# Fofonoff Negative Modes

## Abstract

**:**

## 1. Introduction

^{−1}. Note, however, that our problem is linear and hence the structure of the solution is independent of the velocity magnitude.

_{e}. The southern boundary is at y = 0 and extends from x = 0 to x = x

_{e}while the northern boundary is at y = 1 and extends the same distance in x.

_{e}is different from zero, the streamlines will be forced off latitude circles and relative vorticity will be generated.

_{e}).

## 2. Solutions

^{2}yield shorter wavelengths in x. Note that if the domain in Figure 1 were reflected about y = 0, the solution obtained above would represent the disturbed flow to the east of a narrow island at x = x

_{e}.

## 3. Resonance

_{e}). However, this is generally not a sufficient condition for resonance in all four of the cases described above.

_{e}= 2, resonance occurs when ${K}^{2}/{\pi}^{2}$ = 5. Figure 5a shows the solution for a value of ${K}^{2}/{\pi}^{2}$ very near resonance, i.e., at ${K}^{2}/{\pi}^{2}$ = 5.001. The solution appears to be a pure standing wave and the amplitude of the wave is very large, so much greater than O(1) that the inflow and outflow streamlines that force the solution are not evident. Note that Figure 5b shows a solution that is qualitatively similar to Figure 2b. The entering and exiting streamlines are evident and the amplitude of the excited wave is O(1). In fact, each of the solutions 2 through 4 are non-resonant.

_{e}= 2. Since j is odd and n + j is also odd, each of the four solutions is resonant. Again, the magnitude of the solution at resonance is so large that the details of the forcing inflows and outflows become negligible compared to the free wave excited by the response.

## 4. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Pedlosky, J.; Spall, M.A. The interaction of an eastward-flowing current and an island: Sub- and supercritical flow. J. Phys. Ocean
**2015**, 11, 2806–2819. [Google Scholar] [CrossRef] - Fofonoff, N.P. Steady flow in a frictionless homogenous ocean. J. Mar. Res.
**1954**, 13, 254–262. [Google Scholar] - Ierley, G.R.; Sheremet, V. Multiple solutions and inertial runaway of the wind driven circulation. J. Mar. Res.
**1995**, 53, 703–737. [Google Scholar] [CrossRef] - Fofonoff, N.P. Dynamics of ocean currents. In The Sea: Ideas and Observations; Interscience Publishers: London, UK, 1962; pp. 323–395. [Google Scholar]
- Batchelor, G.K. Steady laminar flow with closed streamlines at large Reynolds numbers. J. Fluid Mech.
**1956**, 1, 177–190. [Google Scholar] [CrossRef] - Pedlosky, J. Geophysical Fluid Dynamics; Springer-Verlag: New York, NY, USA, 1987; p. 710. [Google Scholar]

**Figure 1.**The Fofonoff solution for problem 1 where a uniform flow enters through the western boundary and exits in a restricted range through the eastern boundary for different values of ${K}^{2}$. Panel (

**a**) ${K}^{2}/{\pi}^{2}$ = 0.1; (

**b**) ${K}^{2}/{\pi}^{2}$ = 3.0.

**Figure 2.**As in Figure 1 but for inflow through a narrow gap at the northwest corner and outflow through a similar gap at the northeast corner. Panel (

**a**) ${K}^{2}/{\pi}^{2}$ = 0.1; (

**b**) ${K}^{2}/{\pi}^{2}$ = 3.0.

**Figure 3.**As in the previous two figures but for the case where there is no inflow or outflow. This can be thought of as the pure Fofonoff negative mode. Panel (

**a**) ${K}^{2}/{\pi}^{2}$ = 0.1; (

**b**) ${K}^{2}/{\pi}^{2}$ = 3.0.

**Figure 4.**As in the previous figures but for the case with inflow at (x, y) = (0, 1) and outflow at (x, y) = (x

_{e}, 0). Panel (

**a**) ${K}^{2}/{\pi}^{2}$ = 0.1; (

**b**) ${K}^{2}/{\pi}^{2}$ = 3.0.

**Figure 5.**Panel (

**a**) the solution to problem 1 for $\raisebox{1ex}{${K}^{2}$}\!\left/ \!\raisebox{-1ex}{${\pi}^{2}$}\right.=5.001$ for which near- resonance occurs for n = 1 and j = 4 for x

_{e}= 2; (

**b**) the solution to problem 2 for the same value of K and for which no resonance occurs.

**Figure 6.**Panel (

**a**) the form of the solution for ${K}^{2}/{\pi}^{2}$ = 3.251 for which all examples except problem 4 are resonant, corresponding to n = 1, j = 3. Panel (

**b**) since n + j is even, problem 4 remains non-resonant.

**Figure 7.**The form of the solution for all four cases for ${K}^{2}/{\pi}^{2}$ = 6.251, near the resonance for n = 2, j = 3.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pedlosky, J.
Fofonoff Negative Modes. *Fluids* **2016**, *1*, 13.
https://doi.org/10.3390/fluids1020013

**AMA Style**

Pedlosky J.
Fofonoff Negative Modes. *Fluids*. 2016; 1(2):13.
https://doi.org/10.3390/fluids1020013

**Chicago/Turabian Style**

Pedlosky, Joseph.
2016. "Fofonoff Negative Modes" *Fluids* 1, no. 2: 13.
https://doi.org/10.3390/fluids1020013