# The Seasonal Variability of the Ocean Energy Cycle from a Quasi-Geostrophic Double Gyre Ensemble

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

## 3. Derivation of the Lorenz Energy Cycle

## 4. Results

#### 4.1. The Domain Integrated Lorenz Energy Cycle

#### 4.2. Time Lag in Lower-Layer Energetics

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

`xarray`Python package [60], which we used to post process the model outputs.

## Conflicts of Interest

## Appendix A. Derivation of the Layered Quasi-Geostrophic Potential Vorticity

**Figure A1.**Schematic of a relation between buoyancy (B) and layer interface displacement ($\eta $). The background buoyancy is ${B}_{0}$ defined at $z=H$.

## Appendix B. The Omega Equation with a Temporally Varying Background Stratification

## Appendix C. Decomposing the Mean and Eddy Energetics

## Appendix D. The Three-Layer QG Lorenz Energy Cycle

## References

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**Figure 1.**Vertical structure of the three-layer QG model with a rigid lid and flat bottom. The layer interface displacement (${\eta}_{i}$) is shown in the thin curvy lines and net layer thickness is ${h}_{i}={H}_{i}+{\eta}_{i-1}-{\eta}_{i}$. The stream functions (${\psi}_{i}$) are defined within each layer.

**Figure 2.**Time series of the horizontally averaged barotropic (

**a**) and first-layer TKE (

**b**). Each ensemble member is shown with thin gray lines and standard deviation of the ensemble mean in black shading. The CTRL run is shown with a red dashed line and ensemble mean with a cyan dot-dashed line. The reduced gravity is shown in blue plotted against the right y axis.

**Figure 3.**The summer and wintertime mean and eddy KE and their difference during the last year of output (

**a**–

**f**). Note the differences are plotted on a logarithmic scale. (

**g**,

**h**) Snapshot of eddy PV for summer and winter during the last year of output from the CTRL run. All panels show the variable in the first layer.

**Figure 4.**Time series of each term in the domain-integrated LEC. (

**a**) The mean and eddy KE (black) and APE (red) reservoirs in the units of ×${10}^{13}$ [(J/kg) m${}^{3}$] and stratification of the first layer interface (blue; ${g}_{1}^{\prime}$). MAPE is multiplied by 0.1 to have it fit on the same y axis. (

**b**) The energy fluxes between each energy reservoir and forcing terms due to surface wind stress (${F}_{s}^{{K}^{\#}}$) and temporally varying BPE. (

**c**) Dissipation terms due to horizontal viscosity (${D}_{h}$) and bottom friction (${D}_{b}$). The mean and eddy horizontal dissipation terms are lumped together and fluxes are in the units of ×${10}^{6}$ [(W/kg) m${}^{3}$]. The forcing and dissipation terms are detailed in Appendix D.

**Figure 5.**Time series of the seasonal climatology of energy fluxes between the energy reservoirs (

**a**). (

**b**,

**c**) The LEC diagram for the climatological summer and winter averaged over the last five years of output. The energies are in the units of $\times {10}^{13}$ [(J/kg) m${}^{3}$] and fluxes are in $\times {10}^{6}$ [(W/kg) m${}^{3}$]. The energy exchanges do not exactly cancel out due to each reservoir having temporal variability.

**Figure 6.**Time series of volume-integrated EKE over each layer, and fluxes within and between layers (${\Pi}_{{\mathcal{K}}_{1}\to {\mathcal{K}}_{2}}$, ${\Pi}_{{\mathcal{K}}_{2}\to {\mathcal{K}}_{3}}$) plotted along with the reduced gravity (${g}_{1}^{\prime}$). (

**a**) The EKEs have their temporal mean removed so as to plot against the same y axis. (

**b**) A rolling mean by five time steps (∼29 days) is applied to the time series of the energy fluxes. The energy flux from MKE to EKE within the two bottom layers is summed up (${\Pi}_{{K}_{2,3}^{\#}\to {\mathcal{K}}_{2,3}}$) and conversion from APE is shown as the conversion rate volume integrated over the three layers as the amount that goes into each layer is simply the total conversion weighted by layer thickness (cf. Equations (23), (28), (29) and (32)). The conversion from ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ were in sync with each other (not shown). For further details regarding each term, see Appendix D.

**Table 1.**Parameters used to configure the three-layer QG simulation and dimensionalized characteristic scales. The bottom Ekman number is the ratio between the bottom Ekman-layer thickness and ${\widehat{H}}_{3}$ and bottom friction is $\u03f5=E{k}^{b}/(2R{o}^{m}{\widehat{H}}_{3})$. Beta is dimensionalized as $\beta =\widehat{\beta}U/{L}^{2}$ and the dimensionalized domain size is $4000\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}\phantom{\rule{4pt}{0ex}}(={\widehat{L}}_{0}L)$. The frequency of $Fr$ translates approximately to a 360-day year ($={f}_{Fr}^{-1}L/U$). The prognostic time stepping is determined via the CFL condition within values smaller than ${\delta}_{\widehat{t}}^{\mathrm{max}}$.

Parameter | Notation | Value | Unit |
---|---|---|---|

Number of horizontal grids | N | 1024 | - |

Number of vertical layers | ${n}_{l}$ | 3 | - |

Non-dim. horizontal domain size | ${\widehat{L}}_{0}$ | 80 | - |

Non-dim. horizontal resolution | ${\delta}_{\widehat{x}}$ | ${N}^{-1}{\widehat{L}}_{0}$ | - |

Background Rossby number | $R{o}^{m}$ | $0.025$ | - |

Non-dim. Coriolis parameter | ${\widehat{f}}_{0}$ | ${R{o}^{m}}^{-1}$ | - |

Bottom Ekman number | $E{k}^{b}$ | $0.004$ | - |

Non-dim. surface Ekman pumping | ${\widehat{\tau}}_{0}$ | $0.0001$ | - |

Biharmonic Reynolds number | $R{e}_{4}$ | 4000 | - |

Non-dim. beta | $\widehat{\beta}$ | $0.5$ | - |

Background Froude number | $F{r}_{1}^{m};F{r}_{2}^{m}$ | $0.00409959;0.01319355$ | - |

Amplitude of $F{r}_{i}$ | ${\widehat{A}}_{F{r}_{1}};{\widehat{A}}_{F{r}_{2}}$ | $0.1;0$ | - |

Non-dim. frequency of $F{r}_{i}$ | ${\widehat{f}}_{F{r}_{1}};{\widehat{f}}_{F{r}_{2}}$ | $62.{2}^{-1};62.{2}^{-1}$ | - |

Non-dim. layer thickness | ${\widehat{H}}_{1};{\widehat{H}}_{2};{\widehat{H}}_{3}$ | $0.06;0.14;0.8$ | - |

Non-dim. reduced gravity | $\widehat{{g}_{i}^{\prime}}$ | ${F{r}_{i}}^{-2}\widehat{{H}_{i}^{\u2020}}$ | - |

Non-dim. maximum time stepping | ${\delta}_{\widehat{t}}^{\mathrm{max}}$ | $5\times {10}^{-2}$ | - |

CFL condition | - | $0.4$ | - |

Horizontal velocity | U | $0.1$ | [m s${}^{-1}$] |

Length scale | L | 50 | [km] |

Total layer thickness | H | 5000 | [m] |

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**MDPI and ACS Style**

Uchida, T.; Deremble, B.; Penduff, T. The Seasonal Variability of the Ocean Energy Cycle from a Quasi-Geostrophic Double Gyre Ensemble. *Fluids* **2021**, *6*, 206.
https://doi.org/10.3390/fluids6060206

**AMA Style**

Uchida T, Deremble B, Penduff T. The Seasonal Variability of the Ocean Energy Cycle from a Quasi-Geostrophic Double Gyre Ensemble. *Fluids*. 2021; 6(6):206.
https://doi.org/10.3390/fluids6060206

**Chicago/Turabian Style**

Uchida, Takaya, Bruno Deremble, and Thierry Penduff. 2021. "The Seasonal Variability of the Ocean Energy Cycle from a Quasi-Geostrophic Double Gyre Ensemble" *Fluids* 6, no. 6: 206.
https://doi.org/10.3390/fluids6060206