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# Vertical Velocity 3D Estimates from the Linear Vorticity Balance in the North Atlantic Ocean

Diego Cortés Morales, Alban Lazar

► To cite this version:

Diego Cortés Morales, Alban Lazar. Vertical Velocity 3D Estimates from the Linear Vorticity Balance in the North Atlantic Ocean. Journal of Physical Oceanography, 2024, 10.1175/JPO-D-24-0011.1. insu-04725887

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Distributed under a Creative Commons Attribution 4.0 International License
Vertical Velocity 3D Estimates from the Linear Vorticity Balance in the North Atlantic Ocean

DIEGO CORTÉS-MORALES<sup>a</sup> AND ALBAN LAZAR<sup>a</sup>

<sup>a</sup> LOCEAN, IPSL, Paris, France

(Manuscript received 18 January 2024, in final form 22 June 2024, accepted 28 July 2024)

**ABSTRACT:** Large-scale three-dimensional vertical movements of the world's oceans play a fundamental role in their functioning, yet they are still overlooked. This study aims to enhance our understanding of the relationship between horizontal and vertical flows as a function of depth to derive some major characteristics of the w field from a relatively well-constrained variable. For this purpose, we analyze an eddy-permitting ocean general circulation model simulation, focusing on the linear vorticity balance (LVB) over the North Atlantic basin. For spatial scales greater than 5°, over more than 50-yr average, the LVB holds with less than 10% error over the whole depth range of most of the subtropical upper tachocline-upper ocean and part of the tropical one, away from western boundary currents and zonal tropical flows. The spatial extent of the balance diminishes in the intermediate ocean, particularly in the western half of the basin. Within the deep tachocline, the balance holds over large portions of the abyssal plains. Based on these results, the β-plane geostrophic vertical velocity is derived from the vertical indefinite integral of the geostrophic LVB from the surface. While its time mean aligns only qualitatively with the model reference w, it accurately captures the interannual variability over most of the basin, even close to eastern boundaries. However, it cannot reproduce the variability in the lower tropics, the western boundary, and the North Atlantic Current system. Our results emphasize the dominance of geostrophy in establishing most of the ocean model vertical flow variability.

**KEYWORDS:** North Atlantic Ocean; Thermocline circulation; Vertical motion; Vorticity

# 1. Introduction

One of the most influential theories that explain the ocean's general circulation is the Sverdrup balance, which connects the meridional geostrophic transport with the wind stress at basin scales (Sverdrup 1947). The Sverdrup balance theory describes the time-mean barotropic ocean transport over the subtropical and tropical gyres (STG and TG), assuming a level of no vertical motion. Sverdrup (1947) proposes that the Sverdrup balance should emerge after extended time-averaging. Subsequent research on this theory has suggested that the oceanic response to the atmospheric forcing needed for the Sverdrup balance to hold can occur within the first-mode baroclinic Rossby wave propagation time scales (Tailleux and McWilliams 2001). Hence, the Sverdrup balance approach fails to explain the vertically integrated circulation in regions where the Rossby wave has longer time scales than the studied period, such as in high latitudes. This formulation is also inadequate near the coasts, where the assumption of a level of no vertical motion is no longer valid (Wunsch 2011; Gray and Riser 2014; Thomas et al. 2014).

Recent studies have examined the limitations of the Sverdrup balance through the complete vorticity balance, which includes viscosity, frictional, nonlinear, and bottom pressure torque terms to close the vorticity budget. The importance of

frictional and bottom pressure torque terms in the North Atlantic subpolar gyre interior has been highlighted by Spence et al. (2012), Yeager (2015), and Le Corre et al. (2020). In the Gulf Stream (GS), it is necessary to add the nonlinearities to the bottom pressure torque term to accurately account for the atmospheric forcing (Hughes and de Cuevas 2001; Gula et al. 2015; Schoonover et al. 2016). The global ocean exhibits similar features, as noted by Hughes and de Cuevas (2001) and Sonnewald et al. (2019). Remarkably, the depth dependency of the vorticity balance is little studied. In an ocean general circulation model (OGCM), Lu and Stammer (2004) and Waldman and Giordani (2023) identify the Sverdrupian nature of the center of the oceanic gyres within the upper and intermediate oceans and show that topographic forcing drives the vorticity budget closure in the deep ocean.

To conduct a baroclinic analysis of the basin-scale circulation, the linear vorticity balance (LVB) serves as an appropriate and well-known starting point:

$$ \beta v = f \partial_z w, \quad (1) $$

where f is the Coriolis parameter, β is the meridional gradient of f, and v and w are the meridional and vertical components of the velocity, respectively. When the flow is in LVB, it becomes possible to relate the shear of the vertical velocity field directly and locally to the meridional velocities. When integrated vertically from the surface to a level of no vertical motion, the LVB results in the Sverdrup balance. For a geostrophic and incompressible flow on a β plane, the LVB represents the continuity equation and expresses a local mass balance between the meridional divergent flow and an opposing vertical convergent movement, and vice versa. Figure 1

<footnote><sup>a</sup> Denotes content that is immediately available upon publication as open access.</footnote>

<footnote>Corresponding author: Diego Cortés-Morales, dcortes@locean.ipsl.fr</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. Vertical and geostrophic meridional circulation over a typical Northern Hemisphere STG (northernmost cube) and TG (southernmost cube) within the thermocline. The tendency to reduce the amplitude of the downward (upward) flow in the thermocline must be offset by a divergent (convergent) flow in the STG (TG) to absorb (feed) the Ekman transport to the oceanic surface. Thus, the meridional circulation in the thermocline is determined by the divergence of equatorward transport and the convergence of poleward transport.</figcaption></figure>

illustrates this interpretation schematically for the geostrophic component of two stationary tropical and subtropical circulation gyres in the North Atlantic upper ocean.

Indeed, the inverse dependence on the latitude of the meridional geostrophic transport $v_g$ results in the convergence of the poleward transport within the cyclonic TG thermocline, pushing up the thermocline waters and supplying mass to the diverging Ekman flow, forced by the trade winds, whose intensity increases with latitude. Further north, the intrinsic divergence of the equatorward transport of the STG progressively absorbs the Ekman layer mass excess from the converging flow, leading to downward movements (Tomczak and Godfrey 1994). A comparable logic can be used in the geostrophic deep ocean and the bottom Ekman layer. However, there is no independent forcing like wind stress in the deep ocean to estimate a boundary condition for the ageostrophic component of $w$.

As illustrated for the thermocline, geostrophic vertical movements are essential to global circulation. They redistribute water masses between layers, shaping them (Stramma and England 1999), ventilate the thermocline (Gordon 1981), and also participate in driving the deep circulation (Drake et al. 2020). However, the $w$ field in the open ocean is several orders of magnitude smaller than the horizontal circulation (approximately $10^{-6}$ m s$^{-1}$ in the center of oceanic gyres).

Currently, direct observations of such weak intensities are not yet technically possible (e.g., Comby et al. 2022).

The reconstruction of horizontal geostrophic velocities, derived from observed thermohaline fields, in conjunction with data collected by current meters, expands the possibilities for connecting the three components of ocean circulation.

Many early estimations of $w$ utilized measured profiles of tracers to estimate tracer fluxes (e.g., Stommel and Arons 1959; Robinson and Stommel 1959; Wyrtki 1961; Munk 1966; Wunsch 1984) or the application of mooring measurements of horizontal currents to the continuity equation (Stommel and Schott 1977; Schott and Stommel 1978; Wyrtki 1981; Roemich 1983; Bubnov 1987; Halpern and Freitag 1987; Halpern et al. 1989; Weingartner and Weisberg 1991). These studies reached a consensus on the order of magnitude of $w$ in the different ocean layers ranging from $10^{-5}$ m s$^{-1}$ near the surface, gradually decreasing to $10^{-6}$ m s$^{-1}$ within the upper ocean, and reaching values between $10^{-7}$ and $10^{-8}$ m s$^{-1}$ in the intermediate and deep oceans. However, while the vertical velocity field can be diagnosed in primitive equation numerical models by solving the continuity equation, this approach has been limited by the scarcity of observation-based data at global scale and the potential for prominent instrumental errors when computing the divergence from measured horizontal velocities. One example of retrieving the vertical flow from
observations via continuity equation is presented by Freeland (2013), who calculated observation-based barotropic w at 700 dbar within an approximately 25° × 20° horizontal box. Over the last decade, researchers have explored alternative approaches for assessing large-scale baroclinic vertical velocities. These include the use of observations of isopycnal displacement (Giglio et al. 2013; Christensen et al. 2024), mooring data applied to the momentum and density equations (Sévellec et al. 2015), and biogeochemical tracers (Garcia-Jove et al. 2022). More theoretical approaches have now gained more attention, such as using the Bernoulli function to estimate w (Tailleux 2023). In recent years, several attempts have been made to compute the w field using the omega equation [$f^2(\partial^2 w/\partial z^2) + \nabla_h(N^2 \cdot \nabla_h w) = \nabla \cdot Q$; Giordani et al. 2006], as demonstrated by studies conducted by Buongiorno Nardelli et al. (2018), Ruiz et al. (2019), and Buongiorno Nardelli (2020). Buongiorno Nardelli et al. (2018) developed a three-dimensional w product within the upper ocean (OMEGA3D) from an Argo- and satellite-based dataset (Mulet et al. 2012; Guinehut et al. 2012).

An alternative approach to estimate the w field from observations is theoretically possible within the geostrophic LVB framework, as suggested by Charney (1955). This alternative also offers a simple and intuitive method to understand basin-scale motions in LVB. In the present paper, we aim to develop a comprehensive baroclinic analysis of the LVB and its geostrophic case at all depths, in climatological and interannual scales across the North Atlantic, in an OGCM simulation. This analysis will address the oceanic circulation patterns' ability to react to the surface and bottom boundary forcing, transmitting the forcing to the ocean interior. Last, and from the perspective of future developments based on the use of global geostrophic currents computed from observations, we will assess the feasibility of using an indefinite vertical integration of the geostrophic LVB in order to reconstruct the three-dimensional distribution of the OGCM's vertical flow.

## 2. Methodology and data

### a. The linear vorticity balance

The vorticity balance can be obtained by applying the continuity equation to the vertical component of the curl of the steady-state momentum Navier-Stokes equations under Boussinesq approximation:

$$ 
\begin{aligned}
\partial_t \zeta + \beta v + \mathbf{u} \nabla \zeta &+ [(\partial_x w \partial_z v) - (\partial_y w \partial_z u)] \\
&= (f + \zeta) \partial_z w + \frac{1}{\rho} \nabla \times (\rho A_H \nabla^2 \mathbf{u}_H + \rho A_V \partial_z^2 \mathbf{u}_H),
\end{aligned} 
\quad (2) $$

where $\zeta$ is the relative vorticity; $\mathbf{u}$ and $\mathbf{u}_H$ are the three-dimensional and horizontal velocity fields, respectively; $u$, $v$, and $w$ represent the zonal, meridional, and vertical components of the velocity, respectively; and $A_H$ and $A_V$ are the horizontal and vertical turbulent viscosity coefficients, respectively. The terms on the left-hand side of Eq. (2) are the relative vorticity time variation, the planetary vorticity advection, the relative vorticity advection, and vortex tilting term. On the right side of Eq. (2), the terms are the vortex

stretching term and the horizontal and vertical eddy viscosity terms (Vallis 2006).

The Rossby number ($Ro = V/fL$, where $L$ and $V$ are the characteristic length and horizontal component of the flow) relates the inertial to Coriolis forces. Therefore, this ratio can be used for a magnitude analysis of the vorticity equation. Most term dimensions in Eq. (2) can be expressed in terms of $Ro$. The continuity equation indicates that the vertical velocity scale is related to the horizontal velocity by $V/L = W/D$, where $W$ and $D$ are the vertical component and thickness of the flow. With $T$ being the characteristic time of the flow and expressed as $T = 1/fRo$,

$$ 
\begin{aligned}
& \left[ \partial_t \zeta = \frac{fRo}{T} \right] = f^2 Ro^2; \quad [\mathbf{u} \nabla \zeta] = f^2 Ro^2; \\
& [(\partial_x w \partial_z v) - (\partial_y w \partial_z u)] = f^2 Ro^2; \\
& [\zeta \partial_z w] = f^2 Ro^2; \quad [\beta v] = f^2 Ro; \quad [f \partial_z w] = f^2 Ro.
\end{aligned} 
\quad (3) $$

Considering a low Rossby number ($Ro \ll 1$) describes large-scale ocean flow of $O(100)$ km, the terms explicitly dependent on the Coriolis force are dominant contributors in the balance of forces (Pedlosky 1979). Far from frictional boundaries, the Ekman numbers (horizontal Ek = $2A_H/fL^2$ and vertical Ek = $2A_V/fD^2$) are low ($Ek \ll 1$), and rotation dominates over friction. Therefore, the eddy viscosity-related terms $[(1/\rho)\nabla \times [\rho A_H \nabla^2 \mathbf{u}_H]] = f^2 RoEk; (1/\rho)\nabla \times [\rho A_V \partial_z^2 \mathbf{u}_H] = f^2 RoEk$ can be overlooked. Taking the first-order balance of Eq. (2) while considering the regime flows mentioned above and averaging over a sufficiently long period to neglect time-variation contribution, Eq. (1) emerges.

In this paper, both sides of Eq. (1) are first estimated to assess the extent to which the LVB describes the circulation in the OCGM. Their relative error ($\Delta_{LVB}$) relative to the mean between both terms in Eq. (1) is estimated as follows:

$$ \Delta_{LVB} = \frac{(\beta v - f \partial_z w)}{\frac{1}{2}(|\beta v| + |f \partial_z w|)} \times 100, \quad (4) $$

where $\Delta_{LVB}$ measures the deviation of Eq. (1) from the perfect equilibrium and is zero where the LVB is exactly correct. The regions where $|\Delta_{LVB}|$ is smaller than 10% are considered LVB-valid areas at first order.

The western boundary region (4°-width band), characterized by a high Rossby number, as well as the equatorial band (0°-5°N), where the Coriolis parameter $f$ becomes too small, has been masked and not considered in our quantitative analyses. Additionally, given the significant impact of mixing terms within the mixed layer, we exclude the region between the surface and the maximum mixed layer depth from our analysis when determining the zonal percentage of the basin explained by the LVB.

In this model study, we will divide the tropical and subtropical oceans into three distinct dynamic layers visible in Fig. 2. First, we introduce a new concept: the tachocline. This concept represents a region in a dynamically stratified water column, which separates rapid flowing waters from an almost
<figure><img src="image_4.png" /><figcaption>FIG. 2. Spatial mean of the absolute velocity field over the tropical and subtropical bands (5°–35°N), excluding the 4°-width band along the western boundary as well as the mixed layer. The distance between σ levels on the y axis is proportional to their average width within the basin.</figcaption></figure>

steady region, with velocity amplitude decreasing rapidly with depth. The name “tachocline” is inspired by the construction of concepts such as thermocline, halocline or pycnocline. “Tacho-,” from Greek “tachos,” refers to “velocity, speed,” and “-cline,” from Greek klinein, means “to slope.” Therefore, in Fig. 2, we observe an upper tachocline covering the depth range of the thermocline (not shown) and a lower tachocline within the deep ocean. Between these layers, the intermediate ocean is characterized by minimal and barotropic velocities. In this study, we have identified the maximum mixed layer as the upper boundary condition of the upper tachocline and the isopycnal surface σ 27.1 kg m⁻³, averaged across the subtropical (20°–35°N) and tropical bands (5°–20°N), as the lower boundary. Note that the subtropical upper tachocline extends to approximately σ 27.1 kg m⁻³, while the tropical upper tachocline reaches around σ 26.75 kg m⁻³. In the midlatitudes (35°–50°N), the upper tachocline extends to σ 27.5 kg m⁻³ (not shown). The intermediate ocean is bounded by σ 27.1 and 28 kg m⁻³. Last, the lower tachocline in the deep ocean is bounded from above by σ 28 kg m⁻³ and from below by the seabed.

Given our interest in assessing the horizontal circulation patterns and the extension of the LVB description of each of these three domains, we identified three representative isopycnal surfaces at σ 26, 27.78, and 28.05 kg m⁻³.

b. Vertical velocity estimation from the linear vorticity balance

The current state of the observations at basin scales only allows retrieving geostrophic flows. Although we are assessing the validity of the total LVB, this study aims to evaluate the feasibility of reconstructing w from observation-based data and LVB. Therefore, we derive the three-dimensional w by vertically integrating the geostrophic component of the LVB (GLVB):

$$w_{\text{GLVB}}(z) = w(z_{\text{ref}}) - \int_{z}^{z_{\text{ref}}} \frac{\beta v_g}{f} dz'. \quad (5)$$

In fact, the geostrophic LVB and the $w_{GLVB}$ can be derived from the divergence of the geostrophic flow in a β plane (Pedlosky 1996):

$$\frac{\partial u_g}{\partial x} + \frac{\partial v_g}{\partial y} + \frac{\partial w_g}{\partial z} = 0, \quad (6)$$

where $u_g = -1/f\rho_0\partial_\gamma p$, $v_g = 1/f\rho_0\partial_x p$, and p is the pressure. Considering a β plane, the GLVB is retrieved:

$$\beta v_g = \frac{\partial w_g}{\partial z}, \quad (7)$$

where $w_g$ is the geostrophic component of the vertical flow. According to the GLVB, when accounting for the β-plane effect, the geostrophic meridional transport of the planetary vorticity is compensated by the vortex stretching induced by the vertical transport. By applying the indefinite depth-integral to Eq. (7), we obtain the equation for $w_{GLVB}(z)$ [Eq. (5)]:

$$w_g(z) = w_g(z_{\text{ref}}) - \int_z^{z_{\text{ref}}} \frac{\beta v_g}{f} dz'. \quad (8)$$

To compute $w_g(z_{\text{ref}})$, one method assumes a level of no motion where all velocity vector components are null. However, while recent studies (e.g., Koelling et al. 2020) support this assumption, the depth of this level is uncertain for computations based on data from the real ocean. Alternatively, satellite observations provide variables with extensive spatial and temporal coverages, making them adequate for use as boundary conditions. Therefore, we propose estimating $w_g(z_{\text{ref}})$ at the surface.

The total vertical velocity w is the sum of the ageostrophic $w_{ag}$ and the geostrophic $w_g$ components. Since the vertical velocity must vanish at the ocean surface in a steady state, both components are opposite (Pedlosky 1996). For this purpose, let us recall that $w_{ag}$ is generated by the divergence of the wind-driven ageostrophic flow. When neglecting any other ageostrophic flow, it is therefore sufficient to estimate the Ekman pumping vertical velocity at the surface [$w_{Ek}(z = 0)$]. Within the Ekman layer, dissipation brings the wind-driven
flow to zero at its bottom, allowing us to consider its vertical component as null there (Ekman 1905). Therefore, vertically integrating the Ekman transport over this layer thickness yields

$$w_{Ek}(z=0) = -\frac{1}{\rho_0} \operatorname{curl} \left( \frac{\tau_{surf}}{f} \right), \quad (9)$$

where $z=0$ denotes the surface (Pedlosky 1996), $\rho_0$ represents the water density at the surface, and $\tau_{surf}$ is the wind stress.

The classical view of the Ekman pumping vertical velocity locates it at the bottom of the Ekman layer, neglecting the geostrophic contribution to the horizontal flow (Marshall and Plumb 2007). However, the contribution of the geostrophic flow divergence in the Ekman layer is found to be about 10 (30)% of the Ekman pumping amplitude in the upper 50 (100)-m depth (not shown). In the context of the Ekman pumping described in Eq. (9), e.g., over the downwelling gyre, the Ekman transport convergence within the Ekman layer is balanced by a downward vertical flow from the Ekman layer into the ocean interior, evacuating water masses accumulating due to the Ekman flow.

Thus, we derive the expression for the GLVB vertical velocity (or geostrophic vertical velocity) as a function of depth:

$$w_{GLVB}(z) = w_g(z) = -w_{Ek}(z=0) - \int_z^0 \frac{\beta v_g}{f} dz'. \quad (10)$$

### c. The reference ocean model and the spatial filtering

The OGCM of reference for this study is the Nucleus for European Modelling of the Ocean (NEMO; Madec and the NEMO Team 2017). It embeds an ocean general circulation model [Océan PARallélisé (OPA)] and a sea ice model [Louvain-la-Neuve ice model (LIM)] (see http://www.nemo-ocean.eu/). The dataset used in this study originates from the Oceanic Chaos—Impacts, Structure, Predictability (OCCITENS) simulation (Bessières et al. 2017), conducted over the period 1960–2015, after a spinup of 26 years, at eddy-permitting resolution 1/4° (ORCA025 grid Arakawa type C cell) and forced by Drakkar forcing set (DFS5.2) set (Dussin et al. 2016). The OCCITENS simulation provides monthly 3D estimates of oceanic variables that would be impossible to produce from observations alone, such as the vertical velocity field. Thermohaline and sea surface height fields are used to compute the neutral density field following the Jackett et al. (2006) formula. Furthermore, the wind stress is used to compute $w_{Ek}$ [Eq. (9)]. The geostrophic velocity field is computed using the geostrophic equation applied to the pressure field, which is calculated using sea surface height and integration of the hydrostatic equation. The code used to compute the geostrophic velocities is available online (https://github.com/meom-group/CDFTOOLS).

Previous studies on the validity of the Sverdrup balance, including Thomas et al. (2014), Wunsch (2011), and Gray and Riser (2014), and studies analyzing the complete barotropic vorticity, such as Gula et al. (2015), Le Bras et al. (2019), Sonnewald et al. (2019), and Khatri et al. (2024), have highlighted the significant impact of the model spatial resolution and horizontal low-pass filtering on the role and assessment of each term of the

<figure><img src="image_5.png" /><figcaption>FIG. 3. A 56-yr average LVB validity extent (%) over isopycnal levels in the unmasked region outside the 4° belt from the western boundary, as a function of spatial smoothing size from 1° to 7° kernel.</figcaption></figure>

vorticity balance. To account for the spatial resolution dependency of the LVB, we conducted a low-pass sensitivity analysis with filter kernel sizes ranging from 1° to 7° (Fig. 3). The $w$ in the OGCM is a diagnostic variable derived from the horizontal flow in the continuity equation, exhibiting smaller structures than the horizontal velocity field. Hence, prior to the above analysis, a 1° filter is applied to the $w$ field to remove the spatial high frequency and match its scale to that of the horizontal flow components.

Figure 3 displays the 56-yr average LVB validity extension as a function of depth and filter scale ranging from 1° to 7°. As the filtering kernel size increases, the LVB validity extension for a given isopycnal surface expands within the intermediate and deep oceans. This occurs because the filter ensures that the motion is of a sufficiently large scale for the LVB to remain valid. However, for larger scales, particularly within the upper ocean where the open ocean circulation already exhibits basin-scale features at its native resolution, the filter extends the error near the coasts. Thus, a 5° filter maximizes the LVB validity extension across the water column. It reduces errors along the lateral boundaries, smooths out small-scale structures within the intermediate and deep oceans, and preserves the core circulation patterns. This choice allows us to focus on the large-scale ocean dynamics that the LVB formulation can successfully describe.
<figure><img src="image_6.png" /></figure>

FIG. 4. (a) A 56-yr averaged velocity field from OGCM simulation over $\sigma$ 26 kg m$^{-3}$ representative of the upper tachocline–upper ocean. Vertical velocities are represented with colored-filled contours, and horizontal velocities are represented with streamlines. The black-dashed contours represent the depth of the isopycnal surface (in meters). (b) A 56-yr Ekman pumping vertical velocity [$w_{Ek}$; Eq. (9)] computed from the OGCM simulation wind stress. The schematic streamlines represent the main upper ocean gyres and currents of the North Atlantic Ocean. Adapted from Schmitz and McCartney (1993) and Tomczak and Godfrey (1994). The equatorial band (0°–5°N) has been masked.

### 3. The climatological velocity field in OCCITENS simulation

The 56-yr average of the OGCM vertical and horizontal velocity fields is presented in Figs. 4–6 over three isopycnal surfaces ($\sigma$ 26, 27.78, and 28.05 kg m$^{-3}$), representing the upper tachocline, corresponding to the upper tachocline in the upper ocean, the intermediate ocean, and the lower tachocline in the deep ocean. Figures 4 and 6 also depict the Ekman pumping vertical velocity $w_{Ek}$ and the bottom vertical velocity, respectively, offering insights into how boundary conditions influence the circulation in the ocean interior.

The TG (Fig. 4a), as described by Schott et al. (2004), displays a counterclockwise circulation around the Guinea Dome (Stramma et al. 2005). On the other hand, the STG (Schmitz and McCartney 1993) is characterized by a clockwise circulation. A strong zonal flow is evident in the southern part

of both gyres, the equatorial band and the open ocean Gulf Stream extension (GS extension).

The Ekman pumping vertical velocity $w_{Ek}$ (Fig. 4b) shows an upwelling over the TG, with its maximum occurring near the eastern boundary between 10° and 20°N. Over the STG, the downwelling is centered at 50°W over the southern part of the STG. The Gulf Stream region, including its separation from the coast and its open ocean extension, is characterized by upward movements. The subpolar gyre (SPG) exhibits an upwelling centered in the southeastern part of Greenland with a coastal downwelling along Greenland. These results are consistent with previous computations of $w_{Ek}$ using observed datasets (e.g., Xie and Hsieh 1995).

The vertical velocity field within the upper tachocline–upper ocean (colored-filled contours in Fig. 4a) is largely similar to $w_{Ek}$ spatial patterns (Fig. 4b), evidencing the role of the atmospheric forcing as the primary contributor to the vertical flow within the upper ocean (Huang and Russell 1994). The sign and amplitude of $w_{Ek}$ evidently shape the isopycnal surfaces (Gill 1982). Over $\sigma$ 26 kg m$^{-3}$, the cyclonic circulation of the TG is associated with positive $w$ (Lazar et al. 2002) with maximum upwelling values within the Canary upwelling system (Arístegui et al. 2009). Conversely, the anticyclonic circulation of the STG is characterized by negative velocities, which deepen the isopycnal surfaces. In addition, the Gulf Stream is associated with an overall downwelling along the coast and an upwelling after its separation from the continent, featuring meanders and eddies. The differences in intensity between the upper ocean $w$ and $w_{Ek}$ illustrate the baroclinic nature of the field within the upper tachocline, as depicted in Fig. 2.

Let us examine now the intermediate ocean that we chose to bound by $\sigma$ 27.1 kg m$^{-3}$, representing the bottom of the upper tachocline, and 28 kg m$^{-3}$ where the circulation starts significantly intensifying as shown in Fig. 2. The $\sigma$ 27.78 kg m$^{-3}$ surface (Fig. 5) extends across the entire basin, revealing the cyclonic SPG and the northernmost anticyclonic STG circulation.

<figure><img src="image_7.png" /></figure>

FIG. 5. As in Fig. 4a, but over $\sigma$ 27.78 kg m$^{-3}$, a representative example of the intermediate ocean. The black-dashed contours represent the depth of the isopycnal surface (in km).
<figure><img src="image_8.png" /></figure>

FIG. 6. (a) As in Fig. 4a, but over $\sigma$ 28.05 kg m$^{-3}$, representative of the deep ocean. The black-dashed contours in (a) represent the depth of the isopycnal surface (in km). (b) OGCM simulation vertical velocities on the deepest level. Note that Madec and the NEMO Team (2017) use the topographic-induced vertical velocity (Zhang and Vallis 2007). The contours in (b) indicate the seafloor depth at 4500, 5000, and 5500 m (from thinner to thicker contours). The dashed line in (b) represents the Mid-Atlantic Ridge border used to split the North Atlantic basin into western and eastern subbasins. White hatching masks the regions above $\sigma$ 28.05 kg m$^{-3}$ for comparison purposes.

Comparing the tropical and subtropical bands in Figs. 4b and 5, it is evident that the influence of surface circulation is limited to north of 35°N. South 35°N, $w_{Ek}$ no longer dominates the spatial patterns of the vertical motions in the intermediate ocean. Here, the w field has a spatial scale of a few degrees without coherent large-scale structure, in accordance with Ollitrault and De Verdière (2014). While the magnitude of the w field changes rapidly with depth within the upper tachocline, it remains relatively constant in the intermediate ocean (Fig. 2). Between 40° and 50°N, the vertical flow exhibits values of the same order of magnitude as within the upper tachocline. The equatorward and downward flow are associated with the STG shifted toward the north with depth (Stramma 1984). In subpolar latitudes, the SPG cyclonic circulation is associated with upwelling.

Regarding the deep tachocline circulation in the deep ocean, Fig. 6a displays the three-dimensional flow pathways over $\sigma$ 28.05 kg m$^{-3}$. The western and eastern basins, separated by the Mid-Atlantic Ridge (Fig. 6b), show clockwise basin-scale circulations constrained by the topography. Indeed, the interaction of the deep horizontal circulation with bathymetry generates a vertical velocity field at the bottom (Fig. 6b), expressed as $w_B = -u_B \cdot \nabla D$ (where $u_B$ is the horizontal velocity vector at the seabed and $\nabla D$ is the spatial gradient of the bathymetry) (Zhang and Vallis 2007). Again, the similarities between the deep ocean and bottom vertical velocity fields (Figs. 6a,b) highlight that the deep ocean vertical flow is controlled by the bottom horizontal circulation and the varying bathymetry. This result is consistent with the high spatial correlation between the bottom and deep ocean vertical velocities obtained by Liang et al. (2017). However, note that the OGCM’s reconstruction of the deep circulation has limited confidence. The deep and abyssal flows remain undersampled, with scarce observations in the real ocean (Levin et al. 2019). Therefore, obtaining a clear large-scale picture of the climatological circulation within the deep ocean is challenging, and the patterns and amplitudes shown in Fig. 6a

should be interpreted with caution, as they are specific for the OGCM’s case.

Given these three large circulation patterns, we aim to assess the extent to which the LVB characterizes them and allows a relationship between the meridional and vertical flows.

## 4. Linear vorticity balance validity

First, we analyze the multidecadal climatology of the two LVB terms within the upper tachocline–upper ocean, intermediate ocean, and lower tachocline–deep ocean. This section seeks to evaluate the LVB as a function of depth, quantifying its ability to capture climatological ocean dynamics in layers that are remote from the impact of wind and bathymetry forcings. Moreover, this approach is expected to offer insights into regions within the basin where reconstructing the w field using an indefinite integral of the LVB is feasible, thus enhancing our understanding of circulation patterns. In the final subsection, we will explore the dependency of the extension of LVB validity to the average period.

### a. Upper tachocline–upper ocean

The $|\Delta_{LVB}|$ displayed over $\sigma$ 26 kg m$^{-3}$ in Fig. 7c allows us to unveil where and to what extent this balance characterizes the 56-yr averaged meridional and vertical circulation. The $|\Delta_{LVB}|$ exhibits values smaller than 10%, considered as LVB-valid regions, in the eastern part of the TG where the poleward Mauritian Current (MC; Fig. 4b; Fernandes et al. 2005) and the basinwide equatorward North Equatorial Current (NEC; Fig. 4b; Schott et al. 2002), except for the recirculation core of the gyre, adjacent to the Gulf Stream, with a much weaker flow. For these LVB-valid areas, it appears that the vortex stretching ($\oint_{z}w$; Fig. 7b) is balanced by the planetary vorticity transport ($\beta\nu$; Fig. 7a). It corresponds to the positive (negative) vorticity contribution over poleward (equatorward) flow regions. The sign of the vortex stretching term corresponds to the general downward decrease of the amplitude of the
<figure><img src="image_9.png" /><figcaption>FIG. 7. The 56-yr averaged filtered LVB terms (advection of planetary vorticity and vortex stretching) and (c) $|\Delta_{LVB}|$ [Eq. (1)] over (a),(b) $\sigma$ 26 kg m$^{-3}$. Horizontal velocities are represented with streamlines over (a) (as in Fig. 4). Positive (continuous contours) and negative (dashed contours) vertical velocities have been plotted in (b). Dotted areas in (c) represent regions where the meridional velocity field is in GB with a relative error smaller than 10%. Hatched areas represent the regions where the geostrophic LVB is valid at 10%. The black and green translucent regions indicate the maximum mixed layer mask and the 4°-width band along the western boundary masked region for the zonal percentage extent of valid area computation.</figcaption></figure>

upper tachocline flow (Fig. 2). In the northwestern part of the TG (solid contours in Fig. 7b from 30°–50°W to 10°–15°N), the LVB holds within the NEC exhibiting negative $\partial_z w$ values, unlike the rest of the TG. The divergence of the equatorward flow in shallow layers (not shown) evacuates the downwelling waters from the Ekman layer, changing the sign of the vertical flow. This change displaces the gyre to the north and west with depth, as shown by Elmoussaoui et al. (2005). Note that the Ekman pumping-induced upwelling reaches 35°W at the surface, while the $\sigma$ 26 kg m$^{-3}$ upwelling in the upper tachocline’s tropical gyre extends up to 50°W.

To integrate the LVB from the surface to the depth or sigma of interest, it must hold throughout the entire depth range. Therefore, we now examine the vertical extension of the LVB regions in more detail. Figure 8 shows the zonal fraction of the LVB-valid area at each isopycnal surface and latitude. To address the Eastern Atlantic Currents separately from the contribution of the western boundary currents that reduce this fraction, the North Atlantic basin is divided between western and eastern halves separated by the Mid-Atlantic Ridge (Fig. 6b). To the south, it confirms that the TG departs largely from the LVB, except south of 12°N (Fig. 8b) where the flow, corresponding to the

MC, is essentially meridional. At tropical latitudes, the thermocline (and therefore the upper tachocline) is thin (Salmon 1982), and consequently, the LVB holds over a shallower extension. However, in the North Equatorial Counter Current (NECC; Fig. 4b; Schott et al. 2002), with a predominantly zonal character, the zonal extent LVB’s validity drops to a few spots with a maximum of 60% in the western tropical upper tachocline (Fig. 8a). The LVB validity extends zonally across up to 90% of the basin until $\sigma$ 27.1 in the central region of the subtropical band, between 20° and 30°N. Currents such as the equatorward Canary Current (CC; Fig. 4b; Stramma et al. 2005) and the NEC in the interior of the STG are almost entirely in LVB. At higher subtropical latitudes, between 30° and 35°N, in the vicinity of the Azores Current (AC; Fig. 4b; Klein and Siedler 1989), the LVB accounts for about 80% of the zonal extent of the eastern and western subbasins. The limited extent of LVB validity in the western half of the North Atlantic basin between 35° and 40°N (Fig. 8a) illustrates the importance of nonlinearities in the Gulf Stream. Moving between 40° and 50°N, the LVB also plays a significant role in the region where the poleward flow of the North Atlantic Current (NAC; Fig. 4b; Rossby 1996) is found. It covers up to 90% of the zonal extent near the surface in the
<figure><img src="image_10.png" /></figure>

FIG. 8. Zonal extent (%) of LVB validity ($|\Delta_{LVB}|$ < 10%) at each isopycnal surface level and latitudinal grid point (surface and subsurface tropical bands not represented) considering only the (a) western and (b) eastern parts of the North Atlantic basin defined by the Mid-Atlantic Ridge (Fig. 6b) within the unmasked region. The hatching represents the net equatorward circulation when the meridional circulation is zonally averaged over solely LVB-valid regions. The distance between $\sigma$ levels on the y axis is proportional to their average width within the basin with a zoom between the surface and $\sigma$ 27.1 kg m$^{-3}$. The red arrows indicate the isopycnal levels chosen as representative of the upper, intermediate, and deep oceans.

western basin below and more than 50% around $\sigma$ 27.5 kg m$^{-3}$ in the eastern basin.

In addition, to evaluate the capability of the observation-based geostrophic meridional transport in describing vertical motions, we analyzed the extent where GLVB [Eq. (7)] and the geostrophic balance (GB) can be considered valid ($|\Delta_{GLVB}| = |0.5(\beta v_g - f\partial_z w)|/([\beta v_g] + |f\partial_z w|) \times 100|$, $|\Delta_{GB}| = |0.5(v_g - v_t)/([v_g] + |v_t|) \times 100| < 10\%$) (depicted by hatched and dotted areas in Fig. 7c). This analysis demonstrates that the LVB is geostrophic within the STG and the eastern part of the TG. The overall overlap of the areas where the GLVB and geostrophic balance are valid suggests that while the geostrophic balance is necessary to close the

geostrophic linear vorticity balance, it may not be needed for the total case.

b. *Intermediate ocean*

As the flow decreases below the upper tachocline, one expects the smaller terms of the vorticity balance to become more important compared to the linear terms. However, there is a qualitative agreement between $\beta v$ and $f\partial_z w$ terms over $\sigma$ 27.78 kg m$^{-3}$ (Figs. 9a,b) away from the western boundaries where both terms are mainly positive. Further north, the sign changes, but the terms still agree quite well at first glance. Their values are one order of magnitude larger there than over the tropical and subtropical bands, associated with a
<figure><img src="image_11.png" /><figcaption>FIG. 9. As in Fig. 7, but over σ 27.78 kg m⁻³, a representative example of the intermediate ocean.</figcaption></figure>

more intense circulation. This circulation is predominantly
southwestward between 30° and 50°N, below the NAC, and
northeastward at higher latitudes. They contribute, respec-
tively, with positive and negative vorticities to the LVB.

Figure 9c shows that the LVB holds irregularly across most of the eastern tropical, subtropical, and extratropical bands, away from lateral boundaries. The discrepancy in the eastern boundaries, evident in the upper and intermediate ocean, is likely a result of the diabatic effect of eddy fluxes originating from the eastern boundary currents (Cessi and Wolfe 2013). The intermediate ocean exhibits smaller LVB-valid areas than the upper layers, particularly in the western half of the tropical-subtropical domain, where alternative meridional bands with a relative error larger than 10% are visible. However, the LVB validity extends until reaching the northernmost region of the anticyclonic STG, around 55°N. The zonal extent of LVB validity reaches up to 70% of the basin width away from the masked regions within the subtropical, extratropical, and eastern tropical latitudes (Fig. 8). Moreover, poleward of 50°N, the zonal extent of the LVB drops drastically, accounting for only a maximum of 50% in the eastern half of the basin.

Regarding the nature of the LVB, as expected, one notices
that the geostrophic component of LVB is dominant for the
geostrophic flow north of about 30°N (Fig. 9c). There, the
steeper isopycnal surfaces (contours in Fig. 5) compared with
the subtropical and tropical bands drive more intense geo-
strophic currents.

c. *Lower tachocline-deep ocean*

In the lower tachocline-deep ocean, the validity of the LVB holds over most of the basin except in the Gulf Stream, its adjacent recirculation region, and the narrow regions of the abyssal subbasins within the tropics and midlatitudes (Figs. 8 and 10). The deep ocean upwelling and downwelling horizontal distribution is generally associated with negative and positive vortex stretching values, respectively, in contrast with the sign of this term in the upper ocean. Drawing a parallel with the upper tachocline within the upper ocean, the velocity field in the deep ocean, induced by the interaction of the varying topography and the horizontal circulation ocean bottom, diminishes toward shallower depths (see also Fig. 2). This vertical shear matches the meridional transport direction in the areas where the LVB is valid within the deep ocean. In such proximity to the seabed, the isopycnal surfaces are highly influenced by the topography, promoting larger horizontal density gradients (closer and topography following contours in Fig. 5 compared with Fig. 6a). Note that the GLVB (hatched regions in panel c) characterize large fractions of the deep circulation, particularly those associated with steeper topography.

d. *Sensitivity to averaged period length*

The LVB validity analysis was conducted so far on the longest
multidecadal climatological dataset available to us. However, it
<figure><img src="image_12.png" /><figcaption>FIG. 10. As in Fig. 7, but over σ 28.05 kg m⁻³, a representative example of the deep ocean.</figcaption></figure>

is often not feasible to obtain such a long time frame when using
observations, reanalyses, or shorter runs. One limitation in the
linear vorticity description of the ocean lies in the time required
for the ocean adjustment, determined to first order by the propa-
gation velocity of the first baroclinic Rossby wave, which de-
creases toward higher latitudes (Hughes and de Cuevas 2001;
Wunsch 2011; Thomas et al. 2014). Therefore, a sensitivity study
as a function of the averaging period lengths and latitude is
advocated.

In Fig. 11, the temporal dependence of the LVB validity is
assessed by computing its zonal extent as a function of the
number of years in the climatology for three representative σ
levels in the unmasked region and averaged meridionally over
four different latitudinal bands. Within the upper tachocline,
particularly the subtropical band, a large difference of about
20% is found between 1- and 56-yr climatologies, and each in-
crement up to about 14 years significantly improves the results.
Nevertheless, the extension of the LVB validity shows only a
marginal increase of up to 7 years in the intermediate ocean
and 3 years in the deep ocean. The decay of the extent of LVB
validity in the upper tachocline is more noticeable in the sub-
tropical band than in the tropical band (by a factor of ~4).
However, the decrease in the validity extension within the in-
termediate ocean at shorter time scales is more pronounced in
the tropics than in extratropics.

Our results, based on annual climatologies, assessed the
spatiotemporal extension of the LVB and GLVB throughout
the North Atlantic basin. In the following section, we tackle

the possibility of estimating vertical velocities using GLVB
and the observation-based meridional geostrophic velocity
products by assessing the computation of such geostrophic
component in our reference simulation.

5. Estimation of vertical velocities

a. 56-yr average

Following the methodology presented in section 2, we com-
puted the geostrophic vertical velocity (w<sub>GLVB</sub>) applying Eq. (5)
to the geostrophic meridional velocities and w<sub>Ek</sub> (Fig. 4b) as
the surface boundary condition at an annual frequency. The
multidecadal mean of w<sub>GLVB</sub> is presented in Fig. 12 over the
three representative σ surfaces.

The w<sub>GLVB</sub> field appears to be qualitatively similar to the
reference OGCM output in the open ocean throughout the
water column, as illustrated in Fig. 12. It captures the hori-
zontal patterns across most of the subtropical, extratropi-
cal, and eastern tropical bands, far from the western
boundary and the western part of the tropical band, where
large LVB errors were observed. However, when quantita-
tively evaluating our estimate with the root-mean-square
error (RMSE) normalized by the 56-averaged target model
output (RMSE/w̄; contours in Fig. 12), it shows significant
amplitude biases.

Within the upper tachocline, RMSE/w̄ (contours in Fig. 12a)
shows values below 0.5 in most of the STG and TG, with values
<figure><img src="image_13.png" /></figure>

FIG. 11. The mean zonal extent (%) of LVB validity dependent on the averaged period over the 5°–20°N, 20°–35°N, 35°–50°N, 50°–65°N bands and the full basin over $\sigma$ (a) 26, (b) 27.78, and (c) 28.05 kg m$^{-3}$.

above 1 found in the western boundaries and zonal tropical currents where the flow is not in GLVB. The bias exceeds the 56-yr averaged reference within most of the intermediate ocean (Fig. 12b). Only in some regions of the eastern part of the extratropical bands, $w_{GLVB}$ reproduces the vertical motion of the OGCM’s simulation output with a normalized RMSE smaller than 0.5. These regions in extratropical latitudes coincide with areas described by GLVB (Fig. 9c). The overall coherence of the deep vertical pathways between $w_{GLVB}$ (Fig. 12c) and the reference evidences the lesser influence of the Ekman forcing compared with the bathymetry. However, $w_{GLVB}$ fails in accurately reproducing the amplitude of the OGCM $w$ in most of the basin. Overall, despite the very satisfying qualitative agreement of the reconstruction, the relatively large biases indicate that while the geostrophic component of $w$ establishes the order of magnitude of the total vertical flow, the ageostrophic component (coming from the ageostrophic planetary vorticity advection and the non-linear contribution of the vorticity balance) is necessary to close the vorticity budget.

<figure><img src="image_14.png" /></figure>

FIG. 12. A 56-yr averaged vertical velocity estimate ($w_{GLVB}$) and normalized RMSE [RMSE/$\bar{w}$; 0.1 (white), 0.5 (light gray), 1 (dark gray) contours] of $w_{GLVB}$ estimate with OGCM $w$ output over $\sigma$ (a) 26, (b) 27.78, and (c) 28.05 kg m$^{-3}$ using OGCM data. The black translucent region indicates the maximum mixed layer mask. The equatorial band has been masked.

After analyzing the average of the entire available period, the feasibility of employing the LVB at time scales shorter than the longest multidecadal climatological dataset available enables an exploration of the temporal dimension in the vertical velocity estimates.
<figure><img src="image_15.png" /><figcaption>FIG. 13. Time series of the spatially averaged OGCM *w* (continuous line) and $w_{GLVB}$ estimate (dashed line) over the regions bounded by (a) 55°–30°W, 15°–30°N and (b) 30°–16°W, 8°–16°N at $σ$ 26, 27.78, and 28.05 kg m$^{-3}$. Shading represents the standard deviation of the OGCM *w* output.</figcaption></figure>

b. *Temporal variability*

Finally, we aim to determine the ability of $w_{GLVB}$ to capture the interannual variability. The interannual variability of the $w_{GLVB}$ estimates and the reference over the three representative $σ$ surfaces, considering two spatially averaged example regions bounded by (55°–30°W/15°–30°N) and (30°–16°W/8°–16°N), is displayed in Fig. 13. Notably, the reconstruction accuracy is high in terms of the phase and amplitude of the variations. The correlation coefficients between $w_{GLVB}$ and the model’s $w$ are above 0.9, except for the upwelling over the deep sigma surface. The biases between $w_{GLVB}$ and the OGCM $w$ are now quantified concerning the variability amplitude using the RMSE normalized by the standard deviation (std) of the OGCM $w$ (RMSE/std). The synchronization between the time series and the amplitude of the interannual variability leads to an overall relative minor bias between the $w_{GLVB}$ and OGCM output, except for the abysses. The remarkable reconstruction of the interannual

variability of the total $w$ field by the geostrophic LVB, even at periods shorter than the basin-crossing time scale, suggests that the estimated vertical velocities serve as a signal of propagating Rossby waves. The westward-propagating anomalies of sea surface height and $w$ in the ocean interior within the subtropical band (not shown) are synchronized and exhibit a phase speed of approximately 5 cm s$^{-1}$, consistent with the characteristic phase speed of baroclinic Rossby waves at these latitudes (Chelton et al. 2007; Sanchez-Franks et al. 2021).

Therefore, we now evaluate this accuracy over the entire North Atlantic basin. Figure 14 quantifies the synchronization between the fields by displaying the correlation coefficient and the normalized RMSE (RMSE/std). In the subtropical band, both the upper and intermediate oceans (Figs. 14a,b) exhibit that correlation coefficients between $w_{GLVB}$ and the model’s $w$ are above 0.8, excluding areas close to the western boundary, as well as the eastern tropical and polar bands. However, although the southwestern current between 40° and 50°N
<figure><img src="image_16.png" /></figure>

FIG. 14. Annual correlation coefficient (shading) and normalized RMSE (RMSE/std; white contours for 0.5, black contours for 1) between the $w_{GLVB}$ and the OGCM $w$ output over $\sigma$ (a) 26, (b) 27.78, and (c) 28.05 kg m$^{-3}$. The black translucent region indicates the maximum mixed layer mask. The equatorial band has been masked. (d) Zonal extent (%) of correlation coefficient values above 0.8 at each isopycnal surface level and latitudinal grid point (surface and subsurface tropical bands not represented) considering only the western (upper panel) and eastern (lower panel) parts of the North Atlantic basin defined by the Mid-Atlantic Ridge (Fig. 6b) within the unmasked region. The distance between $\sigma$ levels on the y axis is proportional to their average width within the basin with a zoom between the surface and $\sigma$ 27.1 kg m$^{-3}$.

is in GLVB, the reproduced variability does not synchronize with the reference. In the deep ocean (Fig. 14c), only the sub-tropical band maintains correlation coefficient values above 0.8. Within the upper tachocline, note the high correlation even in regions where the flow deviates significantly from the LVB, such as a large portion of the northeastern STG (Fig. 14a) where the mixed layer reaches this sigma surface, also areas along the Africa coast from 12° to 20°N. High correlation values are also found in areas with large LVB errors in the deep ocean and in the east and subtropical intermediate ocean. The reconstruction of the interannual variability is, therefore, accurate over a much larger part of the ocean than expected.

However, the variability cannot be derived from the GLVB over two extensive regions characterized by low correlation. The first region encompasses the lower tropical band, extending up to 10°N within the upper tachocline and 15°N at the intermediate and deep oceans. The second region includes the flow near the Gulf Stream (between 35° and 55°N in the western basin) and its northeastern extension into the open ocean, reaching the coasts of Northwest Europe (around 50°N in the eastern basin). These two regions exhibit much larger interannual variability

than the region of high correlation. In fact, the ageostrophic circulation accounts for nonnegligible interannual variability, preventing an accurate computation based on the GLVB.

Examining the whole water column (Fig. 14d), high correlation is observed from top to bottom from about 12° to 15°N up to 30°N in the western part and near 45°N in the eastern part. Note that in the western part north 55°N, a narrow region in the open ocean bounded by the Mid-Atlantic Ridge and the mixed layer masked exhibits high values across the water column. Overall, these results evidence the paramount contribution of the geostrophic flow, specifically its meridional component, in dominating the interannual variability of the OGCM vertical circulation over the basin, except for the two regions of high ageostrophic variability: one in the deep tropics and the other along the diagonal band extending from Cape Hatteras and northwest Europe.

One can interpret in two ways the fact that the reconstructed variability is accurate over larger regions than those in LVB. Either the ageostrophic component of $w$ has a negligible contribution to the interannual variability, or it shares the same variability as the geostrophic $w$. Figure 13 shows an almost constant offset between time series across the observed period, suggesting that
the ageostrophic component's influence on the interannual variability of $w$ is negligible within these regions.

**6. Discussion on the relevance of the linear vorticity balance for interpreting the ocean circulation**

The large domain within the upper tachocline in GLVB allowed us to explain the major currents of the North Atlantic by revisiting the principles of the linear theory of large-scale circulation. By flowing equatorward, the geostrophic component of currents like the NEC, the CC, and the AC are divergent, absorbing downwelling Ekman layer waters accordingly. To the contrary, poleward currents like the NECC and the MC converge and provide upper ocean waters to balance the divergent Ekman transport above, feeding the local TG upwelling. Thus, the applicability of the LVB suggests that the direction of the geostrophic meridional component of the flow of the upper tachocline is influenced by both the sign of the divergence of the Ekman transport above and the decrease in the vertical velocity amplitude with depth across the upper tachocline. This latter view corresponds to the classic interpretation of the Sverdrup balance when applied to the upper ocean.

The LVB brings a novel baroclinic analysis of the classical barotropic description of the ocean made by the Sverdrup balance. However, let us recall that the Sverdrup balance needs the LVB and proper boundary conditions. The extensive areas within the upper tachocline identified in LVB are comparable to valid regions observed in Sverdrup balance studies using reanalyses (Wunsch 2011; Thomas et al. 2014) or observation-based dataset (Gray and Riser 2014). Nevertheless, the eastern part of the TG, described by the LVB in this study, does not satisfy the Sverdrup balance according to previous studies such as Wunsch (2011) and Gray and Riser (2014). Such apparent contradiction for flow in LVB occurs when the $w$ field, chosen as the boundary condition to close the Sverdrup balance, is inaccurately estimated. This poorly constrained boundary $w$ field is often at the bottom thermocline (e.g., Kounta et al. 2018), but it can also be the surface $w$ field. The latter generally suffers from considering the Ekman pumping as the sole contributor to the vertical flow in the Ekman layer while neglecting the contribution of the $\beta$-plane geostrophic divergence, as explained in the present study.

The departure from the LVB at the large spatiotemporal scales analyzed in the present paper can be only explained by the importance of nonlinear terms in the vorticity balance [Eq. (2)]. Previous researchers have turned to a barotropic approach to investigate their contribution. Numerous studies have studied the barotropic vorticity balance (BVB) to address the vorticity source and sink mechanisms that the Sverdrupian description of the ocean cannot fully account for.

In both the Gulf Stream region and the SPG, the LVB does not hold along the entire water column (Figs. 7c, 9c, and 10c). In these areas, additional terms are needed to close the vorticity budget. In the Gulf Stream, the bottom pressure torque (BPT) effectively balances the barotropic planetary vorticity advection, with the wind forcing being negligible (e.g., Hughes and de Cuevas 2001; Gula et al. 2015; Schoonover et al. 2016). Regarding the SPG, Le Corre et al. (2020) demonstrate that

the bathymetry forcing (bottom drag curl) and atmospheric forcing, although with lesser influence, are needed to balance the BPT contribution. The observed reduction in the validity extent of the LVB at shorter time scales, visible in Fig. 11, may be attributed to the influence of small-scale nonlinear flows within the open ocean. Similarly, a model study by Khatri et al. (2024) evidences an increase in nonlinear dynamics when diminishing the scale of the low-pass filter. These results are coherent with the fact that oceanic flows' temporal and spatial dimensions often vary with a certain level of proportionality (e.g., Dickey 1991). A key question remains whether OGCM grid's spatial resolution might change the vorticity balance significantly, even after low-pass space filtering. Ongoing research on the vorticity equation in a high-resolution OGCM suggests that the nonlinear advection term is responsible for the first-order balancing of the LVB residual across the upper ocean (J. Gula 2023, personal communication).

Over a substantial portion of the open ocean STG where we have shown that the LVB holds, previous studies (e.g., Khatri et al. 2024; Sonnewald et al. 2019) evidence the necessity of the BPT to close the vorticity budget. To further compare our baroclinic results with barotropic studies, we may also refer to studies considering the depth integration of the vorticity balance (DIVB; see Waldman and Giordani 2023). In Waldman and Giordani (2023), a comparison between the BVB and DIVB approaches supports the equivalence at first order between the BPT term and the bottom vortex stretching ($f_w$; where $w_b$ is the vertical velocity at the bottom of the ocean, being the bottom boundary condition of the $f_0 z w$ term). The mainly geostrophic horizontal velocities at the deep ocean close to the lateral boundaries (dotted surfaces in Fig. 10c), together with the minimal contribution of nonlinear terms to the deep ocean vorticity budget (small $|\Delta_{LVB}|$ in Fig. 10c), allows considering that the BPT is equal to the bottom vortex stretching (Zhang and Vallis 2007). Consequently, a large volume of tropical and subtropical oceans explained by the LVB throughout the water column aligns with the regions where the Sverdrup and topographic Sverdrup (where BPT closes the Sverdrup balance budget; Holland 1967) balances hold within the open ocean (Khatri et al. 2024).

**7. Conclusions**

In this numerical study based on an eddy-permitting OGCM simulation, we have shown that the linear vorticity balance (LVB) approximation, spatially filtered with a 5° low-pass filter and averaged over 56 years, is valid within the upper ocean across most of the STG and eastern TG. The dominance of the LVB diminishes in the intermediate ocean, particularly at tropical latitudes and the western half of the basin. In the deep ocean, the topography-induced circulation is largely explained by the LVB, notably over the abyssal plains. Throughout the entire water column, significant departures from the balance are observed in the Gulf Stream region, the equatorial band, the subpolar gyre, and regions with a mainly zonal flow. Here, the vorticity equation can no longer be assumed linear, and the nonlinear terms, not studied in the present work, become essential to close the vorticity budget. Over the upper ocean, the
mainly geostrophic nature of the LVB determines how the meridional component of the flow responds to the geostrophic vertical divergence. However, the geostrophic LVB holds over more limited extensions within the intermediate ocean and the deep tachocline, reducing its a priori usefulness in deriving the geostrophic *w* from the geostrophic flow below the upper tachocline.

The duration of the averaging period conditions the occurrence of the LVB, although not in a uniform way. A general degradation in the extent of LVB validity occurs when the average period is reduced, with the maximum reduction observed being 20% within the upper ocean. This reduction may be attributed to the enhancement of small-scale structures at shorter time scales. Nevertheless, the increase in the LVB-valid extension becomes marginal after several decades, decreasing to 3 years in the intermediate and deep oceans.

The extensive validity of the geostrophic LVB within the upper and lower tachoclines implies that the linear theory of large-scale circulation offers a simple yet reliable explanation for the direction of the meridional geostrophic flow. In the upper tachocline, a convergent (divergent) flow is needed to feed (absorb) the Ekman layer's upward (downward) pumping. This condition, in turn, establishes the direction of the geostrophic meridional flow since only the poleward (equatorward) flows are convergent (divergent). It corresponds to the mere satisfaction of the continuity equation for geostrophic circulation. A similar interpretation can be applied to circulation in vast portions of the deep ocean tachocline when considering the sign of the bottom vertical velocity.

The large percentage of the basin explained by the geostrophic LVB has supported the effort to develop a methodology for estimating the three-dimensional geostrophic *w* field by vertically integrating the geostrophic meridional velocities from the ocean surface. In this study, we have underlined the critical role of the geostrophic flow divergence across the Ekman layer when computing the Ekman pumping at the surface. In climatological mean, the reconstructed geostrophic *w* (*w*<sub>GLVB</sub>) captures at first order the reference *w* horizontal patterns in the ocean interior below the maximum mixed layer over the tropical, subtropical, and extratropical bands, excluding the equatorial band, the Gulf Stream, and the sub-polar gyre. However, the reconstructed *w* does not accurately match the model's output in terms of amplitude, which prevents its use for quantitative analyses of the role of time-mean vertical currents.

On the contrary, $w_{GLVB}$ effectively captures the interannual variability of the reference $w$ over most of the basin throughout the entire water column far from the Gulf Stream, the North Atlantic Current, and the low tropical band. Therefore, the geostrophic meridional velocities appear to be sufficient to compute the interannual variability of the velocity field to first order over a significant part of the open ocean. The discrepancies between assessing the 56-yr average and interannual variability of the vertical velocity field in the open ocean have revealed the dominant role of the geostrophic component of the vertical velocity in the interannual variability. Conversely, the ageostrophic component plays a minor role in the interannual variability but exhibits an important

contribution to the time-mean amplitude. Nevertheless, in the Gulf Stream, North Atlantic Current, and the low tropics, the ageostrophic circulation mainly controls both the interannual variability and time-mean amplitude of the vertical flow.

It is left to future projects to assess how nonlinear terms could improve the accuracy of the baroclinic vorticity balance in describing the ocean flow. Our OGCM study suggests that the geostrophic LVB could serve as a powerful tool to estimate the interannual variability of the upper tachocline vertical movements using observation-based geostrophic meridional velocities away from the specific regions identified in the OGCM. Consequently, an observation-based *w* product might potentially offer valuable insights into the interannual variations of the physical and biogeochemical advective fluxes in the real-world upper ocean.

*Acknowledgments.* We kindly thank Juliette Mignot, Gilles Reverdin, and Diana Ruiz-Pino who have read and commented on versions of this paper. This work was supported by CNES and LEFE-Mercator.

*Data availability statement.* The outputs of the OCCITENS simulation used in the current study to estimate the neutral density and geostrophic velocity fields, the linear vorticity balance terms, the Ekman pumping vertical velocity, and $w_{GLVB}$ are available upon request. Please contact Thierry Penduff at thierry.penduff@cnrs.fr.

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