As discussed by , in an idealized circular and axisymmetric flow, the nonlinear term (ucg⋅∇)ucg simplifies to the centrifugal acceleration -Vgr2R with Vgr the azimuthal component of the velocity and R the radius of curvature (which coincides with the radial distance to the vortex center in strictly axisymmetric cases). Under these assumptions, Eq. () becomes: 4Vgr2R+fVgr-fVg=0 where Vg is the azimuthal geostrophic velocity, positive for cyclonic eddies and negative for anticyclonic ones. Solving this quadratic equation yields the physically relevant branch of the gradient wind solution: 5Vgr=2Vg1+1+4Vg/(fR)

Equation () provides useful intuition about the conditions under which the cyclogeostrophic balance admits a physical solution. For cyclonic eddies (Vg>0), the term under the square root is always positive, and a real solution exists. In contrast, for anticyclonic eddies (Vg<0) this term becomes negative when |Vg/(fR)|>0.25, in which case no real solution exists. This situation corresponds to the occurrence of inertial instability, indicating a breakdown of the balance assumptions .

While the analytic gradient wind solution is useful for understanding the existence and physical limits of cyclogeostrophic balance, it is restricted to idealized axisymmetric flows. In realistic oceanic conditions – where the flow is neither perfectly circular nor steady – numerical approaches are instead required to solve Eq. (). A widely used strategy is the fixed-point method originally proposed by , which we describe below.

Taking the vector product of k with Eq. () and substituting the geostrophic velocity ug from Eq. (), we obtain: 6ucg-kf∧(ucg⋅∇)ucg=ug Then the iterations proposed by to get the cyclogeostrophic velocity are initialized with ucg(0)=ug and implement as: 7ucg(n+1)=ug+kf∧ucg(n)⋅∇ucg(n)

This approach has traditionally been referred to as the “iterative” method. However, this terminology can be misleading, as other numerical procedures – including our minimization-based formulation (see Sect. ) – are also iterative while relying on fundamentally different update mechanisms. For clarity, we therefore adopt the more precise term “fixed-point” method to describe Eq. ().

As initially mentioned by , these iterations do not always converge; an ad hoc and imperfect stopping strategy is generally implemented to avoid their numerical divergence. A typical case of numerical divergence is when the cyclogeostrophic equation has no solution, as previously discussed in Sect. . From a fixed-point perspective, divergence also occurs whenever the initial guess ucg(0)=ug is not an attracting fixed point of the update map in Eq. (). provide a detailed analysis of the convergence properties of this method in the context of the idealized gradient wind balance.

To mitigate these difficulties, stops the iterations at any grid point i when the residual |ucg,i(n+1)-ucg,i(n)| falls below 0.01 or starts to increase. implements this with two additional ingredients: the initial geostrophic velocity field is projected, with a cubic interpolation, on a grid 3 times finer than the initial one. This is to “improve the computation of the velocity derivatives” in Eq. (). The second modification is in the calculation of the residual norm for each grid point, which includes now the 8 neighboring grid points.

Nonetheless, our own experience indicates that (i) the fixed-point method can fail to converge to the cyclogeostrophic solution (Fig. ) and (ii) the local iteration-stopping strategy can produce noisy or unrealistic velocity fields (Fig. ). These limitations motivate the need for an alternative approach.

We recast the cyclogeostrophic inversion problem in a minimization form, by searching for the velocity field ucg that minimizes the following loss function: 8J(ucg)=∫Ω[Δcg(ucg(x))]2dx where Ω is the 2D spatial domain and Δcg denotes the cyclogeostrophic imbalance function computed locally at each point x=(x,y) in the discretized domain: 9Δcg(ucg)=ucg-kf∧(ucg⋅∇)ucg-ug where ‖⋅‖ is the ℓ2 norm for a 2-component velocity vector: ‖u(x)‖=u(x)2+v(x)2, using the standard notation for the zonal and meridional velocities. In Eq. (), we make it explicit that the loss function is the domain integral of a locally computed norm, although it could equivalently be expressed using an L2 norm over the domain. This explicit form is useful for the discussion in Sect. .

The minimization of Eq. () is performed using gradient descent, i.e. by taking small steps in the direction opposite to the gradient of J: 10ucg(n+1)=ucg(n)-γ∇Jucg(n) where the hyperparameter γ controls the step size. The gradient ∇Jucg(n) is computed automatically using JAX's reverse-mode automatic differentiation: JAX records the computation of J as a sequence of elementary operations with known derivatives and applies the chain rule to construct the corresponding gradient function.

The minimization-based formulation is expected to solve the numerical divergence problem of the fixed-point method; it also provides a measure of the deviation from the cyclogeostrophic solution (when it exists). Where the cyclogeostrophic imbalance Δcg reaches 0, the solution is the cyclogeostrophic velocity. In regions where no exact cyclogeostrophic solution exists, the minimization-based approach – because its update strategy relies on the gradient of a globally evaluated loss involving spatial derivatives – favors a smoother and more coherent estimate of the velocity field, despite the absence of any explicit regularization term. In this sense, it is expected to avoid the unrealistic features that the fixed-point method can generate, since the latter's point-wise update and stopping criterion tend to amplify noise. Interestingly, the cyclogeostrophic imbalance is a straightforward indication of where a cyclogeostrophic velocity can be found, and where it cannot. It is not possible to determine the physical nature of the velocity solution when Δcg does not reach 0. But it is still possible to quantify a deviation from the cyclogeostrophic equilibrium.

Our cyclogeostrophic inversion library, jaxparrow , is implemented with JAX , a Python library developed by Google to perform two main operations on Python functions: acceleration and automatic differentiation. jaxparrow leverages both features. The automatic differentiation capability directly provides the gradient of J, which can be used for gradient-based minimization methods. For the minimization itself, jaxparrow implements Optax , a gradient processing and optimization library specifically developed for JAX.

jaxparrow handles gridded data, making it well-suited for estimating cyclogeostrophic currents from SSH derived from models, Level-4 products, and also 2D Level-3 products. While most altimetry products use Arakawa A-grids, where all quantities are evaluated at the grid center (T point), jaxparrow computes partial derivatives using finite differences on Arakawa C-grids, where the SSH is defined at the grid center, the velocity components at the grid faces, and the vorticity at the grid vertices. As a result, variables must be carefully interpolated when performing numerical computations. Specifically, for the kinematic diagnostics described in Sect. , the velocity components u and v are first interpolated to the T points prior to computing the velocity magnitude, whereas vorticity is calculated directly on the C-grid and then interpolated back to the T points.

To support further evaluation of our minimization-based method and facilitate the integration of cyclogeostrophic currents into a global operational product, our library is easily installable via pip, with its code publicly available on GitHub.

Surface currents derived from SSH using the geostrophic approximation and both minimization-based and fixed-point cyclogeostrophic inversion methods are here qualitatively analyzed (i) with the cyclogeostrophic imbalance from Eq. (), (ii) by observing the velocity and relative vorticity fields, and (iii) through a comparison of EKE.

The measure of the deviation from cyclogeostrophy shows that (i) geostrophy can be a crude approximation of cyclogeostrophy at some locations in space and time and (ii) the minimization-based inversion method is more accurate than the fixed-point method to compute a cyclogeostrophic velocity field. These conclusions are drawn from the examination of Fig.  which presents the time-aggregated deviation from the cyclogeostrophic balance of 3 velocity fields derived from NeurOST SSH, namely the geostrophic field (top) and the cyclogeostrophic solutions from the minimization-based method (bottom left) and the fixed-point method (bottom right). The geostrophic field exhibits large deviations from cyclogeostrophy, with deviations larger than 0.3 ms-1 at nearly 5 % of grid points, hinting that the advective term should not be neglected. The solution of the fixed-point method deviates even further, with differences exceeding 0.35 ms-1 at more than 5 % of points. In contrast, the minimization-based method limits deviations above 0.03 ms-1 to fewer than 5 % of grid points. This suggests that the fixed-point method is less reliable in converging toward a cyclogeostrophic solution, particularly in the western boundary currents, where the minimization-based method shows that a cyclogeostrophic solution exists.

Our implementation of the proposed minimization-based method enables physically consistent estimation of cyclogeostrophic currents on a global scale, including in highly dynamic regions where cyclogeostrophic corrections substantially impact jets and eddies, and where the fixed-point method yields unrealistic physical fields. Figure  presents a global snapshot of the norm of cyclogeostrophic currents derived from NeurOST SSH, along with an enlargement of the Gulf Stream region where relative vorticity and differences compared to geostrophy are also displayed. In the northern meanders of the Gulf Stream jet, cyclogeostrophic corrections are positive and can reach up to +0.2 ms-1, while in the southern meanders they are negative, down to -0.2 ms-1. Similarly, anticyclonic and cyclonic eddies exhibit respective cyclogeostrophic contributions of approximately +0.2 and -0.2 ms-1, corresponding to relative increases of 10 %–50 % in the anticyclonic case and relative decreases of 10 %–50 % in the cyclonic case. Finally, while the minimization-based method allows for the reconstruction of a smooth and physically coherent relative vorticity field, the fixed-point method introduces artifacts in the most dynamic parts of the jet and eddies. As discussed in Sect.  and , the differences are likely linked to the mathematical distinctions between the two approaches.

The EKE computed from the geostrophic and the minimization-based cyclogeostrophic velocities anomalies exhibit differences up to 20 %, essentially at low and middle latitudes. This is shown in Fig. , which presents the relative difference in EKE between cyclogeostrophy and geostrophy, averaged over the whole time period. Positive differences are particularly pronounced near the equatorial band. Regions with intense dynamics such as the western boundary currents are characterized by elongated dipole structures with both positive and negative differences. These reflect a current intensification in anticyclonic eddies detaching poleward and a damping of the current in cyclonic eddies detaching equatorward, in agreement with the magnitude and sign of cyclogeostrophic corrections observed in Fig. . All these observations are consistent with who performed a similar analysis with 1/4° DUACS maps and the historical fixed-point method for cyclogeostrophy over the period 1993–2018. Our results suggest once more that geostrophy can be a crude approximation leading to errors up to 20 % in EKE.

Normalized relative vorticity fields obtained from geostrophy or cyclogeostrophy surface current reconstruction are compared to reference fields derived from eNATL60 total surface currents. To demonstrate the feasibility of performing the cyclogeostrophic inversion in the SWOT swath using our package jaxparrow, the original eNATL60 fields are first interpolated onto the 2 km grid of the SWOT swath before reconstruction.

While normalized relative vorticity fields derived from the cyclogeostrophic balance are generally in better agreement with eNATL60 reference – especially near the cores of persistent anticyclonic eddies – the fixed-point method more frequently exhibits RMSE increases compared to geostrophy, particularly along eddies boundaries where the minimization-based approach continues to outperform geostrophy. This is illustrated in Fig. , which shows a snapshot of the reference normalized relative vorticity field (top-left), the RMSE of the minimization-based reconstruction computed over one month (top-right), and the relative change in RMSE with respect to geostrophy for both the fixed-point method (bottom-left) and the minimization-based method (bottom-right). Figure  also displays the normalized relative vorticity field for the three inversion methods, together with the corresponding surface current velocity fields. Several anticyclonic submesoscale (≤50 km) eddies can be identified in the reference normalized relative vorticity field shown in panel a. Three of these eddies – located North and South of Ibiza, and South of Menorca – are persistent over the full month of the evaluation period (not shown). From panel b, we observe that the RMSE of the minimization-based method exceeds 0.1 only in coastal areas, where the cyclogeostrophic assumption likely breaks down. The relative difference in RMSE with respect to geostrophy in panel d generally indicates a better reconstruction when using the minimization-based approach, particularly in the regions of the three persistent eddies where improvements locally reach 100 %. Conversely, panel c shows that the fixed-point method provides slightly weaker improvements and, more notably, more frequent degradations, with pixel-like patterns similar to the artifacts seen in Fig.  and also noticeable in Fig. .

Consistently with and , these results suggest that cyclogeostrophy should be employed when analyzing high-resolution 2D SSH fields. They also indicate that the minimization-based method may provide more reliable reconstructions than the fixed-based approach in such contexts.

Reconstructed cyclogeostrophic and geostrophic currents are evaluated against drifter-derived velocities using (i) the inversion error defined in Eq. () and binned within 1° latitude ×1° longitude boxes at the global scale, and (ii) a logistic regression modeling the probability for cyclogeostrophy to outperform geostrophy as a function of the cyclostrophic correction magnitude.

When using NeurOST SSH, minimization-based cyclogeostrophic corrections improve surface current estimates, particularly in energetic regions such as western boundary currents, where reconstruction errors are highest. This is illustrated in Figs.  and . Figure  presents global maps of the cyclogeostrophic RMSE obtained from NeurOST SSH (top-left panel) and of the comparison between cyclogeostrophic and geostrophic inversion methods for NeurOST (top-right). Cyclogeostrophic RMSE remains below 0.1 ms-1 across most of the ocean but increases to 0.2–0.5 ms-1 in energetic currents. In these regions, NeurOST-based cyclogeostrophy clearly reduces error, with improvements of up to 10 % in the Gulf Stream and over 20 % in the Kuroshio (see the insets in the top-right panel of Fig. 5, which highlight the error reductions in these western boundary currents). Figure  further illustrates this, showing the probability that cyclogeostrophy outperforms geostrophy as a function of the cyclostrophic correction magnitude. The solid lines correspond to the logistic regression fit, and the shaded envelopes indicate the 95 % confidence bands. We note that these confidence bands are estimated from the whole population of inversion errors, that is why the binned empirical mean probabilities (dots) – which are computed from smaller subsets of data as the cyclostrophic corrections increases – fall outside the bands. Focusing on NeurOST-derived currents (blue), we find that cyclogeostrophy is, on average, consistently a better estimate than geostrophy, and that this probability increases with the magnitude of the cyclostrophic correction, up to 70 % for cyclostrophic corrections of 0.45 ms-1.

In contrast, cyclogeostrophic corrections can degrade performances when applied to DUACS SSH. This is again illustrated in Figs.  and . The bottom-left panel of Fig.  compares cyclogeostrophic and geostrophic inversion methods based on DUACS SSH. Unlike results obtained with NeurOST, regions such as the western boundary currents show a degradation in performance of around 10 % when cyclogeostrophic corrections are applied (see the insets in the bottom-left panel of Fig. 5, which highlight the increased error in these regions). Similarly, the orange line in Fig.  shows the logistic regression fit for improving the reconstruction when using cyclogeostrophy rather than geostrophy for DUACS-based surface currents. Cyclogeostrophy performs worse more often, on average, than geostrophy for cyclostrophic corrections smaller than 0.45 ms-1. These discrepancies could stem from differences in the effective resolution of the SSH products: DUACS may insufficiently capture fine-scale structures, deteriorating the accuracy of cyclogeostrophic corrections in energetic regions.

Importantly, the combination of higher effective resolution SSH fields and cyclogeostrophic inversion yields substantial benefits over the current operational standard. As shown in Fig.  (bottom-right panel), applying minimization-based cyclogeostrophy to NeurOST SSH reduces reconstruction error by 5 %–20 % at mid-latitudes relative to DUACS geostrophy.

These results suggest that cyclogeostrophic corrections will become increasingly relevant as SSH products achieve higher effective resolution – consistent with the findings from – and could significantly benefit future operational surface current products.