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Consistency between the flow at the top of the core and the frozen-flux approximation

机译：核心顶部流动与冻结通量近似值之间的一致性

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The flow just below the core-mantle boundary is constrained by the radial component of the induction equation. In the Alfvén frozen-flux limit, thought to be applicable to the outer core on the decade timescale of interest in geomagnetism, this gives a single equation involving the known radial magnetic field and its secular variation in two unknown flow components, leading to a severe problem of non-uniqueness. Despite this, we have two specific pieces of flow information which can be deduced directly from the frozen-flux induction equation: the component of flow perpendicular to null-flux curves, contours on which the radial magnetic field vanishes, and the amount of horizontal convergence and divergence at local extrema (maxima, minima and saddle points) of the radial magnetic field. To produce global velocity maps, we make additional assumptions about the nature of the flow and invert the radial induction equation for flow coefficients. However, it is not clear a priori that the flows thus generated are consistent with what we know about them along null-flux curves and at local extrema. This paper examines that issue. We look at typical differences between the null-flux curve perpendicular flow component, and convergence and divergence values at extrema, deduced directly from the induction equation and those from the inversions, investigate the effect of forcing the inversions to produce the correct null-flux curve and extremal values, and characterise the uncertainties on the various quantities contributing. Although the differences between the flow values from the induction equation directly and obtained by inversion seem large, and imposing the direct flow information as side constraints during inversion alters the flows significantly, we also show that these differences are within the likely uncertainties. Thus, we conclude that flows obtained through inversion do not contravene the specific flow information obtained directly from the radial induction equation in the frozen-flux limit. This result should reassure the community that frozen-flux flow inversion is a consistent process, even if including the extremal-value and null-flux conditions as additional information on flow inversion is unlikely to be useful. Solving for a time-dependent core-mantle boundary field model and flow simultaneously may be a good way to produce a temporally-varying field model consistent with the frozen-flux constraint; the ability to fit the data with such a model could be used to establish the timescale over which the frozen-flux assumption is valid.
机译：恰好在芯幔边界下方的流动受归纳方程的径向分量约束。在Alfvén冻结通量极限中，据认为适用于地磁感兴趣的十年时间尺度上的外核，这给出了一个包含已知径向磁场及其在两个未知流动分量中的长期变化的方程，从而导致非唯一性问题。尽管如此，我们还是可以从冻结磁通量感应方程式直接推导出两个特定的流信息：垂直于零磁通曲线的流分量，径向磁场消失的轮廓以及水平会聚量在径向磁场的局部极值（最大值，最小值和鞍点）处发散。为了生成整体速度图，我们对流的性质进行了额外的假设，并反演了径向系数方程。但是，尚不清楚先验的是，如此产生的流量是否与我们沿零通量曲线和局部极值所了解的一致。本文研究了这个问题。我们看了零通量曲线垂直流分量与极值处的收敛和发散值之间的典型差异，这些差异直接从归纳方程和反演中推导得出，研究了强迫反演产生正确的零通量曲线的效果和极值，并刻画各种贡献量的不确定性。尽管直接从归纳方程得到的流量值和通过反演获得的流量值之间的差异似乎很大，并且在反演过程中将直接流量信息强加为边约束会显着改变流量，但我们也表明这些差异在可能的不确定性之内。因此，我们得出的结论是，通过反演获得的流量与冻结通量极限中直接从径向感应方程获得的特定流量信息不冲突。该结果应使社区放心，即使包括极值和零通量条件，冻融流量反演也是一个一致的过程，因为有关流量反演的其他信息不太可能有用。同时求解时间相关的芯幔边界场模型和流动可能是产生与冻结通量约束一致的随时间变化的场模型的好方法。使用这种模型拟合数据的能力可用于建立冻结通量假设有效的时间范围。

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