ABSTRACT

We study the following rotational modes of Poincaré Earth models: the tilt-over mode (TOM), the spin-over mode (SOM) and free core nutation (FCN), using first a simple Earth model with a homogeneous and incompressible liquid core (LC) and a rigid mantle (MT). We obtain analytical solutions for the periods of these modes as well as that of the Chandler wobble (CW). We show analytically the distinction between the TOM and the SOM and that the FCN is indeed the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second-order partial differential equation called the Poincaré equation, is used as the independent variable. Analytical solutions are then found for the displacement eigenfunctions in a meridional plane of the liquid core for the aforementioned modes. We next consider a three-layer Earth model similar to above which also includes a rigid inner core (IC). We first show that analytical solutions exist for the period and eigenfunctions of the CW if the IC is locked to the MT, i.e. they have the same wobbling motion. We show that this is significant as it shows that the CW manifests itself for a Poincaré (incompressible and inviscid LC) wobbling Earth model. We further allow for the inner core to wobble independently and compute numerically the periods and displacement eigenfunctions of the TOM, SOM and FCN, as well as those for still another rotational mode, the inner-core wobble (ICW). Next we show that the presence of the characteristic surfaces intercepted by the inner-core, when computing the period and eigenfunctions of the free inner-core nutation (FICN), may be the reason for the slow (or lack of the) convergence of this mode. Finally, we show that even though the wobbling motion of the mantle is ignored when solving for the frequencies of the ICW and the FICN when Sasao's approximation is used, the analytical solutions for both these modes yield periods nearly identical to those in the literature for a similar Earth model with mantle allowed to wobble as well. We infer that the Sasao's approximation, or the severe truncation of the series solution of the field variables, the pressure, the gravitational potential and the components of the displacement vector, may not be adequate to accurately describe the motion in the liquid core during the excitation of the FICN.

摘要

我们首先使用具有均质且不可压缩的液核的简单Earth模型研究PoincaréEarth模型的以下旋转模式：倾斜模式（TOM），旋转模式（SOM）和自由核螺母（FCN）（ LC）和刚性罩（MT）。我们获得了这些模式以及钱德勒摆动（CW）周期的解析解。我们通过分析显示了TOM和SOM之间的区别，并且FCN的确与摇晃地球的SOM处于同一模式。减压作为独立变量，根据该减压，已知矢量动量方程可简化为标量二阶偏微分方程，称为Poincaré方程。然后找到上述模式下液芯子午平面上位移本征函数的解析解。接下来，我们考虑一个类似于上面的三层地球模型，该模型还包括一个刚性内芯（IC）。我们首先表明，如果IC锁定在MT上，则CW的周期和本征函数存在解析解，即它们具有相同的摆动运动。我们证明这是有意义的，因为它表明CW表现为庞加莱（不可压缩且无粘性的LC）摆动地球模型。我们进一步允许内芯独立摆动，并通过数值计算TOM，SOM和FCN的周期和位移本征函数，以及其他旋转模式的内芯摆动（ICW）。接下来，我们显示当计算自由内核章动（FICN）的周期和本征函数时，内核截取的特征表面的存在，可能是此模式收敛缓慢（或缺乏）的原因。最后，我们表明，即使使用Sasao逼近法求解ICW和FICN的频率时，即使忽略了地幔的摆动运动，这两种模式的解析解所产生的周期也几乎与文献中的相同。类似的地球模型，也允许其与地幔一起摆动。我们推断，Sasao近似值或场变量，压力，重力势和位移矢量的分量的级数解的严重截断可能不足以准确地描述激发过程中液芯中的运动FICN。我们表明，即使使用Sasao近似法求解ICW和FICN的频率时，即使忽略了地幔的摆动运动，这两种模式的解析解产生的周期也几乎与文献中相似地球的周期相同。带罩的模型也可以摆动。我们推断，Sasao近似值或场变量，压力，重力势和位移矢量的分量的级数解的严重截断可能不足以准确地描述激发过程中液芯中的运动FICN。我们表明，即使使用Sasao近似法求解ICW和FICN的频率时，即使忽略了地幔的摆动运动，这两种模式的解析解产生的周期也几乎与文献中相似地球的周期相同。带罩的模型也可以摆动。我们推断，Sasao近似值或场变量，压力，重力势和位移矢量的分量的级数解的严重截断可能不足以准确地描述激发过程中液芯中的运动FICN。这两种模式的解析解产生的周期几乎与文献中类似的允许地幔也可摆动的地球模型的周期相同。我们推断，Sasao近似值或场变量，压力，重力势和位移矢量的分量的级数解的严重截断可能不足以准确地描述激发过程中液芯中的运动FICN。这两种模式的解析解产生的周期几乎与文献中类似的允许地幔也可摆动的地球模型的周期相同。我们推断，Sasao近似值或场变量，压力，重力势和位移矢量的分量的级数解的严重截断可能不足以准确地描述激发过程中液芯中的运动FICN。