the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Lagrangian and Eulerian time and length scales of mesoscale ocean chlorophyll from Bio-Argo floats and satellites

### Scott C. Doney

### Alice Della Penna

### Emmanuel S. Boss

### Peter Gaube

### Michael J. Behrenfeld

### David M. Glover

Phytoplankton form the base of marine food webs and play an important role in carbon cycling, making it important to quantify rates of biomass accumulation and loss. As phytoplankton drift with ocean currents, rates should be evaluated in a Lagrangian as opposed to an Eulerian framework. In this study, we quantify the Lagrangian (from Bio-Argo floats and surface drifters with satellite ocean colour) and Eulerian (from satellite ocean colour and altimetry) statistics of mesoscale chlorophyll and velocity by computing decorrelation time and length scales and relate the frames by scaling the material derivative of chlorophyll. Because floats profile vertically and are not perfect Lagrangian observers, we quantify the mean distance between float and surface geostrophic trajectories over the time spanned by three consecutive profiles (quasi-planktonic index, QPI) to assess how their sampling is a function of their deviations from surface motion. Lagrangian and Eulerian statistics of chlorophyll are sensitive to the filtering used to compute anomalies. Chlorophyll anomalies about a 31 d time filter reveal an approximate equivalence of Lagrangian and Eulerian tendencies, suggesting they are driven by ocean colour pixel-scale processes and sources or sinks. On the other hand, chlorophyll anomalies about a seasonal cycle have Eulerian scales similar to those of velocity, suggesting mesoscale stirring helps set distributions of biological properties, and ratios of Lagrangian to Eulerian timescales depend on the magnitude of velocity fluctuations relative to an evolution speed of the chlorophyll fields in a manner similar to earlier theoretical results for velocity scales. The results suggest that stirring by eddies largely sets Lagrangian time and length scales of chlorophyll anomalies at the mesoscale.

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Upper-ocean phytoplankton communities vary on sub-diurnal and sub-seasonal
timescales and submesoscale to mesoscale spatial scales. Fully capturing
this variability is challenging because of the temporal and spatial
limitations of different observational platforms, choices associated with
sampling strategies, and data gaps, creating the need to best leverage a
variety of complementary observing platforms (Chai et al.,
2020). Time derivatives of surface chlorophyll and phytoplankton carbon
provide valuable estimates of the phytoplankton net specific accumulation
rate (*r*) that reflect biological growth and loss processes as well as
physical advection and mixing (e.g. Behrenfeld et
al., 2005). The temporal variability in *r* from Eulerian time series from, for
example, a mooring or high-resolution ship observations at a fixed
geographic location necessarily incorporates a variance component from
advective and mixing divergence. Similar issues arise in the analysis of *r*
from satellite ocean colour data on fixed geographic grids, with the
additional complication of temporal data gaps caused by satellite orbital
dynamics and cloud cover.

In principle, a Lagrangian or water-parcel-following framework isolates net biological growth from horizontal physical transport, allowing more direct comparisons to laboratory and mesocosm biological experiments, theory, and food web models. Analysis of many Lagrangian series reveals sensitivity of phytoplankton community growth rates to environmental conditions experienced (Zaiss et al., 2021), allows for partitioning of chlorophyll (Chl) variance into net community production and advective effects (Jönsson et al., 2011), and reveals how dispersion regulates phytoplankton blooming (Lehahn et al., 2017). Records from surface or mixed-layer drifters with bio-optical sensors are rare and often of short duration (Abbott and Letelier, 1998; Briggs et al., 2018). Alternatively, one can obtain Lagrangian time series by projecting satellite ocean colour data onto surface trajectories (Jönsson et al., 2009), either those from in situ surface drifters or from synthetic particles advected with surface currents from ocean models or satellite altimetry. This approach has yielded important insights into the roles of episodic events in controlling net community production in coastal regions (Jönsson and Salisbury, 2016) and of submesoscale biophysical dynamics at ocean fronts (Zhang et al., 2019). Nevertheless, it ultimately falls victim to the limited spatial information content of any ocean colour product (Doney et al., 2003; Glover et al., 2018).

An alternative, complementary observing strategy involves Bio-Argo floats (Biogeochemical Argo floats, sometimes referred to as BGC-Argo floats), a platform experiencing a rapid growth in deployments for monitoring ocean biogeochemistry and ecosystems (Claustre et al., 2010; Gruber et al., 2010; https://biogeochemical-argo.org/, last access: 27 December 2021). Bio-Argo floats are like traditional Argo floats but equipped with additional sensors to measure variables such as chlorophyll fluorescence, backscatter, and/or nutrient concentrations. The depth resolution of these variables in combination with hydrographic variables allows floats to detect rare or small-scale events, such as wintertime restratification by mixed-layer instabilities (Lacour et al., 2017), subduction of particulate organic carbon (Llort et al., 2018), and upwelling due to rapid evolution of mesoscale eddies (Ascani et al., 2013). However, formally, Bio-Argo floats are only quasi-Lagrangian, reflecting a weighted average of velocities experienced between their parking depth and the surface. To properly sample the evolution of ocean mixed-layer biology, a platform should be nearly Lagrangian with respect to the surface flow. Typically, floats profile every few days, meaning they spend most of their time drifting with more sluggish flows at a parking depth of ∼ 1000 m. However, when vertical shear is weak or the floats profile more frequently, Bio-Argo floats might serve as a viable platform for studying evolution of upper-ocean phytoplankton communities.

In this paper, we seek to understand the Lagrangian statistics (time and length scales) of mesoscale Chl anomalies in a subregion of the North Atlantic Ocean and how these depend on the underlying Eulerian statistics of the Chl field and a water parcel's motion. Because Chl is stirred, we first diagnose the Lagrangian and Eulerian statistics of the velocity field. We take a trajectory-scale perspective, drawing on earlier theoretical (Middleton, 1985) and observational (Lumpkin et al., 2002) studies using the framework of the material derivative to quantify the relative contributions of advective and tendency terms. In particular, we characterize the fields by computing integral time and space scales of autocorrelation functions from floats, surface drifters, and satellite altimetry fields. We also take a local perspective by constructing a quasi-planktonic index (QPI; Della Penna et al., 2015) that quantifies the distance between a float trajectory and synthetic surface trajectories (from altimetric geostrophic currents) over three consecutive profiles. We combined these two perspectives to highlight quasi-Lagrangian behaviour of floats (that affects sampling) by weighting their averaged integral timescales by the inverse-squared median QPI over individual time segments. Similar to the velocity analysis, we compute integral timescales of Chl for floats, ocean colour projected onto surface drifter tracks, and Eulerian fixed-location pixels of ocean colour, and we evaluate them through the framework of the material derivative. Scales of Chl and velocity are compared to assess correspondence. Although submesoscale variability in Chl is of leading importance (Lévy et al., 2018), we focus on mesoscale variance partly because of data restrictions: there is a trade-off between resolving more variance and dealing with increased gaps when working with a finer ocean colour product, and, given the relatively sluggish motion of floats, they may not capture the full spectrum of submesoscale processes. Nevertheless, combined Lagrangian and Eulerian statistics are unknown at any scale, and new findings are still being gleaned about the geostatistics of the mesoscale Chl field and their origin (Eveleth et al., 2021). Our analysis builds on recent regional studies of the spatial geostatistics of satellite ocean colour (Eveleth et al., 2021; Glover et al., 2018) and the seasonal to annual variations in phytoplankton chlorophyll, carbon biomass, and net primary production from Bio-Argo floats (Yang, 2021; Yang et al., 2020).

## 2.1 Material derivative and integral scales

The material derivative of a scalar such as Chl(** x**(

*t*),

*t*) is

where DIFF represents turbulent diffusion, *S*(** x**(

*t*),

*t*) represents sources and sinks along trajectory

**(**

*x**t*), and $\mathrm{d}\mathit{x}/\mathrm{d}t=\mathit{u}$ (Chenillat et al., 2015; Jönsson et al., 2011; van Sebille et al., 2018; d'Ovidio et al., 2013). If Chl were conserved,

*S*would be zero. When

**is inferred from observational data (e.g. a drifter trajectory), a water parcel's motion deviates from the trajectory of an infinitesimally small particle of tracer so that DIFF encompasses unresolved advection that manifests as a diffusion term and is generally nonzero (van Sebille et al., 2018). A scalar or velocity field exhibits decorrelation in space and time, and decorrelations of velocity can be quantified with integral scales**

*u**T*

_{e}and

*L*

_{e}from an Eulerian perspective. A Lagrangian sampling platform moving with the surface flow will experience spatial and temporal velocity decorrelations simultaneously, mixing the field's temporal and spatial information; thus, it will tend to exhibit a shorter decorrelation time compared with an Eulerian observer (

*T*

_{l}≤

*T*

_{e}). To model dispersion of particles advected by the flow

**, one can assume the variances of Chl are equal in each frame so that the ratio of the advective and tendency terms in Eq. (1) scales as follows:**

*u*
where *u*^{′} is a scale for the mesoscale eddy velocities, *L*_{e} and *T*_{e} are the respective
Eulerian length scales and timescales for the mesoscale velocity field, and ${c}^{*}={L}_{\mathrm{e}}/{T}_{\mathrm{e}}$ is an evolution speed for the eddy field.

Philip (1967) argued that the quantity ${T}_{\mathrm{l}}/{T}_{\mathrm{e}}$, a
measure of the difference in Lagrangian and Eulerian perspectives, should
depend only on *α*. For a homogenous and stationary 2-D eddy field,
Middleton (1985) assumed certain functional forms for the
Eulerian energy spectrum and assumed that the distribution of parcel
displacements was stationary and Gaussian to determine the relations

and

Here, $q=\sqrt{\mathit{\pi}/\mathrm{8}}$. To interpret their meaning, consider the
case where *α*≪1. In this case, the tendency term dominates the
advective term or, equivalently, the platform is advected more slowly than
the eddy field evolves and the velocity decorrelation is determined by
Eulerian temporal evolution (*T*_{l}≈*T*_{e}). This renders the
platform like a mooring, and this regime is referred to as the
“fixed-float” regime (terminology as in LaCasce, 2008).
On the other hand, suppose that *α*≫1. In that case, the advective
term dominates or, equivalently, the platform is advected across eddies
faster than they evolve and the Lagrangian decorrelation of velocity is
determined by the temporal imprint of spatial decorrelations (*T*_{l}*<**T*_{e}). This is referred to as the “frozen-turbulence” regime
(as in LaCasce, 2008), related to Taylor's hypothesis
(Taylor, 1938).

Lumpkin et al. (2002) applied Eq. (3) to surface drifters and deep
isopycnal floats, computing Lagrangian integral timescales from those
platforms and computing Eulerian integral timescales from an ocean model.
They found the theoretical model to hold well: the deep floats fell in the
“fixed-float” regime and the surface drifters spanned the two regimes,
with spatial variability accounted for by variability in the kinetic energy
of major current systems in the North Atlantic. As Bio-Argo floats
profile, it is not clear what regime they should experience. Our first step
is to compute Lagrangian velocity scales (*T*_{l}, *L*_{l}) from trajectories
of the Bio-Argo floats and drifters and to evaluate the relations in Eq. (3), where Eulerian velocity scales (*T*_{e}, *L*_{e}) are calculated from
maps of surface geostrophic velocity anomalies from satellite altimetry. If
the flow is dominated by mesoscale balanced motions, flows at parking depth
should mimic those at the surface with a reduction in magnitude (which does
not affect decorrelation time) and a slight decay of high wavenumbers
(Klein et al., 2009; Lapeyre and Klein, 2006),
allowing geostrophic Eulerian scales to be compared to both drifters and
floats.

Such an analysis yields a statistical representation of how an observer
moves horizontally and effectively quantifies particle dispersion governed
by advection; however, it does not directly inform us of the statistics of
how a tracer like Chl is sampled by the moving platform. To do that, we
analyse a scaling of the material derivative using time and length scales of
Chl, which have effects of turbulent diffusion and sources/sinks built in.
While there is not a theoretical relationship equivalent to Eq. (3) for
tracers and no a priori relation between *T*_{l, Chl} and
*T*_{e, Chl}, we first explore the equivalent parameter spaces

and

Here, ${\mathit{\alpha}}_{\mathrm{Chl}}={u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}$, and ${c}_{\mathrm{Chl}}^{\ast}={L}_{\mathrm{e},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$. We envision Eq. (4) as a parallel to Eq. (3) with equivalent interpretation, namely a quantification of how Lagrangian and Eulerian time and length scales of Chl vary as a function of how turbulent velocity fluctuations relate to evolving space–time Chl fields. Additionally, we then admit a scalar variance (angle brackets indicate standard deviation) that may vary by reference frame to obtain the scaling

With this more general approach, we assess the relative magnitude of the
three scaling terms in Eq. (5), which are (from left to right) the Lagrangian
tendency (LAG), Eulerian tendency (EUL), and eddy advection (or stirring;
ADV). The Eulerian scales are derived from satellite ocean colour, and the
Lagrangian scales 〈Chl〉_{l},
*T*_{l, Chl}, and 〈*u*^{′}〉 are derived
from floats (Chl measured by onboard fluorometer) and drifters (Chl from
satellite ocean colour projected onto trajectories). We expect the relative
magnitudes of the Lagrangian and advective terms to be different between
floats and drifters. For both the velocity and Chl analyses, all necessary
scales and the datasets used to estimate them are summarized in Table 1. The
methodology (integral of autocorrelation function) and datasets are
described in Sect. 3.

## 2.2 Quasi-planktonic index (QPI)

Float velocities are estimated by centred differencing positions of
neighbouring profiles. While the preceding material derivative analysis
provides a holistic summary of how a float samples mesoscale fields over
some time window, it is also useful to obtain a more local measure of the
similarity of float and Lagrangian trajectories. We construct a
quasi-planktonic index (QPI) that quantifies the similarity of the float
trajectory to a best-fit synthetic surface trajectory advected by altimetric
total geostrophic currents. This index is similar to the one developed by
Della Penna et al. (2015) but is tailored to evaluate the
centred difference derivatives. At each time step *t*_{i} of a float
trajectory, we advect a disc of particles of radius 0.3^{∘}
both forwards (to *t*_{i+1}) and backwards (to *t*_{i−1}) in time. For each
synthetic trajectory, we compute the distance between itself and the true
float trajectory and choose the trajectory that minimizes the average
distance over the three time steps (*t*_{i−1}, *t*_{i}, *t*_{i+1}), with the
average distance being the QPI (in kilometres). Full details of the
calculation are given in Appendix A. To tie the two frameworks together, we
hypothesize that a Bio-Argo float with a smaller median QPI over some period
of time will behave more like a surface drifter, with a larger *α* and
a smaller ${T}_{\mathrm{l}}/{T}_{\mathrm{e}}$.

## 3.1 Study region

Our study domain approximately corresponds to that of the North Atlantic
Aerosols and Marine Ecosystems Study (NAAMES) field campaign in the
subtropical to subpolar transition region of the North Atlantic Ocean
(Behrenfeld et al.,
2019). The domain boundaries were chosen to encompass the full trajectories
of the Bio-Argo floats that we analyse. The domain includes the typical
spatial extent of the North Atlantic spring bloom and includes the
high-strain and high-eddy-kinetic-energy conditions of the North Atlantic
Current sandwiched between more quiescent subpolar and subtropical
conditions. We tile the domain into 5^{∘} × 5^{∘}
cells as done by Glover et al. (2018) (Fig. 1) and compute
averaged integral scales in each, using satellite pixels or float or drifter
segments whose median latitude and longitude reside within.

## 3.2 Data

### 3.2.1 Floats

We use data from 13 Bio-Argo floats deployed over four cruises during NAAMES
(Fig. 1; floats with prefix “n”). Most of the floats are confined to the
high-strain and high-eddy-kinetic-energy conditions of the North Atlantic
Current (Fig. 1a). In addition to measuring salinity, temperature, and
pressure like standard Argo floats, the NAAMES floats measured backscatter
at 700 nm and chlorophyll fluorescence. Chlorophyll-*a* concentration (Chl) is
derived from fluorescence, is calibrated against discrete high-performance liquid chromatography samples,
and is corrected for non-photochemical quenching (Xing et al.,
2012). All float data are obtained as L2 data from the University of Maine
In-Situ Sound & Color Lab web archive (Haentjens and Boss, 2020) and are distributed as profiles with a
2 m resolution in the vertical between 0 and 500 m (4 m between 500 and 1000 m).
All profile quantities are interpolated to a 1 m grid using a cubic hermite
interpolating polynomial. The Chl data are quality controlled by the UMaine group. For temperature and salinity, when possible, we match the L2
files to profile files in the Argo Global Data Assembly Center (GDAC, Argo, 2021) and keep only samples with adjusted
profiles with a QC flag of 1 (good), 2 (probably good), 3 (possibly bad
after correction; omitted for any “Real Time” profiles), 5 (adjusted), or
8 (estimated) prior to interpolating (see Argo Data
Management Team, 2019). All profiles are visually inspected.

We include five additional floats that were used to inform NAAMES station
sampling but were deployed by other projects (Fig. 1; lovbio^{*} and metbio^{*}
floats). Data were obtained as Sprof files from the Argo GDAC, and only
profiles overlapping in time with NAAMES were retained. While “Delayed
Time” hydrographic profiles were generally available, only “Adjusted” Chl
profiles were available, meaning only an automated set of quality checks
have been applied (Schmechtig et al., 2018). Samples with QC
flags of 1, 2, 3, 5, and 8 are retained, and the profiles are interpolated as
with the NAAMES floats. Throughout, special attention is paid to brief
(∼ 50 d) segments from metbio003d and metbio010d that
exhibited consistently frequent and shallow profiling, as they may exhibit
more closely Lagrangian behaviour.

For all floats, the mixed-layer depth (MLD) is computed as the depth at which
the potential density exceeds that at 10 m by 0.03 kg m^{−3}. To make
float-measured Chl consistent with satellite-measured Chl, we construct a
single time series with a weighted average over one attenuation depth,
utilizing the fact that about 90 % of the satellite-measured Chl signal in
the open ocean comes from a depth of Kd${}_{\mathrm{490}}^{-\mathrm{1}}$, where Kd_{490} is
the diffuse attenuation coefficient at 490 nm (Gordon and
McCluney, 1975). We estimate Kd_{490} following Morel et al. (2007; their Eq. 8):

where we take [Chl] as the mixed-layer average chlorophyll. We then take a weighted vertical average at each time step as follows:

The series is log-transformed and then smoothed with a running 48 h Hamming window to remove sub-daily variability. Float velocities are estimated by centred differencing profile positions, and the QPI is calculated for each profile as described in Appendix A.

### 3.2.2 Satellite data

We use the multi-satellite merged altimetry dataset distributed by
Copernicus Marine Environmental Monitoring Service which includes daily maps of
surface geostrophic velocity and geostrophic velocity anomalies on a
0.25^{∘} grid (Taburet et al., 2019; Copernicus Marine Environment Monitoring Service, 2021). Geostrophic velocities are used for particle
advection when computing the QPI and for comparison to effective float or
drifter velocities. The geostrophic velocity anomalies (from sea level
anomalies relative to a long-term mean) effectively isolate the mesoscale
and are used for computing scales *T*_{e} and *L*_{e}. A high pass filter is
applied to each anomaly time series by Fourier transforming and zeroing out
frequencies lower than (150 d)^{−1} before inverse Fourier transforming
back to the time domain.

We obtain fields of log-transformed, daily, 0.25^{∘}, L3m Chl
fields from GlobColour computed from the Garver–Siegel–Maritorena (GSM) algorithm and blending all
available satellites (ACRI GlobColour Team, 2020). We use a spatially coarse and blended product to
maximize data coverage, with the 0.25^{∘} resolution consistent
with our focus on mesoscale variance. All maps are assumed to correspond to
12:00:00 UTC. The time domain for all satellite data is January 2003–December 2016 and approximately follows the Glover et al. (2018) study.

### 3.2.3 Drifters

The 6 h surface drifter trajectories within the time domain of the
satellite data were obtained from the NOAA Global Drifter Program (Lumpkin and Centurioni, 2019). The
dataset reports velocities obtained by centred differencing positions. To
remove the influence of inertial oscillations and tides, which can decrease
*T*_{l} compared with fluctuations due to mesoscale processes, we filter every
drifter velocity series with an ideal low pass filter by zeroing out all
frequencies lower than $\mathrm{2}/$ (3 IP) in the frequency domain, where IP is the
inertial period corresponding to the trajectory's median latitude. For
series with gaps, the filter is applied to individual segments as long as
they are longer than 20 d. Finally, the filtered time series are
subsampled once per day: at 00:00:00 UTC for comparison with the altimetry
fields or at 12:00:00 UTC for comparison with the ocean colour fields. We
construct Lagrangian time series of Chl by bilinearly interpolating the
daily mesoscale Chl maps onto the subsampled drifter returns.

## 3.3 Subtrahends for chlorophyll

A subtrahend is a field to be subtracted from another. To isolate mesoscale Chl variability, all Chl integral scales are computed from anomalies relative to one of two Chl subtrahends that we construct: a smoothed space–time running filter or a climatology. More details of the methodology and an example time series are given in Appendix B.

The first subtrahend (“smoothed”) is meant to replicate that used by
Glover et al. (2018) so that we can obtain Lagrangian and
Eulerian timescales consistent with their Eulerian space scales. All
GlobColour space–time fields are convolved with a 3-D filter defined as a
31 d Hamming window in time and a 2-D Gaussian in space with a
1^{∘} full width at half maximum (FWHM) and a 2^{∘} cutoff.
For drifters, the anomalies are projected onto the drifter tracks. For
floats, we perform a weighted running average of each float Chl series with
weights defined by a 31 d Hamming window. The objective of the space–time
filtering in Glover et al. (2018) was to isolate mesoscale (and
any resolved submesoscale) variability signals (anomalies) from the lower-frequency seasonal and geographic patterns of bloom formation and decline
that were meant to be captured in the subtrahend. However, the short 31 d
time window may have the undesirable effect of retaining some of the
mesoscale signal in the subtrahend, instead isolating faster processes in
Chl anomalies. Yang et al. (2020) found that phytoplankton
accumulation rates *r* are typically 1–2 orders of magnitude smaller
than growth rates so that intraseasonal *r*^{−1} is generally of order 10 d.
Similar physical timescales are found for lifetimes
(weeks;
Chelton et al., 2011; Gaube et al., 2014) and inverse growth rates
(days to weeks; Smith, 2007) of energy-containing mesoscale
eddies.

Given those concerns, the second subtrahend is meant to strictly isolate
anomalies from a repeating annual cycle (“climatology”). All GlobColour
space–time fields are “stacked” by day of year in a fourth dimension
(for example, 1 January of every year is regarded as having the same time
coordinate) and convolved with a 4-D filter defined as a 31-day-of-year
Hamming window in time, a 2-D Gaussian in space with a 1^{∘}
full width at half maximum and a 2^{∘} cutoff, and a boxcar
function (equal weights) across years, yielding a set of 366 maps. For
drifters, the anomalies are projected onto the drifter tracks. For floats,
the subtrahend is projected onto the float tracks, regressed against float
data to account for different data dynamic ranges, and differenced.

## 3.4 Integral timescales

Our goal is to obtain estimates of the Lagrangian and Eulerian integral timescales of velocity and Chl representative over a 5^{∘} × 5^{∘} bin for each measurement platform (Table 1). For all
platforms (floats, drifters, and satellite-derived fields), all individual time
series of variable *y*(*t*) (zonal [*u*] or meridional [*v*] velocity or Chl anomalies)
are broken into 120 d segments (365 d for Eulerian ocean colour) and
each segment is prewhitened, either by removing the scalar mean (Chl anomaly
series) or a linear trend line fit by regression (velocity series), creating
*y*^{′}(*t*) with zero mean. The latitude–longitude coordinates for each time series
segment are defined as the median latitude and longitude over the segment
length for drifting platforms (float and drifters) or as the pixel latitude and
longitude.

Integral timescales are estimated from autocorrelation functions (ACFs). For
zero-mean data, the discrete autocovariance function (ACVF) at lag *τ*
is

where *N*(*τ*) is the number of data points in ${y}_{i}^{\prime}$ separated by
$\left(n-\mathrm{1}\right)\mathrm{\Delta}\mathit{\tau}\le \mathit{\tau}<n\mathrm{\Delta}\mathit{\tau}$. To arrive at an
ACF, *C*(τ) needs to be normalized by the variance, which
explicitly is $C\left(\mathrm{0}\right)=\stackrel{\mathrm{\u203e}}{y{{}^{\prime}}_{i}^{\mathrm{2}}}$. However, for
unevenly spaced (equivalently gapped) data, dividing by $\stackrel{\mathrm{\u203e}}{y{{}^{\prime}}_{i}^{\mathrm{2}}}$ can lead to ACF values greater than 1. Because ${y}_{i}^{\prime}$ is
stationary, $\sqrt{\stackrel{\mathrm{\u203e}}{y{{}^{\prime}}_{i}^{\mathrm{2}}\left(t\right)}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\stackrel{\mathrm{\u203e}}{y{{}^{\prime}}_{i}^{\mathrm{2}}\left(t+\mathit{\tau}\right)}}=\stackrel{\mathrm{\u203e}}{y{{}^{\prime}}_{i}^{\mathrm{2}}\left(t\right)}$ and
the normalization issue can be avoided by dividing *C*(τ) by
a measure of the variance using only the data points that went into the
calculation at lag *τ*:

where

At this point, we could obtain the integral scale *T* by integrating *R*(τ) to the lag of its first zero crossing, which (for a discrete
ACF) is $T=\sum _{\mathit{\tau}<{\mathit{\tau}}_{\mathrm{0}}}\left[R\left(\mathit{\tau}\right)\right]\mathrm{\Delta}\mathit{\tau}$. Applying this method to all segments *i* from all time
series in some space subset *I* (e.g. a 5^{∘} × 5^{∘} bin), we could then obtain average scales $\stackrel{\mathrm{\u203e}}{T}$
by averaging estimates from all *i*∈*I*. An alternative approach to obtain
a spatially averaged scale would be to first construct a single composite
ACF that is representative of *I* and then integrate that single ACF to its
first zero crossing. To construct such an ACF, we use pairs of points from
all locally prewhitened segments ${y}_{i}^{\prime}\left(t\right)$ to construct a
composite ${R}^{c}\left(\mathit{\tau}\right)={C}^{c}\left(\mathit{\tau}\right)/{J}^{c}\left(\mathit{\tau}\right)$,
where

and

For this composite ACF, ${T}^{c}=\sum _{\mathit{\tau}<{\mathit{\tau}}_{\mathrm{0}}}\left[{R}^{c}\left(\mathit{\tau}\right)\right]\mathrm{\Delta}\mathit{\tau}$. For any dataset involving
satellite ocean colour, *T*^{c} must be used due to the large number of
gaps. For evenly spaced datasets, we find ${T}^{c}\approx \stackrel{\mathrm{\u203e}}{T}$, but
*T*^{c} can be quite different from $\stackrel{\mathrm{\u203e}}{T}$ for the floats with
$\stackrel{\mathrm{\u203e}}{T}$ biased larger. This is because $\stackrel{\mathrm{\u203e}}{T}$ weighs each
segment *i* equally, but segments with shorter median Δ*t* between
profiles contribute more to *R*^{c}(τ) than those with a
longer median Δ*t* by virtue of offering more data pairs. In this
regard, *R*^{c}(τ) is a better estimate of the ensemble ACF,
and *T*^{c} is a better estimate of the ensemble integral scale. However, there
is value in continuing to use $\stackrel{\mathrm{\u203e}}{T}$ when possible, as this
reverse order of operations enables us to construct average scales weighted
by (QPI)^{−2}, allowing us to gauge in parameter space how a smaller
median QPI over the window size affects the turbulence regime experienced by
the floats.

All timescales analysed in this study are derived by averaging in space
(from integrating Eq. 8 and averaging), except any scales involving
satellite ocean colour, where large numbers of gaps require using
space-composited ACFs (from integrating Eq. 9). Each method uses all
segments in the 5^{∘} × 5^{∘} space bins depicted
in Fig. 1. Velocity timescales are computed separately for each component
(zonal and meridional) before taking ${T}_{\mathrm{l}}=\mathrm{0.5}\left(\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{l},u}}+\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{l},v}}\right)$ (similar for *T*_{e}). Velocity standard
deviations are computed by evaluating the segment standard deviation of each
component, averaging them over all segments in a space bin to yield
$\stackrel{\mathrm{\u203e}}{{\mathit{\sigma}}_{u}}$ and $\stackrel{\mathrm{\u203e}}{{\mathit{\sigma}}_{v}}$, and then taking
${u}^{\prime}=\mathrm{0.5}\left(\stackrel{\mathrm{\u203e}}{{\mathit{\sigma}}_{u}}+\stackrel{\mathrm{\u203e}}{{\mathit{\sigma}}_{v}}\right)$. Finally, Lagrangian length scales are computed by multiplying Lagrangian time and velocity scales together as ${L}_{\mathrm{l}}={u}^{\prime}{T}_{\mathrm{l}}$ or
${L}_{\mathrm{l},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}={u}^{\prime}{T}_{\mathrm{l},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}$.

## 3.5 Integral length scales

There are 20 × 20 altimetric geostrophic velocity anomalies per
5^{∘} × 5^{∘} space bin, per map. For each space
bin, we subtract the scalar spatial mean from all points (yielding
*y*^{′}(r)) and compute the distances between all possible
pairs of points to construct a radial (isotropic) ACF for space lag *d* at time
*t*:

where

and

*R*(*d*) is integrated in the same manner as *R*(*τ*) to obtain an integral
length scale *L*. This is repeated for one map per month over the study period
(taken as the 15th of each month), and scales from each map are averaged
to obtain $\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{e},u}}$ and $\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{e},v}}$. As with timescales, scales for velocity are defined as the average of zonal and
meridional scales, ${L}_{\mathrm{e}}=\mathrm{0.5}\left(\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{e},u}}+\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{e},v}}\right)$.

For computational reasons, our estimates of *L*_{e,Chl} in each space
bin come directly from the variograms calculated from daily,
mesoscale-isolating MODIS fields by Glover et al. (2018). While
their analysis used variograms to interpret scale, it is easy to show
(Appendix C) that a measure mathematically identical to integrating the ACF
to its first zero crossing can be derived from the variogram parameters. To
match the length scales defined by Middleton (1985), all
Eulerian integral length scales (velocity and Chl) are multiplied by 2.

## 3.6 Chl frequency spectra

Due to limitations of the data, ACF-derived scales might be biased large
(float scales due to large lag bins necessary for uneven sampling; Table 1)
or short (ocean colour scales due to poor intra-segment estimates of the
mean for sparse segments). Because the power spectrum *P*(*f*) is the Fourier
transform of the ACF, a time series with an exponential ACF has power
spectrum

which depends only on decorrelation time *T*. This spectrum is characterized by
a *f*^{0} power law at low frequencies and a *f*^{−2} power law at high
frequencies, with (2*π**T*)^{−1} setting the transition
frequency. For an independent measure of the power spectrum (that does not
rely on our estimated ACFs), spectra will be calculated on all Chl time
segments (same segments used for ACFs) using the Lomb–Scargle method
(Glover et al., 2011). The frequencies evaluated are the
equivalent of the Fourier harmonics for the segment length, and spectral
estimates are retained for $f\le \mathrm{1}/\mathrm{2}\u2329\mathrm{\Delta}t\u232a$, which
is an effective Nyquist frequency based on the average separation between
data points over the segment. For each platform, valid individual spectra
are averaged together (ocean colour segments with at least 50 % good
samples; float segments with at least 24 profiles).

## 4.1 Velocity analysis

### 4.1.1 Quasi-planktonic index (QPI)

The QPI methodology is illustrated in Fig. 2, where we display two examples
when a float was located near straining maxima at the intersection of
attractive and repulsive flow features. These are regions of rapid tracer
stretching, as evidenced by the elongation of synthetic particle clouds, and
represent a good challenge for a profiling float to keep up with surface
flows. In the case depicted in Fig. 2a where the time spanned by the three
adjacent profiles is only 2 d, the QPI is small at 4.49 km. The
distribution of the QPI for all float profiles is continuous and skewed long,
with a mode centred at about 0–5 km (Fig. 3a). The net velocities
experienced by the floats (by centred differencing their positions) are
well-correlated to the surface geostrophic velocities projected onto the
float track (zonal and meridional correlation coefficients of 0.79 over all
floats) but are systematically smaller by a factor of 2.3 (2.4) for *u* (*v*). The
samples with a QPI *<* 5 km (a threshold for display purposes, chosen as
a compromise between ensuring a small QPI and having a usable amount of
data) tend to fall closer to a one-to-one line (“*x*” symbols in Fig. 3b). The
good correlation suggests that the deeper mesoscale flows that the floats
feel are equivalent barotropic, and the nature of their deviations from a
surface Lagrangian trajectory are primarily in magnitude of displacement,
not in direction. For comparison, net drifter velocities have a slope of
nearly 1 with respect to surface geostrophic currents (Fig. 3c) but are no
better correlated to them (zonal correlation coefficient 0.75; meridional
correlation coefficient 0.72). Scatter about the one-to-one line is partly
due to ageostrophic effects.

The two examples in Fig. 2 differ markedly in their QPI, with a derivative time window increase from 2 to 4 d corresponding to a QPI that is ∼ 6 times larger and a float trajectory that is qualitatively different from surface trajectories. The profiling interval exhibits strong control on the QPI, with the QPI increasing nonlinearly with the derivative time window (Fig. 4). This relationship represents a combination of greater time for the float to experience more sluggish velocities at parking depth (seen in Fig. 3b), greater time afforded to integrate vertical shear, and the manifestation of two-particle dispersion statistics under quasi-geostrophic turbulence due to mismatch between the float and synthetic particle initial locations.

### 4.1.2 Integral timescales

Evaluating *T*_{l} as a function of *u*^{′} (Fig. 5a), drifters (open circles) and
floats (solid circles) largely cluster into two separate regions, with
drifters exhibiting greater velocity variance and a shorter timescale
(∼ 3 d compared with the ∼ 5 d of floats).
However, for space bins with multiple float segments, when we weight the
individual scales by QPI^{−2} (where the QPI is the segment median;
crosses connected by grey lines), we see the cluster of float points moves
towards the cluster of drifter points (shorter *T*_{l} and larger *u*^{′}). In
particular, the two rapid-time-sampling metbio^{*} float segments (triangles in
Fig. 5) reside even closer to the cluster of drifter points than do the
QPI^{−2}-weighted values from their host 5^{∘} × 5^{∘} bin. Nevertheless, they do not exactly reach the drifter
scales of the host bin (open blue and orange circles). Lagrangian integral
length scales (Fig. 5c) generally exhibit a similar clustering of points,
with floats having shorter length scales and with the QPI^{−2}-weighted
points moving towards the cluster of drifter points.

A similar relationship is found when examining ${T}_{\mathrm{l}}/{T}_{\mathrm{e}}$
as a function of ${u}^{\prime}/{c}^{\ast}$ (Fig. 5b). There are two
clusters of points (floats and drifters) which each reside on the
theoretical curve of Middleton (1985), and the
QPI^{−2}-weighted float values move along the curve towards the drifter
values. The same is generally true for ${L}_{\mathrm{l}}/{L}_{\mathrm{e}}$ as a
function of ${u}^{\prime}/{c}^{\ast}$ (Fig. 5d). These results suggest
that drifters are generally in the “frozen-field” regime of turbulence,
whereas floats are primarily in the “fixed-float” regime, effectively acting
as moorings. We found the floats to exhibit a continuum of behaviour, with
segments characterized by a smaller median QPI (trajectories more similar to
a surface Lagrangian trajectory) residing closer to drifters in ${T}_{\mathrm{l}}/{T}_{\mathrm{e}}$ and ${L}_{\mathrm{l}}/{L}_{\mathrm{e}}$
versus ${u}^{\prime}/{c}^{\ast}$ space. Overall, this is consistent
with the currents experienced by the floats being just as well-correlated to
surface geostrophic currents as are the currents experienced by surface
drifters (Fig. 3); this suggests that *T*_{e} calculated from mesoscale
geostrophic velocity anomalies derived from altimetry is a good estimate for
both the surface and deep flow, and the mesoscale currents are generally
equivalent barotropic with only a small horizonal wavenumber attenuation
over depth. Floats generally traverse the eddy field more sluggishly than a
surface parcel (but in a manner still aligned with the surface flow), and
this will impact how they sample a reactive tracer.

## 4.2 Chlorophyll analysis

### 4.2.1 Chlorophyll scales relative to the smoothed subtrahend

Relative to the smoothed subtrahend, both floats and drifters experience
${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}<\mathrm{1}$, even though drifters
move faster than floats, so the Chl field is evolving faster than either
platform moves (Fig. 6). This suggests that the primary balance in the
material derivative should be an approximate correspondence of the Eulerian
and Lagrangian tendency terms, EUL ≈ LAG (Eq. 5). We see that
this is the case, for example, for drifters with ocean colour in the two
space bins that host the rapidly sampling metbio^{*} segments (Fig. 7a, b; see
locations as bolded bins in Fig. 1a–c). The eddy advection term (ADV) is about
25 % the magnitude of either EUL or LAG and is presumably less important.
Drifters sample EUL ≈ LAG, whereas floats sample LAG *<* EUL,
except for the rapidly profiling metbio^{*} segments during which the float is
behaving most like a drifter and EUL ≈ LAG. As the primary
balance is EUL ≈ LAG, ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ for drifters is
approximately 1 in every space bin (Fig. 6a). The ratio is likely also
∼ 1 for floats, but our ratios are biased large because coarse
float time sampling demands using larger lag bins in the ACF (Table 1),
causing the structure of the ACF at small lag to be poorly resolved. We know
that the float scales are biased large because ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ must approach 1 as
${u}^{\prime}\to \mathrm{0}$ (the observer is stationary), but that is not the case. Averaged
float scales weighted by QPI^{−2} are shorter, approaching drifter with
ocean colour scales. For both platforms, because EUL ≈ LAG,
${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}<\mathrm{1}$ everywhere (Fig. 6b). That is because neither platform has enough
time to traverse *L*_{e, Chl} before Chl becomes decorrelated.
This is corroborated by counting the number of ocean colour pixels traversed
by a drifter over *T*_{l, Chl}, which is ∼ 1. The
composite frequency spectra for both platforms (Fig. 8a, b) reveal good
agreement with the model of an exponential ACF with an *e*-folding time
*T*_{l, Chl} of between 0.5 and 1 d (dashed reference curves). The
similar decorrelation time for both platforms is consistent with the
interpretation that EUL ≈ LAG relative to this subtrahend (the
different motions of the two platforms do not matter) and is consistent with
our interpretation that the float *T*_{l, Chl} values computed from the
ACFs are likely biased large and, in reality, are closer to the *T*_{l, Chl} ∼ 1 d of drifters with ocean colour.

### 4.2.2 Chlorophyll scales relative to the climatological subtrahend

Relative to the climatological subtrahend, floats experience ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\le \mathrm{1}$, whereas drifters
experience ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}>\mathrm{1}$. Therefore,
drifters sample Chl space–time fields by traversing mesoscale Chl structures
of diameter *L*_{e, Chl}, whereas floats sample Chl somewhat more
like a fixed observer (Fig. 6c, d). Drifters measure ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}<\mathrm{1}$, whereas floats
measure ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}\approx \mathrm{1}$. Note that this distinction between platforms is similar to what
we saw regarding how they each sample the velocity field (compare Fig. 5b and d with Fig. 6c and d), the meaning of which will be discussed in more detail later.
With ${u}^{\prime}>{c}_{\mathrm{Chl}}^{\ast}$ for drifters (and *u*^{′} occasionally close
to ${c}_{\mathrm{Chl}}^{\ast}$ for floats), we expect effects of advection to
be important. We see this to be the case, for example, in the two space bins
hosting the rapidly profiling metbio^{*} segments (Fig. 7c, d). For drifters
with ocean colour, EUL ≈ ADV and each is about 50 %–60 % of LAG in
those space bins, confirming that all terms are important. For floats, the
ADV term is smaller than it is for drifters (because floats have smaller
velocity variance) except for the rapidly profiling metbio^{*} segments which
have ADV close in magnitude to drifter ADV. The LAG term for floats tends to
be smaller than for drifters, except for the rapidly profiling metbio^{*}
segments, for which the term is large like it is for drifters.

As EUL ≈ ADV for drifters, ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}<\mathrm{1}$. This is because a portion
of the Lagrangian Chl signal is decorrelated by the platform traversing
lateral Chl gradients. For floats, because ADV is less important and LAG is
reduced, ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}\approx \mathrm{1}$, more closely approximating an Eulerian observer. Consistent
with the importance of advection, drifters with ocean colour tend to
experience ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}\approx \mathrm{1}$ or even greater
than 1 in some space bins (Fig. 6d). Drifters can experience
${L}_{\mathrm{l},\mathrm{Chl}}>{L}_{\mathrm{e},\mathrm{Chl}}$ because Lagrangian trajectories are
often aligned with isolines of tracer (Lehahn et al., 2007);
hence, drifters do not move directly down-gradient. Floats, on the other
hand, tend to sit on a linear one-to-one line in ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$ versus ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}$ space (Fig. 6d).
This is because EUL, not ADV, plays a primary role in the decorrelation
because floats move slower than the Chl field evolves, except for the fastest
moving floats which plateau where the drifter points do in Fig. 6d. Note
that averaged scales weighted by QPI^{−2} move towards drifter scales in
both ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ and ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$ versus ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}$ space (Fig. 6c, d), with increased velocity variance and generally (but not exclusively)
shorter *T*_{l, Chl} and longer *L*_{l, Chl}. Note also
that float-based results are not sensitive to ACF methodology (use of Eq. 8 or Eq. 9; Appendix D).

The frequency spectra (Fig. 8c, d) deviate from the theoretical power
spectra corresponding to an exponential ACF for any value of *T*_{l, Chl} (curves with ${T}_{\mathrm{l},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}=\mathrm{5}$ and 10 d are shown,
approximately bracketing the calculated drifter and float scales relative to
this subtrahend) and instead take on an approximate *f*^{−1} slope over
calculated frequencies. Given the role of ADV relative to this subtrahend,
one possibility is that the Lagrangian *f*^{−1} slope represents a
manifestation of an Eulerian wavenumber *k*^{−1} slope associated with a
passive tracer under quasi-geostrophic turbulence (Smith and
Ferrari, 2009) as the platform traverses the approximately frozen field, but
this cannot be confirmed. Regardless, an equal spectral slope for both
platforms is not inconsistent with our ACF-derived timescales but does mean
that we cannot use the spectra to corroborate them.

## 4.3 Biophysical interpretation of time and length scales

To interpret the meaning of these Chl scales, we compare them to the
velocity scales. If stirring by the mesoscale velocity field is a primary
driver of Chl variability, we expect the two variables to have similar
Eulerian scales. Beginning with the smoothed subtrahend, although the
Eulerian space scales of Chl and velocity are similar (Fig. 9b), the
Eulerian timescales are unrelated (Fig. 9a; squares), with *T*_{e, Chl} fixed at ∼ 2 d but *T*_{e} spanning
∼ 5–12 d. Along a trajectory, Chl decorrelates faster than
velocity (Fig. 9c) and over a shorter distance (Fig. 9d) for both
platforms, with floats experiencing longer decorrelation times of each
variable but covering a shorter distance before becoming decorrelated.
Eulerian velocities are calculated from satellite altimetry using a
geostrophic balance, therefore containing mesoscale and larger balanced
flows, and we note that Lagrangian velocities are from drifter trajectories
that were filtered in time to remove fluctuations shorter than 1.5 inertial
periods, a procedure that presumably removes primarily unbalanced motions.
Hence, all velocity signals are likely dominated by balanced, mesoscale
flows. The short time window of the Chl subtrahend (31 d) means that only
rapid fluctuations (relative to inverse growth rates of mesoscale baroclinic
instability or typical inverse phytoplankton accumulation rates) are
retained. While tight coupling between division and loss rates tends to keep
accumulation rates *r* low, abrupt changes in division rates due to rapid
changes in environmental conditions can cause large-amplitude fluctuations
in *r* (Behrenfeld and Boss, 2018). Field studies in the subpolar
North Atlantic observed near-surface accumulation rates of 0.47 d^{−1}
up to 0.77 d^{−1} (Graff and Behrenfeld, 2018),
corresponding to timescales of 1.3–2.1 d. Dynamically, submesoscale
processes have length scales on the order of the deformation radius of the
mixed layer (of order 1 km) and timescales on the order of an inverse inertial
period (of order 1 d) (Mahadevan, 2016), scales smaller than an
ocean colour pixel and shorter than those of mesoscale dynamics. Hence, the
EUL ≈ LAG balance for Chl relative to this subtrahend can be taken
as dominance of ocean colour pixel-scale variability in the retained signal
due to either submesoscale processes or biological sources/sinks.

For the climatological subtrahend, the Eulerian length scales are the same
as relative to the other subtrahend; thus, again, ${L}_{\mathrm{e},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}\approx {L}_{\mathrm{e}}$. However, for this subtrahend, ${T}_{\mathrm{e},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}\approx {T}_{\mathrm{e}}$
(Fig. 9a; circles). This is consistent with mesoscale dynamics setting the
Eulerian statistics of both velocity and Chl. Along a trajectory, Chl
decorrelates more slowly than velocity, and Chl is correlated over a
longer distance than velocity. The Lagrangian timescale for floats is
greater than it is for drifters, as the reduced importance of the ADV
term means that Chl properties are retained longer. The equivalence of Eulerian time and
length scales of velocity and Chl suggests that Chl scales are also
dominated by balanced mesoscale dynamics, via advection, and a physical
interpretation relating mesoscale stirring to Lagrangian Chl scales will be
presented in Sect. 4.5. Because lateral motion and the ADV term are
important, the differing motions of floats and drifters means that they each
sample different Chl signals relative to this subtrahend, with floats
behaving somewhere between an Eulerian and a surface Lagrangian observer.
The exception is the two metbio^{*} segments which sample Chl fields much like
a surface Lagrangian observer would.

## 4.4 Comparison with earlier estimates and a role for biology

Few studies have investigated Eulerian timescales of phytoplankton, and even
fewer works have addressed Lagrangian timescales. Further, comparison of results
across studies is complicated by myriad choices of data processing (e.g.
subtrahends), intrinsic data resolution, and methodology (ACF or otherwise).
In a series of studies, Denman and Abbott (1988, 1994)
analysed a scale-dependent decorrelation time by assessing the spatial
coherence of ocean colour images separated in time and found that Eulerian
decorrelation times are generally less than about a week, being longer for
50–100 km wavelengths and shorter for 25–50 km wavelengths. Wavelengths
smaller than 25 km are decorrelated after about 1 d. Comparing the
cross-coherence of Chl and sea surface temperature (SST), they concluded that physical stirring is the
major driver of Chl variability at the mesoscale, a conclusion shared by
Glover et al. (2018). In a more recent study, Kuhn et al. (2019) assessed Eulerian temporal decorrelation of
numerically modelled biomass and found decorrelation times to generally be
about 15 d (longer in regions of lower eddy kinetic energy and for larger
phytoplankton size classes). Their longer decorrelation times may be
partially due to analysing 3 d averaged model fields. Our
*T*_{e, Chl} values fall into this broad range. The stark
difference in values relative to the two subtrahends may be partially
explained by the wavelength-based analyses of Denman and Abbott (1988), whereby the smoothed subtrahend is emphasizing
pixel-scale (∼ 25 km) variability, contributing to the short
(∼ 1–2 d) *T*_{e, Chl} compared with the climatology
subtrahend (∼ 5–12 d).

Estimates of *T*_{l, Chl} are rarer. Abbott and Letelier (1998) used bio-optical surface drifters in the California
Current and found that *T*_{l, Chl} and *T*_{l} in the open ocean are
both about 2.5 d whereas *T*_{l, SST} is closer to 7 d,
causing them to question the degree to which Chl behaves as a passive
tracer. Boss et al. (2008) found much longer *T*_{l, Chl} of about 2 weeks using a profiling float. We also find *T*_{l}∼ 2–3 d for drifters (longer for floats), but
*T*_{l, Chl} is systematically larger or smaller depending on the
subtrahend (note that Abbott and Letelier, 1998, detrended float segments, whereas Boss et
al., 2008, only removed a scalar mean, with the latter observing that seasonality
dominated their longer decorrelation time). In a series of studies,
Jönsson et al. (2009, 2011) used synthetic particle
trajectories and satellite ocean colour to quantify terms of the material
derivative. They found that the advective term is generally comparable in
magnitude to the Lagrangian tendency and must be included. There are
important differences in the magnitude and sign of each term, where a
near-zero Eulerian tendency can be explained by a large Lagrangian tendency
countering an advective term.

Analyses of Lagrangian series point to an importance of biological sources
and sinks. Advection of phytoplankton across spatial gradients of
environmental conditions will affect dominant phenotypes
(Lévy et al., 2014), with the relative timescales of
physical parameters and physiological acclimation governing species
succession (Zaiss et al., 2021). The LAG term is generally
the largest term in the scaling of Eq. (5) (Fig. 7), meaning that turbulent
diffusion or sources and sinks of Chl are important. Relative to the
smoothed subtrahend, as discussed earlier, the appropriate spatial scales are
not resolved, and an approximate balance of LAG ≈ EUL is achieved.
However, relative to the climatological subtrahend, the magnitudes of LAG and ADV
tend to scale with each other, being largest for drifters and the metbio^{*}
segments and smallest for the unweighted float segments (Fig. 7). By
moving slower than phytoplankton patches, floats underestimate sources and
turbulent diffusion (reducing LAG); thus, there is less Chl
variance apparently advected (reducing ADV, where we interpret platform
speed as the typical speed of velocity fluctuations). Note that this is
consistent with the interpretation posed in Sect. 4.3: effects of turbulent
diffusion and sources and sinks are contained in the information content of
ocean colour space–time fields and will project onto an Eulerian tendency or
an advective term depending on the relative rates at which a phytoplankton
patch and observer move. It remains to be seen which of turbulent diffusion
(DIFF) or sources (*S*) is important in driving the decorrelation, and this will
be addressed in Sect. 4.5.

Finally, comparing Lagrangian and Eulerian timescales of Chl and of phytoplankton biomass (from backscattering) reveals regional discrepancies, confirming that Chl contains an acclimation component; however, each variable has a similar ratio of Lagrangian-to-Eulerian timescales, confirming that they are sampled by a moving observer in the same manner and giving us confidence in our Chl-based biophysical interpretation (Appendix E).

## 4.5 Empirical relationship between chlorophyll Lagrangian and Eulerian scales

The plots of ${T}_{\mathrm{l}}/{T}_{\mathrm{e}}$ and ${L}_{\mathrm{l}}/{L}_{\mathrm{e}}$ as functions of $\mathit{\alpha}={u}^{\prime}/{c}^{\ast}$ (Fig. 5b, d) and ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ and ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$ as functions of ${\mathit{\alpha}}_{\mathrm{Chl}}={u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}$ (Fig. 6c, d) have the same general shape. A similar functional dependence is not surprising given correspondence of Eulerian time and length scales for both velocity and Chl (Fig. 9a, b), with only subtle differences in the Lagrangian time and length scales of each variable (Fig. 9c, d) accounting for differences in the underlying functions. For this reason, we posit that a mesoscale relationship for Chl may take the same form as that Middleton (1985) derived for velocity:

Here, the constant *q*_{Chl} sets the asymptotic value at large
*α*_{Chl}, and the exponent *s* sets how rapidly *G* transitions
between linear and constant behaviour. In the velocity scaling (Eq. 3) of Middleton (1985), the asymptotic value *q* is less than 1 because the
variable being decorrelated is the same variable causing the decorrelation
and *L*_{l} never reaches *L*_{e}. For Chl, one possibility is to set
*q*_{Chl}=1 so that as ADV dominates $\left({u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\to \mathrm{\infty}\right)$ we
have ${L}_{\mathrm{l},\mathrm{Chl}}\to {L}_{\mathrm{e},\mathrm{Chl}}$. However, we have seen that
*L*_{l,Chl} is routinely greater than *L*_{e,Chl}. A better choice (purely
empirically) seems to be ${q}_{\mathrm{Chl}}=\mathit{\pi}/\mathrm{2}$ so that as ADV dominates we
have ${L}_{\mathrm{l},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}/{L}_{\mathrm{e},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}\to \mathit{\pi}/\mathrm{2}$ (grey curves in Fig. 6). For simplicity, we set
*s*=2.

The observation ${L}_{\mathrm{e},\mathrm{Chl}}\approx {L}_{\mathrm{e}}$ shows that the Chl field
is characterized by features with scales similar to mesoscale eddies. To
take this a step further, Glover et al. (2018) showed that, in our study region
of the North Atlantic, the statistical decorrelation length
*L*_{e,Chl} is proportional to a mixing length that quantifies the
distance that a mesoscale eddy could stir a water parcel containing Chl
anomalies, assuming that all Chl anomalies are generated by stirring a mean
gradient (see their Fig. 7). This provides a further clue that mesoscale
stirring might be of primary importance in setting Lagrangian Chl
statistics. The effects of stirring should be most apparent in the asymptotic
limit ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\to \mathrm{\infty}$ as turbulent velocity fluctuations are larger than the
evolution speed of the Chl field. Because Chl isolines twist and deform as
they are strained by the mesoscale field, ${L}_{\mathrm{l},\mathrm{Chl}}={q}_{\mathrm{Chl}}{L}_{\mathrm{e},\mathrm{Chl}}$ can exceed *L*_{e,Chl}. To motivate an empirical
value ${q}_{\mathrm{Chl}}=\mathit{\pi}/\mathrm{2}$ relating the frames, we appeal to an idealized
geometry of mesoscale eddies. Following the convention of Middleton (1985), *L*_{e} equals twice the decorrelation length and is
effectively a mesoscale eddy diameter. From this perspective,
${L}_{\mathrm{e},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}/\mathrm{2}$ serves as a radius of curvature
for the maximal ${L}_{\mathrm{l},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}=\mathit{\pi}{L}_{\mathrm{e},\phantom{\rule{0.125em}{0ex}}\mathrm{Chl}}/\mathrm{2}$ (half a circumference) of a parcel traversing
the perimeter of a mesoscale eddy as it is stirred. Half a circumference may
be a meaningful decorrelation length in the scenario where a mean Chl
gradient is stirred by an eddy of diameter *L*_{e}, partitioning the eddy
into two hemispheres of high and low Chl respectively (e.g. see Fig. 2a of
Gaube et al., 2014) and effectively equating the
Eulerian separation *L*_{e,Chl} in Euclidean space to a trajectory
distance *L*_{l,Chl}. At the other asymptotic limit, if ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\to \mathrm{0}$, turbulent
velocity fluctuations relative to the translation of the Chl field are small
and ${T}_{\mathrm{l},\mathrm{Chl}}\to {T}_{\mathrm{e},\mathrm{Chl}}$ so that ${L}_{\mathrm{l},\mathrm{Chl}}<{L}_{\mathrm{e},\mathrm{Chl}}$. With this interpretation, it is useful to view ADV as a
local stirring of a mean Chl field instead of advection of anomalies over
long distances (although the equivalence of *L*_{e,Chl} and a mixing
length shown by Glover et al., 2018, relates the two perspectives). This is
why the effect of *u*^{′} on ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ matters in its relation to
${c}_{\mathrm{Chl}}^{\ast}$, which is an evolution speed for the scalar field.
Likewise, it is useful to view *u*^{′} as turbulent velocity fluctuations,
which by our filtering are associated with the mesoscale eddy field. When
the observer is a true surface Lagrangian observer, *u*^{′} is properly
captured by the platform's motion; however, as we have shown, for an Argo float,
the effects of stirring are underestimated.

The relationships in Eq. (12) reveal further insight into the processes that
cause the Lagrangian decorrelation of Chl. The turbulent diffusion term from
Eq. (1) (DIFF) can be expressed as a Fickian diffusion with coefficient *K* that is
constant over an integral timescale:

In the Lagrangian frame, the diffusivity *K* scales as $K={\left({u}^{\prime}\right)}^{\mathrm{2}}{T}_{\mathrm{l}}$ (Taylor, 1922), allowing us to scale DIFF as follows:

If we define a parameter *β* as the ratio of the total Lagrangian
tendency to the contribution from turbulent diffusion, we obtain the
following scaling:

This says that *β* is equal to the square of the ratio of the Eulerian
length scale of Chl to the geometric mean of the Lagrangian length scales of
Chl and of velocity. From our Fig. 9d, relative to the climatology
subtrahend, we have ${L}_{\mathrm{l},\mathrm{Chl}}\approx \mathrm{2.5}{L}_{\mathrm{l}}$ so that we can
rewrite *β* as follows:

If we consider the case where LAG is entirely driven by DIFF (*β*=1),
we have

It is worth noting that the asymptotic value ${q}_{\mathrm{Chl}}={L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$ (for ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\to \mathrm{\infty})$,
heuristically argued for above, is $\mathit{\pi}/\mathrm{2}$, which is quite close to
2.5${}^{\mathrm{1}/\mathrm{2}}$. This suggests that, in the limit of large turbulent velocity
fluctuations (where ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}\to {q}_{\mathrm{Chl}})$, we have
*L*_{l,Chl} (and hence *T*_{l,Chl}) determined entirely by turbulent
diffusion. Because ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$ is globally less than or equal to *q*_{Chl} (Fig. 6d; Eq. 12), we can obtain an inequality for *β*, revealing that *β*= LAG $/$ DIFF is globally greater than or equal to 1, or equivalently LAG ≥ DIFF. This must
be true because, from Eq. (1), LAG = DIFF +*S*. As an extension of this result,
it is implied that *S* becomes increasingly important in setting Lagrangian
decorrelation where ${L}_{\mathrm{l},\mathrm{Chl}}\ll {L}_{\mathrm{e},\mathrm{Chl}}$ (as *β*≫1), and this happens in the limit that turbulent
velocity fluctuations are relatively small (${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\ll \mathrm{1}$). Therefore, in short, we have confirmed our earlier
interpretation that when ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\gg \mathrm{1}$, DIFF drives Lagrangian decorrelation
*T*_{l,Chl} as a consequence of mesoscale stirring, and when ${u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\ll \mathrm{1}$, sources *S*
(biological or otherwise) drive Lagrangian decorrelation.

## 4.6 Methodological decisions

Many methodological decisions were made in this study. Here, we discuss some of them and comment on how they may influence our results.

### 4.6.1 Depth reduction of float time series

As described in Sect. 3.2, the Chl time series constructed from the float measurements are weighted by depth to better approximate what satellites see, allowing us to compare scales derived from floats to those derived from ocean colour. We found that using float series calculated as a more traditional (and biophysically meaningful) depth average over the mixed layer (e.g. Yang et al., 2020) yields results that are not appreciably different from those in Figs. 6–9.

### 4.6.2 ACF parameters

Although the methodology is consistent, the segment length and ACF bin sizes
vary for different platforms (Table 1). Lagrangian segments should be kept
as short as possible because a platform may encounter different
environmental (physical or otherwise) conditions as it moves; we followed Lumpkin et
al. (2002) and used 120 d for all variables. For Eulerian series,
this is less of an issue, and, as seasonal variability is removed, the length
of the segment is less likely to have a significant impact on scales. Hence, we used 365–366 d segments for Chl. Regarding temporal ACF
bin sizes, ideally one would use a bin size that matches the sampling
interval because this is the smallest lag that can be resolved. For this
reason, the ACFs based on satellite altimetry, satellite ocean colour, or
drifters use a bin size of 1 d. The floats have a variable profiling
interval (Fig. 4). While they sometimes profile with a frequency of about
once per day, they generally profile less frequently, and we settled on a bin
size of 5 d. The two metbio^{*} float segments are given special attention
because they profiled more frequently; for that reason, we were able to
use a finer bin size of 1 d. As a general statement, choosing a larger bin
size causes the structure (curvature) of the ACF to be poorly resolved at short
lag and biases timescales large, as discussed in Sect. 4.2. A similar
rationale applies to the spatial ACF bin size, where 27.8 km approximately
corresponds to the 0.25^{∘} resolution of the data in the
latitudinal direction. The scales are averaged in (or the ACFs are
composited in) 5^{∘} × 5^{∘} space bins to enhance
the quality of estimates, and this size was chosen to match the grid size of
Glover et al. (2018), who computed variograms of Chl and
found this size good to resolve spatial variability.

### 4.6.3 Ocean colour product

There is a trade-off between resolving more variance and dealing with
increased gaps when moving to a finer-resolution ocean colour product. For
the purpose of this study, we chose to prioritize data coverage, leading us
to select a blended, 0.25^{∘} product, and focus on the
mesoscales. The choice of product is also consistent with the grid size of
the Eulerian velocity field (0.25^{∘} altimetric geostrophic
currents), which is important because we compare the two variables (Fig. 9). In
particular, GlobColour was selected because Zhang et al. (2019)
demonstrated it to resolve realistic Lagrangian behaviour in terms of
(sub)mesoscale dynamics, so we conclude that its space–time information is
biophysically accurate.

We suspect that our findings depend on our choice of ocean colour product and
constitute a representation of mesoscale biophysical dynamics: studies using
submesoscale-resolving velocity and ocean colour data may find different
relationships between Lagrangian and Eulerian scales. In particular, we
suspect that the finding ${T}_{\mathrm{l},\mathrm{Chl}}\le {T}_{\mathrm{e},\mathrm{Chl}}$ is
indicative of mesoscale variance and a consequence, in part, of an
0.25^{∘} ocean colour product. This may be the manifestation of
an observer moving across gradients in the mean chlorophyll field, as would
happen when a mesoscale eddy stirs a horizontal gradient as discussed in
Sect. 4.5, and is consistent with the importance of DIFF – a manifestation of
unresolved advection – in driving the Lagrangian decorrelation over most of
the range of observations instead of *S*. Chlorophyll may actually be conserved
for longer along a trajectory than our results would indicate: if patches
are organized in filaments not fully resolved in a 0.25^{∘}
product, the inability of a drifter-projected time series to resolve
near-constant chlorophyll levels along a filament will result in an early
temporal decorrelation. The result that the ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ ratio is approximately 1
relative to the smoothed subtrahend (where sub-pixel variability probably
dominates) while the ratio is less than 1 relative to the climatology
subtrahend supports this interpretation.

We analysed the Lagrangian and Eulerian statistics (time and length scales) of
velocity and chlorophyll (Chl) as measured by Bio-Argo floats and as
represented by satellite ocean colour (GlobColour) projected onto surface
drifter tracks. Lagrangian statistics of velocity satisfy the Middleton
(1985) relations, with drifters in a frozen field regime
(spatial Eulerian decorrelation drives temporal Lagrangian decorrelation)
and floats in a fixed-float regime (Eulerian tendency drives
decorrelation). However, there is a continuum of behaviour with segments
weighted by the inverse square of the quasi-planktonic index (QPI) – a
metric quantifying the similarity of a float trajectory to a surface geostrophic
trajectory – approaching the frozen-field limit. This is made possible by
the mesoscale flows being approximately equivalent barotropic with small
horizontal wavenumber attenuation over depth. Given the space–time
resolution of our ocean colour product (and typical float time sampling),
both floats and drifters sample anomalies relative to the smoothed
subtrahend as fixed observers, suggesting that Lagrangian Chl variability is dominated by ocean colour pixel-scale
processes at periods shorter than 31 d. Analysis of a finer ocean colour product and a faster sampling
observer is necessary to elucidate biophysical mechanisms (e.g.
submesoscale sources and sinks). However, relative to a climatological
subtrahend, Eulerian decorrelation time and length scales of Chl match those
of velocity, suggesting that mesoscale physical dynamics are important in setting
plankton distributions, as suggested by earlier studies
(Denman and Abbott, 1994; Glover et al., 2018). The ratio
of Lagrangian to Eulerian length scales for chlorophyll, ${L}_{\mathrm{l},\mathrm{Chl}}/{L}_{\mathrm{e},\mathrm{Chl}}$, depends on how fast
a parcel moves relative to how fast the Chl field evolves $\left({u}^{\prime}/{c}_{\mathrm{Chl}}^{\ast}\right)$, following an
empirical curve that appears to have the same functional form as that for
velocity but with the asymptotic value replaced by scalar *q*_{Chl}≥1, a value consistent with stirring by mesoscale eddies. A fundamental
result of this study is that mesoscale Lagrangian time and length scales are
strongly set by the stirring of mesoscale eddies, with turbulent diffusion
generally dominating the decorrelation. As for the biophysical
interpretation of float-measured time series, qualitatively speaking, the
slower horizontal motion of floats relative to surface Chl patch speed means
that advection across mean Chl gradients is reduced and both turbulent
diffusion and intra-patch biological sources are underestimated, both
leading to longer timescales (*T*_{l, Chl}) and smaller Lagrangian
tendency (LAG) and eddy advection (ADV) terms compared with a surface
Lagrangian observer. By choice of data products and filtering, our results
are based on time and length scales representative of mesoscale variances
in velocity and Chl fields. Follow up studies using data resolving
submesoscale variance are warranted.

The quasi-planktonic index (QPI) is a single number that quantifies the
similarity of a float trajectory to the best-fit synthetic trajectory
(computed by the advection of synthetic particles with altimetric total
geostrophic currents) over the temporal footprint of a centred difference
derivative (three total float profiles). The procedure is repeated for each
float profile *i*. We first create a disc of particles about ${\mathit{r}}_{i}=\left({\mathrm{long}}_{i},{\mathrm{lat}}_{i}\right)$ by making a rectangle
about *r*_{i} with particles spaced zonally by ${\mathit{\delta}}_{\mathrm{0}}/\mathrm{cos}\left({\mathrm{lat}}_{i}\right)$ out to ${\mathrm{long}}_{i}\pm {\mathit{\delta}}_{\mathrm{total}}/\mathrm{cos}\left({\mathrm{lat}}_{i}\right)$ and meridionally by *δ*_{0} out to lat_{i}±*δ*_{total} and then retaining those values with a spherical
distance of less than or equal to 0.3^{∘}. We chose ${\mathit{\delta}}_{\mathrm{0}}=\mathrm{0.05}{}^{\circ}$ and ${\mathit{\delta}}_{\mathrm{total}}=\mathrm{0.3}{}^{\circ}$. The disc of
*K* particles is advected by the altimetric total geostrophic currents forward
in time with a fourth-order Runge–Kutta scheme at an hourly time step from
their initial position at *t*_{i} to the first hour past float time
*t*_{i+1}. Velocity fields at intermediate time steps are obtained through
linear interpolation, and particle velocities are updated through bilinear
interpolation in space. For the *k*th trajectory beginning at float time
*t*_{i}, its position at float time *t*_{i+1} is obtained by linearly
interpolating its positions at the two surrounding hourly time steps.
Similarly, the disc of *K* particles is advected backward in time with the same
fourth-order Runge–Kutta scheme and hourly time step out to the first hour
prior to *t*_{i−1}. For the *k*th trajectory beginning at float time
*t*_{i}, its position at float time *t*_{i−1} is obtained by linearly
interpolating its positions at the two surrounding hourly time steps.

The advection gives *K* sets of three positions at each float step *i*. The
penalty function for trajectory *k* at float step *i* measures the average
distance between the *k*th synthetic trajectory and the true float
trajectory over the three time steps that constitute the centred
difference:

where ** r** is the position in latitude–longitude coordinates, and
dist(⋅) is a measure of distance on the sphere.
The QPI for time step

*i*is

and has units of kilometres. The corresponding shadow trajectory is the
three-element trajectory *r*_{k} for the *k* that minimizes *S*.

Synthetic particle trajectories are advected by altimetric geostrophic
currents, which generally approximate the mesoscale flows of interest well.
Della Penna et al. (2015) found that including an Ekman term in
the advecting flow modified only the tail of the distribution of a QPI
calculated against surface drifter trajectories. To gauge how altimetric
geostrophic trajectories approximate surface Lagrangian trajectories, we
also computed a QPI for all drifter returns in our study space–time domain
calculated over Δ*t*=2 d and found that they are larger than expected
(median ∼ 8 km). However, the latitudinal variations in the
drifter QPI are much smaller than latitudinal variations in the float QPI
(the subset with a comparable Δ*t* ∼ 2 d), even though
a geostrophic approximation should perform better at mid-latitudes because
Ekman transports become more important away from the generally balanced,
persistent Gulf Stream and North Atlantic Current and their energetic
eddies. This suggests that the median drifter QPI is large not because of
failing to include an Ekman term but because geostrophic currents from
mapped altimetry data do not resolve inter-swath deformations to the
mesoscale field (Ascani et al., 2013) and underestimate
surface flows when finite differencing (Sudre and Morrow, 2008).
We conclude that the QPI is a useful measure of similarity to surface flow
for floats.

The first subtrahend (“smoothed”) is meant to replicate that used by
Glover et al. (2018) so that we can obtain Lagrangian and
Eulerian timescales consistent with their Eulerian space scales. For
Eulerian data, we perform a 3-D convolution of all GlobColour space–time
fields with a 3-D filter defined as a 31 d Hamming window in time and a
2-D Gaussian in space with a 1^{∘} full width at half maximum and
a 2^{∘} cutoff. Just as the total GlobColour data are
bilinearly interpolated onto drifter tracks, the subtrahend is interpolated
in the same manner, and the two are differenced to obtain Lagrangian Chl
anomalies. Finally, for a float Chl subtrahend, we perform a weighted
running average of each float Chl series with weights defined by a 31 d
Hamming window. This approach to smoothing the float data is less than
desirable because it constitutes a Lagrangian subtrahend, whereas the drifter
anomalies are about an Eulerian subtrahend projected onto a Lagrangian
trajectory. Further, this approach cannot implement an equivalent spatial
smoothing in the subtrahend for the float data because they come from a single
Lagrangian trajectory (see discussion below). The time filter component of
the smoothed subtrahend is effectively a low-pass filter, and, with a
cutoff of 31 d, we found it to retain a sizable portion of the
intraseasonal (and perhaps mesoscale) variance. This is particularly the
case for the float data, where uneven spacing means that the effective
cutoff frequency will vary depending on the sparsity of sampling in a 31 d
window because data points near the centre of the window are weighted more.

As an alternative, the second subtrahend strictly isolates climatological
variability (“climatology”). The spatial footprint of the filter is the
same, but the time filtering is performed about a day-of-year coordinate
rather than an absolute date coordinate (for example, 1 January of every
year is regarded as having the same time coordinate). Specifically, for
Eulerian data, we perform a 4-D convolution of all GlobColour space–time
fields with a 4-D filter defined as a 31-day-of-year Hamming window in time,
a 2-D Gaussian in space with a 1^{∘} full width at half maximum
and a 2^{∘} cutoff, and a boxcar function (equal weights)
across years. Again, the subtrahend is bilinearly interpolated onto the
drifter tracks.

The major difference for the climatological subtrahend concerns how the floats are treated. There are not enough co-located floats to create a climatological subtrahend from float-measured Chl, so we construct a float subtrahend from the filtered GlobColour data (Eulerian subtrahend). Because floats measure Chl year-round, whereas GlobColour contains significant seasonal gaps, we first regress annual and semiannual cosines to the Eulerian subtrahend in each pixel as an interpolant. While the inclusion of higher harmonics would yield a better fit in some regions (especially in regions with complex seasonal cycles such as those with spring and autumn phytoplankton blooms), we stick to only the first two harmonics because gaps in the Eulerian subtrahend of up to 6 months would lead to significant overshoot if higher harmonics were included. The resulting field is then projected onto the float tracks (with a simple nearest-neighbour approach in space and time). Finally, because the dynamic range of GlobColour Chl is different from that of the depth-averaged float Chl series, the projected subtrahend needs to be scaled before it can be removed from the float data (like a bias correction). Thus, as a final step, we regress the projected subtrahend against actual float Chl (all samples across all floats) to obtain a best estimate of seasonal cycle amplitude:

A summary of the subtrahends is provided in Table B1, and a visual example of time series is shown in Fig. B1. Anomalies about the climatology subtrahend contain substantially more low-frequency variance, including intraseasonal variability and interannual variability due to year-to-year variations in phasing and amplitude of blooming. The climatology subtrahend has the added benefit of being applied to the floats and drifters in an equivalent manner as an Eulerian subtrahend projected onto a Lagrangian trajectory. It is fair to question whether the smoothed subtrahend as computed for floats is comparable to the smoothed subtrahend as computed for ocean colour pixels and drifters because the former does not include explicit spatial smoothing, only a combined space–time Lagrangian smoothing. As a test, Fig. B1 additionally includes the Eulerian smoothed subtrahend projected onto float tracks (via the nearest-neighbour interpolant and regressed against float data to correct for dynamic range) for periods when we have overlapping data. In general, the projected Eulerian smoothed subtrahend closely agrees with the float smoothed subtrahend. Exact correspondence between float and satellite Chl is not expected because floats sample a parcel of water on the order of the size of the platform, whereas gridded ocean colour products integrate information in space and time (see discussion in Yang et al., 2020).

Glover et al. (2018) calculate space scales of mesoscale Chl
variability by calculating variograms of mesoscale Chl anomaly fields in
5^{∘} × 5^{∘} space bins. They fit a spherical
variogram model to their calculations by nonlinear regression, which in 1-D is given by

where *c*_{0} is the nugget (unresolved variance), *c*_{∞} is the
sill (approximating the total variance), *a* is the range (closely related to
the decorrelation length), and *δ* is the scale of space separation
being evaluated (Glover et al., 2011, 2018).
Intuitively, *a* is closely related to the integral decorrelation length
(integral of spatial ACF to first zero crossing), but the two are not equal.
The spatial autocorrelation function *R*(δ) is related to
the variogram by

where *C*(δ) is the autocovariance function. Using the
fact that *C*(0)=*c*_{∞}, the space lag *δ*_{0} at
which the ACF *R*=0 is determined by rearranging Eq. (C2) to yield
*c*_{∞}=*γ*(*δ*_{0}), requiring *δ*_{0}=*a*. Thus, the integral of the ACF to the first zero crossing is given by

which, after substituting in Eq. (C1), gives

Glover et al. (2018) report separate ranges *a*_{x} and *a*_{y} from MODIS data in the zonal and meridional directions respectively.
Therefore, we calculate zonal and meridional integral length scales *L*_{x}
and *L*_{y} from Eq. (C4) to define the Chl integral length scale,

In this paper, we analyse chlorophyll, as there is a history of
geostatistical studies of that variable to build upon
(e.g. Denman
and Abbott, 1988, 1994; Doney et al., 2003; Glover et al., 2018; Eveleth et
al., 2021). Chl is a complicated variable, containing a regionally (and
seasonally) strong acclimation signal in addition to a biomass signal
(Behrenfeld et al., 2005) that may imprint on our results.
As a check, we evaluated Eulerian and Lagrangian timescales of phytoplankton
carbon biomass (*C*_{phyto}) and compared them to timescales of Chl.
*C*_{phyto} is derived from float (at 700 nm) or satellite-measured
(at 443 nm) backscattering (following Table 2 in Graff et al.,
2015, and assuming a spectral power law for backscattering of −0.78),
anomalies are computed relative to equivalently constructed subtrahends as
for Chl (as described in Sect. 3.3), and its ACF is integrated as for Chl
(as described in Sect. 3.4). In general, Eulerian and Lagrangian timescales
of Chl are longer than timescales of *C*_{phyto} in the subtropics
(as defined by the −0.10 m mean absolute dynamic
topography contour; Della Penna and Gaube, 2019), are shorter than
timescales of *C*_{phyto} near coasts, and are approximately equal
elsewhere, suggesting that there is a regional acclimation signal built into
Chl that affects its timescales in certain regions of the ocean. However,
importantly for this study, the relevant quantity ${T}_{\mathrm{l},\mathrm{Chl}}/{T}_{\mathrm{e},\mathrm{Chl}}$ is proportional to
${T}_{\mathrm{l},{C}_{\mathrm{phyto}}}/{T}_{\mathrm{e},{C}_{\mathrm{phyto}}}$ everywhere, meaning that, even if the biophysical
drivers of variability in Chl and *C*_{phyto} vary regionally, the variables
are sampled equivalently by a Lagrangian (or quasi-Lagrangian) observer,
giving us confidence in our Chl-based results.

NAAMES float data are available from the University of Maine In-Situ Sound & Color Lab archive (http://misclab.umeoce.maine.edu/floats/, Haentjens and Boss, 2020). Sprof files for the non-NAAMES floats and prof files for the NAAMES floats (hydrographic variables only) are available from the IFREMER Argo Global Data Assembly Center (snapshot from February 2021; https://doi.org/10.17882/42182{#}81474, Argo, 2021). Altimetry data (product SEALEVEL_GLO_PHY_L4_REP_OBSERVATIONS_008_047; https://doi.org/10.48670/moi-00148) use the DT2018 processing (described in Taburet et al., 2019) which was superseded by the DT2021 version at the time of writing and is no longer accessible by URL; however, data can be de-archived by the Copernicus Marine Environmental Monitoring Service (https://marine.copernicus.eu/, Copernicus Marine Environment Monitoring Service, 2021) upon request. GlobColour data (variables CHL1 and BBP using “merged” sensors, L3m, daily binning, 25 km resolution, and the GSM algorithm) are from the R2019 processing available by web or FTP from ACRI-ST, France (https://hermes.acri.fr, ACRI GlobColour Team, 2020). Drifter data are the quality-controlled 6 h product available from the NOAA Global Drifter Program (https://doi.org/10.25921/7ntx-z961, Lumpkin and Centurioni, 2019), where we accessed the ASCII files on 14 September 2020.

DCM and SCD designed the study and DCM performed the analyses. ESB acquired and processed the NAAMES float data, DMG provided the Chl variogram parameters, and ADP helped develop the QPI. DCM prepared the manuscript with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

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

We thank the two referees whose feedback improved this manuscript. Additional support was provided to Scott C. Doney and Darren C. McKee by the University of Virginia.

This research has been supported by the National Aeronautics and Space Administration, Earth Sciences Division (grant nos. 80NSSC18K0018 and 80NSSC20M0202).

This paper was edited by Ciavatta Stefano and reviewed by Bror Jonsson and one anonymous referee.

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