On the Discrete Normal Modes of Quasigeostrophic Theory
On the Discrete Normal Modes of Quasigeostrophic Theory
复制标题
准地转理论的离散简正模态
DOI：
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发表时间：
2021
影响因子：
3.5
通讯作者：
S. Griffies
中科院分区：
文献类型：
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作者：
H. Yassin;S. Griffies
准地转理论的离散斜压模式是不完整的，这种不完整表现为投影过程中信息的丢失。斜压模式的不完整性与平坦边界罗斯贝波问题的两个先前未被注意到的稳态阶跃波解的存在有关。当边界浮力梯度消失时，这些阶跃波是表面准地转波的极限。通过考虑在下边界和上边界具有指定浮力梯度的传统罗斯贝波问题，获得了准地转理论的完整简正模态基础。这些边界浮力梯度的存在激活了先前惰性的边界自由度。这些罗斯贝波具有几个新颖的特性，例如存在没有内部零点的多个模式、具有负范数的有限数量的模式，以及它们的垂直结构形成了能够用可微级数展开表示任何准地转状态的基础。这些属性是存在边界浮力梯度时罗斯贝波问题的庞特里亚金空间设置的结果（与通常的希尔伯特空间设置相反）。我们还研究了准地转垂直速度模式并得出了此类模式的完整基础。这些模式的自然应用是发展考虑地形的地转湍流的弱非线性波相互作用理论。
The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes topography into account.