Community Earth System Model version 1 simulations

The limited observational record presents a major challenge to understanding SST variability, particularly low-frequency phenomena such as the AMV. Long climate model simulations can instead provide a larger subsampling of SST behavior. We use CESM1, a fully coupled general circulation model with land, ice, atmosphere, and ocean components all of nominally 1° horizontal resolution (Hurrell et al. 2013). The preindustrial control experiment featuring aerosol and radiative forcing fixed at 1850 levels is used to investigate natural variability of the climate system without changes in external forcing. We use years 400–2200 in the analysis to avoid contributions from initialization.

In addition to the fully coupled configuration (CESM-FULL), we analyze the slab ocean simulation (CESM-SLAB). CESM-SLAB has identical configurations to CESM-FULL except that the ocean component is a slab ocean model subject only to air–sea heat fluxes and a prescribed flux correction with no interactive lateral or vertical ocean processes. The thickness of the slab at each location is set to the annual mean boundary layer depth from the climatological cycle of CESM-FULL, thus spatially varying but temporally constant (He et al. 2017). MLDs in CESM1 are determined using a maximum buoyancy gradient criterion (Smith et al. 2010). The climatological monthly flux correction term for CESM-SLAB is diagnosed from CESM-FULL, representing the mean heat convergence and divergence induced by ocean dynamics needed to maintain a realistic SST climatology (Bitz et al. 2012). CESM-SLAB provides an opportunity to study SST behavior considering only the local, upper-ocean response to atmospheric forcing, dependent only upon the thermal capacity of the slab ocean. Monthly output from years 200 to 1100 is used in the analysis.

Prior to our analysis, the mean annual cycle is removed from each simulation to compute the monthly anomalies. We exclude grid points where the sea ice coverage exceeds 5% anytime during the simulation to focus on ocean–atmosphere interactions.

a. Formulation

Equation (1) includes both local and nonlocal ocean dynamics. The variability in large-scale geostrophic currents impacts low-frequency SST variability, and its contribution, along with other nonlocal oceanic components, will be inferred from comparison with CESM-FULL (Frankignoul et al. 1998). To retain the SST characteristics primarily dependent on local atmospheric variability we neglect the term F and the geostrophic advection in term B. We neglect the remainder of term B, including the Ekman component that is small away from regions of large SST gradients. The potential contributions of Ekman advection are explored in section 6c. We additionally neglect terms C and E, and anomalous entrainment velocity in term D. These terms are challenging to represent analytically, but could be approximated as part of the stochastic atmospheric forcing since anomalies in MLD and entrainment are primarily driven by short time scale wind and heat flux variability (Frankignoul 1985; Alexander and Penland 1996; Frankignoul et al. 1998).

From this analytical form, the equation is further simplified by combining entrainment and atmospheric damping [$\lambda =({\lambda }_a/\rho C_{p}h)+(\overline{w_e}/h)$]. Equation (3) is integrated with the forward method at a monthly step from t to t + Δt to yield

$T^{\prime }(t+\text{Δ}t)=e^{-\lambda \text{Δ}t}{T}^{\prime }(t)+{\displaystyle {\int }_t^{t+\text{Δ}t}\left[\frac{F^{\prime }}{\rho c_{p}h}+\left(\frac{\overline{w_e}}{h}\right)T_d^{\mathbf{\prime }}\right]e^{-\lambda (t+\text{Δ}t-t^{\prime })}d{t}^{\prime }}\hspace{0.17em}.$(4)

The temperature anomaly below the mixed layer $(T_{\mathit{d}}^{\mathbf{\prime }})$ is computed during the integration using T′ from the prior detrainment at the same depth (Fig. 1). Note that $T_{\mathit{d}}^{\mathbf{\prime }}$ is determined through linear interpolation, and is averaged over the monthly time step:

$(\overline{T_{\mathit{d}}^{\prime }})=\frac{T_{\mathit{d}}^{\mathbf{\prime }}(t)+T_{\mathit{d}}^{\mathbf{\prime }}(t+\mathrm{\Delta }t)}{2}\hspace{0.17em}.$(5)

Example calculation of $T_d^{\mathbf{\prime }}$ for the seasonal mixed-layer cycle (black line) at a subpolar test point (50°N, 30°W). The labeled detrainment times (months, where 1 = January), indicated by ×, are connected to the corresponding entrainment month by the colored line. $T_d^{\mathbf{\prime }}$ is calculated by linearly interpolating between temperature values for the nearest months around the detrainment time.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Example calculation of $T_d^{\mathbf{\prime }}$ for the seasonal mixed-layer cycle (black line) at a subpolar test point (50°N, 30°W). The labeled detrainment times (months, where 1 = January), indicated by ×, are connected to the corresponding entrainment month by the colored line. $T_d^{\mathbf{\prime }}$ is calculated by linearly interpolating between temperature values for the nearest months around the detrainment time.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Example calculation of $T_d^{\mathbf{\prime }}$ for the seasonal mixed-layer cycle (black line) at a subpolar test point (50°N, 30°W). The labeled detrainment times (months, where 1 = January), indicated by ×, are connected to the corresponding entrainment month by the colored line. $T_d^{\mathbf{\prime }}$ is calculated by linearly interpolating between temperature values for the nearest months around the detrainment time.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Assuming the parameters [$({\overline{\lambda }}_a/\rho C_{p}h)$, $({\overline{w}}_{\mathit{e}}/h)$, $(\overline{F^{\prime }}/\rho C_{\mathit{p}}h)$, $(\overline{T_{\mathit{d}}^{\mathbf{\prime }}})$] are constant over the monthly time step (Δt), the integral in Eq. (4) is evaluated to yield

$T^{\prime }(t+\mathrm{\Delta }t)=e^{-\lambda \mathrm{\Delta }t}\hspace{0.17em}{T}^{\prime }(t)+\left\{\frac{1}{\mathrm{\Delta }t}\text{ln}\left[\frac{h(t+\mathrm{\Delta }t)}{h(t)}\right]\overline{T_d^{\mathbf{\prime }}}+\overline{\left(\frac{F^{\prime }}{\rho C_{p}h}\right)}\right\}\left(\frac{1-e^{-\lambda \mathrm{\Delta }t}}{\lambda }\right).$(6)

The stochastic model requires only three seasonally varying input parameters: the atmospheric heat flux feedback strength (λ_a), the amplitude of stochastic forcing (F′), and MLD (h). The latter is obtained from the climatological mean monthly MLD cycle from CESM-FULL (Figs. 2i–l). Note that h is shallow (<100 m) in the summer and deepest in the subpolar region during the winter and spring.

Seasonal mean estimates for (left to right) winter to fall of (a)–(d) stochastic forcing amplitude (in W m⁻²), (e)–(h) atmospheric heat flux feedback strength (in W m⁻² °C⁻¹), and (i)–(l) MLD (in m) from CESM-FULL, where depths greater than 300 m are contoured every 150 m. Note that we force the stochastic model with monthly varying values; seasonal averages are shown here for illustration.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Seasonal mean estimates for (left to right) winter to fall of (a)–(d) stochastic forcing amplitude (in W m⁻²), (e)–(h) atmospheric heat flux feedback strength (in W m⁻² °C⁻¹), and (i)–(l) MLD (in m) from CESM-FULL, where depths greater than 300 m are contoured every 150 m. Note that we force the stochastic model with monthly varying values; seasonal averages are shown here for illustration.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Seasonal mean estimates for (left to right) winter to fall of (a)–(d) stochastic forcing amplitude (in W m⁻²), (e)–(h) atmospheric heat flux feedback strength (in W m⁻² °C⁻¹), and (i)–(l) MLD (in m) from CESM-FULL, where depths greater than 300 m are contoured every 150 m. Note that we force the stochastic model with monthly varying values; seasonal averages are shown here for illustration.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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b. Model hierarchy

We build a hierarchy of stochastic models by successively adding seasonal variation into λ_a, F′, and h and examining their impact on SST variability (Fig. 3a). The hierarchy is divided into two sections. In the lower, non-entraining section (w_e = 0; Levels 1–3), we investigate the effect of including seasonal variation in λ_a and F′. This begins with the canonical stochastic model first proposed by FH77, or Eq. (3) with the entrainment term omitted (Fig. 3; Level 1). Seasonal variation in λ_a (Level 2a) or F′ (Level 2b) is introduced individually (Level 2). As the closest analogy to the slab ocean configuration, the case with seasonal variation in both λ_a and F′, but constant h (Level 3) is compared to CESM-SLAB.

(a) Stochastic model hierarchy, increasing in complexity from bottom to top; h, λ_a, and F′ indicate the MLD, atmospheric heat flux feedback strength, and amplitude of stochastic forcing, respectively. Overbars indicate annual mean values, while m indicates climatological monthly variation. In the entraining model, additional damping from the entrainment (w_e) is included. (b) The same information is summarized in a table, where × denotes that seasonal variation in the corresponding variable is included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) Stochastic model hierarchy, increasing in complexity from bottom to top; h, λ_a, and F′ indicate the MLD, atmospheric heat flux feedback strength, and amplitude of stochastic forcing, respectively. Overbars indicate annual mean values, while m indicates climatological monthly variation. In the entraining model, additional damping from the entrainment (w_e) is included. (b) The same information is summarized in a table, where × denotes that seasonal variation in the corresponding variable is included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) Stochastic model hierarchy, increasing in complexity from bottom to top; h, λ_a, and F′ indicate the MLD, atmospheric heat flux feedback strength, and amplitude of stochastic forcing, respectively. Overbars indicate annual mean values, while m indicates climatological monthly variation. In the entraining model, additional damping from the entrainment (w_e) is included. (b) The same information is summarized in a table, where × denotes that seasonal variation in the corresponding variable is included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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In the upper section (Levels 4–5), we investigate the effect of increasing MLD complexity. First, seasonal variation in h is included (Level 4), followed by the entrainment term (Level 5; w_e ≠ 0). The entraining stochastic model is compared with CESM-FULL to infer the role of ocean dynamics beyond entrainment.

a. Estimating heat flux feedback

A potential source of longer (greater than multiple weeks) atmospheric memory is remote forcing, such as from El Niño–Southern Oscillation (ENSO). To reduce bias due to this persistent source, we first compute the ENSO-related components of heat flux and SSTs, then remove those components prior to estimating λ_a (FK2002; Park et al. 2005). ENSO indices are defined based on the two leading empirical orthogonal functions (EOFs) of the tropical Pacific SST, representative of the central and eastern modes (20°S–20°N, 120°–240°E). The heat flux and SST in the North Atlantic are then regressed on the ENSO indices to get the ENSO-related components, which are subsequently subtracted from the full anomalies of both fields. ENSO was only removed for estimating λ_a, and is retained for all other analyses in this paper.

The separation of λ_a and entrainment damping contrasts with previous work using the stochastic model that estimated an overall damping (λ) using an exponential fit to the SST autocorrelation, an approach that conflates contributions from both atmospheric and oceanic damping (FH77; de Coëtlogon and Frankignoul 2003; Deser et al. 2003). Our approach provides an opportunity to investigate if these previous works overestimated (underestimated) damping due to the atmosphere (ocean), and to further discern how each component impacts the SST persistence.

Our estimates of λ_a from CESM1 agree with previous work based on observations that reported values between 10 and 35 W m⁻² °C⁻¹, sometimes exceeding 50 W m⁻² °C⁻¹ (Figs. 2e–h) (FK2002; Park et al. 2005). The maxima occur along the Gulf Stream, where cold continental winds cause turbulent heat loss from the warmer ocean during the boreal winter, and are stronger than the maximum reported by Park et al. (2005).

b. Estimating atmospheric forcing

More specifically, we estimate F′ using following steps:

1. The stochastic component of $Q_{\text{net}}^{\prime }$ in CESM1 is computed at each grid point using our estimated heat flux feedback ($F^{\prime }=Q_{\text{net}}^{\prime }+{\lambda }_{a}T$).

2. The spatial pattern of forcing (α) is determined using EOF analysis of F′ across the North Atlantic domain (0°–65°N, 80°W–20°E) for each calendar month. The resultant EOFs (in W m⁻²) are the 12 monthly patterns for each mode (n).

3. For each mode, a white noise time series is generated for the length of the simulation (t = 12 × the number of years) and is multiplied by the corresponding amplitude for each month from the EOF pattern. This creates forcing patterns with seasonally varying magnitude that are coherent in space, but stochastic in time.

4. We retain enough modes to explain 90% of the variance for each month and sum the values to reconstruct the monthly varying forcing at each grid point while filtering out small-scale structures. We then amplify the local white noise to boost the variance of stochastic forcing back to 100% of the total variance to ensure the local amplitude of the forcing is replicated. The resultant forcing exhibits larger amplitudes in winter and spring, particularly along the Gulf Stream and Flemish Cap (Figs. 2a–d).

In summary, this formulation of stochastic forcing allows for explicit examination of how spatial structure and seasonal dependence of F′ influence SST variability.

c. Omission of the tropics

The tropics (points equatorward of 20°N) are where two key assumptions for the stochastic model are not appropriate. The first assumption is that the surface heat flux, in particular the solar radiation, is evenly distributed through the mixed layer. This is less valid in the tropics, where the mixed layer is often very shallow and penetration of solar radiation below the mixed-layer base is needed to prevent unrealistically large SST amplitudes (Davis et al. 1981; Hosoda et al. 2016).

The second assumption that atmospheric memory is short relative to the ocean is inappropriate; tropical atmospheric waves and convectively driven fluctuations may have larger persistence than midlatitude synoptic fluctuations, in addition to the influence of remote ENSO forcing (FK2002). Despite ENSO removal, the heat flux feedback estimates in the tropics were weak and insignificant at the 5% level (section 4a), resulting in large SSTs in the stochastic model. This suggests that the sources of tropical atmospheric persistence were insufficiently removed, leading to reduction or cancellation of lag covariance between Q_net and SST. Considering these complications, we omitted points south of 20°N in our regional analyses and AMV index to instead focus our discussion on the extratropics where the stochastic model assumptions hold.

a. A case study in the SPG (50°N, 30°W)

To crystallize our understanding of how seasonal variation and entrainment impact SST variability across time scales, it is useful to first focus on a single location. We thus perform a case study at a point within the SPG (50°N, 30°W; Fig. 4a) selected for three primary reasons:

1. The seasonal cycle is typical of the extratropics for all parameters, with maximum (minimum) values in the winter (summer) (Fig. 4b).

2. The point is away from regions of strong advection, such as the Gulf Stream and the SPG boundary current, allowing for focused analysis on the interplay between the three model parameters and entrainment without confounding variables.

3. Most importantly, it is at the region of maximum loading for the CESM-FULL AMV pattern, providing an opportunity to investigate if the processes included in the stochastic model are adequate to produce realistic low-frequency SST variability where the signal is strongest.

(a) The AMV SST pattern calculated from the CESM-FULL simulation from 20°–60°N, 80°–0°W (dashed box). The yellow star indicates the location of the subpolar gyre (SPG) point for the case study. (b) Seasonal cycles of inputs parameters for the stochastic model at the SPG point, including MLD (blue; from CESM-FULL) as well as the stochastic forcing amplitude (red) and atmospheric heat flux feedback strength (green; both estimated from CESM-SLAB).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) The AMV SST pattern calculated from the CESM-FULL simulation from 20°–60°N, 80°–0°W (dashed box). The yellow star indicates the location of the subpolar gyre (SPG) point for the case study. (b) Seasonal cycles of inputs parameters for the stochastic model at the SPG point, including MLD (blue; from CESM-FULL) as well as the stochastic forcing amplitude (red) and atmospheric heat flux feedback strength (green; both estimated from CESM-SLAB).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) The AMV SST pattern calculated from the CESM-FULL simulation from 20°–60°N, 80°–0°W (dashed box). The yellow star indicates the location of the subpolar gyre (SPG) point for the case study. (b) Seasonal cycles of inputs parameters for the stochastic model at the SPG point, including MLD (blue; from CESM-FULL) as well as the stochastic forcing amplitude (red) and atmospheric heat flux feedback strength (green; both estimated from CESM-SLAB).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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At the point, we integrate the stochastic model starting from January at a monthly step for 10 000 years at each level of the hierarchy. We then examine two aspects to capture how seasonal variation of parameters and entrainment impacts SST variability across time scales:

1. Seasonal-to-interannual persistence of SST using the monthly lagged autocorrelation where lag 0 is February, the month of the deepest MLD. This allows for evaluation of an end-member case where we expect persistence to be longest and facilitates detection of the re-emergence signal (Deser et al. 2003).

2. SST power spectra to examine variability at interannual-to-multidecadal time scales, with a focus on lower frequencies to understand the drivers of AMV.

1) Seasonal variation in forcing and damping

The simplest stochastic model with annually averaged parameters (Level 1 All Constant) produces exponentially decaying SST anomalies (blue line, Fig. 5a), a canonical behavior of the FH77 model. In contrast, autocorrelation at the same point in CESM-SLAB (gray line) exhibits slower decorrelation from April to October. This persistent “shoulder” feature is generated by the weaker stochastic forcing during summer while the heat capacity of the mixed-layer remains fixed due to the constant MLD, and is indeed seen in the stochastic model when the seasonal modulation of the stochastic forcing is included [Level 2b Vary F′ (cyan) and Level 3 Vary F′, λ_a (red lines)]. The inclusion of weaker summertime damping alone does not reproduce this feature (Level 2a Vary λ_a; yellow line), suggesting that seasonal variation in damping is secondary to that of forcing in driving seasonal-to-interannual SST persistence.

(a) SST autocorrelations for the stochastic model at the SPG point with a constant MLD of 54.6 m (consistent with CESM-SLAB) including cases where all parameters are constant (Level 1; blue); seasonal variation only in damping (Level 2a; gold) or forcing (Level 2b; cyan); seasonal variation in both (Level 3; red), and CESM-SLAB (gray) for comparison. The shading indicates the 95% confidence level for each correlation. (b) Autocorrelation for the stochastic models with increasing MLD complexity including seasonally varying MLD (Level 4; magenta), and with entrainment (Level 5; orange). For comparison, CESM-FULL (black) is also included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) SST autocorrelations for the stochastic model at the SPG point with a constant MLD of 54.6 m (consistent with CESM-SLAB) including cases where all parameters are constant (Level 1; blue); seasonal variation only in damping (Level 2a; gold) or forcing (Level 2b; cyan); seasonal variation in both (Level 3; red), and CESM-SLAB (gray) for comparison. The shading indicates the 95% confidence level for each correlation. (b) Autocorrelation for the stochastic models with increasing MLD complexity including seasonally varying MLD (Level 4; magenta), and with entrainment (Level 5; orange). For comparison, CESM-FULL (black) is also included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) SST autocorrelations for the stochastic model at the SPG point with a constant MLD of 54.6 m (consistent with CESM-SLAB) including cases where all parameters are constant (Level 1; blue); seasonal variation only in damping (Level 2a; gold) or forcing (Level 2b; cyan); seasonal variation in both (Level 3; red), and CESM-SLAB (gray) for comparison. The shading indicates the 95% confidence level for each correlation. (b) Autocorrelation for the stochastic models with increasing MLD complexity including seasonally varying MLD (Level 4; magenta), and with entrainment (Level 5; orange). For comparison, CESM-FULL (black) is also included.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Proceeding to longer time scales, all non-entraining stochastic models (Levels 1–4) have SST spectra that broadly resemble first-order autoregressive (AR1) processes without pronounced spectral peaks (Fig. 6a, and magenta curve in Fig. 6b). Adding seasonal variation in damping and forcing slightly increases the overall SST variance, particularly at frequencies between 5 and 20 years (Fig. 6a; Levels 2–3). This frequency dependence suggests that increased summertime persistence in slab-like configurations enhance SST variance beyond seasonal time scales. An important difference between the CESM-SLAB and the stochastic models is that the former includes two-way and nonlocal air–sea interactions, such as the thermally coupled Walker mode described by Clement et al. (2011). These nonlocal processes may provide additional remote forcing, creating larger deviations of the CESM-SLAB spectrum from the AR(1) model. Nevertheless, the total variances for the CESM-SLAB and the stochastic models with fixed MLD are broadly comparable (see the legend of Fig. 6a).

The SST power spectra at the SPG point for stochastic models for the hierarchy (a) Levels 1–3 (i.e., the non-entraining stochastic models with fixed MLD) and (b) Levels 4–5. The total SST variance for each case is indicated in the legend. Each spectral estimate was tapered by 10%, and smoothed with a modified Daniell window of 350 (stochastic model) or 100 (CESM) adjacent bands.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The SST power spectra at the SPG point for stochastic models for the hierarchy (a) Levels 1–3 (i.e., the non-entraining stochastic models with fixed MLD) and (b) Levels 4–5. The total SST variance for each case is indicated in the legend. Each spectral estimate was tapered by 10%, and smoothed with a modified Daniell window of 350 (stochastic model) or 100 (CESM) adjacent bands.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The SST power spectra at the SPG point for stochastic models for the hierarchy (a) Levels 1–3 (i.e., the non-entraining stochastic models with fixed MLD) and (b) Levels 4–5. The total SST variance for each case is indicated in the legend. Each spectral estimate was tapered by 10%, and smoothed with a modified Daniell window of 350 (stochastic model) or 100 (CESM) adjacent bands.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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b. Regional SST analysis

While seasonal MLD variation and entrainment are important processes for realistic representation of SST variability at the SPG point, their applicability over the broader North Atlantic requires further investigation. We expand our analysis by integrating the model for 10 000 years at each point across the Atlantic basin. Parameters estimated at each grid from the CESM-FULL (SLAB) are used for the entraining (non-entraining) stochastic model to facilitate comparison. The same set of white noise time series is used for all integrations, such that the resultant SSTs are driven by spatially coherent stochastic forcing.

We focus our analysis on three key regions of the AMV horseshoe pattern to examine their dynamical origins: the SPG maximum (40°–65°N, 60°–20°W), subtropical gyre minimum (STGw; 20°–40°N, 80°–40°W), and the lobe of elevated values to the east (STGe; 20°–40°N, 40°–10°W,) (Fig. 7a, inset). Autocorrelations and power spectra are computed from the area-weighted average SST over each region to assess if our conclusions apply beyond our test point. While this box-averaged approach undoubtedly conflates regions with very different dynamics, such as the Gulf Stream and subtropical gyre within STGw, our objective here is to obtain a broader understanding of how entrainment may contribute to subsections of the AMV pattern and large-scale, low-frequency SST behavior.

The (a)–(c) autocorrelations and (d)–(f) power spectra for SST anomalies averaged over (a),(d) the SPG, and the subtropical gyre (b),(e) west and (c),(f) east. The corresponding bounding boxes are shown in the inset of (a), over the AMV pattern from CESM-SLAB. The colors corresponding to each model configuration (Level 4 Vary h and Level 5 Entraining) are identical to previous figures.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The (a)–(c) autocorrelations and (d)–(f) power spectra for SST anomalies averaged over (a),(d) the SPG, and the subtropical gyre (b),(e) west and (c),(f) east. The corresponding bounding boxes are shown in the inset of (a), over the AMV pattern from CESM-SLAB. The colors corresponding to each model configuration (Level 4 Vary h and Level 5 Entraining) are identical to previous figures.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The (a)–(c) autocorrelations and (d)–(f) power spectra for SST anomalies averaged over (a),(d) the SPG, and the subtropical gyre (b),(e) west and (c),(f) east. The corresponding bounding boxes are shown in the inset of (a), over the AMV pattern from CESM-SLAB. The colors corresponding to each model configuration (Level 4 Vary h and Level 5 Entraining) are identical to previous figures.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The conclusions at our test point hold for simulated SSTs averaged over the larger SPG domain; entrainment damps SST variability, particularly at interannual-to-decadal frequencies, and its inclusion allows the stochastic model to approximate the SST autocorrelation in CESM-FULL (Figs. 7a,d). One notable difference is that entrainment enhances SPG-averaged SST variability at time scales greater than 20 years [Fig. 7d, Level 5 Entraining (orange) vs Level 4 Vary h (magenta lines)]. This suggests that at a regional scale, the reddening of the spectrum due to longer memory from re-emergence overpowers the entrainment-related damping. Despite this enhancement, the stochastic model’s continued underestimation of the CESM-FULL spectra at low frequencies underscores the importance of low-frequency ocean dynamics beyond entrainment.

Within STGw, damping associated with entrainment yields only modest improvements in modeling the CESM-FULL SST behavior at interannual time scales (Figs. 7b,e). At longer time scales, SST variance is overestimated (underestimated) at periods between 5 and 10 (>10) years. Comparison of spectra at individual grid points reveals that the discrepancy at periods > 10 years is not uniform throughout the domain, but primarily dominated by underestimates within the western Sargasso Sea and south of the Grand Banks that overpower overestimates along the Gulf Stream (not shown). These are regions where we expect a dominant role for horizontal advection, and their collocation with poorer stochastic model performance emphasizes how entrainment alone is insufficient to fully characterize SST behavior in this region (Figs. 7b,e).

The importance of ocean dynamics beyond entrainment becomes increasingly apparent in the STGe. Rebounds in SST autocorrelation associated with re-emergence are inconsistent with the behavior in CESM-FULL (Fig. 7c). Additionally, entrainment-related damping widens the disagreement with CESM-FULL, as the variance is underestimated at periods between 3 and 8 years (Fig. 7f), even if ENSO is removed from CESM-FULL (not shown). This suggests that missing processes from the stochastic model obscure the signal of reemergence and counteract entrainment-related damping of SST variability. One likely candidate is the subduction of anomalies in the seasonal thermocline into the ocean interior (Qiu and Huang 1995; Liu and Huang 2012), which may explain the similar lack of SST reemergence observed in this region (de Coëtlogon and Frankignoul 2003; Hanawa and Sugimoto 2004).

c. The AMV pattern

In this section, we investigate the ability of the stochastic model to reproduce the AMV pattern. We define the extratropical AMV index as the 10-yr low-pass filtered, area-weighted average of SST anomalies in 20°–60°N, 80°–0°W (section 4c). Including tropical points more than doubles the maximum AMV amplitude in the tropical Atlantic, but does not impact our conclusions (not shown). The AMV index is normalized and regressed back to the SST anomalies to obtain the AMV pattern ($°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$), representing the SST anomalies associated with typical AMV fluctuations (one standard deviation).

The stochastic model largely captures the spatial characteristics of AMV and its canonical horseshoe pattern (Fig. 8). Key features of CESM-SLAB pattern, including the maximum loading in the subpolar and tropical regions as well as the connection, albeit weaker, between the two centers along the eastern subtropics, are reproduced by the non-entraining stochastic model (Level 3 Vary F′, λ_a)—the level of hierarchy that corresponds best to CESM-SLAB (Fig. 8a versus Fig. 8c). Despite capturing the general features of the AMV pattern, the amplitude of its centers of action and the AMV index remain underestimated: the variance of the index in the non-entraining model is 43% of CESM-SLAB (cf. Figs. 8a and 8c). A possible source for this discrepancy is the absence of two-way, air–sea feedbacks, such as the wind–evaporation–SST (WES) feedback, which may potentially enhance low-frequency variability (Oelsmann et al. 2020).

Comparison of AMV patterns ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) from the stochastic models with (a) spatially varying, temporally constant h (i.e., non-entraining; Level 3 Vary F', λ_a) and (b) adding seasonally varying MLD and entrainment (Level 5 Entraining). The patterns from (c) CESM-SLAB and (d) CESM-FULL are also included. The variance of the AMV index for each case is indicated in the title. The stippled regions in (a) and (b) indicate where the estimated heat flux feedback is statistically insignificant, and is either set to 0 [in (a)] or replaced with the corresponding CESM-SLAB values [in (b)].

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Comparison of AMV patterns ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) from the stochastic models with (a) spatially varying, temporally constant h (i.e., non-entraining; Level 3 Vary F', λ_a) and (b) adding seasonally varying MLD and entrainment (Level 5 Entraining). The patterns from (c) CESM-SLAB and (d) CESM-FULL are also included. The variance of the AMV index for each case is indicated in the title. The stippled regions in (a) and (b) indicate where the estimated heat flux feedback is statistically insignificant, and is either set to 0 [in (a)] or replaced with the corresponding CESM-SLAB values [in (b)].

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Comparison of AMV patterns ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) from the stochastic models with (a) spatially varying, temporally constant h (i.e., non-entraining; Level 3 Vary F', λ_a) and (b) adding seasonally varying MLD and entrainment (Level 5 Entraining). The patterns from (c) CESM-SLAB and (d) CESM-FULL are also included. The variance of the AMV index for each case is indicated in the title. The stippled regions in (a) and (b) indicate where the estimated heat flux feedback is statistically insignificant, and is either set to 0 [in (a)] or replaced with the corresponding CESM-SLAB values [in (b)].

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The SST maximum in CESM-FULL AMV pattern is weaker than in CESM-SLAB and shifted eastward, away from the sea ice edge, with no change of sign in the western subtropical gyre (cf. Figs. 8c and 8d). The variance of the AMV index in CESM-FULL is 45% of that in CESM-SLAB, reflecting enhanced low-frequency variability in the latter. This comparison underscores how the inclusion of ocean dynamics both alters the spatial pattern and damps its magnitude of variability (Murphy et al. 2021; Patrizio and Thompson 2021).

Including entrainment in the stochastic model (Level 5 Entraining) replicates this behavior by damping the pattern and shifting the subpolar maximum eastward, although too far toward regions of large annual MLD range in the northeast Atlantic (cf. Figs. 8a and 8b). This results in considerable underestimates in the amplitude of low-frequency variability: the entraining model’s AMV index has 51% of the variance in CESM-FULL (cf. Figs. 8b and 8d). Thus, adding entrainment excessively damps SST, necessitating the inclusion of missing dynamics and nonlocal feedbacks, such as lateral advection or WES, to accurately represent the AMV amplitude and pattern.

Considering the importance of spatially coherent forcing in reproducing the AMV pattern, we explore if specific, leading modes of atmospheric variability play a dominant role. Originally, we forced the stochastic model by including enough EOF modes to explain 90% of the variance, with additional corrections to ensure a consistent amplitude. For this additional calculation, we force the stochastic model with only the first two EOFs of each month (which are the NAO and east Atlantic pattern (EAP), respectively), separately and in combination (i.e., EOF1 + EOF2), without variance correction. The first and second modes explain 26.47% and 17.02% of the F′ variance, respectively. We focus our analysis on the entraining case (Level 5) for comparison to CESM-FULL.

Forcing the stochastic model with dominant atmospheric modes alone, whether NAO-like or EAP-like, does not capture the canonical AMV pattern and underestimates its amplitude by an order of magnitude (Figs. 9a–c vs Figs. 8b,d). The first mode’s pattern instead resembles the SST tripole with a negative pole in the subtropics, as expected from NAO forcing (Cayan 1992; Deser et al. 2010). Additionally, the northernmost maximum is displaced too far north (south) for EOF 1 (EOF 2) compared to the CESM-FULL AMV pattern. Applying both modes combined only slightly improves the comparison against CESM-FULL AMV pattern, but the negative subtropical pole persists. Therefore, the higher modes of atmospheric forcing are important for reproducing a single-signed AMV pattern across the basin.

The AMV patterns ($°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model (Level 5) forced with (a) only EOF 1, (b) only EOF2, and (c) both EOFs of the CESM-FULL stochastic forcing (F'). The variance of the AMV index is indicated above each panel.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The AMV patterns ($°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model (Level 5) forced with (a) only EOF 1, (b) only EOF2, and (c) both EOFs of the CESM-FULL stochastic forcing (F'). The variance of the AMV index is indicated above each panel.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The AMV patterns ($°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model (Level 5) forced with (a) only EOF 1, (b) only EOF2, and (c) both EOFs of the CESM-FULL stochastic forcing (F'). The variance of the AMV index is indicated above each panel.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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a. The role of ocean dynamics

While our analyses focus primarily on the contribution of seasonal variation and entrainment to AMV, several implications emerge for the broader role of ocean dynamics in SST variability. Prior to defining this role, we first qualify what falls under the umbrella of ocean dynamics by adopting one of two frameworks.

First, we consider the seasonal mixed-layer cycle and entrainment mechanism as part of the ocean’s local, passive response as opposed to nonlocal, active ocean dynamics. Within this framework, our results suggest that active ocean dynamics are not essential for generating the spatial pattern of AMV. This agrees with Clement et al.’s (2015) statement that the AMV pattern arises from temporally stochastic atmospheric forcing, and further affirms the importance of spatial coherence in heat flux forcing. However, the stochastic model underestimates the amplitude of AMV relative to CESM1, leaving a role for the missing, active oceanic processes or nonlocal air–sea feedbacks.

A surprising result is the stochastic models’ ability to reproduce the SST persistence and spectra in the SPG, a region where active ocean dynamics, such as overturning circulation and horizontal gyre circulation, are thought to play an important role in SST variability (McCarthy et al. 2015; Piecuch et al. 2017; Zhang 2017). Our results indicate that the collocated deep MLD and weak atmospheric damping synergistically enhance the memory of SST anomalies in the SPG (Fig. 2). This enhanced memory due to local conditions may explain the ability of slab simulations to produce multidecadal variability with spatiotemporal characteristics resembling AMV in the fully coupled system, but ultimately driven by different dynamical processes (Garuba et al. 2018; Oelsmann et al. 2020).

However, the SST and AMV variance in SLAB remain overestimated relative to FULL. Previous works have interpreted this difference as indicating the net damping effect of ocean dynamics (Yamamoto et al. 2020; Murphy et al. 2021; Patrizio and Thompson 2021). These are illustrative of a second framework, where all processes missing from SLAB, including seasonal MLD variation and entrainment, are subsumed under the umbrella of ocean dynamics. Our results support these previous assessments up to interannual time scales and suggest that entrainment is the primary damping mechanism rather than advection or nonlocal ocean dynamics. The stochastic model further reveals a frequency-dependent effect, where entrainment enhances (damps) variance at multidecadal (interannual) time scales over key AMV regions. Determination of the contribution of ocean dynamics for AMV is thus sensitive to the inclusion of entrainment and MLD variations.

Ratios of regionally averaged SST spectra underscore the importance of entrainment for understanding SLAB versus FULL differences and potential contributions of nonlocal ocean dynamics (Fig. 10). Previous works have described the greater variance of SST in SLAB simulations relative to both fully coupled models and observations, particularly at interannual time scales in the extratropical Atlantic (Zhang 2017; Garuba et al. 2018; Oelsmann et al. 2020; Murphy et al. 2021; Patrizio and Thompson 2021). A notable exception where both configurations have comparable power is found at periods > 50 years in STGw, suggesting a substantial contribution of ocean dynamics in this region (Fig. 10a). Unlike CESM-SLAB, the entraining stochastic model’s spectra improve agreement with CESM-FULL, particularly at periods up to 20 years, highlighting the importance of including entrainment for representing SST behavior (Fig. 10b). A potential role of nonlocal ocean dynamics is suggested by the underestimates at periods > 20 years remains in the SPG and STGw. Further work is needed to identify the origin of additional of low-frequency variability, potentially through additionally levels in the stochastic model hierarchy that include nonlocal ocean processes.

The log ratio of regionally averaged SST spectra (a) between CESM-SLAB and FULL and (b) between the entraining stochastic model and CESM-FULL in the SPG, western subtropical gyre (STGw), and eastern subtropical gyre (STGe).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The log ratio of regionally averaged SST spectra (a) between CESM-SLAB and FULL and (b) between the entraining stochastic model and CESM-FULL in the SPG, western subtropical gyre (STGw), and eastern subtropical gyre (STGe).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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The log ratio of regionally averaged SST spectra (a) between CESM-SLAB and FULL and (b) between the entraining stochastic model and CESM-FULL in the SPG, western subtropical gyre (STGw), and eastern subtropical gyre (STGe).

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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b. Uncertainties in parameter estimation

Interpretation of our results should include clarification of uncertainties in parameter estimation. We estimated stochastic model parameters from CESM1, with the objective of applying the hierarchy to understand the AMV in SLAB and FULL simulations of this particular model. Differences in the relative importance of these parameters across other models and observations may exist, particularly considering model biases present in CESM1. For example, CESM1 has multidecadal SST variability that is weaker than observations (Kim et al. 2018a; Murphy et al. 2021), indicating that the discrepancy in low-frequency SST variance between the entraining stochastic model and observations is greater than with CESM-FULL. Assuming that the AMV in the limited observational record is robust, our results suggest that either the representation of entrainment, forcing, and damping processes in CESM-FULL is different from observations, or that additional oceanic processes are needed to enhance SST variance at low frequencies. Identifying the contributions of each source to this discrepancy would involve application of the stochastic model hierarchy to other models and observations, and is left to future work.

Considering the importance of the seasonal mixed-layer cycle for recovering realistic amplitudes of SST variability, we examine MLD biases in CESM1 relative to an observational estimate from the Monthly Isopycnal/Mixed-layer Ocean Climatology version 2.2 (MIMOC) (Schmidtko et al. 2013). The winter and spring MLD in CESM1 are largely overestimated around Greenland, north of 50°N (Fig. 11). A deeper bias increases the heat capacity of the mixed layer, weakening both forcing and damping, suggesting that both CESM1 and the stochastic model overestimate SST persistence in this region. In contrast, the shallow MLD bias in the interior of the SPG, a key area for the maxima of the AMV pattern, may potentially explain the lack of power at low frequencies in CESM1 relative to observations (Kim et al. 2018a; Murphy et al. 2021).

Seasonal mean differences in MLD [CESM-FULL minus Monthly Isopycnal/Mixed-layer Ocean Climatology version 2.2 (MIMOC)], where positive values indicate overestimated MLD in CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Seasonal mean differences in MLD [CESM-FULL minus Monthly Isopycnal/Mixed-layer Ocean Climatology version 2.2 (MIMOC)], where positive values indicate overestimated MLD in CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Seasonal mean differences in MLD [CESM-FULL minus Monthly Isopycnal/Mixed-layer Ocean Climatology version 2.2 (MIMOC)], where positive values indicate overestimated MLD in CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Of further relevance to the memory of SSTs is uncertainty in the heat flux feedback estimates. Low-frequency variability is sensitive to the strength of air–sea coupling, and fully coupled model simulations may have heat flux feedbacks that are too strong (Garuba et al. 2018). Our estimates of heat flux feedback in CESM-SLAB and FULL exhibit pronounced differences, where CESM-FULL has stronger damping over the SPG, leading to an underestimate of SST variance by the latter (Fig. 12). A possible explanation is the lack of interactive ocean advection in SLAB, which increases the local SST autocovariance in SLAB, or the denominator in the feedback estimates [Eq. (7)]. The resulting weaker heat flux feedback leads to unrealistically large SPG SSTs in CESM-SLAB. If we integrate the non-entraining stochastic model (Level 3) with FULL heat flux feedback values rather than those from SLAB, the AMV index variance is reduced by 25.7%, highlighting the sensitivity of AMV amplitude to heat flux feedback estimates (not shown).

Differences in seasonally averaged heat flux feedback between CESM-FULL and CESM-SLAB. Stippling indicates where either SLAB or FULL estimates are insignificant.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Differences in seasonally averaged heat flux feedback between CESM-FULL and CESM-SLAB. Stippling indicates where either SLAB or FULL estimates are insignificant.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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Differences in seasonally averaged heat flux feedback between CESM-FULL and CESM-SLAB. Stippling indicates where either SLAB or FULL estimates are insignificant.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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c. Investigating additional ocean processes and limitations

The addition of Q_ek to the entraining model (Level 5+) leads to increases in low-frequency SST variance in the SPG (Fig. 13b), but degrades performance at higher frequencies and in other regions (Fig. 13c vs Fig. 10b). The most noticeable impact is on the AMV pattern: the subpolar maximum is enhanced and shifted southward toward the region with the largest SST gradient near the North Atlantic Current (Fig. 13a). This suggests that Q_ek is a potential source of additional low-frequency variability and slightly improves the ability of the stochastic model to capture the spatial aspects of AMV in the subpolar region.

(a) AMV pattern ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model with Ekman forcing (Level 5+). Log ratios of regionally averaged SST spectra (b) between the entraining stochastic model with (Level 5+) and without (Level 5) Ekman forcing and (c) between the entraining stochastic model with Ekman forcing (Level 5+) with CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) AMV pattern ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model with Ekman forcing (Level 5+). Log ratios of regionally averaged SST spectra (b) between the entraining stochastic model with (Level 5+) and without (Level 5) Ekman forcing and (c) between the entraining stochastic model with Ekman forcing (Level 5+) with CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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(a) AMV pattern ($\text{contour}\hspace{0.17em}\text{interval}=0.025°\mathrm{C}{\sigma }_{\text{AMV}}^{-1}$) for the entraining stochastic model with Ekman forcing (Level 5+). Log ratios of regionally averaged SST spectra (b) between the entraining stochastic model with (Level 5+) and without (Level 5) Ekman forcing and (c) between the entraining stochastic model with Ekman forcing (Level 5+) with CESM-FULL.

Citation: Journal of Climate 36, 4; 10.1175/JCLI-D-22-0309.1

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An additional assumption to interrogate is the fixed seasonal cycle in MLD. Previous studies have noted that MLD anomalies, in concert with anomalous heat fluxes, act to warm the SPG region, playing an important role in modulating AMV (Yamamoto et al. 2020; Li et al. 2020). Interannual MLD variations could thus provide a source of enhanced variability at low frequencies, but could also impact the entrainment mechanism by preventing consistent re-emergence of wintertime anomalies, potentially reducing memory of wintertime SSTs (Buckley et al. 2019). Critical examination of MLD variability’s competing effects on the memory of SST anomalies is needed to constrain its impact on AMV.

As a low-resolution model, CESM1 has limitations in representations of ocean processes, such as biases in the Gulf Stream position and separation and reduced variance due to lack of eddies, as commonly found in other low-resolution climate models (Kirtman et al. 2012). Since the parameters of the stochastic model and inferred role of the ocean dynamics beyond entrainment depend upon CESM1, our results are limited to understanding only SST variability and AMV within this model. Further work involving analysis of higher-resolution models or observations is needed to discern the role of ocean processes misrepresented in CESM1. In addition to resolution dependence, repeating the study for other models can serve to evaluate intermodal consistency in the role of entrainment.