Long period free-oscillations of the Earth

自由振荡提供了对长波地球内部结构最重要的限制之一。事实上，自由振荡周期是用来确定地球球对称密度结构的主要数据，而自由振荡光谱可以用来成像地幔的横向密度变化。后一点与最下层地幔内的对流是否由热浮力或成分浮力驱动的问题特别相关。地球的自由振荡频谱非常多样化，既包括与地震波传播相关的熟悉的“声学模式”，也包括更奇特的长周期模式，如自由核章动、钱德勒摆动和主要局限于外核的低音模式。然而，在这些长周期内，由于地球自转、外核密度分层的性质、粘性或粘弹性耗散以及可能与地球磁场的相互作用，地球的动力学变得非常复杂。这个项目将通过对地球自由振荡的理论和数值研究，扩展我们对固体地球在大约一年到一小时周期内的动力学的理解。如果地球完全是固体并且完全有弹性，那么自由振荡理论就完全被理解了。然而，地球的外核和海洋是流动的，在地震学中，它们通常被认为是不粘的。这种流体区域的包含从根本上改变了与自由振荡相关的特征值问题的性质，而低频频谱的确切性质尚不清楚。此外，有限粘度或粘弹性在这些长周期内的影响很少受到关注，并且与地核和地幔的动态相互作用特别相关。所有这些问题都可能对最长周期自由振荡的数值研究产生重大影响，从而应用于地球结构的研究。该学生将合作开发新的理论和计算方法来研究地球的最长周期自由振荡。一个主要的问题是低频自由振荡频谱的精确性质。由于包含了无粘流体区域，出现在特征值问题中的算子缺乏通常的Fredholm性质，这导致存在一个接近零频率的基本谱。在以前的工作中，有人认为——没有充分的证据——这部分光谱是连续的，而且可观察到的长周期自由振荡，如自由核心章动，嵌入在区域内。如果这是真的，那么与这些振荡的计算和激发有关的问题就很复杂了。进一步的问题是，流体地核内的有限粘度，或者更一般地说，整个地球的粘弹性，如何改变地球的长周期振荡，特别是流体外核内的底音模式。

Free oscillations provide one of the most important constraints on the Earth's internal structure at long wavelengths. Indeed, free-oscillation periods were the main data used in determining the Earth's spherically symmetric density structure, while free-oscillation spectra can be used to image lateral density variations in the mantle. This latter point is of particular relevance to the question of whether convection within the lowermost mantle is driven by thermal or compositional buoyancy. The Earth's free-oscillation spectrum is very diverse, being comprised both of familiar "acoustic modes" associated with seismic wave propagation, and more exotic long period modes such as the free core nutations, the Chandler wobble, and undertone modes that are largely confined to the outer core. At these long periods, however, the dynamics of the Earth is greatly complicated by the Earth's rotation, the nature of the density stratification within the outer core, viscous or viscoelasticity dissipation, and possible interactions with the Earth's magnetic field.This project will extend our understanding of the dynamics of the solid Earth at periods between about one year and one hour, through both theoretical and numerical studies of the Earth's free-oscillations. Were the Earth entirely solid and perfectly elastic, then free-oscillations theory is completely understood. However, the Earth's outer core and oceans are fluid, and within seismology they are often regarded as being inviscid. The inclusion of such fluid regions fundamentally changes the nature of the eigenvalue problem associated with free-oscillations, and the precise nature of the spectrum at low frequencies is not yet understood. Furthermore, the effects of a finite viscosity or viscoelasticity at these long periods has received little attention, and is of particular relevance for dynamic interactions of the core and mantle. All these questions could have significant implications for numerical studies of the longest period free-oscillations, and hence for applications to the study of Earth structure.The student will work collaboratively to develop new theoretical and computational approaches to study the longest period free oscillations of the Earth. A major question is the precise nature of the free-oscillation spectrum at low frequencies. Due to the inclusion of inviscid fluid regions, the operator occurring in the eigenvalue problem lacks the usual Fredholm property, and this leads to the existence of an essential spectrum near to zero frequency. In previous work it has been argued - without full proof - that this part of the spectrum is continuous, and that observable long period free-oscillations such as the free core nutations are embedded within region. If this is true, then there are significant complications related to computation and excitation of these oscillations. A further question is how a finite viscosity within the fluid core, or more generally of viscoelasticity throughout the Earth, modifies the long period oscillations of the Earth, and in particular the undertone modes within the fluid outer core.