A Scale‐Dependent Analysis of the Barotropic Vorticity Budget in a Global Ocean Simulation

The climatological mean barotropic vorticity budget is analyzed to investigate the relative importance of surface wind stress, topography, planetary vorticity advection, and nonlinear advection in dynamical balances in a global ocean simulation. In addition to a pronounced regional variability in vorticity balances, the relative magnitudes of vorticity budget terms strongly depend on the length‐scale of interest. To carry out a length‐scale dependent vorticity analysis in different ocean basins, vorticity budget terms are spatially coarse‐grained. At length‐scales greater than 1,000 km, the dynamics closely follow the Topographic‐Sverdrup balance in which bottom pressure torque, surface wind stress curl and planetary vorticity advection terms are in balance. In contrast, when including all length‐scales resolved by the model, bottom pressure torque and nonlinear advection terms dominate the vorticity budget (Topographic‐Nonlinear balance), which suggests a prominent role of oceanic eddies, which are of O(10–100) $\mathcal{O}(10\mbox{--}100)$ km in size, and the associated bottom pressure anomalies in local vorticity balances at length‐scales smaller than 1,000 km. Overall, there is a transition from the Topographic‐Nonlinear regime at scales smaller than 1,000 km to the Topographic‐Sverdrup regime at length‐scales greater than 1,000 km. These dynamical balances hold across all ocean basins; however, interpretations of the dominant vorticity balances depend on the level of spatial filtering or the effective model resolution. On the other hand, the contribution of bottom and lateral friction terms in the barotropic vorticity budget remains small and is significant only near sea‐land boundaries, where bottom stress and horizontal viscous friction generally peak.


Introduction
Vorticity budget analyses are quite effective for understanding how surface winds drive ocean motions at different length scales.In particular, the classical Stommel model of the wind-driven gyre has provided significant insight into the linear, steady state balance of ocean gyres driven by surface wind stress (Stommel, 1948), (1) Equation 1 shows that, in the absence of bottom stress τ b , the vertical component of the surface wind stress curl, ẑ ⋅ (∇ ∧ τ s ), balances a meridional flow (V is the vertically integrated meridional velocity) through the β-effect (β is the meridional gradient of the planetary vorticity), which is commonly known as "Sverdrup balance" (Sverdrup, 1947).Also, the mass conservation condition requires a return meridional flow in the zonally integrated vorticity balance, which appears to be controlled by bottom friction stress, ẑ ⋅ (∇ ∧ τ b ).The Stommel model effectively explained the east-west asymmetry due to nonzero β and flow intensification at the western boundary in the gyre circulation.In a slight modification, Munk (1950) argued that the ocean flow does not reach the ocean bottom so that bottom friction stress can be replaced with a lateral friction term acting mainly along the western boundary; thus, permitting a return flow along the western boundary.
The Stommel and Munk models apply to a flat bottom ocean or baroclinic flow with zero bottom velocity since neither model accounts for bathymetry.If we take the curl of depth-integrated momentum equations to derive a linear vorticity equation in the presence of a variable topography at z = H(x, y), the resulting vorticity equation has an additional term known as bottom pressure torque (Holland, 1973;Hughes & De Cuevas, 2001), (2) A nonzero bottom pressure torque, J p b , H) = ẑ ⋅ ∇p b ∧ ∇H) , arises due to varying bottom pressure along isobath contours, and the variations in bottom pressure, p b , exert a nonzero torque on fluid lying over a variable topography (Jackson et al., 2006).In essence, Equation 2 implies that the return flow along the western boundary can be balanced by bottom pressure torque, and western boundary currents can be perceived as being largely inviscid because friction is not required to explain a closed gyre circulation (Hughes, 2000;Hughes & De Cuevas, 2001).In addition, potential-vorticity-conserving gyre circulations can arise in regions with closed barotropic potential vorticity isolines, without any need for western intensification (de Miranda et al., 1999;Kiss, 2004;Welander, 1968).In fact, Schoonover et al. (2016) carried out vorticity budget analysis in realistic simulations from three different ocean models and found that bottom pressure torque controls the Gulf Stream flow magnitude along the western boundary; thus, the Gulf Stream is indeed largely inviscid (also see Gula et al., 2015;Le Bras et al., 2019).The three-way balance among ρ o βV (meridional advection of planetary vorticity), bottom pressure torque, and surface wind stress curl is called "Topographic-Sverdrup balance" (Holland, 1967).Notably, from the perspective of energy conservation, friction is ultimately necessary for maintaining an energy equilibrium state in the presence of energy input by wind forcing since bottom pressure torque does not dissipate energy (Jackson et al., 2006).However, in the presence of realistic bottom pressure torques, the role of friction (either bottom or side friction) for establishing basin-scale gyre circulations is no longer fundamental within the vorticity budget framework.
Several works have shown that bottom pressure torque appears as a first-order term in the vorticity budget of the depth-integrated flow and is crucial for understanding the returning boundary flows in gyres (Hughes & De Cuevas, 2001;Le Bras et al., 2019;Lu & Stammer, 2004;Sonnewald et al., 2019;Yeager, 2015).However, there remains significant regional variability in the relative magnitudes of vorticity budget terms.For example, in the North Atlantic Ocean, wind stress curl tends to be more important in controlling the depth-integrated meridional flow in the subtropics (except along the western boundary), whereas bottom pressure torque balances ρ o βV in almost all of the subpolar basin (Le Bras et al., 2019;Sonnewald & Lguensat, 2021;Yeager, 2015).Global analyses from ocean state estimates and in situ observations also show that the Sverdrup balance holds only in the tropics and subtropics (Gray & Riser, 2014;Thomas et al., 2014;Wunsch, 2011).These differences in the interpretation of regional vorticity balances are partly due to the choice of regional boundaries for vorticity budget integration (Sonnewald et al., 2023).For example, bottom pressure torque vanishes when integrated over any area enclosed by an isobath, and the planetary vorticity advection appears to be controlled by wind stress and bottom friction (Kiss, 2004;Stewart et al., 2021).On the other hand, when integrating the vorticity budget over closed streamlines or fixed latitudinal bands, bottom pressure torque appears as the leading-order term (Hughes & De Cuevas, 2001;Stewart et al., 2021).
In addition to the regional variability, spatial resolution in an ocean model affects the interpretation of dominant vorticity balances.In general, Stommel-Munk-type vorticity balances (Equations 1 and 2) apply to large-scale ocean flows (see Section 5.3 in Pedlosky, 1987).Thomas et al. (2014) showed that a linear Sverdrup balance only holds at length scales greater than 5°in ocean models.At relatively small length scales, that is, mesoscales, western boundary currents, and multiple jets, ocean eddies and the associated non-linearities make a notable contribution to the vorticity budget.For example, the nonlinear advection term in the vorticity equation (see Equation 3) can induce narrow and fast western boundary currents in the opposite direction to the wind-driven Sverdrup transport (Fofonoff, 1955).Using an eddy-resolving simulation of the North Atlantic Ocean, Le Corre et al. (2020) showed that bottom pressure torque and curl of nonlinear advection terms appear to be the largest vorticity budget terms.On the other hand, in relatively coarse non-eddy-resolving and eddy-permitting ocean simulations, the nonlinear advection term tends to have a relatively small contribution to the overall vorticity budget (Yeager, 2015), and the meridional flow is mainly controlled by bottom pressure torque and surface wind stress.These differences arise because high resolution models permit the use of lower horizontal viscosity coefficients and can better resolve narrow boundary currents and nonlinear processes than coarseresolution models (Griffies & Hallberg, 2000).Thus, interpretations of vorticity analyses depend on the region of interest, as well as the length scale of interest.
Several model-based vorticity analyses have shown that the relative magnitudes of vorticity budget terms depend on the details of model spatial resolution and associated representation of bathymetry (e.g., Hughes & De Cuevas, 2001;Le Corre et al., 2020;Yeager, 2015).However, a quantitative comparison is not feasible because these studies used different ocean models that significantly differ in terms of numerical methods, sub-grid parameterizations, and other features, each of which can affect the magnitudes of the vorticity terms (Styles et al., 2022).The present study investigates the primary balances in the vorticity budget of the depth-integrated flow in an eddy-permitting global ocean simulation and quantifies the impacts of spatial resolution on dynamical balances.In addition to analyzing the regional variability in vorticity budget terms, we examine how the relative magnitudes of these terms change as a function of length scale, which is achieved by employing a coarse-graining technique (Buzzicotti et al., 2023;Storer et al., 2022).In particular, spatial maps of vorticity budget terms are examined at different coarse-graining length-scales to understand the relative contributions of different processes in controlling the magnitude of planetary vorticity advection.
The methodology is described in Section 2, and the results are in Section 3. Conclusions and broader implications of this study are discussed in Section 4. We offer four appendices that detail the methods used to perform a vorticity budget analysis and coarse-grain terms in that budget.Appendix A presents the mathematical expressions for the vorticity of the depth-integrated flow; Appendix B details the budget terms saved online in the Modular Ocean Model version 6 (MOM6) ocean model and how we then compute the vorticity terms offline; and Appendix C discusses the magnitudes of the vorticity budget terms.Finally, Appendix D compares results from the coarse-graining method to the spatial filtering algorithm of Grooms et al. (2021), revealing that the two approaches agree qualitatively.

Theory of Vorticity Budget Analysis
We analyze the vorticity budget based on the depth-integrated Boussinesq-hydrostatic ocean primitive equations.
Several studies have employed this vorticity budget approach to examine the role of surface wind stress, bottom pressure, and ocean eddies in governing the flow dynamics (e.g., Hughes & De Cuevas, 2001;Le Corre et al., 2020;Yeager, 2015), see Waldman and Giordani (2023) for a recent review.The complete vorticity budget of the depth-integrated flow can be written as (see Appendix A for derivation) where β = ∂ y f is the meridional derivative of the Coriolis parameter, V is the vertically integrated meridional velocity, z = η is the ocean free surface height, z = H is ocean bottom, p b is bottom pressure, ∇ = x ∂ x + ŷ ∂ y is the horizontal gradient operator, and ρ o = 1,035 kg m 3 is the Boussinesq reference density.τ s and τ b are surface wind stress and bottom friction stress vectors, respectively.A and B represent the vertically integrated velocity advection and horizontal viscous friction terms.Q m is the downward mass flux on the ocean surface and U t is the vertically integrated velocity tendency term.By assuming a steady state, negligible horizontal friction, Q m = 0, linearity, and a flat bottom ocean, Equation 3 readily reduces to the Stommel model of wind-driven gyre given by Equation 1.
It is important to note that there are other ways to derive a two-dimensional vorticity equation, for example, compute the curl of the depth-averaged velocity equations (Mertz & Wright, 1992), and the curl of the velocity equations at each depth level and then compute the vertical integral or mean.All these formulations are equally valid and can be used depending on the research problem at hand (these variations on vorticity budgets are reviewed in Waldman & Giordani, 2023).In this study, we only use the vorticity budget formulation in Equation 3, which will be referred to as the "barotropic vorticity budget."We discuss our results in the context of previous studies that used the same formulation.

Diagnosing Vorticity Budget Terms in a Global Ocean Simulation
For the vorticity budget analysis, we employ output from the global ocean-sea ice model GFDL-OM4.0,which is constructed by coupling the MOM6 (Adcroft et al., 2019;Griffies et al., 2020) with the Sea Ice Simulator version 2. GFDL-OM4.0configuration uses a tripolar grid and has nominal 1/4°horizontal grid resolution, which permits mesoscale eddies especially in the lower latitudes, and uses a hybrid z*-isopycnal vertical coordinate, which significantly reduces artificial numerical mixing and the associated biases (Adcroft et al., 2019;Tsujino et al., 2020).The bottom topography is represented by linear piecewise fits, the same as that used by other isopycnal layered models.This approach provides an accurate representation of bottom pressure torques in a manner similar to terrain following models.For the present work, GFDL-OM4.0was forced using the JRA55-do.v1.4 reanalysis product (Tsujino et al., 2018) following the Ocean Model Intercomparison Project protocol (Griffies et al., 2016;Tsujino et al., 2020).Surface wind stress is computed relative to the ocean velocity, and stress is computed between the ice-ocean when ice is present, with the ice experiencing the winds rather than the ocean.In the presence of sea-ice, the net stress on the ocean surface includes both wind-to-ocean stress and ice-toocean stress, weighted by the ice fraction.Bottom frictional stress is computed using a quadratic bottom drag with a dimensionless drag coefficient of 0.003 and a constant background "tide" speed of 0.1 m s 1 .For the horizontal friction, the model uses a biharmonic viscosity taken as the maximum of a dynamic biharmonic viscosity with a coefficient of 0.06 (Griffies & Hallberg, 2000) and a static biharmonic viscosity of the form u 4 Δ 3 where Δ is local grid spacing and u 4 = 0.01 m 1 .The free-slip lateral boundary condition is used for the biharmonic friction operator.Further model configuration details are provided in Adcroft et al. (2019).We initialized the model with temperature and salinity estimates based on the World Ocean Atlas, as done in Tsujino et al. (2020).We ran the model for six repeated cycles of the JRA55-do forcing and analyzed the output from the sixth cycle.This approach aims to reduce the long-term drift in the simulations.The time-mean model output for 60 years  is used for the barotropic vorticity budget analysis.
Since vorticity has a higher-order spatial derivative than velocity, the vorticity field can be very noisy due to strong spatial and regional variability, which is especially enhanced at small length scales.The multi-scale spatial structure of vorticity is revealed by applying a spatial smoothing operation.This operation essentially blurs the data, filtering out small-scale features and allowing us to isolate the larger-scale vorticity patterns (see the maps of relative vorticity of the depth-averaged flow in Figure 1).Smoothing rapidly reduces vorticity amplitudes through cancellation of opposing signs, effectively removing those small-scale features and altering the vorticity budget terms accordingly.Therefore, a closed barotropic vorticity budget is crucial for analyzing vorticity balances across different scales.To diagnose the vorticity budget terms in Equation 3, different terms in the depthintegrated primitive velocity equations from the model are saved as diagnostics, and the curl of these diagnostics is computed to obtain the relevant barotropic vorticity budget terms (see Appendix B for details).
Computing the vorticity budget terms directly from the depth-integrals of velocity equation terms reduces numerical errors due to mathematical manipulations and interpolation, and the vorticity budget closes sufficiently for our purposes.
We point to the particularly difficult task of accurately computing bottom pressure torques using the Jacobian operator, J(p b , H), which generally leads to significant numerical errors in regions of large topographic slopes.To minimize these numerical errors, bottom pressure torque can be computed as the residual of all other vorticity budget terms (Le Bras et al., 2019), or we can locally smooth bottom topography to obtain realistic magnitudes in bottom pressure torque (Le Corre et al., 2020).Our preferred method is to compute the curl of depth-integrated pressure gradient terms from the velocity equations.The same approach holds for the rest of the terms in the barotropic vorticity budget.Hence, to be consistent with the model numerical schemes and minimize the numerical errors in offline calculations, we compute vorticity budget terms directly from the depth-integrated momentum budget diagnostics, an approach used in many previous studies (Bell, 1999;Hughes & De Cuevas, 2001;Yeager, 2015).Since we use the time-mean of vorticity budget terms calculated online every model timestep, our vorticity diagnostics include every modeled timescale and no Reynolds stress terms are required to close the vorticity budget (unlike the situation of offline calculation from time-mean prognostic fields).Note that, for calculating bottom pressure torque, we used the method described in Appendix B2 to minimize numerical errors.
As seen in the spatial maps of the time-mean vorticity budget terms (Figures 2a-2d), planetary vorticity advection, bottom pressure torque, the nonlinear advection curl, and the surface wind stress curl dominate the barotropic vorticity budget in terms of the magnitude.However, the vorticity balance tends to be very region dependent, as different terms dominate in different geographical locations (also see Sonnewald & Lguensat, 2021;Sonnewald et al., 2019).For example, bottom friction and lateral friction stress terms are relatively small in magnitude in most locations (Figures 2e-2f); however, these terms have notable contributions in local balances especially near continental boundaries.Similarly, we observe a drastic change in the relative magnitudes of vorticity budget terms and dominant vorticity balances as we vary the coarse-graining length scale.These characteristics of the vorticity budget terms motivate a length-scale-dependent vorticity analysis considered separately in different ocean regions (e.g., see Le Corre et al., 2020;Palóczy et al., 2020).Note that the remainder of the vorticity budget terms, which are associated with surface mass flux and timetendencies (Figures 2g-2i), have a small contribution.Even so, we include them in the analyses to enable a more accurately closed vorticity budget.
Signs of the barotropic vorticity budget terms can rapidly change spatially (e.g., see spatial variations in bottom pressure torque and nonlinear advection term in the Southern Ocean in Figures 2a-2c).Hence, positive and negative signals tend to cancel when integrated over large domains.For example, the global averages of bottom pressure torque and nonlinear advection vanish and the main balance is between surface wind stress and friction  terms.As a result, a domain-averaged vorticity budget cannot pick up fields that have large magnitudes but with spatially alternating signs.Furthermore, the relative magnitudes of domain-averaged vorticity budget terms can be sensitive to the choice of domain boundaries (Sonnewald et al., 2023;Stewart et al., 2021).The resultant domain-averaged vorticity balance cannot represent the true nature of vorticity dynamics and can lead to incomplete or incorrect interpretations.To overcome these issues, we employ a coarse-graining technique to deduce the dominant vorticity budget terms appearing at different length scales (Buzzicotti et al., 2023).Coarsegraining allows us to examine the local and non-local impacts of different processes as a function of length scale, while maintaining the structure of the patterns corresponding to scales at or larger than the chosen coarse-graining scale.In the present work, we focus on the impacts of the choice of length scale on local barotropic vorticity balances.

The Coarse-Graining Method
Coarse-graining can be used to examine the spatial variability in a multi-dimensional field.For any field, F(x), the coarse-graining produces a filtered field, F ℓ (x), that has variability on scales longer than ℓ, with variability on smaller scales preferentially removed (Buzzicotti et al., 2023).F ℓ (x) is computed as where * is the convolution on the sphere (Aluie, 2019) and G ℓ is a normalized filtering kernel, which is a top-hat filter in this study (see Equation 4in Storer et al., 2022), so that ∫ A G ℓ = 1.Relation 4 basically represents a spatial average of F(x) centered at geographical location x.
In practice, the coarse-graining technique can be applied to the entire globe, which has land/sea boundaries, while preserving the fundamental physical properties, such as the global mean of a field and non-divergence of the velocity in a Boussinesq ocean (Aluie, 2019).Coarse-graining commutes with differential operators so that the coarse-grained equations resemble the original equations and the underlying mathematical properties of the system are preserved across different length scales.There is a deep connection between coarse-graining and the spectral content of a function (e.g., a term in the vorticity budget).In fact, coarse-graining in the manner we are using is a generalized form of Fourier analysis and reduces to it with the choice of a (Dirichlet) convolution kernel (Aluie & Eyink, 2009;Zhao & Aluie, 2023).The sequential coarse-graining we are performing is not a mere signal processing operation that is blurring the signal to remove small scales.Coarse-graining is a mathematically well-defined energy-preserving scale-decomposition method that generalizes Fourier analysis to inhomogeneous fields (Aluie, 2013(Aluie, , 2017;;Eyink, 2005).This method allows for resolving scales locally in space and is closely related to the continuous wavelet transform (Daubechies, 1992;Perrier et al., 1995;Sadek & Aluie, 2018).
Coarse-graining has been successfully used for analyzing the kinetic energy spectrum and inter-scale energy transfers in the oceans (Aluie et al., 2018;Rai et al., 2021;Storer et al., 2022Storer et al., , 2023)).Since the vorticity budget term magnitudes tend to peak around continental boundaries (Figure 2), spatial filtering near boundaries requires additional care so that there are no artificial large signals as a result of the spatial filtering.The coarse-graining technique is well suited for the present analysis as it handles gradients around land-sea boundaries appropriately (see details in Buzzicotti et al., 2023).
Following the steps described in Section 2.2, we compute the barotropic vorticity budget diagnostics, which are then coarse-grained by employing the FlowSieve package (Storer & Aluie, 2023).Prior to coarse-graining, vorticity budget diagnostics were re-gridded from the native tripolar grid (Adcroft et al., 2019) to a uniform 0.25°× 0.25°grid using a conservative regridding method because the current implementation of the FlowSieve package only accepts rectangular latitude-longitude grids.Since we only analyze the vertical vorticity component, the barotropic vorticity budget terms are treated as scalar fields for the purpose of coarse-graining.We use the fixed-kernel method, in which land is treated as ocean with a zero value of every vorticity balance term, to conserve global averages of vorticity terms (Buzzicotti et al., 2023).Coarse-grained diagnostics are then analyzed to identify the dominant vorticity balances as a function of coarse-graining scale, ℓ.
Furthermore, we compute the mean of the absolute values, {|F ℓ |}, for all the vorticity budget terms in different ocean regions and analyze their relative magnitudes as a function of coarse-graining scale, where i is a grid cell index within a region and w i is the associated weight, equal to grid cell area on the uniform 0.25°× 0.25°grid.The regional means of absolute values, {|F ℓ |}, are required to investigate the regional variability and length-scale-dependence in vorticity balances.If we instead preserve the signs of vorticity budget terms while calculating domain-averages, the positive and negative signals will offset each other, potentially resulting in incorrect interpretations of the dominant vorticity balances (see Figure 2).Note that {|F ℓ |} magnitudes decline significantly with increasing the coarse-graining scale (see appendix Figure C1).Thus, we analyze the normalized {|F ℓ |} magnitudes to measure the relative importance of different vorticity budget terms, where j corresponds to a vorticity budget term and {|F ℓ |} j (normalized) measures the relative magnitude a vorticity budget term.

Vorticity Budget Analysis as a Function of Length-Scale
Vorticity budget analyses from relatively coarse ocean models have shown that bottom pressure torque plays a prominent role in regional vorticity balances and in guiding western boundary currents (Hughes & De Cuevas, 2001;Lu & Stammer, 2004;Yeager, 2015;Zhang & Vallis, 2007).On the other hand, more recent studies employed mesoscale eddy-resolving ocean models having horizontal grid spacing of 2-10 km, with these studies emphasizing that bottom pressure torque and nonlinear advection are equally important for regional vorticity dynamics (Le Corre et al., 2020;Palóczy et al., 2020).The present study aims to quantify the impacts of resolution on vorticity balances using a single global ocean simulation.Coarse-grained barotropic vorticity budget terms are examined as a function of coarse-graining scale in different ocean basins to assess the impact of spatial smoothing on the magnitudes of all vorticity terms.

Vorticity Budget in the North Atlantic Ocean
At first, we examine the spatial structure of coarse-grained vorticity budget terms in the North Atlantic Ocean, which has been considered in several works (e.g., Le Corre et al., 2020;Schoonover et al., 2016;Yeager, 2015;Zhang & Vallis, 2007).As seen in Figure 3, all vorticity terms, except the wind stress curl, have pronounced spatial variability and peak near continental boundaries and mid-ocean topographic features.
Coarse-graining has a notable impact on the relative contributions of different vorticity terms.For example, when spatial variations larger than 200 km are retained (Figures 3a1-3g1), planetary vorticity advection (βV ), bottom pressure torque and the curl of the nonlinear advection term (∇ ∧ A), dominate in terms of the magnitude (also see Le Corre et al., 2020).Hence, the local meridional flow is controlled by bottom pressure torque and nonlinear advection (henceforth will be referred to as "Topographic-Nonlinear balance").Surface wind stress, bottom friction, and horizontal friction terms also have large magnitudes around land-sea boundaries; however, their contribution to the local vorticity budget is relatively small.The rest of the vorticity budget terms (surface mass flux and time-tendencies) are negligible in comparison.There appears to be a significant cancellation between bottom pressure torque and ∇ ∧ A at mesoscales and submesoscales (smaller than about 500 km), and their sum is roughly in balance with βV.Our results are consistent with Le Corre et al. ( 2020), who found that bottom pressure torque and ∇ ∧ A signals generally are of opposite signs to each other, so that these two terms compensate for each other (also see Gula et al., 2015).
On the other hand, with coarse-graining at scales 1,000 km and larger (Figures 3a3-3g3), the nonlinear advection term almost disappears, and the dominant balance is then among planetary vorticity advection, bottom pressure torque and wind stress curl.This result suggests that vorticity dynamics at large scales are close to the Topographic-Sverdrup balance, which agrees with vorticity budget analyses from relatively coarse ocean models (Lu & Stammer, 2004;Yeager, 2015).The coarse-graining exercise shows that bottom pressure torque is significant at all length scales, whereas ∇ ∧ A contribution to the barotropic vorticity budget is limited to scales smaller than 1,000 km.These results indicate that the model resolution (or the length scale of interest) is a key parameter while examining relative contributions from different vorticity terms, as physical interpretations of these results depend on the length scale.
For a quantitative investigation on the impacts of coarse-graining on vorticity balances, we compute normalized domain-averaged absolute values of the time-mean budget terms (Figure 3h).Consistent with the results discussed above, for coarse-graining with 200 km length scale (or smaller), bottom pressure torque and ∇ ∧ A are the largest in magnitude vorticity terms and represent more than 60% of the magnitudes of vorticity budget terms.βV is the third largest term and explains about 10%-25% of the signals.As the coarse-graining kernel width increases, ∇ ∧ A signals smooth out, and the primary balance is then among βV, bottom pressure torque, and surface wind stress curl.Together, these three terms capture more than 70% of the vorticity budget at length scales greater than 1,000 km.The rest of the contribution to the vorticity balance is from friction terms, ∇ ∧ τ b /ρ o and ∇ ∧ B, which project on all length scales.Overall, these vorticity analyses show a clear transition from the Topographic-Nonlinear balance to the Topographic-Sverdrup balance as we move from small to large lengthscales.The same results hold even if a different spatial filtering algorithm is used (see Figure D1).

Vorticity Budget Within Closed Gyre Contours
To understand the dominant vorticity balances within subtropical and subpolar North Atlantic gyre circulations, we analyze {|F ℓ |} magnitudes within closed gyre contours (Figure 4).To encompass most of the gyre areas (see gyre contours in Figure 3), we analyze the vorticity budget within the 10 Sv contour of the subtropical gyre (with a maximum barotropic streamfunction of 25.1 Sv) and the 10 Sv contour of the subpolar gyre (with a minimum barotropic streamfunction of 36.2Sv).Even within subtropical and subpolar gyres, the vorticity balance is largely among bottom pressure torque, ∇ ∧ A, and βV when all length scales are included.When only spatial features larger than 1,000 km are retained, there is a relatively small contribution from ∇ ∧ A, and about 70% of the magnitudes of the barotropic vorticity terms are explained with βV, bottom pressure torque, and the surface wind stress curl.However, there is one key difference between the vorticity budgets of subtropical and subpolar gyres.At relatively large length-scales (greater than 500 km), bottom friction and horizontal friction terms, ∇ ∧ τ b /ρ o and ∇ ∧ B, capture about 20% of the signals in the subpolar gyre, whereas their contribution to the vorticity balance in the subtropical gyre is less than 10%.This difference is because a large part of the subpolar gyre is influenced by physical processes occurring near land-sea boundaries.Since bottom and horizontal friction have their peak magnitudes near continental boundaries (see Figures 3e and 3f), they are more important in the vorticity budget of the subpolar gyre than in the subtropical gyre.Furthermore, the pronounced barotropic nature of the subpolar gyre may also amplify contributions from bottom and horizontal friction terms.

Why Does the Nonlinear Advection Term Smooth out at Large Scales?
The nonlinear advection term mainly accounts for the redistribution of vorticity via western boundary currents, transient eddies and standing meanders (Gula et al., 2015), which generally are 1-300 km in size (Chelton et al., 2007;Eden, 2007).Since these nonlinear flow patterns have spatial variations over length scales smaller than about 500 km, the nonlinear term is expected to be weak at large length scales (also see Hughes & De Cuevas, 2001).One can show that the nonlinear advection term has higher-order spatial derivatives than bottom pressure torque and βV (Appendix A2).The spatial spectrum of the nonlinear advection term will therefore contain a greater proportion of variance at high wavenumbers (or small length scales), which will be cut off more rapidly by the coarse-graining low-pass filter with increasing coarse-graining length scale.Consequently, the magnitude of the nonlinear advection term is expected to decrease faster than other vorticity budget terms with increasing coarse-graining length scale.Therefore, at relatively large scales, bottom pressure torque, βV and wind stress curl are expected to be in balance (see Figures 3a1-3c1).
To further investigate the relative importance of the nonlinear advection term and bottom pressure torque at different length scales, we perform a scale analysis (also see Schoonover et al., 2016), where u g is the horizontal geostrophic velocity at the ocean bottom, V is the velocity scale, L h is the horizontal length scale, and L v is the vertical length scale (see Appendix A2 for the definition of A).Equations 7 and 8 imply that the magnitudes of bottom pressure torque and the nonlinear advection term follow 1/ L h and 1/ L 2 h scalings, respectively.Hence, the nonlinear advection term must decay faster than bottom pressure torque when increasing the horizontal length scale.At small scales, the magnitudes of bottom pressure torque and the nonlinear advection term are comparable and are significantly greater than βV (Figures 3 and 4).Therefore, at relatively large length scales, the meridional flow then has to be controlled by a combination of bottom pressure torque and surface wind stress.As seen in Figure 5, the domain-mean absolute values of the nonlinear advection term and bottom pressure torque (in both the subpolar North Atlantic and global ocean) are generally in agreement with these scaling arguments.However, in the subtropical North Atlantic, the decline seems to occur at a relatively slower pace.Overall, the nonlinear term roughly follows ℓ 2 scaling whereas the bottom pressure torque magnitude declines as ℓ 1 .While the qualitative finding of a steeper decline of the advection term is to be expected from Equations 7 and 8, the rough quantitative agreement of the scaling exponents in Figure 5 and Equations 7 and 8 may be partly coincidental.This is because it emerges from the dependence of ℓ on the overlap of the coarse-graining low-pass filter with the shape of the wavenumber spectrum of each vorticity budget term.
At relatively large scales, βV dominates over ∇ ∧ A and the cross-over occurs near 200 km (Figure 5c), which interestingly corresponds to the mesoscale spectral peak in the global kinetic energy spectrum (Storer et al., 2022).Using the scale analysis, we estimate this cross-over length scale, By setting β = 10 11 m 1 s 1 and V = 0.1 m s 1 , we obtain L h = 100 km, which largely agrees with the results from Figure 5. Thus, the contribution of the nonlinear advection term to the barotropic vorticity budget can be neglected at scales larger than 300-400 km, which was also argued by Hughes and De Cuevas (2001).Equation 10 implies a horizontal length scale of ̅̅̅̅̅̅̅̅ V/β √ , which is the same as the inertial western boundary current scale proposed by Fofonoff (1955) and the Rhines scale in geostrophic turbulence (Rhines, 1975).In a sense, all of these different theories predict a length scale beyond which linear flow dynamics takes over nonlinear eddy dynamics, thus the similarity in these different length scales is not surprising.Furthermore, many works have  6) at different coarse-graining scales for the subpolar gyre (within the region of 10 Sv contour) and subtropical gyre (within the region of 10 Sv contour).For brevity, ẑ⋅ is omitted in the legend.investigated the physical processes that determine these length scales, for example, scales of frictional boundary currents, over flat topography (Haidvogel et al., 1992;Ierley & Sheremet, 1995;Kiss, 2002).
One caveat to note is that our analyses use output from a 0.25°ocean model, which does not resolve all mesoscale activity.Hence, the contribution of the nonlinear advection term to the barotropic vorticity budget, especially at mesoscales, is not fully captured.Furthermore, since we coarse-grain the barotropic vorticity budget terms diagnosed on the native model grid, coarse-graining does not remove Reynolds correlations arising from motions at length-scales smaller than the coarse-graining scale.Hence, if we were to calculate the nonlinear advection term in the barotropic vorticity budget using coarse-grained prognostic model diagnostics, such as velocities and layer thicknesses, as a function of coarse-graining scale, the length-scale dependence of the ∇ ∧ A term may differ from the one observed in Figure 5. Consequently, some results may not be directly compared against outputs from coarse non-eddy-resolving ocean models.On the other hand, βV and bottom pressure torque terms are linear and do not suffer from issues related to nonlinear Reynolds stresses.It is important to note that the predicted scaling laws, while valid for isotropic fields in large domains, may not hold for narrow, localized features like boundary currents and discontinuities.This is because these features, due to their significant presence across all length scales, may exhibit a different decrease in amplitudes with increasing coarse-graining scale.Also, the assumption of constant V holds when variations in velocity magnitudes occur at scales significantly larger than the coarse-graining scale, or when the velocity exhibits a linear spatial dependence.However, velocity magnitudes generally decline with coarse-graining (see scaling of β V in Figure 5).

Vorticity Budget in Weddell Sea Region
Topography plays a fundamental role in the Southern Ocean, which comprises highly energetic ocean regions, for example, Weddell Sea and Drake Passage, in terms of flow-topography interactions and mesoscale eddy dynamics (Hughes, 2005;Neme et al., 2023;Rintoul, 2018;Rintoul & Naveira Garabato, 2013;Rintoul et al., 2001).To investigate the roles of topography and nonlinear eddies on local vorticity balances, we repeat the vorticity budget analysis in the Weddell Sea region (Figure 6).For coarse-graining scale of 100-200 km, the main balance is among bottom pressure torque, ∇ ∧ A, and β V.For coarse-grained fields at scales larger than about 1,000 km, the contribution from the nonlinear advection term is minimal, and β V and bottom pressure torque terms explain more than 70% of the signals in the barotropic vorticity balances.
Interestingly, the relative contribution of the surface wind stress curl to the vorticity budget at length scales larger than 1,000 km is much smaller than observed in the North Atlantic Ocean (compare Figures 3h and 6h).This behavior is because the magnitudes of β V and bottom pressure torque are much larger in the Southern Ocean than in the North Atlantic (Figures 2a and 2b), whereas the wind stress curl magnitudes vary little with latitude (Figure 2d).In the Southern Ocean, the presence of prominent topographic features, in conjunction with substantial bottom pressure torque signals associated with strong bottom flows, gives rise to meandering and spatial variations in the flow structure due to topographic steering and potential vorticity conservation (Hughes, 2005;Kiss, 2004).As a consequence, the vorticity balance in this region prominently features substantial bottom pressure torque and β V signals, with wind stress curl playing a secondary role.These results do not imply that the wind component is unimportant in the Weddell Sea region.On the contrary, surface winds are a key driving force   6) at different coarse-graining scales.{|F ℓ |} is computed for the region bounded between 85°S-40°S and 70°W-0°W.Note that ẑ⋅ is omitted in panel titles and legends.
for ocean flows at all length scales.However, for the climatological local vorticity budget and spatial variability in vorticity terms, bottom pressure torque appears to be the primary factor in governing the spatial structure of the depth-integrated meridional flow in the Weddell Sea.

Vorticity Budget in the Equatorial Pacific Ocean
The equatorial Pacific Ocean slightly differs from ocean regions at high latitudes in terms of barotropic vorticity dynamics.Here, the contribution of the nonlinear advection term to the barotropic vorticity budget is relatively small at all length scales (Figure 7).Instead, bottom pressure torque and wind stress curl are the dominant terms that balance β V at all length scales, and these three terms capture more than 80% of the signals.Hence, dynamics in the equatorial Pacific Ocean largely follow the Topographic-Sverdrup balance.These results are in contrast to North Atlantic and Weddell Sea analyses, which indicate significant nonlinear eddy advection contribution to vorticity dynamics at length scales smaller than 1,000 km.

Global Vorticity Budget
To have an understanding of the global picture of vorticity balances, we divide the global ocean into four regions and repeat the vorticity analysis in these four regions (Figure 8).These basins are sufficiently large such that the regional variability (as in Sections 3.1-3.3)becomes less apparent.In general, bottom pressure torque and β V terms are the largest terms, followed by the surface wind stress curl that appears on relatively large scales.These three terms together capture roughly 80% of the signals.As seen in Sections 3.1-3.3,the nonlinear advection term is only important at length scales smaller than about 1,000 km, except in the Indian Ocean sector where, even at length scales of 1,000-2,000 km, the nonlinear advection term is as important as surface wind stress curl and bottom pressure torque.The relatively large contribution of the nonlinear advection in the Indian Ocean could be due to strong western boundary currents (Figure 2c).Also, the magnitudes of bottom pressure torque and bottom friction (Figures 2b and 2e) are weaker in the Indian Ocean than other basins, which would increase the relative magnitude of the nonlinear advection term.Moreover, larger mesoscale eddy length scales in the tropics relative to higher latitudes may enhance the nonlinear advection contribution at large length scales (Chelton et al., 2007(Chelton et al., , 2011)).Similarly, we observe a relatively larger contribution of the nonlinear advection term in the Tropical Pacific-Atlantic region (Figure 8f).In addition, bottom friction and horizontal friction explain about 10%-20% of the signals in the vorticity balance in all four regions.
To further emphasize how spatial smoothing affects the local vorticity balance, we identify grid points at which the largest two or three vorticity budget terms account for 80% of the sum of the magnitudes of all terms in the barotropic vorticity budget.Sonnewald et al. (2019) applied a machine learning algorithm to ECCO global ocean state estimate, which has horizontal grid spacing of 1°, and identified different dynamical regimes using the barotropic vorticity budget framework.However, impacts of the spatial resolution on these dynamical regimes have not been examined before.Here, we analyze point-wise vorticity balances for four coarse-graining scales (Figure 9).First, three vorticity balances stand out, that is, Topographic-Sverdrup balance, Topographic-Nonlinear balance, and Sverdrup balance.The proportion of the global ocean surface area at which these balances are satisfied increases when we increase the coarse-graining scale (see Table 1).In fact, a large part of the global ocean transitions from a Topographic-Nonlinear regime to a Topographic-Sverdrup regime, especially in the Southern Ocean.As the coarse-graining kernel width increases and more length scales are filtered out, the contribution of the nonlinear advection term decreases.In the case of 200 km coarse-graining scale, the vorticity dynamics closely follow Topographic-Sverdrup and Topographic-nonlinear relationships at about 22% and 16% of the global ocean surface area, respectively.On the other hand, these percentages change to 37% and 6%, respectively, at length scales greater than 2,000 km.
In tropical and subtropical oceans (roughly 40°S-40°N), Sverdrup balance holds reasonably well at length scales larger than 1,000 km (Figure 9c), which is in agreement with Gray and Riser (2014); Thomas et al. (2014); Wunsch (2011).However, Sverdrup balance rarely holds at higher latitudes in those regions where topography significantly affects the spatial variability of the depth-integrated meridional flow at large scales.This role for topography is enhanced in such regions due to a relatively weak stratification allowing for strong deep flows.Note that maps of Sverdrup and Topographic-Sverdrup relationships in Figure 9 are not mutually exclusive.If the local vorticity dynamics can be approximated as being in Sverdrup balance (based on the chosen criteria of capturing 80% of the signals in the barotropic vorticity budget), then the dynamics would also be in accord with Topographic-Sverdrup balance.Hence, in the spatial maps shown in Figure 9, Sverdrup balance is a special case of Topographic-Sverdrup balance.At length scales larger than 1,000 km, the barotropic vorticity dynamics can be understood in terms of Topographic-Sverdrup balance in more than 60% of the global ocean.A schematic of different dynamical regimes in the global ocean is shown in Figure 10.  6) at different coarse-graining scales.{|F ℓ |} is computed for the region bounded between 20°S-20°N and 180°W-100°W.Note that ẑ⋅ is omitted in panel titles and legends.
Intriguingly, there is virtually no ocean region in the friction-dominated regime, in which planetary vorticity advection is controlled by bottom friction and horizontal friction.This result suggests that the global ocean is dominated by inviscid processes in terms of barotropic vorticity dynamics.Indeed, there is a large part of the oceans where these simplified vorticity relationships (Topographic-Nonlinear and Topographic-Sverdrup) do not hold and vorticity dynamics are controlled by more than three terms.In these regions, friction can play an important role, for example, by allowing flow across mean potential vorticity contours and altering western boundary current flow and separation (Hughes & De Cuevas, 2001;Jackson et al., 2006).In such situations, the combination of friction with other vorticity budget terms can alter the meridional transport structure and strength, leading to complex vorticity balances that may not be captured by simplified relationships shown in Figure 9. Additionally, Neme et al. (2023) identify the importance of bottom friction for transient vorticity budgets, thus offering a further caveat to the vorticity balances found here, which are based on climatological means .

Discussion and Conclusions
The vorticity budget of the depth-integrated flow is analyzed to understand how bottom pressure torque, surface wind stress curl, nonlinear advection, and friction drive spatial variability in meridional transport in the oceans., which account for at least 80% of the sum of the magnitudes of terms in the barotropic vorticity budget at any grid point.For legend "Other," vorticity balance is complex, and more than three terms are required to account for at least 80% signals in vorticity balances.

Table 1
Percentage of the Global Ocean Surface Area at Which Vorticity Balances Plotted in Figure 9 Satisfy and Account for at Least 80% of the Sum of the Magnitudes of Terms in the Barotropic Vorticity Budget 200 km 500 km 1,000 km 2,000 km Previous studies have shown that interpretations of vorticity budget analyses can significantly change depending on the region of interest and length scale.For example, the classical Sverdrup balance only holds in tropics and subtropics at length scales greater than about 5° (Thomas et al., 2014;Wunsch, 2011).At higher latitudes and in eddy-active regions, bottom pressure torque and nonlinear advection control the spatial variability the depthintegrated meridional flow (Hughes & De Cuevas, 2001;Le Corre et al., 2020;Lu & Stammer, 2004;Yeager, 2015).
The present work investigates the regional variability and length-scale dependence in vorticity budget analyses using the 60-year mean vorticity budget terms from an eddy-permitting global ocean simulation (Adcroft et al., 2019).The time-mean vorticity budget terms are analyzed as a function of spatial-filtering scale by employing a coarse-graining technique (Buzzicotti et al., 2023;Storer et al., 2022).Consistent with previous studies (Hughes & De Cuevas, 2001;Sonnewald et al., 2019), the relative magnitudes of different vorticity budget terms display significant regional variability.In general, depth-integrated meridional velocity is balanced by a combination of the surface wind stress curl, bottom pressure torque, and the curl of the nonlinear velocity advection in the barotropic vorticity budget.The relative importance of these terms is examined by performing vorticity analyses in different ocean regions at different coarse-graining length scales.
We show that Topographic-Svedrup balance, in which βV (meridional gradient of Coriolis parameter × depthintegrated meridional velocity), bottom pressure torque, and surface wind stress curl are in balance (Holland, 1967), applies to vorticity dynamics in the majority of the global ocean.These three vorticity terms capture more than 60% of the signals in the barotropic vorticity budget (Figures 3-9); however, it requires significant spatial coarse-graining, and this simplified balance only holds at length scales larger than about 1,000 km.This result is in agreement with previous studies that employed coarse non-eddy resolving model outputs in their vorticity analyses (Lu & Stammer, 2004;Yeager, 2015).Although bottom pressure torque contribution is significant in all ocean regions that we considered, a simpler Sverdrup balance, in which the depth-integrated meridional transport is driven by surface wind stress curl (Sverdrup, 1947), holds reasonably well in subtropical oceans at length scales greater than 1,000 km (also see Gray & Riser, 2014;Thomas et al., 2014;Wunsch, 2011).On the other hand, at higher latitudes and throughout the Southern Ocean, the contribution of bottom pressure torque for the vorticity balance cannot be neglected, with this importance due to relatively strong deep flows.
In the case of nominal or no coarse-graining (retaining variations on length scales greater than 100 km in the present work), bottom pressure torque and the nonlinear advection term dominate the vorticity budget locally (referred to as "Topographic-Nonlinear" balance here) indicating a prominent role of ocean eddies in vorticity balances.We note that bottom pressure torque and nonlinear advection terms compensate against each other (e.g., see Le Corre et al., 2020), and the residual from these two terms is roughly balanced by planetary vorticity advection.As we increase the length scale of coarse-graining, the nonlinear advection term largely smooths out, and we find a clear transition from Topographic-Nonlinear balance to Topographic-Sverdrup balance in the local vorticity budget (see Figure 9).Hence, the nonlinear advection term contributes to vorticity balances mostly at length scales smaller than 1,000 km, and we offer a scaling argument to explain why it plays a negligible role for larger scale vorticity balances.
By incorporating the coarse-graining method in vorticity budget analysis, we find that the relative magnitudes of vorticity budget terms not only vary regionally but also have a strong length-scale dependence.Although Sverdrup and Topographic-Sverdrup relationships explain the spatial structure of the meridional transport in many places, these relationships only apply to large-scale oceanic flows (larger than about 1,000 km).At relatively small length scales, the contribution of eddies and nonlinear advection to vorticity balance tends to be significant.Hence, the interpretations from vorticity analyses can be completely different depending on the extent of spatial filtering.We present a schematic describing these different vorticity balances (see Figure 10).
The present study only considers time-mean vorticity balances and the temporal variability in local vorticity balances has not been analyzed.Vorticity analyses from seasonal vorticity diagnostics (not shown) closely follow the time-mean results presented in the present work.In temporally varying vorticity diagnostics, we expect similar transitions among different dynamical regimes at different length scales (Figure 9) in barotropic vorticity balances, albeit some regional differences may be present.For example, although the contribution of the friction term is negligible in the time-mean vorticity balances, friction can play an important role in driving transient changes in vorticity balances (Neme et al., 2023).

A1. Depth Integration and Its Curl
To derive the vorticity budget of the depth-integrated flow, we first vertically integrate the velocity Equation A1 from the ocean bottom, z = H(x, y), to the sea surface, z = η(x, y, t), Here, a = ζ ẑ ∧ u ∇ r K w ( ṙ) ∂ r u and b = F (horz diff) .The vertical integral of F (vert diff) , representing the vertical convergence of the viscous flux, vanishes over the entire depth.Note that the vertical viscous flux disappears at both the top and bottom of the ocean.Since we use the depth-integrated velocity equation to derive the vorticity budget, the mathematical manipulations in the following steps remain the same irrespective of the choice of the vertical coordinate in the velocity equation.Thus, for simplicity, the pressure gradient term is just written as ∇p below (note that the geopotential, Φ = gz, does not appear in horizontal pressure gradients), where ∇ = x ∂ x + ŷ ∂ y is the horizontal gradient operator on a fixed depth.We now introduce the shorthand notation and make use of Leibniz's rule on the pressure gradient term to render Here, p s and p b are pressures at the surface and bottom of the ocean, and the terms p s ∇η, p b ∇H are pressure form stresses at the ocean surface and ocean bottom, respectively.We now take the curl of this equation and split the curl of the linear Coriolis term into two terms to obtain We can further manipulate the second term on the right hand side (RHS) by making use of volume conservation for a vertical column of Boussinesq fluid, which is In addition, ocean surface pressure is assumed to be constant, as is the case in the MOM6 configuration used here and often the case in climate models, so that J(p s , η) = 0. Finally, the vorticity budget for the depth-integrated flow (with some rearranging and writing ∫ η H v = V) can be written as Note that there are higher-order derivatives in the nonlinear advection term and bottom pressure torque.Hence, relative to βV, the right-hand side terms have a stronger small-scale spatial variability and relatively larger magnitudes at small length scales.As conjectured by Hughes (2000), the advection term and bottom pressure torque are expected to compensate each other at small length scales, with their residual leading to a relatively large-scale structure in meridional transport (see Figures 3a1-3c1).

Appendix B: Diagnosing Vorticity Budget Terms in MOM6
MOM6 is equipped with online diagnostics sufficient for an offline computation of individual terms in the vorticity Equation A17.We do so by making use of the online depth-integrated velocity budget diagnostics in MOM6.We then take the curl of these diagnostics to obtain the corresponding vorticity budget terms.Actual names of depth-integrated momentum diagnostics and the relevant calculations are shown in Table B1.A more detailed description of velocity and vorticity budget diagnostic calculations in MOM6 is available at Khatri et al. (2023).

B1. Remapping Contribution
In GFDL-MOM6, vertically integrated zonal and meridional momentum budgets can be diagnosed according to Except for the last term on the RHS in Equations B1 and B2, the rest of the terms are names of the MOM6 diagnostics corresponding to vertical-integrals of terms in Equation A1. hf_dudt_2d and hf_dvdt_2d are the depth-averaged velocity-tendency diagnostics, intz_CAu_2d and intz_CAv_2d are the diagnostics for the vertical-integral of ( f + ζ) ẑ ∧ u + ∇K, intz_PFu_2d + intz_u_BT_accel_2d and intz_PFv_2d+ intz_v_BT_accel_2d are the diagnostics for the vertical-integral of ∇p/ρ o , intz_diffu_2d and intz_diffv_2d are the diagnostics for the vertical-integral of F (horz diff) , taux and tauy are the surface wind stress diagnostics, and taux_bot and tauy_bot are the bottom friction diagnostics.The remapping terms correspond to w ( ṙ) ∂ z u, which are not available to be saved as online diagnostics in the current version of MOM6.Thus, the remapping terms are diagnosed offline as a residual in the velocity budget Equations B1 and B2.Refer to the online documentation, mom6-analysiscookbook.readthedocs.io/en/latest/notebooks/Closing_mo-mentum_budget.html, for full details of momentum diagnostics in the MOM6 model.
To compute the contribution of the remapping terms in the vorticity budget, we calculate the curl of the depthintegrated remapping terms diagnosed as residuals from the depth-integrated velocity budget diagnostics.We found that the contribution of the remapping term to the barotropic vorticity budget is minimal, and the vorticity budget closes well even without accounting for the remapping term.This result suggests that the remapping term is not a significant factor in the present analyses.

B2. Bottom Pressure Torque Calculation
From the development in Equations A14-A16, we are required to use the following relation (based on volume conservation) to derive the barotropic vorticity Equation A17.
Generally, the expression on the LHS in Equation B3 results in significant cancellation between the zonal and meridional gradients in the curl operation and the small residual is equal to βV (plus small contributions from nonzero Q m and ∂ t η).However, the analytical result in Equation B3 need not hold in an ocean model, which solves for velocity on a discretized grid.On the MOM6 native grid, cancellation between the zonal and meridional gradients in ∇∧ [f ẑ ∧ ∫ η H u dz] does not occur as expected and the residual, which is due to numerical errors, is at least two orders of magnitudes larger than B1a-B1c).
These numerical errors can lead to spurious forces in vorticity balances and corrupt bottom pressure torques (Styles et al., 2022).These spurious signals arise due to the handling of the Coriolis acceleration and the representation of bathymetry in energy and enstrophy conserving schemes on a discrete C-grid (Arakawa & Lamb, 1981).As a result, a C-grid model does not satisfy discrete versions of the Leibniz's rule, which is used in Equation A13, leading to spurious forces in vorticity balances.MOM6 is discretized using a C-grid and employs a vertical Lagrangian-remap method on a hybrid z*-isopycnal vertical coordinate to simulate the ocean state (Adcroft et al., 2019;Griffies et al., 2020).Hence, bottom pressure torque diagnosed in MOM6 is expected to suffer from these spurious forces (Styles et al., 2022;Waldman & Giordani, 2023).To diagnose physically relevant signals in bottom pressure torque, we need to account for these numerical errors.In some cases, it may be possible to disentangle physical and spurious contributions to vorticity budget terms offline from the knowledge of horizontal velocities and the model grid scale factors in C-grid models.For example, Waldman and Giordani (2023) proposed a method for diagnosing vorticity budget terms in the NEMO ocean model; however, the method does not resolve all numerical issues.
In the present study, we take an alternative approach by making use of the terms leading to a closed momentum budget at every grid point.Thus, if we compute the curls of depth-integrated velocity budget diagnostics, the resultant vorticity budget also closes at every grid point.This closure implies that the sum of numerical errors present in individual vorticity budget terms, diagnosed using the discrete curl operations, must vanish at every grid point.Similar to numerical errors in ∇ ∧ [f ẑ ∧ ∫ η H u dz], we observe unrealistic large signals in B1d).We hypothesize that these large signals are mostly numerical errors due to discretization.Fortunately, the Coriolis acceleration and pressure gradient acceleration are discretized in a consistent manner, so that numerical errors in their curls are roughly equal in magnitude and largely cancel (see Figures B1a, B1d, and B1e).
We make an assumption that spurious signals are only present separately in the curls of depth-integrated Coriolis acceleration and pressure gradient terms.To obtain physically realistic magnitudes and spatial structure of bottom pressure torque, we add the bottom pressure torque relation from Equations A14 and A15 and Equation B3 to obtain Since the sum of the last four terms on the RHS in Equation B4 vanishes analytically (see Equation B3), but not numerically (Figure B1c), the analytical expression B4 computes the curl of the depth-integrated pressure gradient terms, which is bottom pressure torque (by Leibniz's rule assuming uniform surface pressure, see Appendix A1).By using the diagnostic approach of Equation B5, we eliminate the spurious signals in bottom pressure torque because numerical errors in the first two terms on the RHS in Equation B5 largely cancel out.
The spatial structure and magnitudes of diagnosed bottom pressure torque (Figure B1f) agree well with results from Le Corre et al. (2020) (see their Figure 7b), who used a terrain following vertical coordinate C-grid model  (which is partially immune to the numerical issues identified by Styles et al. (2022) and Waldman and Giordani (2023)).Furthermore, there is a fair consistency between the present results and bottom pressure torque diagnosed using B-grid model outputs (Hughes & De Cuevas, 2001;Yeager, 2015), which also do not suffer from numerical issues present in C-grid models (Styles et al., 2022).
Our diagnostic approach assumes that numerical errors in ∇ ∧ [f ẑ ∧ ∫ η H u dz] and ∇ ∧ [ 1 ρ o ∫ η H ∇p dz] are exactly equal in magnitude and opposite in sign, which need not be true in general.Numerical errors may also be present in nonlinear advection, bottom stress, and horizontal friction in the barotropic vorticity budget.However, accelerations from the pressure gradient and Coriolis appearing in the velocity equation are at least two orders of magnitude larger than the rest of the terms (Figure B2).Therefore, it is safe to assume that numerical errors are contained in pressure gradient and Coriolis acceleration, with the diagnostic approach of Equation B5 being a practical diagnostic method.6) at different filter scales.{|F ℓ |} is computed for the region bounded between 30°N-70°N and 80°W-0°W.Note that ẑ⋅ is omitted in panel titles and legends.

Figure 1 .
Figure 1.Spatial maps of the vertical component of relative vorticity (units are in s 1 ) computed using the time-mean (1958-2017) depth-averaged velocity.The plotted vorticity maps are coarse-grained to (a) 200 km, (b) 1,000 km horizontal length scale (using the FlowSieve package, Storer & Aluie, 2023).Note the different color ranges used for the two panels.

Figure 3 .
Figure 3. Vorticity budget analysis for the North Atlantic Ocean (a-g) Time-mean (1958-2017, indicated with overbars) spatial maps of barotropic vorticity budget terms (units are in m s 2 ) as a function of the coarse-graining scale; (h) Normalized magnitudes of the absolute budget terms (see Equation6) at different coarse-graining scales.{|F ℓ |} is computed for the region bounded between 30°N-70°N and 80°W-0°W.Green contours indicate the subpolar and subtropical gyres as in Figure4.Note that ẑ⋅ is omitted in panel titles and legends.

Figure 4 .
Figure 4. Vorticity budget analysis for North Atlantic gyres (a) Time-mean (1958-2017, indicated with overbars) barotropic streamfunction computed as ∫ x x w Vdx; (b)-(c) Normalized magnitudes of the absolute budget terms (see Equation6) at different coarse-graining scales for the subpolar gyre (within the region of 10 Sv contour) and subtropical gyre (within the region of 10 Sv contour).For brevity, ẑ⋅ is omitted in the legend.

Figure 6 .
Figure 6.Vorticity budget analysis for the Weddell Sea region (a-g) Time-mean (1958-2017, indicated with overbar) spatial maps of barotropic vorticity budget terms (units are in m s 2 ) as a function of the coarse-graining scale; (h) Normalized magnitudes of the absolute budget terms (see Equation6) at different coarse-graining scales.{|F ℓ |} is computed for the region bounded between 85°S-40°S and 70°W-0°W.Note that ẑ⋅ is omitted in panel titles and legends.

Figure 7 .
Figure 7. Vorticity budget analysis for an oceanic region in the equatorial Pacific (a-g) Time-mean (1958-2017, indicated with overbar) spatial maps of barotropic vorticity budget terms (units are in m s 2 ) as a function of the coarse-graining scale; (h) Normalized magnitudes of the absolute budget terms (see Equation6) at different coarse-graining scales.{|F ℓ |} is computed for the region bounded between 20°S-20°N and 180°W-100°W.Note that ẑ⋅ is omitted in panel titles and legends.

Figure 8 .
Figure 8. Vorticity budget analysis for the global ocean (a) Extent of four ocean basins (b)-(f) Normalized magnitudes of the absolute budget terms (see Equation 6) at different coarse-graining scales.{|F ℓ |} is computed separately for the basins shown with different colors in panel (a) and the hatched region covers tropical Atlantic-Pacific Ocean (15°S-15°N).Note that ẑ⋅ is omitted in the legends.

Figure 9 .
Figure 9. Global map of leading vorticity balances with different levels of coarse-graining (a) 200 km (b) 500 km (c) 1,000 km (d) 2,000 km.Different colors indicate balance among different vorticity terms (see legend), which account for at least 80% of the sum of the magnitudes of terms in the barotropic vorticity budget at any grid point.For legend "Other," vorticity balance is complex, and more than three terms are required to account for at least 80% signals in vorticity balances.

Figure B1 .
Figure B1.Time-mean (1958-2017) of terms in Equation B4: (a) Vertical component of the curl of depth-integrated planetary vorticity advection, ∇ ∧ [f ẑ ∧ ∫ η H u dz] , in model diagnostics (terms in second and third lines on the right hand side (RHS) in Equation B5).(b) βV + fQ m /ρ o f∂ t η.(c) Sum of fields shown in panels (a) and (b).(d) Vertical component of the curl of depth-integrated pressure gradient, ∇ ∧ [ 1 ρ o ∫ η H ∇p dz] , in model diagnostics (terms in the first line on the RHS in Equation B5).(e) Sum of fields shown in panels (a) and (d).(f) Sum of fields shown in panels c and d to compute bottom pressure torque.No coarse-graining (or regridding) was applied and the plotted diagnostics are on the model native grid (units are in m s 2 ).However, for a better visualization, plotted diagnostics were smoothed by averaging over neighboring four grid points to remove grid-scale noise (used GCM-Filters package Loose et al., 2022).

Figure D1 .
Figure D1.Vorticity budget analysis for the North Atlantic Ocean (a-g) Time-mean (1958-2017, indicated with overbars) spatial maps of filtered barotropic vorticity budget terms (used GCM-Filters package, units are in m s 2 ) as a function of filter scale; (h) Normalized magnitudes of the absolute budget terms (see Equation6) at different filter scales.{|F ℓ |} is computed for the region bounded between 30°N-70°N and 80°W-0°W.Note that ẑ⋅ is omitted in panel titles and legends.

Table B1
Method for the Computations of Vorticity Budget Terms Using Depth-Integrated Momentum Budget Diagnostics (D = H + ηIs the Full Depth of the Ocean) in Modular Ocean Model Version 6 Δx) , where Δx is the zonal grid spacing andρ o = 1,035 kg m 3 ∂ x [umo_2d/ ρ o Δy)] ∂ y [vmo_2d/ ρ o Δx)] (following Equation A16) ẑ ⋅ (∇ ∧ U t ) ∂ x [D × hf_dvdt_2d] ∂ y [D × hf_dudt_2d]Note.The contribution from remapping in ∇∧ A can be computed as discussed in Appendix B1. "intz" and "2d" in diagnostic names indicate vertical-integral; for example, intz_diffv_2d is the vertical-integral of diffv diagnostic.Note that hf_dvdt_2d and hf_dudt_2d are the depth-averaged velocity-tendency diagnostics, thus requiring multiplication by the ocean depth, D, in ∇ ∧ U t calculation. o