Estimating Ocean Heat Uptake Using Boundary Green's Functions: A Perfect-Model Test of the Method

Abstract

Ocean heat uptake is caused by “excess heat” being added to the ocean surface by air-sea fluxes and then carried to depths by ocean transports. One way to estimate excess heat in the ocean is to propagate observed sea surface temperature (SST) anomalies downward using a Green's function (GF) representation of ocean transports. Taking a “perfect-model” approach, we test this GF method using a historical simulation, in which the true excess heat is diagnosed. We derive GFs from two approaches: (a) simulating GFs using idealized tracers, and (b) inferring GFs from simulated CFCs and climatological tracers. In the model world, we find that combining simulated GFs with SST anomalies reconstructs the Indo-Pacific excess heat with a root-mean-square error of 26% for depth-integrated changes; the corresponding number is 34% for inferred GFs. Simulated GFs are inaccurate because they are coarse grained in space and time to reduce computational cost. Inferred GFs are inaccurate because observations are insufficient constraints. Both kinds of GFs neglect the slowdown of the North Atlantic heat uptake as the ocean warms up. SST boundary conditions contain redistributive cooling in the Southern Ocean, which causes an underestimate of heat uptake there. All these errors are of comparable magnitude, and tend to compensate each other partially. Inferred excess heat is not sensitive to: (a) small changes in the shape of prior GFs, or (b) additional constraints from SF₆ and bomb ¹⁴C.

Key Points

• Green's functions (GFs) broadly reconstruct simulated ocean heat uptake when constrained by simulated observations in a historical simulation

• Errors in the GF method arise from insufficient observational constraints and forced changes in ocean transports

• GFs underestimate the Southern Ocean heat uptake when using sea surface temperature anomalies as boundary conditions

Plain Language Summary

Ocean warming is caused by “excess heat” being added to the ocean surface by air-sea fluxes and then carried to depths by ocean currents. Tracking global ocean warming is important for monitoring climate change. However, a substantial amount of ocean warming occurs at depths, where temperature measurements are scarce. A workaround is to treat well-observed surface ocean warming as a heat source, and propagate it downward using ocean currents. This method of estimating the interior ocean warming is called the Green's function (GF) method. But how accurate is the GF method? Here, we address this question by treating a computer simulation of the historical ocean as the real world, and comparing simulated ocean warming (as the “truth”) with that estimated using the GF method. We find that the GF method broadly reconstructs simulated ocean warming. The results contain some inaccuracies because: (a) neither computer simulations nor observations give an accurate estimate of ocean currents, and (b) part of the surface temperature changes cannot be treated as a source of the interior ocean warming.

1 Introduction

Imbalance in Earth's top-of-atmosphere radiative forcing leads to accumulation of “excess heat” in the climate system. Over 93% of the excess heat is stored in the ocean, causing ocean warming and sea-level rise (Meyssignac et al., 2019). Excess heat invades the ocean from the surface, like a drop of dye spreads in a water tank. This process can be conveniently described using a mathematical tool called Green's functions (GFs). Here, we examine how accurately excess heat in the ocean can be estimated using GFs.

A change in ocean heat content can be understood in terms of excess and redistributed heat content. Excess heat is defined as the change (warming or cooling) that is added to the ocean by air-sea fluxes, and then carried to depths by ocean transports. Redistributed heat, on the other hand, is defined as the change of the pre-existing heat in the ocean (i.e., spatial redistribution). Isolating excess heat is useful because: (a) excess heat change can be constrained by observations of transient tracers in the ocean (see Section 6), and (b) excess heat change dominates global/basin integrated ocean heat content change. Under CO₂ forcing, climate models show that excess heat largely accumulates in the North Atlantic and the Southern Ocean, while redistributed heat tends to accumulate at low latitudes (Gregory et al., 2016; Newsom et al., 2022). Excess and redistributed heat are both theoretical constructs; neither of them is directly observable in the ocean.

Excess heat at depths can be estimated by propagating its surface “source” downward using boundary GFs of the tracer equation (Holzer & Hall, 2000). We refer to this method as the GF method. The source or boundary condition (BC) of excess heat is often computed from observed sea surface temperature (SST) in the literature (e.g., Messias & Mercier, 2022; Zanna et al., 2019). Boundary GFs represent the ocean's surface-to-interior transport. They can be derived from: (a) simulating idealized tracers in a model (e.g., Khatiwala et al., 2005; Zanna et al., 2019) or (b) solving an inverse problem using tracer observations (e.g., Holzer et al., 2010; Khatiwala et al., 2009). The GF method adds useful information to the in-situ estimate of ocean heat content change. The two estimates are directly comparable for basin integrals; regionally, their differences indicate redistributed heat in the ocean.

We refer to GFs derived from model simulations as “simulated GFs,” and GFs inferred from tracer observations as “inferred GFs.” In practice, both types of GFs are at best approximations of the real-world ocean transport, due to various assumptions, simplifications and trade-offs. Simulated GFs are often coarse grained in space and time, hence they do not fully capture the covariance between the true GFs and surface BCs. In addition, simulated GFs rely on a model's ocean transports, but no model is perfect. Inferred GFs, on the other hand, do not rely on a model; but they too are inaccurate, because observations are insufficient constraints.

As well as the GFs, surface BCs are not perfectly known for estimating excess heat. SST anomalies, as used by Zanna et al. (2019), are contaminated by redistributed temperatures which are not BCs of excess heat. This error affects both types of GFs.

The accuracy of the GF method for estimating excess heat in the ocean has not been examined in the literature. In this study, we address this problem using a HadCM3 historical simulation (1860–2008). We treat this simulation as the real world, and compare excess heat diagnosed in it (as the “truth”) with that estimated using simulated/inferred GFs. This approach is useful because it allows a separation of excess and redistributed heat and a quantification of different errors, both of which are not accessible in observations.

Because our historical simulation agrees well with observations for large-scale ocean heat uptake, our error estimates are relevant to applying the GF method to the real world. Importantly, our result pinpoints the main error sources in the GF method, and provides a quantitative benchmark for each of them. Nonetheless, we expect that at least some of our error estimates are HadCM3 specific, especially since HadCM3 is a coarse resolution model and not constrained by observations. Future studies with high-resolution models or ocean state estimates would be useful to provide a more robust error estimate.

Setup of the HadCM3 historical simulation and definitions of excess and redistributed heat are explained in Section 2. In Section 3, we explain how to solve the passive tracer equation using GFs. Section 4 explains the method of simulating GFs. Section 5 evaluates excess heat estimates based on simulated GFs. The same pattern is repeated in Sections 6 and 7, but for inferred GFs. Finally, a summary is given in Section 8 and discussions in Section 9.

2 Historical Simulation and Temperature Tracers

2.1 Setup of Historical Simulation

HadCM3 is an Atmosphere-Ocean General Circulation Model (AOGCM) that has been used extensively for climate studies (Gordon et al., 2000). The HadCM3 atmosphere model is based on the UK Met Office Unified Model, with a horizontal resolution of 2.5° × 3.75° and 19 vertical layers. The HadCM3 ocean model is based on the Cox (1984) model with a horizontal resolution of 1.25° × 1.25° and 20 vertical levels (vertical resolution is enhanced near the surface). Horizontal eddy mixing of tracers in the HadCM3 ocean is parameterized using the Gent and Mcwilliams (1990) and Redi (1982) schemes.

We run a pre-industrial control experiment and a historical experiment in parallel with HadCM3. Both experiments start from a pre-industrial state at 1860 and run to 2008. (This choice omits the ocean's slow response to the global cooling before 1860 cf., Gebbie and Huybers (2019).) The historical experiment is conducted by adding historical effective radiative forcing Q_ERF (space and time dependent) to the sea-water surface. To compute Q_ERF the atmosphere model ECHAM6.3 (Giorgetta et al., 2013) is forced with time-dependent historical changes in all forcing agents and fixed pre-industrial SSTs and sea ice concentrations, following the design of the piClim-histall experiment (Pincus et al., 2016). This ECHAM6.3 simulation is used in Gregory et al. (2020) to compute the global-mean Q_ERF. Note that we choose to force HadCM3 with Q_ERF instead of adding forcing agents to its atmosphere; Appendix A explains the motivation for this choice.

2.2 Evolution Equations of Temperature Tracers

2.2.1 Historical and Control Temperatures

Evolution of ocean potential temperature Θ in the control and historical experiments can, in general, be written as

(1)

Θ₀ is the pre-industrial state at 1860. Ψ is the source/sink of Θ; it is zero everywhere except at the surface (ignoring geothermal heat flux). Φ is the ocean transport operator that evolves an ocean tracer field forward in time. A simple form of Φ can be given as

(2)

χ is the concentration of a tracer. v is a 3D velocity vector. κ_δ and κ_z are isopycnal and vertical eddy diffusivities, respectively. ∇_δ computes the lateral gradient of a scalar on isopycnal surfaces. Note that the Φ operator in modern ocean models is more complex than Equation 2.

The control and historical Θ fields are different because the two experiments have different Φ and Ψ. The ocean transport operator Φ is different because global warming affects ocean transports in many ways; for example, a reduction in high-latitude convection. The surface source Ψ = Q_ctrl/(ρ₀c_pdz₁) in the control experiment, where Q_ctrl is the net surface heat flux (W m⁻²) under the pre-industrial condition. In the historical experiment Ψ = (Q_ctrl + Q_ERF + Q′)/(ρ₀c_pdz₁); the two additional terms come from: (a) the historical forcing (Q_ERF) and (b) climate feedbacks (Q′) in response to the forcing. ρ₀c_pdz₁ is the top layer thermal inertia (J K⁻¹ m⁻²), wherein ρ₀ is reference density, c_p specific heat capacity, and dz₁ top layer thickness.

2.2.2 Linear Equations of Temperature Evolution

Φ is a non-linear operator when applied to Θ, because Φ itself depends on Θ; for instance, v in Equation 2 is a function of Θ. To facilitate a linear decomposition of temperature change, we define two linear versions of Φ, denoted as L_ctrl and L_hist, using velocities and diffusivities from the control and historical experiments, respectively. Unlike the Φ operator, which is a function of ocean states, the L operator is a pre-computed quantity, for example, an array of coefficients. The evolution equation of Θ in the control experiment can be rewritten as

(3)

Similarly, the evolution equation of Θ in the historical experiment can be rewritten as

(4)

It is important to note that L_ctrl and L_hist are linear operators when applied to any tracer fields. For instance, we have L_hist(Θ_hist) − L_hist(Θ_ctrl) = L_hist(Θ_hist − Θ_ctrl). The same does not hold for the Φ operator because it is nonlinear in Θ. In Section 2.2.3, we will use the linearity of L_hist to derive the governing equation of redistributed temperature. L_ctrl and L_hist both have time-varying coefficients due to variability and change in ocean transports.

2.2.3 Excess and Redistributed Temperatures

Temperature anomaly in the historical simulation relative to the control can be written as the sum of two passive tracers, Θ_e and Θ_r

(5)

The excess temperature Θ_e is the part of Θ_a driven by changes in surface heat fluxes (Equations 3 and 4 have different right-hand-side forcing terms). Its evolution equation is defined as

(6)

We implement Equation 6 by simulating Θ_e as a passive tracer in the historical simulation from 1860 to 2008. The redistributed temperature Θ_r is the part of Θ_a driven by changes in ocean transports (Equations 3 and 4 have different transport operators). Its evolution equation can be derived by combining Equations 3-6 and making use of the linearity of L_hist.

(7)

Note that changes in ocean transports acting on Θ_ctrl are the source term of Θ_r. For convenience, we compute Θ_r as Θ_a − Θ_e (Equation 5) instead of using Equation 7. In the historical simulation, Θ_e and Θ_r are both affected by unforced variability and forced climate change. We focus on multi-decadal changes in Θ_e and Θ_r to highlight the role of forced response.

The key difference between Θ_e and Θ_r is that Θ_e only comes from the surface, while Θ_r has sources throughout the volume of the ocean. The global volume integral of Θ_r is zero, because the effect of L integrates to zero over the global ocean. (Θ_r defined in Gregory et al. (2016) does not have exactly zero volume integral.)

2.2.4 Converting Temperature to Heat Content

We compute ocean heat content anomaly as . r is a 3D position vector of the ocean, an arbitrary control volume, and ρ₀c_p ≡ 4 × 10⁶ J K⁻¹ m⁻³. Applying the same procedure to Θ_e and Θ_r results in excess heat content and redistributed heat content , respectively .

2.3 Evaluation of Historical Simulation

The HadCM3 historical simulation captures the surface and depth integrated ocean warming in observations reasonably well (Figure 1). The global mean SST in HadCM3 generally follows that in HadISST (Rayner et al., 2003), but it does not capture the early 21st century warming hiatus in HadISST (Figure 1a). HadCM3 also tends to overestimate the surface cooling after volcanic eruptions compared to observations (Figure 1a); this is a common feature among CMIP5 models (D. M. Smith et al., 2016; Marotzke & Forster, 2015). In both HadCM3 and observations of Cheng et al. (2017), the global integrated (0–2,000 m) increases by about 300 ZJ in 2008 relative to 1946–1955 (1 ZJ = 1 × 10²¹ J); more than half of that is stored in the Indo-Pacific (Figures 1b–1d). HadCM3 does not capture the observed plateauing of increase after the 1963 Mount Agung eruption (Figures 1b and 1c).

Figure 1

Surface and depth integrated ocean warming in the HadCM3 historical simulation. (a) Global averaged sea surface temperature. (b) Global integrated ocean heat content. Panels (c and d) are the same as panel (b), but for basin integrals. Global and basin integrals are calculated for the 0–2,000 m layers. All quantities are shown as anomalies relative to the 1946–1955 average. For comparison, observational estimates from Rayner et al. (2003) (HadISST) and Cheng et al. (2017) (heat content) are also plotted. 1 ZJ = 1 × 10²¹ J.

We compare and simulated in HadCM3 with those in Bronselaer and Zanna (2020) (BZ2020). BZ2020 infers by scaling the pattern of anthropogenic carbon in the ocean; is then derived by subtracting inferred from observed change.

HadCM3 and BZ2020 have similar patterns of changes during 1951–2008 (Figure 2 left column, changes are integrated over 0–2,000 m). In both of them, tends to accumulate in the subtropical gyres and the North Atlantic, but features little signal at low latitudes. HadCM3 has larger changes than BZ2020 in the North Atlantic and the Arctic. This is partly due to different definitions of . The of BZ2020 is defined in a fixed-circulation scenario, which has a smaller Q_ERF + Q′, hence a smaller , than a free-circulation scenario (i.e., the HadCM3 simulation) at northern high latitudes (Winton et al., 2013). is less coherent in space than in both HadCM3 and BZ2020; the two data sets show very different changes in the subpolar North Atlantic and the Southern Ocean (Figure 2, right column).

Figure 2

Linear trends of excess and redistributed heat content (0–2,000 m integrated) during the 1951–2008 period. (a and b) Bronselaer and Zanna (2020) (based on anthropogenic carbon). (c and d) HadCM3 historical simulation. 1 GJ = 1 × 10⁹ J.

2.4 Heat Uptake by Control Ocean Transport

How important is the control ocean transport L_ctrl in shaping the regional pattern of ? We investigate this question using the pseudo excess temperature

(8)

is evolved by the control transport L_ctrl, as if heat uptake does not affect ocean transports. This is in contrast with Θ_e which is evolved by the historical transport L_hist. We implement Equation 8 by simulating as a passive tracer in the control experiment from 1860 to 2008. Pseudo excess heat content is denoted as .

Changes of , and in 1999–2008 relative 1946–1955 are shown in Figure 3 (a change is denoted as “Δ”). is much less uniform than and across latitudes (Figures 3a and 3b), highlighting the role of in shaping the patterns of . A similar result is found in Zika et al. (2021) but on a shorter timescale (2006–2017).

Figure 3

Excess heat content change resulting from (1) historical and (2) control ocean transports. These two quantities are denoted as (red line) and (blue line), respectively. Total heat content change ( = +) is shown as black lines. (a and b) Zonal-and-depth integrated change (0–2,000 m). (c and f) Depth distribution of panels (a and b). A change is calculated as the difference between 1999–2008 and 1946–1955. In panels (e and f), contours indicate and ; shading indicates minus . Contour levels are 10, 30, and 45 in panel (c); 5, 15, and 25 in panel (d); 10, 20, 30, 50, and 70 in panel (e); and 5, 10, 15, and 25 in panel (f). 1 PJ = 1 × 10¹⁵ J. 1 TJ = 1 × 10¹² J.

The latitude distributions of and are very similar, especially in the southern subtropics (Figure 3, compare red and blue lines). This suggests that the patterns of is mostly driven by the climatological ocean transport (i.e., L_ctrl), not its transient response (i.e., differences between L_hist and L_ctrl). A similar conclusion was found in several climate models under 1% increase of the atmospheric CO₂ concentration (Couldrey et al., 2021; Gregory et al., 2016). Differences between and are most evident at northern mid latitudes, where is redistributed equatorward relative to in 0–200 m (Figures 3e and 3f shading). This redistribution pattern implies a weakening of the poleward ocean transport in the historical simulation.

3 Formulating Tracer Evolution Using Green's Functions

Evolution of a passive tracer in the historical simulation is governed by

(9)

r is a 3D position vector of the ocean and t is time. X is the concentration of a passive tracer and Ψ is its 3D source/sink. All tracers studied here have no source/sink in the ocean interior, therefore we set Ψ to zero everywhere below the surface. At the surface, , where is air-sea tracer fluxes, and dz₁ the top layer thickness. ( has a unit of K m s⁻¹ for Θ_e.)

3.1 Concentration GF Formulation

The general solution of Equation 9 is given in Holzer and Hall (2000). When X has zero initial conditions, the solution of Equation 9 can be written as a superposition of all tracer pulses emitted from the surface. For concentration BCs, the superposition is given as

(10)

X^s is X at the surface. r_s is a surface position vector. G_c is the GF of Equation 9 that propagates concentration BCs. Ω denotes the global ocean surface. The two integrals sum up X(r, t) emitted from X^s anywhere in Ω and anytime prior to t.

Formally G_c is defined as a special solution of Equation 9 that satisfies

(11)

δ is the Dirac delta function.

3.2 Interpretations of Concentration GFs

G_c can be interpreted from two perspectives. When we fix the surface coordinate (r_s, t_s), G_c(r, t) is a time-evolving 3D field in the ocean. The 3D field depicts how a tracer injected at (r_s, t_s) spreads in the ocean subject to zero concentration BCs at all other times and surface locations. The BCs remove any tracer that surfaces after t_s, hence we have , where τ = t − t_s is elapsed time. This perspective is useful for probing GFs from forward simulations in an ocean model (see Section 4).

When we fix the field coordinate (r, t), G_c(r_s, t_s) is a time-evolving 2D map of the ocean surface. It shows how sensitive X(r, t) is to individual pulses in its surface history X^s. Holzer and Hall (2000) interpreted the 2D map as a measure of how a tracer injected at (r, t) surfaces in the time-reversed flow after t − t_s. This perspective is useful for inferring GFs from tracer data (see Section 6). Causality requires that G_c = 0 whenever t <t_s.

G_c(r, t | r_s, t_s) is also referred to as a “joint water-mass composition and transit-time distribution” which measures the fraction of water at (r, t) that has made its last surface contact at location r_s and time t_s (Haine & Hall, 2002). Since all the water at (r, t) can be traced back to the surface eventually, the following must be satisfied.

(12)

G_c has been used to study the transit-time distribution of the ocean (e.g., Ito & Wang, 2017; Maltrud et al., 2010; Peacock & Maltrud, 2006) and to estimate the ocean's uptake of anthropogenic carbon and heat (Gebbie & Huybers, 2019; Khatiwala et al., 2009; Newsom et al., 2020; Zanna et al., 2019).

3.3 Air-Sea Flux GF Formulation

The solution of Equation 9 can also be written in terms of the air-sea tracer flux , when X has zero initial conditions.

(13)

G_f is the boundary GF that propagates the surface source/sink into the ocean. Formally G_f is defined as a special solution of Equation 9 that satisfies

(14)

Note that Equation 14 does not impose zero concentration BCs as Equation 11, hence tracers are not removed when they surface. G_f has been used to probe atmosphere transports (Holzer, 1999) and to define a tracer age (Holzer & Hall, 2000).

3.4 Limitation of Boundary GFs

We want to stress that G_c and G_f are both boundary GFs; that is they only account for tracers emitted from the surface. The redistributed temperature Θ_r cannot be accounted for using G_c or G_f because it has non-zero source below the surface (Equation 7).

4 Simulating GFs in an Ocean Model

4.1 Approximations of Simulated GFs

The boundary GFs, G_c and G_f, can be generated for an ocean model by simulating passive tracers in it. By definition, we need to compute a GF for every possible (r_s, t_s), which is computationally demanding. To reduce computational cost, we make the following approximations.

First, we assume that ocean transports are constant. Taking G_c as an example, this assumption means: (a) G_c is the same for L_ctrl and L_hist, (b) G_c does not depend on t_s, hence G_c(r, t | r_s, t_s) = G_c(r, t − t_s | r_s, 0). Note that G_c(r, t − t_s | r_s, 0) only needs to be solved once (at t_s = 0) for every r_s. The constant-transport assumption neglects variability and forced-change in ocean transports; we refer to the resulting errors as a “unforced-transport error” and a “forced-transport error,” respectively.

Second, we assume that the boundary terms, X^s and , are dominated by large-scale patterns, hence we can approximate tracers emitted from them using coarse-grained GFs. Specifically, we derive GFs using surface patches defined in Figure 4. For G_c, we divide the global ocean into 27 regions based on the climatological surface densities in HadCM3, similar to Khatiwala et al. (2009). For G_f, we divide the global ocean into 20° latitude bands for each basin (20 patches in total). This step greatly reduces the dimension of GFs at the surface (the dimension of r_s is about 1 × 10⁴ in a 1° × 1° model). X^s and are averaged onto the corresponding patches when convolved with GFs.

Figure 4

Surface patches for simulating boundary Green's functions G_c and G_f. Shading indicates the patch index. G_c propagates concentration boundary conditions, while G_f propagates surface sources/sinks.

Finally, we approximate the Dirac delta function in Equations 11 and 14 using a boxcar (rectangular) function with a unit height. The boxcar function lasts for 1 year after it is activated, so that the resulting GFs capture the effect of ocean transports averaged over a year, not that of a particular month. Using surface patches and the boxcar function neglects the covariance between GFs and the boundary terms within patches and years. We refer to this error as a “patch error.”

4.2 Defining Simulated GFs

GFs resulting from the above approximations are referred to as simulated GFs. Formally simulated concentration GF G_c (denoted as ) is defined as

(15)

The boxcar function M_t(t, 0) is activated when t > 0 and switched off after 1 year. t = 0 is assigned to the beginning of the control experiment. M_c is a mask function that returns unity if r is within the surface patch p and zero otherwise. Ω denotes the global ocean surface. Similarly, simulated air-sea flux GF G_f (denoted as ) is defined as

(16)

M_f is a different mask function than M_c, because we use different surface patches to define and (Figure 4). and are solved by integrating Equations 15 and 16, respectively, in the control experiment for 200 years (from 1860).

4.3 Estimating Tracers Using Simulated GFs

Simulated GFs can be used to compute the boundary response of a passive tracer following

(17)

(18)

All quantities in Equations 17 and 18 are defined on a yearly grid. and are X^s and averaged onto the corresponding patches, respectively. p is surface patch index and l is year number. To reduce the unforced-transport error, we simulate GFs from four initial conditions (1860, 1870, 1880, and 1890 of the control experiment), and use their ensemble means (denoted by “[ ]”) in Equations 17 and 18.

We introduce as a shorthand for . Using this notation, the estimate of Θ_e can be written as (substitute Θ_e for X in Equation 17), where is Θ_e at the surface averaged onto patches. Similarly, the estimate of can be written as .

4.4 Error Definitions

is inaccurate because of the patch, unforced-transport and forced-transport errors. We quantify these errors using Equations 19-21. Because is not affected by forced change in ocean transports, we use it to isolate the patch and unforced-transport errors. We note that different members of give similar estimates of for large-scale changes examined in Section 5; the spread is <20% compared to the ensemble mean in most regions (not shown). Similarly, Maltrud et al. (2010) suggested that the first moment of (i.e., mean age) can be robustly estimated using just a few members. We therefore assume that the estimate of , that is , is dominated by the patch error (Equation 19, without “^” is the model truth). The change of the estimate resulting from replacing with (a single realization) gives the unforced-transport error (Equation 20). A larger ensemble of is useful to refine the unforced-transport error; however, it is unlikely to change the error substantially (e.g., by a factor of two).

(19)

(20)

Because Θ_e is evolved by the historical ocean transport, we use it to compute the forced-transport error. The estimate of Θ_e, that is , is affected by the patch and forced-transport error; the patch error is solved in Equation 19, therefore the forced-transport error is given as Equation 21.

(21)

In practice, there is also a “BC error” when estimating Θ_e using . This is because surface excess temperature is poorly known. Using SST anomaly to approximate is not accurate because is contaminated by redistributed temperature. We compute the BC error by replacing with in the estimate (Equation 22).

(22)

We compute the errors for the estimate (Equation 18) in a similar way as for the estimate (Equation 17). The only difference is that the boundary term is the same for and Θ_e in the estimate by definition.

4.5 Surface Concentration BCs

, and are supplied as anomalies relative to 1860–1880 when evaluating Equation 17. This step is to exclude a shock in (+0.15°C in global mean) shortly after the start of the historical simulation. Because does not show a similar behavior, we suspect that the shock is due to an abrupt change in ocean transports. If not removed, the shock would cause a warm bias in the estimate of Θ_e, because is derived from the control experiment. Since this warm bias can be removed easily, we do not count it as a forced-transport error.

A comparison between , and is shown in Figure 5 for the 1999–2008 average. (black line) consistently has less warming than (red line) in the Southern Ocean and the North Atlantic (north of 40°N) by as much as 2°C (Figures 5a and 5b). This difference is likely caused by a reduction of convection and a slow down of the Atlantic meridional overturning circulation, as shown in Gregory et al. (2016).

Figure 5

Surface temperature anomaly and surface excess temperature ( and ). is the same as except that it is evolved by the control ocean transport instead of the historical one. Values shown are differences between 1999–2008 and 1860–1880. (a and b) Zonal average over a basin. (c and d) Spatial map.

(red line) and (blue line) are very similar at most latitudes (Figures 5a and 5b). The exception is the North Pacific and the North Atlantic, where is much warmer than . This implies a reduction of the ocean's surface-to-interior transport in those regions during the historical simulation (because global warming stratifies the ocean and thus inhibits heat uptake).

4.6 Potential Nonlinear Errors

Equation 19 assumes that the function Φ is strictly linear when operating on passive tracers in models. This is not necessarily true because some models use flux-limited transport schemes, which makes Φ nonlinear even when operating on passive tracers. This nonlinear error is included in the error computed from Equation 19, but it is likely small compared to the patch error.

Excess temperature Θ_e can alternatively be defined as a dynamical tracer that affects ocean transports (i.e., replacing L_hist with Φ in Equation 6). This definition leads to a set of “dynamical” GFs, as opposed to “passive” GFs of Section 3 (their distinctions are further discussed in Appendix B). The GF estimate of the dynamical Θ_e (Equation B1) contains a nonlinear error because the dynamical ocean response is not a linear function of the forcing. In contrast, the GF estimate of the passive Θ_e (Equation 6) does not have a nonlinear error, because Equation 6 is strictly linear. The errors introduced in Section 4.1 can all be eliminated by simulating GFs for Equation 6 at very fine space and time resolution. The nonlinear error, however, cannot be eliminated by any means.

5 Estimating Excess Heat Using Simulated GFs

In this section, we examine how well simulated GFs can reproduce excess heat changes in the historical simulation. The inaccuracy is partitioned into the patch, unforced-transport, forced-transport and BC errors (Equations 19-22).

We focus on three metrics when comparing the model truth with the GF estimates: (a) global/basin volume integral (0–2,000 m), (b) zonal-and-depth integral (0–2,000 m), and (c) depth distribution of (b) (0–1500 m). Metric (a) is shown as anomalies relative to 1946–1955. Metrics (b) and (c) are shown as changes between 1999–2008 and 1946–1955 (denoted using “Δ”). The root-mean-square error (RMSE) of the GF estimate (total error) and the RMS value for each error source (Equations 19-22) are listed in Figures 6-8. Each realization of GFs gives an unforced-transport error; we report the unforced-transport error averaged over four realizations. Since our metrics are all extensive quantities, we also report their normalized RMSEs; that is the ratio between RMSE and root-mean-square magnitude (RMSM).

Figure 6

Estimating the global/basin integrated (0–2,000 m) excess heat and in the historical simulation using simulated Green's functions (GFs) (Sections 5.1 and 5.2). Black and gray lines show and in HadCM3, respectively. is the same as except that it is evolved by the control ocean transport. Blue and green lines are the estimates of and , respectively. The estimates of and are identical and both shown by red lines. The root-mean-square magnitude (RMSM) of the model truth and the root-mean-square errors (RMSEs) of different GF estimates are listed. The RMS values of the patch, unforced-transport and forced-transport errors are listed for the and estimates from left to right. The two estimates have different RMSEs because and are different in the model truth.

Figure 7

Estimating latitude distribution of excess heat change and in the historical simulation using (Section 5.1). (a and b) Zonal-and-depth integral (0–2,000 m). (c–f) Depth distribution of panels (a and b). In all panels, black and gray lines show and in HadCM3, respectively; blue and green lines show the estimates of and , respectively. Shading in panels (c and d) indicates errors in the estimate of (the patch error). Shading in panels (e and f) indicates errors in the estimate of minus the patch error (the forced-transport error). For each metric, the root-mean-square magnitude (RMSM) of the model truth and the root-mean-square error (RMSE) of the estimate are listed, along with the RMS values of the patch, unforced-transport and forced-transport errors. All changes are calculated as differences between 1999–2008 and 1946–1955.

Figure 8

Estimating latitude distribution of excess heat change and in the historical simulation using (Section 5.2). (a and b) Zonal-and-depth integral (0–2,000 m). (c–f) Depth distribution of panels (a and b). In all panels, black and gray lines show and in HadCM3, respectively; the estimates of and are identical, and shown by red lines. Shading in panels (c and d) indicates errors in the estimate of (the patch error). Shading in panels (e and f) indicates errors in the estimate of minus the patch error (the forced-transport error). For each metric, the root-mean-square magnitude (RMSM) of the model truth and the root-mean-square error (RMSE) of the estimate are listed, along with the RMS values of the patch, unforced-transport and forced-transport errors. The two estimates have different RMSEs because and are different in the model truth. All changes are calculated as differences between 1999–2008 and 1946–1955.

5.1 Concentration GF Estimate

In this subsection, we evaluate the GF estimate of and based on Equation 17 (referred to as the estimate). is the same as except that it is evolved by the control ocean transport (see Section 2.4). is simulated in the control experiment. The BCs and are diagnosed in HadCM3 (i.e., BCs are perfectly known). Θ_e and are converted to excess heat and , respectively, following the procedure of Section 2.2.4. We exclude and resulting from the Arctic patch to be consistent with Zanna et al. (2019).

5.1.1 Patch and Unforced-Transport Errors

We start with . The estimate of is inaccurate because of the patch error. For all metrics, the RMS value of the unforced-transport error is less than 1/3 of the patch error (compare numbers in Figures 6 and 7). The global integrated (black line) increases by about 300 ZJ over 1860–2008, of which two thirds are stored in the Indo-Pacific and one third in the Atlantic (Figure 6). For this metric, the estimate (blue line) reproduces the model truth well, with a RMSE of 16 ZJ for the global integral and 8 ZJ for basin integrals (Figure 6).

The zonal-and-depth integrated (black line) has a RMSM of 13.6 PJ m⁻¹ in the Indo-Pacific and 6.2 PJ m⁻¹ in the Atlantic, averaged over latitudes (Figures 7a and 7b). The estimate of this metric (blue line) has an error of 24% in the Indo-Pacific and 42% in the Atlantic (Figures 7a and 7b). The patch error is most evident in the North Atlantic (underestimate) and in the Southern Ocean (overestimate) (Figures 7a and 7b, compare black and blue lines).

The latitude-depth pattern of (black contour) has a RMSM of 14.9 TJ m⁻² in the Indo-Pacific and 5.2 TJ m⁻² in the Atlantic, averaged over latitudes and 0–1,500 m (Figures 7c and 7d). The estimate of this metric (blue contour) has an error of 26% in the Indo-Pacific and 44% in the Atlantic (Figures 7c and 7d). The patch error is strongest in the upper 200 m (Figures 7c and 7d shading). Below that, the estimate follows the model truth broadly, except in the Atlantic around 60°N (Figures 7c and 7d, compare black and blue contours).

5.1.2 Forced-Transport Error

We next examine the estimate of . By definition, this estimate is inaccurate because of the patch and forced-transport errors; we subtract the patch error (Section 5.1.1) from the total error to derive the forced-transport error (Equation 21). Global warming stratifies the ocean and weakens the surface-to-interior transport. The forced-transport error arises because does not capture this weakening effect, hence tends to overestimate warming at depths. This is evident in Figure 6; the estimate (green line) overestimates (gray line) in both global and basin integrals. The overestimate is strongest at northern mid latitudes (Figures 7e and 7f shading).

In the North Atlantic, the forced-transport error is associated with a 1 Sv slowdown of the overturning circulation at 45°N after 1960 (not shown). In contrast, the overturning circulation shows little change compared to the control in the North Pacific, implying the forced-transport error there is associated with parameterized transports. Interestingly, the forced-transport error is nearly zero in the Southern Ocean. This is probably because the Southern Ocean circulation has a strong wind-driven component (Marshall & Speer, 2012), hence is less sensitive to surface warming compared to the North Atlantic circulation.

The forced-transport error is more than twice as large as the patch error for global/basin integrated (Figure 6), and about the same size as the patch error for zonal integrated (Figure 7). The patch and forced-transport errors partially compensate each other for zonal integrated in some regions (Figure 7, compare middle and bottom row shading), hence the RMSE of the estimate is only slightly larger than that of the estimate in Figure 7.

5.2 Air-Sea Flux GF Estimate

In this subsection, we evaluate the GF estimates of and based on Equation 18 (referred to as the estimate). is simulated in the control experiment. The BCs are diagnosed in HadCM3 as (Q_ERF + Q′)/(ρ₀c_p). is the same as by definition, therefore the estimates of and are identical. Θ_e and are converted to excess heat and , respectively, following the procedure of Section 2.2.4.

The estimate has smaller RMSEs than the estimate for global/basin integrals (Figure 6) and zonal-and-depth integrals (Figures 7 and 8, top row). In particular, the unforced- and forced-transport errors of the estimate are much smaller than those of the estimate. Note that the basin integrated and are largely determined by their surface fluxes, which are directly supplied to the estimate.

In contrast, the estimate is less accurate than the estimate for the latitude-depth patterns of and in the Indo-Pacific (compare Figure 7 with Figure 8 for (c) and (e)). This is because the estimate has a larger patch error, especially in 0–200 m (compare shading in Figure 7c with Figure 8c). In the Atlantic, the and estimates have similar RMSEs for the latitude-depth patterns of and , but the forced-transport error is smaller in the estimate, especially below 200 m (compare shading in Figure 7f with Figure 8f).

5.3 GF Estimate in a Real-World Application

In this subsection, we simulate a real-world application of the GF method in the model world. Specifically, we estimate excess heat in the historical simulation using: (a) simulated and (b) derived from HadCM3. This setup corresponds to Zanna et al. (2019) who reconstructed the real-world by combining: (a) observed and (b) derived from an ocean model. To distinguish the -based estimate (examined in Section 5.1) from the -based estimate (to be examined below), we refer to the latter as the estimate. The estimate suffers an additional BC error compared to the estimate, because of the differences between and (see Section 4.5).

5.3.1 BC Error

The BC error is the largest error for the latitude-depth pattern of (Figures 9e and 9f); it is as large as the forced-transport error for basin integrated (Figures 9a and 9b) and depth integrated (Figures 9c and 9d). For zonal-and-depth integrated , the BC error causes an underestimate in most of the Atlantic and south of 40°S of the Indo-Pacific (Figures 9c and 9d, compare orange and green lines). In the Southern Ocean, the underestimate caused by the BC error partially compensates the overestimate caused by the patch and forced-transport errors, reducing the total error there.

Figure 9

Estimating excess heat in the historical simulation using and (Section 5.3). This estimate is referred to as the estimate. (a and b) Basin-volume integral. (c and d) Zonal-and-depth integrated change (0–2,000 m). (e and f) Depth distribution of panels (c and d). In all panels, black lines are the model truth, orange lines are the estimate, and green lines are the estimate in Figures 6 and 7. Shading in panels (e and f) indicates the boundary condition (BC) error (Equation 22). For each metric, the root-mean-square magnitude (RMSM) of the model truth and the root-mean-square errors (RMSEs) of the two Green's function estimates are listed, along with the RMS values of the patch, forced-change and BC errors.

5.3.2 Total Error

When all error terms are considered, the estimate (orange line) reconstructs the model truth (black line) with an error of 48% and 39% for basin integrated in the Indo-Pacific and Atlantic, respectively (Figures 9a and 9b). In the Indo-Pacific, the total error is 26% for zonal-and-depth integrated and 39% for its depth distribution (Figures 9c and 9e). These numbers are larger in the Atlantic, at 37% and 68%, respectively (Figures 9d and 9f). For depth-integrated , the largest error occurs at mid and high latitudes, for example, an overestimate in the North Pacific (Figures 9c and 9d, compare black and orange lines).

6 Inferring GFs From Tracer Data

6.1 Introducing the Inverse Problem

The GF model Equation 10 connects X at (r, t) with its surface history X^s via G_c at r. This forms a constraint on G_c at r for every pair of X(r, t) and X^s in observations. In this section, we introduce a method to infer G_c from such constraints. This problem is the “inverse” of the forward problem discussed in Section 5, which uses X^s and G_c at r to estimate X(r, t). Inferring G_c is useful as it only requires tracer data; for example, one can use it to estimate the real-world G_c from observed tracers.

Observations are insufficient constraints on G_c. We assume that ocean transports are constant to reduce the number of unknowns in G_c. This allows us to rewrite G_c(r, t, | r_s, t_s) as G_c(r, 0, | r_s, t_s − t) and rewrite Equation 10 as

(23)

n is an index for different observations; t_n is the time of the nth observation. For example, one can assign n = 1 to CFC-11 observed at year 1994, n = 2 to CFC-12 at year 2000 and n = 3 to CFC-12 at year 1994. Note that G_c in Equation 23 depends on t_s − t_n, not t_n alone, which is different from G_c in Equation 10. Causality requires that G_c(r, 0 | r_s, τ) = 0 for τ > 0, where τ = t_s − t_n.

6.2 Maximum Entropy Method

At every r, N observations of X_n and impose N constraints on G_c via Equation 23. In practice, N is much smaller than the number of unknowns in G_c; the latter is at least the number of locations in r_s. Among infinitely many G_c that satisfy constraints, we choose the one that is the most “similar” to an initial guess of G_c (denoted as G_pr). This method is called the Maximum Entropy (MaxEnt) method and was first applied to infer G_c by Khatiwala et al. (2009) and Holzer et al. (2010). Formally, the above procedure can be cast as a constrained optimization problem, and solved using the method of Lagrangian multipliers.

Given N observations of tracers at r, X_n(r, t_n), n = 1, …, N, the MaxEnt estimate of G_c is a function of and G_pr

(24)

Z is a normalization factor to ensure that G_ME integrates to unity over the global ocean surface (Ω) and all τ values, which is required by Equation 12. To determine the N unknowns (a₁, …, a_N), we substitute G_ME and into Equation 23

(25)

and solve for a₁, …, a_N that minimize the misfit between X_n and . Formally, the desired a_n is given as

(26)

where x, x′ and a are column vectors of X_n, and a_n, respectively. ‖ ⋅ ‖ is the 2-norm of a vector (i.e., ). Each row of x′(a) and x are normalized so that model-data misfits of different tracers are comparable. The regularization term is included to prevent overfitting, because there are errors in observations and in Equation 23. We set λ to unity based on the L-curve method (Hansen & O’Leary, 1993). Derivation of Equation 24, the L-curve method, and how we choose λ are described in Appendix C.

There are other methods to infer G_c from observations. For example, Gebbie and Huybers (2010) and DeVries and Primeau (2011) estimate the operator L (Equation 9) from observations. Once L is derived, one can use it to calculate G_c analytically.

6.3 Transient Tracers in the Ocean

6.3.1 Introducing CFCs, SF₆, and Bomb Δ¹⁴C

Observations of CFC-11, CFC-12, and SF₆ (Fine, 2011) are often used as data constraints in the MaxEnt method (i.e., X_n in Equation 23). CFCs and SF₆ are man-made chemical tracers that have been released into the atmosphere since the 1930s and gradually taken up by the ocean. CFCs and SF₆ are stable in the oxygenated open ocean. Once entering the ocean, they are advected and diffused by ocean transports, like passive tracers.

We also explore the use of bomb ¹⁴C as data constraints in the MaxEnt method. ¹⁴C is commonly expressed as Δ¹⁴C, which is the deviation of the ¹⁴C/¹²C ratio relative to a standard value. ¹⁴C is naturally generated in the atmosphere by cosmic rays. The ¹⁴C content of a water parcel decays with a half-life of 5,730 years once it is out of contact with the atmosphere. During the 1950s and 1960s, the nuclear weapon tests dramatically increased Δ¹⁴C in the atmosphere. This “bomb Δ¹⁴C” signal invades the ocean in a way similar to CFCs and SF₆.

6.3.2 Spatial Distribution

We use results from a historical simulation of CESM2 (Danabasoglu et al., 2020) to demonstrate passages of CFCs, SF₆ and bomb Δ¹⁴C in the ocean (Figure 10). The CESM2 historical simulation is conducted under the CMIP6 protocol (Eyring et al., 2016; Orr et al., 2017). We derive bomb Δ¹⁴C as anomalies in Δ¹⁴C relative to its 1850–1870 climatology. Measurements of CFCs, SF₆ and Δ¹⁴C from historical cruises are made available as gridded and profile data by Global Ocean Data Analysis Project (GLODAP; Key et al., 2004; Olsen et al., 2016).

Figure 10

Transient tracers in the ocean from the CESM2 historical simulation. (a) Sea surface concentrations of CFC-11, CFC-12, and SF₆ (solid lines) and their atmosphere mixing ratios (dashed lines). Both quantities are shown as global mean. Dashed lines are multiplied by arbitrary scaling factors. (b and c) CFC-11 at year 1994 at the surface and the 150°W section (shading and black contours). For these two metrics, CFC-12 and SF₆ have similar patterns compared to CFC-11, but with different magnitudes. Panels (d–f) are the same as panels (a–c), but for bomb Δ¹⁴C. CFC-11 and bomb Δ¹⁴C in HadCM3 (Section 6.4) and observations are shown as red and green contours, respectively, in the bottom row. 1 nmol = 1 × 10⁻⁹ mol.

Both CFC-11 and bomb Δ¹⁴C invade the ocean from the surface, similar to how excess heat is carried to depths, for example, at the 150°W section (Figures 10c and 10f, shading). A major difference between CFC-11 and ¹⁴C is that the latter has a much longer air-sea equilibration timescale than the former (10 years vs. weeks) (Broecker & Peng, 1974). This has two consequences for CFC-11 and bomb Δ¹⁴C in the ocean. First, the surface CFC-11 (solid line) follows its atmospheric history (dashed line) closely for global mean, while the surface bomb Δ¹⁴C shows a slower increase and decay compared to its atmospheric history (Figures 10a and 10d). Second, and more importantly, the surface CFC-11 and bomb Δ¹⁴C have very different patterns in the ocean (compare Figures 10b and 10e), because the pattern of bomb Δ¹⁴C is more affected by ocean transports (due to slow air-sea equilibration). CFC-12 and SF₆ have similar air-sea equilibration timescales and spatial patterns as CFC-11.

6.4 Simulated Tracer Observations

To derive G_ME as one would do with observations, we include CFCs, SF₆, and bomb Δ¹⁴C in the historical simulation (see Appendix D for details). The resulting CFC-11 and bomb Δ¹⁴C are similar to the gridded GLODAPv1 observations. Taking the 150°W section as an example, this is evident as a good agreement between red and green contours in Figures 10c and 10f. In polar regions, tracers in HadCM3 tend to penetrate to greater depths than those in CESM2, implying that HadCM3 has a stronger convection than CESM2 there. To isolate the forced-transport error, we also simulate CFCs, SF₆, and bomb Δ¹⁴C in the control experiment. In Section 7, we use simulated observations in the historical and control experiments to estimate and , respectively.

6.5 A Baseline Setup for Computing G_ME

It is important to note that G_ME is not uniquely defined, but depends on the choice of data constraints and priors. Because we want to test the application of G_ME in the real world, we construct a G_ME using HadCM3 equivalents of real-world observations. We refer to this G_ME as G_MEb.

6.5.1 Data Constraints

We use four tracers simulated in HadCM3 to compute G_MEb; they are CFC-11 and CFC-12 at year 1994 and climatological temperature and salinity. These four tracers are available in observations from the gridded GLODAPv1 data (Key et al., 2004). We choose this data set because it has a nearly global coverage, hence one could compute G_ME everywhere in the ocean.

For climatological temperature and salinity, we repeat their surface BCs in time, and truncate the time integral in Equation 25 from (−∞, t_n] to [t_n − 7,999, t_n] years. The 8,000-year limit is an upper bound of the timescale to tracer equilibrium in the global ocean under concentration BCs. One can set t_n to an arbitrary number for climatological tracers because their X_n and are both constant in time.

6.5.2 Space and Time Average

All data on the HadCM3 grid are averaged onto a 10° × 10° grid before solving for G_ME. Because every interior point has a G_ME, the spatial averaging reduces the total number of G_ME to be solved. Despite the low resolution, the coarse grid can still capture most of spatial variability in the surface BCs of CFCs, SF₆ and bomb Δ¹⁴C, because they all exhibit coherent spatial structures (Figures 10b and 10e). On the time dimension, we focus on annually averaged quantities as for simulated GFs (Sections 4 and 5). After the coarse-grained averaging, G_ME(r, 0 | r_s, τ) now becomes a 10° × 10° resolution 2D map defined on a yearly grid for a given r.

6.5.3 Computing Prior GFs

We use simulated in a 1,000-year control run of the FAMOUS climate model as G_pr for G_MEb.

(27)

where p(r_s) returns the index of surface patch that r_s is in and A(⋅) returns the surface area of a patch given its index. FAMOUS is a low resolution version of HadCM3, which uses most of the HadCM3 codes, but runs about 10 times faster (R. S. Smith et al., 2008). The FAMOUS is generated using the same procedure and surface patches as described in Section 4. Because deep oceans ventilate on millennial timescales, G_pr derived from Equation 27 does not satisfy Equation 12 at every r. We fill in the rest of G_pr by assuming that the fraction of water ventilated at τ ≤ −1,000 years is uniformly distributed over a 7,000-year period and the global ocean surface

(28)

A₀ is the surface area of the global ocean . We simply set G_pr to zero for τ ≤ −8,000 years.

7 Estimating Excess Heat Using Inferred GFs

In this section, we examine how well inferred GFs G_ME can reproduce excess heat change in the historical simulation. We derive G_ME by updating a prior estimate of G_c to fit simulated tracer observations (Section 6.2).

7.1 Error Definitions

The G_ME estimate of is inaccurate for the following reasons. First, the G_MEb estimate suffers an “information error,” because observations are insufficient constraints on G_c. Although this problem is regularized by the MaxEnt method, it is likely that G_ME still differs from the true G_c in many aspects. The G_ME estimate also suffers the patch, unforced-transport and forced-transport errors like the estimate discussed in Section 5. We assume that the former two error sources are small for the G_ME estimate because: (a) the G_ME estimate resolves surface BCs at 10° × 10° resolution; and (b) the unforced-transport error is found small for the estimate. We partition errors in the G_ME estimate into the information, forced-transport and BC errors as below, similar to Section 4.4.

(29)

(30)

(31)

We compare the model truth with the G_ME estimates using the same metrics as Section 5. They are: (a) global/basin volume integral (0–2,000 m), (b) zonal-and-depth integral (0–2,000 m), and (c) depth distribution of (b) (0–1,500 m). All metrics are showed as anomalies relative to the 1946–1955 average. A change (denoted using “Δ”) is calculated as the difference between 1999–2008 and 1946–1955.

7.2 Evaluating a Baseline Estimate

In this subsection, we evaluate the G_MEb estimate of and . This estimate is calculated from Equation 23, wherein we replace G_c with G_MEb. We use G_MEb derived from the historical and control experiments to estimate and , respectively. is the same as except that it is evolved by the control ocean transport (see Section 2.4). G_MEb is a particular G_ME constrained by simulated observations in HadCM3 (see Section 6.5). The BCs and are diagnosed in HadCM3 (i.e., BCs are perfectly known). Note that Section 5 uses the same to estimate and , which is different from here.

7.2.1 Information Error

The G_MEb estimate (blue line) reproduces the global/basin integrated in HadCM3 (black line) well (Figure 11), with an error of 25% for the global ocean, 27% for the Indo-Pacific and 38% for the Atlantic. (A percentage error is calculated as the ratio between RMSE and RMSM.) A constant 50 ZJ offset between the G_MEb estimate and the model truth is evident in Figure 11a after 1965 (compare blue and black lines).

Figure 11

Estimating global/basin integrated (0–2,000 m) excess heat and in the historical simulation using G_MEb (Section 7.2). Black and gray lines show and in HadCM3, respectively. Blue and green lines are the G_MEb estimates of and , respectively. is the same as except that it is evolved by the control ocean transport. The root-mean-square magnitude (RMSM) of the model truth, the root-mean-square errors (RMSEs) of the G_MEb estimate (first number) and the prior estimate (second number), and the RMS values of the information and forced-transport errors are listed.

The G_MEb estimate broadly captures the latitude-depth pattern of in the Indo-Pacific and the Atlantic, with a greater error in the latter (Figures 12a–12d compare black and blue lines/contours). The error for depth integrated is 25% and 35% in the Indo-Pacific and the Atlantic, respectively. In both basins, is underestimated by 0–5 PJ m⁻¹ at most latitudes, except south of 50°S where it is overestimated (Figures 12a and 12b compare black and blue lines). The overestimate is evident over the 0–1,500 m depths, while the underestimate mostly comes from the 0–400 m depths (Figures 12c and 12d shading). For these zonal integrated metrics, the G_MEb estimate has a similar accuracy compared to the estimate (Section 5.1) in both basins.

Figure 12

Estimating latitude distribution of excess heat change and in the historical simulation using G_MEb (Section 7.2). (a and b) Zonal-and-depth integral (0–2,000 m). (c–f) Depth distribution of panels (a and b). In all panels, black and gray lines show and in HadCM3, respectively; blue and green lines show the G_MEb estimates of and , respectively. Shading in panels (c and d) indicates errors in the G_MEb estimate of (the information error). Shading in panels (e and f) indicates errors in the G_MEb estimate of minus the information error (the forced-transport error). For each metric, the root-mean-square magnitude (RMSM) of the model truth, the root-mean-square errors (RMSEs) of the G_MEb estimate (first number) and the prior estimate (second number), and the RMS values of the information and forced-transport errors are listed. All changes are calculated as differences between 1999–2008 and 1946–1955.

7.2.2 Forced-Transport Error

The forced-transport error causes an overestimate in the G_MEb estimate, especially at northern mid latitudes (Figures 12e and 12f, shading), similar to that in the estimate. The forced-transport error is more than twice as large as the information error for the global and the Atlantic integrated (Figure 11), while it is about the same size as the information error for zonal integrated (Figure 12). In the Atlantic, the underestimate caused by the information error is partially compensated by the overestimate caused by the forced-transport error, reducing the total error there (except south of 50°S) (Figure 12b).

7.2.3 Effects of Data Constraints

How do data constraints improve on the initial guess G_pr? We examine this question by comparing RMSEs between the G_MEb estimate and the G_pr estimate. The G_pr estimate is calculated using the same equation as the G_MEb estimate, except replacing G_MEb with G_pr. The G_MEb estimate has a smaller RMSE compared to the G_pr estimate for all the metrics examined in Figures 11 and 12 (shown by numbers in the legends). The reduction of RMSE is between 20% and 40% (the number is different for different metrics). The exceptions are the global and the Atlantic integrated , for which the G_MEb estimate has a greater RMSE than the G_pr estimate (Figure 11c). We suspect that this increase of RMSE is related to the forced-transport error, because the same behavior is not found for .

7.3 GF Estimate in a Real-World Application

In this subsection, we simulate a real-world application of the GF method in the model world. Specifically, we estimate excess heat in the historical simulation using: (a) simulated and (b) G_MEb derived from simulated observations. This calculation can be repeated using the real-world and observations. To distinguish the -based G_MEb estimate (examined in Sections 7.2) from the -based G_MEb estimate (to be examined below), we refer to the latter as the estimate. The estimate suffers an additional BC error compared to the G_MEb estimate, because of the differences between and . The BC error of the estimate is similar to that of the estimate in Section 5.3 (compare Figures 9 and 13). In particular, the BC error is at least as large as the information and forced-transport errors for all metrics examined in Figure 13.

Figure 13

Estimating excess heat in the historical simulation using G_MEb and (Section 7.3). This estimate is referred to as the estimate. (a and b) Basin-volume integral. (c and d) Zonal-and-depth integrated change (0–2,000 m). (e and f) Depth distribution of panels (c and d). In all panels, black lines are the model truth, orange lines are the estimate, and green lines are the G_MEb estimate in Figures 11 and 12. Shading in panels (e–f) indicates the boundary condition (BC) error (Equation 31). For each metric, the root-mean-square magnitude (RMSM) of the model truth and the root-mean-square errors (RMSEs) of the two Green's function estimates are listed, along with the RMS values of the information, forced-change and BC errors.

When all errors are considered, the estimate reconstructs the model truth with an error of 50% for basin integrated and 40% for zonal-and-depth integrated (Figures 13a–13d, RMSEs of orange lines). In the Indo-Pacific the error is largest around 40°S, while in the Atlantic the error is of similar magnitude across latitudes (Figures 13c and 13d, compare black and orange lines). It is important to note that the G_MEb estimate is more accurate than the estimate for all metrics examined here. This highlights the need to reduce the BC error when applying inferred GFs to estimate the real-world excess heat.

7.4 Sensitivity of the G_MEb Estimate

In this subsection, we examine how sensitive the G_MEb estimate is to the choice of data constraints and priors. For each sensitivity experiment, we focus on two metrics: (a) basin integral and (b) zonal-and-depth integrated change. Both metrics are calculated for only and integrated over the 0–2,000 m layers.

7.4.1 Constraints From SF₆ and Bomb Δ¹⁴C

In the first experiment, we add SF₆ and bomb Δ¹⁴C at year 1994 as additional constraints, while keeping other settings unchanged. Adding SF₆ alone or SF₆ and Δ¹⁴C together has little impact on the G_MEb estimate (not shown). For instance, the RMSE change due to SF₆ is less than 2% for all the metrics.

How much can SF₆ and bomb Δ¹⁴C improve on the G_MEb estimate if it has little prior knowledge of ocean transports? We examine this question by replacing the FAMOUS prior in G_MEb with a uniform prior; the resulting G_ME is referred to as G_MEu. The uniform prior is defined as

(32)

where A₀ is the surface area of the global ocean. The uniform prior assumes that a water parcel, regardless of its interior location, contains equal amounts of water from all surface locations and times over the previous 8,000 years. When the uniform prior is used, we relax the regularization parameter λ (see Section 6.2) from unity to 0.1 so that more modification to the prior is allowed compared to when the FAMOUS prior is used.

The G_MEu estimate of (red line) is about 50% lower than the G_MEb estimate (gray line) in both the Indo-Pacific and the Atlantic (Figure 14). The G_MEu estimate is improved by adding bomb Δ¹⁴C (green line) as a constraint, but not by adding SF₆ (blue line) (Figure 14). This is because bomb Δ¹⁴C and CFCs have very different surface BCs (Figures 10b and 10e), which provides additional constraints (equations) to the inverse problem. In contrast, the surface BCs are very similar between SF₆ and CFCs. The improvement due to bomb Δ¹⁴C is much greater in the Indo-Pacific than in the Atlantic. This is probably because the surface BCs of Θ_e and bomb Δ¹⁴C are more alike in the Indo-Pacific than Atlantic (compare Figures 10e and 5d). In particular, the pattern of surface Θ_e peaks in the North Atlantic, whereas the pattern of surface bomb Δ¹⁴C is at its minimum in that region.

Figure 14

Sensitivity of the G_MEu estimate to additional constraints from SF₆ and bomb Δ¹⁴C. (a and b) Basin integrated . (c and d) Zonal-and-depth integrated . The model truth is shown in black lines. The G_MEu estimates constrained by different tracers are color coded. For comparison, the G_MEb estimate is included as gray lines. The root-mean-square magnitude (RMSM) of the model truth and the root-mean-square errors (RMSEs) of different estimates are listed.

7.4.2 Perturbing Prior GFs

In the second experiment, we replace the FAMOUS prior in the G_MEb estimate with Inverse Gaussian (IG) distributions of different shape, following Holzer et al. (2018). The IG distribution is the analytical form of G_c for constant 1D flow; a narrower IG distribution implies that the flow has a higher Peclet number (Waugh & Hall, 2002). The method of constructing the IG prior is described in Appendix E. The three IG priors tested here are called IG-0.5, IG-1.0, and IG-1.5.

Replacing the FAMOUS prior with the IG priors leads to a change in RMSE of less than 20% for all metrics examined in Figure 15. Among the three IG priors, IG-1.0 (corresponds to a Peclet number of one) gives the closest estimate compared to the FAMOUS prior. The RMSE of the G_MEb estimate (first column) is always reduced compared to that of the G_pr estimate (second column) regardless of which prior is used (Figure 15 numbers in the legends), except for the Atlantic integral. This highlights the constraints of CFCs and climatological temperature and salinity on the passage of excess heat in the ocean.

Figure 15

Sensitivity of the G_MEb estimate to the choice of prior (G_pr). (a and b) Basin integrated . (c and d) Zonal-and-depth integrated . The model truth is shown in black lines. The G_MEb estimates with different G_pr are color coded. For each G_pr, the RMSEs of the G_MEb and G_pr estimates are listed from left to right. IG-0.5, IG-1.0, and IG-1.5 are approximations of the FAMOUS prior using Inverse Gaussian forms of different shape.

7.4.3 Perturbing Time of Constraints

In the third experiment, we alter the year of data constraints in the G_MEb estimate from 1994 to 1984, 1989, 1999, and 2004, while keeping other settings unchanged. The resulting change in RMSE is about 2% in the Indo-Pacific and 5% in the Atlantic (Figure 16 RMSEs in the legends).

Figure 16

Sensitivity of the G_MEb estimate to the time of data constraints. (a and b) Basin integrated . (c and d) Zonal-and-depth integrated . The model truth is shown in black lines. The G_MEb estimates with different data years are color coded. The prior estimate is shown in gray lines. The root-mean-square magnitude (RMSM) of the model truth and the root-mean-square errors (RMSEs) of different estimates are listed.

8 Summary

8.1 Excess Heat and Green's Functions

The ocean stores over 93% of the “excess heat” that has entered the climate system in recent decades (Meyssignac et al., 2019). This excess heat is added to the ocean surface by air-sea fluxes (warming or cooling) and carried to depths by ocean transports. One method to estimate excess heat is to propagate its surface BCs downward using a GF representation of ocean transports. The GFs can be derived from: (a) simulating idealized tracers in a model (“simulated GFs”) or (b) solving an inverse problem using tracer observations (“inferred GFs”) (Holzer et al., 2010; Khatiwala et al., 2009; Zanna et al., 2019). The BCs are often derived from SST anomaly in the literature.

8.2 Errors in the GF Method

The GF estimate of excess heat is inaccurate for the following reasons.

1. Patch error: Simulated GFs are coarse grained in space and time, hence they partially neglect the covariance between the true GFs and surface BCs.

2. Transport error: Simulated/inferred GFs do not resolve time-varying ocean transports due to unforced variability and forced change.

3. Information error: Observations are insufficient constraints for inferring GFs.

4. BC error: SST anomalies are contaminated by redistributed changes.

5. Model error: Modeled ocean transports encoded in simulated GFs are different from those of the real world.

8.3 HadCM3 Perfect-Model Test

How different errors affect the accuracy of the GF method has not been examined in the literature. Here, we investigate this question using a historical simulation (1860–2008) conducted in the HadCM3 AOGCM. We treat this simulation as the real world, and compare excess heat diagnosed in it (as the “truth”) with that estimated using simulated/inferred GFs. Details on how different errors are computed are given in Sections 4.4 and 7.1. We focus on evaluating derived from GFs instead of GFs themselves, because not every detail in GFs matters for estimating .

8.4 Estimating Excess Heat Using Simulated GFs

We generate simulated GFs in a 200-year pre-industrial control experiment of HadCM3.

8.4.1 How Accurate Is the Method?

The simulated GFs reconstruct in the Indo-Pacific with a RMS error of 48% for the volume integral and 26% for zonal-and-depth integrated changes; the corresponding numbers are 39% and 37% in the Atlantic, respectively (including all errors except the model error). The volume integral is most affected by the forced-transport and BC errors; the patch error is <1/3 of the BC error in terms of the RMS value. The zonal-and-depth integral is affected by the patch, forced-transport and BC errors to a similar degree; the BC error is slightly larger than the other two in the Indo-Pacific. The unforced-transport error is <1/3 of the patch error for all metrics examined here. Results of this subsection are summarized in Figure 9.

8.4.2 Underestimated or Overestimated?

The patch error causes an underestimate of in the North Atlantic, and an overestimate of south of 40°S. The forced-transport error causes an overestimate of at most latitudes, especially in the northern subtropics. The BC error causes an underestimate of in the Southern Ocean. This underestimate partially cancels out the patch and forced-transport errors, reducing the total error in the Southern Ocean. Note that the degree to which this error compensation would work may be different in the real world and in other models.

8.5 Estimating Excess Heat Using Inferred GFs

We compute inferred GFs by using HadCM3 equivalents of the GLODAPv1 data as constraints to update a prior estimate of GFs. The GLODAPv1 data consist of CFC-11 and CFC-12 at year 1994 and climatological temperature and salinity (Key et al., 2004).

8.5.1 How Accurate Is the Method?

The inferred GFs reconstruct in the Indo-Pacific with an error of 50% for the volume integral and 34% for zonal-and-depth integrated changes; the corresponding numbers are 44% and 42% in the Atlantic, respectively (including all errors). The volume integral is most affected by the BC error; the information and forced-transport errors are about 2/3 of the BC error (in terms of the RMS value) in the Indo-Pacific. The zonal-and-depth integral is affected by the information, forced-transport and BC errors to a similar degree; although the BC error is slightly larger than the other two in the Indo-Pacific. Results of this subsection are summarized in Figure 13.

8.5.2 Underestimated or Overestimated?

The information error causes an underestimate of at most latitudes (except south of 50°S). The forced-transport and BC errors have the same effects as discussed with simulated GFs. In the Atlantic, the information error partially compensates the forced-transport error, reducing the total error there. It is unclear whether the same compensation would occur in the real world or in other models. Removing the BC error improves the estimate with inferred GFs significantly.

8.5.3 Sensitivity to Data Constraints and Priors

The estimate of from inferred GFs is not sensitive to: (a) shifting the data year by ±10, (b) small changes in the shape of prior GFs, or (c) adding 1994 SF₆ and bomb ¹⁴C as additional constraints, although bomb ¹⁴C (but not SF₆) helps when a less informative prior is used.

9 Discussions

9.1 Model Error of Simulated GFs

Because we use GFs simulated in HadCM3 to estimate excess heat in the HadCM3 world, our results do not include the model error. To explore this error, one could perturb simulated GFs to generate an ensemble of estimates (Zanna et al., 2019). An alternative would be to use more than one AOGCM; by treating one of them as though it were perfect, one could make an estimate with the GFs of another. This approach would not include the effect of errors common to all models.

9.2 Is Air-Sea Flux GF a Better Option?

As well as concentration BCs, one can propagate surface heat fluxes to estimate using simulated GFs (with a different configuration). In HadCM3, we find that this method gives a better estimate of than propagating concentration BCs. However, observations of surface heat flux are not adequate for the purpose of estimating . For example, the Objectively Analyzed air-sea Fluxes (Yu et al., 2008) are not available before 1985 and do not have the accuracy to resolve the global mean energy imbalance.

9.3 Simulated GFs Versus Ocean Model

Evolution of passive tracers in the model world can be studied using simulated GFs as well as ocean models. Ocean models are more accurate than simulated GFs for this regard, because they do not suffer the patch and transport errors. In addition, GFs are computationally expensive to derive.

Nonetheless, simulated GFs are useful for the following purposes. First, simulated GFs encapsulate the effect of a model's ocean transports in a form that can be easily shared within the community. Especially, GFs are much easier to use than 3D ocean models. Second, simulated GFs can be used to quantify the surface sources and timescales of a tracer response (e.g., Marzocchi et al., 2021; Wu et al., 2021; Zanna et al., 2019).

9.4 Improving Simulated GFs

Simulating GFs with finer surface patches can reduce the patch error. At the limit that every grid box is a patch, the patch error is completely eliminated. What is the best strategy to simulate GFs given a limited amount of computer time? Air-sea fluxes and surface concentrations of a tracer often exhibit low-dimensional structures in space. Designing patches around these structures can reduce the patch error at low computational cost (see Appendix F for two examples). On the time dimension, simulating GFs starting from various years in a historical simulation (e.g., Marzocchi et al., 2021) can reduce the unforced- and forced-transport errors. For instance, a set of GFs per year can capture time variation of ocean transports on interannual and longer timescales. Simulating GFs starting from every 10 years of the historical run would be less accurate, but more appealing computationally. Sensitivity tests to find the optimal time interval for time-dependent GFs would be useful.

9.5 Improving Inferred GFs

To reduce the information error, one could add observations in the GLODAPv2 data set (Olsen et al., 2016) as additional constraints. At present, CFCs only constrain GFs at multi-decadal and shorter lead times, limited by their surface histories. It is important to maintain observations of transient tracers in the ocean, so that new observations can be added to constrain GFs over longer lead times in the future.

9.6 Excess Temperature BCs

To derive excess temperature BCs, one could combine modeled patterns of surface excess temperature and observed global-mean SST anomalies to form hybrid excess temperature BCs. These new BCs may help reduce the contamination of redistributive cooling in the SST BCs (e.g., Figure 5). Note that there are uncertainties in the modeled patterns of excess temperature because of the spread in the modeled surface heat fluxes (e.g., in the CMIP6 ensemble).

Notation

r

3D position vector of the ocean

r_s

2D position vector of the ocean surface

t

time variable in general

t_s

time variable of surface source

Q_ctrl

Net surface heat fluxes in the pre-industrial control experiment

Q_ERF

Effective radiative forcing of the historical experiment

Q′

Changes in surface heat fluxes due to climate feedbacks

Θ_ctrl

Ocean potential temperature in the pre-industrial control experiment

Θ_hist

Ocean potential temperature in the historical experiment

Φ

3D ocean transport operator in general

L_ctrl,Φ

saved from the pre-industrial control experiment

L_hist,Φ

saved from the historical experiment

Θ_a

Historical ocean temperature anomaly, Θ_a = Θ_hist − Θ_hist

Θ_e

Excess temperature tracer evolved by L_hist

Excess temperature tracer evolved by L_ctrl

Θ_r

Redistributed temperature tracer, Θ_a = Θ_e + Θ_r

Historical ocean heat content anomaly, extensive form of Θ_a

Excess heat content evolved by L_hist, extensive form of Θ_e

Excess heat content evolved by L_ctrl, extensive form of

Redistributed heat content, extensive form of Θ_r

X

Concentration of a tracer, X could be Θ_a, Θ_e, , etc.

X^s

X at the surface

X^s averaged onto a yearly grid and surface patches

G

Boundary GF of tracer transport equation in general

G_c

G defined for surface concentration BCs

G_f

G defined for air-sea tracer fluxes or surface sources/sinks

Simulated G_c defined for yearly- and patch-averaged surface conditions

Simulated G_f defined for yearly- and patch-averaged surface sources/sinks

G_ME

Maximum entropy estimate of G_c in general

G_pr

Prior estimate of G_c used in G_ME

G_MEb

G_ME constrained by GLODAPv1 data and the FAMOUS prior

G_MEu, G_ME

constrained by GLODAPv1 data and the uniform prior

Acronyms

GF

Green's function

BC

boundary condition

MaxEnt

maximum entropy

IG

Inverse Gaussian

RMSE

root-mean-square error

RMSM

root-mean-square magnitude

Acknowledgments

The authors are grateful to Samar Khatiwala for discussions of the maximum entropy method and the suggestion about testing anthropogenic radiocarbon, to Thorsten Mauritsen for providing the ECHAM6.3 surface forcing data set, and to Elaine McDonagh, Laure Zanna, Heather Graven, and other colleagues at TICTOC meetings for useful discussions. This project has received funding from the UK Natural Environment Research Council (NERC) Grant NE/P019099/1 for the TICTOC project and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 786427, project “Couplet”). The authors thank Tom Haine and two anonymous reviewers for their careful evaluation of the manuscript. Their comments and suggestions help to improve the manuscript significantly. The authors acknowledge the World Climate Research Program's Working Group on Coupled Modelling, which is responsible for CMIP, for producing and making available their model output.

Appendix A: Forcing the Historical Simulation

We did not conduct the historical simulation by prescribing CO₂ and other forcing agents in the atmosphere. This is because HadCM3 does not include an up-to-date treatment of anthropogenic aerosol forcing, especially aerosol-cloud interaction. (In the historical experiments of Stott et al. (2000), the effect of tropospheric aerosol on cloud reflectivity was approximately represented by prescribed perturbations to surface albedo.) In addition, since our focus is on heat uptake, it is convenient to prescribe the heat flux which is added to the ocean. The use of surface forcing omits any non-radiative effects of forcing agents and their radiative effects directly in the atmosphere. But those effects are relatively minor and not important in this work. As shown in Section 2.3, the historical simulation is sufficiently close to observations for the purposes of this study.

Appendix B: Passive and Dynamical Θ_e Definitions

In Section 2, we define excess temperature Θ_e as a passive tracer evolved by the pre-computed transport operator L_hist (Equation 6). Alternatively, we can define Θ_e as a dynamical tracer that affects ocean transports.

(B1)

Φ is the ocean transport operator in general (Equation 2). Φ(Θ_e) is a nonlinear function of Θ_e because Θ_e affects Θ which affects Φ. Integrating Equation B1 in the historical simulation gives the same evolution of Θ_e as that derived from Equation 6. The two definitions, however, correspond to different constructions of GFs.

The GFs of Equation 6 are “passive” GFs; they are evolved by the same L_hist operator (see Equation 14) regardless of the time and location of the Θ_e perturbation, because L_hist is a pre-computed quantity. To simulate GFs for Equation 6 one only needs pre-computed velocities and diffusivities in principle. In contrast, the GFs of Equation B1 are “dynamical” GFs; they are evolved by different Φ operators, because Φ depends on the location and time of the Θ_e perturbation. To simulate GFs for Equation B1 one needs a full ocean model to interactively compute Φ for every Θ_e perturbation.

Because Φ depends on the forcing, the solution of Equation B1 cannot be written as a superposition of impulse responses scaled with the forcing. That is, the GF estimate of the dynamical Θ_e has a nonlinear error. The GF estimate of the passive Θ_e, however, does not have a nonlinear error, because L_hist is independent of the forcing (i.e., Equation 6 is strictly linear). The unforced- and forced-transport errors (Section 4.1) arise because we neglect the time-evolution of GFs to reduce computational time, not because the GF method is inherently inaccurate.

Appendix C: Maximum Entropy Method

C1 Problem Formulation

Inferring G_c(r, 0 | r_s, τ) from X_n(r, t_n) and is an underdetermined problem. Among infinitely many estimates of G_c that satisfy Equation 23, we choose one based on the principle of Minimum Discrimination Information (MDI) (Kullback, 1959). Recall that G_c is like a Probability Distribution Function (PDF) (Section 3). The principle of MDI states that: to update a prior PDF using new facts (i.e., constraints), the new PDF should be chosen such that it is the “nearest” to the prior PDF. The distance between two PDFs is measured using relative entropy in MDI.

Applying the principle of MDI, we determine the best estimate of G_c by solving the following constrained optimization problem. The Lagrangian function of the problem is given as

(C1)

G is an unknown PDF. G_pr is a prior PDF. D_KL(G | G_pr) is the Kullback-Leibler divergence, or the relative entropy, between G and G_pr

(C2)

a_n is the Lagrange multiplier. N is the total number of constraints. Each constraint takes the form . is an estimate of X_n using G, calculated by substituting G into Equation 23.

(C3)

C2 Exact Solution

The optimization problem is solved by setting and simultaneously. Because is a function of the function G (i.e., is a functional), we use the Euler-Lagrange equation to solve . First, we substitute Equations C2 and C3 into Equation C1 and rewrite as

(C4)

where

(C5)

Next, we solve ∂f/∂G = 0. This yields G_ME in Equation 24, where G_ME is a function of N unknowns a₁, ⋯, a_N. Substituting G_ME into Equation 23 turns into a function of the N unknowns a₁, ⋯, a_N as well. Finally, a₁, ⋯, a_N are determined by using N constraint equations derived from setting . Collecting X_n, and a_n into column vectors x, x′, and a, respectively, the N constraint equations can be written as

(C6)

where a can be solved using standard numerical routines, for example, fsolve in MATLAB.

The principle of MDI is also known as the principle of Maximum Entropy (MaxEnt) (Jaynes, 1957), which maximizes the negative relative entropy subject to constraints. Here, we refer to the procedure of deriving Equation 24 as the MaxEnt method following Khatiwala et al. (2009).

C3 Least Squares Solution

In practice, an exact fit between X_n and (i.e., Equation C6) is not desirable, because there are errors in Equation 23 and in observations. Solving for a using Equation C6 sometimes results in an overfitted G_ME that is difficult to interpret physically. For instance, a G_ME may have extremely large values at just a few r_s and τ locations, as opposed to a much broader distribution in G_pr derived from models. These extreme cases often come with large a_n values, which modify G_pr via the exponential function in Equation 24.

To avoid overfitting, we relax the equality constraints and solve for a in a least squares sense with Tikhonov regularization (Tikhonov & Arsenin, 1977). This gives a_n in Equation 26. The regularization term penalizes large a_n values that cause G_ME to deviate from G_pr substantially.

We determine the λ value using the L-curve method (Hansen & O’Leary, 1993). A smaller λ value corresponds to a smaller model-data misfit and a larger value. Setting λ = 0 gives a without the regularization (i.e., Equation C6). The L-curve method finds the smallest λ allowed before any further decrease of λ leads to a rapid increase of . The L-curve method requires repeating the minimization process for different λ values, which is computationally prohibitive if carried out at each interior location. We choose λ by applying the L-curve method to the centers of the subtropical gyres, where ocean tracers tend to accumulate. We find that the optimal λ value is between 0.1 and 10 in those locations; λ values within that range give a similar model-data misfit and a similar value. Based on this evidence, we set λ to unity globally in this study.

Appendix D: Simulating Tracer Observations

We include CFCs, SF₆ and bomb Δ¹⁴C in the HadCM3 historical simulation. All these tracers are simulated in the ocean with zero initial conditions and prescribed surface concentration boundary conditions (BCs) from 1860 to 2008. The BCs are derived by interpolating monthly outputs of the CESM2 historical simulation to each timestep of HadCM3 linearly. CESM2 is chosen here because it is the only model available to us that includes CFCs, SF₆ and Δ¹⁴C in a historical simulation.

For simplicity, we choose not to simulate the air-sea gas transfer of chemical tracers in HadCM3, which is different from the CMIP6 biogeochemical protocol (Orr et al., 2017). This is because the MaxEnt method is not concerned with the air-sea gas transfer. G_ME is determined by the relationship between and X_n, which is only affected by ocean transports (e.g., L_hist or L_ctrl).

The radioactive decay of ¹⁴C can be accounted for by adding an exponential decay term in Equation 23 (see Holzer et al., 2010). Because the history of bomb Δ¹⁴C is very short compared to its half-life, we neglect its radioactive decay in this study. The method of simulating Δ¹⁴C (hence the ¹⁴C/¹²C ratio) as a tracer was first proposed by Toggweiler et al. (1989).

Appendix E: Inverse Gaussian Prior

Following Waugh and Hall (2002) the Inverse Gaussian (IG) distribution is given as

(E1)

Γ > 0, Λ > 0, and Γ³/(2Λ²) are the mean, the width and the shape parameter of the IG distribution, respectively. Equation E1 is the analytical form of G_c for 1D flow with constant velocity and diffusivity; Γ²/Λ² is the Peclet number of the flow (Waugh & Hall, 2002).

We generate an ensemble of IG priors by fitting a scaled IG distribution to the FAMOUS prior (Equation 27) with varying Λ/Γ ratios.

(E2)

A higher Λ/Γ ratio corresponds to a more diffusive flow, in which water parcels are ventilated over a wider range of timescales. The Λ/Γ ratio is zero for pure advective flow.

For every r_s and a given Λ/Γ ratio, we choose Γ such that the resulting IG prior has the same mean age as the FAMOUS prior for 0 ≥ τ > −1,000 years. The mean age between τ₁ and τ₂ is calculated as

(E3)

b(r, r_s) determines the fraction of water at r that is formed at r_s over all τ values (i.e., the denominator of Equation E3); we set this parameter to be the same as the one in the FAMOUS prior.

We generate the IG prior for the Λ/Γ ratio of 0.5, 1.0, and 1.5 to cover its likely range in the ocean (Waugh et al., 2006). These priors are referred to as IG-0.5, IG-1.0, and IG-1.5, respectively.

Appendix F: Alternative Patch Designs

Instead of decomposing BCs into pulses in the lon-lat space, one could project BCs onto Empirical Orthogonal Functions (EOFs), and construct GFs based on the leading EOFs. (EOFs are the optimal coordinates to capture the spacetime variability of a field.) This method has one limitation: the sources of a tracer are often interpreted in terms of water-mass formation sites, but EOFs do not always project back to isolated regions in the lon-lat space.

One could also reduce the patch error by prescribing a spatial pattern within every patch when simulating GFs. Such a pattern can be derived from long-term trends for surface excess temperature or long-term averages for surface heat fluxes, for instance. In this way, the covariance between the true GFs and BCs are better incorporated into simulated GFs than assuming that BCs are uniform within every patch. However, because tracers exhibit different patterns in their BCs, there is no universal pattern that would work for every tracer.

Open Research

Data Availability Statement

Outputs from the historical simulation are published at https://doi.org/10.5281/zenodo.6790458. Simulated tracer Green's functions are published at https://doi.org/10.5281/zenodo.6792335. The CESM2 data are available at https://esgf-node.llnl.gov. For the use of the HadCM3 model, contact [email protected].