在波导内沿着水平方向传输,模拟得到的波参数与理论值相近.重力波包在温度波导中传播时伴随着能量泄漏,且能量泄漏的速率随时间变缓,最终总有部分能量被限制在波导区域.重力波在水平方向上传播几百公里后,依然维持着良好的波结构,同时数值模拟也给出了重力波在波导区域内能量密度的时空分布.     

Global pattern of the ionospheric response to large-scale internal gravity waves

The global pattern of the ionospheric response to large-scale acoustic gravity waves (LS AGW) has been constructed on the basis of an analysis of the large data set available during the 22 March 1979, magnetic storm. Ground-based ionospheric measurements and in-situ satellite measurements from Cosmos-900 were used in this study together with the Joule heating distribution in the high-latitude ionosphere specifically taken at the maxima of two substorms. The characteristics of the reconstructed planetary pattern of the LS AGW have been analysed in detail. It has been established that the LS AGW effects in the ionosphere in terms of both universal and local time were determined by the pattern of high-latitude atmospheric heating, and that the wave front of the LS AGW during both substorms covered practically all local times, i.e. all longitudes. In addition, it was established that one of the sources of the LS AGW was the thermospheric heating in the day-side cusp region. The local time dependence of the amplitude of the AGW effect in both maximum height, h_mF2, and critical frequency, f_OF2, has been reconstructed for the mid-latitude F2 layer. The AGW effects were clearly separated from the electric field effects related to turnings of the interplanetary magnetic field (IMF) B_Z. In the day-time, electric field effects prevailed over the AGW effects, but during the night-time the amplitudes of these two effects were comparable. In contrast to the common view, f_OF2 variations after the AGW passage had a quasi-sinusoidal character both in the day-time and in the night-time. In the night-time ionosphere a high degree of symmetry was observed for the AGW effects in Northern and Southern hemispheres. During the day-time a significant asymmetry was observed in the American longitudinal sector which was related largely to the peculiarities of the heating pattern in the high-latitude ionospheres of the Northern and Southern hemispheres. These observations demonstrate the complexity of the response of the ionosphere at all latitudes to heating of the auroral region.     

Observations indicating parametric instabilities in internal gravity waves at thermospheric heights

Abstract

Theoretical studies predict a parametric instability of finite-amplitude internal gravity waves which hitherto has been observed only in laboratory experiments. The occurrence of this process in the atmosphere is of basic interest because finite-amplitude gravity waves, which are almost ubiquitous especially at upper atmospheric heights, would produce unstable flows even at large Richardson numbers. Maximum entropy power spectra of a strong internal gravity wave in the thermosphere, which was generated by a volcanic eruption and detected on records of the Doppler shift of high-frequency radio waves, in fact show good agreement with the spectra of synthetic Doppler records obtained from a calculated unstable gravity wave. The frequencies and wavenumbers observed in the gravity wave domain satisfy in particular the theoretically predicted resonance conditions. The observed Doppler records also show two significant lines in the acoustic domain which probably result from a nonlinear interaction with the basic gravity wave. It is suggested that acoustic double peaks, which are commonly observed in high-frequency Doppler spectra in the presence of nearby thunderstorms, represent parametric instabilities of internal gravity waves generated by penetrative cumulus convection.     

Towards petascale simulation of atmospheric circulations with soundproof equations

This paper highlights progress with the development of a petascale implementation of general-purpose high-resolution (nonoscillatory) hydrodynamical simulation code EULAG [Prusa et al. 2008, Comput. Fluids 37, 1193]. The applications addressed are anelastic atmospheric flows in the range of scales from micro to planetary. The new modeldomain decomposition into a three dimensional processor array has been implemented to increase model performance and scalability. The performance of the new code is demonstrated on the IBM BlueGene/L and Cray XT4/XT5 supercomputers. The results show significant improvement of the model efficacy compared to the original decomposition into a two-dimensional processor array in the horizontal — a standard in meteorological models.     

Numerical comparisons of the damping of internal gravity waves in stratified fluids

Summary Comparisons are made of the attenuation of progressive internal waves in stratified fluids consisting of (a) a representation of a single thermocline, (b) a weak exponential density distribution and (c) a two-layer system. Effects of the earth's rotation are neglected, as also are heat and salt diffusion. Significant differences are found in the decay characteristics of the three density structures. Internal waves are most strongly damped in the layered system (c) whilst the calculated attenuation rates are least for the thermoclinic structure (a). All the internal modes investigated are, however, shown to be more strongly damped than the associated surface wave.     

Weakly non-linear analysis of small-amplitude internal gravity waves forced by isolated topography

Asymptotic methods and numerical simulations are used to examine the evolution of an internal gravity wave packet comprising a continuous spectrum of horizontal wavenumbers and propagating upwards in a continuously stratified shear flow. In the multiple-scale framework for a horizontally localized wave packet generated by stratified flow over a localized mountain range with multiple peaks, there are in general two horizontal scales: the “fast” scale which is defined by the oscillations within the packet, i.e. the number of peaks, and the “slow scale” which is defined by the horizontal extent of the packet, i.e. the width of the mountain range. The focus here is on the specific case of an isolated mountain where the spectrum of horizontal wavenumbers is centred at zero and the multiple-scaling procedure is thus simplified by the absence of the fast spatial scale. The background flow is vertically sheared and critical-level interactions occur. The time frame within which non-linear critical-level effects become significant is determined by the magnitude of the non-linear terms in the governing equations. With the isolated mountain forcing this time frame is significantly longer than in the case of a multiple-peak mountain range forcing and it depends on the horizontal scale of the forcing, as well as on the amplitude. At leading-order, the non-linear asymptotic solution approaches a steady state in the outer region at late time, but the zero-wavenumber component of the solution continues to evolve with time in the vicinity of the critical level.     

A nonhydrostatic primitive-equation model for studying small-scale processes: An object-oriented approach

A nonhydrostatic model for simulating small-scale processes in the ocean is described. The model is developed using the object-oriented approach. The system is modeled as a set of cooperating objects to manage both the behavioral and information complexity associated with modeling oceanic processes. Objects are storage variables that are created based on classes. A class defines the variables and routines that are members of all objects of that class. The program accesses data stored in these objects using the defined interfaces. Because both data and function are accessed through objects, the model is better organized than one written in a procedural language. The program is easier to understand, debug, maintain, and evolve. Abstraction of the data in the nonhydrostatic model is implemented in both C++ and Matlab. Three examples obtained from the Matlab version of the code illustrate the capabilities of the model in cases where nonhydrostatic effects are important. The model successfully simulates nonhydrostatic atmospheric lee waves, internal waves at a discharge plume, and internal solitary waves generated by tidal flow over a sill. These examples show that the model is capable of studying strongly nonlinear, nonhydrostatic flow processes.     

Island sheltering of surface gravity waves: model and experiment

A field experiment is used to evaluate a numerical model of the sheltering of gravity waves by islands offshore of the Southern California region. The sheltering model considered here includes only the effects of island blocking and wave refraction over the island bathymetry. Wave frequency and directional spectra measured in the deep ocean (unsheltered region west of the islands) were used as input to the sheltering model and compared with coastal observations. An airborne L-band synthetic aperture radar was used to image the directional properties of the waves in the deep ocean. In addition to the unsmoothed spectra, a unimodal directional spectrum model obtained from fits to the radar spectra was also employed to suppress the high noise level of this system. Coastal measurements were made in about 10 m depth at Torrey Pines Beach with a high resolution array of pressure sensors. The model predictions and data at Torrey Pines Beach agree well in a limited frequency range (0.082 to 0.114 Hz) where the unimodal deep ocean model is appropriate. The prediction that unimodal northern swell in the deep ocean results in a bimodal directional spectrum at Torrey Pines Beach is quantitatively verified. The northern peak of the bimodal spectra is due to waves coming through the window between San Clemente and San Miguel-Santa Rosa Islands. The southerly peak is due to wave refraction over Cortez and Tanner Banks. For lower frequency waves, the effects of strong refraction in the island vicinity are shown qualitatively. Refraction can theoretically supply up to approximately 10% of the deep ocean energy that is otherwise blocked at this site. The modifications of the island shadows due to wave refraction become theoretically negligible for wave frequencies 0.11Hz. Also, local wave generation effects, which are not included in this sheltering model, are shown to be occasionally important for waves with frequencies 0.12Hz.     

A numerical experiment on the general circulation of the middle atmosphere with a three-dimensional model explicitly representing internal gravity waves and their breaking

The general circulation of the middle atmosphere is simulated by means of a three-dimensional primitive equation model which covers from the south pole to the north pole but is limited to a ten-degree sector in the latitudinal direction; cyclic conditions are imposed at the east—west lateral boundaries. The model is capable of explicitly representing internal gravity waves of zonal wavelength greater than a few hundred kilometers with the use of a one-degree mesh, but planetary-scale waves were excluded. No parameterization is employed for subgrid-scale eddy viscosity (or diffusivity).With the assumption of a simple external-heating function corresponding to solstice conditions, a time integration was performed for about thirty days from the motionless state. During the whole period, random forcings were imposed on each grid of the lowest level in order to generate small-scale upwardly propagating internal gravity waves.The experiment has shown that small-scale waves were indeed excited, propagated upward, broke up near the mesopause, and greatly changed the thermally induced zonal mean motion and temperature fields in the upper mesosphere and lower thermosphere. As a result, important features of the general circulation at those levels, such as reversals of the zonal motion and the latitudinal gradient of zonal mean temperature were reproduced.     

Variations in the characteristics of acoustic gravity waves according to simulation data

The characteristics of different-scale acoustic gravity waves (wavelengths of 100–1200 km, periods of 10–50 min) under different geophysical conditions have been studied using a numerical model for calculating the vertical structure of these waves in a nonisothermal atmosphere in the presence of an altitudedependent background wind and in a situation when molecular dissipation is taken into account. It has been established that all considered acoustic gravity waves (AGWs) effectively reach altitudes of the thermosphere. The character of the amplitude vertical profile depends on the AGW scales. The seasonal and latitudinal differences in the AGW vertical structure depend on the background wind and temperature. A strong thermospheric wind causes the rapid damping of medium-scale AGWs propagating along the wind. Waves with long periods to a lesser degree depend on dissipation in the thermosphere and can penetrate to high altitudes. A change in the geomagnetic activity level affects the background wind vertical distribution at high latitudes, as a result of which the AGW vertical structure varies.     

A semi-implicit semi-Lagrangian global nonhydrostatic model and the polar discretization scheme

The Global/Regional Assimilation and PrEdiction System(GRAPES) is a newly developed global non-hydrostatic numerical prediction model,which will become the next generation medium-range opera-tional model at China Meteorological Administration(CMA).The dynamic framework of GRAPES is featuring with fully compressible equations,nonhydrostatic or hydrostatic optionally,two-level time semi-Lagrangian and semi-implicit time integration,Charney-Phillips vertical staggering,and complex three-dimensional pre-conditioned Helmholtz solver,etc.Concerning the singularity of horizontal momentum equations at the poles,the polar discretization schemes are described,which include adoption of Arakawa C horizontal grid with ν at poles,incorporation of polar filtering to maintain the computational stability,the correction to Helmholtz equation near the poles,as well as the treatment of semi-Lagrangian interpolation to improve the departure point accuracy,etc.The balanced flow tests validate the rationality of the treatment of semi-Lagrangian departure point calculation and the polar discretization during long time integration.Held and Suarez tests show that the conservation proper-ties of GRAPES model are quite good.     

热源激发重力波特征以及波流作用的数值模拟研究

本文在二维等温可压大气中引入了一个随时间和空间变化的热源扰动,分别以静止风和中纬1月份月平均向东的纬向风急流为背景,对不同背景下热源激发的重力波的传播详细过程及其特性进行了数值模拟研究.热源激发出来的重力波在初始阶段有很宽的频谱范围,随后由于重力波的传播效应,水平波长和垂直波长分布范围随时间都有所减小.顺风传播的重力波的小尺度和低频部分会容易被急流吸收,从而加强了对流层急流,而逆风传播的重力波更容易上传,会导致中间层区域向西的背景风增强.这体现了低层大气急流对中间层大气风场结构的影响.热源的尺度直接决定激发波的尺度;激发波的垂直尺度和时间尺度对热源的变化比其水平尺度更敏感.     

The decay of gravity waves

Summary The decay of waves of small amplitude in a viscous liquid of finite depth is investigated. It is shown that the various modes may be conveniently classified and that in many limiting cases the modes of aperiodic decay are of particular importance. Detailed numerical results are given for these modes.     

On the instability of internal gravity waves propagating at small but finite angles to the vertical

The question of the instability of internal gravity waves (IGWs) propagating at small angles to the vertical is re-visited. The case of an IGW of finite-amplitude propagating at a very small but finite angle to the vertical is considered. This angle serves as a small parameter in the problem, and the instability of such an IGW is investigated by using the Fourier method and the Sivashinsky integral relations. The analysis undertaken confirms the existence of short-wave instability for small IGW amplitudes and for an arbitrarily small value of their propagation angle to the vertical. For small viscosity and thermal conductivity of the fluid medium, the growth rate of the most unstable mode is proportional to the square of the amplitude of the IGW. The results obtained may be of interest for interpreting the results of observations and confirming the existence of turbulence in the middle atmosphere.     

Earlier days of gravity waves revisited

The means whereby the author came to be involved in the study of atmospheric gravity waves, and then came to involve others in that study, are outlined. In particular, events leading up to, during and following the International Symposium on Fluid Mechanics in the Ionosphere, of July 1959, are described.     

A note on the general concept of wave breaking for Rossby and gravity waves

A recently proposed general definition of wave breaking is further discussed, in order to deal with some points on which misunderstanding appears to have arisen. As with surface and internal gravity waves, the classification of Rossby waves into breaking and not breaking is a generic classification based on dynamical considerations, and not a statement about any unique signature or automatically recognizable shape. Nor is it a statement about passive tracers uncorrelated with potential vorticity on isentropic surfaces. A strong motivation for the definition is that proofs of the nonacceleration theorem of wave, mean-flow interaction theory rely, explicitly or implicitly, on a hypothesis that the waves do not break in the sense envisaged.The general definition refers to the qualitative behaviour of a certain set of material contours, namely those, and only those, which would undulate reversibly, with small slopes, under the influence of the waves' restoring mechanism, in those circumstances for which linearized, nondissipative wave theory is a self-consistent approximation to nonlinear reality. The waves' restoring mechanism depends upon the basic-state vertical potential density gradient in the case of gravity waves, and upon the basic-state isentropic gradient of potential vorticity in the case of Rossby waves. In the usual linearized theory of planetary scale Rossby waves on a zonal shear flow, the relevant material contours lie along latitude circles when undisturbed.     

A global experimental model for gravity tides of the Earth

The long-term, continuous and high-quality tidal gravity data, recorded with the superconducting gravimeters (SGs) at 19 stations in the Global Geodynamics Project (GGP), were simultaneously used to investigate the global pattern of the tidal gravity variations. The atmospheric effects were removed from the gravity observations by using the simultaneous pressure records at the stations. A total of six global co-tidal models were employed to remove the loading effects of oceanic tides. The resonance parameters of the Earth's free core nutation (FCN), as well as the spheroidal constant components in the gravimetric factors of waves O₁ and M₂, were accurately retrieved. As a result, a global experimental model for gravity tides (GEMGT) was developed, considering the nearly diurnal resonance and the latitude-dependence of the gravimetric amplitude factors. The final results indicate that the mean discrepancy of the four main tidal waves (i.e. O₁, K₁, M₂ and S₂) between the GEMGT and SG observations is less than 0.2% on average. The GEMGT is in good agreement with the theoretical models based on the inelastic non-hydrostatic equilibrium Earth models [Dehant, V., Defraigne, P., Wahr, J., 1999. Tides for a convective Earth. J. Geophys. Res. 104, 1035–1058; Mathews, P.M., 2001. Love numbers and gravimetric factor for diurnal tides. J. Geodetic Soc. Jpn. 46 (4), 231–236] with a mean discrepancy less than 0.15%. However, the GEMGT is in closer accordance with the theoretical model given by Mathews [Mathews, P.M., 2001. Love numbers and gravimetric factor for diurnal tides. J. Geodetic Soc. Jpn. 46 (4), 231–236] for the diurnal tides while it is in closer agreement with one obtained by Dehant et al. [Dehant, V., Defraigne, P., Wahr, J., 1999. Tides for a convective Earth. J. Geophys. Res. 104, 1035–1058] for the semi-diurnal tides.