Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

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.     

冬季平流层波动模型的分岔特性

从描述波流相互作用的Ｈｏｌｔｏｎ－Ｄｕｎｋｅｒｔｏｎ简称Ｈ－Ｄ)模型出发，应用延拓方法求解常微分方程的分岔问题，研究冬季平流层波动模型的分岔特性．给出了大气行星波２与流相互作用的底部边界强迫波、底部边界平均纬向风场、风切变等参数的分岔特性，同时给出了波１与流相互作用的底部边界强迫波的分岔特性的结果．     

On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

Abstract

A spectral low-order model is proposed in order to investigate some effects of bottom corrugation on the dynamics of forced and free Rossby waves. The analysis of the interaction between the waves and the topographic modes in the linear version of the model shows that the natural frequencies lie between the corresponding Rossby wave frequencies for a flat bottom and those applying in the “topographic limit” when the beta-effect is zero. There is a possibility of standing or eastward-travelling free waves when the integrated topograhic effect exceeds the planetary beta-effect.

The nonlinear interactions between forced waves in the presence of topography and the beta-effect give rise to a steady dynamical mode correlated to the topographic mode. The periodic solution that includes this steady wave is stable when the forcing field moves to the West with relatively large phase speed. The energy of this solution may be transferred to the steady zonal shear flow if the spatial scale of this zonal mode exceeds the scale of the directly forced large-scale dynamical mode.     

Acceleration of mean zonal flows by planetary waves

The mechanism of acceleration of the mean zonal flow by a planetary wave is explained intuitively by considering the wave drag which a corrugated bottom feels when it excites the wave. The explanation is justified by solving the problem of vertical propagation of a planetary wave packet and the second order mean motion induced around it. The discussion is slightly extended to the case of small damping, to illustrate in a compact form the fact that the mean zonal acceleration is determined by a forcing due to wave transience plus that due to wave dissipation.The mean flow induced by a steady, dissipating planetary wave is discussed, and it is shown that it depends largely on the dissipation scale-height of the wave whether the northern region is heated or cooled. For example, if the wave velocity-amplitude increases upward in spite of dissipation, the induced easterly flow increases with height and the temperature of the northern region increases relative to that in the southern region. A similar point has been made byDunkerton (1979) in connection with westerly flows induced by Kelvin waves.The Lagrangian-mean motion induced by a planetary wave is briefly discussed in connection with the mechanism of acceleration of the mean zonal flow, in the case of a slowly varying wave packet. Further, in order el elucidate the effects of wave dissipation and time dependence of wave amplitude, the results obtained for a steady, dissipating wave and for a growing baroclinic wave are mentioned.     

The nonlinear development of disturbances in a zonal shear flow

Abstract

A study is made of the nonlinear stability of a weakly supercritical zonal shear flow in the β-plane approximation. The dynamics of initially small disturbances are examined. The main nonlinear effects are associated with the rearrangement of the critical layer. It is shown that as the wave grows in amplitude, linear regimes of the critical layer (viscous and nonstationary) change over to a nonlinear regime while the exponential law of disturbance growth becomes a power-law.     

A study of the Joule and Lorentz inputs in the production of atmospheric gravity waves in the upper thermosphere

First results of a modelling study of atmospheric gravity waves (AGWs) are presented. A fully-coupled global thermosphere-ionosphere-plasmasphere model is used to examine the relative importance of Lorentz forcing and Joule heating in the generation of AGWs. It is found that Joule heating is the dominant component above 110km. The effects of the direction of the Lorentz forcing component on the subsequent propagation of the AGW are also addressed. It is found that enhancement of zonal E × B forcing results in AGWs at F-region altitudes of similar magnitudes travelling from the region of forcing in both poleward and equatorward directions, whilst enhancement of equatorward meridional E × B forcing results in AGWs travelling both poleward and equatorward, but with the magnitude of the poleward wave severely attenuated compared with the equatorward wave.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

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.     

Effects of wave-wave and wave-mean flow interactions on the evolution of a baroclinic wave

Abstract

The weakly nonlinear evolution of a free baroclinic wave in the presence of slightly supercritical, vertically sheared zonal flow and a forced stationary wave field that consists of a single zonal scale and an arbitrary number of meridional harmonics is examined within the context of the conventional two-layer model. The presence of the (planetary-scale) stationary wave introduces zonal variations in the supercriticality and is shown to alter the growth rate and asymptotic equilibrium of the (synoptic-scale) baroclinic wave via two distinct mechanisms: The first is due to the direct interaction of the stationary wave with the shorter synoptic wave (wave-wave mechanism), and the second is due to the interaction of the synoptic wave with that portion of the mean field that is corrected by the zonally rectified stationary wave fluxes (wave-mean mechanism). These mechanisms can oppose or augment each other depending on the amplitude and spatial structure of the stationary wave field. If the stationary wave field is confined primarily to the upper (lower) layer and consists of only the gravest cross-stream mode, conditions are favorable (unfavorable) for nonzero equilibrium of the free wave.

In addition to the time dependent heat flux generated by baroclinic growth of the free wave, its interaction with a stationary wave field consisting of two or more meridional harmonics generates time dependent heat fluxes that vary with period of the free wave. However, if the stationary wave field contains several meridional harmonics of sufficiently large amplitude, the free baroclinic wave is destroyed.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

Seasonal variabilities of low-latitude mesospheric winds

Observations of mesospheric winds over a period of four years with the partial reflection radar at Tirunelveli (8.7°N, 77.8°E), India, are presented in this study. The emphasis is on describing seasonal variabilities in mean zonal and meridional winds in the altitude region 70–98 km. The meridional winds exhibit overall transequatorial flow associated with differential heating in the Northern and Southern Hemispheres. At lower altitudes (70–80 km) the mean zonal winds reveal easterly flow during summer and westerly flow during winter, as expected from a circulation driven by solar forcing. In the higher altitude regime (80–98 km) and at all altitudes during equinox periods, the mean zonal flow is subjected to the semi-annual oscillation (SAO). The interannual variability detected in the occurrence of SAO over Tirunelveli has also been observed in the data sets obtained from the recent UARS satellite mission. Harmonic analysis results over a period of two years indicate the presence of long-period oscillations in the mean zonal wind at specific harmonic periods near 240, 150 and 120 days. Results presented in this study are discussed in the context of current understanding of equatorial wave propagation.     

Oscillating Flow of a Compressible Fluid through Deformable Pipes and Pipe Networks: Wave Propagation Phenomena

Similarly to blood pulse propagation in the artery system, oscillating flow can propagate as a wave in fluid-saturated pipes, networks of pipes or, by extension, in porous media, if the fluid is compressible and/or the pipes are elastically deformable. First, propagation of flow waves generated in a semi-infinite pipe by harmonic pressure oscillations at the pipe entrance is analyzed. The dispersion equation is derived, allowing determination of the phase velocity and quality factor as functions of frequency. Wave reflections at the end of a finite-length pipe and ensuing interferences between forward and backward traveling waves are then examined. Because of fluid storage in the pipe, the amplitude of the AC volumetric fluxes entering and exiting the pipe at its upstream and downstream ends are not equal. Thus, two different, upstream and downstream, frequency-dependent, AC hydraulic conductivities are introduced. Superposed on the classic viscous-inertial flow transition (controlled by the value of the pipe radius), these complex-valued parameters show another transition between an interference-free regime at low frequencies and a strong interference regime above a critical frequency that roughly scales as the pipe length. Because of attenuation, the flow wave interferences tend to gradually weaken with increasing frequencies. Finally, the single pipe model is used to investigate fluid flow waves through pipe networks with results very similar to those described above. The flow waves analyzed here are akin to the Biot’s slow P waves and their propagation properties could affect seismic soundings in some geological settings.     

Modelling the response of a double-barred sandy beach system to time-varying wave angles

Sandy beaches typically have one or more shore-parallel bars with superimposed smaller-scale three-dimensional (3D) bars. Knowledge of their morphodynamic behaviour under more realistic wave conditions is limited. This study investigates the response of beaches with two shore-parallel bars to sinusoidally time-varying angles of incidence, using a non-linear morphodynamic model. Different periods and amplitudes of this sinusoidal variation are considered, as well as different time-mean wave angles. For time-invariant and normally incident waves, results show that alongshore rhythmic 3D bars form in the domains of inner and outer shore-parallel bars. The 3D bars in the inner domain are coupled at half the outer-bars wavelength. This phase coupling breaks up when the wave angle varies in time. Initially, regular 3D bars form in the inner domain (free behaviour), which become irregular when 3D bars develop in the outer domain (forced behaviour). The heights of the 3D bars oscillate with time, reaching maximum values when the forcing period is comparable to the system adjustment time scale (∼ 10–20 days). For a time-varying wave angle around an oblique mean, alongshore migrating 3D bars emerge in both inner and outer domains. In contrast, for an oblique (constant) wave angle, 3D bars only form in the inner domain and they hardly migrate alongshore. For any forcing period, the dominant response period of the oscillating bar heights is at half the forcing period when waves are (on average) normally incident, and it equals the forcing period when waves are on average obliquely incident. Compared with time-invariant angles, heights of inner and outer 3D bars are (on average) smaller and larger, respectively, when the angle varies with time, particularly for forcing periods in the order of the system adjustment time scale. Increasing the amplitude of the time-varying wave angle weakens bar growth. Explanations of these results are also provided.     

Formation of one-dimensional waves for resonant flow in a channel

When a forcing moves in a shallow channel at a velocity near the phase velocity for linear long waves, energy cannot escape from the forcing at the linear group velocity and nonlinear effects become important in describing the resulting flow. This flow is termed resonant or transcritical. It has been found both experimentally and numerically that large amplitude upstream propagating waves are generated by the forcing. These waves are straight crested, even though the forcing is two-dimensional. It is shown that these upstream waves become straight crested due to geometrical effects aided by the presence of side walls. Using energy conservation, approximate values of the amplitude of the upstream waves are obtained which are compared with recent experimental and numerical results.     

The nonlinear stabilization of a zonal shear flow instability

Abstract

The development of initially small perturbations in a weakly supercritical zonal shear flow on a β-plane is studied. Two different scenarios of evolution are possible. If the supercriticality is sufficiently small, the growth of a perturbation is stopped in the viscous critical layer regime; for this case the evolution equation (corrected by the inclusion of a quintic nonlinearity) is derived. At greater supercriticality the nonlinearity cannot stop the growth of the perturbation in a linear (viscous or unsteady) critical layer regime, and the evolution is more complicated. Transition to a nonlinear critical layer regime leads to a reduction in the growth rate and to a slowing (but not a stopping) of the increase in amplitude, A. These are connected to the formation of a plateau (S=constant) of width L=O(A ^½) in the profile of absolute vorticity, S. Careful analysis reveals that the growth in amplitude ceases only when the whole instability domain (where the slope of unperturbed S-profile is positive) becomes covered again by the plateau.     

纬向非均匀基流对大气长波调整的作用

大气长波的发展和演变影响着大气的可预报性,并对提高天气预报和气候预测水平有重要的意义.在影响大气长波演变的因子中,除波与波非线性相互作用外,基流的作用也非常重要.本文利用非均匀基本场下Rossby波运动方程,通过数值求解,分析了基本场结构和初始场对Rossby波演变的影响,揭示了纬向非均匀基本场对长波调整的作用.研究结果表明:基流纬向非均匀时,线性Rossby波也会出现长波调整现象,基流随纬向变化是长波发生调整的又一个重要机制;大气长波调整对波动的初始振幅不敏感,但基本场振幅影响着长波调整能否出现和出现的时间;基本场纬向平均西风基流的大小除影响波动传播的速度和方向外,还影响长波调整出现的时间和规律;长波调整的出现还与基本场和初始场的结构有关,不同基本场时,波动是否发生调整、向高波数还是向低波数调整都决定于基本场结构,相同基本场时,不同初始结构的波动也有着不同的演变过程.     

A primitive equation,solar driven,perturbation model of the thermospheres of mars and venus

Abstract

A primitive equation, solar driven, thermospheric model is derived which has applications to the neutral gas components on Mars and Venus. The full effects of molecular viscosity and thermal conductivity are included, necessitating the development of a combined analytic and numerical solution technique. The model is applied to Venus in order to understand how thermospheric rotation, if present, would affect the dynamics. Results indicate that rotation periods of eight days or less should be observable. Application of the model to Mars indicates that the perturbation solar heating and the atmospheric response have primarily a diurnal component for which typical temperature and zonal wind maximum amplitudes are 20 K and 30 m/sec respectively. Because of uncertainty in the solar heating efficiency, calculations were made varying this parameter by an order of magnitude. The results imply that the response due to solar forcing alone is probably too small to account for observed concentrations of the minor constituents CO and O. An upper limit estimate is made of the upward propagation of wave energy from the lower atmosphere and the resulting response of the thermosphere.     

On the interaction between a radiatively damped planetary wave and the zonally averaged circulation in the middle atmosphere

The interaction between a planetary wave damped by cooling to space and the zonally averaged circulation in the middle atmosphere is examined for a steady-state situation in middle latitudes. Quasi-geostrophic scaling of Type 2 is assumed (i.e. the space scales are planetary and the superrotation is small).A set of mean equations is derived for this scaling which is complementary to the set of perturbation equations previously studied. The mean equations show that a planetary wave induces a mean meridional circulation which is balanced by an eddy momentum forcing function and a mean diabatic heating which is balanced by an eddy heat flux forcing function. The vertical eddy fluxes enter the forcing at the same order as the horizontal eddy fluxes.An analytical wave solution is found for the case of an atmosphere in constant superrotation. The eddy fluxes and forcing functions are evaluated for this special case. It is found that they are very sensitive to the values of the radiative damping coefficient and the superrotation. Since the damping coefficient depends on the ozone concentration and the intensity of the solar ultraviolet flux, the results suggest that changes in these quantities can strongly modify the wave-mean flow interaction in the middle atmosphere. Possible implications for climate change are discussed.     

On the generation and propagation of internal solitary waves in the southern Andaman Sea: A numerical study

Numerous internal solitary waves(ISWs) have been observed in the southern Andaman Sea. In this study, the two-dimensional Massachusetts Institute of Technology general circulation model is applied to investigate the dynamics of ISWs and explore the effects of the bottom topography and tidal forcing on the generation and propagation of ISWs in the southern Andaman Sea. The results show that the large-amplitude depression ISWs are mainly generated via the oscillating tidal flow over the sill of the Great Channel, and the generation of ISWs is subject to the lee wave regime. The Dreadnought Bank cannot generate ISWs itself; however, it can enhance the amplitudes of eastward-propagating ISWs generated from sill A, owing to constructive interference of internal tide generation between the sill of the Great Channel and the Dreadnought Bank. The eastward-propagating ISWs generated by the eastern shallow sill near the continental shelf can propagate to the shelf, where they evolve into elevation waves because of the shallow water. Sensitivity runs show that both the semidiurnal and diurnal tides over the sill of the Great Channel can generate ISWs in this area. However, the ISWs generated by diurnal tides are much weaker than those generated by semidiurnal tides. Mixed tidal forcing has no significant effect on the generation of ISWs.