# A String Function for Describing the Propagation of Baroclinic Anomalies in the Ocean

## Summary (3 min read)

### 1. Introduction

- An important consideration in large-scale ocean dynamics are the spatial variations in topography and the Coriolis parameter f.
- In other work (Tyler and Käse 2000a, hereafter TK1) the authors introduced the ‘‘string function’’ for the case of a homogeneous one-layer model and in a companion paper (Tyler and Käse 2000b), validated these theoretical results using results from a primitive equation model.
- In summary, two points are noteworthy: first, the geostrophic momentum is properly viewed as having a flow part and a propagation part and, second, the velocity of the propagation can be calculated from a scalar string function that encapsulates beta, topographic, and finite amplitude effects simultaneously.
- It is, however, useful for the discussion in this paper to describe results for a separate baroclinic mode and then include consideration of the barotopic coupling terms.
- The authors briefly discuss other previous work, restricting ourselves mostly to those works with which they will later make comparisons.

### 2. Formulation

- Let us consider the two-layer model shown in Fig.
- The authors will consider dynamics described by the shallowwater, hydrostatic momentum equations in layer integrated flux form.
- When r1 ± r2 the starred term remains and the equation above, which the authors will still refer to as describing the barotropic mode, shows coupling with the baroclinic effects.
- A similar description can be given for (22).

### 3. Case of geostrophic flow

- In later sections the authors will add in the nongeostrophic flow effects.
- Note the similarities in these definitions; m̃ always refers to a mass anomaly due to the alterations in layer thick- nesses, while the string function cm 5 c2/ f with the shallow water wave speed c chosen appropriately for the model.
- This is very much like that for the one-layer model except that the propagation is less sensitive to topography in the open ocean and is typically much slower.
- While the ratio c m/ may increase entering shallow water, suggestingcmo an increase in nonlinearity, increasing topographic slope increases the background string velocity and may predominate to make the disturbance less nonlinear.
- This will make the asymmetries between the propagation of positive and negative m̃ more complicated.

### 4. Case including nongeostrophic flow

- Because (25) does not involve R1 explicitly, using (6) it can be cast in a form similar to that for the one-layer case described in detail in TK1.
- Hence, the nongeostrophic effects for the two-layer baroclinic case are similar to those in the one-layer case and the criteria derived in TK1 can be applied.
- For scales larger than the Rossby radius (5cm/ f ) the dispersion term (involving ]ts2) and the nonlinear advection term (involving ] j) can be neglected relative to ] tm̃ and the nonlinear string velocity contribution, respectively.
- That is, the dynamics are controlled by advection (propagation really) of mass by the string ve- locity, and advection of angular momentum by the flow velocity.

### 5. Barotropic coupling

- As the authors described in the introduction, in the general two-layer case involving finite amplitudes and topography the barotropic and baroclinic modes remain linearly coupled.
- The two evolution equations, (20) and (23), would need to be solved simultaneously.
- U must be prescribed and the constraints given by the equations governing the barotropic dynamics have not yet been imposed.
- In this case the two string velocity terms must balance.
- The authors emphasize that several assumptions have been made in this section that limit the validity of the results presented.

### 6. Summary of theoretical results

- Hence, it is only through the different definitions (see Table 1) for the mass disturbance m̃ and the shallow water wave speed c that the models differ from one another regarding the generalized beta drift described by the terms on the left of (35).
- Changes in m̃ following the string velocity are due to the convergences in the various momentum fluxes shown on the right.
- For many cases these right-side terms vanish and the evolution of m̃ is determined entirely by cm.
- The momentum terms on the right of (35), in sequence, are SF, which in the case of the one-layer model is the Ekman momentum (5 f 21t 3 z), while in the baroclinic models it is the fraction of the geostrophic barotropic momentum in the lower layer [5(h2/h)Sg];.
- When the authors assume that s2 is geostrophic to first order, the friction term can be transformed into a Laplacian diffusion term for m̃, and for scales larger than the Rossby radius, SI becomes negligible.

### 7. Comparisons with previous results and discussion

- In the small amplitude limit, it is simple to show in a manner similar to that done in TK1 that the predictions given by the string function are consistent with the theory for linear topographic and beta Rossby waves.
- Interpreting (36) in their context, the authors can say that the term involving H1 in (36) is due to propagation of m̃ by the background strings , while the m̃2 term iscmo due to the propagation by the nonlinear component of the strings m.c̃ Cushman-Roisin et al. also extended the analysis of westward drift to a two-layer flat-bottom case.
- They found equations describing the drift in each layer that agree with those for their two-layer case in the flat-bottom limit when barotropic effects are discarded.
- The numerical results of Swaters (1998) showed speeds 20% less than the Nof speed and this difference was attributed to the interaction between the dome and the surrounding topographic Rossby wave field.

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### Cites background from "A String Function for Describing th..."

...Circular eddies of different size (Nof (1984)) as well as the leading edge of a continuous dense bottom current (Tyler & Kæse 2001; Wåhlin 2002) move with the same speed, which is proportional to the slope of the bottom and the reduced gravity, and inversely proportional to the Coriolis parameter....

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### "A String Function for Describing th..." refers background in this paper

...Earlier work describing the westward propagation of eddies due to the beta effect (e.g., Flierl 1977; McWilliams and Flierl 1979; Nof 1981, 1983a; Killworth 1983; Cushman-Roisin 1986; Shapiro 1986) have been reviewed and a general formulation for the westward drift has been given (Cushman-Roisin et al....

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...We close this section by drawing attention to a case previously discussed by Shi and Nof (1994) in a similar theoretical context....

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...Earlier work describing the westward propagation of eddies due to the beta effect (e.g., Flierl 1977; McWilliams and Flierl 1979; Nof 1981, 1983a; Killworth 1983; Cushman-Roisin 1986; Shapiro 1986) have been reviewed and a general formulation for the westward drift has been given (Cushman-Roisin et…...

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "A string function for describing the propagation of baroclinic anomalies in the ocean" ?

Important recent papers ( Dewar 1998, Killworth et al. 1997 ) address this topic ; yet further work to settle this issue is required. Of course the results the authors have presented apply to a two-layer ocean, not one that is realistically stratified ; any comparisons are intended only to motivate further research rather than to explain the observations. Still, it is premature to offer this as an explanation for the observed phase speeds before further fundamental studies of the string function are given. 4. As can be seen, the observed phase speeds are typically a couple to several times greater than that theoretically expected, and the discrepancy shows an increase poleward.