完整Coriolis力作用下带有外源强迫的非线性ZK方程

在正压流体中,从包含完整Coriolis参数的准地转位涡方程出发,在弱非线性长波近似下,采用多时空尺度和摄动方法,推导出大气非线性Rossby波振幅演变满足带有外源强迫的二维Zakharov-Kuznetsov(ZK)方程.然后利用Jacobi椭圆函数展开法,求解了ZK方程的椭圆正弦波解和孤立波解.分析结果表明,Coriolis参数的水平分量将影响二维Rossby波传播的频率特征,而外源不仅对二维Rossby波动的传播的频率有影响,对振幅也产生一个调制作用.     

一类完整Coriolis力作用下的高阶非线性Schr?dinger方程的推导

在正压流体中,从含有完整Coriolis力的准地转位涡方程出发,采用摄动展开的方法推导了,一类新的高阶非线性Schr?dinger方程,用于描述地球流体力学中的非线性调制Rossby波.从方程中,讨论了调制波列.结果表明,完整Coriolis力下的水平分量和地形会影响均匀Rossby波调制不稳定,并且不稳定.区域也会随着改变.此外,均匀基本流也是影响Rossby孤立波调制不稳定性的的重要因素.     

完整Coriolis力作用下带有外源强迫的非线性KdV方程

利用摄动方法,从描写既有Coriolis力垂直分量又含有水平分量的位涡方程出发,给出了近赤道非线性Rossby波所满足的具有外源强迫的非线性KdV方程,并利用Jacobi椭圆函数展开法,求解了改进后的非线性KdV方程的行波解及孤立波解.通过分析KdV方程的行波解,指出Coriolis力的水平分量和外源对Rossby波动的影响.     

完整Coriolis力作用下的非线性Rossby波

在正压流体中,利用摄动方法从描写既有Coriolis参数的垂直分量又含有水平分量的位涡方程出发,推导了近赤道非线性Rossby波振幅演变所满足的非线性mKdV方程,并利用Jacobi椭圆函数展开法,求解了推广后的非线性mKdV方程的行波解及孤立波解.通过分析其方程的行波解及孤立波,表明地球旋转的水平分量对Rossby波动产生一定的影响.     

一类描述大气中重力孤立波的新模型

从两层流体浅水波方程出发,运用尺度分析与扰动方法,建立了一类新的模型(mKdV-BO模型)来描述大气中的重力孤立波。前人建立的KdV模型和BO模型适合描述经向和纬向扰动较弱时重力孤立波的生成和演化,而该模型的非线性更强,适合描述经向、纬向扰动较强时重力孤立波的生成与演化。通过运用试探函数法获得了模型的代数孤波解,并分析了孤立波的生成条件与传播速度。新模型的建立对于进一步解释大气中列队雷雨阵的形成机制,探讨大气中的强对流天气如飑线的形成等具有重要意义。 关键词：重力孤立波;试探函数法;列队雷雨阵     

旋转层结流体中垂直切变基本流、β效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法(多尺度方法),从位涡守恒方程出发,分析旋转层结大气中基本流有垂直切变以及层结效应对β效应、地形效应和强迫耗散共同作用下的Rossby波的影响,得到一个非标准形式的非线性Schr?dinger方程,而在水平波数小于3时该方程有包络孤立波解;又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外,本文还应用常数变异法求解了非齐次的Bessel方程,得到包络Rossby孤立波的经向结构.     

完整Coriolis力与弱地形作用下的非齐次mKdV-Burgers方程

本文研究了中高纬度含有完整Coriolis力的准地转位涡方程问题.利用时空伸缩变换方法,获得了描述Rossby波的振幅形态满足非齐次mKdV-Burgers方程的结论.方程的近似解表明,弱地形效应对Rossby波的振幅产生强迫作用,并推广了文献[12]中的结果.     

旋转层结流体中垂直切变基本流、$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法（多尺度方法）, 从位涡守恒方程出发, 分析旋转层结大气中基本流有垂直切变以及层结效应对$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波的影响, 得到一个非标准形式的非线性Schr\"{o}dinger方程,而在水平波数小于3时该方程有包络孤立波解; 又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外, 本 文还应用常数变异法求解了非齐次的Bessel方程, 得到包络Rossby孤立波的经向结构.     

有限振幅超长波的发展及对流层大气环流的指数循环

本文由一个描写超长波运动的准地转三层模式,利用多尺度分析法,得到了有限振幅超长波不稳定和西风不稳定的非线性理论.当考虑到超长波扰动和基本西风气流的非线性相互制约后,同时得到了超长波振幅和西风强度作周期性振荡(周期为2—6周),与超长波尺度和初始振幅有关.流场演交表现了类似低指数的阻塞形势和高指数的纬圈环流的转换.超长波发展过程中的热量输送,维持了指数循环的能量转换.因此本文证明了指数循环是大气的超长波和基本纬圈气流相互制约的非线性反馈过程.     

低纬地区非线性孤立波存在性讨论

本文讨论了静力平衡下的绝热自由大气非线性重力惯性波的孤立渡解存在性,设运动在y方向是均匀的,又采用了β平面近似,同时忽略了动量方程中垂直平流项,我们导出了孤立波解的解析解表达式,通过对波速c的讨论,我们指出:在中纬地区,很难形成孤立波;在低纬地区(热带地区),由于科氏力很小,层结稳定度较弱有可能产生孤立波。最后我们讨论了孤立波的波速c与科氏力f,β和稳定参数σ_8之间关系。     

Weak Formulation of Free-Surface Wave Equations

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, [equation] where the an(t) are time-dependent coefficients. For a tank with vertical sides the E_n are eigenfunctions of the eigenvalue problem, [equation] on the tank side walls. Evolution equations for the a_n(t) can be obtained by taking inner products of the lin-earised equation of motion, [equation] with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.     

孤立子型热源强迫与大气和海洋流场的奇异响应

运用非线性动力学分析方法,对一个广义的地球流体动力学正压准地转模式中,局地热源强迫对大气和海洋流场产生的响应结果,进行了解析研究.发现热源扰动和响应流场之间存在很好的对应关系:两者具有类似的分布模态,且孤立子型的响应流场要比孤立子型的热源强迫范围大得多,即"狭窄"的热源扰动可能导致"宽广"的流场响应;强迫热源的特殊结构有可能导致大气或海洋流场出现奇异响应,从而使大气或海洋环流发生异常(如大气阻塞现象);中、高纬地区的大气和海洋流场对热源强迫的响应要比低纬地区显著得多.上述结果与中、低纬大气试验和观测资料的研究结果相符,可部分解释地球流体中,局地热源异常可能导致的环流异常现象.     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

A fully coupled method for simulation of wave-current-seabed systems

Coastal flow involves surface wave propagation, current circulation, and seabed evolution, and its prediction remains challenging when they strongly interact with each other, especially during extreme events such as tsunami and storm surge. We propose a fully coupled method to simulate motion of wave-current-seabed systems and associated multiphysics phenomena. The wave action equation, the shallow water equations, and the Exner equation are respectively used for wave, current, and seabed morphology, and the discretization is based on a second-order, flux-limiter, finite difference scheme previously developed for current-seabed systems. The proposed method is tested with analytical solutions, laboratory measurements, and numerical solutions obtained with other schemes. Its advantages are demonstrated in capturing interplay among wave, current, and seabed; it has the capability of first-order upwind schemes to suppress artificial oscillations as well as the accuracy of second-order schemes in resolving flow structures.     

The oscillations of a body with an orbital tethered system

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.     

Nonlinear inertial effects on the instability of a single long wave of finite amplitude at the interface of two viscous fluids

The evolution of a single long wave of finite amplitude at the interface of two immiscible fluids of different viscosities and densities, between two horizontal plates is solved, using a boundary layer flow approximation for the equation of motion in each fluid layer. It is found that when the nonlinear inertial effects are taken into account in a moderate manner, at least in the frame of the boundary layer approximation, the initial unperturbed flow with smooth interface is stable to a single wave perturbation at the interface, even in the presence of adverse density and viscosity stratifications. However, when the nonlinear effects are increased in a specific way, and the magnitudes of the parameters involved are kept within the order of magnitude established for the present theory, an unstable flow configuration can be obtained.     

Exact geophysical waves in stratified fluids

When studying water waves travelling over an inviscid fluid at the Earth's surface there are additional Coriolis and centrifugal forces which influence the motion of the fluid particles. In particular, for waves propagating near the Equator the geophysical wave problem can be modelled by the so-called f-plane approximation. In this paper, we provide an explicit exact solution to the edge wave problem for stratified geophysical flows in the equatorial f-plane approximation.