Stratified Fluid Of Finite Depth Research Articles

An equivalent mass source method for internal waves generated by a body moving in a stratified fluid of finite depth

An equivalent mass source is modeled as excited sources of internal waves generated by the displacement of fluid and the wake of vortex shedding or turbulence due to the body moving in a stratified fluid, a mathematical model for the vertical displacement of internal waves generated by such an equivalent mass source is presented by use of both the associated eigenvalue problem and the Fourier transform method, and the method for estimating the speed, length and diameter of the equivalent mass source is proposed. Calculations based on the proposed method are performed for the generation of internal waves by a towed sphere in a stratified fluid, and the numerical results are in good agreement with the available experimental results of Robey in the kinematic wave patterns and the maximum peak-to-peak amplitude of the internal waves. Such a method may have potentially important applications.

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

Abstract The evolution equation for the second-order internal solitary waves in stratified fluids, that are incomepressible inviscid ones of mixed stratification in which the density varies with depth in lower layer but keeps uniform in upper layer with a finite depth, is derived directly from the Euler equation by using the balance between the nonlinearity and the dispersion with the scaling μ= O(ɛ), and by introducing the Gardner-Morikawa transformation, asymptotic expansions and the matching of the solutions for the upper and lower layers of fluid via the computer algebraic operation. The evolution equation and its solution derived for the first-order wave amplitude are consistent with the classical ones, and the desired equation governing the second-order amplitude is reduced as follows ∂ f 2 ∂ τ - c 1 ∂ f 2 ∂ ξ + α ∂ ( f 1 f 2 ) ∂ ξ + β ∂ ∂ ξ ∫ - ∞ ∞ f 2 ( ξ ' ) g ( ξ - ξ ' ) d ξ ' = G 2 ( f 1 , c 1 , κ 1 ; c 0 , ϕ ) Where g ( ξ ) = - 1 2 π ∫ - ∞ ∞ k coth ( k l ) e j k ξ d k and G 2 ( f 1 , c 1 , κ 1 ; c 0 , ϕ ) is the inhomogeneous term, f 1 first-order wave amplitude.

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

Internal Waves Behind a Local Disturbance upon its Weakly Non-Stationary Motion in Stratified Fluid of Finite Depth

Internal Waves Behind a Local Disturbance upon its Weakly Non-Stationary Motion in Stratified Fluid of Finite Depth

The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid

The problem of lateral heat/buoyancy transport in localized turbulent convection dominated by rotation in continuously stratified fluids of finite depth is considered. We investigate the specific mechanism of the vortex-dominated lateral spreading of anomalous buoyancy created in localized convective regions owing to outward propagation of intense heton-like vortices (pairs of vortices of equal potential vorticity (PV) strength with opposite signs located at different depths), each carrying a portion of buoyancy anomaly. Assuming that the quasi-geostrophic form of the PV evolution equation can be used to analyse the spreading phenomenon at fast rotation, we develop an analytical theory for the dynamics of a population of three-dimensional hetons. We analyse in detail the structure and dynamics of a single three-dimensional heton, and the mutual interaction between two hetons and show that the vortices can be in confinement, splitting or reconnection regimes of motion depending on the initial distance between them and the ratio of the mixing-layer depth to the depth of fluid (local to bulk Rossby radii). Numerical experiments are made for ring-like populations of randomly distributed three-dimensional hetons. We found two basic types of evolution of the populations which are homogenizing confinement (all vortices are predominantly inside the localized region having highly correlated wavelike dynamics) and vortex-dominated spreading (vortices propagate out of the region of generation as individual hetons or heton clusters). For the vortex-dominated spreading, the mean radius of heton populations and its variance grow linearly with time. The law of spreading is quantified in terms of both internal (specific for vortex dynamics) and external (specific for convection) parameters. The spreading rate is proportional to the mean speed of propagation of individual hetons or heton clusters and therefore depends essentially on the strength of hetons and the ratio of local to bulk Rossby radii. A theoretical explanation for the spreading law is given in terms of the asymptotic dynamics of a single heton and within the frames of the kinetic equation derived for the distribution function of hetons in collisionless approximation. This spreading law gives an upper ‘advective’ bound for the superdiffusion of heat/buoyancy. A linear law of spreading implies that diffusion parameterizations of lateral buoyancy flux in non-eddy-resolving models are questionable, at least when the spreading is dominated by heton dynamics. We suggest a scaling for the ‘advective’ parameterization of the buoyancy flux, and quantify the exchange coefficient in terms of the mean propagation speed of hetons. Finally, we discuss the perspectives of the heton theories in other problems of geophysical fluid dynamics.

Point sink flow in a linearly stratified fluid of finite depth

The evolution of selective withdrawal through a point sink of horizontally unbounded, linearly stratified fluid of finite depth is studied as an initial-value problem. Following the initiation of discharge from the sink, internal gravity wave modes propagate radially upstream to change the flow pattern. These modes are called cylindrical modes. We first consider the case of F→0 (where F is the Froude number) to get linearized governing equations, and seek a linear asymptotic solution for large times t* after starting the discharge, of the cylindrical modes in a stratified fluid where viscous and diffusive effects are negligible. The obtained solution shows that the strength of the modal front grows like t*1/3, unlike the case of two-dimensional modes whose strength at the front is kept constant. Numerical calculations are also performed to study the case of F>0. The results then indicate that all the cylindrical modes can propagate upstream for any F except infinity. The steady-state withdrawal-layer thickness and time to steady state are also investigated over the full parameter range considering the viscous and diffusive effects. The obtained results are then compared with both the analytical and experimental results of prior works.

Sub-inertial dynamics of density-driven flows in a continuously stratified fluid on a sloping bottom. I. Model derivation and stability characteristics

A theory is developed describing the sub–inertial baroclinic dynamics of bottom–intensified density–driven flows within a continuously stratified fluid of finite depth with variable bottom topography. The evolution of the density–driven current is modelled as a geostrophically balanced homogeneous flow, which allows for finite–amplitude dynamic thickness variations and for which the pressure fields in each layer are strongly coupled together. The evolution of the overlying fluid is governed by baroclinic quasi–geostrophic dynamics describing a balance between the production of relative vorticity, the vortex–tube stretching/compression associated with a deforming gravity current height, and the rectifying influence of a background topographic vorticity gradient. The model is derived as a systematic asymptotic reduction of the two-fluid system in which the upper fluid is described by the Boussinesq adiabatic equations for a continuously stratified fluid and a lower homogeneous layer described by shallow–water theory appropriate for an f -plane with variable bottom topography. The model is shown to possess a non–canonical Hamiltonian formulation. This structure is exploited to give a variational principle for arbitrary steady solutions and stability conditions in the sense of Liapunov. The general linear stability problem associated with parallel shear flow solutions is examined in some detail. Necessary conditions for instability are derived. The instability is convective in the sense that it proceeds by extracting the available gravitational potential energy associated with the lower–layer water mass sliding down the sloping bottom. For the normal–mode instability problem, a semicircle theorem is derived. The linear stability characteristics are illustrated by solving the normal–mode equations for a simple linearly varying lower–layer height profile. In the overlying fluid, the unstable normal modes correspond to amplifying bottom–intensified topographic Rossby waves.

Waves due to initial disturbances at the inertial surface in a stratified fluid of finite depth

This paper is concerned with a Cauchy‐Poisson problem in a weakly stratified ocean of uniform finite depth bounded above by an inertial surface (IS). The inertial surface is composed of a thin but uniform distribution of noninteracting materials. The techniques of Laplace transform in time and either Green′s integral theorem or Fourier transform have been utilized in the mathematical analysis to obtain the form of the inertial surface in terms of an integral. The asymptotic behaviour of the inertial surface is obtained for large time and distance and displayed graphically. The effect of stratification is discussed.

Radiating solitary waves of a model evolution equation in fluids of finite depth

The evolution equation proposed by Kubota et al. [J. Hydronautics 12 (1978) 157–165] and Joseph [J. Phys. A 10 (1977) 225–227] for long waves in stratified fluids of finite depth is considered. Under a small third-order-derivative dispersive perturbation, solitary-wave solutions to this model equation become weakly nonlocal — they develop oscillatory tails of infinite extent and exponentially small amplitude. Here, these tails are calculated asymptotically following a perturbation procedure in the wavenumber domain.

On stationary equivalent modons in an eastward flow

Modons correspond to isolated dipole vortex solutions of the quasigeostrophic equations. They have been proposed as prototype models for some geophysical (and plasma) vortices. The classical modon solution on a β plane does not permit a Rossby wave field in the exterior or far-field region of the modon. However, it is qualitatively known that the gravest mode associated with a normal mode decomposition of a stationary modon in a continuously stratified fluid of finite depth necessarily contains a Rossby wave tail in the downstream region if the background flow is eastward. The same effect can be formally recreated in an equivalent-barotropic model of a stationary modon embedded in a constant eastward zonal flow. An analytical solution to this problem satisfying the correct upstream radiation condition is presented and its dynamical characteristics are discussed.

Total Positivity and the Stability of Internal Waves in Stratified Fluids of Finite Depth

Solitary internal waves in stratified fluids of finite depth are investigated. Using a general criterion for the stability of travelling-wave solutions of one-dimensional nonlinear long-wave models and ideas from the theory of totally positive operators, the stability of the solitary-wave solutions of the intermediate long-wave equation is demonstrated

On periodic and solitary wavelike solutions of the intermediate long-wave equation

The intermediate long-wave (ILW) equation is a weakly nonlinear integrodifferential equation which governs the evolution of long internal waves in a stratified fluid of finite depth. It reduces to the Korteweg–de Vries (KdV) and to the Benjamin–Ono (BO) equations for shallow and large depths respectively. Solitary wave solutions of the ILW equation are well known, however analytic expressions for periodic solutions of the same equation do not seem to exist. Such expressions are derived in this paper and a remarkable property discovered for these periodic waves is that they can be represented as an infinite sum of spatially repeated solitons. Thus, nonlinear periodic solutions of the ILW equation are obtained by linear superposition of solitons.

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.

Intermediate asymptotic behavior of the far field of internal waves in a layer of stratified fluid lying on a homogeneous layer

A study is made of the field of internal waves excited by fixed and moving sources in a layer of stratified fluid of finite depth h lying on a homogeneous layer of thickness H. It is shown that in the limit H → ∞ the asymptotic behavior of the field of the moving source can be expressed in terms of a Fresnel integral. For finite H the field asymptotic behavior expressed in terms of the Fresnel integral is intermediate – it is valid at distances h ≪ r ≲ L; the asymptotic behavior expressed in terms of the Airy function is true for r ≳ L, with the intermediate scale L = const H3/h2. In the case of a fixed source the intermediate asymptotic behavior can be expressed in terms of a parabolic cylinder function, and the farfield behavior can be expressed in terms of the square of the Airy function. The transition between the intermediate and far-field asymptotic behaviors is investigated numerically.

Laboratory study on the two‐dimensional flows of stratified fluids over barriers

Abstract This paper describes an experimental study of a stratified fluid of finite depth flowing over obstacles which induce flow seperation and turbulence on the lee side, in which the inviscid model is no longer useful. Various properties of the flow field, such as the development of lee waves behind the obstacle, the blocking effect upstream, the effect of viscosity, and in particular the criterion for the onset of gravitational instability in the lee‐wave field, are observed and analyzed. The results show that lee waves produced by obstacles in a stratified flow depend on the internal Froude number, the ratio of the height of obstacles to the channel depth, the ratio of the height to the half width of obstacles, and the Reynolds number of the flow. Therefore, it can be found that the existence of upstream influence and the flow seperation induced by the obstacle have great effect in some flow conditions on the development of the lee‐wave field.

Direct and inverse scattering problems of the nonlinear intermediate long wave equation

The inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed. The equation describes long internal gravity waves in a stratified fluid of finite depth, and reduces to the Korteweg–de Vries equation as shallow water limit and the Benjamin–Ono equation as deep water limit. Both limits of the method and novel aspects of the theory are also discussed.

On the development of plane waves generated by a periodic pressure applied to the flow of a continuously stratified fluid

In the linear formulation, an investigation is made into the development of undamped (in time) plane waves generated by a. harmonically varying pressure applied to the free surface of an initially undisturbed flow of a continuously stratified fluid of finite depth. The cases of a homogeneous fluid and two-layer fluid are considered in [1–3]. Nonstationary waves in a continuously stratified flow generated by a time-independent pressure were investigated in [4].

Exact One-and Two-Periodic Wave Solutions of Fluids of Finite Depth

Exact one- and two-periodic wave solutions of a nonlinear integro-differential equation describing the propagation of waves in a stratified fluid of finite depth is presented. The solution reduces to the Korteweg-de Vries (KdV) periodic wave solution in the shallow water limit and to the Benjamin-Ono (B-O) periodic wave solution in the deep water limit.

Exact multi-soliton solution for nonlinear waves in a stratified fluid of finite depth

The exact multi-soliton solution of a nonlinear integro-differential equation describing the propagation of waves in a stratified fluid of finite depth is presented. The solution reduces to the Korteweg-de Vries (KdV) soliton solution is the shallow water limit and to the Benjamin-Ono (BO) soliton solution in the deep water limit. The asymptotic form of the solution for large times is derived.

Steady stratified flows over a long obstacle

The nonlinear problem of the flow of a stratified fluid of finite depth over an obstacle is investigated. The flow is assumed to be steady and two dimensional. The effects of viscosity are neglected and the pressure field is assumed to be hydrostatic. Analytic solutions are obtained for compressible flows with constant entropy profiles far upstream. Solutions for large Froude number flows are also obtained. A numerical procedure is introduced which permits the calculation of all possible steady flow fields with or without closed streamlines for any given upstream velocity and entropy (or density) profiles. Calculations are carried out for upstream data obtained from actual measurements in the Rocky Mountains and comparison is made between the solutions corresponding to the compressible and incompressible models.