# The Effects of Mesoscale Ocean–Atmosphere Coupling on the Quasigeostrophic Double Gyre

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## Abstract

**:**

## 1. Introduction

## 2. A Quasigeostrophic Model with SST

#### 2.1. Formulation

- An evolution equation for surface buoyancy (18), to which is added biharmonic diffusion $\kappa {\nabla}_{h}^{4}{b}_{+}$ with no-flux boundary conditions;

#### 2.2. Configuration

**I**- Non-interactive wind stress ${\tau}_{+}={\rho}_{a}{C}_{D}\left|{u}_{a}\right|{u}_{a};$
**II**- Wind stress of the form (1), i.e., depending only on the ocean surface velocity;
**III**- Wind stress of the form (2), i.e., depending only on the SST;
**IV**- Wind stress of the form (8), i.e., depending on both SST and ocean surface velocity.

**II**does not track an SST perturbation; it sets ${\psi}_{+}={\psi}_{1}$ rather than the SST-dependent approximation (17). Simulations are spun up from rest until the kinetic energy reaches a statistical equilibrium, typically several decades.

## 3. Results

**I**to case

**IV**. The QG SST perturbation (Figure 1e,f) covers a range of about 30 ${}^{\circ}$K, which corresponds roughly to the observed SST range of the polar/sub-polar gyre system of the Atlantic and Pacific. An asymmetry about $y={L}_{y}/2$ is visible on some snapshots of relative vorticity and SST perturbation in Figure 1. This asymmetry will be discussed in more detail later in this section.

**I**) only injects energy at the largest scale (notice that for each case, the value of the power input at the largest scale has been divided by 10 to fit the vertical axis on Figure 2b). The cospectrum breaks the total wind power input into contributions from different spatial scales; and the total power input is obtained by summing the cospectrum. Negative values indicate that the wind is removing energy from the oceans at a particular length scale.

**II**) has two effects on the energy input: it decreases the power input of the largest scale and it removes energy from the smaller scales. The peak of this energy sink occurs at a wavelength of about 300 km, slightly larger than the wavelength of the largest internal Rossby radius. In contrast, Figure 2b shows that the SST dependent formulation does not remove energy at the smaller scales and only affects directly large-scale wind power input. In this case, the spectral decomposition of power input for independent snapshots have a non-zero contribution from the smaller scales. However, these contributions cancel each other when the time-mean is applied.

**II**). In contrast, the addition of SST dependence to the wind stress formulation breaks the aforementioned symmetry of the QG equations. In the following, we show that the SST dependent formulation (2) induces a systematic asymmetry leading to a southward slanting and shift of the inter-gyre jet.

**III**), and with an initial condition taken from a previous simulations, but transformed according to $y\to {L}_{y}/2-y$ (flipped north–south) and $\psi \to -\psi $ (reversed). If a northward-shifted regime exists for the SST dependent case, then this simulation should remain northward-shifted, perhaps after a period of adjustment. Figure 4 shows a Hovmöller diagram similar to the ones of Figure 3 but for the simulation with sign reversed and flipped initial conditions. The southern gyre rapidly shrinks and strengthens, as seen in the rapid increase of mean negative PV, while the northern gyre slowly dilates and weakens, as seen in the slow reduction of mean positive PV. After approximately 25 years, the asymmetry of the initial condition has reversed, and the simulation has returned to a southward-shifted state.

**I**) where the initial condition was flipped and reversed. Unlike the simulation with SST dependent wind stress, this simulation remained in a northward-shifted state for the length of the simulation. In principle, the simulations with SST independent wind stress can transition between northward-shifted and southward-shifted states, but this happens on a very long-term scale. In contrast, the simulations with SST dependent wind stress have a clear preference for a southward-shifted state.

#### Physical Mechanism

## 4. Discussion and Conclusions

**III**). The time-averaged eddy-scale wind power input for the southern half was 7.6 MW (megawatts), and for the northern half was $-6.2$ MW, leading to near-cancellation. The real ocean gyres are not symmetric so the effects from the subpolar and subtropical gyres are not likely to cancel as completely in the real ocean, or in a particular region of the ocean as in the study of Byrne et al. [9].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PV | Potential vorticity |

SST | Sea surface temperature |

QG | Quasigeostrophic |

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**Figure 1.**Snapshots from the four configurations. The top four panels show surface vorticity ${\nabla}_{h}^{2}{\psi}_{1}$ (units: s${}^{-1}$) from the simulations with (

**a**) non-interactive wind stress (

**I**); (

**b**) wind stress depending only on ocean surface velocity (

**II**); (

**c**) wind stress depending only on SST (

**III**); and (

**d**) wind stress depending on both SST and ocean surface velocity (

**IV**). The bottom two panels show the QG SST perturbation (units: ${}^{\circ}$C) from the simulations with (

**e**) wind stress depending only on SST (

**III**); and (

**f**) wind stress depending on both SST and ocean surface velocity (

**IV**).

**Figure 2.**(

**a**) Time series of total horizontal "kinetic energy" ${\left|\mathit{u}\right|}^{2}/2$; and (

**b**) cospectra of wind power input; the plot is normalized such that total power (watts) is obtained by summing up the cospectrum. The wind power input at the largest scale is denoted by × symbols, and has been divided by a factor of 10 to fit on the same axes. Symbols

**I**–

**IV**indicate the wind stress formulation used in the simulations.

**Figure 3.**Temporal evolution of the distribution of surface pseudo PV (${Q}_{1}$) from the simulations with different wind stress formulations

**I**–

**IV**. The northern 20% and southern 20% of the domain have been excluded from these statistics in order to highlight the asymmetry concentrated around the inter-gyre jet. Temporal evolution of the distribution of potential vorticity (

**up**) and of the mean positive and (minus) mean negative potential vorticity (

**down**).

**Figure 4.**Hovmöller diagram of the global distribution of surface pseudo PV (${Q}_{1}$) from a simulation using Wind

**III**, restarted from sign reversed and flipped initial condition. Temporal evolution of the distribution of potential vorticity (

**up**) and of the mean positive and (minus) mean negative potential vorticity (

**down**).

**Figure 5.**Time-averaged meridional profiles at 150 km east of the western boundary, centered on the instantaneous jet maximum, from a simulation with wind stress depending only on SST (

**III**). (

**a**) zonal velocity (m/s), high-pass filtered SST perturbation ${(\Delta T)}^{\prime}$ (scaled by a factor of 10; ${}^{\circ}$C), and zonal wind stress ${\tau}^{x}$ (scaled by 0.12; Pa); and (

**b**) surface pseudo PV ${Q}_{1}$ (scaled by ${10}^{-4}$; s${}^{-1}$) and Ekman pumping ${w}_{+}^{E}$ (scaled by ${10}^{-4}$; m/s).

Parameter | Value | Description |
---|---|---|

L | 3072 km | Square domain size |

$\Delta x$ | 3 km | Horizontal grid spacing |

H | 4.5 km | Total depth |

n | 10 | Number of vertical layers |

$\Delta t$ | 15 min | Time step |

${N}_{+}$ | 0.01 s${}^{-1}$ | Surface buoyancy frequency |

$\Delta N$ | 0.004 s${}^{-1}$ | Top to bottom difference in buoyancy frequency |

${N}_{a}$ | 0.0026 s${}^{-1}$ | Abyssal buoyancy frequency |

${l}_{z}$ | 1 km | Decay scale of the stratification |

${f}_{0}$ | 1 × ${10}^{-4}$ s${}^{-1}$ | Coriolis frequency |

β | 2 × ${10}^{-11}$ (ms)${}^{-1}$ | Coriolis parameter gradient |

${\rho}_{0}$ | 1000 kg·m${}^{-3}$ | Reference ocean density |

${\rho}_{a}$ | 1 kg·m${}^{-3}$ | Reference atmosphere density |

${\nu}_{4}$ | 2 × ${10}^{8}$ m${}^{4}$·s${}^{-1}$ | Biharmonic viscosity coefficient |

κ | 4 × ${10}^{8}$ m${}^{4}$·s${}^{-1}$ | Biharmonic SST diffusion coefficient |

${C}_{D}$ | 1.3 × ${10}^{-3}$ | Wind stress drag coefficient |

g | 9.8 m·s${}^{-2}$ | Gravitational acceleration |

${\tau}_{0}$ | 0.12 Pa | Background wind stress amplitude |

d | 4.5 m | Depth of the bottom Ekman layer |

${L}_{f}$ | 40 km | High-pass SST filter length scale |

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**MDPI and ACS Style**

Grooms, I.; Nadeau, L.-P.
The Effects of Mesoscale Ocean–Atmosphere Coupling on the Quasigeostrophic Double Gyre. *Fluids* **2016**, *1*, 34.
https://doi.org/10.3390/fluids1040034

**AMA Style**

Grooms I, Nadeau L-P.
The Effects of Mesoscale Ocean–Atmosphere Coupling on the Quasigeostrophic Double Gyre. *Fluids*. 2016; 1(4):34.
https://doi.org/10.3390/fluids1040034

**Chicago/Turabian Style**

Grooms, Ian, and Louis-Philippe Nadeau.
2016. "The Effects of Mesoscale Ocean–Atmosphere Coupling on the Quasigeostrophic Double Gyre" *Fluids* 1, no. 4: 34.
https://doi.org/10.3390/fluids1040034