This study considers
year-to-year and decadal variations in as well as secular trends
of the sea–air

The atmospheric

In order to understand the future climate trajectory, we therefore need to quantitatively understand the carbon response of the natural systems.
For example,
how will secular trends towards higher wind speeds in the Southern Ocean
affect the sea–air

Suitable observational data therefore need to provide sufficient spatial and temporal detail
and span several decades.
Regarding ocean

Therefore, the surface-ocean

With regard to the aim of understanding how the oceanic carbon cycle
may respond to decadal and secular climatic changes as laid out above, however,
the current

The second limitation arises from the fact
that very few

As a contribution to overcome these two limitations, this study has a 2-fold aim:

First, extending the CarboScope

Second,
we have combined this multi-linear regression
with an additive auto-regressive correction into a “hybrid” mapping,
inheriting the complementary advantages of
both auto-regressive and regression-based

After describing the mapping methods (Sect.

The

Illustration of the quantities involved in the mixed-layer scheme
(time series panels)
and the calculations done to connect them
(thick-framed boxes).
At the arrows on the right of each calculation box,
we give its most important environmental input fields
(see Table

Input data sets.

SST: sea surface temperature; MLD: mixed-layer depth; LOCEAN: Laboratoire d'océanographie et du climat: expérimentations et approches
numériques;
NCEP: National Centers for Environmental Prediction;
SOCAT: Surface Ocean CO

Parameterizations of sea–air gas exchange

Mapping runs used in this study.
The main results are given in bold;
the other runs are used to assess uncertainty
(“uncertainty cases”; Sect.

Spatial and temporal interpolation between the very inhomogeneously sampled data
is implemented in the following way.
By choosing a set of spatial patterns of adjustment that
are centred at all the individual ocean pixels
but simultaneously affect the respective neighbouring pixels within some correlation radius (to be detailed below),
in conjunction with additional Bayesian terms in the cost function that penalize
large adjustments to the adjustable parameters,
the parameter fields (the ocean-internal DIC flux field
or the fields of sensitivities, respectively; see below) are forced to be smooth.
These smoothness constraints spread the information from data-covered pixels
to neighbouring unconstrained pixels

The four mapping variants used here (Table 2) differ in the choices of the prior for

Information flow between the presented mapping runs.
Thick-framed boxes denote calculations,
with arrows denoting their input and output data sets (mostly spatio-temporal fields).
The violet, green, orange, and blue arrows represent the spatio-temporal sea–air

The starting variant
has a general set of (many) patterns of adjustment, allowing
an arbitrary smooth spatio-temporal internal DIC flux field

The interannual term

Spatially, the level of smoothness in all three terms
corresponds to a priori correlation length scales
of about

As symbolized by the capitalized superscript “ADJ”, the a priori uncertainties in the seasonal Fourier terms of

In order to stabilize the secular trend in the early decades
(as discussed in Sect.

In the third variant,
the ocean-internal DIC flux is represented as

sea surface temperature (SST),

its temporal change (

squared wind speed (

The simultaneous use of SST and

To prevent confusion, we point out that the multi-linear regression as introduced here
is set up in terms of the
ocean-internal DIC flux

All the explanatory fields

In order to avoid influences of the spin-up transient on the regression coefficients
(estimated sensitivities),
the regression terms (first line of Eq.

The final variant aims to combine
the temporal extrapolation capability of the multi-linear regression (Sect.

As the essential change,
the interannually varying result of the multi-linear regression
(Sect.

As a merely technical change,
the a priori uncertainties in the mean flux

In contrast to

As in

Potentially, we might expect to need a loop
with further pre-mappings, each getting its

All other mapping runs of this study use
the same spatio-temporal linearization fields

As in

In contrast to

As in

In order to cover the entire calculation period since 1951,
we now use SST from
Hadley EN.4.2.1

Compared to

In order to explore how robust the results of the
multi-linear regression (Sect.

– using SST from NOAA_ERSST v5

– using wind speeds from NCEP reanalysis

– additional regression term based on (

– additional regression term based on

– additional regression term based on decadally smoothed

– removing any decadal variability and secular trends
from the explanatory fields

– shorter spatial correlation lengths for the sensitivities

– a priori uncertainty in the sensitivities increased by a factor of 4 (i.e. the strength of the mathematical regularization is reduced);

– dividing mixed-layer depth by 2;

– multiplying mixed-layer depth by 2 (lacking a clear uncertainty range of mixed-layer depth, MLDq2 and MLDx2 represent a rather strong change, maybe already outside the actual uncertainty);

– weaker gas exchange by scaling the gas transfer velocity field such that
its global mean matches the lower limit of the range

– stronger gas exchange (analogously, using upper limit);

– replacing the

– replacing the

– the explanatory variables used individually (i.e. the regression terms of the remaining two were omitted);

– addition of Chl

– replacing

– replacing

– using the same regression terms as in the base case but restricting the time period of regression to the same years as used for RegrAddChl_98r19, RegrHeat_85r09, and RegrCurl_88r18, respectively.

Uncertainties in the hybrid mapping (Sect.

In order to test
whether the multi-linear regression against explanatory variables
(Sect.

The main results of this study are of two different types:

From the

The

The multi-linear regression attempts to trace the interannual variations
in the surface-ocean carbon system (and hence in the sea–air

Left: estimated contributions of the three explanatory variables
in the multi-linear regression
(as well as the prior, plotted here without its mean)
to the ocean-internal

When disregarding the secular increase in the ocean carbon sink,
the largest year-to-year variations in the regionally integrated sea–air carbon flux
are found in the
tropics (Fig.

In the high-latitude bands (top and bottom left panels of Fig.

The year-to-year anomalies from the

The estimated sensitivities of the ocean-internal DIC flux (

Estimated sensitivities of the ocean-internal DIC flux

We start with

In the rest of the ocean, the absolute value of the sensitivity

The estimated sensitivity

Positive interannual sensitivity to SST would also be compatible with a nutrient effect.
Upwelling and mixing-in from below both decreases SST and increases the availability of nutrients.
Thus, negative anomalies in SST tend to be associated with higher biological production and thus enhanced removal of carbon (negative anomalies in

As the statistical inference by our regression can only respond to the sum of all contributing processes,
we therefore cannot draw specific conclusions from the estimated

Higher wind speeds are estimated to be associated with more positive ocean-internal DIC fluxes
(stronger sources into or weaker sinks out of the mixed layer)
along the Equator in the Pacific;
in the eastern upwelling zones of the North Pacific, South Pacific, and South Atlantic; and in circumpolar bands in the high latitudes of both hemispheres
(red and yellow areas in Fig.

In contrast, higher wind speeds tend to be associated with more negative ocean-internal DIC fluxes
(i.e. weaker sources or stronger sinks)
at the western extratropical fringes of all ocean basins (blue areas).
In these regions of mode water formation, higher wind speeds
lead to more subduction of anthropogenic

We note again that the sensitivities discussed here
are those of the ocean-internal DIC sources and sinks

Even though the sea–air

The results of the multi-linear regression are only meaningful if the regression actually possesses some predictive skill to bridge unconstrained periods. Only then can they be considered to represent generalizing relationships.

In order to test this, we performed runs with artificial data gaps
of 5 years length (Sect.

Interannual variations in the sea–air

As demonstrated by Fig. S1 in the Supplement,
this predictive skill generally holds for all parts of the ocean and other 5-year data gaps.
This means that no particular

After presenting the interannual sensitivities from the multi-linear regression,
we now turn to interannual flux variations as estimated by the hybrid mapping
involving an additional interannually varying correction
(Sect.

Yearly sea–air

The most prominent feature of interannual variability is the secular trend towards more

In addition to the variations in the sea–air

Estimated

In the intermediate and high latitudes (top and bottom panels of Fig.

Although the hybrid mapping (blue) has the same interannual degrees of freedom
(i.e. the same flexibility) as the explicitly interannual mapping,
it does not always bring the fluxes
back to the explicitly interannual result (green),
especially in the region south of the tropical Pacific (Fig.

In view of applying the multi-linear regression as a prior of the hybrid mapping,
its predictive skill (Sect.

In light of climate change,
quantitative information about the secular flux trend is relevant.
Unfortunately,
as discussed in more detail in the appendix
(Sect.

Statistical inferences by multi-linear regression are at the risk of overfitting,
i.e. adjustment of coefficients to follow minor signals in the data, or even noise.
If that was the case, the estimated sensitivities would not
reflect underlying biogeochemical processes.
Various findings indicate however
that the results of the presented multi-linear regression do reflect actual signals:

The patterns of the sensitivities (Fig.

Test regression runs only using one of the explanatory variables (RegrOnlySST, RegrOnlydSSTdt, RegrOnlyU2) yield sensitivities very similar to the base case using all explanatory variables (Supplement Fig. S2). This indicates that the regression terms are essentially mutually independent, such that each explanatory variable picks up a more or less unique portion of the signals contained in the data.

The regression possesses predictive skill (Sect.

The estimates are relatively robust against alternative data sets for the explanatory variables (cases RegrSSTNOAA and RegrU2NCEP in Supplement Fig. S4). This corroborates the notion that the regression is likely not dominated by any particular feature in these fields.

The regression hardly responds to a 4-fold change in the regularization strength (case RegrLoose in Supplement Fig. S4). In the overfitting regime, one would expect a substantial dampening effect when the regularization is stronger.

The regression results are quite robust against further changes in the set-up (Supplement Figs. S4, S5, and S6).

Clearly, for any given spatial area,
the presence of

As seen in Fig.

Amplitudes of variability in the sea–air

The situation is different in the tropical Pacific (Fig.

To elucidate the ability of the multi-linear regression to capture year-to-year anomalies,
we compare it with other

Could alternative or additional explanatory variables help to
capture a larger fraction of variability by the multi-linear regression?

As the explanatory variables of the base case are all physical variables,
we tested using chlorophyll

Conceivably, more general non-linear relationships between

Using heat flux as an explanatory variable instead of

Replacing

In our formulation of the regression (Eq.

We assessed this by the uncertainty case RegrNoDecad,
where any decadal variability (including any secular trend)
has been removed from the three explanatory variables.
As this case can only pick up year-to-year signals to constrain the sensitivities,
any changes compared to the base case may indicate such potential timescale conflicts.
In most regions, this is not evident (Supplement Fig. S6).
Exceptions are the southern Pacific and the tropical Indian (for the wind-speed sensitivity

An alternative way to assess the impact of secular trends in the explanatory variables
is the uncertainty case RegrAddpaCO2
having an additional regression term proportional to decadally smoothed atmospheric

We note that in ocean areas with data periods of a few years only, a possible timescale dependence will not affect the sensitivities themselves, but it may still affect secular trends in the fluxes if sensitivities estimated for year-to-year variations are applied to secular trends in the explanatory variable. We do not have a means to detect whether this is the case.

Errors in the sea–air

Luckily, the interannual variability in the sea–air

The estimated ocean-internal DIC flux

Freshwater fluxes dilute not only alkalinity but also DIC,
in equal proportions.
At the same time, the sensitivities of

Alkalinity variations related to mixing from below
are linked to DIC variations as well
because deep waters are rich in both DIC and alkalinity,
compared to the mixed layer.
In contrast to the freshwater effects, however,
the regression terms

We note that the spurious compensatory contributions to

The interannual variations estimated before the

In this study, we considered the interannual variability in the
sea–air

According to our multi-linear regression,
interannual variability in the tropical Pacific
is dominated by a positive correlation of ocean-internal DIC fluxes to

In the eastern upwelling zones as well as in circumpolar bands in the high latitudes of both hemispheres,
we find a positive sensitivity to wind speed,
compatible with the entrainment of carbon-rich water during wind-driven deepening of the mixed layer.
To the extent that this sensitivity inferred from year-to-year variations also applies
to secular trends, the wind trend in the Southern Ocean (south of 45

As a

Here we discuss the global total of the sea–air

Figure

Mean global sea–air

The spread between
the flux estimates from other

The comparison between the results of the hybrid mapping and further literature values
is hampered as

The components of the contemporary net sea–air

From the increase in the anthropogenic carbon inventory in the ocean
between the extensive ocean surveys in 1994 and 2007,

The

Figure

Regarding the 1960–2019 secular sink trend,
our estimate from the hybrid mapping
(1) is not able to add much independent information and
(2) even slightly overestimates this trend relative to OCIM used in the prior:

According to Fig.

Figure

Secular linear trend of the global sea–air

Looking at the linear trend over the better-constrained, more recent period 1990–2019
(Fig.

The level of constraint in the trend over the different periods
is corroborated by the “zero-prior” mapping not using the secular trend from OCIM as a prior
(Fig.

Interannually filtered sea–air fluxes as in Fig.

As the better-constrained trend over the recent decades (after about 1992)
is essentially the same as that in the prior of the explicitly interannual mapping,
the flat multi-decadal trend of the zero-prior mapping in the early decades is very unlikely to be true.
This illustrates that a prior with the correct secular trend (such as the OCIM result used here)
is indeed needed to extrapolate the ocean

The sea–air CO

The supplement related to this article is available online at:

CR designed and developed the

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank all
contributors to the SOCAT database,
which is the basis of this work.
We are grateful to Jim Orr for kindly providing the
code of the

Corinne Le Quéré received funding from the Royal Society (grant no. RP/R1/191063) and the Natural Environment Research Council Sonata project (NE/P021417/1). Tim DeVries acknowledges support from NSF grant OCE-1948955. The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Jack Middelburg and reviewed by two anonymous referees.