# Dynamics and Thermodynamics of the Boussinesq
North Atlantic Eddy Kinetic Energy Spectral Budget

Takaya Uchida, Quentin Jamet, Andrew C Poje, Nicolas Wienders, Luolin

Sun, William K Dewar

► To cite this version:

Takaya Uchida, Quentin Jamet, Andrew C Poje, Nicolas Wienders, Luolin Sun, et al.. Dynamics and Thermodynamics of the Boussinesq North Atlantic Eddy Kinetic Energy Spectral Budget. Journal of Advances in Modeling Earth Systems, 2025, 17 (4), 10.1029/2024MS004781. hal-05450740

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
RESEARCH ARTICLE

10.1029/2024MS004781

**Key Points:**

* Continuous wavelet transform is used to diagnose the eddy-kinetic energy (EKE) spectral budget

* Vertical advection intermittently modulates the direction and strength of EKE spectral cascade throughout the North Atlantic

* EKE input from eddy advection of potential temperature tends to counteract that of practical salinity and dynamic enthalpy

**Correspondence to:**

T. Uchida,
takachanbo@gmail.com

**Citation:**

Uchida, T., Jamet, Q., Poje, A. C., Wienders, N., Sun, L., & Dewar, W. K. (2025). Dynamics and thermodynamics of the Boussinesq North Atlantic eddy kinetic energy spectral budget. *Journal of Advances in Modeling Earth Systems, 17*, e2024MS004781. https://doi.org/10.1029/2024MS004781

Received 18 OCT 2024

Accepted 11 MAR 2025

**Author Contributions:**

**Conceptualization:** Takaya Uchida

**Data curation:** Quentin Jamet, Nicolas Wienders, William K. Dewar

**Formal analysis:** Takaya Uchida

**Funding acquisition:** Andrew C. Poje, Nicolas Wienders, William K. Dewar

**Investigation:** Takaya Uchida, Quentin Jamet, Andrew C. Poje, Nicolas Wienders, Luolin Sun

**Methodology:** Takaya Uchida, William K. Dewar

**Software:** Takaya Uchida

**Validation:** Takaya Uchida

**Visualization:** Takaya Uchida

**Writing – original draft:** Takaya Uchida

**Writing – review & editing:**

Takaya Uchida, Quentin Jamet, Andrew C. Poje, William K. Dewar

© 2025 The Author(s). Journal of Advances in Modeling Earth Systems published by Wiley Periodicals LLC on behalf of American Geophysical Union. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

<figure><img src="image_2.png" /></figure>

# Dynamics and Thermodynamics of the Boussinesq North Atlantic Eddy Kinetic Energy Spectral Budget

Takaya Uchida (内田貴也)<fnref n="1" />,²,<fnref n="3" /> ID, Quentin Jamet⁴,⁵, Andrew C. Poje⁶ ID, Nicolas Wienders<fnref n="1" /> ID, Luolin Sun (孙洛琳)²,⁷ ID, and William K. Dewar²,⁷ ID

**Abstract** Statistical characterization of oceanic flows has been a long standing issue; such information is invaluable for formulating hypotheses and testing them. It also allows us to understand the energy pathways within the ocean, which is highly turbulent. Here, we apply the wavelet approach to wavenumber spectral analysis, which has recently been proved to be beneficial in quantifying the spatially heterogeneous and anisotropic nature of oceanic flows. Utilizing an eddy-rich ensemble simulation of the North Atlantic, we are able to examine the spectral transfers of eddy kinetic energy (EKE) and effect of potential energy, here defined via dynamic enthalpy, on the EKE spectral budget. We find that vertical advection of EKE modulates the up- and down-scale direction and strength of EKE spectral flux throughout the North Atlantic domain. The vertical eddy buoyancy flux tends to be small below the mixed layer, suggesting that the flow is largely adiabatic. In maintaining this adiabatic nature, the eddy advection of dynamic enthalpy and practical salinity tend to partially compensate for the eddy advection of potential temperature; this partial cancellation between temperature and salinity is similar to the thermodynamic spice variable.

**Plain Language Summary** Turbulence is defined as a regime where the flow is irregular and chaotic. In other words, it cannot be analytically described nor understood. Oceanic flows fall under this category and we depend on statistical methods to characterize and quantify them. Here, we focus on the North Atlantic Ocean and apply a framework which allows us to examine the statistical features of oceanic turbulence in a spatially local manner, for example, near the Gulf Stream or afar from it. We find that the vertical up- and down-heaval of currents and transport of temperature and salt modulate the turbulent characteristics.

## 1. Introduction

It has been known for some time that turbulence in the real ocean is fundamentally three-dimensional and vertical motion can have first-order effects on the dynamics and tracer transport in highly stratified regions with fronts, jets and vigorous eddy activity (e.g., Balwada et al., 2018; Maze et al., 2013; Pollard & Regier, 1992; Rossby et al., 1985; Siegelman et al., 2020; Uchida et al., 2019; Yu et al., 2019). Yet, the quasi-geostrophic (QG) “dogma” that somehow oceanic flows can be described quasi two-dimensionally everywhere and thus expectations on up-scale energy cascade be universally met sets the oceanographic Overton window to this day. Notable examples are studies where observed wavenumber spectral slopes of sea-surface height or kinetic energy (KE) are ascribed to QG inertial-range predictions (e.g., Le Traon et al., 2008; Richman et al., 2012; Stammer, 1997; Storer et al., 2022; Vergara et al., 2019; Xu & Fu, 2012). The problem here is two folds: (a) different dynamics can lead to similar spectral slopes and the agreement between observations and dimensional analysis may be coincidental especially under states of non-equilibrium (Alexakis & Biferale, 2018; Callies & Ferrari, 2013; Maltrud & Vallis, 1991; Mininni et al., 2009; Valadão et al., 2024; Yassin & Griffies, 2023; Zhou, 2021; Zhu et al., 2024), and (b) studies corroborating the QG-like characteristics often stop short with examining only the horizontal spectral flux of KE (Ajayi et al., 2021; Arbic et al., 2013; Balwada et al., 2022; Khatri et al., 2018; Schubert et al., 2023; Scott & Wang, 2005; Storer et al., 2023). Furthermore, there is a subtle yet note worthy discrepancy between the theory of geostrophic turbulence, which is a theory on eddies

<footnote n="1">¹Center for Ocean-Atmospheric Prediction Studies (COAPS), Florida State University, Tallahassee, FL, USA, ²CNRS, INRAE, IRD, Grenoble INP, Institut des Géosciences de l'Environnement, Université Grenoble Alpes, Grenoble, France,</footnote>

<footnote n="3">³Now at Climate Dynamics Laboratory, Moscow Institute of Physics and Technology, Moscow, Russia, ⁴Service Hydrographique et Océanographique de la Marine (SHOM), Brest, France, ⁵Institut National de Recherche en Sciences et Technologies du Numérique (INRIA), ODYSSEY Group, Ifremer, Plouzané, France, ⁶Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, NY, USA, ⁷Department of Earth, Ocean and Atmospheric Science, Florida State University, Tallahassee, FL, USA</footnote>
(Vallis, 2006), and spectral analyses of oceanic flows, which in practice often take the total (mean + eddy) flow (e.g., Capet et al., 2008).

The rambling above is not to diminish the stupendous success the QG framework has had in elucidating eddy-
mean flow processes in geophysical flows (Charney, 1948; Eady, 1949; Held et al., 1995; Phillips, 1954,
1956, 1990; Vallis, 2006). There is no doubt that QG dynamics constitute an important part of the full dynamics
(e.g., Carli et al., 2024; Z. Chen et al., 2024; Deremble et al., 2023; Early et al., 2011; Grooms et al., 2011;
Lapeyre & Klein, 2006; Smith & Marshall, 2009; Thiry et al., 2024; Tulloch et al., 2011; Zhang et al., 2024).
Given the Boussinesq hydrostatic primitive equations of the ocean, however, vertical advection should also be
considered and is indeed a key component if one is to close the momentum and tracer budgets in the ocean (Y.
Chen et al., 2024; Griffies et al., 2015; Liang et al., 2015; Su et al., 2018; Uchida et al., 2020). Namely, the vertical
spectral flux of KE also contributes to the spectral cross-scale transfer of KE (Delpech et al., 2024; Lind-
borg, 2006; Oks et al., 2017; Srinivasan et al., 2023).

Our study is motivated by Uchida, Jamet, Poje, et al. (2024) in particular; they found that vertical advection of eddy kinetic energy (EKE) non-negligibly affected the direction of up- and down-scale cascade of EKE particularly in the separated Gulf Stream region even at mesoscale-permitting model resolution (Θ(1/12°)). Here, we expand on their results and examine the spatial variability in EKE cascade across the entire North Atlantic domain. We also quantify how the influx from potential energy affects the EKE budget. Potential energy, here, is defined via dynamic enthalpy; it is a natural and dynamically consistent extension of gravitational potential energy for a non-linear equation of state (EOS) for density (Eden, 2015; Jamet et al., 2021; Nycander, 2011; Young, 2010). The fact that a background stratification cannot be easily defined in primitive equations urges us to investigate the effect of thermodynamics and go beyond the QG framework.

In defining the eddies, we take the single-model (as opposed to multi-model) ensemble approach where the mean flow is defined by the ensemble mean and the eddies become the fluctuations about this mean (R. Chen & Flierl, 2015; Uchida, Deremble, & Penduff, 2021; Uchida, Jamet, et al., 2021). This leads to a parameter-free definition of eddies and the ensemble-based Reynolds' decomposition commutes with all space-time derivatives (Jamet et al., 2022). Furthermore, we use continuous wavelet transforms (CWTs) to execute spatially local "pseudo-Fourier" spectral analyses of the eddies. Wavelets have been widely adopted to spectrally analyze non-stationary, anisotropic and inhomogeneous phenomena in the broader field of fluid mechanics (Dolcetti & García Nava, 2019; Farge, 1992; Maurya et al., 2024; Meneveau, 1991), climate science (Lau & Weng, 1995; B. Wang & Wang, 1996; Weniger et al., 2017), seismology (Diallo et al., 2006; Ricker, 1940), and magnetohydrodynamics (Farge & Schneider, 2015; Horbury et al., 2008; Titus et al., 2023). CWTs allow for non-periodicity in the data, and have been demonstrated to successfully extract local and anisotropic features in spatially heterogeneous oceanic flows (Uchida, Jamet, Poje, et al., 2024).

The manuscript is organized as follows: We describe the ensemble data set and wavelet-based spectral approach
in Section 2. Results are presented in Section 3. We end with a Discussion in Section 4.

**2. Data and Methods**

We provide an overview of the ensemble data set and wavelet-based spectral budget of eddy kinetic energy (EKE) with an emphasis on the impact of potential energy on EKE. Mathematical notations are summarized in Table 1 and details on CWT are provided in Appendix A.

**2.1. Eddying Ensemble Simulation of the North Atlantic**

Our data set is a recently developed 48-member eddying (1/12°) ensemble of the North Atlantic Ocean partially
air-sea coupled via an atmospheric boundary layer model (CheapAML; Deremble et al., 2013; Jamet et al., 2019,
2020) using the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall
et al., 1997). This approach is used to mitigate the suppression of thermodynamic air-sea exchanges and surface
ocean dynamics caused by directly prescribing an atmosphere. In CheapAML, atmospheric surface temperature
and relative humidity are prognostically computed over the ocean but are restored toward prescribed values over
land. Other variables, such as 10-m winds, downward longwave radiation, solar shortwave radiation, and
<table><caption>**Table 1**
Notation of the Variables and Description</caption><thead><tr><th>Mathematical notation</th><th>Description</th></tr></thead><tbody><tr><td>u = ue<sub>1</sub> + ve<sub>2</sub></td><td>Horizontal velocity vector</td></tr><tr><td>v = u + wε<sub>3</sub></td><td>Three-dimensional (3D) velocity vector</td></tr><tr><td>u = &lt;u&gt; + u'</td><td>Ensemble-based Reynolds' decomposition</td></tr><tr><td>(·)</td><td>Fast Fourier transform (FFT)</td></tr><tr><td>(·)</td><td>Continuous wavelet transform (CWT, Appendix A)</td></tr><tr><td>˜b(Θ, S, Φ) = −g (˜b(Θ, S, Φ) − ρ<sub>0</sub>) = b(r, z, y, x)</td><td>Tilde notation to distinguish a thermodynamic function (˜p; Jackett & Mcdougall, 1995) from fields in space-time. ρ<sub>0</sub> = 999.8 kg m<sup>−3</sup></td></tr><tr><td>˜b# = ˜b(Θ), (S), Φ) (≈ 〈 ˜b(Θ, S, Φ) 〉)</td><td>Buoyancy assessed at ensemble-mean potential temperature and practical salinity</td></tr><tr><td>˜b&#x20d7; = ˜b − ˜b# (≈ b' = b − 〈b〉)</td><td>Buoyancy fluctuations</td></tr><tr><td>˜H&#x20d7; = g<sup>−1</sup> ∫<sub>θ<sub>0</sub></sub><sup>Φ</sup> ˜b&#x20d7; dΦ<sup>†</sup></td><td>Mean total dynamic enthalpy (MTDE)</td></tr><tr><td>˜h&#x20d7; = ˜h − ˜H</td><td>Dynamic enthalpy fluctuations</td></tr><tr><td>A<sub>K</sub> = A<sub>h</sub> + A<sub>v</sub> (= A<sup>m</sup> + A<sup>e</sup>)<br>= − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> ∇ · (vu<sup>'</sup>)⟩ + ⟨v<sup>†</sup> ∇ · (vv<sup>'</sup>)⟩]<sup>2</sup> x<sub>0</sub><sup>2</sup>k<br>= − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> [∇ · (vu)]<sup>'</sup>⟩ + ⟨v<sup>†</sup> [∇ · (vv)]<sup>'</sup>⟩ − ⟨u<sup>†</sup> ∇ · (v<sup>'</sup>⟨u⟩)⟩ − ⟨v<sup>†</sup> ∇ · (v<sup>'</sup>⟨v⟩)⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Total spectral transfer of EKE; it is re-arranged to achieve machine precision (cf., Uchida, Jamet, Poje, et al., 2024)</td></tr><tr><td>A<sub>h</sub> = − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> ∇<sub>h</sub> · (uu<sup>'</sup>)⟩ + ⟨v<sup>†</sup> ∇<sub>h</sub> · (uv<sup>'</sup>)⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Horizontal spectral transfer of EKE</td></tr><tr><td>A<sub>v</sub> = − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> (wu<sup>'</sup>)<sub>z</sub>⟩ + ⟨v<sup>†</sup> (wu<sup>'</sup>)<sub>z</sub>⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Vertical spectral transfer of EKE</td></tr><tr><td>A<sub>K</sub><sup>e</sup> = − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> ∇ · (v<sup>'</sup>u<sup>'</sup>)⟩ + ⟨v<sup>†</sup> ∇ · (v<sup>'</sup>v<sup>'</sup>)⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Eddy spectral transfer of EKE</td></tr><tr><td>A<sub>K</sub><sup>m</sup> = − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> ∇ · (⟨v⟩u<sup>'</sup>)⟩ + ⟨v<sup>†</sup> ∇ · (⟨v⟩v<sup>'</sup>)⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Mean spectral transfer of EKE</td></tr><tr><td>MtE<sub>K</sub> = − 1/C<sub>Ξ</sub> R [⟨u<sup>†</sup> ∇ · (v<sup>'</sup>⟨u⟩)⟩ + ⟨v<sup>†</sup> ∇ · (v<sup>'</sup>⟨v⟩)⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Spectral shear production</td></tr><tr><td>H<sub>K</sub> = − 1/C<sub>Ξ</sub> R [⟨v<sup>†</sup> · ∇h<sup>†</sup>⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Spectral work on EKE by dynamic enthalpy</td></tr><tr><td>S<sub>K</sub> = θ<sub>K</sub> + θ<sub>K</sub> = 1/C<sub>Ξ</sub> R [⟨v<sup>†</sup> · ˜H<sub>Θ</sub>(v) ∇Θ'⟩ + ⟨v<sup>†</sup> · ˜H<sub>S</sub>(s) ∇S'⟩]<sup>x</sup> x<sub>0</sub><sup>2</sup>k</td><td>Spectral EKE source due to tracer advection</td></tr><tr><td>ε<sub>K</sub> = ∫<sub>k>k'</sub> A<sub>K</sub>(k<sup>‡</sup>) dk<sup>‡</sup></td><td>Total spectral flux of EKE</td></tr><tr><td>ε<sub>h</sub> = ∫<sub>k>k'</sub> A<sub>h</sub>(k<sup>‡</sup>) dk<sup>‡</sup></td><td>Horizontal spectral flux of EKE</td></tr><tr><td>ε<sub>v</sub> = ε<sub>K</sub> − ε<sub>h</sub> (= ∫<sub>k>k'</sub> A<sub>v</sub>(k<sup>‡</sup>) dk<sup>‡</sup>)</td><td>Vertical spectral flux of EKE</td></tr></tbody></table>
<table><caption>**Table 1**
*Continued*</caption><thead><tr><th>Mathematical notation</th><th>Description</th></tr></thead><tbody><tr><td>ε<sub>K</sub><sup>c</sup> = ∫<sub>k</sub><sup>k<sup>+</sup></sup> A<sub>K</sub><sup>c</sup>(k<sup>+</sup>) dk<sup>+</sup></td><td>Eddy spectral flux of EKE</td></tr><tr><td>ε<sub>K</sub><sup>m</sup> = ε<sub>K</sub> - ε<sub>K</sub><sup>c</sup> ( = ∫<sub>k</sub><sup>k<sup>+</sup></sup> A<sub>K</sub><sup>m</sup>(k<sup>+</sup>) dk<sup>+</sup>)</td><td>Mean spectral flux of EKE</td></tr><tr><td>ε<sub>c</sub> = −⟨S<sub>c</sub>(u') : T<sub>c</sub>(u, u')⟩</td><td>Horizontal spectral flux of EKE based on Aluie et al. (2018) (cf., Equation 10)</td></tr><tr><td>η<sub>K</sub> = ∫<sub>k</sub><sup>k<sup>+</sup></sup> H<sub>K</sub>(k<sup>+</sup>) dk<sup>+</sup></td><td>Cumulative input from eddy advection of dynamic enthalpy</td></tr><tr><td>χ<sub>K</sub> = ∫<sub>k</sub><sup>k<sup>+</sup></sup> S<sub>K</sub>(k<sup>+</sup>) dk<sup>+</sup></td><td>Cumulative input from eddy advection of active tracers</td></tr><tr><td>Θ<sub>K</sub> = ∫<sub>k</sub><sup>k<sup>+</sup></sup> δ<sub>K</sub>(k<sup>+</sup>) dk<sup>+</sup></td><td>Cumulative input from eddy advection of potential temperature</td></tr><tr><td>Σ<sub>K</sub> = ∫<sub>k</sub><sup>k<sup>+</sup></sup> S<sub>K</sub>(k<sup>+</sup>) dk<sup>+</sup></td><td>Cumulative input from eddy advection of practical salinity</td></tr></tbody></table>
precipitation, are prescribed everywhere. The North Atlantic domain was configured to be zonally re-entrant so as to reduce the memory allocation in producing the simulations (cf., Figure 4). We will use the outputs from 1967 where variables from the MITgcm diagnostics package were saved as instantaneous snapshots permitting us to close the momentum budgets to machine precision (cf., Jamet et al., 2021; Uchida, Jamet, Dewar, et al., 2024; Uchida, Jamet, Poje, et al., 2024). Namely, no temporal averaging was applied to the outputs. Additionally, we restrict our analyses to below the mixed layer ($z = -452$ m) to avoid energy input from surface convection and wind stress ($\mathcal{X}' \sim 0$ in Equation 2 and $D_K \sim 0$ in Equation 5; Section 2.2) but shallow enough to detect the signal of a wind-driven gyre (Uchida, Jamet, Poje, et al., 2024).

The additional ensemble dimension allows for a parameter-free definition of eddies for an arbitrary variable $\psi$

$$ \psi_i = \psi_i - \langle \psi \rangle \quad (1) $$

where the subscript i is the member of the ensemble. $\langle \psi \rangle$ ($\stackrel{\text{def}}{=} \frac{1}{N} \sum_{i=1}^{N} \psi_i$) is the ensemble mean where $N = 48$. A key strength of the ensemble dimension being orthogonal to the space-time dimensions is that the ensemble-based Reynolds' decomposition commutes with space-time derivatives and leads to an exact expression of eddy-mean flow decomposition.

## 2.2. Wavelet-Based Spectral EKE Budget

We follow Uchida, Jamet, Dewar, et al. (2024, their Eq. (14) and (A15)) in formulating the EKE budget where available potential energy (APE) is defined by dynamic enthalpy (Eden, 2015; Nycander, 2011; Young, 2010); this reads as

$$ \langle \mathcal{K} \rangle_t + \langle \mathbf{v} \rangle \cdot \nabla \langle \mathcal{K} \rangle \approx - \langle \mathbf{v}' \cdot \nabla (\mathcal{K} + \phi^\dagger + h^\dagger) \rangle + \langle \mathbf{u}' \cdot \mathcal{X}' \rangle - \langle \mathbf{v}' \mathbf{u}' \rangle \cdot \nabla \langle \mathbf{u} \rangle + \tilde{H}_{\Theta}|_{\langle \Theta \rangle} \langle \mathbf{v}' \cdot \nabla \Theta' \rangle + \tilde{H}_{S}|_{\langle S \rangle} \langle \mathbf{v}' \cdot \nabla S' \rangle, \quad (2) $$

where $\langle \mathcal{K} \rangle \stackrel{\text{def}}{=} \langle |\mathbf{u}'|^2 \rangle / 2$ is EKE, $\Theta$ is potential temperature, $S$ is practical salinity and $\mathcal{X}'$ is the diabatic term of external forcing and mixing due to the K-profile parametrization (KPP; Large et al., 1994). Dynamic enthalpy is

$$ \tilde{h}(\Theta, S, \Phi) = \int_{\Phi_0}^{\Phi} \frac{\tilde{b}(\Theta, S, \Phi^{\ddagger})}{g} d\Phi^{\ddagger}, \quad (3) $$

where $\Phi = \Phi_0 - gz$ and density in defining buoyancy follows Jackett and Mcdougall (1995), which takes potential temperature and practical salinity as functional inputs. The superscript $\ddagger$ indicates a dummy variable for integration and thermodynamic variables are denoted with a tilde, $\tilde{\cdot}(\cdot)$, to distinguish them from variables in space-time (Young, 2010). $\tilde{H} \stackrel{\text{def}}{=} g^{-1} \int_{\Phi_0}^{\Phi} \tilde{b}^{\ddagger} d\Phi^{\ddagger}$ is the mean total dynamic enthalpy (MTDE), and $\tilde{b}^{\ddagger} \stackrel{\text{def}}{=} \tilde{b}(\langle \Theta \rangle, \langle S \rangle, \Phi) (\approx \langle \tilde{b}(\Theta, S, \Phi) \rangle)$ is buoyancy assessed at $\langle \Theta \rangle$ and $\langle S \rangle$. $\tilde{h}^{\ddagger} \stackrel{\text{def}}{=} \tilde{h} - \tilde{H}$ is the dynamic enthalpy fluctuations; we remind the reader that $\tilde{h}^{\ddagger}$ does not correspond to eddy APE (Uchida, Jamet, Dewar, et al., 2024, their Appendix). $\tilde{H}_{\Theta}|_{\langle \Theta \rangle}$ and $\tilde{H}_{S}|_{\langle S \rangle}$ are partial derivatives with respect to $\Theta$ and $S$ assessed at $\langle \Theta \rangle$ and $\langle S \rangle$.

The most unfamiliar term perhaps in Equation 2 is how the exchange between potential and kinetic energy is described, which is often summarized as a vertical flux of buoyancy, $\langle w' b^{\ddagger} \rangle$, where $\tilde{b}^{\ddagger} \stackrel{\text{def}}{=} \tilde{b} - \tilde{b}^{\ddagger} (\approx b' = \tilde{b} - \langle b \rangle)$. Under a non-linear EOS for density, Uchida, Jamet, Dewar, et al. (2024) demonstrated that the energy exchange may be better represented by $\tilde{H}_{\Theta}|_{\langle \Theta \rangle} \langle \mathbf{v}' \cdot \nabla \Theta' \rangle + \tilde{H}_{S}|_{\langle S \rangle} \langle \mathbf{v}' \cdot \nabla S' \rangle$ for cases where the background stratification cannot be easily defined as in the QG sense. The two are related as

$$ \tilde{H}_{\Theta}|_{\langle \Theta \rangle} \langle \mathbf{v}' \cdot \nabla \Theta' \rangle + \tilde{H}_{S}|_{\langle S \rangle} \langle \mathbf{v}' \cdot \nabla S' \rangle \approx \langle w' b^{\ddagger} \rangle + \langle \mathbf{v}' \cdot \nabla h^{\ddagger} \rangle, \quad (4) $$
assuming that the eddy flux of potential temperature and practical salinity dominate over the numerical dissipation and external forcing of ensemble-mean potential temperature and practical salinity. Equation 4 states that the vertical buoyancy flux can be decomposed into its contribution from advection of potential temperature and practical salinity, the two actual prognostic tracers in MITgcm, and of dynamic enthalpy, which captures the non-linearity in EOS. While this manipulation may seem redundant, it allows us to diagnose how potential energy affects the EKE budget in more detail. Uchida, Jamet, Dewar, et al. (2024) found Equations 2 and 4 to approximately hold throughout the modeled North Atlantic domain. The advantage of Equation 2 is that it is not only dynamically but also thermodynamically consistent.

Re-writing Equation 2 using CWTs and explicitly separating out the terms responsible for spectral transfer yields (cf., Table 1)

$$ \begin{aligned} \underbrace{\langle \check{u}'^* \cdot \check{u}'_t \rangle}_{C \Xi T_K(x_0^2 k)^{-1}} & \approx - \underbrace{\langle \check{v}^* \cdot \overline{\nabla \phi}^\dagger \rangle}_{C \Xi P_K(x_0^2 k)^{-1}} - \underbrace{\langle \check{v}'^* \cdot \overline{\nabla h}^\dagger \rangle}_{C \Xi H_K(x_0^2 k)^{-1}} + \underbrace{\langle \check{u}'^* \cdot \overline{\mathcal{X}'} \rangle}_{C \Xi D_K(x_0^2 k)^{-1}} \\ & \quad - \underbrace{[\langle \check{u}'^* \cdot \overline{\nabla \cdot (vu')} \rangle + \langle \check{v}'^* \cdot \overline{\nabla \cdot (vv')} \rangle]}_{C \Xi A_K(x_0^2 k)^{-1}} \\ & \quad - \underbrace{[\langle \check{u}'^* \cdot \overline{\nabla \cdot (v'\langle u \rangle)} \rangle + \langle \check{v}'^* \cdot \overline{\nabla \cdot (v'\langle v \rangle)} \rangle]}_{C \Xi \mathrm{MtE}_K(x_0^2 k)^{-1}} \\ & \quad + \underbrace{\langle \check{v}'^* \cdot \overline{\widetilde{H}_\Theta |_{\langle \Theta \rangle} \nabla \Theta'} \rangle + \langle \check{v}'^* \cdot \overline{\widetilde{H}_S |_{\langle S \rangle} \nabla S'} \rangle}_{C \Xi S_K(x_0^2 k)^{-1} = C \Xi (\vartheta_K + \zeta_K)(x_0^2 k)^{-1}} \end{aligned} \quad (5) $$

where * is the complex conjugate and the breve, $\check{\cdot}$, denotes the CWT coefficients (cf., Appendix A). We find that Equation 5 has the input from dynamic enthalpy, $H_K$ and from active tracers $S_K(=\vartheta_K + \zeta_K)$, explicitly written out, a difference from Uchida, Jamet, Poje, et al. (2024, their Eq. (A.12)) where they focused solely on kinetic energy. $\vartheta_K$ is the EKE input from eddy advection of potential temperature and $\zeta_K$ is from salinity. Somewhat contrary to Molemaker and McWilliams (2010) and J. Wang et al. (2024) where they find spectral cascades of APE, input from potential energy in Equation 5 presents purely as a source of EKE and not a driver of cross-scale energy cascade. This hinges on the fact that APE in the Lorenz sense is a squared property in buoyancy (Lorenz, 1955) while dynamic enthalpy is a first-order property in buoyancy and the contributions from advection of potential temperature and practical salinity ($S_K$) cannot be written in the form of a divergence.

The spectral transfer of EKE can be decomposed into its horizontal and vertical components, $A_K = A_h + A_v$ where

$$ A_h(\gamma, \phi, k) = -\frac{1}{C_\Xi} \Re \left[ \langle \check{u}'^* \overline{\nabla_h \cdot (uu')} \rangle + \langle \check{v}'^* \overline{\nabla_h \cdot (uv')} \rangle \right] x_0^2 k, \quad (6a) $$

$$ A_v(\gamma, \phi, k) = -\frac{1}{C_\Xi} \Re \left[ \langle \check{u}'^* \overline{(wu')_z} \rangle + \langle \check{v}'^* \overline{(wv')_z} \rangle \right] x_0^2 k, \quad (6b) $$

We can also decompose the spectral transfer into its contribution from eddy and mean advection of EKE, $A_K = A_K^e + A_K^m$ where

$$ A_K^e(\gamma, \phi, k) = -\frac{1}{C_\Xi} \Re \left[ \langle \check{u}'^* \overline{\nabla \cdot (v'v')} \rangle + \langle \check{v}'^* \overline{\nabla \cdot (vv')} \rangle \right] x_0^2 k, \quad (7a) $$
$$A_K^m(\gamma, \phi, k) = -\frac{1}{C_\Xi} \Re \left[ \langle \tilde{u}^* \tilde{\nabla} \cdot (\langle \overline{\nu} \rangle u') \rangle + \langle \tilde{v}^* \tilde{\nabla} \cdot (\langle \overline{\nu} \rangle v') \rangle \right] x_0^2 k. \quad (7b)$$

Integrating the spectral transfers in wavenumber translates to the spectral flux of EKE

$$\varepsilon_K(\gamma, \phi, k) = \int_{k > k^\dagger} A_K(k^\dagger) \, dk^\dagger, \quad (8a)$$

$$\varepsilon_h(\gamma, \phi, k) = \int_{k > k^\dagger} A_h(k^\dagger) \, dk^\dagger, \quad (8b)$$

$$\varepsilon_K^e(\gamma, \phi, k) = \int_{k > k^\dagger} A_K^e(k^\dagger) \, dk^\dagger, \quad (8c)$$

$$\varepsilon_K^m(\gamma, \phi, k) = \int_{k > k^\dagger} A_K^m(k^\dagger) \, dk^\dagger, \quad (8d)$$

respectively. The cumulative energy input from the thermodynamics can also be quantified as

$$\eta_K(\gamma, \phi, k) = \int_{k > k^\dagger} \mathcal{H}_K(k^\dagger) \, dk^\dagger, \quad (9a)$$

$$\chi_K(\gamma, \phi, k) = \int_{k > k^\dagger} S_K(k^\dagger) \, dk^\dagger, \quad (9b)$$

$$\Theta_K(\gamma, \phi, k) = \int_{k > k^\dagger} \theta_K(k^\dagger) \, dk^\dagger, \quad (9c)$$

$$\Sigma_K(\gamma, \phi, k) = \int_{k > k^\dagger} \zeta_K(k^\dagger) \, dk^\dagger. \quad (9d)$$

Another method that has gained traction in the field of oceanography is the spatial-filtering approach (e.g., Aluie et al., 2018; Schubert et al., 2020; Contreras et al., 2023; Storer et al., 2023; Tedesco et al., 2024, and references therein). The horizontal spectral flux of EKE corresponding to Equation 8b becomes (Aluie et al., 2018; Uchida, Jamet, Poje, et al., 2024)

$$\begin{aligned} \varepsilon_{\ell}(\ell) &= -\langle \mathbf{S}_{\ell}(\mathbf{u}') : \mathbf{T}_{\ell}(\mathbf{u}, \mathbf{u}') \rangle \\ &= -\left\langle \left(\overline{\overline{u}u'}^{\ell} - \overline{u'}\overline{u'}^{\ell}\right) \overline{u'}_{x}^{\ell} + \left(\overline{\overline{u}v'}^{\ell} - \overline{u'}\overline{v'}^{\ell} + \overline{v}u'}^{\ell} - \overline{v'}\overline{u'}^{\ell}\right) \frac{\overline{u'}_{y}^{\ell} + \overline{v'}_{x}^{\ell}}{2} + \left(\overline{\overline{v}v'}^{\ell} - \overline{v'}\overline{v'}^{\ell}\right) \overline{v'}_{y}^{\ell} \right\rangle, \end{aligned} \quad (10)$$

where $\overline{\cdot}^{\ell}$ is the filtering operator with length scale $\ell$, $\mathbf{T}_{\ell}(\mathbf{u}, \mathbf{u}) = \overline{\overline{u} \otimes \overline{u}^{\ell}} - \overline{u}^{\ell} \otimes \overline{u}^{\ell}$ is the sub-scale stress tensor, $\otimes$ is the outer product operator, and: is the tensor inner-product operator.

$\mathbf{S}_{\ell}(\mathbf{u}) = [\nabla_h \otimes \overline{u}^{\ell} + (\nabla_h \otimes \overline{u}^{\ell})^{\mathrm{T}}]/2$ the horizontal strain tensor, and the superscript T is the transpose operator.

We will use $\varepsilon_{\ell}$ later on as a point of reference to compare $\varepsilon_h$. The spatial filtering was applied directly on the curvilinear grid of MITgcm with a horizontally isotropic Gaussian filter, which approximately commutes with the space-time derivatives (Grooms et al., 2021). We note that given the non-linearity in EOS, adopting the spatial-filtering approach to the thermodynamics would require applying the filtering to the potential temperature and practical salinity fields themselves for thermodynamic consistency, for example, $\tilde{b}(\overline{\Theta}^{\ell}, \overline{S}^{\ell}, \Phi) (\neq \tilde{b}(\overline{\Theta}, \overline{S}, \Phi)^{\ell})$;

$\tilde{b}(\overline{\Theta}^{\ell}, \overline{S}^{\ell}, \Phi) - \tilde{b}(\overline{\Theta}^{\ell}, \overline{S}^{\ell}, \Phi) (\neq \tilde{b}^{\ell})$ and $\tilde{H}(\overline{\Theta}^{\ell}, \overline{S}^{\ell}, \Phi)_{\Theta}|_{\overline{\Theta}^{\ell}} (\neq \tilde{H}(\overline{\Theta}^{\ell}, \overline{S}^{\ell}, \Phi)_{\Theta}|_{\overline{\Theta}^{\ell}})$ etc.; this
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Isotropized (azimuthally integrated) terms on the right-hand side of the spectral EKE budget (Equation 5) in the Gulf Stream extension at the depth of $z = -452$ m. All terms except for tendency, pressure work and diabatic terms on 1 January 1967 (a, b). Horizontal spectral transfer is shown in the dashed magenta curve and vertical buoyancy flux as the dotted-dashed brown curve. The 95% bootstrapped uncertainty is shown in colored shadings. The seasonally averaged terms are shown in panels (c, d) for JFM, and (e, f) for JAS. The right column zooms in on the spectral transfer terms $A_K$ and $A_h$. The vertical spectral transfer $A_v$ is added in orange dotted curves.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Isotropic (azimuthally integrated) spectral flux of EKE on January 1 and averaged over JFM and JAS of 1967 at the depth of $z = -452$ m. Total 3D EKE spectral flux $\epsilon_K$ is shown in solid blue, total 2D flux $\epsilon_h$ in dashed magenta, eddy 3D flux $\epsilon_K^e$ in dotted black, eddy 2D flux $\epsilon_h^e$ in dotted-dashed neon-green curves. The 95% bootstrapped uncertainty is shown in colored shadings.</figcaption></figure>

introduces a myriad of additional terms to the EKE budget corresponding to length scale $\ell$ due to non-commutability and we shall not delve further into it here.

### 3. Results

Wavelet-based spectral analyses are presented every 5° in latitude-longitude while the spatial-filtering results are presented as continuous maps. CWTs were taken over $10^\circ \times 10^\circ$ boxes but the spatially local nature of wavelets minimizes the artifact of overlapping subdomains. Following (Equation A4), the interpretation of CWTs is that they indeed provide spectral estimates at any given single grid point $\gamma$. To emphasize this point, we will be providing spatial maps of EKE budget terms about a spatial scale as scatter plots. We start by documenting the instantaneous EKE budget terms relevant to spectral fluxes and potential energy on 1 January 1967 at 294°E, 35°N, a location closest to what was examined by Uchida, Jamet, Poje, et al. (2024) in the Gulf Stream extension. The location is marked in Figure 4a with an arrow. Using the same model outputs, Uchida, Jamet, Poje, et al. (2024) demonstrated that the wavelet-based spectral EKE budget could be closed to machine precision so here, we will only present the terms of interest.

Figure 1 shows that the vertical eddy buoyancy flux tends to be relatively small (dotted-dashed brown curve), implying that the flow is largely adiabatic. In maintaining this adiabatic nature, there is a large cancellation between input from eddy potential temperature advection $\theta_K$ and the sum of eddy practical salinity and dynamic enthalpy advection ($\zeta_K + H_K$); the latter two tend to energize the flow (solid curves in black and cyan) and vice versa (solid red curve). Zooming in on the advection terms, the total three-dimensional (3D) spectral transfer $A_K(>0)$ implies a forward cascade of EKE while the two-dimensional (2D) spectral transfer $A_h(\le 0)$ an inverse EKE cascade (solid blue and dashed magenta curves in Figure 1b). This becomes clearer when the spectral transfers are integrated over wavenumber to give the spectral flux due to 3D advection and 2D advection by the total (mean+eddy) flow ($\epsilon_K > 0$ and $\epsilon_h < 0$; Figure 2a).

Figures 1c–1f documents the budget terms computed every 15 days and seasonally averaged for 1967; the 15-day intervals are there to reduce temporal autocorrelation. Winter was defined as the months of January, February and March (JFM), and summer as July, August and September (JAS). The uncertainty decreases significantly by taking a time average. The contribution from vertical advection somewhat cancels out when averaged over time; $A_K$ and $\epsilon_K$ become more aligned with $A_h$ and $\epsilon_h$ respectively (Figures 1 and 2). There is a relative increase in forward cascade of EKE when vertical advection of EKE is included on January 1 and when averaged over summer ($\epsilon_h < \epsilon_K$) but it decreases when averaged over winter. In relation to the vertical eddy buoyancy flux, $A_K$ and $A_h$ tend to be positive at scales smaller than where the buoyancy flux reaches maxima ($\sim 300$ km) and negative at larger scales. Comparing the spectral flux due to 3D advection by the total and

eddy flow ($\epsilon_K$ and $\epsilon_K^e$), the latter tends to be smaller than the former across all spatial scales ($\epsilon_K^e < \epsilon_K$; solid blue and dotted black curves in Figure 2). In other words, the advection of EKE by the mean flow $A_K^m (=A_K - A_K^e)$ tends to transfer EKE toward smaller scales. Overall, there is not a strong seasonal variation below the mixed layer in the budget terms but the spectral estimates in summer tend to be smoother in wavenumber than during winter. When cumulatively integrated over wavenumber, however, the difference is highlighted where we find an increase in forward EKE cascade during winter at scales below 300 km ($\epsilon \gtrsim 0$) while an inverse EKE cascade in summer ($\epsilon \lesssim 0$).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Isotropic (azimuthally integrated) vertical EKE spectral flux, $\epsilon_v$, for the every 15 days that go into estimating the wintertime average (a). The confidence intervals are not shown for visual purposes. Panels (b and c) are the same as Figure 1 but for 15 February 1967.</figcaption></figure>

It is interesting that the forward cascade due to vertical advection of EKE on January 1 cancels out when averaged over winter. Figure 3a documents the vertical EKE cascade $\epsilon_v$ for the days that go into estimating the wintertime average; we find that there is an episodic strong event of inverse cascade $\epsilon_v < 0$ on February 15, which when averaged over wintertime leads to a cancellation of the contribution from January 1. Given the non-linear, chaotic and unequilibrated nature of the system (Figures 3b and 3c), it is difficult to identify what led to this burst of inverse cascade but it is “the” strength of the ensemble framework where we can diagnose the spectral characteristics of eddies on any given single day; the fact that the direction and strength of EKE cascade temporally vary is a fundamental aspect of eddies and should not be overlooked.

### 3.1. Spatial Maps of EKE Cascade

Figure 4 shows the spatial map of instantaneous EKE spectral flux on 1 January 1967. There is a general agreement between the wavelet and spatial-filtering approach for the horizontal EKE spectral flux to cascade upscale at scales about the length scale of 400 km ($\epsilon_h < 0, \epsilon_\ell < 0$) and downscale about 120 km ($\epsilon_h > 0, \epsilon_\ell > 0$) over the modeled domain (Figures 4a–4d); this provides confidence to the wavelet approach. The scales of 120 and 400 km were both chosen to be large enough so as to reduce the impact of numerical dissipation but to sandwich the scale where the spectral transfer $A_K$ tends to reach its local maxima (Θ(300 km), Figure 1; Uchida, Jamet, Poje, et al., 2024). The difference between the total and horizontal EKE spectral flux, which corresponds to the vertical EKE spectral flux ($\epsilon_v = \epsilon_K - \epsilon_h$), tends to be positive right off the Gulf Stream separation (localized
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Instantaneous spectral EKE flux on 1 January 1967 at the depth of z = -452 m. Horizontal spectral flux of EKE based on wavelets $\epsilon_h$ at scales about (a) 400 km and (b) 120 km, and (c, d) spatial filtering $\epsilon_l$. The black arrow identifies the location analyzed in Figures 1 and 2. (e, f) Difference between the total 3D and 2D spectral flux of EKE based on wavelets $\epsilon_K - \epsilon_h (=\epsilon_v)$. For the wavelet approach, subdomains with the ocean covering them over 70% were used and subdomains which had land cells are shown as triangles. They are shown in scatter plots as the wavelet spectral estimates are indeed at each grid point in space.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Same as Figure 4 but for the annual mean of 1967 where the fluxes were computed every 15 days.</figcaption></figure>

west of 300°E) but is overall negative in the basin interior (Figures 4e and 4f); the fact that $ε_v$ is non-zero throughout the entire domain and has similar magnitudes to $ε_h$ indicate that the vertical advection of EKE non-trivially affects the direction and strength of EKE cascade.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Annual-mean isotropic spectral EKE flux at the depth of $z = -452$ m about the length scales of 400 and 120 km. (a, b) Difference between the total and eddy spectral flux of EKE based on wavelets $\epsilon_K - \epsilon_K^e (=\epsilon_K^m)$. (c, d) Difference between the horizontal total and eddy spectral flux $\epsilon_h - \epsilon_h^e (=\epsilon_h^m)$. (e, f) Difference between the 3D eddy and horizontal eddy spectral flux $\epsilon_K^e - \epsilon_h^e (=\epsilon_v^e)$.</figcaption></figure>

When annually averaged, the horizontal EKE spectral flux, $\epsilon_h$, demonstrates a more coherent signal of inverse cascade about the length scale of 400 km and forward cascade about 120 km (Figures 5a–5d). The persistence of localized patches of forward cascade in the equatorial region and separated Gulf Stream below the mixed layer is
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Instantaneous cumulative spectral input from DE on 1 January 1967 at the depth of *z* = −452 m. χ<sub>K</sub> based on wavelets at scales about (a) 400 km and (b) 120 km. Contribution of eddy advection of potential temperature (Θ<sub>K</sub>; c) and practical salinity (Σ<sub>K</sub>; d) to χ<sub>K</sub> about 400 km. η<sub>K</sub> based on wavelets at scales about (e) 400 km and (f) 120 km.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Same as Figure 7 but for the annual mean of 1967 where the terms were computed every 15 days.</figcaption></figure>

consistent with Aluie et al. (2018, their Figs. 9 and 10). Vertical EKE spectral flux, $\epsilon_v$, increases a forward cascade right along the western boundary but reinforces the inverse cascade across the basin interior about 400 km (Figure 5e). About the scale of 120 km, the vertical flux shows an inverse cascade in the separated Gulf Stream region (Figure 5f). Seasonal spatial maps are given in Appendix B.
We end this section by documenting the annual mean of other spectral flux properties due to advection of EKE by the 3D mean flow $\epsilon_K^m (=\epsilon_K - \epsilon_K^e)$, horizontal mean flow $\epsilon_h^m (=\epsilon_h - \epsilon_h^e)$, and vertical eddy flow $\epsilon_v^e (=\epsilon_K^e - \epsilon_h^e)$. It is interesting that about the scales of 400 km, the EKE spectral flux by the 3D mean flow overall induces an inverse cascade ($\epsilon_K^m < 0$; Figure 6a) but the spectral flux by the horizontal mean flow forces a forward cascade in the Gulf Stream extension ($\epsilon_h^m > 0$; Figure 6c). The fact that the horizontal mean flow in the Gulf Stream extension, a region with vigorous fronts, generates a forward EKE cascade is perhaps not surprising. The vertical mean flow, however, compensates for this and the 3D mean flow results in an inverse cascade at many locations. The spectral flux by vertical eddy motion generally yields an inverse cascade across all scales ($\epsilon_v^e < 0$; Figures 6e and 6f).

## 3.2. Spatial Maps of EKE Input From Thermodynamics

Cumulative net energy input due to eddy advection of potential temperature and practical salinity, $\chi_K$, on 1 January 1967 is spatially heterogeneous (Figures 7a and 7b). There is a higher tendency for $\chi_K$ to energize EKE at the larger scales (~400 km) than smaller (~120 km) in the gyre interior. The contributions from potential temperature $\Theta_K$ and practical salinity $\Sigma_K$ tend to counteract each other throughout the domain (Figures 7c and 7d); $\Theta_K$ generally wins over leading to the spatial distribution we observe in $\chi_K$ (Figures 7a and 7c). This relation of $\Theta_K$ dominating over $\Sigma_K$ did not change at scales about 120 km (not shown). This partial compensation between potential temperature and practical salinity is analogous to spice (Legg & McWilliams, 2000; McDougall et al., 2021). Advection of dynamic enthalpy, $\eta_K$, tends to counteract $\chi_K$ (top and bottom rows of Figure 7). Because $\chi_K$ is dominated by $\Theta_K$, what we find is something peculiar; dynamic enthalpy itself is a function of temperature but when advected by the eddies, the two oppose each other in affecting EKE. The overall pattern is that $\eta_K$ energizes EKE in the separated Gulf Stream and equatorial regions (15°S-10°N). $\chi_K$ energizes the gyre interior (10°N-35°N, 300°E-335°E) at scales about 400 km and acts as a sink of energy about 120 km; this pattern is robust and persists when averaged over the year of 1976 (Figure 8).

## 4. Conclusions and Discussion

Given the nature of spatial heterogeneity and aperiodicity of oceanic flows, we have opted for using continuous wavelet transforms (CWTs) instead of the canonical Fourier approach for spectral analyses. Namely, notable practical strengths of CWTs are that: (a) no windowing needs to be applied and thus does not skew the data, and (b) provides a spatially local “pseudo” Fourier transform. The mathematical justification for using CWT for spectral analyses hinges on the fact that it satisfies a Parseval's equality (Equation A4) and, from a practical point of view, that we can close the spectral budget (Uchida, Jamet, Poje, et al., 2024). Furthermore, in contrast to the spatially-filtering approach where each term in the EKE budget would need to be recursively filtered for length scale $\ell$ in order to close the budget spectrally, our approach benefits from the fact that we are able to spectrally close the budget by taking the CWT once per each term in the budget (Uchida, Jamet, Poje, et al., 2024); this is particularly noticeable when the thermodynamics are considered under a non-linear EOS.

Using the wavelet approach, we have examined the dynamical and thermodynamical characteristics of the Boussinesq and hydrostatic North Atlantic EKE spectral budget where the eddies were defined as fluctuations about an ensemble mean. The usage of ensemble simulations has allowed us to diagnose the instantaneous along with time-averaged properties of the budget specific to the year of 1967. It is interesting that right off of the Gulf Stream separation, the direction of EKE cascade demonstrates a pronounced seasonality (Figures 1c-1f and 2; Figure B1). This may be related to the seasonal cycle in the destabilization point (Sánchez-Román et al., 2024, their Fig. 4) and its relation to EKE cascade will be an avenue for future investigation. We find that advection of EKE by vertical motion and the mean flow non-trivially affects the magnitude and direction of up- and down-scale cascade of EKE throughout the North Atlantic basin. Vertical advection generally enhances an inverse EKE cascade at scales above the Rossby radii, which may indicate a barotropization of the flow (Figures 5e and 6e; Vallis, 2006), and a forward cascade about and smaller than the Rossby radii (Figure 5f). Interestingly, vertical advection of EKE by the eddy flow contributes to an inverse cascade even about 120 km ($\epsilon_v^e < 0$; Figure 6f). The barotropization of KE has been numerically documented in the open Agulhas Current region (Tedesco et al., 2022), and generally favors the QG understanding of the ocean. Interestingly, the horizontal mean flow $\langle u \rangle$ drives a forward cascade of EKE in the Gulf Stream extension (Figure 6b), which may be associated with frontogenetic processes, while the vertical mean flow counteracts this and the three-dimensional mean flow $\langle v \rangle$
results in an inverse cascade of EKE (Figure 6a). These two factors, vertical motion and mean flow, are often overlooked in oceanographic studies where they only examine the horizontal advection of total (mean +eddy) KE (e.g., Arbic et al., 2013; Aluie et al., 2018; Storer et al., 2023, and references therein). Nonetheless, distinct behaviors between horizontal and vertical spectral cascade of total KE have been previously documented in the California Current region [Capet et al. (2008, their Fig. 5); Srinivasan et al. (2023, their Fig. 14); Delpech et al. (2024, their Fig. 2)]. A more detailed analysis of EKE cascades in the vertical direction of the ocean is warranted and will be left for consideration elsewhere.

The mutual two-way coupling between the mean flow and eddies has only recently garnered attention in the theory of geostrophic turbulence (Deremble et al., 2023; Grooms et al., 2011). This recent development is important since as it stood, in the multiscale asymptotic sense, mesoscale dynamics fed off of the planetary-scale dynamics but there was little feedback from the meso- to planetary-scale (Pedlosky, 1984). We hope our approach and results shed some light on this coupling process in a realistic setting. It is likely that the impact of vertical advection is underestimated at the spatial resolution of 1/12° (Balwada et al., 2018; Capet et al., 2008; Chassignet & Xu, 2017; Lévy et al., 2012; Uchida et al., 2019). It would be interesting to repeat our analyses at submesoscale-permitting resolution $\Theta(1/50°)$ (cf., Contreras et al., 2023; Uchida, Jamet, Poje, et al., 2024, their Supplementary Material Figs. S2 and S3).

A final interesting, and somewhat surprising result is that a phenomenon analogous to the counterbalance observed in the thermodynamic spice variable affects the EKE budget; energy input from eddy potential temperature advection compensates for that of practical salinity. We found this relation to be remarkably robust in space-time across the North Atlantic domain (Figures 7 and 8). The fact that eddy temperature advection generally wins over eddy salinity advection is consistent with previous studies where they found salinity to have minor contribution to eddy potential energy (Guo et al., 2022; Luecke et al., 2017). Future work may involve relating spice to the EKE budget and using spice as a proxy to identify regions where the thermodynamics are energizing the flow. In-situ temperature and salinity measurements of the ocean are much more readily available than velocity fields (cf., Johnson et al., 2016; T. Wang et al., 2022).

## Appendix A: Continuous Wavelet Transform

Given a function of two spatial dimensions, $f(x)$, its CWT for location $\gamma(\in \mathbb{R}^2)$ is given by (C. Chen & Chu, 2017; Daubechies, 1992; Torrence & Compo, 1998)

$$ \tilde{f}(s, \varphi, \gamma) = \int_{x} f(x) \frac{1}{s^{\xi}} \left( \mathbf{R}^{-1} \cdot \left( \frac{\mathbf{x} - \mathbf{\gamma}}{s} \right) \right) dx. \quad (A1) $$

$s$ defines the non-dimensionalized length scales of the wavelet. $\mathbf{R}^{-1}$ is the inverse of the rotation matrix

$$ \mathbf{R}^{-1} = \begin{pmatrix} \cos(\varphi) & \sin(\varphi) \\ -\sin(\varphi) & \cos(\varphi) \end{pmatrix}, \quad (A2) $$

for rotation through an angle $\varphi$. The mother Morlet wavelet is (Gabor, 1946; Morlet et al., 1982)

$$ \xi(\mathbf{x}) = (e^{-2\pi i k_0 \cdot \mathbf{x}} - c_0) e^{-\frac{\kappa_0 \cdot \mathbf{x}}{2c_0}}, \quad (A3) $$

where we have prescribed $x_0 = 50$ km, $k_0 = (x_0^{-1}, 0)$ and $c_0 = 0$.

As discussed in detail by Perrier et al. (1995), Uchida et al. (2023), Uchida, Jamet, Poje, et al. (2024, and references therein), it is possible to utilize the CWT coefficients to formulate a spatially local Parseval's equality in wavenumber and angle

$$ \int_{x} f(\mathbf{x})g(\mathbf{x}) \, d\mathbf{x} = \frac{1}{C_{\Xi}} \int_{\gamma} \int_{\varphi} \int_{k} \int_{\gamma} \int_{\varphi} \int_{k} \tilde{f}^{*\xi} g x_0^2 k \, d\gamma \, d\varphi \, dk, \quad (A4) $$
where $k = (sx_0)^{-1}$, $C_Ξ = \int_k |k|^{-2\hat{\Xi}^*\hat{\Xi}} dk$ and $\hat{\Xi}$ is the Fourier transform of the mother wavelet (Equation A3). Equation A4 identifies $C_Ξ^{-1} \Re[f^*g] x_0^2 k$ as the cross spectral density where $\Re[\cdot]$ is the real part. The MITgcm outputs on a curvilinear grid were first interpolated onto a monotonic Cartesian grid and land cells were treated as missing data when taking the CWT (Equation A1). Missing data were not substituted and the spatial integration in Equation A1 was taken using `nansum`. The data were not windowed in contrast to Fourier analyses given the locality of the wavelet functions themselves. A Python implementation of CWT is available via Uchida and Dewar (2022).

## Appendix B: Spatial Maps of Seasonality in Cascades

Spatial maps of the difference in seasonally averaged EKE cascade between winter (JFM) and summer (JAS) are provided for the year of 1967. We find that overall, there is more forward cascade in the horizontal component during winter ($\epsilon_h^{JFM} - \epsilon_h^{JAS} > 0$; Figures B1c and B1d) while the vertical component induces a stronger inverse cascade during winter ($\epsilon_v^{JFM} - \epsilon_v^{JAS} < 0$; Figures B1a and B1b). This is consistent with Figures 2b and 2c. In other words, the horizontal and vertical EKE cascade have somewhat opposing seasonal tendencies.

<figure><img src="image_11.png" /><figcaption>**Figure B1.** Difference in EKE cascade between winter and summer ($\epsilon_{JFM}^{JFM} - \epsilon_{JAS}^{JAS}$) about scales of 400 and 120 km. Top row corresponds to the total vertical EKE cascade, $\epsilon_v$, and bottom row to the total horizontal EKE cascade, $\epsilon_h$.</figcaption></figure>
## Data Availability Statement

The MITgcm outputs were read using the `xmitgcm` Python package (Abernathey et al., 2022) and postprocessed with the `xgcm` Python package (Abernathey et al., 2021). We used the `fastjmd95` Python package for offline thermodynamical diagnostics (Abernathey & Busecke, 2020). Fourier and wavelet transforms were taken using the `xrft` and `xwavelet` Python packages respectively (Uchida & Dewar, 2022; Uchida, Rokem, et al., 2021). The `gcm-filters` Python package was used to take the spatial filtering (Grooms et al., 2021; Loose et al., 2022). The ensemble outputs are available via the Florida State University (FSU) cluster (http://ocean.fsu.edu/~qjamet/share/data/twa_Uchida/).

## Acknowledgments

We would like to dedicate this study to William Kurt Dewar who sadly passed away in 2024. He is dearly missed. His genuine enthusiasm and excitement towards science were truly an inspiration for all. The passing of Bill marks the end of an era; an era where giants like Bill could naturally speak and breathe the language of GFD and mathematics. The uniqueness of Bill was that he did it with such elegance, humility and a great sense of humor. Simply put, it was always fun interacting with him, whether it be discussing ocean dynamics and equations at the office or life philosophy over pints of IPA. As the leading author and early-career scientist, I am grateful that I got to spend some quality time with Bill during my time in Grenoble and Tallahassee; he will forever be my aspiration as a scientist, mentor and good friend. We thank the editor Stephen Griffies, associate editor Ian Grooms, along with two anonymous reviewers and Hemant Khatri for their reviews. This study is a contribution to the "Assessing the Role of forced and internal Variability for the Ocean and climate Response in a changing climate" (ARVOR) project supported by the French "Les Enveloppes Fluides et l'Environnement" (LEFE) program. W. Dewar was supported through National Science Foundation (NSF) Grants OCE-1829856, OCE-1941963, and OCE-2023585, and the French "Make Our Planet Great Again" (MOPGA) program managed by the Agence Nationale de la Recherche under the Programme d'Investissement d'Avenir, reference ANR-18-MPGA-0002. T. Uchida acknowledges support from the NSF grant OCE-2123632. A. Poje acknowledges support from the NSF Grant OCE-2123633. High-performance computing resources on Cheyenne (https://doi.org/10.5065/D6RX99HX) used for running the Chaocean ensemble of the North Atlantic were provided by NCAR's Computational and Information Systems Laboratory, sponsored by NSF, under the university large allocation UFSU0011. We would like to thank Edward Peirce and Kelly Hirai for maintaining the FSU cluster on which the data were analyzed.

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