  Long-term stability stands as a crucial requirement in data-driven
medium-range global weather forecasting. Spectral bias is recognized as the
primary contributor to instabilities, as data-driven methods difficult to learn
small-scale dynamics. In this paper, we reveal that the universal mechanism for
these instabilities is not only related to spectral bias but also to
distortions brought by processing spherical data using conventional
convolution. These distortions lead to a rapid amplification of errors over
successive long-term iterations, resulting in a significant decline in forecast
accuracy. To address this issue, a universal neural operator called the
Spherical Harmonic Neural Operator (SHNO) is introduced to improve long-term
iterative forecasts. SHNO uses the spherical harmonic basis to mitigate
distortions for spherical data and uses gated residual spectral attention
(GRSA) to correct spectral bias caused by spurious correlations across
different scales. The effectiveness and merit of the proposed method have been
validated through its application for spherical Shallow Water Equations (SWEs)
and medium-range global weather forecasting. Our findings highlight the
benefits and potential of SHNO to improve the accuracy of long-term prediction.
