若干地形下内波传播及内潮生成问题的理论解
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摘要
海洋中的湍混合包括机械混合和对流混合，混合的强弱是控制大洋环流的重要因素，其中机械混合的强弱取决于外界机械能的供给，风和潮汐是重要的机械能之源。内波是外界提供的机械能与混合之间联系的桥梁之一，一方面风和潮汐通过激发内波，将能量向深海和大洋内区传递，另一方面，内波的不稳定和破碎直接将机械能串级到小尺度湍流混合中去。开展倾斜地形上内波的传播以及内潮的产生等方面的研究对海洋混合过程的参数化以及混合对大洋环流的影响等方面的研究具有重要的科学意义。本文在二维线性理论框架下研究了内波在若干地形上的传播特征以及这些地形上内潮的产生。
针对海洋中的一些地形，如陆架陆坡地形、海山地形、海沟地形，提出了一组坐标变换，该变换可以将这些海底地形变换成平底，从而使得在这些特征地形上内波的传播和内潮的产生问题得到很大的简化。
利用提出的变换讨论了内波在线性斜坡地形、凸地形和凹地形上的传播特征，分析了入射波能通量在不同模态上的重新分布。变换的使用使得内波在给定地形上的反射波可以解析的表示出来，从而摆脱了只能使用射线追踪方法研究内波在倾斜地形上的反射问题的束缚。研究发现：1、内波在线性斜坡地形上传播时，入射波能通量在模态间的重新分布特征不仅与入射波的模态有关，还与入射波波射线的斜率与地形斜率的比值有关；无论亚临界线性斜坡地形还是超临界线性斜坡地形，其传到陆架区的能通量和反射回深海的能通量(仅对超临界线性斜坡地形)总是集中在某个或某两个模态附近；对于超临界线性斜坡地形，随着入射波波射线斜率向地形斜率的逐渐逼近，向高模态散射的能通量越来越大，向低模态散射的能通量越来越小。2、内波在凸地形上传播时，能通量在模态上的分布范围比较宽，但是同样在低模态和高模态上存在明显的峰值。3、内波在凹地形上传播时，能通量在模态上的分布范围也比较宽，但与凸地形和超临界线性斜坡地形相比，能通量在高模态上的分布没有明显的峰值。4、从总的能通量来讲，内波在凸地形和凹地形上传播时，向高模态散射的能通量与向低模态散射的能通
若十地形下内波传播及内潮生成问题的理论解
量近似相等，这意味着二者在向高模态散射能通量方面具有同等效率。5、凸地
形上临界点附近的反射，反射波的波射线能够延伸到深海和陆架，凹地形上临界
点附近的反射，反射波的波射线很快遇到地形发生二次反射，一般来讲二次反射
后回到深海的波动其速度剪切并不很强，但是在进行二次反射之前的区域，速度
剪切仍然很强，因而内波在凹地形上的反射依然能够在地形附近区域，产生强的
速度剪切区，诱发混合。
利用提出的坐标变换研究了若干亚临界地形上(如陆架陆坡地形、海山地形)
内潮的产生。由于非线性边界条件的使用，前人处理有限地形上内潮的产生问题
只能求助于射线理论，本文提出的坐标变换将若干海底地形变换成平底，从而使
得利用特征值方法研究这些地形上的内潮生成问题成为可能。所用的求解方法为
将解和强迫项在变换平面内进行付里叶级数展开，得到一维的波动方程，通过设
行波解的方法求解该一维波动方程，所有模态解的叠加即为内潮生成问题的解。
坐标变换将倾斜地形变换到了平底，同时也改变了强迫源强度的分布特征，使得
变换之后强迫源强度的分布与原始分布有很大的不同。在亚临界地形上，随着地
形的斜率向临界斜率逼近，对应强迫源的强度也逐渐变大，趋于无穷，因而从临
界点出发的波射线上内潮的能量最大，剪切最强，容易诱发混合。数值模拟和外
海实验发现内潮多从地形的波折处产生，这从本文得到的坐标变换之后的强迫源
的强度分布中可以得到解释。亚临界陆架陆坡地形和海山地形上内潮生成问题的
解也显示内潮的能量束从地形斜率最大的区域出发。
The ocean circulation is controlled by diapycnal mixing which consists of mechanical mixing and convective mixing. The intensity of mechanical mixing depends on the amount of mechanical energy offered by external processes, among which wind stress and tidal generation force are major sources. Internal waves are a kind of bridge relating the diapycnal mechanical mixing to external sources of mechanical energy: first, internal waves are driven by wind stress and barotropic tide so that large amount of mechanical energy are transported to abyssal ocean with propagation of internal waves; second, the mechanical energy can directly be cascaded to mixing through instability and breaking of internal waves. So it is important and essential to further investigate the generation of internal tides and the propagation of internal waves over bottom topographies.
A sets of transforms are introduced in this dissertation, by using of which some kinds of topographies, such as continental slope and ridge, can be converted into flat bottom. The processing simplifies the generation of internal tides and reflection of internal waves over those kinds of topographies.
Propagation of internal waves over linear slope, convex slope and concave slope are discussed by using of the transforms introduced in this dissertation. The transforms let the reflected waves from those topographies can be expressed in analytical forms. It is found that: (1) for linear slope, redistribution of incoming energy flux in modenumber space depends on both the modenumber of incident waves and the ratio of the slope of incident wave ray to slope of topography; both the transmitted and reflected energy flux(only for supercritical linear slope) focus near one or two modenumbers; for supercritical linear slope, the energy flux scattering to higher modenumbers becomes larger and the energy flux to lower modenumbers becomes smaller as the slope of incident wave ray comes near to slope of topography; (2) for convex slope, energy flux is redistributed in wide-range modenumbers and also show peaks on lower-modenubers and higher-modenumbers; (3) for concave slope, energy flux is also redistributed in wide-range modenumbers, but does not show peaks on higher-modenumbers as supercritical linear slope and convex slope do; (4) the total energy flux scattering to higher modenumbers is approximately equal to
the total energy flux to lower modenumbers for internal waves propagating over both convex slope and concave slope, which means the concave slope is the same efficient to convex slope in scattering energy flux to higher modenumbers; (5) for convex slope, the wave ray reflected from near-critical topography can extend to deep ocean and shallow shelf, while for concave slope, the wave ray reflected from near-critical topography can only extend to limited distance, then meet the topography and be reflected again; generally the shear of the internal waves reflected twice from concave slope is not enhanced while the enhanced shear does also appear before second reflection takes place, the enhanced shear can bring about intense mixing, so internal waves reflected from concave slope can also produce marked mixing near critical slope.
Internal tides generated over some kinds of topographies (such as continental slope and shelf, ridge topography) are investigated by using of the transforms introduced in this dissertation. Due to nonlinear bottom boundary condition, the generation of internal tides over finite topography can only be deal with by using of ray-tracing method beforetime. The transforms introduced in this dissertation make it possible use eigenvalue method to investigate the generation of internal tides over finite topographies. One dimensional equation can be obtained through extending the stream function and forcing term into fourier series, and this equation can be solved by assuming the traveling-wave solution. Besides converting the topographies to flat bottom, the transforms also change the distribution of forcing. For subcritical topography, the inten
针对海洋中的一些地形，如陆架陆坡地形、海山地形、海沟地形，提出了一组坐标变换，该变换可以将这些海底地形变换成平底，从而使得在这些特征地形上内波的传播和内潮的产生问题得到很大的简化。
利用提出的变换讨论了内波在线性斜坡地形、凸地形和凹地形上的传播特征，分析了入射波能通量在不同模态上的重新分布。变换的使用使得内波在给定地形上的反射波可以解析的表示出来，从而摆脱了只能使用射线追踪方法研究内波在倾斜地形上的反射问题的束缚。研究发现：1、内波在线性斜坡地形上传播时，入射波能通量在模态间的重新分布特征不仅与入射波的模态有关，还与入射波波射线的斜率与地形斜率的比值有关；无论亚临界线性斜坡地形还是超临界线性斜坡地形，其传到陆架区的能通量和反射回深海的能通量(仅对超临界线性斜坡地形)总是集中在某个或某两个模态附近；对于超临界线性斜坡地形，随着入射波波射线斜率向地形斜率的逐渐逼近，向高模态散射的能通量越来越大，向低模态散射的能通量越来越小。2、内波在凸地形上传播时，能通量在模态上的分布范围比较宽，但是同样在低模态和高模态上存在明显的峰值。3、内波在凹地形上传播时，能通量在模态上的分布范围也比较宽，但与凸地形和超临界线性斜坡地形相比，能通量在高模态上的分布没有明显的峰值。4、从总的能通量来讲，内波在凸地形和凹地形上传播时，向高模态散射的能通量与向低模态散射的能通
若十地形下内波传播及内潮生成问题的理论解
量近似相等，这意味着二者在向高模态散射能通量方面具有同等效率。5、凸地
形上临界点附近的反射，反射波的波射线能够延伸到深海和陆架，凹地形上临界
点附近的反射，反射波的波射线很快遇到地形发生二次反射，一般来讲二次反射
后回到深海的波动其速度剪切并不很强，但是在进行二次反射之前的区域，速度
剪切仍然很强，因而内波在凹地形上的反射依然能够在地形附近区域，产生强的
速度剪切区，诱发混合。
利用提出的坐标变换研究了若干亚临界地形上(如陆架陆坡地形、海山地形)
内潮的产生。由于非线性边界条件的使用，前人处理有限地形上内潮的产生问题
只能求助于射线理论，本文提出的坐标变换将若干海底地形变换成平底，从而使
得利用特征值方法研究这些地形上的内潮生成问题成为可能。所用的求解方法为
将解和强迫项在变换平面内进行付里叶级数展开，得到一维的波动方程，通过设
行波解的方法求解该一维波动方程，所有模态解的叠加即为内潮生成问题的解。
坐标变换将倾斜地形变换到了平底，同时也改变了强迫源强度的分布特征，使得
变换之后强迫源强度的分布与原始分布有很大的不同。在亚临界地形上，随着地
形的斜率向临界斜率逼近，对应强迫源的强度也逐渐变大，趋于无穷，因而从临
界点出发的波射线上内潮的能量最大，剪切最强，容易诱发混合。数值模拟和外
海实验发现内潮多从地形的波折处产生，这从本文得到的坐标变换之后的强迫源
的强度分布中可以得到解释。亚临界陆架陆坡地形和海山地形上内潮生成问题的
解也显示内潮的能量束从地形斜率最大的区域出发。
The ocean circulation is controlled by diapycnal mixing which consists of mechanical mixing and convective mixing. The intensity of mechanical mixing depends on the amount of mechanical energy offered by external processes, among which wind stress and tidal generation force are major sources. Internal waves are a kind of bridge relating the diapycnal mechanical mixing to external sources of mechanical energy: first, internal waves are driven by wind stress and barotropic tide so that large amount of mechanical energy are transported to abyssal ocean with propagation of internal waves; second, the mechanical energy can directly be cascaded to mixing through instability and breaking of internal waves. So it is important and essential to further investigate the generation of internal tides and the propagation of internal waves over bottom topographies.
A sets of transforms are introduced in this dissertation, by using of which some kinds of topographies, such as continental slope and ridge, can be converted into flat bottom. The processing simplifies the generation of internal tides and reflection of internal waves over those kinds of topographies.
Propagation of internal waves over linear slope, convex slope and concave slope are discussed by using of the transforms introduced in this dissertation. The transforms let the reflected waves from those topographies can be expressed in analytical forms. It is found that: (1) for linear slope, redistribution of incoming energy flux in modenumber space depends on both the modenumber of incident waves and the ratio of the slope of incident wave ray to slope of topography; both the transmitted and reflected energy flux(only for supercritical linear slope) focus near one or two modenumbers; for supercritical linear slope, the energy flux scattering to higher modenumbers becomes larger and the energy flux to lower modenumbers becomes smaller as the slope of incident wave ray comes near to slope of topography; (2) for convex slope, energy flux is redistributed in wide-range modenumbers and also show peaks on lower-modenubers and higher-modenumbers; (3) for concave slope, energy flux is also redistributed in wide-range modenumbers, but does not show peaks on higher-modenumbers as supercritical linear slope and convex slope do; (4) the total energy flux scattering to higher modenumbers is approximately equal to
the total energy flux to lower modenumbers for internal waves propagating over both convex slope and concave slope, which means the concave slope is the same efficient to convex slope in scattering energy flux to higher modenumbers; (5) for convex slope, the wave ray reflected from near-critical topography can extend to deep ocean and shallow shelf, while for concave slope, the wave ray reflected from near-critical topography can only extend to limited distance, then meet the topography and be reflected again; generally the shear of the internal waves reflected twice from concave slope is not enhanced while the enhanced shear does also appear before second reflection takes place, the enhanced shear can bring about intense mixing, so internal waves reflected from concave slope can also produce marked mixing near critical slope.
Internal tides generated over some kinds of topographies (such as continental slope and shelf, ridge topography) are investigated by using of the transforms introduced in this dissertation. Due to nonlinear bottom boundary condition, the generation of internal tides over finite topography can only be deal with by using of ray-tracing method beforetime. The transforms introduced in this dissertation make it possible use eigenvalue method to investigate the generation of internal tides over finite topographies. One dimensional equation can be obtained through extending the stream function and forcing term into fourier series, and this equation can be solved by assuming the traveling-wave solution. Besides converting the topographies to flat bottom, the transforms also change the distribution of forcing. For subcritical topography, the inten
引文
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