Two-dimensional ideal magnetohydrodynamic waves on a rotating sphere under a non-Malkus field: I. Continuous spectrum and its ray-theoretical interpretation

Two-dimensional ideal incompressible magnetohydrodynamic (MHD) linear waves at the surface of a rotating sphere are studied as a model to imitate the outermost layer of the Earth's core or the solar tachocline. This thin conducting layer is permeated by a toroidal magnetic field the magnitude of which depends only on the latitude. The Malkus background field, which is proportional to the sine of the colatitude, provides two well-known groups of branches; on one branch, retrograde Alfvén waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfvén waves undergo a gradual transition into slow MR waves. In the case of non-Malkus fields, we demonstrate that the associated eigenvalue problems can yield a continuous spectrum instead of Alfvén and slow MR discrete modes. The critical latitudes attributed to the Alfvén resonance eliminate these discrete eigenvalues and produce an infinite number of singular eigenmodes. The theory of slowly varying wave trains in an inhomogeneous magnetic field shows that a wave packet related to this continuous spectrum propagates toward a critical latitude corresponding to the wave and is eventually absorbed there. The expected behaviour whereby the retrograde propagating packets pertaining to the continuous spectrum approach the latitudes from the equatorial side and the prograde ones approach from the polar side is consistent with the profiles of their eigenfunctions derived using our numerical calculations. Further in-depth discussions of the Alfvén continuum would develop the theory of the “wave–mean field interaction” in the MHD system and the understanding of the dynamics in such thin layers.

本研究以旋转球面表面的二维理想不可压缩磁流体动力学（MHD）线性波动为模型，用以模拟地球外核最外层或太阳差旋层（solar tachocline）。该薄导电层被仅随纬度变化幅值的环形磁场贯穿。与余纬正弦成正比的马尔库斯（Malkus）背景场可产生两组知名的波动分支：其中一支中，逆行阿尔文波（Alfvén wave）在磁场幅值降低时会逐渐演变为快磁罗斯贝（MR）波；另一支中，顺行阿尔文波则会逐步过渡为慢MR波。针对非马尔库斯磁场情形，本研究表明对应的本征值问题可生成连续谱，而非阿尔文波与慢MR波的离散模。由阿尔文共振引发的临界纬度会消除这些离散本征值，并产生无穷多奇异本征模。非均匀磁场中的缓变波列理论显示，与该连续谱相关的波包会向对应于自身的临界纬度传播，并最终在该处被吸收。预期的行为模式为：属于连续谱的逆行波包从赤道侧趋近临界纬度，而顺行波包则从极地方向趋近，这一结果与我们通过数值计算得到的本征函数分布特征相符。对阿尔文连续谱展开进一步深入探讨，将有助于完善磁流体动力学系统中的“波-平均场相互作用”理论，并增进对这类薄导电层动力学过程的理解。