赤道热斑的大尺度动力学
Large-scale dynamics of equatorial thermal spots
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中文总结 AI 辅助
本研究针对旋转流体薄球层模型，解析求解了赤道热斑的静止与自相似解，发现热斑运动方向取决于形状参数，并揭示了半径演化的幂律标度规律。
中文摘要 AI 辅助
针对受科里奥利力和浮力作用的不可压缩旋转流体薄球层模型，我们找到了沿赤道移动的静止热斑形式的解析解以及自相似解。研究了决定热斑形状的参数对其传播速度和方向的影响。研究发现，热斑的运动速度和方向不仅取决于其与背景流的热对比符号，还取决于决定其形状的半轴之比。特别是，当赤道半轴与经向半轴之比 $b/a$ 等于 $\pi/2$ 时，热斑静止不动。如果热斑沿赤道轴拉长到 $b/a>\pi/2$ 的程度，则“冷”热斑向西移动，“热”热斑向东移动。在 $0.747\leq b/a<\pi/2$ 的条件下，情况相反：“热”热斑向西移动，“冷”热斑向东移动。模型中的自相似解严格以圆形热斑的形式实现，其半径按幂律变化，标度指数为 $k$，该指数由热斑轮廓上的跨锋浮力梯度行为决定。我们注意到，在恒定浮力梯度的假设下实现 $k=1/4$ 的机制，而 $k=1/6$ 的机制对应于总浮力的守恒。
英文摘要
For a model of a thin spherical layer of an incompressible rotating fluid subjected to the action of Coriolis and buoyancy forces, both analytical solutions in the form of stationary thermal spots moving along the equator and self-similar solutions are found. The influence of the parameters determining the shape of the spots on their propagation speed and direction is studied. It is found that the speed and direction of motion of the spots depend not only on the sign of their thermal contrast with the background flow, but also on the ratio of the semi-axes determining their shape. In particular, if the ratio of the equatorial semi-axis to the meridional one $b/a$ is equal to $π/2$, the spots are at rest. If the spots are elongated along the equatorial axis to such an extent that $b/a>π/2$, then ``cold'' spots move westward and ``hot'' spots move eastward. Under the condition $0.747\leq b/a<π/2$, the situation is reversed: ``hot'' spots move westward and ``cold'' spots move eastward. Self-similar solutions in the model are realized strictly in the form of circular thermal spots, and their radius varies according to a power law with a scaling exponent $k$, which is determined by the behavior of the cross-frontal buoyancy gradient on the spot contour. We note that the regime $k=1/4$ is realized under the assumption of a constant buoyancy gradient, while the regime $k=1/6$ corresponds to the conservation of total buoyancy.
