Water wave scattering by bottom undulations in an ice-covered two-layer fluid

The scattering of plane surface waves by bottom undulations in an ice-covered ocean modelled as a two-layer fluid consisting of a layer of fresh water of lesser density above a deep layer of salt water, is investigated here by using a simplified perturbation analysis. In such a two-layer fluid there exist waves of two different modes, one with higher mode propagates along the interface and the other with lower mode propagates along the ice-cover. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes so that transfer of wave energy from one mode to another takes place. The first-order reflection and transmission coefficients of two different modes are obtained due to incident waves of again two different modes by employing Fourier transform technique in the mathematical analysis. For sinusoidal bottom topography these coefficients are depicted graphically against the wavenumber. These figures show how the transfer of energy from one mode to another takes place.     

The effect of large-scale bottom irregularities on the dynamics of Rossby waves in the ocean

This paper discusses, in terms of the geometrical optics approximation, how large-scale bottom irregularities influence the propagation of Rossby waves in the ocean. To describe the major peculiarities of the phenomenon, a two-layer model is applied, with the depth of the upper layer being considerably smaller than that of the lower layer. However, even with the bottom topography being allowed for, the wave motion is described by two Rossby wave modes, namely, a barotropic mode and a baroclinic mode. It is demonstrated that barotropic mode transformation caused by large irregularities of the sea-floor may lead to wave interaction, resulting in their anomalous distribution. Translated by Vladimir A. Puchkin.     

Coastal trapped waves in a two-layer ocean: Wave properties when the density interface intersects a sloping bottom

Properties of coastal trapped waves when the pycnocline intersects a sloping bottom are studied using a two-layer model which has slopes in both layers. In this system there is an infinite discrete sequence of modes, and four different sorts of waves exist: the barotropic Kelvin wave, the upper shelf wave, the lower shelf wave and the internal Kelvin-type wave. They all propagate with the coast to their right in the Northern Hemisphere. The upper and lower shelf waves are due to the topographic-effect on the upper-layer and lower-layer slopes, respectively. Their motions are dominant in the respective layers being accompanied by significant interface elevations. The properties of the upper (lower) shelf wave are almost unaffected by the existence of a lower-layer (upper-layer) slope. The motion of the internal Kelvin-type wave is confined to the region around the line where the density interface intersects the bottom slope.The modes, except that with the fastest phase speed (the barotropic Kelvin wave), are assigned mode numbers in order of descending frequency. Characteristics of Mode 1 change with wavenumber; the upper shelf wave for small wavenumbers and the internal Kelvin-type wave for large wavenumbers (high frequencies). The higher modes of Mode 2 and above can be classified into the upper and lower shelf waves.     

Stability of a barotropic jet on a sloping bottom

Linear stability of a barotropic jet on a sloping bottom with and without a side boundary is examined. When a sloping bottom and a side boundary are absent, a symmetric jet generally has two unstable modes: a symmetric mode and an antisymmetric mode. In the presence of a sloping bottom or a side boundary, they are modified and lose their symmetry.The presence of a side boundary does not produce substantial change in the stability characteristics, except that it stabilizes the flow to some degree. In the presence of a sloping bottom, the following features are noted; 1) when the direction of the jet is opposite to the propagation direction of topographic Rossby waves, the change of a preferred mode occurs at a certain slope, 2) when the direction of the jet is opposite to 1), with a side boundary, the dispersion relations change from unstable mode type to shelf wave type at a certain slope, accompanied by kissing.     

Waves and currents near the continental shelf edge

The abrupt depth increase which characterises the edge of many continental shelves determines a reduced horizontal length scale and a localised transition from shelf seas to the deep ocean. Particular forms of motion which may arise from the steep slopes include topographically guided currents along the slope, shelf-break upwelling, topographic Rossby waves and internal lee waves in the tidal current. The ocean/shelf mismatch may lead to a clear separation of water types, substantial reflection (from the shelf-edge neighbourhood) of all oceanic and shelf motions with periods greater than a few hours, and interaction between barotropic and baroclinic motions. Unstable longshelf currents, interleaving water masses, strong internal tides and internal waves, and narrow canyons enhance mixing across the shelf edge.     

Stability of planetary waves in a two-layer system

A stability of planetary waves on an infinite beta-plane is investigated in an idealized two-layer fluid system for the large local Rossby numberM. When a primary wave is barotropic, two kinds of barotropic instability modes are found. One of them was previously discussed byGill (1974). When a primary wave is baroclinic, two different kinds of modes that enable barotropic and baroclinic energy transfers are found. The one that has the larger growth rate gains its energy mainly from the mean shear of the primary wave when the internal rotational Froude numberF is smaller than 1/2. WhenF is larger than 1/2, however, the available energy conversion of the primary wave is dominant. This mode has a fairly large part of its energy in the barotropic motion although the primary wave is purely baroclinic.The effect ofO(M ^–2) corrections is found to have a stabilizing influence on all symmetrical modes. The geophysical applications of the present analysis are suggested in the context.     

Calculation of Interannual Variations of Sea Level in the Subtropical North Pacific

The interannual variations of sea level at Chichi-jima and five other islands in the subtropical North Pacific are calculated for 1961–95 with a model of Rossby waves excited by wind. The Rossby-wave forcing is significant east of 140°E. Strong forcing of upwelling (downwelling) Rossby wave occurs during El Niño (La Niña) and warm (cold) water anomaly in the eastern equatorial Pacific. The first and second baroclinic modes of Rossby wave are more strongly generated than the barotropic mode in the study area. A higher vertical mode of Rossby wave propagates more slowly and is more decayed by eddy dissipation. The best coefficient of vertical eddy dissipation is determined by comparing the calculated sea level with observation. The variation in sea level at Chichi-jima is successfully calculated, in particular for the long-term change of the mean level between before and after 1986 with a rise in 1986 as well as the variations with periods of two to four years after 1980. It is concluded that variations of sea level at Chichi-jima are produced by wind-forced Rossby waves, the first baroclinic wave primarily and the barotropic wave secondly. The calculation for other islands is less successful. Degree of the success in calculation almost corresponds to a spatial difference in quantity of wind data, and seems to be determined by quality of wind data.     

Effect of stratification on long period trapped waves on the shelf

The effect of stratification on very long-period waves trapped on a straight continental shelf of constant depth is examined for a two-layer model. There are 4 modes in this system. The characteristics of the mode with the largest phase velocity can be approximated by the barotropic mode. The mode corresponding to the barotropic shelf-wave mode is modified by the baroclinic motions significantly, and in the limit of very narrow shelf width, the mode characteristics are transformed from those of the barotropic shelf-wave to the baroclinic Kelvin wave if the long-shore wave length is larger than the internal deformation radius. In this case, the stratification has an apparent effect of increasing phase velocity of barotropic shelf-waves. The remaining two modes are dominated by baroclinic motions with significant contribution from barotropic motions: among which the one has a shelf-wave characteristics for small values of the shelf width and approaches the mode corresponding to the baroclinic Kelvin wave in shallower water for large shelf width and the other is a stationary mode. If the long-shore wave length is much shorter than the internal deformation radius, the motions in the upper and lower layers are decoupled: the surface and bottom modes analogous to those discussed byRhines (1970) appears.If the interface is deeper than the shelf depth, the stationary mode is absent and the characteristics of the third mode approaches those of the baroclinic double Kelvin wave mode as the shelf width increases.     

Northward transport of pacific summer water along the Northwind Ridge in the Western Arctic Ocean

The recent sea-ice reduction in the Arctic Ocean is not spatially uniform, but is disproportionally large around the Northwind Ridge and Chukchi Plateau compared to elsewhere in the Canada Basin. In the Northwind Ridge region, Pacific Summer Water (PSW) delivered from the Bering Sea occupies the subsurface layer. The spatial distribution of warm PSW shows a quite similar pattern to the recent ice retreat, suggesting the influence of PSW on the sea-ice reduction. To understand the regionality of the recent ice retreat, we examine the dynamics and timing of the delivery of the PSW into this region. Here, we adopt a two-layer linearized potential vorticity equation to investigate the behavior of Rossby waves in the presence of a topographic discontinuity in the high latitude ocean. The analytical results show a quite different structure from those of mid-latitude basins due to the small value of β. Incident barotropic waves excited by the sea-ice motion with large annual variation can be scattered into both barotropic and baroclinic modes at the discontinuity. Since the scattered baroclinic Rossby wave with annual frequency cannot propagate freely, a strong baroclinic current near the topographic discontinuity is established. The seasonal variation of current near the topographic discontinuity would cause a kind of selective switching system for shelf water transport into the basin. In our simple analytical model, the enhanced northward transport of summer water and reduced northward transport of winter water are well demonstrated. The present study indicates that these basic dynamics imply that a strengthening of the surface forcing during winter in the Canada Basin could cause sea-ice reduction in the Western Arctic through the changes of underlying Pacific Summer Water.     

基于类海啸波型的岛礁水动力特性数值模拟研究

基于非静压单相流模型NHWAVE建立了高精度二维数值波浪水槽,采用日本2011年实测真实海啸波型系统研究了海啸波在岛礁上传播变形的规律,并且分析了波高、礁坪淹没水深和礁前斜坡坡度等因素对孤立波和真实海啸传播变形的影响。结果表明,相比孤立波,类海啸波的波长明显大于孤立波波长,在测点处引起的水面变化持续时间更长,同等波高情况下真实海啸波型比孤立波能够携带更多的能量,与岛礁的相互作用也更为复杂,在礁坪上形成的淹没水深约为孤立波的两倍。礁前斜坡坡度和礁坪淹没水深均对类海啸波的反射和透射系数有显著影响。随着礁前斜坡坡度的增加,反射系数和透射系数均逐渐增加。随着礁坪淹没水深的增加,反射系数逐渐减小,而透射系数逐渐增大。但是,反射系数和透射系数均随着入射波高的增加而逐渐减小。     

Generation of spatial internal waves in a rotating ocean by moving non-stationary disturbances

In a general linear statement, the kinematic structure of spatial internal waves generated by a uniformly moving area of oscillating surface pressures in a continuously stratified ocean of constant depth is studied. The earth's rotation effects are considered. Possible types of ocean wave fields with a constant Brunt-Väisälä frequency are examined. The wave regimes for individual modes of internal and gyroscopic waves are classified on the basis of estimating the integrals asymptotically.Translated by V. Puchkin.     

Response of a two-layer ocean to typhoon passage in the western boundary region

Effect of the typhoon passage on the western boundary region of a two-layer ocean with bottom topography is studied. The ocean is initially at rest and is set in motion by a typhoon passing parallel to the west coast. Equations that represent barotropic and baroclinic modes of motions are solved numerically by means of the method of finite differences. Motions of the barotropic mode are assumed to be horizontally non-divergent. In this mode, an elongated vortex is produced by the typhoon and propagates toward the south after passage of the typhoon. Behavior of the vortex may be interpreted as continental shelf waves. It is found that the formation and propagation of continental shelf waves are hardly affected by the density stratification. As for the baroclinic response, the typhoon causes considerable interface displacements along its track. The interface displacements are associated with geostrophic motions and remain for long time, though they are formed on the continental slope. Besides the large scale baroclinic response, internal Kelvin waves are induced along the artificial east wall.     

Onset of coastal upwelling in a two-layer ocean by wind stress with longshore variation

On the assumption that motions of the barotropic mode are horizontally nondivergent, action of the wind stress with longshore variation on a two-layer ocean adjacent to the meridional east coast is studied. Only the equatorward wind stress is considered. Along the east coast, upwelling is induced by the direct effect of the coast and is confined in a narrow strip with the width of the order of the internal radius of deformation. The upwelling propagates poleward with the internal gravity wave speed. Coastal upwelling induced by the wind stress with longshore variation may be interpreted as the generation and propagation of internal Kelvin waves. Associated with the coastal upwelling, the equatorward flow in the upper layer and the poleward flow in the lower layer are formed as an internal mode of motions. When the bottom topography with the continental shelf and slope is taken into account, occurrence of the poleward undercurrent is delayed by a few days because of the generation of continental shelf waves. And, after the forcing is stopped, the shelf waves propagate poleward away from the upwelling region and the poleward undercurrent fully develops. At the margin of the continental shelf, another upwelling region is induced and propagates poleward.     

Generation of internal waves by barotropic tidal flow over a steep ridge

A three-dimensional nonhydrostatic numerical model is used to study the generation of internal waves by the barotropic tidal flow over a steep two-dimensional ridge in an ocean with strong upper-ocean stratification. The process is examined by varying topographic width, amplitude of the barotropic tide, and stratification at three ridge heights. The results show that a large amount of energy is converted from the barotropic tide to the baroclinic wave when the slope parameter, defined as the ratio of the maximum ridge slope to the maximum wave slope, is greater than 1. The energy flux of internal waves can be normalized by the vertical integral of the buoyancy frequency over the ridge depths and the kinetic energy of the barotropic tides in the water column. A relationship between the normalized energy flux and the slope parameter is derived. The normalized energy flux reaches a constant value independent of the slope parameter when the slope parameter is greater than 1.5. It is inferred that internal wave generation is most efficient at the presence of strong upper-ocean stratification over a steep, tall ridge. In the Luzon Strait, the strength of the shallow thermocline and the location of the Kuroshio front could affect generation of internal solitary waves in the northern South China Sea.     

Influence of Approach Channel on Wave Propagation with Varying Incident Angles

- The variation of the amplitude of waves with varying incident angles when waves propagate through a typical approach channel is discussed by a numerical calculation method, the result of which shows that the influence of the channel on wave propagation is obvious. When the wave propagation direction is in coincidence with the channel axis, the wave amplitude ratio will decrease with the increase of propagation distance. When the incident angle is 15 - 30 , there appears an area of larger wave amplitude ratio on the side slope facing the waves, but at the another side, the wave amplitude ratio is generally small, indicating that the channel has a shielding effect. When waves propagate across the channel perpendicularly, the wave amplitude ratio can be calculated with the shallow water coefficient.     

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.     

港池中波浪折射、绕射的数值模拟方法

从流体运动方程和动量方程出发,引入海底摩擦力和港池不完全反射作用,推导出了Berkhoff折射、绕射方程。在深海部分解析方法用精确解(含待定系数)、在复杂地形用数值离散方法求解,中间过度段用的光滑匹配将离散数值(有限元)和解析解(精确解)一同求解。通过极值问题建立泛函,利用泛函的驻定性将海岸(港湾)问题进行数值离散,建立了可行的数值模拟模型。     

Far fields of internal gravity waves in a stratified liquid of varying depth

Asymptotic representations of solutions describing the far fields of internal gravity waves in a stratified medium of varying depth are constructed. The effect of space-frequency cutoff of the wave field for a real oceanic shelf is revealed. Depending on frequency characteristics of the wave field and bottom topography, far fields of internal waves either are located in a certain confined space domain (trapped waves) or propagate in the absence of turning points over sufficiently large distances when compared with the sea depth (progressive waves). The space domain where the progressive waves penetrate is fully determined by the presence of turning points whose locations depend on the medium stratification and inhomogeneities of bottom topography.     

Reflection of a long wave from an underwater slope

Reflection of long sea waves from an underwater slope described by a power law is studied within the shallow water theory. The slope is connected with the flat bottom. This model allows us to estimate the roles of a pointwise reflection from the inflection point of the bottom profile and distributed reflection at the underwater slope. The case of the underwater slope described by the so-called nonreflecting beach (h(x) ∼ x ^4/3, where h is the depth of the basin and x is the coordinate) when the wave is reflected only from the inflection point (pointwise reflection) is specially considered. The reflection and transmission coefficients over the bottom topography were calculated, and it was shown that the sum of the squared absolute values of these values differs from unity for all profiles except the nonreflecting one. This difference is related to the distributed re-reflections (resonances) over the underwater slope that lead to the deviations in the wave height from the known Green’s law.     

Interannual Variations of Sea Level at the Nansei Islands and Volume Transport of the Kuroshio Due to Wind Changes

Interannual variations of sea level at the Nansei Islands and volume transport of the Kuroshio during 1967–95 are calculated by integrating variations carried by windforced Rossby waves. Effects of eddy dissipation and ocean ridges are considered. Ridge effect is inferred by comparing between the calculated and observed sea levels. The calculation is satisfactory to sea levels and Kuroshio transport for the whole period. They are mostly caused by Rossby waves forced by wind and modified by the ridges, and are due to barotropic wave primarily and the first baroclinic wave secondly. The calculated Kuroshio transport well represents variations of several-year scales with maximums in respective duration of the large meander (LM) of the Kuroshio, as well as bi-decadal variation that transport was small during the non-LM period of 1967–75 and large during the LM-dominant period of 1975–91. Mean volume transport of the subtropical gyre is estimated at 57 Sv (1 Sv = 10⁶ m³s^–1) and divided by the Nansei Shoto Ridge into those of the Kuroshio in the East China Sea (25.5 Sv) and a subsurface current east of this ridge (31.5 Sv). The Subtropical Countercurrent and a southward deep current east of the Izu-Ogasawara Ridge are estimated at 16 Sv and 7 Sv, respectively. The calculated transports of the Kuroshio and other subtropical currents reach maximums at every El Niño event due to strong excitement of upwelling barotropic Rossby wave.