Parameterizing Eddy Heat Flux in a 1/10 Ocean. Simulation: A Wavenumber Perspective

Transcription

1 Parameterizing Eddy Heat Flux in a 1/10 Ocean Simulation: A Wavenumber Perspective Alexa Griesel Sarah T. Gille Janet Sprintall Julie L. McClean Scripps Institution of Oceanography, La Jolla Mathew E. Maltrud Los Alamos National Laboratory Corresponding author address: Alexa Griesel, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA , USA. agriesel@ucsd.edu

2 Abstract The skill of an eddy heat flux parameterization based on the downgradient diffusion of temperature is assessed as a function of wavenumber in the Southern Ocean of the 1/10 degree Parallel Ocean Program. Coherence analysis suggests that the meridional eddy heat flux, v T, is coherent with its diffusive parameterization, y T, on lengthscales larger than 50 while the divergence of the eddy heat flux, (u T ), is coherent with the Laplacian of mean temperature, 2 T, on scales less than 10. Eddy heat fluxes can be partitioned into rotational and divergent components; only the divergent component influences the heat budget. The curl and divergence of the eddy heat flux provide measures of the rotational and divergent components. At the scales at which most of the eddy heat flux energy occurs the rotational component accounts for more than 95% of the total flux, increasing to more than 99% with increasing lengthscale. As a result, the significant coherence of v T with y T at large scales is due to the rotational component, and the coherence between (u T ) and 2 T is low since it is associated with very small and noisy features. Diffusivities of about 500 m 2 s 1 were found within the ACC, with highest values of up to 2000 m 2 s 1 in the Aghulas Retroflection area close to the surface. Diffusivities estimated from the eddy heat flux showed a high wavenumber dependence and were as high as 5000 m 2 s 1, reflecting the presence of the rotational component. The rotational component can be approximated by assuming it to consist of geostrophic flow along temperature contours. However, this rotational component only accounts for a fraction of the total rotational component, leaving a residual that is still more rotational than divergent. 1

3 1. Introduction One of the most challenging problems of ocean and climate modeling is the representation and parameterization of mesoscale eddies. Eddy fluxes u q in coarse resolution models are parameterized assuming flux-gradient relationships: they are related to mean quantity gradients q, such as temperature, isopycnal thickness (Gent and McWilliams 1990, henceforth GM) or potential vorticity (Rhines and Young 1982), involving an eddy diffusivity κ: u q = κ q (1) Here, the overbar is an average in time, the prime denotes the deviation from the time mean and the brackets are an average in space. The eddy fluxes in equation (1) are comprised of distinct rotational and divergent components (Lau and Wallace 1979; Marshall and Shutts 1981, henceforth MS). The term that enters the actual tracer balance equation is the divergence of the eddy flux ( ) t q eddies = u q = κ q, (2) which specifies how the eddy fluxes influence the large scale quantity. The rotational component of the eddy flux does not contribute to the local heat budget since it transports as much heat into a region as out of it (Jayne and Marotzke 2002) and does not play any role in equation (2). Hence, to assess how well eddies may be represented by a diffusive parameterization in climate models and to compute diffusivities, the divergence of the eddy flux rather than the raw eddy flux should be employed. However, equation (2) shows this involves the computation of two derivatives and is challenging since the diffusivity appears within the divergence operator. Patterns of diffusivities calculated from the divergence of the eddy heat flux are noisy with no clear sign or order of magnitude of the diffusivities (Nakamura and Chao 2000). In some 2

4 studies, some spatial smoothing of the divergence improved the results (Bryan et al. 1999; Tanaka et al. 2007), but correlations of the divergence of the eddy flux with the parameterization were still found to be low (Gille and Davis 1999; Bryan et al. 1999; Solovev et al. 2002). An alternative approach is to separate the rotational and divergent components from the raw eddy fluxes in equation (1) through a Helmholtz decomposition (e.g. Lau and Wallace 1979; Roberts and Marshall 2000; Drijfhout and Hazeleger 2001; Jayne and Marotzke 2002). The total horizontal eddy heat flux u T is given by u T = u T div + u T rot, (3) where the divergent heat flux is the gradient of a scalar potential Φ u T div = h Φ, (4) and the rotational heat flux u T rot can be written with a streamfunction Θ: u T rot = k h Θ. (5) Taking the divergence of the total horizontal eddy heat flux (3), and using (4) and (5) gives: h 2 Φ = h u T (6) and h 2 Θ = k h u T. (7) One can solve for the potentials by applying a Laplacian inverter (e.g. Jayne and Marotzke 2002). The Helmholtz decomposition however depends on the choice of boundary conditions and is therefore not unique (Fox-Kemper et al. 2003). It is also computationally fairly expensive. Rather than performing the mathematical Helmholtz decomposition, MS identified a rotational component using dynamical arguments. They considered the eddy potential energy 3

5 (EPE) equation in the quasigeostrophic approximation and neglected eddy advection of EPE. If the deviation of the mean flow from the mean temperature gradients is small, the eddy heat flux separates into (a) a rotational component that largely balances the mean flow advection of EPE and represents the spatial growth and decay of the eddies, and (b) a residual component that balances the conversion of EPE to kinetic energy and can be used to assess a downgradient parameterization and compute diffusivities. The MS decomposition has been applied to the atmosphere and ocean in several studies (Illari and Marshall 1983; Marshall 1984; Cronin and Watts 1996; Eden et al. 2006a,b; Eden 2006), although none of these studies investigate correlations between the residual fluxes and the mean quantity gradients nor do they consider how this correlation might change when averaged over different spatial scales. Eden et al. (2006a,b) diagnosed thickness diffusivities κ in an eddying model for the North Atlantic and Southern Ocean using both the MS approach and also a more generalized approach without any assumption about the mean flow (Medvedev and Greatbatch 2004). The MS method does not guarantee that the residual flux is rotation-free and the question remains how well it balances the conversion of EPE to kinetic energy in the energy balance equation. All these methods that attempt to isolate the divergent component have their drawbacks, and the fact remains that generally it is the raw eddy fluxes that are most readily available from observations and eddying models. Thus, we pursue the concept that even though there is not necessarily a strong association locally between mean gradients and the raw eddy fluxes, the raw eddy fluxes averaged over large enough scales may be expected to have a net component down the mean gradient resulting in the release of potential energy. In general, the zonal integral of the rotational part of the meridional eddy heat flux with the streamfunction Θ from equation (5) is simply x 2 x 1 v T rot = x 2 x 1 x Θdx = Θ(x 2 ) Θ(x 1 ). The zonal integral is zero over any closed 4

6 latitude circle. It is less obvious what the integral yields when the integration is performed over arbitrary distances x 2 x 1. In the simplest scenario, when eddies are purely rotational, the cumulative zonal integral alternates between positive and negative values. For the integral of the total (rotational plus divergent) meridional eddy heat transport, one might expect the divergent component to dominate provided that the averaging lengthscale was sufficiently large compared with the eddy length scales. Hence, raw eddy heat fluxes averaged over appropriate length scales might still be used to assess parameterizations and compute diffusivities. Indeed, spatial averaging of the eddy fluxes seems to improve correlations of the eddy fluxes with the downgradient parameterization in some studies (e.g. Rix and Willebrand 1996). Eddy fluxes present a conundrum: If diffusivities or correlations are computed from the eddy fluxes, then the presence of the rotational flux biases the analyses. In contrast, if eddy fluxes are computed from the divergence of the flux, correlation patterns are very noisy and some degree of spatial smoothing of the fields usually appears necessary to obtain meaningful correlations. It remains unresolved whether the skill of downgradient parameterizations and diffusivities should be evaluated by employing the divergence of the fluxes, the raw fluxes themselves averaged over appropriate spatial scales, or the fluxes minus a rotational component. In this study we ask how well eddy heat fluxes can be represented by a downgradient diffusion as a function of averaging lengthscale. Is there a minimum averaging lengthscale for the raw eddy heat flux that eliminates the rotational component and leads to significant correlation with mean gradients? Do the divergent components of the eddy heat flux and the total eddy heat flux correlate differently with the diffusive parameterization as a function of averaging lengthscale? Why is the divergent component so noisy? How do diffusivities vary as a function of averaging lengthscale when computed from the total eddy heat flux or the divergent com- 5

7 ponent? To address these questions we first employ a spatial coherence analysis between the eddy heat fluxes and the temperature downgradient parameterization on the one hand, and between the divergence of the heat flux and the Laplacian of temperature on the other hand. Here, we consider only the downgradient diffusion of temperature on horizontal surfaces. This is a straight forward first step to introduce concepts and mirrors z-coordinate model structure. We expect that the results will not change qualitatively in a more sophisticated isopycnal diffusion scheme. Our results will show that the rotational component of the eddy heat flux dominates over all lengthscales. We then follow the ideas of MS and assess the extent to which eddies are geostrophic. We expect that if most of the eddy heat flux can be explained locally through geostrophy then, because of the alignment of temperature contours and dynamic topography, the rotational part should be large and there should be no connection between the time mean temperature gradient and the total eddy heat flux. However, it turns out that it is not possible to extract most of the rotational component using the MS method since it makes up more than 95% of the total eddy heat flux. The analysis is conducted with output from the 1/10 horizontal resolution Parallel Ocean Program (POP) in the Southern Ocean (90 S - 35 S). Eddies appear to be crucial in determining the stratification of the Southern Ocean and the transport of the Antarctic Circumpolar Current (ACC) as well as indirectly affecting the Atlantic overturning circulation by modifying the return flow of surface waters into the Atlantic (see Rintoul et al. 2001, and references therein). Recent studies have focused on the role that Southern Ocean eddies play in the zonally or streamline averaged picture of the ACC and the associated Southern Ocean overturning cell (Karsten et al. 2002; Marshall and Radko 2003; Olbers and Visbeck 2005). However, in a zonally-averaged framework, the rotational part of the eddy heat flux cancels out, and many 6

8 aspects of the three-dimensional eddy field are masked. In particular, the spatial distribution of eddy coefficients is of interest for modeling variations in, for example, eddy heat transport, mixed-layer depths and Antarctic Intermediate Water formation in climate models. The paper is organized as follows: Section 2 describes the POP model. Section 3 evaluates the spatial distribution of the model s eddy heat flux and diffusivities. Section 4 provides the spatial coherence analysis for the eddy heat flux and its divergence with the parameterizations. In Section 5 the rotational component of the eddy heat flux based on MS is analyzed as a function of averaging lengthscale. Section 6 presents the discussion and conclusions. 2. Model description We use the 1/10 POP model because the model is sufficiently eddy resolving in the sense that between about 50 S and 50 N it resolves the first baroclinic Rossby radius, which is smaller than typical eddy length scales (Smith et al. 2000). The model reproduces the mesoscale sea surface height variance of the Southern Ocean (McClean et al. 2006). It uses a Mercator grid in the Southern Hemisphere with a latitudinal grid spacing of 1/10 cos(φ), where φ is latitude, leading to horizontal resolutions between 2 and 9 km for the Southern Ocean. There are 40 depth levels in the vertical, ranging from 10 m at the surface to 250 m at depth, with a maximum depth of 5500 m. Details of the bathymetry can be found in Maltrud and McClean (2005). The horizontal diffusion of tracers and momentum is biharmonic with coefficients varying with the cube of the average grid length: ( ) 3 ( ) 3 dl dl ν = ν 0 ; κ = κ 0, (8) dl 0 dl 0 where dl = A and A is the area of the grid cell. Equatorial values are ν 0 = m 4 s 1 7

9 and κ 0 = m 4 s 1. The biharmonic mixing is scale selective and acts to dampen short scales and noise. For vertical mixing in the model, the K-profile parameterization of Large et al. (1994) is used, with background mixing coefficients ranging from m 2 s 1 near the surface to m 2 s 1 at depth, to parameterize mixing associated with breaking of internal waves. Viscosity values are an order of magnitude higher. Large values of diffusivity and viscosity are used to simulate convection. Tracer advection is discretized with a centered difference scheme that does not introduce any numerical diffusion. The tracer time step is 6.3 min. Maltrud and McClean (2005) discuss the forcing and initial conditions. The model was spun up for 15 years ( ) and then run post-spinup from Wind, airsea heat fluxes and evaporation are calculated from NCEP/NCAR reanalysis products (6-hour fields averaged to one day) in combination with monthly data from the International Satellite Cloud Climatology project (for shortwave flux and cloud fraction). Precipitation comes from the monthly microwave sounding unit (Spencer 1993) and the Xie and Arkin (1997) dataset. Fields are then interpolated to the tracer time step. Since no sea ice model is used, under-ice sea surface temperature and salinity are additionally restored to climatology (Steele et al. 2001) in regions known to be ice covered. In the open ocean, only sea surface salinity is restored to climatological data with a restoring time scale of 6 months to avoid model drift. Figure 1 shows the vertically- and zonally-integrated meridional eddy heat transport (PW) over each grid cell in the Southern Ocean as calculated from the deviation of the annual 1998 mean heat flux. In Sections 3, 4 and 5 we examine the eddy heat flux in different regions of the Southern Ocean (shown in Figure 1) that are characterized by different eddy heat flux intensities. Region 360 encircles the Southern Ocean and largely encloses the ACC. The spatial 8

10 extent of the ACC is indicated in Figure 1 by the two circumpolar contours of annual 1998 mean surface temperature. Region AGR includes the intense eddy activity in the Agulhas Retroflection. Region SSP in the Southern South Pacific is within the ACC and also shows relatively high eddy activity. Region SPA is relatively quiescent and is in the South Pacific subtropics outside the ACC. Region 360 has 3600 gridpoints in longitude and 50 gridpoints in latitude. Regions AGR, SSP and SPA have 600 points in longitude and 100 points in latitude. There are no bathymetric features in any subregion above the depth of 918 m. In Regions 360, AGR, SSP and SPA monthly averages of all fields from model year 1998 were archived (Section 3 and 4). Region WDR west of Drake Passage has 200 grid points in longitude and 184 gridpoints in latitude. Region WDR was chosen because daily averages are available for model years 1998, 1999 and 2000, and is used to calculate the rotational component of the eddy heat flux based on MS (Section 5). 3. Eddy heat fluxes In this section we characterize the eddy heat transport and its downgradient parameterization in the model. We start by looking at the meridional eddy heat flux in the zonal integral and then examine its horizontal and wavenumber distribution. We will estimate diffusivities and address the relevant issues concerning the separation into rotational and divergent components. a. Spatial distribution Figure 2a shows the zonally- and vertically-integrated total, mean and eddy meridional heat transport. North of 55 S the southward eddy heat transport exceeds the northward mean heat 9

11 transport so that the total heat transport is mostly southward. South of 55 S both eddy and mean meridional heat transports are southward and a maximum total heat transport of 0.3 PW occurs at 57 S. The maximum eddy heat transport of about 0.5 PW is at 40 S. Figure 2b shows that about half to two thirds of the southward eddy heat transport north of about 45 S is accomplished within the top 300 m, whereas south of 45 S most of the heat transport occurs within the layers below 300 m. The maximum eddy heat transport is comparable to other estimates from eddy resolving or permitting models (e.g. Thompson 1993; Jayne and Marotzke 2002). Observational estimates of eddy heat fluxes are sparse and difficult to interpret. Analyses based on hydrographic data obtain maximum heat transports between 0.36 and 0.7 PW (deszoeke and Levine 1981; Macdonald and Wunsch 1996; Sloyan and Rintoul 2001; Ganachaud and Wunsch 2000; Gille 2003). From altimeter data, Keffer and Holloway (1988) obtained 0.7 PW at about 53 S and Stammer (1998) obtained a maximum of about 0.3 PW at 40 S. Figure 3a and Figure 3c show the meridional eddy heat transport and the divergence of the eddy heat transport and Figures 3b and 3d their respective downgradient parameterizations for Region SSP (Figure 1). The eddy heat flux terms (Figures 3a,c) appear to show little correlation with the diffusive downgradient parameterizations in (Figures 3b,d). Not even the signs of the terms coincide. Since the meridional temperature gradient is northward (positive) over most of the region, the meridional eddy heat flux is both temperature upgradient and downgradient. The pattern of positive and negative heat flux cannot be explained solely as an artifact of the rotational flux since the divergence of the eddy heat flux also alternates between positive and negative values (Figure 3c). Decorrelation scales indicate that the typical length scales for eddies in the model Southern Ocean are km, or approximately at 55 S (McClean et al. 2006). For Region SSP, the length scales for the meridional eddy heat fluxes are around 10

12 1-2 (Figure 3a), while for the divergence they are about 0.5 (Figure 3c). The characteristic length scales for both the divergence (Figure 3c) and the Laplacian (Figure 3d) are smaller than for the meridional eddy heat flux (Figure 3a) and meridional temperature gradient (Figure 3b). This is expected: taking an extra derivative in the space domain is equivalent to multiplying by the wavenumber in the wavenumber domain and therefore shifts energy to higher wavenumbers and whitens the spectrum. Wavenumber spectra of the fluxes and related quantities allow us to identify the wavelengths where most of the energy is concentrated. Spectra were calculated for Region 360 to assess the largest possible zonal wavelengths for each latitude segment and then averaged over 50 latitude grid points. Figure 4 shows wavenumber spectra for eddy heat fluxes, derivatives of the eddy heat fluxes, and the diffusive parameterizations of these quantities. Left-hand panels (Figure 4a-e) represent diffusive terms. Right-hand panels (Figure 4f-j) are eddy flux terms. If fluxes were exactly diffusive for all wavenumbers, then the spectra in the left column would match the spectra in the right column. The zonal spectra of the zonal derivatives of quantities should show a shift of the energy toward higher wavenumbers as compared to the spectra for the quantities themselves. This is observed in the spectrum for the zonal derivatives of the temperature where the energy is shifted toward smaller wavelengths (Figure 4a) so that for xx T (Figure 4b), most energy lies at wavelengths smaller than 10 longitude. While it is not necessary that a similar shift occurs for zonal spectra of meridional derivatives, we do in fact see a similar, though less pronounced, shift (Figures 4c,d). The spectrum for the Laplacian of temperature (Figure 4e) is the sum of the spectra for xx T and yy T (Figures 4b,d) and hence also shows energy concentrated at wavelengths shorter than 10. Most of the meridional eddy heat flux (Figure 4h) occurs between lengthscales of 1 and 11

13 10 longitude, longer than the first baroclinic Rossby radius (which is < 0.2 at 60 S ). The spectrum for the zonal eddy heat flux (Figure 4f) is more red with a substantial part of the energy occuring on lengthscales between 1 and 100 longitude, which define here the mesoscale, where most of the eddy heat transport takes place. Here, the submesoscale will be lengthscales shorter than 1. The derivatives of the eddy heat flux (Figures 4g,i) also indicate a shift to small wavelengths relative to the eddy heat flux (Figures 4f,h). The spectrum for the divergence of the eddy heat transport (Figure 4j) has most energy at wavelengths shorter than 10 longitude, ( ) ( ) as do its components x ut and y vt (Figures 4g,i). However, the spectral amplitudes of the divergence of the eddy heat flux (Figure 4j) have a very similar distribution with wavenumber but are only about 10% of the spectral amplitudes of the zonal and meridional derivatives (Figures 4g,i). This indicates that their values in the space domain are similar in magnitude but opposite in sign: if eddy velocities are along temperature contours, then v T x T 2 and u T y T 2 and (u T ) ( ) ( ) = x ut + y vt 0, implying that most of the eddy heat transport is geostrophic. We will come back to this point in Section 5. b. Diffusivity estimates Here we estimate diffusivities from linear fits of the eddy heat flux to mean temperature gradients and discuss how they depend on different averaging lengthscales. First, we define the brackets in equation (1) to represent zonal averages over 360. As discussed in the introduction, this has the advantage that the resulting diffusivities reflect only the divergent component of the total flux. However, this also means that we assume that the effective diffusivity κ can vary with latitude but must be constant in longitude. We also assume diffusivity is constant in depth, either over the whole water column ( m) or for a subsurface depth range (300 m m). 12

14 The fit for the whole water column (dashed line in Figure 5a) shows that diffusivities increase equatorward from values below 500 m 2 s 1 between 65 S and 50 S to about 2500 m 2 s 1 at 35 S. The high meridional diffusivities and heat transport between S reflect the intense eddy activity of the Agulhas Retroflection and Argentine Basin (as illustrated in Figure 1). For gridpoints below 300 m (dotted line in Figure 5a) diffusivity estimates are much lower between 45 S and 35 S, illustrating that the high southward meridional eddy heat transport in the Agulhas Retroflection and Argentine Basin occurs mainly close to the surface. This is also suggested in Figure 2b, where south of about 37 S the net eddy heat transport is northward below 300 m. The difference between the dashed and dotted lines in Figure 5a implies that the assumption that diffusivity is constant with depth is inadequate, at least for the upper ocean, and that diffusivities are elevated close to the surface as compared to the oceanic interior. The magnitudes of the diffusivities, their equatorward increase and their decrease with depth in Figure 5a are consistent with recent estimates of Marshall et al. (2006) and Eden (2006). Figure 5b shows the zonally and vertically integrated meridional eddy heat transport that arises from the parameterization with the m depth fit as compared to the model heat transport. Overall the two agree within the 95% confidence limit indicating that most of the heat transport can be explained by the parameterization in the zonally averaged and vertically integrated view. The parameterized heat transport underestimates the true model meridional eddy heat transport between 60 S and 50 S and overestimates it north of 40 S. Effective diffusivity may vary in the zonal direction as well as in the meridional direction. We can reduce the zonal averaging lengthscale, bearing in mind that if we do not consider a full 360 zonal distance, the rotational component of the flux may contribute to the computation of diffusivities. Performing the same fit for the basin scale (meaning averaging zonally over 13

15 about 90 longitude for the South Atlantic Ocean, 120 longitude for the South Indian Ocean and 150 longitude for the South Pacific Ocean) yields diffusivities similar to those obtained with the zonal average in Figure 5a and a similar reasonable fit to the model eddy heat transport (not shown). However, in the smaller 60 longitude subregions shown in Figure 1 a good linear fit between the model and the parameterized meridional eddy heat transport is no longer found. Eden et al. (2006a) suggested distinguishing between the parts of the eddy heat flux that are directed across and along mean temperature contours when computing eddy diffusivities. Therefore we computed diffusivities from a gridpoint-by-gridpoint correlation of cross-mean temperature contour eddy heat flux with the magnitude of the mean temperature gradients. They are horizontally extremely variable (not shown), and even after they have been smoothed with a boxcar filter of 5, there are still large spatial variations, alternating between m 2 s 1 and m 2 s 1. This again demonstrates the presence of a strong rotational flux, leading to diffusivity estimates that have no relation to the mean heat budget (see also Eden 2006). These diffusivities are very similar to those calculated from the gridpoint-by-gridpoint correlation of the meridional eddy heat flux and the meridional mean temperature gradient, reflecting the fact that the mean temperature contours are mainly zonally oriented. Does the decrease in the goodness of the fit with decreasing averaging lengthscale depend on the presence of the rotational part? We have initially hypothesized that the impact of the rotational part could be reduced with increased averaging, and furthermore that the divergent part of the eddy heat flux will be more correlated to mean temperature gradients than the rotational part. However, when we eliminate the rotational component by examining only the divergence of the eddy heat flux and the Laplacian of temperature we do not find a significant correlation. This warrants further investigation. Rather than averaging over a hierarchy of different filter 14

16 scales, in the following section we use coherence analysis to isolate the relation between u T and T (or (u T ) and 2 T ) as a function of wavenumber. 4. Coherence analysis Coherence analysis is used to estimate the scale dependence of the correlation between the eddy heat flux and its parameterization. We assess the coherence for two different cases: Case FG (for Flux and Gradient) considers the meridional eddy heat flux v T and the meridional mean temperature gradient - y T ; Case DL (for Divergence and Laplacian) considers the divergence of the eddy heat flux (u T ) and the Laplacian of the mean temperature - 2 T. In Case FG the rotational part of the eddy heat flux is present. Case DL does not include the rotational component and represents the terms as they appear in the temperature balance equations. Comparing the two cases allows us to assess the extent to which the rotational part of the eddy heat flux plays a role as a function of length scale. The squared coherence between two fields is the squared amplitude of the cross-spectrum of the two fields divided by the product of the power spectral amplitudes for each field. It represents the fraction of the spectral energy of one field attributable to a second field through a linear relation. Thus, if all of the eddy heat flux at a specific wavelength could be represented by its diffusive parameterization, then the coherence at that wavelength would be one. Translated to the space domain this would mean that the two fields were perfectly correlated when bandpass filters were applied that retain variability at that specific scale. If there is no correlation, then coherence is zero, or below the significance level. As an example, if the fields are highly correlated in space when they are simply averaged over 10 degrees in longitude, then, in the wavenumber domain, the integral of coherence will be high 15

17 over wavelengths greater than 10 degrees. a. Zonal coherence for different regions and depth levels Figure 6a shows the zonal one-dimensional squared-coherence for Region 360 over the 360 zonal segment for Case FG, and Figure 6b shows the same for Case DL, at different depth levels. Coherence was computed by Fourier transforming 360 long records at each of 50 latitudinal grid points and then averaging over the 50 records and band-averaging over 9 zonal wavenumbers. The wavelength dependence of the squared coherence is very different for the two cases. The Case DL coherence (Figure 6b) exceeds the 95% significance level for zonal wavelengths of 10 or smaller. There is no significant coherence for Case DL for wavelengths greater than 10 at any depth level. The maximum coherence at 318 m occurs at 0.3 (3 gridcells). The maximum shifts to smaller wavelengths at 918 m and to larger wavelengths at 579 m. Depending on depth, between 20% and 50% of the divergence of the eddy heat flux can be explained with a diffusive parameterization at wavelengths smaller than 10 longitude. In contrast, the coherence of Case FG (Figure 6a) is small for short wavelengths, increases somewhat for wavelengths larger than about 2 until a sudden increase occurs at wavelengths of about 50 or larger. More than 40% of the total eddy heat flux can be represented as a diffusive process when fields are filtered in space over lengthscales of at least 50, at least for depth levels below the surface. However, as we will see below, high coherence at large scales does not imply that by averaging over long lengthscales we have isolated the divergent part of the eddy heat transport. A coherence analysis for a specific region assumes that the same linear relationship exists between the two variables over the whole region, which implies a spatially uniform diffusivity 16

18 over that region. However, diffusivity could be highly variable in space. Figure 6c shows the zonal one-dimensional coherence at 318 m for Case FG for all the subregions (Figure 1), using a zonal segment length of 60 which is the zonal extent of the regions. All regions show low coherence at small wavelengths that increases somewhat for wavelengths of 2 or larger consistent with Figure 6a. The zonal extent of the regions is not large enough to detect the sudden increase in coherence for wavelengths larger than 50 as found in Figure 6a. Case DL coherence in Figure 6d exhibits the general pattern as in Figure 6b, with high coherence at small lengthscales reaching a maximum below 10 lengthscale. Coherence decreases for lengthscales longer than the lengthscale at which most of the eddy heat flux occurs. The similarities between Figures 6a,b and 6c,d imply that the wavelengths with low coherence in Figure 6a,b can most likely not be attributed to the neglect of spatial variations in the diffusivity and must instead have an alternate explanation. In order to interpret the coherence patterns for Cases FG and DL we need an estimate of the rotational and divergent components of the flux. The curl and the divergence are unique measures of how rotational and divergent the eddy heat flux is. Figure 7 shows the spectra of the curl of the eddy heat flux normalized by the sum of the spectra of the curl and divergence of the eddy heat transport. This ratio is a measure of the fraction of the total flux due to the rotational component as a function of wavenumber. More than 95% of the eddy heat flux is rotational at different depth levels (Figure 7a) and for all regions at 318 m depth (Figure 7b). The divergent eddy heat transport is largest between 1 and 10 where most of the eddy energy is located but amounts to less than 2% for Region 360 and no more than 5% for Region SPA at a depth of 318 m. It is below 1% for Region SPA at 318 m. The divergent eddy heat transport decreases with increasing length scale and approaches zero for length scales larger than about 25. The 17

19 coherence for Case DL (Figures 6b,d) is consistent with the idea that the divergent component of the eddy heat flux decreases with increasing wavelength. On the gridscale, at lengthscales smaller than the scale of the maximum coherence in Case DL, biharmonic diffusion eliminates grid scale noise and smooths out gradients. This destroys coherence on very small scales. Our results demonstrate that the rotational component dominates for all scales and that there is no lengthscale at which one can expect the divergent component to dominate. The conclusion to draw for the significant coherence for Case FG on large scales is that it must be due to the rotational component. b. Diffusivities as a function of wavenumber One dimensional diffusivities as a function of wavenumber can be computed from κ = 1 Y X, (9) Ny Ny 1 XX where X and Y are zonal Fourier transforms and N y is the number of meridional gridpoints over which results are averaged (N y =50 for Region 360 and 100 for all other regions). Figure 6e shows diffusivities for Case FG as computed with X = v T and Y =- y T, and Figure 6f shows results for Case DL with X = (u T ) and Y =- 2 T for Region 360. In general, for both cases, diffusivities decrease with depth. Diffusivities for Case FG however are wavenumber dependent reaching maxima for wavenumbers between 0.1 and 1, corresponding to wavelengths between 10 and 1 longitude, and reaching values of up to 5000 m 2 s 1 at the surface (Figure 6e). This wavenumber band corresponds to the band where most of the energy of the meridional eddy heat flux lies (see Figure 4h). These high values are consistent with the gridpoint by gridpoint diffusivity estimates in Section 3 and are due to the ro- 18

20 tational component. Diffusivities for Case DL are less wavenumber dependent (Figure 6f). They are about 1000 m 2 s 1 for the surface, about 500 m 2 s 1 for 318 m and below 500 m 2 s 1 for 918 m. Even though coherence at wavelengths larger than about 2 is low for Case DL at all depths, the diffusivity estimates yield values consistent with other studies (Marshall et al. 2006; Eden 2006) and with the estimates in Section 3. Below 2 lengthscale a diffusive parameterization with the diffusivities from Figure 6f explains up to 50% of the variance. We expect the phase of the coherence for Case DL to be zero in the waveband where coherence is significant as then a positive divergence is associated with a temperature downgradient flux, consistent with potential energy being released. Note that the negative of the Laplacian of mean temperature is used to calculate coherence. However, the phase is π/2 (Figure 8b) which makes the coherence difficult to interpret. For Case FG, the phase is zero where coherence is significant (Figure 8a), meaning that the eddy heat flux, that is almost purely rotational at these wavelengths, is down the meridional temperature gradient. Note that the negative of the meridional temperature gradient is used to calculate coherence. The diffusivities estimated from equation (9) in the different subregions at 318 m depth are large in regions with intense eddy activity and small in regions with low eddy activity (Figures 6g,h). There is much less wavenumber dependence in Case DL (Figure 6h) compared to Case FG (Figure 6g), bearing in mind that the Case FG diffusivities reflect the rotational component. Diffusivities are highest for Region AGR with its very intense eddy activity, with values between 1000 and 2000 m 2 s 1 in Case DL and up to 5000 m 2 s 1 in Case FG. Diffusivities are 500 m 2 s 1 for Region 360 and Region SSP (which both include the eddy activity within the ACC) for Case DL and up to 1000 m 2 s 1 for Case FG. Diffusivities are very low for the Region SPA, below 100 m 2 s 1 for both cases. In this region, eddy activity is very low. 19

21 5. The separation of a rotational component for Region WDR The results from Section 4 showed that the rotational component of the eddy heat flux dominates over all lengthscales and increases to almost 100% with increasing lengthscale. In this section we analyze data from Region WDR west of Drake Passage (see Figure 1) for which daily fields are available. The intent is to separate a rotational component from the eddy heat flux using the MS method and to reassess the coherence. In the previous sections we calculated the eddy heat fluxes as a deviation from the 1998 annual mean. In order to evaluate possible sensitivities to the duration of the averages, we here used the deviation from a three-year mean for the years Time averaging proves not to influence the findings shown here and results based on one year are not qualitatively different from the results derived from three-year averages. a. The MS method Instead of specifying boundary conditions for the Helmholtz decomposition (equations (3)- (5)), one can choose the rotational streamfunction Θ based on physical considerations: if eddies are predominantly geostrophic and if we neglect the influence of salinity, the eddy velocities are aligned with temperature contours u h T and u T = γ 1 2 ht 2. (10) Thus, we can choose Θ to be γt 2 /2 and the rotational component is directed along the contours of eddy variance T 2. The factor γ needs to be horizontally uniform for the eddy heat flux to be divergence-free. It can be determined in different ways. Equation (10) specifies the amount of geostrophic eddy heat flux when γ is determined directly from the correlation of the eddy heat 20

22 flux with gradients of eddy variance under an f-plane approximation, and neglecting salinity. MS (see also Eden et al. 2006b) chose γ based on consideration of the steady-state eddy variance equation in the quasigeostrophic approximation, assuming no external sources or sinks of heat and neglecting the eddy advection of temperature. MS then approximated the mean flow to be along mean temperature contours [ ] dψ u = k h dt T [ k h γ MS T ], (11) where Ψ is the mean velocity streamfunction. The horizontal eddy heat flux can be written as the sum of the rotational flux and a residual flux u T res = u T k h Θ. (12) If Θ = γ MS T 2 /2 the residual horizontal eddy heat flux u T res is related to the conversion of eddy potential energy to eddy kinetic energy (w T z T ). This motivates the choice to parameterize u T res as a diffusive process κ h T. For a positive κ, eddies release potential energy and w T z T is negative. The MS rotational eddy heat flux (superscript MS in the following) ( u T ) MS rot ( [dψ ] ) T 2 = k h, (13) dt 2 is purely horizontally non-divergent. It is associated with the spatial growth and decay of the eddies and, under the above approximations, balances the mean flow advection of eddy potential energy. It is parallel to the eddy potential energy contours and circulates around the mean temperature contours. The residual flux u T MS res however is not rotation-free. 21

23 b. Results Figure 9 shows the zonal squared coherence of the eddy heat transport with the gradient of eddy variance as a function of wavenumber at 318 m (Figure 9a) and 918 m (Figure 9b). It specifies the extent to which there is a linear relation between the eddy heat flux and the gradients of eddy variance at every wavenumber and hence the amount of rotational flux as defined by equation (10). Here the factor γ determined directly from equation (10) turns out to be essentially the same as γ from the MS method. Coherence has a minimum at around a lengthscale of 1 and increases with increasing lengthscale over the mesoscale (Figure 4h). At lengthscales of 2 or larger the rotational contribution as defined by equation (10) seems to level off at about 60%-70% of the total flux at a depth of 318 m (Figure 9a). Deeper, at 918 m, where geostrophy is a better approximation, 70%-90% of the total flux can be explained by the rotational component as defined by equation (10). The gray line in Figure 9 indicates the amount of rotational flux measured uniquely by the normalized curl of the eddy heat flux. Evidently, the residual eddy heat flux as defined by equation (12) still makes up about 20% of the total eddy heat flux, whereas the rotation-free component of the eddy heat flux, as measured by the curl and divergence, makes up less than 1% of the total eddy heat flux (gray line in Figure 9). Note that the increase in coherence from Figures 9a,b coincides with the decrease in coherence for Case DL (dotted line in Figure 9). The analyses in this section illustrate the difficulty of subtracting a rotational component that captures most of the rotational eddy heat flux, since the residual will likely still have a large rotational component that dominates over the divergent part. Hence, one cannot expect the residual to be correlated any better with the mean temperature gradients or yield physically more reasonable diffusivities than the raw eddy fluxes. 22

24 6. Conclusions The goal of this paper has been to assess the relationship between eddy heat transport and a downgradient diffusive parameterization as a function of lengthscale in a high resolution ocean model. We used the curl and divergence of the eddy heat flux as a unique measure of the rotational and divergent components respectively. The divergent component accounts for no more than 5% of the total eddy heat transport decreasing towards zero with increasing lengthscale. Hence, the rotational component dominates all lengthscales and cannot be eliminated simply by averaging over long enough length scales. A coherence analysis was performed, distinguishing between two cases: Case FG assessed the relation of the total eddy heat flux, which includes a rotational component that is irrelevant for the mean tracer budget, and the downgradient parameterization; Case DL considered the divergence of the eddy heat fluxes. The divergent part of the flux (Case DL) was not coherent with the Laplacian of mean temperature except on spatial scales smaller than 10. The divergence is small and noisy which makes obtaining significant correlations with a diffusive parameterization difficult. Unfortunately, averaging in space does not improve the correlation since the relative role of the divergence decreases with increasing lengthscale as compared to the curl. Coherence for Case FG occurred on scales larger than 50 but not on scales smaller than 10. These results imply that the significant coherence must be due to the rotational component. The divergent component will not be significant no matter how long the lengthscale over which the raw eddy fluxes are averaged. We used the ideas of MS to separate a rotational component from the total eddy heat flux. This rotational component again constituted a very large portion of the total heat flux and increased with depth, reflecting the extent to which eddies are along contours of eddy variance. 23

25 This rotational part of the flux increased with increasing wavelength, just as did the curl, but it accounted for no more than 80% of the total eddy heat flux, leaving 20% of the flux as a residual. This residual is not purely divergent and it is 4 to 5 times larger than we would expect the divergent component to be. We conclude that the residual still appears to be dominated by the curl. Our results imply that the advective terms in the eddy variance equation are so dominant that it is difficult to extract the part that balances the conversion term w T z T. This is also in agreement with Wilson and Williams (2004), who diagnosed fluxes of potential vorticity in an idealised eddy resolving model and found that they are controlled by the advection of the mean flow as well as the eddy advection. This result is not surprising. In fact, at present, it is difficult to measure the vertical velocity needed to compute the conversion term from observations. Diffusivities were estimated using a variety of approaches. The diffusivities estimated from the zonally-averaged heat transport and mean temperature gradient were of O(100) m 2 s 1 for latitudes south of 50 S, increasing equatorward to a maximum of up to 2800 m 2 s 1. This is in agreement with the surface eddy diffusivity estimates by Marshall et al. (2006) who used streamline coordinates. In our zonally averaged view, the high diffusivities equatorward of 50 S reflect the intense eddy activity in the Agulhas Retroflection and the Argentine Basin. Diffusivities as estimated from Case DL were essentially wavenumber independent, whereas Case FG diffusivities had a large wavenumber dependence. Maximum diffusivities of up to 5000 m 2 s 1 close to the surface were found in Case FG in the waveband between 1 and 10 which is where most of the energy of the total heat flux lies. The eddy heat flux is essentially rotational in this waveband, implying that these large diffusivities are due to the rotational component. Case DL seems to give physically plausible diffusivities that are consistent with recent estimates using the divergence of the eddy buoyancy flux in a high resolution ocean model (Tanaka et al. 2007). 24

26 Diffusivities of about 500 m 2 s 1 were found within the ACC west of Drake Passage and were highest in Region AGR, being up to 2000 m 2 s 1 close to the surface. The lowest diffusivities of about 200 m 2 s 1 were found for Region SPA, north of the ACC. Diffusivities decreased with depth for all regions. However, coherence is low in Case DL and very little of the eddy heat flux can be explained with a temperature downgradient parameterization using these diffusivities. The low variance explained is consistent with other studies (Gille and Davis 1999; Roberts and Marshall 2000; Solovev et al. 2002). Some studies have speculated that low correlation between the eddy heat flux and its parameterization might be the result of poor eddy statistics with insufficient averaging in time (e.g. Rix and Willebrand 1996). Here, we do not find any sensitivity of the coherence to averages over longer periods of time, and we suggest that the main reason for noisy patterns and low correlations is the dominance of the rotational component over all space scales. Some of the main caveats of this study are the neglect of salinity and the very simple model of horizontal diffusion. Further analysis of this issue would benefit from examining fluxes along isopycnals (e.g. Gent and McWilliams 1990; Visbeck et al. 1997). We expect that our results will not change qualitatively once the heat flux is considered along and across mean isopycnals. The curl will still dominate over the divergence on all scales but the coherence for Case DL might be larger and the phase relationship improved. Our results demonstrate that one needs to be cautious when interpreting correlations between eddy heat flux and temperature gradients. Neither high correlations nor diffusivity estimates need be associated with the relevant divergent component, even when fluxes are averaged over large spatial scales. Our findings underscore the importance of using the divergence of the eddy heat flux, despite its noisiness. 25

27 Acknowledgments. This research was supported by the National Science Foundation grants OCE , OCE /OCE , OPP , by the National Aeronautics and Space Administration Contract EOS/ and the Office of Science (BER), U.S. Department of Energy, Grant No. DE-FG02-05ER The global simulation was carried out as part of a Department of Defense High Performance Computing Modernization Program (HPCMP) grand challenge grant at the Maui High Performance Computing Center (MHPCC). 26

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33 List of Figures 1 The vertically- and zonally-integrated meridional eddy heat transport over each gridcell in the model Southern Ocean (PW). The eddy heat flux is defined here as v T = vt vt, with the overbar denoting the annual 1998 mean and the prime the deviation from it. Note the nonlinear color key. Also shown are subregions for which analyses are done in the manuscript and two circumpolar contours of 1998 annual mean temperature at the surface (green) which approximately indicate the location of the ACC Zonally- and vertically-integrated meridional heat transports. a) The eddy component v T (solid line), total heat transport vt (dashed line) and mean heat transport vt (dotted line). b) Eddy heat transport for depth intervals m (solid line), m (dashed line) and m (dotted line) a) ρc p v T in 10 6 W m 2 ; b) ρc p y T in 200 W s m 4 ; c) ρc p (u T ) in 10 W m 3 and d) ρc p 2 T in 10 2 W s m 5 for Region SSP at a depth of 318 m Variance preserving power spectral densities (psd) in the zonal direction as a function of zonal wavenumber (or, equivalently, zonal wavelength) for Region 360, averaged over 50 latitude grid points and band-averaged over 9 points. a) ρc p x T b) ρc p xx T c) ρc p y T d) ρc p yy T e) ρc p 2 T f) ρc p u T ( g) ρc p x u T ) ( h) ρc p v T i) ρc p y v T ) j) ρc p (u T ) at a depth of 318 m. Also shown are the 95% confidence intervals (gray shading), a decorrelation scale of 5 gridpoints in latitude was assumed, corresponding to about 40 km

34 5 a) Diffusivities κ y as determined from linear fit for depth intervals m (40 depth levels) and m (24 depth levels) as a function of latitude. b) Modelled eddy heat transport (solid line) and eddy heat transport as computed from diffusivities from the m fit and meridional mean temperature gradient (dashed line). Also shown are the 95% confidence intervals (gray shading) a),b): The zonal squared coherence for Region 360 for different depth levels a) Case FG b) Case DL. c), d): The zonal squared coherence for the different regions at 318 m for c) Case FG d) Case DL. e), f):diffusivity for Region 360 for different depth levels for e) Case FG f) Case DL. g), h):diffusivity for the different regions at 318 m for g) Case FG h) Case DL. Spectra and cross-spectra were averaged over 50 latitude grid points for Region 360 and averaged over 100 latitude grid points for all other regions and then band-averaged over 9 gridpoints. Spectra were also averaged over 60 longitude segments for Region 360 in c),d),g),h). The horizontal black lines are the 95% confidence levels The spectrum of the curl of the horizontal eddy heat transport, normalized by the sum of the spectra of the curl and the divergence of the horizontal eddy heat transport P SD( x v T y u T )/ ( P SD( x u T + y v T ) + P SD( x v T y u T ) ), where PSD is the power spectral density. a) Region 360: 5 m (solid), 318 m (dashed) and 918 m (gray). b) 318 m: Region AGR (solid), Region SSP (dashed) and Region SPA (gray) Phase of coherence (black line) with 95% confidence intervals (gray shading) and 4 the squared coherence (gray line) for Region 360 at a depth of 318 m. a) Case FG b) Case DL

35 9 Squared coherence between u T and y T 2 (solid line), v T, x T 2 (dashed line) and Case DL (dotted line) in Region WDR. Also shown is the spectrum of the curl of the eddy heat flux normalized by the sum of the spectra of the curl and the divergence of the eddy heat flux (gray line). a): 318 m. b): 918 m. The zonal extent of Region WDR is 20. Spectra and cross spectra were averaged over 184 latitude grid points and band-averaged over 9 points. The horizontal black lines are the 95% confidence levels

36 FIG. 1. The vertically- and zonally-integrated meridional eddy heat transport over each gridcell in the model Southern Ocean (PW). The eddy heat flux is defined here as v T = vt vt, with the overbar denoting the annual 1998 mean and the prime the deviation from it. Note the nonlinear color key. Also shown are subregions for which analyses are done in the manuscript and two circumpolar contours of 1998 annual mean temperature at the surface (green) which approximately indicate the location of the ACC. 35