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The flow just below the core-mantle boundary is constrained by the radial component of the induction equation. In the Alfv´ en 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.

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