A supplementary note on the completeness of the Rossby normal modes in a rectangular basin

In previous studies of the completeness of Rossby normal modes in a closed basin, proofs have been obtained by rather technical methods based on the Green's function for the Laplacian operator or based on the theory of compact self-adjoint operators. This short note (1) gives a much simpler proof by the use of only the norm relations between the sine functions and the Rossby normal modes and (2) clarifies the relations between the two different sets of eigenfunctions and between the ordinary norm and the Dirichlet norm, though the validity of the present method is restricted to a special one-dimensional case. The expansion theorem is also derived from the completeness theorem with the aid of transformation laws, where the assumption about the smoothness of the expanded function is weaker than in the elementary proof of Masuda (1987a). Both the set of Rossby normal modes and that of sine functions are shown to be complete for the Dirichlet norm as well.     

南海一个罗斯贝标准模态的特征与归因

南海是一个准封闭海盆,其本征值问题是理解南海动力学的重要内容。本文利用一种新的泛函工具—多尺度子空间变换,从卫星观测资料中分离得到一个南海本征模态,即罗斯贝标准模态的近似场。发现该模态的周期为6个月左右、波长近250 km,在深水海盆向西传播,这与寿命为3个月左右的南海中尺度涡群体活动特征相一致。在此基础上,本文通过Liang-Kleeman信息流这一严格建立在第一性原理上的定量因果分析工具,探究南海两个最重要外部强迫,即黑潮入侵与南海季风对该罗斯贝标准模态的影响。结果表明二者与该模态均有较强因果关系,但分别影响模态的不同阶段：黑潮入侵主要影响其1/2π和3/2π位相,季风的作用则体现在3/4π位相。二者共同作用,调制该模态在近一个周期内的变化。进一步研究发现,黑潮入侵的过渡态是影响该模态的关键,这时黑潮在南海的分支与流套强度相当,有利于吕宋海峡西部形成不同极性涡旋的排列,从而影响南海内部罗斯贝标准模态。对南海季风而言,冬季风与夏季风的成熟阶段是影响该模态的重要时期,但并非整个南海的季风都发挥作用,泰国湾是季风改变南海罗斯贝标准模态的关键区域,这表明局地的强迫对激发全局模态起作用。     

Tidal oscillation of water in a rotating rectangular basin of variable depth

The free oscillation of water in a rotating rectangular basin of variable depth is discussed. The depth is assumed to decrease from the center to the edges according to a paraboloidal law. The solution is obtained in terms of double series of zonal harmonics. Numerical solutions were worked out. Complete sets of modes will be evaluated by an electronic computer.     

Effect of ice cover on oscillations of fluid in a closed basin

Within the framework of the linear theory of long waves, the problem of the effect of ice cover on seiche oscillations of fluid in a two-dimensional constant-depth basin is solved. The eigenfrequencies and eigenfunctions of seiche oscillations are obtained for different boundary conditions at ice edges: rigid coupling and free edges. The forced oscillations of fluid and ice under the action of a moving disturbance of atmospheric pressure are investigated. The change in the stress of ice bending is considered and it is shown that the coast ice can be broken.     

Effect of hydrodynamic properties of the sea bottom on the tidal dynamics in a rectangular basin

Expressions derived for the friction coefficient in an oscillatory rotating turbulent bottom boundary layer (BBL) over rough, incompletely rough (smoothly rough), and smooth underlying surfaces are incorporated as an individual module into a two-dimensional nonlinear tidal model, and the standard version of the model and its modified analogue are used to discuss the titular subject. It is established that the dynamics of tides in the Taylor basin can change noticeably under the effect of hydrodynamic properties of the sea bottom. Such changes occur mainly in the influence domains of amphidromies. In the remaining parts of the basin, relative changes in the amplitudes and phases of tidal sea-surface level elevations do not exceed ±10% and ±10°, respectively. The largest discrepancies of tidal characteristics take place in the cases of the incompletely rough and smooth sea bottoms; the smallest discrepancies, in the case of the rough sea bottom. Estimates for the changes in tidal characteristics that are caused by the usually neglected effects of rotation and phase difference between the bottom friction and the tidal velocity at the upper BBL boundary are presented as well.     

带丁坝型半封闭矩形海湾中M_2分潮潮波和潮流特性分析

将黄渤海海域概化为矩形海湾,山东半岛概化为垂直于海岸的巨型丁坝,形成一个带丁坝型半封闭矩形海湾。利用DELFT3D-FLOW计算模块在上述海湾中进行了M2分潮数值模拟,对其无潮点和辐射状潮流场的特性进行了分析。研究发现:考虑科氏力影响和巨型丁坝反射作用,坝前形成了明显的无潮点,但在等水深条件下并未形成辐射状潮流场,而叠加上倾斜海底地形后其得以形成;无潮点和辐射状潮流场顶端位置受水深影响明显,随着平均水深的增大,无潮点将向湾口和湾中轴线方向偏移,而辐射状潮流场顶端则向湾口方向偏移。研究结果有助于加深对带丁坝型半封闭矩形海湾中驻潮波系统形成机制和动力特性的理解和认识。     

Energy dissipation in sloshing waves in a rolling rectangular tank — III. Results and applications

The hydrodynamic aspects of the motion of a viscous fluid having a free surface in a rolling tank have been investigated. In a sequence of three papers, an analytical technique together with a numerical solution method will be presented to describe the sloshing phenomenon accurately and efficiently. This first paper introduces a linear theory of viscous liquid sloshing and formulates a boundary value problem subject to appropriate conditions. Viscosity is included in the problem formulation and its effects are properly accounted for. The second paper will describe a solution of the problem in function space by the truncation of infinite series. Boundary conditions are satisfied through the use of Fourier series expansions. However, the no-slip condition at the side walls can also be treated in a least-squares sense.Among the results that will be reported in the third paper are the effects of viscosity on liquid sloshing phenomenon and the dependence of viscous dissipation on the Reynolds and Froude numbers. Furthermore, the influence of the tank aspect ratio on viscous dissipation has been explored. These results demonstrate some unknown features of the functional relationships that exist between the dissipated energy and the Reynolds and Froude numbers. Similarly, the dependence of the dissipated energy on the aspect ratio has been analytically studied. The results obtained agree with the physical laws for the range of parameters investigated.     

Transmission and reflection of the Rossby waves in a stratified ocean with bottom topography

Transmission and reflection coefficients are calculated for Rossby waves incident on a bottom topography with constant slope in a continuously stratified ocean. The characteristics of the coefficients are interpreted in terms of the quasigeostrophic waves on the slope. In the parameter range where only the barotropic Rossby waves can propagate in the region outside the slope, the bottom trapped wave plays the same role as the topographic Rossby wave in a homogeneous ocean, and hence the transmission is weak unless phase matching takes place. When both of the barotropic and baroclinic Rossby waves can propagate outside the slope, the total transmission can be strong. The bottom trapped wave affects the transmission and reflection, and it leads to the possibility that the Rossby wave is transmitted as a mode different from the incident mode. When the number of the wavy modes on the slope is smaller than that of the Rossby wave modes outside the slope, strong reflection occurs.The results for an ocean with linear distribution of the squared Brunt-Väisälä frequency are compared to those in a uniformly stratified ocean. The weakening of the stratification near the bottom is almost equivalent to reducing the effect of the slope.     

Hadley and Rossby regimes in a simple model of convection of a rotating fluid

This paper analyzes the properties of solutions to the equations describing the motion of a stratified fluid in the class of velocity and temperature fields linear in coordinates. For an ideal fluid, these equations, on the one hand, are exact for the corresponding hydrodynamic problem and, on the other hand, are identical to the equations of motion for a heavy top. In a conservative case, the equations of motion of a top share common solutions with the equations of geophysical fluid dynamics and reproduce motions similar to those existing in the theory of the large-scale atmospheric circulation. This study considers the effects of viscosity and heat conduction in the fluid, which are, in a sense, similar to the effect of friction in the case of a top. The influence of deflections of the vectors of gravity and external rotation from their standard directions for a plane-parallel atmosphere is also considered. The regimes of motions that are described by the starting equations and approximations commonly used to model the atmospheric general circulation (the quasi-geostrophic approximation) are analyzed. It is shown that these equations correctly describe the Hadley and Rossby circulation regimes and transitions between them that are observed in numerical and laboratory experiments. Particular attention is given to the consistency between different regimes of the exact equations and their quasi-geostrophic approximations, which is manifested for small Rossby numbers and is generally absent for large Rossby numbers. The asymptotic behaviors of the curves of transition between the Hadley and Rossby regimes under the conditions of breaking the external symmetry of flows are obtained. These asymptotics explain the corresponding transition boundaries for the regimes observed in the known experiments in annuluses.     

Transmission and reflection of planetary and topographic Rossby waves in a two-layer ocean

Transmission and reflection problems when kissing≓ occurs among planetary and topographic Rossby waves in a two-layer ocean are studied. The slope parameterS(=dh ₂/dx, whereh ₂is the thickness of the lower layer) is assumed to have constant values in the regionsx 0 andxL and to vary linearly with the increase ofx in the region0xL (refer to Fig. 2 in the text). Furthermore, a wave is entered fromx=– and kissing is assumed to occur in the region (0<)x _axx_b(<L).It is found that a wave of the same type as the incident wave is mainly transmitted when the width of the region in which kissing occurs,L _kiss(=tx _b–x_a), is smaller than _kiss=2/(¦K¦+ _y/2), whereK is a representative wavenumber in the regionx _ab, _y is they-component of , and is the frequency. WhenL _kiss is larger than _kiss, on the other hand, the main wave transmitted is of a different type to the incident wave. As an application, transmission and reflection problems of planetary Rossby waves are considered, and it is shown that when an external (internal) planetary Rossby wave is entered, an internal (external) one can be transmitted due to the effect of kissing.     

Interaction between Rossby Waves and a Jet Flow: Basic Equations and Verification for the Antarctic Circumpolar Current

Izvestiya, Atmospheric and Oceanic Physics - Abstract—The article focuses on the interaction of Rossby waves in the ocean with zonal jet flows. A new approach is proposed to show that...     

Potential Vorticity in the Ocean: Ertel and Rossby Approaches with Estimates for the Lofoten Vortex

Izvestiya, Atmospheric and Oceanic Physics - The potential vorticity (PV) in the ocean is considered, including the history of the term. Various aspects and basic formulas used to calculate the PV...     

Solving the equation for viscous incompressible liquid in a rectangular domain for small reynolds numbers

An algorithm is given for an analytical solution to the problem of the eigenvalue and eigenfunction for the Navier-Stokes equation within a rectangle under the adherence condition at its boundaries. The resulting system of orthonormal functions can be used for correctly deriving the divergence-free component of flow at the free surface of laboratory currents. This system makes it possible to apply the spectral approach to the solution of the Navier-Stokes equation, strictly calculate the primary regimes, investigate their stability, and calculate the secondary currents for small above-criticalities for the conditions of a laboratory experiment on simulating geophysical flows.     

A unified, polynomial approach for the approximate solution of thermoelastic problems in rectangular plates

The present paper, of a tutorial nature, solves two different thermoelastic problems using a unified approach: (a) thermoelastic stresses and deflections in thin, rectangular plates with edges elastically restrained against rotation when small-deflection theory is used and stress resultants in the middle plane of the plates are neglected; and (b) thermoelastic plane stress situations in rectangular plates with stress-free edges subjected to a two-dimensional temperature distribution.The first problem is attacked using polynomial coordinate functions which identically satisfy the governing boundary conditions, and the Ritz method is used to determine the plate response. When the flexibility coefficients approach zero (clamped edge conditions) the calculated results practically coincide with the exact solution for the test case treated in this paper. On the other hand this degenerate solution turns out to be a very convenient approximation for obtaining a solution to Airy's problem described in the second place and which is also solved in a straightforward manner using a variational approach.     

Assimilation of simulated altimeter data in a two-layer linear Rossby wave model using the variational method

The variational assimilation method has been examined for ability of reconstructing mesoscale features in altimeter data using a simple dynamic model. A one-dimensional, two-layer Rossby wave model in a cross-track channel has been chosen. The simulated data are constructed from a theoretical solution, which is composed of any combination of two normal vertical (barotropic and baroclinic) modes. The data are collected along tracks and with repeat periods similar to those of the Geosat altimeter. The phase space of control variables is composed of initial and boundary conditions. A cost function is defined to measure differences between the simulated data and the model solution. Regularization (smoothing) terms are also included in the cost function in the form of secon-order spatial and time derivatives of the solution. In this paper, two potential problems existing in the altimeter data assimilation are addressed: one is low cross-track resolution, and the other is vertical projection of the data measured at the sea surface. A succesful metho is developed for reconstructing Rossby waves with wavelengths as short as twice the track intervals for any combination of two vertical modes. A key component to efficient assimilation is a preparation step prior to the actual variational assimilation: a uniform ratio of pressure amplitudes in the two layers is included as an optimization parameter. Starting with the first guess from the preparation step, the variational method is carried out based on adjoint equations without such constraint. Separation of the control variables into the two subsets of the initial and the boundary conditions is found useful. Characteristics of the Hessian matrix are related to the performance of this technique. The method developed for the linear system implies steps to be included in data assimilation for nonlinear meanders and eddies in a major current system as well.     

渤海夏季第一斜压罗斯贝变形半径的计算与分析

利用渤海2006年夏季大面观测的CTD温盐资料,分别运用WKB近似和数值方法计算了渤海夏季第一斜压罗斯贝变形半径和第一斜压重力波相速度。结果显示,两种不同算法得出的斜压罗斯贝变形半径和斜压重力波相速度的分布格局相似,与渤海等深线分布大体一致。数值解法下,在北戴河和长兴岛附近海域,斜压罗斯贝变形半径较大,最大为4.72km,对应的斜压重力波相速度最大为0.44m/s;在莱州湾海域,斜压罗斯贝变形半径较小,最大仅为1.76km,对应的斜压重力波相速度最大为0.16m/s;在渤海中部斜压罗斯贝变形半径和斜压重力波相速度的等值线发生明显弯曲,在靠近渤海海峡内侧出现一个极小值,而与之对应在渤海中部以西出现一个极大值,导致变形半径的等值线呈NE-SW向倾斜的"S"状。渤海夏季第一斜压罗斯贝变形半径在WKB近似下的结果较低于数值解法求出的结果,这主要是因为,WKB近似依赖于浮性频率对水深的积分,适用于浮性频率垂向变化较弱的情况,渤海夏季浮性频率垂向变化较大,WKB近似得出的结果误差较大。     

青岛-石岛近海反气旋中尺度涡旋存在证据及数值模拟

系统整理的历史实测海流和温盐资料显示出青岛-石岛近海水域全年都存在着中尺度反气旋式涡旋运动.利用二维数值模式,首次模拟出该涡旋的存在,并对这一涡旋的形成机制进行了探讨,数值模拟结果和实测吻合良好.数值计算表明:在冬季盛行风的作用下,山东半岛南岸出现减水,迫使北上的黄海暖流中的一部分向西北方向延伸,在到达山东半岛近海后向右转向,形成东北向沿岸流,这样,与绕过山东半岛大约沿40～50m等深线南下的黄海沿岸流作用,在青岛-石岛近海形成一个中尺度反气旋涡旋.夏季由于偏南风的作用,致使山东半岛南部近海浅水区域海水都向东北方向流动,与沿黄海冷水团锋面南下的黄海沿岸流相作用,由于流速切变也形成了上述涡旋.     

青岛─石岛近海反气旋中尺度涡旋存在证据及数值模拟

系统整理的历史实测海流和温盐资料显示出青岛─石岛近海水域全年都存在着中尺度反气旋式涡旋运动．利用二维数值模式,首次模拟出该涡旋的存在,并对这一涡旋的形成机制进行了探讨,数值模拟结果和实测吻合良好．数值计算表明：在冬季盛行风的作用下,山东半岛南岸出现减水,迫使北上的黄海暖流中的一部分向西北方向延伸,在到达山东半岛近海后向右转向,形成东北向沿岸流,这样,与绕过山东半岛大约沿40～50m等深线南下的黄海沿岸流作用,在青岛─石岛近海形成一个中尺度反气旋涡旋．夏季由于偏南风的作用,致使山东半岛南部近海浅水区域海水都向东北方向流动,与沿黄海冷水团锋面南下的黄海沿岸流相作用,由于流速切变也形成了上述涡旋．