A kinetic theory for internal waves in a randomly stratified fluid

We discuss the transport of energy of internal waves propagating in a stratified unbounded fluid with randomly varying buoyancy frequency N of the form $\text{N}{}^2\text{= N}{}_0{}^2\text{[1 + ?Ξ(}\textcolor{red}{}\text{)]}$. Here N₀ = constant, 0 < ? ? 1 and Ξ is a zero-mean stationary random function of $\textcolor{red}{}\text{= (x,z)}$ where x and z are respectively horizontal and vertical coordinates. In the limit of small ?, a linear kinetic equation (transport equation) is derived which describes the space—time evolution of the mean wave energy density in such a medium. When it is integrated over all wave number space, the kinetic equation implies the total conservation of wave energy. The approach used is reminiscent of the nonlinear wave interaction theories of Phillips (1960), Benny (1962) and Hasselmann (1966) and others, in which the random microstructure Ξ would be regarded as a nonpropagating (zero-frequency) random wave field. Our analysis is based on the Eulerian equations of motion in which no a priori assumptions are made regarding the scattered wave field — on the contrary, it is a deduction of the theory presented here that only the propagating internal wave modes participate in the energy exchange processes. In particular we do not assume that the internal wave field constitutes a homogenous assembly of random wave packets evolving in time alone — unlike much of the earlier work — and this enables us to treat the scattering of individual internal waves by the microstructure.The kinetic equation is used to determine the energy transmitted through and reflected by a horizontally oriented random slab which models a layer of microstructure of finite thickness in the ocean. Specifically, we show that significant reflection can occur when Ψ(2k_0z) is sufficiently large, where Ψ is the vertical wavenumber spectrum of the microstructure fluctuations Ξ(z) and k_0z is the vertical wavenumber of the incident wave. We also show that the reflection coefficient increases monotonically with increasing frequency, which is in qualitative agreement with recent measurements at site D(39°N, 70°W) which indicate that in regions where density inhomogeneities are present, the vertical coherence decreases with increasing frequency. Actual numerical estimates for the reflection coefficient are obtained for vertical microstructure data from station P(50°N, 145°W). It is found that for intermediate wavelengths — 0(10²m) — and a broad band of frequencies (0.6 ? ω/N₀ < 1), the reflection coefficient is greater than 0.5. Finally, the qualitative behaviour of the kinetic equation for two-dimensional microstructure is examined in the geometric optics limit: wavelength much less than (the integral) correlation scale. In this case the integro-differential kinetic equation reduces to a Fokker—Planck diffusion equation. From the latter we infer that at high frequencies, a wave packet becomes incoherent after propagating a distance that is less than a typical correlation scale associated with the fluctuations Ξ.     

Measurements of internal waves and turbulence in two-dimensional stratified shear flows

A turbulent stratified shear flow is generated in a towing tank by towing a grid or a circular cylinder through a tank of stratified salt water. The internal waves and turbulence generated in these flows are visualized with shadowgraphs and measured with quartz-coated hot-film probes (up to four probes for velocity fluctuations) and single-electrode conductivity probes (up to four probes for salinity fluctuations) which are towed at the same speed as the obstacle. The velocity and salinity signals are recorded on magnetic tapes. A portion of these signals is processed directly-on-line with a digital computer. From these shadowgraphs and probe measurements, we observe that

1. Far downstream of the obstacle where the turbulence has already subsided, the stratified fluid always has a layered structure. This layered structure persists for a long time, and is a result of the convection of turbulently mixed layers by the mean flow. These results indicate that in the regions of a stably stratified atmosphere and ocean where the turbulence has subsided, one could often find layered structure.
2. There are spectral peaks and valleys in the measured velocity and salinity autospectra when the stratifications are sufficiently strong. Under certain conditions, these spectral peaks tend to lift up the spectral curves to show substantialf ^?5/3 subranges, although the turbulence Reynolds numbers are too low for the flows to have recognizable inertial subranges. This anomalousf ^?5/3 subrange demonstrates the pitfalls of using spectral measurements in thef ^?5/3 subrange to predict the turbulent energy dissipation rate through the Kolmogorov hypothesis.
3. A diagnostic method is developed for distinguishing internal waves from turbulence, utilizing their phase characteristics. The phase characteristics can be conveniently examined from the cospectra and quadrature spectra measurements of: (a), two vertically separated velocity probes; (b), two vertically separated density probes; and (c), a velocity probe and a density probe. This method is demonstrated to be useful in the laboratory and can be applied directly to atmospheric and oceanic measurements to distinguish internal waves from turbulence.
4. From the coherency measurements, it is found that the entire turbulent stratified wake is actually whipping up and down at a frequency corresponding to the Brunt-Väisälä frequency. This indicates that similar stratified shear flows in the atmosphere and in the ocean, such as the jet streams in the atmosphere and the Cromwell current in the ocean, may oscillate vertically, which in turn can induce horizontal oscillation and meandering.

     

On the three-dimensional internal waves excited in the flow of a linearly stratified Boussinesq fluid

Three-dimensional flow of a linearly stratified Boussinesq fluid is studied numerically. The flow is assumed to be confined in a rectangular channel and internal waves are excited by bottom topography. Near resonance of the first vertical internal wave mode, it was found that the reflection of the internal wave at the sidewall is ‘abnormal’ in the sense that the reflection angle is larger than the incident angle and a third wave perpendicular to the sidewall is generated. The waves become straight crested (two-dimensional) as this third wave becomes longer. The whole mechanism is similar to the ‘Mach reflection’ observed in the general stratified fluid in which the usual solitary waves are generated. In the case of the linearly stratified Boussinesq fluid, the abnormal reflection occurs even though the wave near the sidewall has a sinusoidal profile and not a sech² profile. This suggests that the abnormal reflections similar to Mach reflection always occur when the wave amplitude is large enough, irrespective of the wave profile.     

Transverse upwelling in a long narrow lake,with applications to Babine Lake and Lake Michigan

Abstract

The wind generation of transverse thermocline motions in an infinitely long two‐layer channel is studied theoretically. Winds both along and across the channel are considered. Horizontal momentum transfer is parameterized by a constant coefficient of eddy viscosity. For a channel narrow compared with its internal Rossby radius of deformation, the transverse motions are uni‐modal and generated most efficiently by long‐shore winds. Theoretical results in this narrow channel limit agree well with observations made at Babine Lake, British Columbia. For a channel wide compared with its Rossby radius, the response is multi‐modal, especially for cross‐channel winds. For a model Lake Michigan, computed interfacial displacements due to a steady wind either across or along the channel are small compared with the observed displacements. However, a semi‐diurnal wind (in either direction) is in near resonance with the seventh transverse mode. Thus multi‐modal displacements as large as those observed could possibly be generated by semi‐diurnal winds across or along the channel.     

Wind stress over water waves: Field experiments on lake of Geneva

Summary A fixed platform (Fig.3), installed 100 m from the shoreline in 3 m water depth, was instrumented with velocity, temperature and wave-height sensors. 132 data (10 minutes averages) were analysed to calculate the wind stress; from these, 99 data were used to investigate the vertical distribution of the wind stress; all data are presented with Table 1.It was postulated that the total stress, _t being constant with height, is made up additively of two components, the wave-supporting stress, _w , and the turbulent stress, _c ; see Eq. 1. The vertical distribution of these two components is shown schematically in Fig. 1.The total stress, _t , evaluated outside the zone of wave influence, is given in the classical way with Fig. 4. The wave-supporting stress, _w (z), was evaluated from the data according to a relation proposed by Kitaigorodskii et al. (1984); it is given with Fig. 5. A height-dependency is clearly evident. The turbulent stress _c (z), was evaluated with data of the velocity gradient; it is given with Fig. 6. A height-dependency is not evident.The field data from the lake of Geneva give evidence that the additive relation of Eq. 1 seems to be justified.With 6 Figures     

Internal gravity waves in a stably stratified boundary layer

Often, a combination of waves and turbulence is present in the stably stratified atmospheric boundary layer. The presence of waves manifest itself in the vertical profiles of variances of fluctuations and in low-frequency contributions to the power spectra. In this paper we study internal waves by means of a linear stability analysis of the mean profiles in a stably stratified boundary layer and compare the results with observed vertical variance profiles of fluctuating wind and temperature along a 200 m mast. The linear stability analysis shows that the observed mean flow is unstable for disturbances in a certain frequency and wavenumber domain. These disturbances are expected to the detectable in the measurements. It is shown that indeed the calculated unstable frequencies are present in the observed spectra. Furthermore, the shape of the measured vertical variance profiles, which increase with height, is explained well by the calculated vertical structure of the amplitude of unstable Kelvin-Helmholtz waves, confirming the contribution of waves to the variances. Because turbulence and waves have quite distinct transport properties, estimates of diffusion from measurements of variances would strongly overestimate this diffusion. Therefore it is important to distinguish between them.     

The linear steady response of a stratified baroclinic atmosphere to elevated diabatic forcing

Abstract

A parameter study is presented of the linear steady response to an elevated diabatic forcing in a Boussinesq baroclinic atmosphere. The model is two‐dimensional on the vertical plane; the basic wind is perpendicular to this plane with horizontal and vertical shear. The intensity of the circulation is sensitive to the strength of the baroclinic zone only for weak static stability. A scale analysis supports this conclusion: the importance of baroclinic effects depends on the ratio of the aspect ratio of the circulation to the aspect ratio of the baroclinic basic state. Baroclinicity also leads to a tilt of the circulation due to enhanced horizontal temperature advection, and the corresponding vertical flux of horizontal momentum. The magnitude of the latter can be large when the Richardson number approaches its critical value from above, i.e. for slightly symmetrically stable basic flows; this is true even with viscosity. The resonant limit where the Richardson number approaches the critical value for symmetric instability is also examined. For non‐zero dissipation, the critical value is smaller than that of the inviscid limit.     

Diagnosis of the forcing of inertial-gravity waves in a severe convection system

The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms—linear and nonlinear terms, and diabatic heating—on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.     

The response of an open stratified bay to wind forcing

Abstract

Current and temperature measurements collected during the summers of 1974, 1975 and 1979 are used to investigate the wind‐induced response of St Georges Bay, Nova Scotia. A multivariate‐frequency response analysis shows that temperature and the along‐bay component of current in the lower layer are coherent with the local wind stress at periods of 2–6 d, with the wind stress accounting for 35–65% of the observed variances. Winds are also coherent with the surface currents but account for only 20–25% of the variance. Two dynamically different regions are identified. Near the entrance to the bay, cross‐bay wind stresses (τ_x) produce Ekman drift with compensating flow in the lower layer. In the interior of the bay, the near‐bottom currents are coherent with along‐bay (τ_y) wind stresses and are directed upwind. These currents are topographically steered and a response to surface pressure gradients set up by the wind. The surface gradients are believed to be part of the wind‐induced set‐up within the southern Gulf of St Lawrence.     

Modulation of short waves by long waves

The amplitude, wavelength, and frequency of short waves in the presence of waves of a longer scale vary in a manner that is related in phase to the long-wave profile. The purpose of this study is to observe and quantify the change in the variance of short-wave slope that occurs as a result of the change in short-wave position along a coincident long wave, during the active generation of the short-wave field by wind. To this end, measurements of wave-slope time series are made in a laboratory environment where the long-scale waves are generated mechanically and the short scale are generated primarily by air flow. The frequency variation of the short waves, as measured along the long-wave profile, is described by considering the waves to be linearly advected by the longer waves. The peak-to-peak variation along the long-wave profile of the short-wave slope variance for a given frequency band is commonly found to be 10% of its mean value. The magnitude of the excursions become smaller as short-wave frequency increases, and larger as wind speed increases. The maximum value of the short-wave slope variance generally leads the long-wave profile curve by 45 ° to 180 °.     

Generation of internal waves over a shelf

Experiments are performed in a 13-m cylindrical tank to study the generation of interfacial internal waves by barotropic sinusoidal waves passing over a slope. At each tidal cycle, there are two waves generated, one propagating onshore and the other propagating offshore. The amplitude of the waves increases with increasing forcing and evolves as nonlinear waves if the shelf width is smaller than the wavelength of the baroclinic tide. Rotation does not modify the generating mechanism but the amplitude of the generated waves decreases with increasing rotation rate; also no internal waves are generated when the forcing period is larger than the inertial period, and at high rotation rate, there are only dispersive waves propagating from the shelf break region. The experiments covered a large range of internal Froude number, Rossby number and temporal Rossby number and compare well with in situ observations.     

层结大气中重力惯性波的发展

应用WKB方法讨论了非均匀层结大气条件下重力惯性波的发展。分析表明,在水平方向上,当波动由层结不稳定度小的地方向不稳定度大的地方传播时,波将发展,从而修正了文献[2]的结论。     

A study of internal waves in a wind tunnel

The combined use of a stratified flow wind tunnel and of periodic sampling methods in low Reynolds number flows allows the recovery of the instantaneous dynamics of internal waves. Several detailed examples are given of the thermal structure of large propagating and breaking internal waves and Kelvin-Hemholtz waves. Preliminary measurements of the stability of finite waves as a function of Richardson number are also reported.     

Instability of internal waves near a critical level

The three-dimensional stability problem is investigated for a family of velocity and density profiles similar in form to those expected for large-amplitude internal gravity waves near a critical level. These profiles exhibit regions of high shear and stable stratification alternating with regions of weak shear and unstable stratification. Analytical solutions are given for inviscid, neutral modes that are similar to those found under neutral conditions with stable stratification. Neutral modes form closed streamline patterns centered at locations of maximal shear, and are not strongly influenced by nearby regions of unstable stratification. Unstable modes are computed numerically. It is shown that the instability mechanism for these wave-like flows fundamentally three-dimensional in character and exhibits both shear and convective dynamics. For flows with parameter values below the neutral curves, unstable modes oriented in the streamwise direction undergo shear instability, while modes oriented orthogonally are convectively unstable. In addition to their intrinsic physical relevance, the results of this study have important implications for the physics and the numerical modeling of breaking internal gravity waves. Two-dimensional models will bias the breaking dynamics by eliminating the possibility for convection oriented in the transverse plane.     

The temperature structure in a stably stratified internal boundary layer over a cold sea

Data from the Öresund experiment are used to investigate the structure of the stably stratified internal boundary layer (SIBL) which develops when warm air is advected from a heated land surface over a cooler sea. The present study is based on a theory developed by Stull (1983a, b, c). He proposed that the turbulence and the mean structure of the nocturnal boundary layer is controlled by the time-integrated value of surface heat flux and that the instantaneous heat flux is of less importance.Dimensional arguments are used to define simple, physically consistent, temperature, velocity and length scales. The dimensionless surface heat flux has a high value immediately downwind of the shoreline and it decreases rapidly in magnitude with increasing distance from the coast. Farther away, it is essentially constant. The dimensionless potential temperature change exhibits an exponential profile. It is estimated that turbulence accounts for 71% of boundary-layer cooling while clear-air radiational cooling is responsible for the remaining 29%.Finally it is found that theoretical predictions for the height of the SIBL are in a good agreement with observations.     

Wind forcing of the ocean and the Atlantic meridional overturning circulation

Wind forcing of the oceans is analysed for a millennium-length period of a simulation with a global coupled model. The time mean value of the zonal wind energy input to the oceans was found to be 0.97?TW, similar to other estimates, with maximum inputs in the Southern Ocean and Equatorial Pacific Ocean. The meridional wind energy input was also evaluated. The time series of the zonal wind energy input consisted of white noise, with marked multiannual and multidecadal variability, and had a range of 1.6 between minimum and maximum values. Seasonal variations in the wind energy input were most marked in the Pacific Ocean. Composites of the various climatic terms involved in the wind energy input, for maximum and minimum values of the input time series, identified distinct differences in anomaly values, particularly over the Southern Ocean. The temporal variability of the Pacific equatorial wind energy input was clearly identified with ENSO events. The model reproduced the observed structure of the Atlantic meridional overturning circulation surprisingly well. The time series of this circulation exhibited interannual to centennial variability. Correlations, both instantaneous and lagged, failed to identify any meaningful relationship between the temporal variability of the circulation and the wind energy input to the Southern Ocean. The optimum correlation was found between time smoothed versions of the time series for this circulation and the heat input to the North Atlantic Ocean, implying, as noted elsewhere, that this heat input is the principal driver of the temporal variability of the circulation.     

Linear baroclinic instability of a nonzonal flow with special regard to thermal internal forcing

The linear baroclinic instabilities of a basic geostrophic shear flow in a stratified, thermally active ocean are considered, including the β-effect. It is shown that the internal heat production, even if it introduces only minor changes into the geostrophic balance, may deeply modify the stability properties of the stationary solutions. If density, geostrophic basic flow, and heat production are all regarded as arbitrary functions of depth only, and if a linear nonzonal assumption is made about the geostrophic pressure field, it can be shown how the geostrophic equilibrium is altered and how the internal heating affects both the directions of propagation and the growth rates of quasi-geostrophic disturbances. A semicircle theorem valid for every direction of propagation is proved, by generalizing well-known results obtained for adiabatic geostrophic flows.Finally, it is found that the modified instability can be described adequately by a directional Richardson number related to the perturbed stationary geostrophic flow.     

Evidence of long lee waves in Southern Alberta

A brief review of past studies of the of the order of 50—100 km or longer, chinook and a summary of some recent although resonance may be present satellite and visual observations of the in some cases. Both satellite and visual chinook arch cloud are presented as observations indicate the frequent ocevidence of the occurence of long currence of the wave during the colder gravity wave phenomena in the lee of period of the year. Taken together, the Rockies of Southern Alberta and these data imply that the wave pheno‐Northern Montana. A quasi‐stationary menon is potentially important in cloud band that extends several de‐ terms of local surface weather and grees of latitude along the lee of the vertical momentum transport and that mountains appears to be associated it should be subjected to more detailed with a single long wave with a length theoretical and observational studies.     

The influence of bottom topography on internal seiches in stratified media

Standing internal waves, so-called seiches, are ubiquitous in reservoirs and lakes. Although the stratification in such basin is often continuous, the modeling of seiches has been confined mostly to two-layer models. Such models are unable to give reliable insights into the vertical structure of the seiches, which might be crucial for the understanding of vertical mixing in natural water basins. To obtain this kind of information a two-dimensional computer model has been developed, which takes both continuous stratification and bottom topography into account. The results of this model are presented. The computed seiche modes reveal that (1) several large-scale modes can exist with similar eigenfrequencies, (2) the modes have a tendency to develop narrow jets and (3) only the lowest modes are strongly influenced by the bottom topography.