Small-Amplitude Disturbances in a Radiating and Scattering Grey MediumIII. Gravitational Effects on the Solutions of Given Real Wave Number k

We study the fundamental modes of radiation hydrodynamic linear waves that arise from one-dimensional small-amplitude initial fluctuations with wave number k in a radiating and scattering grey medium by taking into account the gravitational effects. The equation of radiative acoustics is derived from three hydrodynamic equations, Poisson’s equation, and two moment equations of radiation, by assuming a spherical symmetry for the matter and radiation and by using the Eddington approximation. We solve the dispersion relation as a quintic function of angular frequency ω, the wave number k being a real parameter. Numerical results reveal that wave patterns of five solutions are distinguished into three types: the radiation-dominated, type 1, and type 2 matter-dominated cases. In the case of no gravitaional effects (Kaneko et al., 2005), the following wave modes appear: radiation wave, conservative radiation wave, entropy wave, Newtonian-cooling wave, opacity-damped and cooling-damped waves, constant-volume and constant-pressure diffusions, adiabatic sound wave, cooling-damped and drag-force-damped isothermal sound waves, isentropic radiation-acoustic wave, and gap mode. Meanwhile, the gravitaional effects being taken into account, the growing gravo-diffusion mode newly arises from the constant-pressure diffusion at the point that k agrees with Jeans’ wave number specified by the isothermal sound speed. This mode changes to the growing radiation-acoustic gravity mode near the point that k becomes Jeans’ wave number specified by the isentropic radiation-acoustic speed. In step with a transition between them, the isentropic radiation-acoustic wave splits into the damping radiation-acoustic gravity mode and constant-volume diffusion. The constant-volume diffusion emerges twice if the gravitational effects are taken into account. Since analytic solutions are derived for all wave modes, we discuss their physical significance. The critical conditions are given which distinguish between radiation-dominated and type 1 matter-dominated cases, and between type 1 and type 2 matter-dominated cases. Waves in a self-gravitating scattering grey medium are also analyzed, which provides us some hints for the effects of energy and momentum exchange between matter and radiation.     

A new form of generalized Boussinesq equations for varying water depth

A new set of equations of motion for wave propagation in water with varying depth is derived in this study. The equations expressed by the velocity potentials and the wave surface elevations include first-order non-linearity of waves and have the same dispersion characteristic to the extended Boussinesq equations. Compared to the extended Boussinesq equations, the equations have only two unknown scalars and do not contain spatial derivatives with an order higher than 2. The wave equations are solved by a finite element method. Fourth-order predictor–corrector method is applied in the time integration and a damping layer is applied at the open boundary for absorbing the outgoing waves. The model is applied to several examples of wave propagation in variable water depth. The computational results are compared with experimental data and other numerical results available in literature. The comparison demonstrates that the new form of the equations is capable of calculating wave transformation from relative deep water to shallow water.     

A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave motions

A new three-dimensional, non-hydrostatic free surface flow model is presented. For simulating water wave motions over uneven bottoms, the model employs an explicit project method on a Cartesian the staggered gird system to solve the complete three-dimensional Navier–Stokes equations. A bi-conjugated gradient method with a pre-conditioning procedure is used to solve the resulting matrix system. The model is capable of resolving non-hydrostatic pressure by incorporating the integral method of the top-layer pressure treatment, and predicting wave propagation and interaction over irregular bottom by including a partial bottom-cell treatment. Four examples of surface wave propagation are used to demonstrate the capability of the model. Using a small of vertical layers (e.g. 2–3 layers), it is shown that the model could effectively and accurately resolve wave shoaling, non-linearity, dispersion, fission, refraction, and diffraction phenomena.     

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.     

Scour at the head of a vertical-wall breakwater

This paper presents the results of an experimental investigation on the near-bed flow patterns, the bed shear stress amplification and scour around the head of a vertical-wall breakwater, using regular waves. The Keulegan-Carpenter number (KC), based on the diameter of the breakwater head, is found to be the major parameter that governs the flow and the equilibrium scour depth. Basic flow structures are identified as function of KC. The scour depth is found to increase with increasing the Keulegan-Carpenter number. The necessary extent of the conventional stone protection is studied. An empirical formula is worked out for the width of the protection layer as function of KC. Also, the effects of head shape, the angle of attack and the presence of a co-directional current are investigated. The results indicate that the scour depth is increased considerably in the presence of a current. Likewise, the scour depth is increased when the head shape is changed from a round shape to a sharp-edged one. It is found that the angle of attack is also an influencing factor as regards the scour depth.     

Near-infrared imaging in H2 of molecular (CO) outflows from young stars

We have imaged several known molecular (CO) outflows in H₂ v=1-0 S(1) and wide-band K in order to identify the molecular shocks associated with the acceleration of ambient gas by outflows from young stars. We detected H₂ line emission in all the flows we observed: L 1157, VLA 1623, NGC 6334I, NGC 2264G, L 1641N and Haro 4-255. A comparison of the H₂ data with CO outflow maps strongly suggests that prompt entrainment near the head of a collimated jet probably is the dominant mechanism for producing the CO outflows in these sources.     

The densest prestellar condensations: H2CO Studies ofρ Oph B1

We present new images of the well-known molecular outflow and Herbig-Haro complex L 1551-IRS 5. Deep, high-resolution images of the central region of the flow in [SII] 6716,6731 and H (6565 Å) are complemented by a mosaic of much of the CO outflow in H₂ v=1-0 S(1). While the optical data trace the intermediate-to-high excitation shocks in the flow (v _shock > 30 – 50 km s^–1), the near-IR data reveal the lower-excitation, molecular shocks (v _shock 10–50 km s^–1). In particular, the H₂ data highlight the regions where the flow impacts and shocks ambient molecular gas.     

Spontaneous Emission of Gravity Waves by Interacting Vortex Dipoles in a Stratified Fluid: Laboratory Experiments

Results from a new series of experiments on the geophysically important issue of spontaneous emission of internal gravity waves during unsteady interactions of vortical structures are presented. Vortex dipoles are a common element of a quasi-two-dimensional turbulent flow. Vortex dipoles perform translational motion and can collide with other vortices. During collision events the flow is unsteady and unbalanced and a further adjustment process associated with these events can therefore result in the spontaneous emission of gravity waves. Our laboratory experiments demonstrate that gravity waves are emitted when two translating vortex dipoles interact (collide) in a layered fluid, in accord with the current theoretical results. The emission was evident both in a two-layer system and in a fluid with a linear distribution of density with depth. The waves were generated during the period of deceleration of the secondary dipoles which constitute a vortex quadrupole emerging immediately after the collision of the primary dipoles.     

二维不均匀介质中点源P-SV波响应的有限差分近似算法

本文从一阶方程组形式的波动方程出发,发展了一种计算二维不均匀介质中点源P-SV波响应的近似方法。该方法通过引入线分布的应力作为震源,利用二维有限差分方法计算出线源响应,然后再经过波形校正和几何扩散校正得出相应的近似点源响应。通过把波形和振幅与精确解比较表明,该方法具有较好的精度。由于有限差分方法对于介质中速度和密度的分布没有特殊要求,另一方面,本文所给出的震源可以适用于位错点源、爆炸源或集中力源,因此上述方法十分适合于研究横向不均匀介质中的近场强地运动、爆炸振动或地震勘探等问题。