Likely weakening of the Florida Current during the past century revealed by sea-level observations

The Florida Current marks the beginning of the Gulf Stream at Florida Straits, and plays an important role in climate. Nearly continuous measurements of Florida Current transport are available at 27°N since 1982. These data are too short for assessing possible multidecadal or centennial trends. Here I reconstruct Florida Current transport during 1909–2018 using probabilistic methods and principles of ocean physics applied to the available transport data and longer coastal sea-level records. Florida Current transport likely declined steadily during the past century. Transport since 1982 has likely been weaker on average than during 1909–1981. The weakest decadal-mean transport in the last 110 y likely took place in the past two decades. Results corroborate hypotheses that the deep branch of the overturning circulation declined over the recent past, and support relationships observed in climate models between the overturning and surface western boundary current transports at multidecadal and longer timescales.


Supplementary Note 1 A note on symbols and notation in this document
In the Methods section, where I develop the Bayesian algorithm, I exhaust most of the letters in the Roman and Greek alphabets. In what follows here, I present more equations, which are informative for interpreting results in the main text. To avoid making these equations cumbersome, I must reuse some of the letters from Methods, but for different purposes. To reduce confusion, I have structured the text of this section and the Methods so that there is no crossover in symbol meaning between the text in the two documents; that is, symbols and letters as defined in the text here are not referred In these cases, I give the best estimate of the trend followed by a ± value that represents twice the estimated standard error. Here I describe how the standard errors are estimated.
Typically, when estimating a trend from ordinary least squares, one assumes that residuals are independent and identically distributed (iid) and errors are uncorrelated (e.g., white noise). Yet, many geophysical time series do not behave as white-noise processes, but rather exhibit temporal autocorrelation 1, 2 , which is sometimes called long-range dependence or persistence or memory [3][4][5] .
If autocorrelation is not taken into account, then standard errors on trends will be underestimated.
To account for autocorrelation of the residuals, I use the method of surrogate data 6 . Given a data time series x(t), I compute the best estimate of the trend through the data using ordinary least squares. I then remove the trend from the data, leaving the residual series x (t). Next, I randomly generate a large number (e.g., 10 3 ) of synthetic time series x i (t) based on x (t), such that each x i (t) has the same Fourier amplitudes as x (t) but randomized (scrambled) phase. I compute linear trends in eachx i (t), resulting in a histogram of the possible apparent trends, or stochastic trends, in a random stationary process with the same basic timescales and amplitudes of variation and the same effective degrees of freedom as the original data series. I take the standard deviation of all of thex i (t) trends as the estimated standard error on the original x(t) trend.

Supplementary Note 3 Simulating the hypothetical transport of the Antilles Current
The question arose in the text as to whether the probable weakening of the Florida Current transport over the past century is partly balanced by compensating changes in the Antilles Current transport.
Direct measurements of the Antilles Current are short and do not allow for a direct observational assessment. However, it is possible to estimate a range of possible stochastic (or random) transport trends, given the time-series properties of the available data.
Following Emery and Thomson 8 , the integral timescale τ of a discrete time series is defined as, where C k = C (k∆t) is the autocorrelation function of the time series with itself at lag k∆t for a time increment ∆t. These statistics form the basis of simple simulation experiments of the Antilles Current transport.
I assume that Antilles Current transport behaves as random stationary red noise that can be modeled as an autoregressive process of order 1, Here y k is Antilles Current transport at time step k, ϕ is the lag-1 autocorrelation coefficient, and ε k ∼ N (0, σ 2 ) is stationary white noise with zero mean and variance σ 2 . To simulate this process, values for ϕ and σ 2 must be assigned based on the Antilles Current transport observations.
Based on the properties of an autoregressive processes, the autocorrelation of y k is C k = ϕ k .
Using this form of C k in Eq. (S1) and evaluating the sum of the geometric series gives, Rearranging to solve for ϕ, and using τ = 19.0 days and ∆t = 1 day from Meinen et al. 7 , gives, The variance of y k is var (y k ) = σ 2 (1 − ϕ 2 ). Rearranging to solve for σ 2 , setting ϕ = 0.9 based on Eq. (S4), and using var (y k ) = (7.5 Sv) 2 from Meinen et al. 7 gives, Using these values for ϕ and σ 2 , I run simulation experiments to quantify the possible range of stochastic trends in Antilles Current transport as a function of timescale. For time-series lengths between 1 and 150 y, I generate random values for ε k in Eq. (S2) to yield 1,000 separate synthetic daily series of surrogate Antilles Current transport. I compute the linear trend in each of these 1,000 surrogate time series for each specified time-series length. This allows me to populate a histogram of the trends possible for a stationary random red-noise process with the same variance and integral timescale as the Antilles Current transport data. Shading in Supplementary Figure 11 represents the 95% confidence interval (2.5th and 97.5th percentiles) from these simulations as a function of timescale, which are ±2.9 and ±1.2 Sv century −1 for periods of 50-and 100-y, respectively.
Supplementary Note 4 Computing the trend in surface heat flux implied by a trend in seasurface temperature What follows in this section is based on the forms of air-sea fluxes described by Large and Yeager 9 (their section 2.1). Most symbols used in this section are theirs.
I start by establishing basic definitions, after Large and Yeager 9 . The total air-sea heat flux Q as (positive into the ocean) is given by a sum of contributions, where Q S is shortwave solar radiation, where Q I is the insolation and α is the surface albedo; Q L is the net longwave flux, where Q A is the longwave energy received from the atmosphere, SST is sea-surface temperature, and σ = 5.67 × 10 −8 W m −2 K −4 is the Stefan-Boltzmann constant; Q E is the latent turbulent flux, where Λ v = 2.5 × 10 6 J kg −1 is the latent heat of vaporization, ρ a = 1.22 kg m −3 is air density near the surface, q (z q ) is the specific humidity of air at a height z q above the surface, C E is the transfer coefficient for evaporation, q sat is the specific humidity of air at saturation, and ∆ U is the near-surface wind speed; and Q H is the sensible turbulent flux, where c p = 1000.5 J kg −1 K −1 is the specific heat of air, C H is the transfer coefficient of sensible heat, and θ (z θ ) is air temperature at a height z θ above the surface. Note that I ignore heat flux due to precipitation, since its contribution is often small and uncertain.
The transfer coefficients of latent and sensible heat are functions of the drag coefficient C D , and where U N (10 m) is variable wind speed at 10 m under neutral stability. Note that I assume stability in defining C H in Eq. (S12); see Large and Yeager 9 for more details on forms of C H .
The specific humidity at saturation q sat is a function of SST, where q 1 = 0.98 × 640380 kg m −3 and q 2 = −5107.4 K are the coefficients.
All of the above is as in Large and Yeager 9 . I now use these forms to consider an infinitesimal perturbation dSST in sea-surface temperature. The resulting perturbation dQ as in surface heat flux is exactly, where SST indicates that values are evaluated at the background average SST value. Contributions from Q S fall away because they have no explicit dependence on SST. The partial derivatives on the right-hand side of Eq. (S15) are, and whence, For a finite but small, linear SST perturbation, Eq. (S19) will hold approximately. Therefore, I use Eq. (S19) to estimate the Q as trend implied by the SST trend observed over the warming hole.
Values for Λ v , ρ a , q 1 , q 2 , σ, and c p are given above. Based on examination of a global oceanic state estimate 10  From basic conservation principles, if a surface heat flux Q as acts on the ocean surface, and all of the heat gained is stored locally in the ocean, then the heat budget is, where Θ is the depth-averaged ocean (potential) temperature, ρ is seawater density, C p is the specific heat of seawater (distinct from c p , which is the specific heat of air in the past section), and H is the depth of the water column. Equivalently, taking a time derivative, Hence, the right-hand side takes the form of a temperature acceleration. Setting ∂Q as ∂t equal to 16 ± 11 W m −2 century −1 for the warming-hole region from the previous section, choosing typical round numbers of ρ = 10 3 kg m −3 and C p = 4 × 10 3 J kg −1 • C −1 , and selecting H = 2.5 × 10 3 m as a representative depth for the northern North Atlantic Ocean, I obtain a range for the temperature acceleration ∂ 2 Θ ∂t 2 of 5.0 ± 3.5 • C century −2 , which equates to a warming of 3.1 ± 2.1 • C over a 110-y period.
These acceleration and warming numbers apply to the subpolar warming-hole region, which has a surface area of about 5.3×10 12 m 2 ( Figure 7). In other words, these are the ocean temperature changes that would be experienced in that region due to the surface heat flux in the absence of any lateral redistribution of heat by circulation and mixing. However, given the focus of this paper, it is instructive to consider whether the heat gain is stored not locally over the warming hole, but rather more broadly across the northern North Atlantic and Arctic Ocean, from 27 • N to Bering Strait. In the main text, I explain that the weakening of Florida Current transport and the surface heat flux trend resulting from the cooling of subpolar sea-surface temperatures over the warming hole must be physically consistent with two simple conservation principles: the sum of changes in all volume transports at 27 • N must equal zero (mass conservation), and the trend in ocean heat transport across 27 • N must match to lowest order the trend in surface heat flux over the warming-hole region (heat conservation). I express these requirements for mass and heat conservation respectively as, and Here ψ F , ψ D , and ψ T are the volume transports across 27 • N (positive northward) by the Florida Current, the deep branch of the overturning circulation, and thermocline recirculation, respectively, and Θ F , Θ D , and Θ T are corresponding representative ocean temperatures in Florida Straits, the deep ocean ( 1000 m), and the interior upper ocean ( 1000 m), respectively. As before, Q as is surface heat flux over the warming hole, ρ ocean density, and C p specific heat capacity of seawater, and here A is the surface area of the warming hole, so that AQ as is the total surface heating of the control volume. Primes are used here to indicate linear trends whereas overbars represent time means. Note that in Eq. (S23), I ignore the time-tendency (local storage) term and heat transport by currents acting on temperature anomalies. These assumptions are discussed in more detail below.
This linear system can be rearranged to solve for ψ D and ψ T in terms of ψ F and SST , viz., and, where Eq. (S19) was used to substitute Γ SST for Q as . Based on examination of climatological temperature from the World Ocean Atlas 13 along 27 • N (Supplementary Figure 14), I assume that