# Submesoscale Turbulence over a Topographic Slope

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

^{−1}. Typically, resolving submesoscale eddies requires the horizontal resolution on the order of one tenth of the Rossby deformation radius [5]. The model domain used in this study is 640 km in the meridional direction and 320 km in the zonal direction with a horizontal resolution of 1 km ×1 km. This scale is much smaller than the Rossby deformation radius, $\lambda =NH/f$, where N is the buoyancy frequency ($\sim 4\times {10}^{-3}$ s

^{−1}). These values produce a $\lambda $ that varies between 10 km and 30 km, depending on the depth. In the vertical direction we have 60 layers evenly spaced from the surface to a maximum depth of $H=600$ m, giving a vertical resolution of 10 m. Density is a linear function of the potential temperature ($\theta $) with a constant thermal expansion coefficient $\alpha =1\times {10}^{-4}$ (°C)

^{−1}. The initial θ (°C) profile is a function of latitude and depth,

^{−2}and a standard deviation of $\sigma =40$ km. The momentum input by the wind stress is balanced by a linear bottom friction with a constant bottom drag coefficient, $r=1.1\times {10}^{-3}$ m s

^{−1}. In this model, horizontal and vertical viscosities are set to be 1 and ${10}^{-5}$ m

^{2}s

^{−1}respectively. Horizontal and vertical temperature diffusion coefficients are 10 and ${10}^{-5}$ m s

^{−1}, respectively. To simulate vertical mixing in the ocean surface boundary layer, the K-profile parametrization (KPP) method [43] is employed.

## 3. Results

## 4. Discussion

#### 4.1. Spectral Slope

#### 4.2. Interior PV Gradients

#### 4.3. Interaction between Mesoscale and Submesoscale

## 5. Conclusions

- Surface vorticity characteristics are modified by the presence of a sloping bottom. Most of the persistent eddies near the surface are cyclonic.
- The sloping bottom modifies the spectra of near-surface vertical velocities and kinetic energies. Velocity spectra exhibit spatial asymmetries between the northern and southern flanks of the domain.
- Interaction of wind forcing, mean flow and a topographic slope can lead to substantial PV gradients over relatively short distances, which are generated by the Ekman transport at the sea surface and at the ocean bottom. These are reflected in the characteristics of the submesoscale motions, which are more active in low PV regions.
- Regions of enhanced submesoscale vertical velocity are associated with areas of larger mesoscale eddy kinetic energy due to enhanced stirring and the subsequent filamentation and frontogenesis associated with these larger-scale motions.
- These results are not consistent with the spectral predictions of SQG theory, which is perhaps not surprising due to strong interior PV gradients, but points to a critical role for ageostrophic velocities generated at both surface and bottom boundary layers.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Overview map of the Weddell Sea sector of the Southern Ocean. (

**b**) An enhanced view of the red box in panel (

**a**), where bathymetry (m) is given in color. Circles in panel (

**b**) correspond to a single hydrographic transect collected by an ocean glider in January 2012. (

**c**) Vertical, cross-slope section of cross-transect (along-slope) velocity, v; the zero contour is given in white. (

**d**) Rossby number approximated by ${v}_{x}/f$, where x is the cross-slope direction of the same transect in panel (

**c**). Tick marks at the top of panel (

**c**) indicate the surfacing positions of each glider dive. See Thompson et al. [23] for further details.

**Figure 2.**Schematic overview of the model configuration for the five simulations described in Table 1. Panels (

**a**–

**e**) correspond to experiments (1)–(5). Experiment ID is indicated in each panel. Colors and contours show the zonally-uniform initial temperature profile. The temperature is relaxed to these initial values at the northern and southern boundaries. The thick black curve marks the bathymetry, while the circle over each panel marks the wind orientation: down-front (dots) or up-front (crosses). The blue curve on the top of panel (

**a**) shows the surface wind stress profile, with a peak value ${\tau}_{0}=0.05$ N m

^{−2}.

**Figure 3.**(

**a**) Growth of total kinetic energy in Experiment DF-S. (

**b**) Snapshots at day 900 for Experiment DF-S (Table 1) surface potential temperature (°C) at 10 m depth. (

**c**) Same as (

**b**), for vertical velocity w (${10}^{-4}$ m s

^{−1}) at 30 m depth.

**Figure 4.**Near-surface Rossby number, $Ro=\zeta /f$, at 10 m depth for the five experiments. Panels (

**a**–

**e**) correspond to experiments (1)–(5) described in Table 1, and experiment ID is indicated. The left-hand plot in each plan shows the zonally-averaged root mean square (RMS) $Ro$ averaged over a period of 200 days. The right-hand plot is a snapshot of surface $Ro$ at day 900.

**Figure 5.**Spectra of surface horizontal kinetic energy (10 m depth, left panels) and vertical velocity (30 m depth, right panels) averaged from day 800 to 1000. (

**a**) Kinetic energy spectra and (

**b**) vertical velocity spectra in Experiment DF-Fl. (

**c**) Kinetic energy spectra and (

**d**) vertical velocity spectra in Experiment DF-S. (

**e**) Kinetic energy spectra and (

**f**) vertical velocity spectra in Experiment DF-N. Dotted lines represent ${k}^{-1}$ and ${k}^{-2}$ spectral slope, provided for reference. Blue lines represent the southern flank of the domain from −300 km $<y<$−100 km. Red lines represent the middle of the domain (frontal region) from −100 km $<y<$ 100 km. Black lines represent the northern flank of the domain from 100 km $<y<$ 300 km. Spectra in Experiments UF-S and UF-N are similar to those in the control Experiment DF-Fl, and are not shown in this figure.

**Figure 6.**Probability density function (PDF) for (

**a**) surface Rossby number, and (

**b**) Rossby number at 180 m depth averaged from day 800 to 1000 for Experiments (1)–(5). Values of PDF skewness are labeled using the same color for each experiment.

**Figure 7.**PV cross section in the middle of the domain (x = 0) averaged from day 800 to 1000. Values are displayed in $log10$ scale. Black contours show the mean potential temperature and also indicate isopycnal surfaces. Panels (

**a**–

**e**) correspond to experiments (1)–(5) described in Table 1, and experiment ID is indicated.

**Figure 8.**Upper panels are PV snapshots on the 12 °C (

**a**), 14 °C (

**b**), and 18 °C (

**c**) isopycnal surfaces at day 900 for Experiment DF-S. White areas indicate the isopycnal surface intersecting with the topographic slope or the surface. Lower panels (

**d**–

**f**) are the corresponding time and zonal averaged PV fluxes. PV fluxes are calculated using snapshots between day 800 and 1000.

**Figure 9.**Zonal and time averaged EKE ($\rho ({\overline{{u}^{\prime 2}+{v}^{\prime 2}}}^{x,t})/2$) from day 800 to 1000. Values are displayed in $log10$ scale. Contour lines show zonal and time averaged zonal velocity (

**u**). Black line represents positive values. Gray line represents negative values. Panels (

**a**–

**e**) correspond to experiments (1)–(5) described in Table 1, and experiment ID is indicated.

**Figure 10.**Zonal and time averaged vertical kinetic energy ($\rho {\overline{{w}^{2}}}^{x,t}/2$) from day 800 to 1000 for for five experiments. Values are displayed in $log10$ scale. Contour lines show zonal and time averaged zonal velocity (

**u**). Black line represents positive values. Gray line represents negative values. Panels (

**a**–

**e**) correspond to experiments (1)–(5) described in Table 1, and experiment ID is indicated.

**Figure 11.**Vertical structure of $Ro$ at day 900 for Experiment DF-S (

**a**–

**c**) and Experiment UF-S (

**d**–

**f**). Cross sections at 10 m, 150 m, and 300 m depth are shown. White areas in (

**c**,

**f**) are associated with topographic slope interception.

**Table 1.**Simulation configurations. The five experiments correspond to the schematics in Figure 2. The identifying characteristics include the wind and topography orientation. The front velocity is diagnosed at the location where $\left|\overline{u}(y)\right|$ is greatest, when the simulation has reached a statistically-equilibrated state.

Experiment Number | Experiment ID | Surface Wind Orientation | Shelf Location | Frontal Zonal Velocity (m s^{−1}) |
---|---|---|---|---|

1 | DF-Fl | down-front | flat bottom | 0.1895 |

2 | DF-S | down-front | south | 0.4047 |

3 | DF-N | down-front | north | 0.4086 |

4 | UF-S | up-front | south | −0.2589 |

5 | UF-N | up-front | north | −0.3694 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lazar, A.; Zhang, Q.; Thompson, A.F.
Submesoscale Turbulence over a Topographic Slope. *Fluids* **2018**, *3*, 32.
https://doi.org/10.3390/fluids3020032

**AMA Style**

Lazar A, Zhang Q, Thompson AF.
Submesoscale Turbulence over a Topographic Slope. *Fluids*. 2018; 3(2):32.
https://doi.org/10.3390/fluids3020032

**Chicago/Turabian Style**

Lazar, Ayah, Qiong Zhang, and Andrew F. Thompson.
2018. "Submesoscale Turbulence over a Topographic Slope" *Fluids* 3, no. 2: 32.
https://doi.org/10.3390/fluids3020032