Consistency between the flow at the top of the core and the frozen-flux approximation Kathy Whaler, University of Edinburgh Richard Holme, Liverpool University

Background Induction equation in the frozen-flux approximation Some flow information can be derived directly from this: (i) At local extrema of B r (ii) On null-flux curves, NFC (contours on which B r =0) where v 1 is the flow component perpendicular to the NFC

Tests Are flows calculated by inverting the induction equation consistent with the direct information? How much are the flows modified if the direct information is imposed as side constraints during inversion? How do we calculate the uncertainties on the various quantities involved?

Method Use the ufm model of Bloxham and Jackson (1992) for Invert secular variation coefficients at 1970, 1975 and 1980 for a steady flow (also a tangentially geostrophic flow for 1980) Approximately 75 extrema at each epoch Find a similar number of points along NFCs Compare direct and inverted values of  H.v and v 1

Extrema comparisons EpochActualPredicted Comparison between and (in nT/yr) at the extrema of B r. Left hand set of columns is for a damping parameter of 5 x 10 -5, the right hand set for

NFC comparisons EpochDirect v rms Inversion v rms  v rms (  v/v) rms TG Comparison between rms values of v 1 (in km/yr) on NFCs. Left hand steady flow columns for a damping parameter of 5 x 10 -5, right hand for For the 1980 tangentially geostrophic flow, damping parameters are (left) and (right).

Summary Differences between direct and inversion estimates of quantities are as large as the quantities themselves Suggests inversion flows are incompatible with underlying frozen-flux assumptions Can small changes to the inversion flow pattern bring it into agreement?

Imposing frozen-flux constraints Add as side constraints using a Lagrange multiplier Constrain at extrema rather than When satisfied at the 90% level, fit to the secular variation coefficients is considerably degraded, and complexity of the flow significantly increased

Both constraints simultaneously Very difficult to satisfy both sets of constraints simultaneously –Consider small null-flux patch beneath the North Atlantic –Both B r and are positive within it, implying convergence –But, on the NFC bounding it, because is positive, the flow is anti-parallel to, i.e. outward, and hence divergent Quantify uncertainties to assess how serious discrepancies are

Uncertainties rms uncertainty on depends strongly on truncation level 1381 nT/yr for degree 10, nT/yr for degree 14; take 1500 nT/yr as representative rms uncertainty on is 207 nT/km for degree 14 – many extrema not well constrained Two uncertainties combined at the 1 standard deviation level – constraints fail at only 5 out of 75 extrema, well below 1 / 3 expected by chance Uncertainty on B r significantly smaller than on other quantities

Grey shaded areas enclose 1 standard deviation uncertainties on extremal positions; contours are ± 500, 1000 and 1500 nT/yr of. Thus grey shaded areas should be cut by a contour.

Flow component uncertainties Several sources of uncertainty On the direct value of v 1 - calculate pointwise On the direction of the normal to the NFC – can’t straightforwardly calculate an rms value, so estimate its standard deviation at individual points Very variable, depending on how ‘flat’ B r is at the point On the flow components obtained from inversion – use the covariance matrix

Flow component uncertainties NFC v1v1 Inversion flow and its error ellipse Component of inversion flow in direction of v 1 and its uncertainty All inverse flow components within green cone possible

Example v 1 NFC results V 1 (km/yr)MinimumMaximum Direct-5.91 ± 3.40 Unconstrained inversion-4.18 ± Extremal constrained inversion-5.95 ± NFC constrained inversion-5.40 ± Direct3.68 ± 2.08 Unconstrained inversion8.69 ± Extremal constrained inversion18.94 ± NFC constrained inversion3.73 ±

Summary v 1 NFC results Approximately 2 / 3 of the unconstrained inversion flow components match the direct components to within the sum of the various uncertainties Almost all the NFC constrained flow components match the direct components Only about 40% of the extremal constrained flow components match the direct components

Summary Differences between aspects of the direct and (unconstrained) inverse flow are within the uncertainties No need to build constraints into inversion ufm field model used here is not itself consistent with frozen-flux – may be affecting results Newer satellite based models may reduce the secular variation, and hence flow, uncertainties