Ocean Model Response to Stochastically Perturbed Momentum Fluxes

Terence J. O’Kane aCSIRO Environment, Hobart, Tasmania, Australia

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Russell Fiedler aCSIRO Environment, Hobart, Tasmania, Australia

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Mark A. Collier bCSIRO Environment, Aspendale, Victoria, Australia

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Vassili Kitsios bCSIRO Environment, Aspendale, Victoria, Australia
cLaboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering Monash University, Clayton, Victoria, Australia

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Abstract

Recent studies of various stochastic forcing and subgrid-scale parameterization schemes applied to climate and atmospheric models have revealed a diversity of model responses. These responses include degeneracy in the response to different forcings and compensating model errors. While stochastic parameterization of the ocean eddies is an active area, this has mainly involved idealized models with fewer studies employing ocean general circulation models. Here we examine the sensitivity of a low-resolution climate model to stochastic forcing of the momentum fluxes restricted to regions of the three-dimensional ocean where an eddy-resolving ocean model configuration has high variability. We consider the changes in the modeled energetics of low-resolution simulations in response to increased stochastic forcing. We find that as forcing amplitudes are increased there is enhanced conversion of transient to seasonal potential energy. Additionally, there is a systematic redistribution from seasonal to small-scale transient kinetic energy. Our approach has zero mean noise such that the total kinetic energy spectra remain largely unchanged even as small-scale eddy kinetic energy is increased in the targeted regions. However, we also show that strong stochastic forcing, particularly when applied in the tropics, can induce substantial changes to the ocean steady state that are undesirable. These changes include overly strong vertical mixing leading to unrealistic increases in ocean heat content and latitudinally dependent changes to sea level. We show that judicious selection of the magnitude and spatial extent of the stochastic forcing is required for desirable results. Our results point to the importance of a comprehensive evaluation of ocean model responses to stochastic parameterizations.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Terence J. O’Kane, terence.okane@csiro.au

Abstract

Recent studies of various stochastic forcing and subgrid-scale parameterization schemes applied to climate and atmospheric models have revealed a diversity of model responses. These responses include degeneracy in the response to different forcings and compensating model errors. While stochastic parameterization of the ocean eddies is an active area, this has mainly involved idealized models with fewer studies employing ocean general circulation models. Here we examine the sensitivity of a low-resolution climate model to stochastic forcing of the momentum fluxes restricted to regions of the three-dimensional ocean where an eddy-resolving ocean model configuration has high variability. We consider the changes in the modeled energetics of low-resolution simulations in response to increased stochastic forcing. We find that as forcing amplitudes are increased there is enhanced conversion of transient to seasonal potential energy. Additionally, there is a systematic redistribution from seasonal to small-scale transient kinetic energy. Our approach has zero mean noise such that the total kinetic energy spectra remain largely unchanged even as small-scale eddy kinetic energy is increased in the targeted regions. However, we also show that strong stochastic forcing, particularly when applied in the tropics, can induce substantial changes to the ocean steady state that are undesirable. These changes include overly strong vertical mixing leading to unrealistic increases in ocean heat content and latitudinally dependent changes to sea level. We show that judicious selection of the magnitude and spatial extent of the stochastic forcing is required for desirable results. Our results point to the importance of a comprehensive evaluation of ocean model responses to stochastic parameterizations.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Terence J. O’Kane, terence.okane@csiro.au

1. Introduction

The question of how to incorporate the effects of unresolved turbulent motions, and their role in determining large-scale dynamics, represents a common problem in large-eddy simulations (LES) of nonlinear fluids. This issue is particularly crucial for simulating geophysical flows. In ocean and climate modeling it is typical to employ deterministic methods. Simulations over long time periods are often required to evaluate such approaches, since it is only the statistical effects of the subgrid scales (eddies) on the retained large scales (mean flow) that can be approximated empirically. Due to the computational cost, this means often reduced-resolution model configurations are adopted. Furthermore, it is well known that small-scale errors grow rapidly on (finite) time scales determined by their initial spatial structure. This means that even small random errors will quickly become organized by the model dynamics and undergo rapid growth and project onto large-scale mean features of the flow. The structure and growth rate of small-scale errors is not confined to subgrid parameterizations but is relevant to all aspects of simulating and predicting geophysical flows (Kalnay 2003).

A foundational understanding of subgrid parameterizations to correct biases in the small-scale energy spectra of LES has deep roots in statistical dynamics. As discussed by O’Kane and Frederiksen (2008a), and recently reviewed by Zhou (2021), fundamental insights into stochastic–dynamic parameterization were pioneered by the efforts of a key group working on turbulent energy closures for ensemble weather prediction. Specifically, in the work of Epstein (1969), Fleming (1971a,b), Epstein and Pitcher (1972), and Pitcher (1977) third- and higher-order cumulants are discarded in order to directly forecast mean and variance information via statistical dynamical prognostic equations and stochastic perturbations to velocity tendencies. However, it was the seminal work of Kraichnan (1976) that marked the arrival of the modern theory of eddy viscosity and stochastic backscatter (i.e., scale-selective injection and/or drain of energy with a predetermined functional form). Since then, there have been ongoing efforts over several decades to establish a rigorous mathematical basis for subgrid-scale parameterizations based on statistical mechanics and dynamics, including formal renormalization methods (Frederiksen 1999; O’Kane and Frederiksen 2008b), stochastic approximations (Zidihkeri and Frederiksen 2008), and the subsequent identification of universal scaling laws for subgrid dynamics in atmospheric and oceanic flows (Kitsios et al. 2016). Various approaches to incorporating stochastic kinetic energy (KE) backscatter have for some time now been applied to reduce systematic model errors in operational weather prediction and atmospheric models (Berner et al. 2012; Franzke et al. 2015; Berner et al. 2017).

One unavoidable consequence of the addition of stochastic forcing to a nonlinear system is that, typically as the amplitude of the noise increases with wavenumber, the small scales become more isotropic weakening phase relationships. In this case, whereas the amplitude of the small-scale transient kinetic energy spectrum is increased, structure may be lost. Additionally, it is often unclear the spatiotemporal scales at which the model will organize the noise and hence there is no a priori way to determine the coherent response to the forcing. Simply put, it remains unclear as to how any given nonlinear dynamical system will respond to a particular application of stochastic forcing.

Stochastic forcing can act in many ways to modify the dynamics of a nonlinear system. Examples include regime transitions in simple scalar systems such as the stochastically forced double well potential (Miller et al. 1985). In two-dimensional turbulence, weak stochastic forcing of a particular large-scale mode or particular small wavenumber has been shown to be able to initiate large energy transfers from small to large scales via the inverse energy cascade (Bouchet and Simonnet 2009; Nadiga and O’Kane 2017). More generally, it has for some time now been recognized that stochastic forcing of the ocean surface fluxes interacting with nonlinearities in the climate model equations can lead to enhanced variability and changes in the climatological state (Zavala-Garay et al. 2003; Beena and von Storch 2009; Williams 2012). This is even the case for isotropic random perturbations with zero mean. Williams et al. (2016) examined the response to zero mean multivariate stochastic perturbations to the temperature tendencies in the three-dimensional ocean. They considered both isotropic temporally uncorrelated and temporally correlated noise forcing. The amplitude of the forcing was calculated from a higher-resolution (1/3° horizontal resolution; 40 vertical levels) climate model. This was then used to stochastically force the ocean temperature tendencies of a very low-resolution (2.5° latitude × 3.75° longitude; 20 vertical levels) model. They found a stronger response occurred for correlated noise, with significant warming of the upper ocean and cooling at depth, such that an overall significant loss of global ocean heat content occurred. They argued that perturbed temperature tendencies resulted in reduced biases and improved ocean temperature and salinity fields both at the surface and at depth, as well as improvements in the variability of the strength of the global ocean thermohaline circulation. One notable study by Brankart (2013) considered perturbations to horizontal density gradients via jointly perturbing both temperature and salinity. They found that the uncertainties arising in the calculation of large-scale horizontal density gradients could be simulated using stochastic noise processes to approximate subgrid scale temperature and salinity fluctuations. These uncertainties were primarily due to the unresolved scales in the temperature and salinity fields, and amplified via the nonlinearity of the seawater equation of state.

Andrejczuk et al. (2016) applied a systematic approach to stochastic parameterization comparing three methods including a surface flux parameterization similar to Williams (2012); perturbations of the equation of state as in Brankart (2013); and perturbing the parameterized tendencies of mixing and viscosity inspired by an approach used in atmospheric models (Buizza et al. 1999; Palmer et al. 2009). They applied uniformly distributed random perturbations to the parameterized tendencies of the equation of state at all vertical levels and to surface heat and freshwater fluxes. Comparing model performance in terms of biases in initialized ensemble seasonal forecasts, they found that, although ensemble spread was increased in the eddying regions and western boundary currents, there was little evidence of improved forecast skill. Brankart (2013) and Andrejczuk et al. (2016) also found that, depending on the structure of the spatial correlation, spatially correlated noise applied to the density may become unstable. Williams et al. (2016) showed that temporally correlated noise increased the effect of stochastic parameterization but not the qualitative impact. Zanna et al. (2019) estimated major sources of subgrid-scale uncertainties in the climate system analyzing a large database of (ocean) model configurations at horizontal resolutions spanning 0.25°–2.8°. In particular, they note the potential for unresolved stochastic temperature and salinity fluctuations to significantly change the large-scale density across the Gulf Stream front with resultant major changes to the large-scale transport. Stanley et al. (2020) formulated a theoretical explanation for the rectified effect of subgrid-scale variability on the resolved density field. They showed that temperature fluctuations dominate salinity fluctuations such that density errors are, to leading order, proportional to the product of a subgrid-scale temperature variance and a second derivative of the equation of state.

In an approach with certain similarities to that presented here, Cooper and Zanna (2015) applied stochastic forcing to the velocity components of a 30-km resolution shallow water ocean model simulation of a chaotic barotropic double gyre optimized to have the same climatological mean and variance as that of a model with a higher resolution of 7.5 km. They applied linear stochastic forcing with a spatially varying but time constant amplitude independent of the large-scale flow, with no spatial correlation and a fixed time scale in terms of a 5-day lag-covariance of the velocity. The parameterization coefficients were derived from a single realization integration of the high resolution model to provide an optimization target. They reported that the optimized stochastic parameterization correcting the model variance was associated with the spatial pattern of eddy-decorrelation time scales rather than the corresponding pattern associated with the amplitude of the variance. Using a double gyre configuration of the NEMO ocean model, Perezhogin (2019) demonstrated a stochastic KE backscatter (SKEB) parameterization was able to improve the circulation by increasing the small-scale KE to better resolve the barotropic inverse energy cascade. He found that the SKEB solution more closely resembled “synthetic” turbulence as it was lacking the filaments found in higher-resolution reference simulations. Importantly, he reported that it was possible to reduce the amount of undesirable (grid scale) noise in the solution by application of a time-correlated stochastic process. Storto and Andriopoulos (2021) examined the utility of stochastically perturbed parameterization tendencies (SPPT), stochastically perturbed parameters (SPP) and SKEB schemes in a limited area NEMO configuration with the aim of increasing ensemble spread at various scales and diagnostics of relevance to hybrid-covariance data assimilation.

Here our approach will be to apply stochastic perturbations to the horizontal momentum fluxes in conjunction with the GM (Gent and McWilliams 1990) mesoscale eddy parameterization. Kjellsson and Zanna (2017) discuss the motivation for incorporating some form of KE backscatter to alleviate the known GM-induced spurious dissipation of energy at the small scales and return KE to the resolved flow. Bachman (2019) applied a version of this approach in a simpler quasigeostrophic model setting computing the magnitude of energy backscatter required to balance the instantaneous energy dissipation by the GM parameterization. This dissipation in part arises from the fact that the GM parameterization is derived for eddy–mean interactions but in practice is applied to the instantaneous field. Recent scale-aware parameterizations have been implemented where the backscatter is formulated as a negative viscosity (energy injection) on the barotropic component of the flow alone (Jansen et al. 2015; Grooms et al. 2015). In addition to the aforementioned KE backscatter parameterizations relevant recent studies with specific application to problems in ocean modeling include those of Zanna et al. (2019), Porta Mana and Zanna (2014), Grooms et al. (2015), Grooms and Kleiber (2019), Jansen and Held (2014), Jansen et al. (2015, 2019). Many of these studies (e.g., Pearson et al. 2017) argue for the need to inject energy at the small scales of the resolved flow even at eddy-resolving resolutions.

Hewitt et al. (2020) recently noted that “there is currently no parameterization implemented in climate models that mimics the important transfer of KE from small to large scales, namely KE backscatter, which affects the large-scale flow.” As a step toward better understanding that mechanism, we are interested in the response of an ocean general circulation model (GCM) to stochastic perturbations to those regions of the global ocean where eddies are observed to be present but are absent in the model due to insufficient horizontal resolution. There remains much uncertainty as to how best to apply stochastic forcing to the three-dimensional state of a particular ocean or climate GCM and what the modeled response might be. It is also unclear if such stochastic parameterizations should be applied in conjunction with existing schemes.

Here we apply stochastic perturbations to the horizontal momentum flux in a general circulation ocean––sea ice model configuration with resolution typical for climate simulations. Specifically, a high-resolution 0.1° eddy-resolving reference calculation forced by nominal year surface boundary conditions is used to determine regions of high eddy variance that are unresolved in the low-resolution model. A low-resolution 1° control simulation and a series of simulations with stochastic perturbations of increasing amplitude added to the horizontal momentum flux are conducted. The perturbation amplitudes are applied as a fraction of the variance of the high-resolution reference eddy variability. We describe the model configurations and construction of the stochastic forcing in section 2. Results for a range of diagnostics are presented in sections 3 through 8 followed by a summary and discussion in section 9.

2. Experimental design and model configuration

a. Model configurations

We employ the ACCESS-OM community model (Kiss et al. 2020) driven by JRA55-do repeat year forcing (Stewart et al. 2020) at two horizontal resolutions, i.e., nominally 1° and 0.1°. These models have been configured with model parameters as consistent as possible to assist in studies of resolution dependence. Away from the continental shelf and equatorward of 50°, the 0.1° model resolves the first baroclinic deformation radius indicating some degree of representation of a transient mesoscale eddy field, whereas the 1° does not. The low- and high-resolution models have different vertical resolutions where the vertical grid in the ACCESS-OM2 1° configuration has 50 levels and 2.3 m spacing at the surface, increasing smoothly to 219.6 m by the bottom at 5363.5 m, whereas the ACCESS-OM2 0.1° configuration has 75 levels and 1.1 m spacing at the surface, increasing smoothly to 198.4 m by the bottom at 5808.7 m. Kiss et al. (2020) provide a detailed description of the model parameters and performance of ACCESS-OM2 at three horizontal resolutions (i.e., 1°, 0.25° and 0.1°). The initial state from which to start all experiments is taken from World Ocean Atlas 2013 version 2 (WOA13) (https://www.nodc.noaa.gov/OC5/woa13/). Each of the stochastically perturbed simulations are then started from this initial state, and run to a subsequent equilibrated state, defined by stable values of the annual average global KE, typically achieved after ≈150 years. The simulations are continued for another two decades where the final decade of each of the 170-yr simulations was used to examine the climatological (mean) ocean states, energetics and transports.

The applied GM scheme (Gent and McWilliams 1990) acts in combination with stochastic perturbations to the momentum flux as a subgrid-scale parameterization for mesoscale eddies in the 1° model configurations. No subgrid mesoscale parameterization is applied in the eddy resolving high resolution ACCESS-OM2 0.1°. The 1° model uses the GM parameterization to represent the quasi-Stokes transport associated with mesoscale eddies whereas the neutral-direction diffusive tracer transport is parameterized by a neutral diffusivity (Redi 1982). The diffusivity associated with the GM skew-diffusive flux is depth independent but flow-dependent and is the product of an inverse time scale, a squared length scale, and a grid scaling factor [see section 3.3 of Griffies et al. (2005)]. The length scale is 50 km. The inverse time scale is an Eady growth rate determined from the horizontal density gradient averaged between 100 and 2000 m using a constant buoyancy frequency of 0.004 s−1. The Eady growth rate is subject to a limiter and is smoothed both vertically and horizontally, and vertically averaged in the mixed layer. The GM diffusivity is scaled in proportion to how well the numerical grid resolves the first baroclinic Rossby radius (or the equatorial Rossby radius within 5° latitude; Hallberg 2013) and is limited to the ranges 50–600 m2 s−1. Additional details regarding parameterizations including neutral tracer diffusion, restratification in the surface mixed layer due to submesoscale eddies, horizontal friction, bottom and background vertical viscosities, Rayleigh damping and so on can be found in section 2.1.4 of Kiss et al. (2020).

b. Stochastic forcing

To generate the three-dimensional mask used to specify the forcing amplitudes, model climatological root-mean-square (rms) values are calculated from daily velocities over the final decade of a long control simulation of the ACCESS-OM 0.1° high-resolution model reference calculation. Masks are then generated for weighting the stochastic perturbations to the velocity tendencies based on thresholded rms values greater than 0.15 m s−1. This threshold was chosen empirically as it isolates the eddying regions in the high-resolution three-dimensional ocean simulation. For zonal velocity tendencies, the number of surface grid points in the mask is 22 K (22 000) or 32% of the total number of grid points at the surface. The total number of grid points at all levels in the global ocean u mask is 240 K or 9% of the total ocean model grid. Similarly for the meridional velocities υ the number of surface grid points in the mask is 14 K or 10% at the surface and for all levels in the global ocean the total number of grid points in the mask is 156 K or 6%. Surface values of the zonal and meridional injection velocity amplitudes are shown in Fig. 1 specifically, the amplitude (mean) for the meridional and zonal velocity tendency forcing at the surface and also for the zonal velocities down to 300 m depth along the equator. The stochastic forcing is applied at each model time step. The mask considered here is fixed in time and has no seasonal variations, which might potentially lead to difficulties for regions such as the Antarctic Circumpolar Current (ACC). That said, the implementation is autonomous allowing for easy implementation of time-dependent masks and scaling based on the current velocity field. These adaptations would enable easy application of flow-dependent spatiotemporal correlation scales and amplitude thresholds.

Fig. 1.
Fig. 1.

Masks for stochastic perturbations to the velocity tendencies based on the rms over the last decade of the 0.1° reference calculation thresholded to values greater than 0.15 m s−1.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

The instantaneous u zonal and υ meridional velocity tendencies (u/t,υ/t) [Eq. (2)] of the low-resolution model are then perturbed by the addition of a random fluctuation (ϵu, ϵυ) and scaled by a factor γ% to be either 10%, 20%, 50%, or 100% of the rms of the ACCESS-OM 0.1° model (see Table 1). No temporal or spatial correlations between the random perturbations to the u and υ velocity tendencies are considered and the w vertical velocity tendencies are not directly perturbed. The amplitudes (Uamp, Vamp) at each grid point (x, y) and model level l are determined by the mask. Scaling is converted to appropriate units, i.e., the velocity injection to the tendencies at any instant is
ϵu=γ%×Uamp(x,y,l)×fu(x,y,l,t)/dt,
ϵυ=γ%×Vamp(x,y,l)×fυ(x,y,l,t)/dt,
where fu and fυ are independent samples both uniformly distributed over [−1, 1] and dt is the time step. We apply a uniform random distribution in common with many other studies of stochastic forcing including operational systems as described by Buizza et al. (1999) for example. Here we are assuming that alternate specification of, for example, a Gaussian distribution will not lead to substantively different results as it is the amplitude of the forcing that determines the model response. A Gaussian distribution, however, has potential for very large perturbations to be applied at any given grid point increasing the likelihood of numerical instability and model failure.
Table 1

Model configuration and amplitude of stochastic forcing as a percentage of the standard deviation from the ACCESS-OM2-0.1 high-resolution reference simulation.

Table 1

Different random numbers are applied at each grid point of the horizontal velocities and at each level with amplitudes determined by the mask restricting the stochastic forcing to regions of the coarse-resolution models where the variance is unrealistically low and vertical velocities poorly resolved. As previously noted by Cooper (2017), “a potential weakness of the parameterization implemented here, is that the vertical velocity, diagnosed from model variables, is not accurate. Random variations of velocity, along with conservation of mass, induce diagnosed vertical velocities that are too large.” This effect is evident in the instantaneous surface velocities for the stochastic-100 strong forcing case (Fig. 2) indicating that for very strong forcing, inaccuracies due to increased vertical velocities (shear) are present and that a threshold for physically realistic simulations has been exceeded. The ranges of the monthly mean vertical velocities were found to be comparable across experiments with a maximum increase of around 50% in the surface vertical velocities relative to the control occurring only for the largest amplitude forcing considered (i.e., stochastic-100). For the weaker forcing cases stochastic-10, stochastic-20, and stochastic-50 the structure of the tropical velocities (e.g., the instability waves) remains spatially coherent.

Fig. 2.
Fig. 2.

Instantaneous zonal surface (u) velocities (m s−1) for the access-om2–0.1, 1° control, and stochastic-10 and stochastic-100 amplitude perturbation cases on 1 Jan of year 170.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

We enforce divergence free conditions via the barotropic scheme for the total horizontal velocity tendency, rather than only for the stochastic tendency terms separately, using a time step of 90 min and a barotropic substep time of 67.5 s (80 steps). Specifically, the noise is implemented in modifications to the ocean-momentum-source module with the divergence free condition enforced via the ocean-barotropic modules in the MOM5 codes. We now include the modules the modified input.nml namelist in the github repository https://github.com/russfiedler/MOM5/tree/perturb. This approach is different to the stochastic GM scheme of Andrejczuk et al. (2016) where the divergence-free condition may potentially be violated for the parameterized bolus velocities for the tracer equations. Note also the studies of Juricke et al. (2017) and Grooms and Kleiber (2019), who further examined the impact of stochastic perturbations to the GM parameterization including taking bolus velocity divergence-free conditions into account by design. We saw no evidence of numerical instability. The uncorrelated noise is applied to the u and υ tendencies as a forcing as would typically be implemented as part of a stochastic backscatter scheme. For the largest amplitude case (stochastic-100 shown in Fig. 2) the instantaneous horizontal velocities are noisier and more isotropic than the 1° control however, these effects will not be apparent for daily or longer averages.

Here we consider the simplest choice for perturbing the momentum fluxes. In this way, the tensorial flux form of the momentum equations in a curvilinear z-coordinate system (Madec and The NEMO Team 2016) are now given by
ut=[f+1e1e2(υe2iue1j)]υ1e1e2[(e2u2)i+(e1υu)j]1e3(wu)k1e1i(ps+phρo)+ϵu+(subgridterms+surfaceforcing),
υt=[f+1e1e2(υe2iue1j)]u1e1e2[(e2uυ)i+(e1υ2)j]1e3(wυ)k1e2j(ps+phρo)+ϵυ+(subgridterms+surfaceforcing),
where (i, j, k) are orthogonal curvilinear coordinates on the sphere associated with the positively oriented orthogonal set of unit vectors (i, j, k) such that k is the local upward vector and (i, j) are two vectors orthogonal to k along geopotential surfaces. Here f is the Coriolis parameter, ps and ph are the surface and hydrostatic pressure, respectively. Here (λ, φ, z) define the geographical coordinate system where position is defined by the latitude φ (i, j), the longitude λ(i, j) and the distance from Earth’s center a + z(k) and where a is Earth’s radius and z the altitude above a reference sea level. The local deformation of the curvilinear coordinate system is then given by e1, e2, and e3, three scale factors defined as
e1=(a+z)[(λicosϕ)2+(ϕi)2]1/2,
e2=(a+z)[(λjcosϕ)2+(ϕj)2]1/2,
e3=(zk).
In Fig. 2 we show an instantaneous zonal velocities for the 1 January of the last simulation year for the 0.1° reference, 1° control, stochastic-10 and stochastic-100 cases. For surface u there is clear evidence of gridscale noise in particular regions of the 100% forcing amplitude where these perturbations are acting to make the field more isotropic. This noise is not evident for perturbation amplitudes of 10% (shown), 20%, or 50% (not shown) and suggests an upper bound on the noise to be between 50% but much less than 100% amplitude. The amplitude of the instantaneous velocities in the stochastic forcing cases does not ever exceed that of the 0.1° simulation. Understanding the relative importance of the instantaneous tendencies is important. Given that our scheme uses a zero mean stochastic force, direct comparison of its mean to the time averaged terms in the momentum equation is not as relevant. However, comparison of the variance of the forcing to the terms in the energy budget equations are indeed appropriate and insightful. This is the focus of the following sections.

3. Velocity field structure

We next make direct comparison of the 1.0° control and stochastic simulations to the ACCESS-OM2-0.1 eddy resolving case after application of a low-pass (12-month) filter. The low pass is applied to the high-resolution model to extract the components of the signal that are potentially representable by the low-resolution model and to see if the application of noise is degrading those resolved scales. In both the large-scale zonal and meridional velocities we see that, while there are indeed some inherent differences between the large scales of the 0.1° and 1.0° control simulations, the broad-scale features are quite similar in both range and spatial patterns. Differences between the 0.1° reference and 1° simulations remain consistent with those of the 1° control with the application of noise, even for the stochastic-100 case. Therefore, the noise is largely modifying the small scales of the velocities. In Figs. 3 and 4 we show the surface zonal and meridional velocity for the January average. The panels show differences between the 0.1°, the control 1°, and the respective stochastically perturbed simulations.

Fig. 3.
Fig. 3.

(left) High- and low-resolution zonal surface velocities and differences with stochastically perturbed simulations (center) after application of a 12-month low-pass filter to the high-resolution ACCESS-OM2-0.1 prior to interpolation onto a common 1° grid. (right) Differences are also shown between the 1° control and stochastically perturbed simulations. Climatological January averages are shown. Note that the leftmost color bar is associated with the left and center columns, and the rightmost color bar is associated with the right column. This is indicated by the boxes.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Fig. 4.
Fig. 4.

As for Fig. 3, but for meridional surface velocities.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Of immediate note for the zonal velocities, is the good correspondence between the broad features of the high-resolution reference and low-resolution control simulations. In the tropics however, the zonal equatorial currents in the 1° control substantially differ from those of the 0.1° simulation despite both models being driven with the same surface forcing. To be expected is the absence of high-amplitude, small-scale features in the low-resolution control simulation, and in particular in the Antarctic Circumpolar Current (ACC) and in the midlatitude boundary current regions such as the Kuroshio and Gulf Stream. In comparison to the ACCESS-OM2-1 control, it is the tropics, and in particular the Indonesian Throughflow (ITF) and Indian Ocean that respond most immediately to the applied stochastic forcing. The responses seen in the stochastic-10 and stochastic-20 simulations are in the tropical instability waves in the equatorial Pacific and Atlantic, an equatorward displacement of the current associated with the ITF and similarly with the Indian ocean storm track extending from the Western Australian coast (Chapman et al. 2020). As the amplitude of the perturbations is increased, we continue to see a strong response in the tropics but also responses in the Kuroshio and Gulf Stream associated with a poleward displacement of their separation and extensions.

For the meridional velocity there is a similar close correspondence between the broad scale structures of the respective high-resolution reference and low-resolution control simulations. Significant differences, however, are apparent in the tropics and in particular associated with the Pacific tropical instability waves. This is evident in the strong response in the equatorial Pacific at 240° longitude. In comparing the control and stochastic simulations, it is evident that, even for very weak stochastic forcing, these waves are easily excited by noise. As the strength of the perturbation amplitude is increased there emerge responses at the midlatitudes and in particular located wherever major topographic features are present. For example, in the ACC of the Southern Ocean we see significant shifts in the meridional velocities in the vicinity of the east Pacific Rise. This region has previously been noted as one where intrinsic variability can be excited by reanalyzed synoptic-scale atmospheric surface (10 m) winds alone (O’Kane et al. 2013). In the Northern Hemisphere, for the higher-amplitude perturbations, there are significant responses across the entire North Atlantic and a westward shift in the Kuroshio separation.

4. Transient kinetic energy and baroclinic instability

The primary reason to perturb the velocity tendencies in conjunction with a GM scheme in a low-resolution model is to account for the understrength transfer of energy from baroclinic to barotropic processes (Kjellsson and Zanna 2017; Bachman 2019; Jansen et al. 2019; Grooms and Kleiber 2019). The total energy tendency, first defined by Orlanski and Cox (1973) and here in the form described in appendix C of Oey (2008), can be written as
ddt(EKE+EPE)=(vp/ρ0¯)+BT+BC+KH.
Here
EKE=12(u2¯+υ2¯),
EPE=g22N2ρ2¯ρ02,
BT=(u2¯u¯x+υ2¯υ¯y+uυ¯u¯y+uυ¯υ¯x),
BC=g2ρ02N2(uρ¯ρ¯x+υρ¯ρ¯y),
KH=(wu¯u¯z+wυ¯υ¯z),
where ρ is the density of seawater, p the pressure, and N2 the buoyancy frequency. In general, the overline (i.e., u¯) can refer to the time mean but here will indicate the monthly climatology with a prime (i.e., u'), denoting anomalies about the climatology. For the respective terms in Eq. (4a), EKE is the transient or eddy KE, and EPE the transient potential energy. BT and BC are the barotropic and baroclinic conversion terms. For BT positive, energy is drained from the mean horizontal shears to the eddy field. For BC positive, energy is drained from the horizontal density gradients, equivalent to the mean available potential energy, to the eddy field. Contributions from the mean vertical shears and Reynolds stresses in the vertical plane are included in the Kelvin–Helmholtz (KH) instability. In order for conservation of energy transfers, release of mean KE (i.e., positive BT and KH) must be accompanied by capture of potential energy (i.e., negative BC). The divergence (i.e., pressure work) term (vp/ρ0¯) vanishes if integrated over a closed domain. We also can define an additional exchange term, that, if positive, describes the drain of energy from EPE into EKE:
PKC=gρ0(ρw¯).
In the results to follow, KH contributions will not be explicitly considered.
Following Oliver et al. (2015), we now reconsider the time-mean transient (eddy) KE redefining Eq. (4b) in joules (J) within a volume V in the modified form
EKE=12Vρ(u2¯+υ2¯)dV.
Following O’Kane et al. (2013), the transfer rate of mean to transient potential energy representing baroclinic instabilities, in joules per second (J s−1), is now given by
GPE=gVuρ¯ρ¯x+υρ¯ρ¯yρ˜zdV,
where g is the acceleration due to gravity and ρ˜ is a reference state for the ocean approximated by the zonally and meridionally averaged density.

For a detailed examination of the energetics in physical space, we consider the transient KE [Eq. (6)] and potential energy transfer [Eq. (7)]. In Fig. 5, we show global maps of January anomalous surface EKE (ζSurface), and differences between 0.1°, 1° control, and stochastically forced simulations. Of note are the very large differences between the 0.1° reference and 1° control simulations where high EKE values are observed in the southwestern Indian Ocean (10°–30°S, 60°–120°E) and in the western tropical Pacific (10°–30°N, 120°E–180°). In contrast, regions of high EKE in the 0.1° reference calculation, largely located in the Southern Ocean and western boundary current regions, are entirely absent in the 1° control. While EKE is also present in the southwestern Indian Ocean and western Pacific, it is 10%–20% weaker than for the 1° control simulation. The Southern Ocean and in particular the ACC is a known region of high eddy variability (Fig. 5). As expected, the 0.1° reference calculation shows regions of high EKE throughout the ACC whereas for the 1° control simulation EKE is weaker with no values above 1011 J observed. In general, as the amplitude of the stochastic forcing increases so does EKE. The difference plots (Fig. 5) show that stochastic forcing generally increases EKE in the regions targeted by the mask and that these increases may be as much as 10%–20% of the control simulation values (i.e., 101–102 J). For the ACC, initial responses are collocated about large topographic features. Here the stochastic-100 simulation displays comparable values and spatial distributions of EKE to the 0.1° reference calculation, and similarly for the boundary current regions. Outside of the tropics, the differences between 0.1° reference and stochastic-100 are small and mostly confined to coastal regions of the NH at latitudes > 60°S. However, there is excessive EKE amplification evident in the tropics that may not be desirable if the goal is to replicate the spatial distribution of EKE in the 0.1° reference simulation. This effect could be eliminated by modifications to the mask such that it is applied taking into account the first baroclinic Rossby radius of deformation.

Fig. 5.
Fig. 5.

Climatological averages for all Januaries over the last decade of simulations for anomalous KE[log10 (J)], and differences between stochastically forced and control 1° simulations. All calculations are done on the low-resolution 1° grid. Note the range for the full field calculations, i.e., (1011–1014 J), in rows 1 and 2, where white shading refers to values less than 1011 J and not zero values, and has been chosen to accentuate the differences. The colorbar is the log10 of the field.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Considering baroclinic instability in the form of the transfer of mean to transient potential energy (GPE), we again focus on the January average over the last simulated decade. In Fig. 6, substantial differences between the 0.1° reference calculation and the 1° control are evident. Relative to the high-resolution reference simulation, the regions of large negative values (Fig. 6 row 1) in the low-resolution control in the southwestern Indian and western Pacific oceans are associated with substantive transfers of GPE to sustain the strong EKE values (Fig. 5), evident in the same regions. The substantive differences between the low- and high-resolution reference calculations in the tropics and ACC are associated with the drain of mean to anomalous potential energy, which is subsequently converted to increase small-scale EKE. This is somewhat consistent with the mechanism described by Kjellsson and Zanna (2017) and outlined by Bachman (2019).

Fig. 6.
Fig. 6.

As in Fig. 5, but for the transfer rate of mean to anomalous PE[log10(J s−1)]. The colorbar is the log10 of the field.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Relative to the low-resolution control, for GPE (Fig. 6, rows 2 and 3) we see even weak stochastic forcing (stochastic-10 and -20) is able to enhance both positive and negative GPE transfers in the low resolution model associated with the tropical instability waves and eddying regions of the high-resolution reference ocean. In the ACC, the addition of even very weak stochastic forcing (stochastic-10) excites structures located over topographic features whose values are comparable to those in the 0.1° reference calculation. These structures become larger in extent and magnitude as the amplitude of the stochastic forcing increases and for similar 1° MOM configurations have previously been shown to be consistent with Rossby waves, which can also be excited by the addition of noise directly to the surface forcing (O’Kane et al. 2013). Additional increases to the stochastic forcing amplitude, while further amplifying transfers in the regions targeted by the mask, results in additional potential energy drains. This is particularly the place in the subtropics, as GPE is converted to EKE.

In Fig. 7, we consider the zonal average GPE in the tropics for the January average over the last simulated decade. The 1° control shows large-scale structures to 2000 m in depth north of 10°N with absolute values exceeding 104 J s−1 with lower values extending only to 1000 m depth south of 10°S. The values within +10° and −10° latitude are small scale and weak. With the application of stochastic forcing, GPE values increase everywhere with larger-scale structures appearing in the regions poleward of 10° latitude and at greater depth in the south. With increasing stochastic forcing, GPE values and structures within 5° of latitude in the equatorial regions are amplified and are reaching larger values than the 0.1° reference simulation with stochastic-100. As the latitude increases there are some structural differences between the 1° simulations and the 0.1° reference calculation but the overall effect of increasing noise is to amplify the existing structure of the climatological zonal GPE.

Fig. 7.
Fig. 7.

Zonal-average GPE in the tropics averaged for January over the last decade of simulations. Calculations are on a common vertical grid. The colorbar is the log10 of the field.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

5. Kinetic energy spectra

For coarse-resolution finite-difference models of the ocean and climate, the resolved small-scale total, mean, and eddy kinetic energies are all systematically underestimated with respect to an eddy-resolving simulation. To increase the small-scale kinetic energy to match that of the eddy-resolving simulation requires some form of energy injection, which is a motivation for many of the aforementioned studies employing stochastic backscatter methodologies. In Fig. 8 we compare globally averaged total, mean, and eddy KE spectra calculated from surface velocities for the 1° control and 0.1° reference simulations. From this figure we clearly see that the required correction to the kinetic energies of the coarse-resolution control simulation to match that of eddy resolving reference case would necessitate a large injection of both mean and transient KE across the entire wavenumber spectrum. Here we only consider the case where the applied stochastic forcing has zero mean such that the total KE of the coarse-resolution 1° control simulation is largely conserved. This represents the minimal case where we are interested in the energy redistribution and model response due to stochastic forcing independent of the additional amplifying effects of energy injection.

Fig. 8.
Fig. 8.

Globally averaged (a) total, (b) mean, and (c) eddy KE spectra calculated from surface velocities for the 1° control and 0.1° reference simulations. The spectra are calculated from temporal averages over 10 years of surface velocity data.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

The question arises as to the mechanism by which the energetics and temperature in the model respond to increasingly larger-amplitude stochastic perturbations. As the stochastic forcing applied to the velocity tendencies has zero mean, one might naively expect that energy could be simply redistributed across scales and not injected as would be the case if the system was linear. To better understand the redistribution of energy and the source of the increased transients in our low-resolution ocean GCM, we next consider the KE spectra averaged across the global ocean.

Specifically, we consider total KE and its component parts in terms of the triple decomposition (Hussain and Reynolds 1970; Kitsios et al. 2010):
ζ(x,t)=ζ(x)+ζ˜(x,t)+ζ(x,t),
where ζ¯(x)=ζ(x)+ζ˜(x,t) (climatology), ζ˜(x,t) (seasonal = climatology minus mean), ζ'(x, t) (anomalies about the climatology), and the mean ζ(x)=(1/T)0Tζ(x,t)dt, where T is the length of the time series. Here we depth average the layers that lie within the first 1000 m, then regrid the velocity fields of the depth averaged ocean onto a Gaussian grid, where velocities over continents were given a value of zero. Spherical harmonic transforms were then calculated from this Gaussian grid. The use of a Gaussian grid addressed the issue of the model grid having cells changing in size as they near the poles. The degree width of a given wavenumber n is equal to 120/n where the corresponding length scale then is dependent on the line of latitude. Different approaches to calculating ocean spectra are detailed by Kjellsson and Zanna (2017) and Aluie et al. (2018). In the top row of Fig. 9 we show the total KE and its constituent components (i.e., mean, seasonal, and anomalous KE). We show spectra calculated from velocities depth averaged to 1000 m, noting that investigations at various depth levels in the upper ocean reveal a qualitatively similar picture. The total and mean KE spectra are closely matched for the control and all stochastically forced models indicating that the total energy remains largely conserved regardless of the strength of the forcing. The interesting result is that as the forcing amplitude is increased, energy is transferred from the seasonal cycle (Fig. 9, top row, second panel from the right) to generate transients (Fig. 9, top row, last panel on the right) with transient KE being preferentially generated at the small scales, i.e., total wavenumbers n ∈ [10, 100].
Fig. 9.
Fig. 9.

Global KE spectra calculated from velocities depth averaged from 0 to 1000 m for the 1° simulations. Here n defines the total wavenumber. (top) Total KE and the constituent terms of the triple decomposition and (second row) the ratio of stochastically forced and control simulations. (third row) Terms of the triple decomposition normalized by the total KE. (bottom) Ratio of anomalous to seasonal KE for each of the 1° simulations. The legend in the top-right panel is applicable to all other panels.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

This observation is made even clearer when we consider the total, mean, seasonal, and anomalous KE as a ratio between forced and control simulations (Fig. 9, second row). For the ratio of forced to control total, we see additional redistribution of KE from the large scales to the small scales relative to the control as the stochastic forcing amplitude increases. In contrast, for the mean and seasonal KE there is uniform transfer of KE to the transient anomalous KE across all scales but in particular to the smallest resolved scales. This transfer to the transients exhibits a cusp like functional form reminiscent of stochastic backscatter subgrid terms first described by Kraichnan (1976) for homogeneous turbulent flows. Similar observations were made when considering the KE transfers from the mean field to the transients at a given level (not shown). Our results are consistent with those of O’Kane and Frederiksen (2008b) (see their Figs. 1 and 6c,d), who showed that coherent eddy formation in the large-scale flow came at the expense of the small-scale mean-field energy. The transfers from the large-scale mean flow to generate small scale transients is even more clearly demonstrated when the mean, seasonal, and anomalous KE is normalized by the total KE at each wavenumber (Fig. 9, third row). For scales smaller than total wavenumber n = 50, significant reductions in mean and seasonal KE occur relative to the total KE for the stochastic-50 and stochastic-100 simulations. For n > 10, this energy is preferentially redistributed to the smaller scales; however, some of the mean and seasonal KE is uniformly transferred to the large-scale structures (i.e., n ≤ 10). The extent to which stochastic forcing initiates energy transfers from the seasonal to the transients is revealed in the ratio of anomalous to seasonal KE (Fig. 9, bottom row).

The results described in sections 4 and 5 are somewhat consistent with those of Bachman (2019) and Kjellsson and Zanna (2017). They described the resolution dependence of the mechanism by which baroclinic flows are organized into barotropic flows and that more baroclinic flow is present at lower horizontal resolutions. They suggested that the effect of mesoscale eddies could potentially be parameterized by enhancing the PE to KE conversion. Here we have seen that application of noise to regions of the global ocean where transient EKE is unresolved does in fact lead to increased conversion of PE to generate large increases in the small scale EKE. In the following sections we will examine the wider effect of these noise induced changes in the energetics on the climatological ocean state. Hewitt et al. (2020) also discuss the general finding that GM mesoscale eddy parameterizations mimic baroclinic instability acting as a net sink of available potential energy [see also Zanna et al. (2020)].

6. Sea surface temperature and mixed layer depth

Given the observed transfer of energy from the mean and seasonal spectra to generate anomalous KE, the question arises as to the spatial imprint on the dynamically active regions. We begin by first comparing January SST climatologies for the low-resolution control simulation, the high-resolution reference calculation, and differences between the forced simulations and control (Fig. 10). We first notice that agreement between the high- and low-resolution simulations is high, in part expected due to the common surface boundary conditions. With the application of stochastic forcing, we see the initial response in the midlatitude boundary current regions of the North Pacific and Atlantic, once again notably in the regions associated with the Kuroshio Extension and Gulf Stream separation.

Fig. 10.
Fig. 10.

(top left),(middle left) In the black box we show high- and low-resolution reference calculations of SST. (bottom left) Difference between 1° control and 0.1°. (center),(right) Differences w.r.t. the stochastic simulations for the climatological January over the last decade of simulations. The two top-left and middle-left panels have different colorbars than the differences shown in the bottom-left panel and the center and right columns.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

For the stochastic-50 simulation, the Northern Hemisphere responses are revealed as largely meridional displacements to the aforementioned boundary currents and in the Atlantic to the gyre circulation encompassing the North Atlantic drift and Canaries Current. In the western Pacific, we see localized warming along the Alaska and California Currents. In the Southern Hemisphere, there is cooling in the East Australian Current, the South Equatorial, Mozambique, and Agulhas Currents, the Falklands Current, and regions in the ACC. At 100% amplitude stochastic forcing (stochastic-100) there is further amplification of the aforementioned responses but with additional cooling in the western equatorial Pacific. In contrast to substantial cooling (up to 4°C) in the western equatorial Atlantic, warming is evident all along the eastern coast of South America. The general patterns of warming and cooling in the Southern Hemisphere are less representative of meridional displacement of currents and more indicative of changes to mixing processes. This is indeed shown to be the case in examination of the January climatological mixed layer depth (MLD) (Fig. 11). Of note is the substantial difference in MLD between the high-resolution reference and low-resolution control simulations at the Kuroshio Extension in the North Pacific at around 40°N.

Fig. 11.
Fig. 11.

As for Fig. 10, but for January mixed layer depths.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

With increased forcing amplitudes up to 50%, the meridional displacement of the currents in the Northern Hemisphere are also shown to be accompanied by substantive changes in MLD. For stochastic-100 there is noticeable degradation of the structure of the Kuroshio rather than the meridional displacement observed for stochastic-20 and -50. In the Southern Hemisphere, the cooling observed in the Southern Ocean is now revealed to occur primarily due to substantial increases in MLD of over 200 m at locations where significant topographic features are located. This could simply be due to an enhancement of mixing over topography and or an enhancement of the eddy-topographic force due to increased momentum fluxes. Considering the responses in July at the height of the austral winter (Fig. 12), substantive increases in MLD are observed throughout the ACC and, for maximum amplitude stochastic-100, at the Tasman Front extending from the Australian coast to the west of New Zealand and in the southern Atlantic in the region of the Brazil–Malvinas Confluence.

Fig. 12.
Fig. 12.

As for Fig. 10, but for July mixed layer depths.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

For the perturbed experiments, the deepening of the mixed layer is consistent with the amplitude of the noise. We only see anomalously deeper mixed layer depths at the mid- to higher latitudes of the summer hemispheres due to the applied forcing. In the tropics we see widespread deepening of the mixed layer by 10–20 m only for the strongest forcing.

7. Sea level, temperature, and ocean heat content

Regions of substantive surface cooling can also be accompanied by subduction of large amounts of heat and local increases in sea level. This is exactly the case where the surface cooling previously observed in the equatorial oceans for large-amplitude stochastic forcing (Fig. 10), is shown to be associated with increases in sea level of over 20 cm (Fig. 13, last panel) and anomalous temperature increases of more than 4°S at the thermocline (Fig. 14, last panel). There is also a corresponding increase in global volume annual averages of ocean temperature. At year 170, there are increases of approximately 4% and 12% for the stochastic-50 and stochastic-100 simulations respectively, relative to the initial state. Decreases in sea level occur in the midlatitudes south of 30°S and at the high latitudes in the sea ice zones. These regions are however not associated with substantive surface (Fig. 10) or subsurface (Fig. 14) cooling.

Fig. 13.
Fig. 13.

As for Fig. 10, but for January sea level.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Fig. 14.
Fig. 14.

As for Fig. 10, but for January zonally averaged temperatures.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Hewitt et al. (2020) recently undertook an assessment of the impact of ocean resolution on the mean state and variability in CMIP (3, 5, and 6) class Earth system models. Their discussion notes both positive and negative biases for heat uptake in the upper ocean across this class of models. To better understand changes in sea level, in Fig. 15, we show ocean heat content (OHC) annually averaged through time and spatially averaged globally and for the Atlantic, Pacific, Southern, and Indian Oceans, both integrated and by depth contrasting the 1° control and stochastic-100 simulations. For the global ocean, the control simulation exhibits warming above 1000 m and cooling below, and noticeably at depths around 4000 m. The strongly forced simulation requires around 150 years to reach a global steady state. Relative to the control, the stochastic-100 simulation is characterized by substantially larger increases in OHC, in particular above 2000 m, with accompanying increases in the mean thermocline depth from ≈1500 to ≈2500 m. Below 2000 m the rate of increasing cooling in stochastic-100 is substantially less than observed in the control such that the change in volume summed global OHC, reaching around 100 × 1022 J after 40 years before equilibrating at 250 × 1022 J after 150 years, represents an increase of 0.17% globally on the control steady state. However, this transient response to strong stochastic forcing is large in comparison to the global warming signal, also noted by Williams et al. (2016). That said, the physical processes involved in global warming are not comparable to the mechanisms described here.

Fig. 15.
Fig. 15.

Annually averaged ocean heat content (OHC) in terms of the difference with respect to the initial state for (top to bottom) the global, Atlantic, Pacific, Southern, and Indian Oceans by depth and volume integrated. We show only the 1° control and stochastic-100 simulations.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

The change to OHC caused by strong uncorrelated transient (eddy) noise is of the same magnitude but opposite sign to that observed by Williams et al. (2016) employing strong temporally correlated noise perturbations applied to the temperature tendencies. Where Williams et al. (2016) also observed warming in the top 1000–2000 m, they observed proportionally much larger cooling at depths between 3000 and 4000 m, to the extent that there was a net cooling of the global ocean. In contrast, our results show the warming rate of the upper ocean is substantially larger than the rate of cooling at depth with changes (units of 1022 J) in the Atlantic (≈55), Southern (≈40), and Indian (≈40) Oceans at year 170, and where nearly half of the total warming occurs in the Pacific ≈115, mostly concentrated at the equator (see also Fig. 14). As noted earlier, all our simulations have reached an approximate steady state for global OHC after a transient period of ≈150 years. For the stochastic-100 case, the very large increases in sea level in the tropics are clearly unrealistic with large amounts of heat being subducted into the tropical upper ocean, and in particular the Pacific. Here it is an indication that very strong stochastic forcing can lead to deleterious impacts on the modeled steady state dependent on the variables considered. We are at present unclear as to the causes of the reduced SSH at high latitudes in regions outside of the mask. For the high-latitude Southern Ocean austral winter, we see increased mixing and OHC occurring simultaneously with reductions in SSH of up to 20 cm, which is surprising.

8. Ocean transports

Finally, we are interested to see what impact the described changes in the climatological state have on ocean transports. In Fig. 16 we consider time series of the transports for Drake Passage, the Atlantic meridional overturning circulation at 26°N (AMOC26°N), and North Atlantic Deep Water (NADW), and in Fig. 17 the meridional overturning circulation (MOC) and overturning streamfunction on potential density surfaces. The MOC consists of the shallow tropical wind-driven cells, the Deacon cell driven by the Southern Hemisphere subpolar westerlies, the Antarctic Bottom Water (AABW) cell south of 60°S, the Southern Ocean abyssal cell, and the North Atlantic Deep Water (NADW) cell. For an overview of the global overturning circulation and inverse based transport estimates we refer to Lumpkin and Speer (2007).

Fig. 16.
Fig. 16.

Comparison of transports for (top) Drake Passage, (middle) AMOC26°N, and (bottom) NADW with observational estimates included.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

Fig. 17.
Fig. 17.

Overturning is computed on density surfaces, integrated zonally around the globe, and averaged over the last 10 years of simulation. The contour interval is 2 Sv, and the density axis is nonuniform. The panels shown correspond to the (a) 0.1°, (b) 1° control, (c) stochastic-50, and (d) stochastic-100 simulations.

Citation: Journal of Climate 36, 6; 10.1175/JCLI-D-21-0796.1

The MOC is defined using the streamfunction ψ in units of Sverdrups (1 Sv ≡ 106 m3 s−1 or approximately 109 kg s−1). It is the zonally integrated and vertically accumulated meridional volume transport, which in depth coordinates is defined as
ψ(z)=zoxwxeυ(x,z)dxdz.
The ocean model generates mass transport (kg s−1) as a function of horizontal location and depth, and so these can be used in place of volume in Eq. (9) to generate a MOC in the preferred mass units. To obtain the transports for the MOCs in a basin defined as a function of depth and latitude, the full depth model mass transports are used and integrated between east and west boundaries (coast to coast). It is therefore possible to define them globally, or for particular ocean domains.

To obtain the strength of overturning for a particular common index and to compare to generally very limited observations, we follow Bi et al. (2013) and simply identify the maximum value within a particular depth and latitude box. The ranges of these boxes are

  • North Atlantic Deep Water (NADW): 30°–50°N, 500–3000 m

  • Atlantic meridional overturning streamfunction (AMOC26°N): 25.5°–26.5°N, 0–6000 m

  • Antarctic Bottom Water (AABW): 90°–60°S, 0–3000 m

where the AMOC26°N is identified at the two latitudes 25.5° and 26.5°N that span the required 26°N.

Drake Passage transport (Fig. 16) is a proxy for the strength of the ACC. Here it is calculated using monthly averaged 3D ocean horizontal mass transports from which the eastward component is integrated along a single line from the southern tip of South America to the northern tip of the Antarctic Peninsula and to the ocean bottom. Here the ACC strength for the 0.1° high-resolution reference simulation lies on average between 140 and 150 Sv and between 150 and 160 Sv in the 1° control increasing to a maximum of between 155 and 165 Sv for stochastic-100 with a near-linear response to increases in stochastic forcing amplitude. All simulations are within observational estimates of the observed Drake Passage transport values, which range between 123 ± 10.5 Sv (Whitworth and Peterson 1985) and 173 Sv (Donohue et al. 2016). We caution that the normal year forcing does not allow for realistic variations of the transports, which may lead to a substantial underestimation of seasonal and interannual variability. In Fig. 16 it can be clearly seen that the year-to-year variability is very similar, especially for the mostly laminar 1° simulation, where internally driven dynamics remain small. This may bias the comparison to observational estimates [as shown in Fig. 16 of O’Kane et al. (2021)].

The 0.1° high-resolution reference transport for the AMOC cell is centered about the estimated observed transport of 17.2 Sv at 26°N (McCarthy et al. 2015) and within the observed range of seasonal variations between 10 and 25 Sv from the RAPID-WATCH (Smeed et al. 2015). However, the 1° control reveals a much too weak AMOC26°N transport with seasonal fluctuations of between 3.5 and 9.5 Sv. Stochastic forcing acts to increase the transport by up to 3.75 Sv to maximum steady state values of 12.5 Sv (stochastic-100). The 1° control NADW intensity averages between 7.5 and 12 Sv whereas the high-resolution reference ranges between 16 and 25 Sv. The stochastic-50 and stochastic-100 simulations both generate seasonally varying values of between 10 and 15 Sv comparable to the observed values ranging about ≈15 Sv (Lumpkin et al. 2008; Ganachaud 2003).

In Fig. 17, the overturning streamfunction is shown (0.1°, 1° control, stochastic-50, and stochastic-100). The mean overturning is calculated on potential density surfaces referenced to 2000 dbar. The AMOC corresponds to the positive cell with a maximum value close to 1036.5 kg m−3. Again, the high-resolution reference calculation is closer to the observed values with the effect of increasing stochastic forcing to push the overturning closer to the 0.1° reference simulation. The AABW corresponds to the negative cell at ≈1037 kg m−3 in the Southern Ocean. Observed values of the AABW cell transports range from 5.6 ± 3.0 Sv reported by Lumpkin and Speer (2007) to values of 9.77 ± 3.7 Sv reported in the Weddell Sea (Sloyan and Rintoul 2001; Naveira Garabato et al. 2002; Talley 2013). For AABW, isosurface fluxes and resolved bathymetry around the Antarctic Peninsula drive production of dense shelf water, hence it is not expected that stochastic forcing as applied here will be effective. Despite this, it does appear that the impact of increased noise is usually to increase the average negative transport. We also note that the overturning streamfunctions for the 0.1° high-resolution reference and 1° low-resolution control simulations are very similar to those reported by Kiss et al. (2020) where interannually varying JRA-55 forcing was used.

Thus, we see that the addition of noise is to most often adjust the transports of the low-resolution noneddying model where the negative transports are becoming increasingly more negative, and positive transports becoming increasingly more positive as the forcing amplitude is increased. Despite having the same repeat forcing applied as the 1° simulations, the 0.1° reference calculation transports display less regular seasonal variability. We ascribe this to the presence of randomly generated eddies with deep vertical extent in the high-resolution reference model but we have not undertaken an assessment of this hypothesis.

9. Summary and discussion

We have implemented a simple stochastic forcing in regions associated with fast ocean transients (eddies), via random perturbations to the horizontal momentum fluxes. The statistics of the transients were calculated from the velocities of a high-resolution, eddy-resolving ocean model ACCESS-OM2-0.1. After thresholding, a three-dimensional mask was generated whereby stochastic noise, representative of subgrid transients, perturb a low-resolution, 1° non-eddy-resolving variant of the same ocean–sea ice model configuration. Four variants of the stochastically forced 1° ACCESS-OM2 model were considered, with varying amplitudes of the noise relative to the high resolution reference calculation applied. All low-resolution model configurations were run to an approximate steady state (≈150 years) before calculation of the statistics of their respective climatological states. This approach was shown to lead to a range of varied regional responses in the simulated climatological steady intrinsic ocean states where particular choices of perturbation amplitudes have certain advantages for some variables in some areas, but may also lead to certain areas having the largest degradations.

Spectra from a triple decomposition revealed that, despite having zero-mean, random noise forcing was able to initiate a redistribution of KE largely from seasonal variations to generate large amplitude small scale anomalous transient kinetic energies. The various simulations span a range of amplitudes determined by a high-resolution eddy resolving simulation. The model redistributes KE from the seasonal mean to generate increased anomalous EKE preferentially at the small scales. The rate of energy transfer increases nearly linearly with the amplitude of the applied forcing while the total KE spectra remains largely unchanged for the range of forcing amplitudes considered. Transient KE present in the high-resolution reference simulation Antarctic Circumpolar Current, largely absent in the 1° control simulation, was able to be approximated to a large degree in the stochastically forced simulations for sufficiently strong noise amplitudes.

Surface temperature responses were largely consistent with increases in mixed layer depths and meridional displacement of Northern Hemisphere boundary currents. Specifically, SST, SSH, and MLD differences in the northern (boreal) winter in the regions of the Kuroshio and Gulf Streams are reflecting a meridional displacement of the WBCs. This is reflected in the dipolar structures in the differences between stochastic-50 and -100 perturbed and 1° control simulations. During the austral summer, the midlatitude MLD in the 1° control is biased 10–20 m shallower with respect to the 0.1° reference climatology. Here differences between stochastic-50 and stochastic-100 and 1° control simulations show the MLD increased by over 200 m in localized regions of the ACC where significant topographic features are present leading to increased SSH and reduced SST directly over those structures. For the boreal summer (JUL) there is no widespread impact on the extratropical NH MLD. For the SH winter there is a significant deepening of the MLD in the ACC with localized impacts on the EAC, Brazil-Malvinas confluence, and other boundary currents extending to 30°S. In Fig. 12 there is also evidence of a poleward shift in the ACC front at 40°–50°S, 60°–150°E in the differences between stochastic-50 and stochastic-100 perturbed and 1° control simulations.

Increased sea levels at the equator were found to be largely in response to injection of heat into the equatorial Pacific at the thermocline and into the mixed layer. For stochastic-100, the overall impact on SSH is dominated by the very large increase in OHC in the tropics between ±15° latitude with corresponding widespread increases in the MLD of between 10 and 20 m. While consistent warming was observed at all depths, by far the majority of the OHC warming occurred in the equatorial Pacific upper ocean. This warming was clearly excessive for large-amplitude noise cases but may be addressed by applying latitudinal weighting to the mask dependent on the Rossby radius of deformation much as we have done for our modified GM scheme (Gent and McWilliams 1990; Hallberg 2013). As the noise amplitude is increased there is observed a proportional strengthening of the AMOC26°N and NAWDW. Only Drake Passage transport moved farther from the high-resolution reference calculation but curiously closer to the most recent observational estimates. While the maximum amplitudes of OHC differences between control and stochastic forcing experiments were comparable to those observed by Williams et al. (2016) using perturbed temperature tendencies, stochastic perturbations to the momentum fluxes produced global OHC warming whereas perturbed temperature tendencies produced cooling of the total OHC. Both responses can be at least as large in amplitude as the observed anthropogenic global warming signal.

The transfer of energy from baroclinic to barotropic processes is seen to be consistent with the mechanism proposed by Kjellsson and Zanna (2017) and Bachman (2019) where the addition of stochastic forcing at high latitudes can increase the small-scale KE to levels comparable to those observed at higher horizontal resolutions. For the tropics we find that EKE is very readily increased to excessive levels as the forcing amplitude is increased. Our findings are also consistent with the idealized study of Cooper (2017). However, our results do not show that application of large-amplitude perturbations to the momentum fluxes leads to significant increases in the amplitude of instantaneous vertical velocities. Rather we find that the vertical velocities become very noisy and decorrelated acting to enhance mixing and leading to large increases in OHC and deleterious impacts to sea level. Therefore, forcing amplitudes need to be chosen judiciously.

For data-driven approaches, the question arises as to how to specify the stochastic forcing in a low-resolution climate model without also running a high-resolution simulation to provide the required target data. This is the approach used in large-eddy simulation (LES) where the largest scales are resolved and the smallest scales parameterized. Recently LES subgrid closures for parameterizing ocean mesoscale eddies in the quasigeostrophic turbulence regime have been proposed [see Pearson et al. (2017), Kitsios et al. (2016), and references therein] inspired by inertial range energy/enstrophy cascades in homogeneous isotropic (Kraichnan 1967, 1976; Leith 1967, 1996) and inhomogeneous (Frederiksen 1999; O’Kane and Frederiksen 2008b) two-dimensional turbulence. In quasigeostrophic turbulent flows, the effect of finite resolution on the resolved flow is to remove the eddy–eddy interactions by which energy is transferred from the retained to the subgrid scales resulting in an accumulation of energy in the retained small-scale flow such that the inertial range kinetic energy spectrum is flattened. As the tail of the kinetic energy spectrum is raised, energy is spuriously drained from the large to small resolved scales in a process known as the tail wagging the dog effect. To correct for the missing subgrid terms, a higher order dissipation operator is required to drain energy from the small retained scales and an injection of energy from the subgrid to the large scales (i.e., stochastic backscatter). However, as has been discussed previously (Pearson et al. 2017; Kjellsson and Zanna 2017) and observed in this current study, for low-resolution ocean models there occurs suppressed transfer of energy from baroclinic (available potential energy) to barotropic (eddy kinetic energy) processes resulting in a systematic under representation of small-scale kinetic energy. As ocean model resolution is increased and resolved eddies provide additional dissipation such that eddy viscosity and diffusivity parameters may be reduced without compromising numerical stability, energy can be transferred more freely from baroclinic to barotropic processes and the small-scale kinetic energy is increased. Various approaches have been employed to correct overdamping of the resolved small-scale low-resolution ocean model flow. However, the mechanism by which overdamping of the small scales occurs is largely through numerical diffusion and not the impact of finite resolution as described by quasigeostrophic turbulence theory (Kraichnan 1976; Frederiksen 1999). Despite this, the idea of parameterizing unresolved eddy mixing using stochastic parameterizations remains attractive, particularly as it potentially allows targeting the k−3 power law in the mesoscales consistent with theory (Kraichnan 1967) and modeling (Khatri et al. 2021) as one possible approach to avoiding the need to first run a high-resolution model.

That said, the motivation for this study is to better understand oceanic responses and sensitivity to stochastic forcing and in particular to understand how noise modifies the model energetics leading to systematic changes in the state vector. This sensitivity study shows the effect of stochastic forcing of the momentum flux in terms of the redistribution of energy and the corresponding impact on various standard ocean metrics. The point is to examine the broad response not to make a best choice. In fact, a more complex approach to stochastic subgrid-scale parameterization is almost certainly required rather than the simple approach taken here. Based on the results of this study, for the current implementation of stochastic forcing plus GM as a parameterization for mesoscale eddy variability, whereas the transfer of potential to eddy kinetic energy is increased, this occurs at the expense of generally increased biases in terms of MLD, OHC, and SSH dependent on the amplitude of the applied noise.

Acknowledgments.

The authors thank the editor and three anonymous reviews for their constructive comments that substantially improved this manuscript. The authors further acknowledge support from National Computational Infrastructure (NCI) Australia. This study would not be possible without the combined efforts of the Consortium for Ocean–Sea Ice Modelling in Australia (COSIMA) in developing the model configurations used in this study.

Data availability statement.

All model configurations are available from the COSIMA github repository https://github.com/COSIMA/access-om2 with code modifications available here https://github.com/russfiedler/MOM5/tree/perturb. Calculation of the global overturning streamfunction uses Python jupiter notebooks and utilities supplied by COSIMA and available at https://github.com/COSIMA/cosima-recipes. The data used and additional analysis codes are available on request.

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  • Aluie, H., M. Hecht, and G. K. Vallis, 2018: Mapping the energy cascade in the North Atlantic Ocean: The coarse-graining approach. J. Phys. Oceanogr., 48, 225244, https://doi.org/10.1175/JPO-D-17-0100.1.

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    • Export Citation
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