On the contribution of Rossby waves driven by surface buoyancy fluxes to low-frequency North Atlantic steric sea surface height variations

Peter Kowalski

Department of Physics, Imperial College, London, UK

Keywords:Sea surface height Rossby waves Subpolar North Atlantic 1.5-Layer ocean

ABSTRACT Previous studies have shown that wind-forced baroclinic Rossby waves can capture a large portion of lowfrequency steric sea surface height (SSH) variations in the North Atlantic.In this paper, the classical wind-driven Rossby wave model derived in a 1.5-layer ocean is extended to include surface buoyancy forcing, and the new model is then used to assess the contribution from buoyancy-forced Rossby waves to low-frequency North Atlantic steric SSH variations.Buoyancy forcing is determined from surface heating as freshwater fluxes are negligible.It is found that buoyancy-forced Rossby waves are important in only a few regions belonging to the subtropicalto-midlatitude and eastern subpolar North Atlantic.In these regions, the new Rossby wave model accounts for 25%–70% of low-frequency steric SSH variations.Furthermore, as part of the analysis it is also shown that a simple static model driven by local surface heat fluxes captures 60%–75% of low-frequency steric SSH variations in the Labrador Sea, which is a region where Rossby waves are found to have no influence on the steric SSH.

1.Introduction

In the North Atlantic, variations in sea surface height (SSH) on interannual and longer timescales have been shown to be primarily due to variations in steric SSH (e.g., Piecuch and Ponte, 2011 ; Forget and Ponte, 2015 ; Polkova et al., 2015 ), which is defined as

whereDis the ocean depth,ρ0a characteristic ocean density, and Δρ=ρ?ρ0the density anomaly.Many studies have thus linked lowfrequency steric SSH variations in the North Atlantic to wind-forced baroclinic Rossby waves (e.g., Sturges et al., 1998 ; Cabanes et al.,2006 ; Zhang and Wu, 2010 ; Polkova et al., 2015 ; Zhang et al., 2016 ;Calafat et al., 2018 ), which can take many years to cross an ocean basin and are therefore thought to influence climate on decadal timescales(e.g., Schneider and Miller, 2001 ).With regards to buoyancy-forced Rossby waves, Piecuch and Ponte (2012) showed that they significantly influence the phase of the steric SSH in the tropical South Atlantic; however, their contribution to low-frequency North Atlantic steric SSH variations is yet to be explored.

In the literature, the most widely used Rossby wave model for the steric SSH is the wind-forced model derived in a 1.5-layer ocean (e.g.,Schneider and Miller, 2001 ; Zhang and Wu, 2010 ).In this framework,the ocean is divided into two layers of constant density with the interface between the layers taken to represent the depth of the pycnocline, which is the depth in the permanent pycnocline where the vertical density gradient is a local maximum (e.g., Feucher et al., 2016 ).Furthermore, the ocean dynamics are governed by the upper-ocean linear vorticity balance in a wind-driven ocean that is in geostrophic and hydrostatic balance, but with only the upper layer in motion.In this paper, we extend this framework to include surface buoyancy forcing by allowing the upper-layer density to vary with space and time.This leads to a new model in which the steric SSH can be decomposed into a static component that is influenced by local surface buoyancy fluxes,and a component that is associated with wind- and buoyancy-forced baroclinic Rossby waves.We use the latter component to explore the role of buoyancy-forced Rossby waves in low-frequency North Atlantic steric SSH variability.

This paper is organized as follows: In Section 2 , we describe the(model) dataset used in this study (the “observations ”) and methodology.In Section 3 , we examine the role of Rossby waves forced by surface buoyancy fluxes by comparing low-frequency SSH variations predicted by the wind- and buoyancy-forced Rossby wave model with those of the“observed ”.This section also includes a summary of the findings along with some suggestions for future work.The new model framework and derivation of the governing equations can be found in the appendix, as well as an assessment of two models driven by local surface heat fluxes that are used in the analysis of the Rossby wave model (see supplementary materials).

2.Data and methodology

The dataset that we use is the ECCO v4r1 ocean state estimate( Forget et al., 2016 ) produced by the Estimating the Circulation and Climate of the Ocean (ECCO) consortium ( Wunsh and Heimbach, 2007 ).The data cover the period 1992–2011.Furthermore, the horizontal resolution of this particular solution in the North Atlantic is 0.25° with 50 vertical levels of varying thickness, and the temporal resolution is monthly.For a detailed description of the ECCO v4r1 ocean state estimate, see Liang et al.(2017) , who used this version of the solution to study bidecadal changes in ocean heat content.Data for all the relevant variables (e.g., surface heat fluxes, density, and wind-stress) used in this study are readily available in ECCO v4r1.

The model steric SSH, denoted byηρ, is computed using Eq.(1) withρ0= 1027 kg m?3.We form time series of anomalies forηρ, and the forcing terms in the Rossby wave model, by removing the time mean,linear trend, and seasonal cycle.The resulting time series for the forcing terms are used to derive model-simulated steric SSH anomalies, which are denoted byηS.The time series ofηSandηρare then low-pass filtered to remove any signals shorter than 1 year to focus on interannual-todecadal timescales.

We compareηSandηρusing the skill metric,

where ?.? denotes time averaging.S→100% indicates thatηSis very close toηρin both phase and amplitude, whileS <0 indicates thatηSmay capture the phase but overestimates the magnitude ofηρ, or that it is a poor match.In all figures showing forecast skill, the regions in whichS <0 are therefore masked.Note also that correlations betweenηSandηρare statistically significant at the 90% confidence level in the regions whereS>10% (not shown).

3.Response of steric SSH to buoyancy-forced Rossby waves

As shown in the appendix, in a 1.5-layer geostrophic and hydrostatic ocean the steric SSH can be written as

whereηLis the change in height due to local surface buoyancy fluxes(Eq.(A.20)) andηP, which is associated with variations in upper-ocean pressure (Eq.(A.14)), is the change in height due to Rossby waves(Eq.(A.17)).In this section, we setηL= 0 and focus on the contribution fromηPtoηS.The resulting Rossby wave equation forηSis

wheret,x,yrefer to time and the distances in the east–west and north–south directions, respectively;g′is the reduced gravity;CRis the Rossby wave phase speed and in this equation it is negative;WEis the local Ekman pumping that is derived using the wind-stressτfrom ECCO v4r1;Bis a source of buoyancy; and note that we have added dissipationε, which represents, among other possible mechanisms, destabilization of long Rossby waves by baroclinic instability ( Lacasce and Pedlosky, 2004 ).Following Piecuch and Ponte (2012) , we assume thatBrepresents mixed-layer fluxes of heat and fresh water; however, similar to Cabanes et al.(2006) , an analysis of the variance ofBshows that freshwater fluxes are negligible over the entire North Atlantic (not shown).Therefore,

whereQnetis the surface heat flux in ECCO v4r1,αT= 2 × 10?4K?1is the coefficient of thermal expansion, andcp= 4028 J kg?1K?1is the specific heat capacity.Thus, buoyancy-forced Rossby waves in our study correspond to those driven by local surface heat fluxes.

The analytical solution to Eq.(4) is obtained by integrating along Rossby wave characteristicsx?CRt= constant (e.g., Fu and Qiu, 2002 ;Cabanes et al., 2006 ).The resulting solution after replacing ?B∕ρ0with Eq.(5) is

where

andηS(xE,y,t) is the eastern boundary steric SSH, which is computed using the ECCO model for the observed steric SSHηρ(see Section 2 )at the eastern boundaryxE.Briefly, we performed model simulations in which we varied the position ofxEand found that moving it closer to the coastlines reduces the skill of the model in the eastern part of the basin (not shown).The reduction in skill could be due to the inclusion of SSH anomalies that are associated with other factors instead of westward-propagating baroclinic Rossby waves (e.g., topography or coastally trapped waves, as suggested by Zhang et al., 2016 ).

We assess the contribution from buoyancy-forced Rossby waves by comparing the forecast skill ( Eq.(2) ) derived whenFQ= 0 andFQ≠0 in Eq.(6) .Following previous studies (e.g., Zhang and Wu, 2010 ;Zhang et al., 2016 ), we letg′,ε,andCRdepend on latitude for simplicity.To derive estimates for these parameters, we first findg′,ε,andCRat each grid point that yield the largest values for forecast skill whenFQ= 0 andFQ≠0 , and then for each parameter we calculate the zonal average over both experiments but only using values from regions where forecast skill is greater than 10%.In these experiments,g′ranges from 0.01 m s?2to 0.06 m s?2andεranges from (0.01 yr )?1to (16 yr )?1(e.g.,Zhang and Wu, 2010 ; Zhang et al., 2016 ), and forCRwe first obtain an estimate at each latitude using the Liouville–Green approximation (e.g.,Piecuch and Ponte, 2012 ),

and then allowCRto vary around this estimate.In the above equation,β= df∕dy,Nis the Brunt–V?is?l? frequency computed at each grid point using the time-mean density in ECCO v4r1, and<.>denotes the zonal average at each latitude.As seen in Fig.1 ,g′, which is a measure of the ocean stratification, decreases roughly monotonically from the tropics to the poles ( Fig.1 (a));εin the tropics is much larger than in the subtropical-to-subpolar North Atlantic ( Fig.1 (b)); andCR( Fig.1 (c)) is very similar to that obtained by Zhang and Wu (2010) and Zhang et al.(2016) .Note also that the values for forecast skill derived by finding the optimalg′,ε,andCRat each grid point (not shown) are similar to those that are obtained using the parameter values in Fig.1 .Finally, similar to Chelton and Schlax (1996) , we find that there is a discrepancy betweenC( Eq.(10) ) andCR( Fig.1 (c)), the latter of which is the phase speed of our ECCO model for the observed steric SSHηρ.In the literature, this difference has been linked to factors such as the inclusion of relative vorticity and Doppler shift by the depth mean flow(e.g., Tulloch et al., 2009 ; Samelson, 2010 ; Klocker and Marshall, 2014 ),effects which are absent in Eq.(10) and our Rossby wave model.

Fig.1.(a) Reduced gravity g ′.(b) Dissipation rate ε .(c) Rossby wave phase speed C R .

Fig.2.(a) ΔS = S 2 ? S 1 , where S 1 is the forecast skill of the wind-forced Rossby wave model ( Eq.(6) with F Q = 0 ) and S 2 is the forecast skill of the new Rossby wave model ( Eq.(6) ).(b) ΔS > 0 in the regions where S1 > 0 .

3.1. Results and discussion

Fig.2 (a) shows ΔS=S2?S1, whereS1is the forecast skill of the classical wind-forced Rossby wave model ( Eq.(6) withFQ= 0 ) andS2is the forecast skill of the new Rossby wave model ( Eq.(6) ) .It can be seen that buoyancy-forced Rossby waves (FQ) do not make a positive contribution to forecast skill everywhere in the North Atlantic (e.g., ΔS<0 in the western subpolar North Atlantic from 50°N to 55°N).Comparing this figure (or Fig.2 (b)) with Fig.B.1(b), it is also evident that in the majority of the basin ΔShas the same sign asSρC,which is the forecast skill of the model for the upper-ocean density content (Appendix B in supplementary materials).Thus, since buoyancy forcing enters Eq.(4) via this model, extending it to include other important factors may subsequently result in enhanced predictions on the basis of baroclinic Rossby waves.Potential factors can be found in Buckley et al.(2014) , who showed that variations in North Atlantic heat content, which is closely tied to the density content, can also be attributed to Ekman transport, with bolus transports and diffusion also playing a key role in the subpolar North Atlantic.

We now focus on the regions where baroclinic Rossby waves make a positive contribution to forecast skill, which are those where both ΔSandS1are positive.As seen in Fig.2 (b), the contribution from buoyancy-forced Rossby waves is negligible in these regions ( ΔS≈0 ),with the exception of some parts of the subtropical-to-midlatitude and eastern subpolar North Atlantic where ΔSranges from 10% to 25% (e.g.30°W ? 60°W, 30°N ? 43°N and 20°W ? 40°W, 55°N ? 60°N).Their effect in the latter regions can also be seen by comparingS1in Fig.3 (a) withS2in Fig.3 (b), and note further thatS2varies between 25% and 70%; however, in most of the regionsS2≈45% (e.g., the eastern subpolar North Atlantic).

With regards to previous studies that have investigated the role of wind-driven Rossby waves in North Atlantic steric SSH variability, our results ( Fig.3 (c)) are most similar to those obtained by Zhang et al.(2016) , who also included dissipation and eastern boundary steric SSH anomalies in the solution to their wind-driven Rossby wave model (both are omitted by Zhang and Wu (2010) , whereas Cabanes et al.(2006) include eastern boundary forcing but omit dissipation).In particular, we obtain similar values for forecast skill on the eastern side of the basin; however, our values on the western side of the tropical-to-midlatitude North Atlantic are generally higher.This difference may be partly due to the different dataset used; Zhang et al.(2016) compared their model-simulated SSH with SSH from satellite altimetry, whereas we compared ours with a model steric SSH derived using data from the ECCOv4 ocean state estimate, and is therefore not a pure observation.Note also that as in Cabanes et al.(2006) and Zhang et al.(2016) , the effect of eastern boundary steric SSH anomalies, which in our model can be determined by isolatingηBin Eq.(6) , is confined to the regions near the eastern boundary.Although we do not show figures for this result, this effect ofηBcan be easily deduced from either Fig.3 (c) or Fig.3 (d).

Fig.3.(a) Forecast skill S 1 of the wind-forced Rossby wave model ( Eq.(6) with F Q = 0) in the regions where both S 1 > 0 and ΔS > 10% . (b) Forecast skill S 2 of the new Rossby wave model ( Eq.(6) ) in the regions where both S 1 > 0 and ΔS > 10% .(c) S 1 > 0 , and (d) S 2 in the regions where S 1 , ΔS > 0.

As in previous studies (e.g., Cabanes et al., 2006; Zhang and Wu, 2010; Zhang et al., 2016) , we find that the skill of the classical wind-driven Rossby wave model ( Fig.3 (c)) is poor in the Mid-Atlantic Ridge, the eastern subpolar North Atlantic from 50°N to 55°N, and the Labrador Sea, and we also obtain the same result with the new Rossby wave model (not shown; however, note that the regions withS2<0 are roughly the same as those whereS1<0 in Fig.3 (c)).In the Mid-Atlantic Ridge and the eastern subpolar North Atlantic, topography has been shown to play a key role ( Osychny and Cornillion, 2004 ; Zhang et al.,2016 ); and in the Labrador Sea, which is a region with weak vertical density gradients (e.g., Buckley et al., 2019 ), studies suggest that local surface heat fluxes have a large influence on the steric SSH (e.g.,Piecuch and Ponte, 2011 ).We indeed find that a simple static model driven by local surface heat fluxes, which arises in our model framework by assuming that the vertical stratification is weak (Appendix C in supplementary materials), captures 60% ? 75% of low-frequency steric SSH variations in the Labrador Sea (Fig.C.1(b)).

Finally, there are many studies (e.g., Schneider and Miller, 2001 ;Capotondi and Alexander, 2001 ; Fu and Qiu, 2002 ) that show winddriven Rossby waves play a key role in the North Pacific; however, the role of buoyancy-forced Rossby waves is yet to be explored.It would thus be interesting to conduct a similar study for the North Pacific.

Acknowledgements

The author is grateful to Arnaud Czaja and the two anonymous reviewers for their constructive comments on the manuscript.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.aosl.2022.100153 .