# A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography

## Abstract

**:**

## 1. Introduction

- Long Rossby waves are “steered” by the bottom topography in precisely the same manner as the time-mean surface streamlines. In the limit in which the surface flow is relatively unaffected by the bottom topography, so are the long Rossby waves. This concept is made rigorous through comparison of the mathematical equations for long Rossby waves in the present continuously-stratified model with variable bottom topography and the the equivalent equations for long Rossby waves in a two-layer model, linearised about a state of rest.
- The result that long Rossby waves propagate quasi-zonally breaks down catastrophically wherever $f/H$ contours close, irrespective of the stratification. This is demonstrated through the derivation of an integral constraint in which a weighted integral of the dominant Rossby propagation term vanishes over any area enclosed by an $f/H$ contour. Such behaviour has been studied in the analogous two-layer model [16] and has been shown to result in the long Rossby waves partially “jumping” across the closed $f/H$ contour.
- Following the approach of Salmon [15], a nonlinear long Rossby wave equation can be derived which demonstrates, in this model, that the long Rossby wave speed is Doppler shifted by the depth-mean flow. For realistic ACC parameters, the latter term dominates and causes eastward propagation relative to the sea floor, at speeds consistent with the observed eastward propagation of Southern Ocean surface anomalies.

## 2. Planetary Geostrophic Equations

## 3. Application to the Southern Ocean

#### 3.1. Interior Dynamics

#### 3.2. Boundary Conditions

## 4. Steady State

#### 4.1. Characteristics

#### 4.2. An Illustrative Solution

## 5. Linear Rossby Waves

#### 5.1. Linear Wave Equations

#### 5.2. Relation to the Two-Layer Model

#### 5.3. Shallow Pycnocline Limit: Topographic Shielding and Rossby Wormholes

## 6. Nonlinear Rossby Wave Equation

## 7. Conclusions

- Long Rossby waves propagate along the same path as followed by the mean surface geostrophic flow, characteristics that are intermediate to f and $f/H$ contours. For realistic Southern Ocean parameters, these characteristics are nearly zonal, with only slight deflections over major topographic features, aside from the Kerguelan Plateau which represents a more substantial obstacle.
- The quasi-zonal propagation of long Rossby waves breaks down catastrophically in regions of closed $f/H$ contours where, by analogy with the simpler two-layer model, the long Rossby waves can be expected to partially jump across the closed $f/H$ contour.
- In the absence of topographic variations, the Rossby propagation speed consists of an intrinsic Rossby speed, slightly modified from the classical Rossby speed to account for finite ocean depth, and Doppler shifting by the depth-mean flow, consistent with an earlier result obtained by Salmon [15]. This Doppler shift dominates for realistic Southern Ocean parameters, consistent with the observed eastward propagation of Southern Ocean anomalies in surface altimetric observations.

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

ACC | Antarctic Circumpolar Current |

JEBAR | Joint Effect of Baroclinicity and Relief |

## Appendix A. Derivation of Streamfunction of the Depth-Integrated Flow

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**Figure 1.**Schematic diagram illustrating the key ingredients of the model. In the ocean interior, the ansatz that the potential vorticity is a linear function of buoyancy imposes a known exponential decay of buoyancy (thick solid contours) with depth. The solution is therefore determined by conservation of buoyancy along the sea surface, ${b}_{s}(x,y)$, and along the sea floor, ${b}_{b}$ (see Section 3). The total ocean depth is $H(x,y)$ and the path of the surface streamlines in equilibrium is given by the characteristic function, $\varphi (x,y)$ (thin solid contours, see Section 4).

**Figure 2.**Contours of: (

**a**) $f/H$; (

**b**) γ; (

**c**) $\varphi $ with ${H}_{\mathit{ref}}=4\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$; (

**d**) $\varphi $ with ${H}_{\mathit{ref}}=6\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$. The contours of $\varphi $ represent the characteristics along which the surface geostrophic flow is directed in the absence of forcing (Section 4) and also along which long Rossby waves propagate (Section 5).

**Figure 3.**Illustrative steady-state solution with ${H}_{ref}=4\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$. The figure shows buoyancy at the sea surface, $\overline{{b}_{s}}$, which is a prescribed function of the characteristic function, $\varphi $, and also a vertical section of the buoyancy, b, at the longitude 150W. The buoyancy is related to (neutral) density anomalies, $\delta \rho $, by $b=-g\phantom{\rule{0.166667em}{0ex}}\delta \rho /{\rho}_{0}$, where g is the gravitational acceleration, and so the values should be multiplied by a factor of roughly $-{10}^{2}$ to convert to (neutral) density anomalies in $\mathrm{kg}\xb7{\mathrm{m}}^{-3}$.

**Figure 4.**Flow fields in the steady-state solution with ${H}_{ref}=4\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$. (

**a**) depth-integrated streamfunction, $\overline{\psi}$ (Sv); (

**b**) geostrophic streamlines, $\overline{p}/{\rho}_{0}$, at $0\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$ (${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-2}$); (

**c**) geostrophic streamlines, $\overline{p}/{\rho}_{0}$, at $2\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$ (${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-2}$); (

**d**) geostrophic streamlines, $\overline{p}/{\rho}_{0}$, at $4\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$ (${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-2}$).

**Figure 5.**Graph of $coth(H/2{H}_{\mathrm{strat}})-2{H}_{\mathrm{strat}}/H$ versus $H/2{H}_{\mathrm{strat}}$.

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Marshall, D.P. A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography. *Fluids* **2016**, *1*, 17.
https://doi.org/10.3390/fluids1020017

**AMA Style**

Marshall DP. A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography. *Fluids*. 2016; 1(2):17.
https://doi.org/10.3390/fluids1020017

**Chicago/Turabian Style**

Marshall, David P. 2016. "A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography" *Fluids* 1, no. 2: 17.
https://doi.org/10.3390/fluids1020017