Our intent is to forecast SV by performing the 4DEnVar data assimilation using a numerical geodynamo code and existing data with respect to the poloidal scalar potential of the geomagnetic field at the core–mantle boundary (CMB) and the toroidal and poloidal components of the core surface flow. From the variational data assimilation experiments with an inertiafree MHD dynamo model, Li et al. (2014) found that the magnetic field below the CMB is hard to reconstruct only from the magnetic data taken outside the core due to the diffusiondominant Ekman boundary layer. In agreement with their proposal, we use the preliminary estimated core surface flow as part of observation vectors in our data assimilation as well as the geomagnetic data. Inclusion of the core surface flow in data vectors results in indirect inclusion of SV data in the data assimilation.
In this section, we first describe the assimilation theory (“Data assimilation theory” section) and details of our dynamo model and how to adjust dimensionless time to the actual time (“Geodynamo simulation: parameters and scaling of time” section). We then briefly discuss the nonlinearity of our dynamo model (“Nonlinearity of the numerical geodynamo: error growth efolding time” section). Next, we explain details of preparation of the observational data and how to convert dimensionless simulation outputs to variables comparable to real data (“Data 1: poloidal scalar potential at the CMB obtained from the MCM model” section and “Data 2: core surface flow” section) and finally describe the way of practical implementation of the data assimilation (“Implementation of assimilation” section).
Data assimilation theory
We consider the minimization of the following cost function:
$$V\left( {\varvec{x}_{0} } \right) = \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left[ {\varvec{y}_{k}  \varvec{h}_{k} \left( {\varvec{x}_{k} } \right)} \right]^{\text{T}} {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{h}_{k} \left( {\varvec{x}_{k} } \right)} \right],$$
(1)
where \(\varvec{x}_{k}\) is the state vector of a dynamo model at time \(t_{k}\), \(\varvec{y}_{k}\) denotes the observation vector, \({\mathbf{R}}_{k}\) is the covariance matrix of observation noise, and \(\varvec{h}_{k}\) is an observation operator which converts a state vector \(\varvec{x}_{k}\) to observable variables for the comparison with \(\varvec{y}_{k}\). Given the dynamo model, \(\varvec{x}_{k}\) is uniquely determined from the initial state \(\varvec{x}_{0}\). This allows us to represent \(\varvec{x}_{k}\) as a function of \(\varvec{x}_{0}\), that is, \(\varvec{x}_{k} = \varvec{f}_{k} \left( {\varvec{x}_{0} } \right)\). Defining a function \(\varvec{g}_{k}\) as \(\varvec{g}_{k} \left( {\varvec{x}_{0} } \right) = \varvec{h}_{k} \left( {\varvec{f}_{k} \left( {\varvec{x}_{0} } \right)} \right)\), the cost function in Eq. (1) can be rewritten as follows:
$$V\left( {\varvec{x}_{0} } \right) = \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\varvec{x}_{0} } \right)} \right]^{\text{T}} {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\varvec{x}_{0} } \right)} \right].$$
(2)
The minimization of this cost function is achieved by an iterative algorithm based on the 4DEnVar method (Liu et al. 2008). At the \(m\)th iteration, we approximate the cost function by using an ensemble of the simulation outputs \(\left\{ {\varvec{x}_{0:K,m}^{\left( 1 \right)} \ldots ,\varvec{x}_{0:K,m}^{\left( N \right)} } \right\}\), where \(N\) is the size of ensemble and \(\varvec{x}_{0:K,m}^{\left( n \right)} , n \in \left\{ {1, \ldots ,N} \right\}\) is the sequence of vectors \(\varvec{x}_{0,m}^{\left( n \right)} , \varvec{x}_{1,m}^{\left( n \right)} , \ldots , \varvec{x}_{K,m}^{\left( n \right)}\). This ensemble is calculated by MHD dynamo simulations from the initial conditions \(\left\{ {\varvec{x}_{0,m}^{\left( 1 \right)} , \ldots ,\varvec{x}_{0,m}^{\left( N \right)} } \right\}\), which are prepared so that the ensemble mean, \(\left( {{{\varSigma }}_{n = 1}^{N} \varvec{x}_{0,m}^{\left( n \right)} } \right)/N\), is equal to the \(m\)th estimate \(\bar{\varvec{x}}_{0,m}\). At the \(m\)th iteration, we seek \(\varvec{x}_{0}\) that minimizes Eq. (2), which turns out to be \(\bar{\varvec{x}}_{0,m + 1}\), with given \(\bar{\varvec{x}}_{0,m}\) and \(\left\{ {\varvec{x}_{0:K,m}^{\left( 1 \right)} , \ldots ,\varvec{x}_{0:K,m}^{\left( N \right)} } \right\}\). As an important first step of the 4DEnVar, we express \(\varvec{g}_{k} \left( {\varvec{x}_{0} } \right)\) in terms of the firstorder Taylor expansion,
$$\varvec{g}_{k} \left( {\varvec{x}_{0} } \right) \approx \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right) + \varvec{G}_{k} \left( {\varvec{x}_{0}  \bar{\varvec{x}}_{0,m} } \right),$$
(3)
where \(\varvec{G}_{k}\) is the Jacobian of \(\varvec{g}_{k}\) at \(\bar{\varvec{x}}_{0,m}\). We then approximate \(\varvec{x}_{0}\) as a weighted sum of the ensemble members. Now we define the following matrices \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m}\) and \({\hat{\mathbf{\varGamma }}}_{k,m}\) for convenience:
$${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m} = \frac{1}{{\sqrt {N  1} }}\left( {\varvec{x}_{0,m}^{\left( 1 \right)}  \bar{\varvec{x}}_{0,m} \cdots \varvec{x}_{0,m}^{\left( N \right)}  \bar{\varvec{x}}_{0,m} } \right),$$
(4)
$$\begin{aligned}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} = \frac{1}{{\sqrt {N  1} }}\left( {\varvec{g}_{k} \left( {\varvec{x}_{0,m}^{\left( 1 \right)} } \right)  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right) \cdots \varvec{g}_{k} \left( {\varvec{x}_{0,m}^{\left( N \right)} } \right)  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)} \right).\end{aligned}$$
(5)
This allows us to write \(\varvec{x}_{0} = \bar{\varvec{x}}_{0,m} + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m} \varvec{w}\), where \(\varvec{w}\) consists of weight for each ensemble member. \(\bar{\varvec{x}}_{0,m}\) is the mean of \(\varvec{x}_{0,m}^{\left( n \right)} \left( {n = 1, \ldots ,N} \right)\). Using Eqs. (4) and (5), the function \(\varvec{g}_{k} \left( {\varvec{x}_{0} } \right)\) in Eq. (3) can then be expressed:
$$\varvec{g}_{k} \left( {\varvec{x}_{0} } \right) \approx \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right) + \varvec{G}_{k} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m} \varvec{w} \approx \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right) + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} \varvec{w}.$$
(6)
Note here that the Jacobian \(\varvec{G}_{k}\) disappears in the expression of \(\varvec{g}_{k} \left( {\varvec{x}_{0} } \right)\) with the aid of the relationship derived from Eq. (3), \(\varvec{g}_{k} \left( {\varvec{x}_{0,m}^{\left( n \right)} } \right)  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right) \approx \varvec{G}_{k} \left( {\varvec{x}_{0,m}^{\left( n \right)}  \bar{\varvec{x}}_{0,m} } \right), \left( {n = 1, \ldots ,N} \right)\). Equation (6) allows us to circumvent direct calculation of the Jacobian \(\varvec{G}_{k}\). On the other hand, the linear approximation Eq. (3) (and resulting Eq. (6)) imposes us on two requirements:

(I)
the assimilation window indexed by \(k = 1, \ldots ,K\) is so short that nonlinearity of \(\varvec{g}_{k} \left( {\varvec{x}_{0} } \right)\) is negligible (or weak);

(II)
the deviations, \(\varvec{x}_{0,m}^{\left( n \right)}  \bar{\varvec{x}}_{0,m}\) (\(n = 1, \ldots ,N)\) in Eq. (4), are small enough.
For the first requirement, we discuss the nonlinearity of our dynamo model using the error growth rate (Hulot et al. 2010) in “Nonlinearity of the numerical geodynamo” section later, while we see that ensembles shrinking through iterations meet the second requirement in “Numerical experiments” section. From Eqs. (2) and (6), we introduce the following objective function;
$$\hat{J}_{m} \left( \varvec{w} \right) = \frac{{\sigma_{m}^{2} }}{2}\varvec{w}^{\text{T}} \varvec{w} + \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)  {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} \varvec{w}} \right]^{\text{T}} {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)  {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} \varvec{w}} \right],$$
(7)
where \({{\sigma }}_{m}\) is a parameter, which is fixed to \(\sigma_{m} = 1\) in this study, while we decrease elements of \({\mathbf{R}}_{k}\) at each step. This cost function is minimized provided that:
$$\hat{\varvec{w}}_{m} = \left( {\mathop \sum \limits_{k} \left[ {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}
{\Gamma}}_{k,m}^{\text{T}} {\mathbf{R}}_{k}^{  1} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} } \right] + \sigma_{m}^{2} \varvec{I }} \right)^{  1} \mathop \sum \limits_{k} \left( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma }}_{k,m}^{\text{T}} {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)} \right]} \right).\varvec{ }$$
(8)
The \(\left( {m + 1} \right)\)th estimate \(\bar{\varvec{x}}_{0,m + 1}\) is then obtained as
$$\bar{\varvec{x}}_{0,m + 1} = \bar{\varvec{x}}_{0,m} + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m} \hat{\varvec{w}}_{m} ,$$
(9)
and we proceed to the next iteration. The first term of the righthand side in Eq. (7) is added to ensure robustness. This iterative application of Eq. (8), which is similar to the iterative ensemble Kalman smoother algorithm (Gu and Oliver 2007; Bocquet and Sakov 2013), minimizes Eq. (2) in the subspace spanned by the ensemble members (Nakano 2020). After obtaining \(\bar{\varvec{x}}_{0,m + 1}\) it is necessary to perform MHD dynamo simulations with a set of initial conditions \(\left\{ {\varvec{x}_{0,m + 1}^{\left( 1 \right)} , \ldots ,\varvec{x}_{0,m + 1}^{\left( N \right)} } \right\}\) to renew the ensemble members for the (\(m + 1\))th iteration. See Appendix A for how to prepare the set of initial conditions from \(\bar{\varvec{x}}_{0,m + 1}\), \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{0,m}\) and \(\hat{\varvec{w}}_{m}\).
At the final (5th is chosen in this study) iteration, we also estimate the bias and trend components which correspond to model error in the dynamo model, by minimizing the following function:
$$\begin{aligned} \hat{J}_{m} \left( {\varvec{w},\varvec{b},\varvec{a}} \right) & = \frac{{\sigma_{m}^{2} }}{2}\varvec{w}^{\text{T}} \varvec{w} + \frac{1}{2}\varvec{b}^{\text{T}} {\mathbf{P}}_{b}^{  1} \varvec{b} + \frac{1}{2}\varvec{a}^{T} {\mathbf{P}}_{a}^{  1} \varvec{a} \\ & \quad + \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)  {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} \varvec{w}  \varvec{b}  k\varvec{a}} \right]^{\text{T}} \\ & \quad \times {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,m} } \right)  {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Gamma } }}_{k,m} \varvec{w}  \varvec{b}  k\varvec{a}} \right], \\ \end{aligned}$$
(10)
where \(\varvec{b}\) denotes the bias component, while \(\varvec{a}\) is the coefficient for the trend component. The bias and trend terms correspond to the offset and the linear departure in time between the observation and the model, respectively. Here, we assume that the observation is mostly explained by the dynamo model output and that the bias and trend components are minor. We thus select \({\mathbf{P}}_{a}\) and \({\mathbf{P}}_{b}\) as:
$${\mathbf{P}}_{a} = {\mathbf{P}}_{b} = 10^{  4} {\mathbf{R}}_{k} .$$
(11)
Large norms of \({\mathbf{P}}_{b}^{  1}\) and \({\mathbf{P}}_{a}^{  1}\) suppress intensities of \(\varvec{b}\) and \(\varvec{a}\) while minimizing Eq. (10). \(\varvec{w},\varvec{b},\varvec{a}\) that minimize \(\hat{J}_{m} \left( {\varvec{w},\varvec{b},\varvec{a}} \right)\) in Eq. (10) can be obtained in a similar manner to Eq. (8) (see Appendix B for details). The minimization of Eq. (10) gives the approximate minimum of the following cost function:
$$V\left( {\varvec{x}_{0} } \right) = \frac{1}{2}\mathop \sum \limits_{k = 1}^{K} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\varvec{x}_{0} } \right)  \varvec{b}  k\varvec{a}} \right]^{\text{T}} {\mathbf{R}}_{k}^{  1} \left[ {\varvec{y}_{k}  \varvec{g}_{k} \left( {\varvec{x}_{0} } \right)  \varvec{b}  k\varvec{a}} \right].$$
(12)
The final estimate and prediction are obtained by the following equation:
$$\bar{\varvec{g}}_{k,M} = \varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,M} } \right) + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varGamma } }}_{k,M} \hat{\varvec{w}}_{M} + \hat{\varvec{b}} + k\hat{\varvec{a}},\varvec{ }$$
(13)
where \(M\) indicates the final step, i.e., \(M = 5\), and \(\hat{\varvec{w}}_{{_{M} }}\), \(\hat{\varvec{b}}\) and \(\hat{\varvec{a}}\) are solutions to Eq. (10). Then we can obtain the final estimate of Eq. (13) by the sum of a single MHD simulation starting from \(\bar{\varvec{x}}_{0,M}\), the weighted sum of the Mth ensemble members, and the trend and bias terms. Note that we can use Eq. (13) not only for the final estimate within the assimilation window, but also for the estimate in the forecast period outside the assimilation window when the future extensions of \(\varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,M} } \right)\) and \(\varvec{g}_{k} \left( {\varvec{x}_{0,M}^{\left( n \right)} } \right)\), \((K < k)\), are available, which requires only additional dynamo runs for \(\varvec{g}_{k} \left( {\bar{\varvec{x}}_{0,M} } \right)\) and all the ensemble members. Figure 1 shows how to prepare an SV model for IGRF13 by our assimilation scheme and Eq. (13), where the future extensions of the ensemble members (the gray area after “Release of IGRF”) generate a future prediction (the blue line after “Release of IGRF”) by the weighted sum of the ensemble members via Eq. (13).
Note that the firstorder approximation of the function \(\varvec{f}_{k}\) allows us to approximate the state at an arbitrary time \(t_{{k^{\prime}}}\) (\(0 \le k^{\prime} \le K\)) in a similar manner to Eq. (6):
$$\varvec{x}_{{k^{\prime}}} \approx \varvec{f}_{{k^{\prime}}} \left( {\bar{\varvec{x}}_{0,m} } \right) + \varvec{F}_{{k^{\prime}}} \left( {\varvec{x}_{0}  \bar{\varvec{x}}_{0,m} } \right) \approx \bar{\varvec{x}}_{{k^{\prime},m}} + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{{k^{\prime},m}} \varvec{w} ,$$
(14)
where \(\varvec{F}_{k}\) is the Jacobian of \(\varvec{f}_{k}\) at \(\bar{\varvec{x}}_{0,m}\) and \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{{k^{\prime},m}}\) is a matrix obtained by replacing 0 by \(k'\) in Eq. (4) as follows:
$${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{{k^{\prime},m}} = \frac{1}{{\sqrt {N  1} }}\left( {\varvec{x}_{{k^{\prime},m}}^{\left( 1 \right)}  \bar{\varvec{x}}_{{k^{\prime},m}} \cdots \varvec{x}_{{k^{\prime},m}}^{\left( N \right)}  \bar{\varvec{x}}_{{k^{\prime},m}} } \right).$$
(15)
The state at \(t_{{k^{\prime}}}\) can be estimated from the following equation:
$$\bar{\varvec{x}}_{{k^{\prime},m + 1}} = \bar{\varvec{x}}_{{k^{\prime},m}} + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}_{{k^{\prime},m}} \hat{\varvec{w}}_{m} ,$$
(16)
without any modification to \(\hat{\varvec{w}}_{m}\) in Eq. (8), which is the same as used in Eq. (9). To obtain \(\bar{\varvec{x}}_{{k^{\prime},m + 1}}\) by Eq. (16) means that we minimize the cost function Eq. (2) (or (12)) using the firstorder Taylor expansion of \(\varvec{g}_{{{{\Delta }}k}} \left( {\varvec{x}_{{k^{\prime}}} } \right)\) at \(\bar{\varvec{x}}_{{k^{\prime},m}}\),
$$\varvec{g}_{k} \left( {\varvec{x}_{0} } \right) = \varvec{g}_{{{{\Delta }}k}} \left( {\varvec{x}_{{k^{\prime}}} } \right) \approx \varvec{g}_{{{{\Delta }}k}} \left( {\bar{\varvec{x}}_{{k^{\prime},m}} } \right) + \varvec{G}_{{{{\Delta }}k}} \left( {\varvec{x}_{k'}  \bar{\varvec{x}}_{k',m} } \right),$$
(17)
instead of Eq. (3), where \({{\Delta }}k = k  k'\), and \(\varvec{g}_{{{{\Delta }}k}}\) and \(\varvec{G}_{{{{\Delta }}k}}\) are associated with the inverse function of \(\varvec{f}_{{  {{\Delta }}k}}\) when \({{\Delta }}k < 0\). This enables us to choose arbitrary time, \(t_{{k^{\prime}}}\), at which we restart MHD dynamo simulation after the data assimilation, for the purpose of better accuracy of the linear approximation at times close to \(t_{{k^{\prime}}}\). For the future forecast, single MHD or kinematic dynamo (KD; meaning steady flow) simulation can also estimate future magnetic fields, running them from the initial condition optimized at arbitrary time \(t_{{k^{\prime}}}\) (Eq. 16) by data assimilation. We discuss differences in forecasts by MHD and kinematic dynamo simulation from different \(t_{k}^{'}\) in “Numerical experiments” section.
Geodynamo simulation: parameters and scaling of time
We adopt the geodynamo simulation code established by Takahashi (2012, 2014) for our data assimilation and forecast processes. In this code, the velocity and magnetic fields are decomposed into the poloidal and toroidal parts and spatially expanded in terms of spherical harmonics in the tangential directions, while combined compact finite differencing and the Crank–Nicolson scheme are adopted for discretization in the radial and time coordinates, respectively. The temperature and composition are treated separately (Takahashi 2014). We select the following dimensionless parameters for the geodynamo simulation: the Ekman number is \(E = 3 \times 10^{  5}\), the magnetic Prandtl number is \(Pm = 2.0\), the Prandtl number is 0.1, the compositional Prandtl (or Schmidt) number is 1.0, the modified thermal Rayleigh number is 500, and the modified compositional Rayleigh number is 2000. The magnetic Reynolds number (\(R_{m}\)) of this dynamo is 181 with the standard deviation of 15, which is evaluated from the initial ensemble members for our data assimilation (see “Numerical experiments” section for details of the ensemble).
Since all the parameters and variables are dimensionless in the numerical geodynamo model, the simulation outputs should be scaled for comparison with the actual data of the geomagnetic main field and the core surface flow. A time scale of the geomagnetic secular variation, \(\tau_{\text{SV}}\), can be characterized by simple dependence of spherical harmonic degree, \(l\) (Lhuillier et al. 2011a). Christensen and Tilgner (2004) obtained a value of \(\tau_{\text{SV}} = 535 {\text{years}}\) by fitting \(\tau_{\text{SV}} /l\) (\(l \ge 2\)) to the time scale of secular variation for each spherical harmonic degree, \(\tau_{l}\) (see Christensen and Tilgner 2004, for the definition). In the same manner, we estimate the secular variation time scale of our numerical geodynamo model to be \(\tau_{\text{SV}}^{*} = 0.0411\), scaled by the viscous dissipation time; we first estimate the mean \(\tau_{l}\) of our numerical geodynamo from a longtime dynamo run (about one magnetic diffusion time, i.e., 26,000 years in actual time) and then evaluate \(\tau_{\text{SV}}^{*}\) by fitting \(\tau_{\text{SV}}^{*} /l\) to the estimated \(\tau_{l}\) for \(l \ge 2\). Figure 2 shows the estimated \(\tau_{l}\) and the fitting result. Hence, we convert the dimensionless time of the numerical geodynamo into the actual time by using \(\tau_{\nu } \equiv \tau_{\text{SV}} /\tau_{\text{SV}}^{*} = 13,049 \;{\text{years }}\) throughout this study.
Nonlinearity of the numerical geodynamo: error growth efolding time
Since the 4DEnVAR approach relies heavily on the linear assumption Eq. (3) and resulting Eq. (6), we should be careful about its validity in the problem in concern, i.e., the data assimilation and forecast of the geomagnetic field for IGRF13SV. We here investigate the timescale of nonlinearity of our numerical geodynamo by introducing the error growth rate (e.g., Hulot et al. 2010), which measures the rate at which small error in a dynamo state vector grows with time. It is known that the error growth rate is stable once a dynamo model and parameters are provided, independent of the magnitude of inserted error and types of variables into which error is contaminated (Lhuillier et al. 2011b). Its inverse, i.e., the efolding time of error growth \(\tau_{e}\), therefore, can be interpreted as a limit of the period over which data assimilation is feasible with the nonlinear dynamo model; when the length of assimilation window is comparable with \(\tau_{e}\), small error in the initial state is critical to the state at the end of the window, and one cannot expect to obtain reasonable initial condition. The linear assumption for 4DEnVar (Eq. 3) clearly requires that the assimilation window should be shorter than \(\tau_{e}\) of our numerical geodynamo model.
We estimate the efolding time \(\tau_{e}\) of our numerical geodynamo by following the method in Lhuillier et al. (2011b). We first choose ten different dynamo state vectors from a dynamo run for about one magnetic diffusion time. We then add a small perturbation to the poloidal components of the magnetic field over the first eight harmonic degree by the expression:
$$\tilde{S}_{l}^{m} \left( {t_{p} ,r} \right) = \left( {1 + \alpha_{l}^{m} \epsilon } \right)S_{l}^{m} \left( {t_{p} ,r} \right),$$
(18)
where \(\tilde{S}_{l}^{m}\) and \(S_{l}^{m}\) are the poloidal scalar functions of the magnetic field with degree \(l\) and order \(m\) of spherical harmonics for the perturbed and original initial conditions at a chosen time \(t_{p}\), respectively. \(\alpha_{l}^{m}\) is a coefficient obeying the standard normal law and ϵ is the relative magnitude of perturbation. We choose ϵ = 10^{−10} in this study. MHD dynamo simulations are performed from both original and perturbed initial conditions and the discrepancy between them are measured by the magnetic error at the CMB,
$${{\Delta }}y_{l} \left( t \right) = \sqrt {\frac{1}{2l + 1}\mathop \sum \limits_{m} \frac{{\left( {\tilde{S}_{l}^{m} \left( {t,r_{\text{CMB}} } \right)  S_{l}^{m} \left( {t,r_{\text{CMB}} } \right)} \right)^{2} }}{\langle {S_{l}^{m} \left( {t,r_{\text{CMB}} } \right)^{2} \rangle }}} ,$$
(19)
where \(t\) is time scaled by the magnetic diffusion time \(\tau_{\eta }\), \(r_{\text{CMB}}\) is the radial coordinate of the CMB, and \(\langle \cdot \rangle\) denotes the time average. For each pair of dynamo runs, the error growth rate \(\lambda\) and its inverse, the efolding time \(\tau_{e} = \lambda^{  1}\), are estimated by fitting the model of \(\lambda t + a\), where \(a\) is the offset, jointly to \(\log {{\Delta }}y_{l} \left( t \right)\) (\(1 \le l \le 8\)). The time window for the fitting is manually determined, by simultaneously considering the first eight \(\Delta y_{l} \left( t \right)\). After the same procedure for ten chosen times of \(t_{p}\), we average ten \(\lambda\) s and \(\tau_{e}\) s for our final estimates.
Figure 3 shows results of estimation of \(\lambda\) and \(\tau_{e}\) for our geodynamo model. As shown in Fig. 3a, the error growth rate (slope of the blue line) was obtained by the linear regression over the first eight spherical harmonic degrees (the gray lines). As in Fig. 3b, we finally evaluate \(\lambda = 187 \pm 36\) [\(1/\tau_{\eta }\)] and \(\tau_{e} = \left( {5.37 \pm 1.02} \right) \times 10^{  3} \left[ {\tau_{\eta } } \right]\), where the uncertainties are given by two standard deviations. Using the scaling law, \(\tau_{\eta } = P_{m} \tau_{\nu } = 2 \cdot 13,049 {\text{years}}\), \(\tau_{e}\) of our numerical dynamo is found to be \(\tau_{e} = 140 \pm 27\) years. Then we should choose the assimilation window shorter than ~ 110 years to avoid the effect of nonlinearity of our geodynamo model (Takahashi 2012, 2014). We later select a maximum assimilation window of 10 years and a prediction period of 5 years, which seem short enough to circumvent the nonlinearity effect and to hold Eq. (3).
Hulot et al. (2010) claimed that \(\tau_{e}\) of the Earth’s dynamo system appears to be about 30 years from the linear relationship between \(\tau_{e} /\tau_{\text{SV}}\) and \(R_{m}\), provided that \(R_{m} > 100\), \(E < 10^{  9}\), and \(\tau_{\text{SV}} = 535 {\text{years}}\) (Christensen and Tilgner 2004) for the Earth. Although our \(\tau_{e} (\sim 140\) years) is much longer than \(\tau_{e} (\sim\) 30 years) by Hulot et al. (2010), it is found that \(\tau_{e}\) highly depends on numerical dynamo models even under the similar dynamo parameter settings (e.g., see Fig. 3 in Hulot et al. (2010) for the comparison between \(\tau_{e}\) by Hulot et al. (2010) and that by Olson et al. (2009) under \(E = 3 \times 10^{  4}\)). We, therefore, think our long \(\tau_{e}\) is a characteristic of our geodynamo model. We here conclude that our assimilation window of maximum 10 years is short enough to adopt linear approximation of Eq. (3) and resulting Eq. (6) from the perspective of \(\tau_{e}\) of our numerical geodynamo.
Data 1: poloidal scalar potential at the CMB obtained from the MCM model
We use a version of the MCM model (Ropp et al. 2020) which spans from 2001.5 to 2019.25, referred to as the MCM3 model in this paper hereafter, to construct data vectors for our data assimilation. The MCM3 model has been built using hourly mean observatory data together with CHAMP and SwarmA satellite magnetic data. These data have been selected depending on their local time and for magnetically quiet periods (c.f., Lesur et al. 2008). The model is made of a series of snapshot models, 3 months apart, that have been built through a Kalman filter approach combined with a correlationbased modeling technique for the analysis step (Holschneider et al. 2016). Each snapshot model is parameterized in terms of spherical harmonics and includes static core field and secular variation contributions. Lithospheric, external and induced fields are also coestimated as well as crustal offsets at each observatory. Details of the modeling technique are described in Ropp et al. (2020). The Gauss coefficients of the core field and its secular variation can be downward continued to the CMB for use, e.g., for information on the flow in the liquid outer core.
Assuming the mantle an electrical insulator, we convert the Gauss coefficients, \(g_{l}^{m} \left( t \right)\) and \(h_{l}^{m} \left( t \right)\), given by the MCM3 model to the poloidal scalar potential, \(S_{l}^{\text{mc}} \left( t \right)\) and \(S_{l}^{\text{ms}} \left( t \right)\), at the CMB, respectively, as
$$\begin{aligned}S_{l}^{\text{mc}} \left( t \right)& = \frac{1}{l}\frac{1}{{r_{\text{oc}}^{2} }}\left( {\frac{{r_{e} }}{{r_{\text{oc}} }}} \right)^{l + 2} g_{l}^{m} \left( t \right), \\ S_{l}^{\text{ms}} \left( t \right)& = \frac{1}{l}\frac{1}{{r_{\text{oc}}^{2} }}\left( {\frac{{r_{e} }}{{r_{\text{oc}} }}} \right)^{l + 2} h_{l}^{m} \left( t \right), \end{aligned}$$
(20)
where \(t\) is the dimensional time, \(r_{e} = 6371.2 {\text{km}}\) and \(r_{oc} = 3485 {\text{km}}\) are the radii of the Earth and outer core, respectively, and \(S_{l}^{\text{mc}} \left( t \right)\) and \(S_{l}^{\text{ms}} \left( t \right)\) are poloidal scalar functions for cosine and sine terms, respectively. One of data to be assimilated is \(S_{l}^{m} \left( t \right)\) which denotes \(S_{l}^{\text{mc}} \left( t \right)\) and/or \(S_{l}^{\text{ms}} \left( t \right)\), and the truncation of spherical harmonics is \(L_{\text{MCM}} = 14\). In the assimilation procedure, however, we use only coefficients with degree up to 13 assigning very small weight to coefficients for \(l = 14\). The magnetic data are constructed in the form of
$$d_{{S_{l}^{m} }} \left( t \right) = \frac{{S_{l}^{m} \left( t \right)}}{{{S_{1}^{0}}_{\text{ref}} }},$$
(21)
where \({S_{1}^{0}}_{\text{ref}}\) is a certain reference value for \(S_{1}^{0}\). For the observation, we use \(S_{1}^{0}\) at the start time of assimilation window as \({S_{1}^{0}}_{\text{ref}}\) throughout this study. For the numerical geodynamo, we substitute a typical \(S_{1}^{0}\) value of 0.3 into \({S_{1}^{0}}_{\text{ref}}\) in Eq. (21) to generate vectors comparable with the magnetic data vectors. We prepare the data vector by a 0.25year interval. From the outputs of the numerical geodynamo, the poloidal scalar potentials are normalized according to Eq. (21) after the conversion of the dimensionless time to the actual time.
Data 2: core surface flow
We also include the core fluid velocity field slightly below the CMB in our data assimilation. In the present numerical simulation of geodynamo, the noslip condition for the velocity field is imposed at the CMB (Takahashi 2012). It, therefore, is pertinent to adopt core surface flows estimated with the effect of viscous boundary layer at the CMB (Fig. 4a). Hence, we employ a method of Matsushima (2015). Before recalling the method, we define the velocity field, \(\varvec{V}\), and the magnetic field, \(\varvec{B}\), in the outer core. \(\varvec{V}\) can be expressed in terms of the poloidal and toroidal scalar functions as
$$\varvec{V}\left( {r,\theta ,\phi ,t} \right) = \nabla \times \nabla \times \left( {\hat{\varvec{r}}U} \right) + \nabla \times \left( {\hat{\varvec{r}}W} \right),$$
(22)
$$U\left( {r, \theta ,\phi ,t} \right) = \mathop \sum \limits_{l = 1}^{L} \mathop \sum \limits_{m = 0}^{l} \left\{ {U_{l}^{\text{mc}} \left( {r,t} \right)\cos m\phi + U_{l}^{\text{ms}} \left( {r,t} \right)\sin m\phi } \right\}P_{l}^{m} (\cos \theta ),$$
(23)
$$W\left( {r, \theta ,\phi ,t} \right) = \mathop \sum \limits_{l = 1}^{L} \mathop \sum \limits_{m = 0}^{l} \left\{ {W_{l}^{mc} \left( {r,t} \right)\cos m\phi + W_{l}^{ms} \left( {r,t} \right)\sin m\phi } \right\}P_{l}^{m} (\cos \theta ),$$
(24)
where \(\left( {r,\theta ,\phi } \right)\) is the spherical coordinates, \(\hat{\varvec{r}}\) is a radial unit vector, and \(P_{l}^{m}\) is Schmidt quasinormalized associated Legendre function with degree \(l\) and order \(m\). Near the core surface, the radial component of the velocity field, \(V_{r}\), is likely to be much smaller than the horizontal component, \(\varvec{V}_{h}\). Therefore, in core surface flow modeling, only the horizontal component, \(\varvec{V}_{h} = \hat{\varvec{\theta }}V_{\theta } + \hat{\phi }V_{\phi }\), is computed:
$$V_{\theta } \left( {r,\theta ,\phi ,t} \right) = \mathop \sum \limits_{l = 1}^{L} \mathop \sum \limits_{m = 0}^{l} \left[ {\left\{ {\frac{1}{r}\frac{{\partial U_{l}^{mc} }}{\partial r}\cos m\phi + \frac{1}{r}\frac{{\partial U_{l}^{ms} }}{\partial r}\sin m\phi } \right\}\frac{{dP_{l}^{m} }}{d\theta } + m\left\{ {  \frac{{W_{l}^{mc} }}{r}\sin m\phi + \frac{{W_{l}^{ms} }}{r}\cos m\phi } \right\}\frac{{P_{l}^{m} }}{\sin \theta }} \right],$$
(25)
$$V_{\phi } \left( {r,\theta ,\phi ,t} \right) = \mathop \sum \limits_{l = 1}^{L} \mathop \sum \limits_{m = 0}^{l} \left[ {m\left\{ {  \frac{1}{r}\frac{{\partial U_{l}^{mc} }}{\partial r}\sin m\phi + \frac{1}{r}\frac{{\partial U_{l}^{ms} }}{\partial r}\cos m\phi } \right\}\frac{{P_{l}^{m} }}{\sin \theta }  \left\{ {\frac{{W_{l}^{mc} }}{r}\cos m\phi + \frac{{W_{l}^{ms} }}{r}\sin m\phi } \right\}\frac{{dP_{l}^{m} }}{d\theta }} \right],$$
(26)
where \(\hat{\varvec{\theta }}\) and \(\hat{\phi }\) are unit vectors in the \(\theta\) and \(\phi\)directions, respectively. Hence, Matsushima (2015) computed \(\bar{U}_{l}^{m} = \partial U_{l}^{m} /r\partial r\) and \(\bar{W}_{l}^{m} = W_{l}^{m} /r\). A boundary condition, \(U_{l}^{m} \left( {r_{o} } \right) = 0\), leads to
$$\frac{1}{{r_{o}  \xi_{2}^{*} }}\frac{{\partial U_{l}^{m} }}{\partial r} \approx \frac{1}{{r_{o}  \xi_{2}^{*} }}\frac{{U_{l}^{m} \left( {r_{o} } \right)  U_{l}^{m} \left( {r_{o}  \xi_{2}^{*} } \right)}}{{\xi_{2}^{*} }} =  \frac{1}{{r_{o}  \xi_{2}^{*} }}\frac{{U_{l}^{m} \left( {r_{o}  \xi_{2}^{*} } \right)}}{{\xi_{2}^{*} }},$$
(27)
where \(r_{o}\) is the outer radius of the rotating spherical shell in numerical MHD dynamo model, and \(\xi_{2}^{*}\) is a depth from \(r = r_{o}\). Therefore, taking into account the length scale \(r_{oc}  r_{ic}\) and the velocity scale \(\left( {r_{oc}  r_{ic} } \right)/\tau_{\nu }\) (see Appendix C), we have
$$U_{l}^{m} \left( {r_{o}  \xi_{2}^{*} } \right) =  \frac{{r_{o}  \xi_{2}^{*} }}{{r_{oc}  r_{ic} }}\frac{{\xi_{2}^{*} }}{{r_{oc}  r_{ic} }}\frac{{\bar{U}_{l}^{m} \left( {r_{oc}  \xi_{2} } \right)}}{{\left( {r_{oc}  r_{ic} } \right)/\tau_{\nu } }},$$
(28)
and
$$W_{l}^{m} \left( {r_{o}  \xi_{2}^{*} } \right) = \frac{{r_{o}  \xi_{2}^{*} }}{{r_{oc}  r_{ic} }}\frac{{\bar{W}_{l}^{m} \left( {r_{oc}  \xi_{2} } \right)}}{{\left( {r_{oc}  r_{ic} } \right)/\tau_{\nu } }}.$$
(29)
In the method of Matsushima (2015),

(0)
At the CMB \(\left( {r = r_{oc} } \right)\), geomagnetic secular variations are caused only by magnetic diffusion due to the noslip boundary condition;
$$\frac{{\partial B_{r} }}{\partial t} = \frac{\eta }{r}\nabla^{2} \left( {rB_{r} } \right),$$
(30)
where \(\eta\) is the magnetic diffusivity.

(1)
Inside the viscous boundary layer \(\left( {r = r_{b1} = r_{oc}  \xi_{1} } \right)\), the viscous force plays an important role in force balance as
$$\hat{\varvec{r}} \cdot \nabla \times \left( {  2{\varvec{\Omega}} \times \varvec{V} + \rho^{  1} \varvec{J} \times \varvec{B} + \nu \nabla^{2} \varvec{V}} \right) = 0,$$
(31)
where \({\varvec{\Omega}}\) is the rotation rate of the mantle, \(\rho\) is the density of the core fluid, \(\varvec{J}\) is the electric current density, and \(\nu\) is the kinematic viscosity. The magnetic diffusion, motional induction, and advection are assumed to contribute to temporal variations of the magnetic field;
$$\frac{{\partial B_{r} }}{\partial t} = \frac{\eta }{r}\nabla^{2} \left( {rB_{r} } \right)  \left( {\varvec{V} \cdot \nabla } \right)B_{r} + \left( {\varvec{B} \cdot \nabla } \right)V_{r} ,$$
(32)
where core fluid is assumed to be incompressible, \(\nabla \cdot \varvec{V} = 0\).

(2)
Below the boundary layer \(\left( {r = r_{b2} = r_{oc}  \xi_{2} } \right)\), core flow is assumed to be tangentially magnetostrophic (Matsushima 2020) as
$$\hat{\varvec{r}} \cdot \nabla \times \left( {  2{\varvec{\Omega}} \times \varvec{V} + \rho^{  1} \varvec{J} \times \varvec{B}} \right) = 0.$$
(33)
Temporal variation of the magnetic field is assumed to be caused by the motional induction and advection, and the magnetic diffusion is neglected, as in the frozenflux approximation (e.g., Holme 2015) as
$$\frac{{\partial B_{r} }}{\partial t} =  \left( {\varvec{V} \cdot \nabla } \right)B_{r} + \left( {\varvec{B} \cdot \nabla } \right)V_{r} .$$
(34)
The radial component of the electric current density, \(J_{r}\), is likely to be much smaller than the horizontal component, \(\varvec{J}_{h}\), near the core surface due to the electrically insulating mantle. Following Shimizu (2006), we assume that the electric field does not contribute to the electric current density, and we have
$$\varvec{J}_{h} = \sigma \left( {\varvec{V} \times \varvec{B}} \right)_{h} \approx \sigma B_{r} \varvec{V}_{h} \times \hat{\varvec{r}},$$
(35)
where \(\sigma\) is the core electrical conductivity. Hence, the tangentially magnetostrophic constraint is given as
$$\nabla_{h} \cdot \left( {2\varOmega \cos \theta \bar{V}_{h} + \rho^{  1} \sigma B_{r}^{2} \bar{V}_{h} \times \hat{r}} \right) = 0.$$
(36)
The ratio of the Lorentz to the Coriolis forces is the Elsasser number given by
$$\varLambda = \frac{{\sigma B_{r}^{2} }}{{2\rho \varOmega \left {\cos \theta } \right}}.$$
(37)
We can have a local value of \(\varLambda \approx 0.62\), using actual physical parameters for the outer core, \(\sigma = 1 \times 10^{6} {\text{S m}}^{  1}\), \(\varOmega = 7.29 \times 10^{  5} {\text{rad s}}^{  1}\), \(\rho = 1.1 \times 10^{4} {\text{kg m}}^{  3}\), and \(B_{r} = 1 {\text{mT}}\). In the numerical MHD dynamo model, we obtain \({{\varLambda }}^{ *} = 1.778\) from the initial ensemble members for our data assimilation (see “Numerical experiments” section for details of the ensemble), which is approximately three times larger than \(\varLambda\) for the actual outer core. Such an Elsasser number larger than 1 means relative importance of the Lorentz force in our numerical geodynamo. Hence, we consider that the tangentially magnetostrophic constraint for core surface flow is appropriate. The Elsasser number \(\varLambda^{*} = 1.778\) leads to \(B_{r} \sim 0.1\; {\text{mT}}\) in our numerical geodynamo model, where we adopt \(\sigma = 1.277 \times 10^{5} {\text{S m}}^{  1}\) and \({{\varOmega }} = 4.05 \times 10^{  8} {\text{rad s}}^{  1}\) derived from \(\tau_{\nu } = 13049\) years (see Appendix C). This shows that the magnetic field in our numerical geodynamo model is rather weak compared to that of the actual Earth, possibly due to the small \(\varOmega\). This is the reason why we use a relative magnetic field \(d_{{S_{l}^{m} }} \left( t \right)\) defined in (21). To overcome this problem, we should perform more realistic numerical simulation of geodynamo for a much smaller Ekman number corresponding to much larger \(\varOmega\), although this remains a problem to be solved in the future.
We compare the poloidal and toroidal scalar potentials, \(U_{l}^{m}\) and \(W_{l}^{m}\), for the core flow field with those for the numerical flow field, up to \(l = 14\) (the same as the truncation degree of the MCM3 model, and also practically used up to degree 13 in the assimilation by controlling the weight). The core flow data, \(U_{l}^{m}\) and \(W_{l}^{m}\), are constructed from the main field and SV at 0.25year intervals in the MCM3 model. To make the dimensionless numerical velocity output comparable with the core flow data, we scale the simulated velocity using the relation of \(\tau_{\text{SV}} /\tau_{\text{SV}}^{*} = 13,049 {\text{years}}\) and the kinematic viscosity, which is derived from Ekman number and magnetic Prandtl number adopted in the numerical dynamo. See “Appendix C” for more details of scaling of the dimensionless velocity field. The left column of Fig. 4c shows examples of core flows at \(r_{b2} = 3428.5 {\text{km}}\), which is \(\xi_{2} = 56.5 {\text{km}}\) below the Earth’s core radius of \(r_{oc} = 3485.0 {\text{km}}\), corresponding to a numerical grid point in the radial direction, \(r_{89}^{*}\), for the epochs of 2007.0, 2010.0 and 2013.0 calculated from the MCM3 model.
Implementation of assimilation
We implement the data assimilation for the following data vectors:
$$\varvec{y}_{k} = \left[ {\begin{array}{*{20}c} {\varvec{d}_{S} \left( {t_{k} } \right)} \\ {\varvec{d}_{U} \left( {t_{k} } \right)} \\ {\varvec{d}_{W} \left( {t_{k} } \right)} \\ \end{array} } \right],$$
(38)
where \(\varvec{d}_{S}\) denotes the magnetic data vector composed of the scaled poloidal component specified in Eq. (21), while \(\varvec{d}_{U}\) and \(\varvec{d}_{W}\) represent the velocity data vectors. Subscripts U and \(W\) denote the poloidal and toroidal scalars of the velocity field, respectively. In this study, \(\varvec{d}_{U}\) and \(\varvec{d}_{W}\) consist of \(U_{l}^{m}\) and \(W_{l}^{m}\) at \(r_{b2} = 3428.5\) km, respectively. We consider the observation error covariance matrix, \({\mathbf{R}}_{k}\) in Eqs. (2) and (12), as having a simple timeindependent form:
$${\mathbf{R}}_{k} = {\mathbf{R}} = \left[ {\begin{array}{*{20}c} {\alpha_{S}^{2} {\mathbf{R}}_{S} } & 0 & 0 \\ 0 & {\alpha_{UW}^{2} {\mathbf{R}}_{U} } & 0 \\ 0 & 0 & {\alpha_{UW}^{2} {\mathbf{R}}_{W} } \\ \end{array} } \right],$$
(39)
where \({\mathbf{R}}_{S}\), \({\mathbf{R}}_{U}\), and \({\mathbf{R}}_{W}\) are the diagonal covariance matrices with each degree dependence. Weights for data sets are controlled by the two scalar coefficients of \(\alpha_{S}\) and \(\alpha_{UW}\). We simply adopt a timeindependent form of \({\mathbf{R}}_{k} = {\mathbf{R}}\), since the MCM3 model is a qualitycontrolled smooth model. We prepare \({\mathbf{R}}_{S}\) by the expression \({\left\{ {\varvec{R}_{S} } \right\}}_{l} = {{\text{C}}_{\text{S}}}_{{l}^{m}} \left[ {{\sigma_{G}^{2}}_{l = 1} \left( {l + 1} \right)^{  1} } \right]\), where \(\left\{ {{\mathbf{R}}_{S} } \right\}_{l}\) is the diagonal element of \({\mathbf{R}}_{S}\) for degree \(l\), \(\left( {l + 1} \right)^{  1}\) is the degree dependence of the variance of Gauss coefficients based on the theory by Lowes (1975), \({\sigma_{G}^{2}}_{l = 1}\) is the averaged actual variance of Gauss coefficient at degree \(l = 1\) from the MCM3 model, and \({{\text{C}}_{\text{S}}}_{l}^{m} \left[ \cdot \right]\) is a function for conversion of the covariance of Gauss coefficients to that of the poloidal scalar potentials at the CMB for degree \(l\) and order \(m\). Similarly, we prescribe \({\mathbf{R}}_{U}\) and \({\mathbf{R}}_{W}\) by the expressions \(\left\{ {{\mathbf{R}}_{U} } \right\}_{l} = {C_{U}}_{l}^{m} \left[ {{\sigma_{U}^{2}}_{l = 1} \left( {2l + 1} \right)/\left( {l^{2} + l} \right)} \right]\) and \(\left\{ {{\mathbf{R}}_{W} } \right\}_{l} = {C_{W}}_{l}^{m} \left[ {{\sigma_{W}^{2}}_{l = 1} \left( {2l + 1} \right)/\left( {l^{2} + l} \right)} \right]\), respectively, where \(\left\{ {{\mathbf{R}}_{U} } \right\}_{l}\) and \(\left\{ {{\mathbf{R}}_{W} } \right\}_{l}\) are expressions similar to \(\left\{ {{\mathbf{R}}_{S} } \right\}_{l}\), \(\left( {2l + 1} \right)/\left( {l^{2} + l} \right)\) is the degree dependence of the variance of Schmidt quasinormalized poloidal and toroidal scalar potentials of the core surface flow based on discussion in Holme (2015). \({\sigma_{U}^{2}}_{l = 1}\) and \({\sigma_{W}^{2}}_{l = 1}\) are the averaged actual variances of the Schmidt quasinormalized poloidal and toroidal scalar potentials of the core surface flow at degree \(l = 1\), respectively. They are calculated from the observation error of the Gauss coefficients in the MCM3 model. \({{\text{C}}_{\text{U}}}_{l}^{m} \left[ \cdot \right]\) and \({{\text{C}}_{\text{W}}}_{l}^{m} \left[ \cdot \right]\) are functions converting the covariances of the poloidal and toroidal scalar potentials for the velocity field at the CMB to covariances of those dimensionless at the radial grid \(r_{89}^{*}\) in our numerical dynamo model with the velocity scaling by \(\left( {r_{\text{oc}}  r_{\text{ic}} } \right)/\tau_{\nu } = 0.1736 {\text{km year}}^{  1}\) (see Appendix C), respectively. \({{\text{C}}_{\text{S}}}_{l}^{m} \left[ \cdot \right]\), \({{\text{C}}_{U}}_{l}^{m} \left[ \cdot \right]\), and \({{\text{C}}_{W}}_{l}^{m} \left[ \cdot \right]\) also include the conversion from the Schmidt quasinormalization to the normalization adopted in our MHD dynamo model (Takahashi 2012, 2014), since we prepare \(\varvec{y}_{k}\) (Eq. 38) directly comparable to the outputs of our numerical geodynamo model.