Nonlinear Rossby adjustment in a channel
Nonlinear Rossby adjustment in a channel
复制标题
通道中的非线性罗斯贝调整
DOI：
10.1017/s0022112099005042
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发表时间：
1998
影响因子：
3.7
通讯作者：
L. Pratt
中科院分区：
文献类型：
--
作者：
K. Helfrich;Allen C. Kuo;L. Pratt
对于初始深度不连续性较大的情况，解决了通道中均匀流体的罗斯贝调整问题。我们首先分析经典的溃坝问题，其中不连续一侧的深度为零。这种情况的近似解可以通过假设半地转动力学并使用特征方法来构建。该理论得到了完整浅水方程数值解的补充。流动的发展和最终的平衡体积传输由罗斯贝变形半径与通道宽度的比率（唯一的无量纲参数）控制。大坝被摧毁后，旋转的流体沿着通道的干燥部分溢出，形成稀薄的侵入物，对于北半球的旋转，该侵入物靠在右侧的墙壁上（面向下游）。随着河道宽度的增加，前缘（沿右侧壁）的速度超过非旋转情况下的侵入速度，达到上游盆地线性开尔文波速的3.80倍的极限值。在通道的左侧，流体在某个点与侧壁分离，当通道宽度接近无穷大时，该点的速度减小到零。对于宽度小于变形半径的情况，演化流的数值计算显示出与半地转理论的良好一致性。对于较大宽度的跨河道加速度，半地转近似中不存在，会降低一致性。沿着通道的最终平衡传输是根据半地转理论确定的，并且发现对于通道宽度大于大约一个变形半径的情况，偏离非旋转结果。对于大于大约四个变形半径的通道宽度，旋转将传输限制为恒定的最大值。然后对大坝下游的初始流体深度非零的情况进行数值检查。主要的稀薄侵入现在被开尔文激波或孔所取代，其速度远小于零深度侵入速度。冲击波要么直接穿过通道，要么仅附着在右侧墙壁上，具体取决于通道宽度和附加参数（初始深度差）。对于固定的下游深度，右侧壁上的冲击速度和振幅随着河道宽度的增加而增加到高于非旋转值。然而，旋转会降低非旋转情况下给定振幅的冲击速度。我们还发现了孔产生庞加莱波共振的证据。将激波特性与旋转激波理论进行比较，发现除了激波上位涡度的变化（对模型耗散非常敏感）之外，定性一致。在主导激波后面，除了产生强烈的非线性横向振荡和沿着最初位于大坝上游的流体的右侧通道壁快速平流外，流动的演变方式与线性理论所描述的方式大致相同。随着初始深度差的减小，最终稳态传输从零上游深度情况开始减小。
The Rossby adjustment problem for a homogeneous fluid in a channel is solved for large values of the initial depth discontinuity. We begin by analysing the classical dam break problem in which the depth on one side of the discontinuity is zero. An approximate solution for this case can be constructed by assuming semigeostrophic dynamics and using the method of characteristics. This theory is supplemented by numerical solutions to the full shallow water equations. The development of the flow and the final, equilibrium volume transport are governed by the ratio of the Rossby radius of deformation to the channel width, the only non-dimensional parameter. After the dam is destroyed the rotating fluid spills down the dry section of the channel forming a rarefying intrusion which, for northern hemisphere rotation, is banked against the right-hand wall (facing downstream). As the channel width is increased the speed of the leading edge (along the right-hand wall) exceeds the intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid separates from the sidewall at a point whose speed decreases to zero as the channel width approaches infinity. Numerical computations of the evolving flow show good agreement with the semigeostrophic theory for widths less than about a deformation radius. For larger widths cross-channel accelerations, absent in the semigeostrophic approximation, reduce the agreement. The final equilibrium transport down the channel is determined from the semigeostrophic theory and found to depart from the non-rotating result for channels widths greater than about one deformation radius. Rotation limits the transport to a constant maximum value for channel widths greater than about four deformation radii. The case in which the initial fluid depth downstream of the dam is non-zero is then examined numerically. The leading rarefying intrusion is now replaced by a Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion speed. The shock is either straight across the channel or attached only to the right-hand wall depending on the channel width and the additional parameter, the initial depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed downstream depth, increase above the non-rotating values with increasing channel width. However, rotation reduces the speed of a shock of given amplitude below the non-rotating case. We also find evidence of resonant generation of Poincaré waves by the bore. Shock characteristics are compared to theories of rotating shocks and qualitative agreement is found except for the change in potential vorticity across the shock, which is very sensitive to the model dissipation. Behind the leading shock the flow evolves in much the same way as described by linear theory except for the generation of strongly nonlinear transverse oscillations and rapid advection down the right-hand channel wall of fluid originally upstream of the dam. Final steady-state transports decrease from the zero upstream depth case as the initial depth difference is decreased.