The langevin model for turbulent diffusion

Abstract

The Lagrangian approach to turbulent diffusion is considered through an improved Langevin model approach. The improved model partially accounts for the interaction between pressure and viscous stresses, an interaction not included in previous studies. The model is used to obtain one and two particle dispersion relations and it permits calculation of Eulerian-Lagrangian scale relations. Turbulence Reynolds number and time span limits for Richardson's law of diffusion are obtained. Non-stationarity of the acceleration between two particles is discussed as it relates to Richardson's law and the conclusions are considered in view of both laboratory and geophysical fluid dynamical data.     

The viscous damping of internal waves on the rotating earth

Summary A formula is derived for the spatial attenuation of the two possible modes of oscilation of a two-layer, rotating system. If the long-wave approximation is made, it is found with reference to the internal mode that this formula disagrees with an earlier result obtained byRattray [8]²).     

The wave-induced boundary layer under long internal waves

The boundary layer formed under the footprint of an internal solitary wave is studied by numerical simulation for waves of depression in a two-layer model of the density stratification. The inviscid outer flow, in the perspective of boundary-layer theory, is based on an exact solution for the long wave-phase speed, yielding a family of fully nonlinear solitary wave solutions of the extended Korteweg–de Vries equation. The wave-induced boundary layer corresponding to this outer flow is then studied by means of simulation employing the Reynolds-averaged Navier–Stokes (RANS) formulation coupled with a turbulence closure model validated for wall-bounded flows. Boundary-layer characteristics are computed for an extensive range of environmental conditions and wave amplitudes. Boundary-layer transition, identified by monitoring the eddy viscosity, is correlated in terms of a boundary-layer Reynolds number. The frictional drag is evaluated for laminar, transitional, and turbulent cases, and correlations are presented for the friction coefficient plus relevant measures of the boundary-layer thickness.     

Space-Time scales of internal waves

Abstract

We have contrived a model E(αω) α μ^?1ω^?p+1(ω ²?ω _i ²)^?+ for the distribution of internal wave energy in horizontal wavenumber, frequency-space, with wavenumber α extending to some upper limit μ(ω) α ω ^r-1 (ω ²?ω _i ²)^½, and frequency ω extending from the inertial frequency ω _i to the local Väisälä frequency n(y). The spectrum is portrayed as an equivalent continuum to which the modal structure (if it exists) is not vital. We assume horizontal isotropy, E(α, ω) = 2παE(α₁, α₂, ω), with α₁, α₂ designating components of α. Certain moments of E(α₁, α₂, ω) can be derived from observations. (i) Moored (or freely floating) devices measuring horizontal current u(t), vertical displacement η(t),…, yield the frequency spectra F _(u,η,…)(ω) = ∫∫ (U ², Z ²,…)E(α₁, ∞₂, ω) dα₁ dα₂, where U, Z,… are the appropriate wave functions. (ii) Similarly towed measurements give the wavenumber spectrum F _(…)(α1) = ∫∫… dα₂ dω. (iii) Moored measurements horizontally separated by X yield the coherence spectrum R(X, ω) which is related to the horizontal cosine transform ∫∫ E(α1, α2 ω) cos α₁ Xdα₁ dα₁. (iv) Moored measurements vertically separated by Y yield R(Y, ω) and (v) towed measurements vertically separated yield R(Y, α₁), and these are related to similar vertical Fourier transforms. Away from inertial frequencies, our model E(α, ω) α ω ^?p-r for α ≦ μ ω ω ^r, yields F(ω) ∞ ω ^?p, F(α₁) ∞ α₁ ^?q, with q = (p + r ? 1)/r. The observed moored and towed spectra suggest p and q between 5/3 and 2, yielding r between 2/3 and 3/2, inconsistent with a value of r = 2 derived from Webster's measurements of moored vertical coherence. We ascribe Webster's result to the oceanic fine-structure. Our choice (p, q, r) = (2, 2, 1) is then not inconsistent with existing evidence. The spectrum is E(∞, ω) ∞ ω ^?1(ω ²?ω _i ^{2 ?1}, and the α-bandwith μ ∞ (ω ²?ω _i ²)⁺ is equivalent to about 20 modes. Finally, we consider the frequency-of-encounter spectra F([sgrave]) at any towing speed S, approaching F(ω) as S ≦ S _o, and F(α₁) for α₁ = [sgrave]/S as S ≧ S _o, where S _o = 0(1 km/h) is the relevant Doppler velocity scale.     

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.     

Wind-generated internal waves and intertial-period motions

Summary The ocean is subdivided into a homogeneous upper layer (Ekman layer) and a continuously stratified lower layer. Horizontal velocities which in analogy to Fredholm's solution contain inertial oscillations, are generated in the upper layer by the tangential stress of the wind field varying both in space and time. The spatial structure of the wave field corresponds to that o the stress field. If the surface remains at rest (=0), vertical motions result from the horizontal oscillations. At the bottom of the Ekman layer the spatial structure of these vertical motions is proportional to divergence and rotation of the wind stress. Internal waves of the lower layer which primarily have periods approaching the inertial-period are generated by the vertical velocity field at the bottom of the Ekman layer (Ekman suction). Their structure is essentially more complicated than near the surface. Beats are a common phenomenon due to the superposition of internal modes.

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Windbedingte interne Wellen und Trägheitswellen

Zusammenfassung Das Meer wird in eine homogene Deckschicht (Ekmansche Reibungsschicht) und eine kontinuierlich geschichtete innere Region unterteilt. Die raum-zeitlich variable tangentiale Schubspannung erzeugt in der Deckschicht Horizontalgeschwindigkeiten, denen analog der Fredholmschen Lösung Trägheitswellen aufgeprägt sind. Ihre räumliche Struktur ist mit der des Schubspannungsfeldes identisch. Wenn keine Stauerscheinungen auftreten (=0), werden durch diese Horizontalbewegungen Vertikalbewegungen bedingt. An der Untergrenze der Ekman-Schicht ist deren räumliche Struktur der Divergenz und Rotation des Schubspannungsfeldes proportional. Das Vertikalgeschwindigkeitsfeld an der Untergrenze der Deckschicht erregt interne Wellen in der unteren Region, die vorwiegend Perioden in der Nähe der Trägheitsperiode aufweisen. Ihre Struktur ist wesentlich komplizierter als in der Deckschicht. Insbesondere zeigen sich Schwebungen.

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Ondulations internes dues au vent et mouvements dont la période dépend de l'inertie

Résumé L'Océan se divise en une couche supérieure homogène (couche d'Ekman) et une couche inférieure stratifiée d'une façon continue. Les vélocités horizontales, affectées, par analogie avec la solution de Fredholm, d'oscillations dues à l'inertie, prennent naissance dans la couche supérieure par suite de l'effort tangentiel du champ du vent, variable à la fois dans l'espace et dans le temps. La structure du champ du vent, dans l'espace, correspond à celle du champ de l'effort. Si la surface reste calme (-0) les mouvements verticaux sont dus aux oscillations horizontales. Au fond de la couche d'Ekman, la structure dans l'espace de ces mouvements verticaux est en rapport avec la divergence et la rotation de l'effort du vent. Les ondulations internes de la couche inférieure qui ont, à l'origine, des périodes proches de celle due à l'inertie, sont engendrées par le champ de la vitesse verticale au fond de la couche d'Ekman (succion d'Ekman). Leur structure est beaucoup plus compliquée que près de la surface. Les battements sont un phénomène commun du à la superposition de modes internes.

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The turbulent dispersion of surface drifters by water waves: experimental study

Ocean Dynamics - Dispersion of a passive tracer by water waves is of significant importance for many scientific and technological problems including bio-diversity of marine life, ecological impact...     

Contribution to the statistical dynamics of turbulent diffusion

Zusammenfassung Bei Studium der turbulenten Diffusion passiver Teilchen gehen wir aus statistischer Dynamik des Prozesses hervor, die auf dem zu einem unendlichen System von Gleichungen führenden Formalismus der Verteilungsfunktionen beruht. Wir behandeln in Kürze das Problem der Abgeschlossenheit, das man sowohl für die Turbulenz-Dynamik, als auch für die turbulente Diffusion übereinstimmend formulieren kann. Verschieden sind nur die Ausgangs-Bewegungsgleichungen. Wir studieren die statistische Struktur des entsprechenden Operators der Übertragung (kinetisches Modell) und suchen den Zusammenhang zwischen dem Modell des Übergangs zum Gleichgewichtszustand für die grundlegende kinetische Gleichung und die Struktur des Übergangs für die Einteilchen-Dichte, Schließlich untersuchen wir ein einfaches stochastisches Modell der in der Übertragungs-Gleichung der Diffusion enthaltenen Korrelations-Funktion.     

The incidence of internal waves onto a thin submerged barrier

Abstract

The problem of oblique incidence of internal ocean waves on a thin submerged ocean barrier is considered when the ocean has exponential density stratification. A Wiener-Hopf approach is used combined with numerical evaluation of series. Results for the reflected energy are obtained and reveal a complex dependence on incidence and barrier height. Application of this model to waves incident on the Mid-Atlantic ridge suggests that the ridge almosts isolates first mode energy on one side of the ocean from the other side. In certain circumstances there, is a surprising appearance of “barrier” waves. These waves are closely confined to the barrier and propagate along it.     

On topographic generation and coupling of internal waves

Abstract

For a two-dimensional geometry the internal wave eigenfunctions are constructed as combinations of non-uniform wave trains to satisfy variable boundary conditions. All the notable features of the wave-topography interaction problem can be derived from the form of in homogeneous phase function; hence its construction is the central theme of this paper. It is shown that the general solution can be expanded in many (perhaps an infinite number) complete sets of eigenfunctions, but that the application of physical constraints (if available) narrows the choice to at most two sets, in terms of which the general topographic generation and coupling problem can be solved. The general method is illustrated by several special cases, which indicate that coupling is relatively weak for sub-critical slopes but strong for critical and also for supercritical slopes.     

The influence of bed forms on reference concentration and suspension under waves and currents

Hydrodynamic, suspension and bed-form measurements were made 2 km off the Dutch coast near Noordwijk aan Zee in ∼14 m water depth for a period of 32 days in 2003. Tidal currents were just able to suspend sand at the bed at peak spring tide but most suspension and transport occurred as a result of the combination of waves and currents. Burst-average (17 min) sand concentration profiles (C¯-profiles) from an acoustic backscatter instrument were used to define the (varying) location of the sea-bed, following the method used by Green et al. [Green, M.O., Dolphin, T.J., Swales, A., Vincent, C.E., 1999. Transport of mixed-size sediments in a tidal channel. Coastal Sediments ‘99, edited by N.C. Kraus, and W.G. McDougal, ASCE, Long Island, New York, pp. 644–658]. Reference concentrations at the sea-bed (C₀) and at 1 cm (C₁) were examined in relation to both the hydrodynamic conditions and the type of bed forms present. The C₀ predictive equations of Green and Black [Green, M.O., Black, K.P., Suspended sediment reference concentration under waves: field measurements and critical analysis of two predictive models, Coastal Engineering, 38, 115–141, 1999](short-wave ripples) and Nielsen [Nielsen, P., Suspended sediment concentrations under waves, Coastal Engineering, 10, 23–31, 1986](all bed forms; includes ripple steepness), both of which require knowledge of the bed-form type, were not as successful in explaining the variance in our C₀ data as a regression of C₀ against the skin-friction Shields parameter θ′_cw that ignored bed-form type (73% of variance explained). The values of the reference concentration C₁ were compared with the Lee et al. [Lee, G.-H., Dade, W.B., Friedrichs, C.T., Vincent, C.E., Examination of Reference Concentration Under Waves and Currents on the Inner Shelf., Journal of Geophysical Research, 109, 1–10, 2004] equation which predicts C₁ from the product of the Shields parameter and the inverse Rouse parameter; 51% of the variance in C₁ was explained.     

The influence of background winds and attenuation on the propagation of atmospheric gravity waves

The influence of background winds and energy attenuation on the propagation of atmospheric gravity waves is numerically analyzed. The gravity waves, both in the internal and ducted forms, are included through employing ray-tracing method and full-wave solution method. Background winds with different directions may cause ray paths of internal gravity waves to be horizontally prolonged, vertically steepened, reflected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionospheric height with less energy attenuation. The propagation status of gravity waves with different periods and phase speeds is classified into the cut-off region, the reflected region and the propagating region. All the three regions are influenced significantly by winds. The area of the reflected region reduces when gravity waves propagate in the same direction of winds and expands when propagating against wind. In propagating region, the horizontal attenuation distances of gravity waves increase and the arrival heights decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival heights increase when gravity waves propagate against winds. The results for ducted gravity waves show that the influence of winds on waves of lower atmospheric modes is not noticeable for they propagate mainly under mesosphere, where the wind field is relatively weak. However, strong winds at thermospheric height lead to considerable changes of dispersion relation and attenuation distance of upper atmospheric modes. Winds against the wave propagating direction support long-distance propagation of G mode, while the attenuation distances decrease when winds blow in the same direction of the wave. The distribution of TIDs observed by HF Doppler array at Wuhan is compared with the simulation of internal gravity waves. The observation of TIDs shows agreement with our numerical calculations.     

Random model of turbulent diffusion and latent parameters of the process

Zusammenfassung Wir studieren die Folgen eines Modells der unter der Voraussetzung konstruierten turbulenten Diffusion, dass die zuf?llige FunktionX(X ₀,t) eine Markowsche Funktion ist und dass der Mittelwert der Konzentration von der Beimengung der zweiten Kolmogorowschen Gleichung entspricht. Das Verlassen dieser Vorstellung führt zur Schlussfolgerung, dass man den Zustand des „Mikroobjekts” (Teilchens) für jedent-Wert als eine Klasse von Gr?ssenwerten betrachten kann, die es gleichzeitig prinzipiell m?glich ist mit beliebiger Genauigkeit zu messen. Eine Abwendung von der statistischen Relation der Unbestimmtheit wird durch die Existenz von verborgenen Parametern bei den diffusions- und Turbulenzprozessen verursacht. ihr Vorkommen ruft jedoch keinen übergang zum Kausalmodell des Prozesses hervor; die FunktionX(X ₀,t) (Lagrangesche Charakteristik vom Fliessen einer unkompressibeln Flüssigkeit) bleibt immer eine zuf?llige Funktion. Die Klasse der m?glichen Wahrscheinlichkeitswerte des übergangsp bildet jedoch keinen Markowschen Prozess im entsprechenden Phasenraum.

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Address: Boční II, Praha 4-Spořilov.     

海洋中尺度涡与内波的地震图像

海洋反射地震通常用于调查、研究海底地质构造,勘探油气与天然气水合物资源.近期研究表明多道反射地震方法也可以对水柱的热盐细结构成像.中尺度涡与内波是重要的物理海洋现象,但是常规的物理海洋调查是在间隔若干公里的离散测站上进行的,水平分辨率较低,因此对中尺度涡的结构与内波的横向分布了解较差.本文利用在大西洋东部、南海采集的地震数据给出了低频反射地震可以对中尺度涡与内波清晰成像的新的证据.反射地震方法较传统海洋观测手段,具有明显的优势,主要体现在高的水平分辨率和短时间内对整个海水剖面进行成像方面．从地震剖面上,能够清楚地观测到中尺度涡、内波造成的反射特征变化,从而有助于改进对能量在不同尺度的海水运动之间传递过程的认识.     

The influence of the time change of static stability and wind shear on baroclinic waves

The effect of the time change of static stability and wind shear on a two-level quasigeostrophic baroclinic wave is considered. First, by keeping the wind shear as a constant the effect of the increase of static stability is that the amplitude of the temperature wave reaches a maximum first while that of the stream field is still amplifying and, as that of temperature wave reaches a minimum, that of the stream field becomes a maximum, and both are in phase. Next, by keeping the static stability as a constant, a life cycle of disturbance associated with the time change of wind shear is obtained. In this case the maximum amplitude of the wave appears at about 6 days and the life cycle is about 11 days for reasonable values of the model parameters. Finally, both effects are considered. The results show that as the wind shear decreases, the static stability increases, and the percentage change of wind shear is larger than that of static stability.This paper is not an entirely convincing analysis of the finite amplitude dynamics of an unstable wave in the two-level model, but rather a pedagogically useful approximate theory in which the retention of some terms rather than others is justified in view of the plausibility of the results.     

Problem of transient turbulent convection and a semiempirical profile of turbulent heat exchange coefficient in freshwater bodies

A problem of convective cooling of both freshwater and saltwater lake is formulated. A transient one-dimensional turbulent model with variable layer depth is proposed to describe the process of freezing. The dependence of layer depth on time is determined by a universal equation describing the propagation of Deardorff convective layer. A combination of the main form of equations of Kolmogorov–Obukhov theory and available experimental data is shown to allow the construction of a universal profile of turbulent exchange coefficient, depending on the height.     

Temperature and heat flux spectra in the turbulent buoyancy subrange

The physical nature of motions with scales intermediate between approximately isotropic turbulence and quasi-linear internal gravity waves is not understood at the present time. Such motions play an important role in the energetics of small scales processes, both in the ocean and in the atmosphere, and in vertical transport of heat and constituents. This scale range is currently interpreted either as a saturated gravity waves field or as a buoyancy range of turbulence.We first discuss some distinctive predictions of the classical (Lumley, Phillips) buoyancy range theory, recently improved (Weinstock, Dalaudier and Sidi) to describe potential energy associated with temperature fluctuations. This theory predicts the existence of a spectral gap in the temperature spectra and of an upward mass flux (downward buoyancy and heat fluxes), strongly increasing towards large scales. These predictions are contrasted with an alternate theory, assuming energetically insignificant buoyancy flux, proposed by Holloway.Then we present experimental evidences of such characteristic features obtained in the lower stratosphere with an instrumented balloon. Spectra of temperature, vertical velocity, and cospectra of both, obtained in homogeneous, weakly turbulent regions, are compared with theoretical predictions. These results are strongly consistent with the improved classical buoyancy range theory and support the existence of a significant downward heat flux in the buoyancy range.The theoretical implications of the understanding of this scale range are discussed. Many experimental evidences consistently show the need for an anisotropic theory of the buoyancy range of turbulence.