# 12.14.10: Notes on literature concerning deep recirculation gyre¶

## Lozier and Riser (1989)¶

This paper studies “the western boundary in an eddy-resolving general circulation model” and “assess the boundary’s contribution to the overall gain/loss cycle of potential vorticity for the wind-driven surface layer and the deep ocean”. The interesting aspect of the paper is that the authors use both an Eulerian and Lagrangian analysis of the flow.

The model is composed of three isopycnal layers and is forced by a constant wind stress that produces a double-gyre ocean as in Holland and Rhines (1980). The main differences with our simulation are 1) the wind is not periodic in time and 2) the interfacial stress is not zero and largely responsible for the mean motion in the middle and bottom layers.

In the middle layer, potential vorticity (PV) is homogenized even if there is a strong eastward flow in the middle of the homogenized domain; this is an example where a strong Eulerian-mean flow is not associated with a strong cross-flow gradient of Eulerian-mean PV. According to the authors, this homogenization is expected as the layer satisfies the criteria of Rhines and Young (1982). What is different in the middle layer of our simulation so that PV is not expected to homogenize? Within the western boundary of PV, PV changes a lot due to dissipation (frictional boundary), while outside the boundary, PV stays nearly constant (inertial boundary). In this layer, the eddies contribute little.

In the bottom layer, in the interior, eddy flux is important as the mean flow crosses PV contours. The Eulerian-mean PV equation is then the turbulent Sverdrup equation. The dynamics of the interior in this layer is the closest to the dynamics of our middle layer.

They do not discuss the PV dynamics in the interior of the middle layer. See Rhines and Young (1982)?

## Rhines and Young (1982)¶

“There is a threshold of energy which must be surpassed if the flow is to produce closed PV-contours”.

“The corresponding [eddy] stresses can transport momentum both laterally (i.e. within density surfaces), through Reynolds stresses, and vertically (i.e. across density surfaces), through form drag. In fact, Rhines and Holland argue that for large-scale mean flows (i.e. flows in which the relative vorticity is negligible compared with the planetary vorticity and vortex stretching) the latter, vertical processe dominates.”

The zero flow solution “can be avoided if the geostrophic contours and steamlines close upon themselves in the interior of the basin. [...] It is in these regions, isolated from lateral boundaries, that we expect strong interior flows to develop. [...] In order that the interior geostrophic contours close, however, the forcing in the upper ocean must deform the density interfaces sufficiently to overcome the β effect in the layers below.”acroread

## Lozier and Riser (1990)¶

The author uses the same simulation as in Lozier and Riser (1989).

In the upper layer, the balance in the eastern half is between the wind curl and the meridional mean flow and trajectories are typically a wave motion superimposed onto a southward drift. In the western half, the turbulent Sverdrup balance dominates.

In the middle and bottom layer, the trajectories of parcel in the eastern half are wave-like but without any net displacement and any change of PV.

The rest of the study is about the exchange of parcels between the two gyres. This is not what I study. (NOTA: There are recirculation gyres in their model but they strongly associated with the dynamics of the boundary; I am interested instead of recirculations that exist independent on boundaries.

The authors study the effect of eddies in an abyssal layer generated by the instabilities of a surface jet in modifying the meridional transport of mass. This is not about the deep recirculation gyre

## Qiu et al. (2008)¶

The authors study the northern recirculation gyre below the Kuroshio extension.

Their estimate of the transport is from the Eulerian mean. If the Eulerian mean flow is indeed due to eddies, that transport is likely to be different from the Lagrangian transport.

Their Fig. 10 shows the Eulerian-mean geostrophic circulation along one isopycnal superimposed onto the Eulerian-mean PV field. The PV field is still dominated by the β effect so that the recirculation gyres have to be associated with flow across these mean PV contours, which fits well our Lagrangian analysis.

See papers that study the dynamics of the southern Recirculation Gyre of the Kuroshio Extension but mostly of the Gulf Stream.

## Jayne et al. (2009)¶

The authors review observations of the Kuroshio Extension and its recirculation gyres. See Waterman (2009) for their dynamics:

“A variety of mechanisms have been offered for producing the intense recirculation zones found in the vicinity of western boundary currents. A hypothesis is that they result from the need of the inertial, baroclinic western boundary current to rid itself of anomalous potential vorticity which it acquired at other latitudes, prior to separating from the coast.Three classes of theories appear in the literature to explain how this occurs. The first are inertial theories in which time-mean recirculation gyres can arise from the steady-state time- mean advection of potential vorticity alone (e.g.Fofonoff, 1954; Marshall and Nurser, 1986; Greatbatch, 1987; Cessi, 1990; Nakano et al., 2008).The second group are theories in which the eddy fluxes resulting from a directly prescribed vorticity forcing generate mean rectified flows through eddy-mean flow and eddy-eddy interactions (e.g. Haidvogel and Rhines, 1983; Cessi et al., 1987; Malanotte- Rizzoli et al., 1995; Berloff, 2005). Finally, the third group of theories derives from studies of unstable jets in which the generation of mean recirculations results from the combination of eddy effects (arising from jet instabilities) and inertial effects(e.g.Spall, 1994;Jayne et al., 1996; Beliakova, 1998;Jayne and Hogg, 1999;Mizuta, 2009). These theories and their relevance to the Kuroshio Extension are discussed in more depth by Waterman (2009).”

Again, their transport estimate is from the Eulerian-mean flow which can be different in the presence of eddies from the Lagrangian-mean transport. Is there any tracer study of the recirculation gyres?

• See McDowell et al. (1982): North Atlantic potential vorticity and its relation to the general circulation

In this paper, the authors show that mean PV is not everywhere parallel to latitude circles so that dissipation is not always needed for water to move meridionally (any update on that paper because this is pretty old?)

• There are the observations by Bower and colleagues but, dynamically, the problem is more complex as it involves the source of deep water to the North and the Deep Western Boundary Current.

• Pickart (PhD Thesis, 1987)?

## Waterman (PhD Thesis, 2009)¶

Eddies within the eastward offshoot of the western boundary currents are important in

1. modifying the amplitude and structure of the eastward jet

2. generating recirculaiton gyre on both sides of the jet

3. driving deep mean flows

“Hogg (1983, 1985, 1993) examined the issue of eddy effects on the deep circulations from the point of view of vorticity dynamics, and suggested that (given relatively large error bars) lateral relative vorticity and thickness eddy fluxes appear to have gradients of the proper sign and strength to drive a deep recirculation of the right order of magnitude.”

4. modulating the low-frequency variability of the system

About previous work on the dynamics:

“[R]ecirculations can arise from the steady-state inertial terms, or the rectification of eddy fluxes, or potentially both. Steady-state inertial theories show that closed recirculation gyres are steady solutions to the nonlinear equations of motion forced by a balance between the inertial term (the mean advection of PV) and dissipation. At the same time, time-dependent numerical simulations demonstrate that zonal flows and closed recirculations can be generated solely from rectification effects through nonlinear eddy-mean flow and eddy-eddy interactions. Finally, rectified mean flows can also result from forcing by an unstable jet. In this case, mean recirculations to the north and south of the jet are produced by eddies, generated by the jet’s instability, acting to smooth the PV anomalies associated with the jet, and in the process produce homogenized regions in which essentially inertial recirculations can develop. It is interesting that recirculations generated in this way in barotropic models are able to predict recirculation strength quite accurately in spite of their reliance on the barotropic instability mechanism.”

Her first theoretical study is that of eddy-driven recirculations from a localized, transient forcing, that is, the time mean-flow response of a barotropic and equivalently barotropic fluid subject to a simple vorticity forcing that is localized in space and oscillatory in time. Her study is relevant to “the phenomenon of zonal jet formation, observed to occur spontaneously in turbulent β-plane flows (e.g. Rhines, 1975, 1977; Williams, 1978). This process has recently received renewed interest given the discovery of deep zonal jets in ocean observations (Maximenko et al., 2005) and ocean GCMs (Richards et al., 2006). Finally, the problem can be considered a small contribution towards improving our understanding of the varied effects of eddies on the large-scale and time-mean state, critical in our search for better ways to parameterize unresolved eddy effects in general circulation models.”

“at least in the weakly nonlinear limit, it is necessary for the forcing to have a finite length scale in order to generate rectified flow in the far field, a lesson that provides us with a new understanding of the rectification mechanism.” Yes, that is for the Eulerian-mean flow but what about the Lagrangian-mean flow?

“It implies that the mechanism that is generating the rectified flow is occurring inside the forcing region (where the Green’s function and particular solution differ) and not by the waves in the far field (where they do not).” I am not sure about that.

“a resonant-like response is achieved when the forcing length scale L, is well matched to the spectrum of free Rossby waves the forcing can excite. [...] Specifically, by corresponding to the wave in the available Rossby wave spectrum with zero zonal group velocity and maximized meridional group velocity, it is the wave that is best suited to producing zonal asymmetry in zonal energy radiation and a large meridional component of energy radiation, both of which are necessary for effective rectification.” That is a nice result: the rectification (in the Eulerian-mean) depends on the population of waves that are excited. What I also show is that for the same population, the rectification in the Lagrangian-mean depends also on the level of dissipation.

With increasing stratification, the Eulerian-mean flow increases in strength and then decays to nearly-zero. In that latter regime, the forcing field fails to excite waves that can propagate outside the forcing domain. This seems contradictory to a previous statement that says that the rectification does not occur locally, outside the domain.

As for the stratification, the strength of the rectification peaks for a value of a zonal background flow. The background flow plays a role mainly by changing the phase and group speed of the waves.

In a more nonlinear regime, the interaction between the Eulerian-mean flow and the waves becomes important and “[d]espite this complexity however, the net effect of [that interaction] will be to always counteract or reduce the ability of the waves/eddies to rectify. It is this counteracting effect that results in the saturation in the mean-flow strength at large forcing amplitude that is observed.”

Boning and Budich (1992) references Boning and Cox (1988); they also mention that when higher resolution is used, there is no change in the meridional heat transport (see Drijfhout and Walstein (1998), Fanning and Weaver (1997) and Drijfhout (1994) for a more recent discussion of this matter).

Figueroa and Olson (1994) references Boning and Cox (1988)

Drijfhout (1994) references Boning and Cox (1988)

Bryan (1996) references Boning and Cox (1988)

Ladd and Thompson (1998) references Boning and Cox (1988)

Jayne and Marotzke (2002) references Boning and Cox (1988)

## Yang (1996)¶

This paper references Lozier and Riser (1989).

## Other papers on the water exchange across the Gulf Stream¶

Song et al. (1995)

Bower and Lozier (1994)