Author Correction: Vertical migration by bulk phytoplankton sustains biodiversity and nutrient input to the surface ocean

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

q r (t) = q 1 + q 2 e −pt (S1) The coefficients q 1 , q 2 and p are set according to whether the cell rests in the light-limited part of the water column (where a(q, I) I <1), or in the sun-lit surface layer. Under light limitation, quota dependency includes the adsorption-Droop term q/q + Q 0 , leading to U − pq at the right hand side of Eq. (2) with p=P max α(1 − I/I α )I and q 1 =U/p. From the initial condition q r (t = 0)=q(0), we have q 2 =q(0) − q 1 . Under light saturation, the subsistence quota contribution in (q+Q 0 )P max can be shifted to the uptake term such that the coefficients read p=P max , q 1 =(U −Q 0 p)/p, and again q 2 =q(0) − q 1 . However, the resulting exponential depletion of internal nutrient stores will quickly lower light affinity to the point that light becomes limiting (a(q, I) I <1). Without refurnishing supplies, thus U=0, q 1 ≤ 0, and regardless of the degree of light limitation, internal stores are depleted proportional to the specific production rate (p). The exponential decrease of q r (t) in Eq. (S1) reflects a resource depletion acting through time and proportional to the current value of q r (t) itself. Intracellular depletion of quota is faster at high q, when both production and nutrient demand are high.
(2) As cells move vertically, rates of both net primary production P and nutrient uptake U in Eq. (1)-(2) vary, and our approach requires an explicit formulation for their dependence on ambient conditions. Nutrient uptake rate U depends linearly on depth-dependent ambient nutrient concentration, because we omitted an explicit account of uptake saturation and down-regulation at high internal nutrient stores. Instead, we imposed the boundary condition q < q max , which suffices to capture spatio-temporal trends in q, because our approach neglects temporal changes in the ambient nutrient pool.
The dependence of phytoplankton physiology on migration history demands a shift in the coordinate system, from grid-based (Eulerian) to individual-based (Lagrangian). In this Lagrangian view, ambient conditions such as nutrient availability and light vary along the vertical pathways of individual cells. Nutrient uptake U then depends exponentially on vertical position z, U=u * e k N ·(z−z N ) , in accordance with the often observed steep rise in nutrient concentration around the chemocline position z N (see Fig. S2). At the chemocline center, U is by construction given by a global constant u * . Similarly, light availability, I, varies for a cell migrating within the chemocline, but in the opposite direction: I=I N e −k I (z−z N ) with date-and site-dependent irradiance I N at the chemocline center (z=z N , Sec. S4) and k I described in Eq. (S12) .
For a compact description of variations in both chemical and light conditions we introduce the variable ζ (t)=e −k I (z(t)−z N ) , which represents the experienced light level relative to the level at the chemocline center z N (i.e. ζ equals one at z(t)=z N ). As shown in Fig. 1 and Fig. S2, the slope of the chemocline is much steeper than that of the photocline. The relation k N =3k I provides a good fit to the NO 3 data and enables to use the position variable ζ also for describing variations in the uptake rate (U=u * ζ −3 , with u * denoting the nutrient uptake rate at the central chemocline). The new variable ζ not only represents the vertical position, but can also be substituted for time t for a cell assumed to move at constant vertical speed v. This movement can be either downward, dz/dt=v, or upward, dz/dt=−v, so that In the light-limited part of the water column (where a(q, I) I <1), the balance equation can be transformed into a differential equation in the relative light level ζ using Eq. (S2) : The rescaled photosynthesis rate, p ± =±p * /k I v, is dimensionless, and can be interpreted as potential production relative to a typical rate of change in light availability; the rescaled uptake u ± =±u * /k I v has the units of a nutrient quota. Eq. (S3) can be solved analytically, where the integration constant q 2± is determined by the initial condition q ± (ζ (0))=q(0). The special function called exponential integral Ei(ζ ) can be well approximated by series expansions, and is made available within many numerical packages. Unlike case (1) for resting cells, vertical migration leads to great variations in the specific exponential argument of Eq. (S4) over time. If the swimming is downward directed (p ± =+p * /k I v), the positive specific argument p ± describes an accelerating rise in internal nutrient quota during the passage as photosynthesis steadily declines while ambient nutrient levels increase.
On the contrary, for upward swimming (p ± =−p * /k I v), the specific rate becomes negative and this depletion rate increases with light availability ζ towards the surface. This change of ζ in turn depends on the vertical swimming speed v and light extinction coefficient k I defined in Eq. (S10) . Although both factors negatively affect the specific coefficient in the exponential function, their net effect on the argument in the exponential function is positive, because the two factors enhance differences in ζ (or ζ (t) − ζ (0)) along the trajectory of the cell.
(2b) When a cell enters the surface layer where light is saturating, a I Iq/(q + Q 0 ) >1, the balance equation with solution q s± (ζ (t)) in analogy to the light limited solution Eq. (S4) Again, the coefficient q 2± makes sure that q s± (0) matches the initial value of the intracellular free quota:

S2 Near-surface resting
Our algorithm for simplicity includes the scenario that cells rest near the surface but not at or below the deeper chemocline because at very high nutrient concentration and virtually no light, stopping has practically the same effect as extending migration deeper. A near-surface resting period is added, if the free quota at the upper turning point (z=z C − δ z) exceeds the threshold Q 0 . Resting time T r is then estimated using Eq. (S1) as the time needed for the free quota q to reach the threshold Q 0 , thus T r =log(q/2Q 0 )/p, with rescaled photosynthesis rate p introduced in Eq. (S1) . Near-surface resting modifies the biomass distribution function of active movers, φ (z), formulated in Eq. (5) . The algorithm adds a contribution φ r (z), which like φ (z) arises from the solution of the one-dimensional diffusion equation. Dirac-type initial conditions at z=z C − δ z generate a Gaussian function, with diffusion length δ (Sec. S5) The ratio of T r over total migration time, w r =T r /(2T +T r ), becomes the relative weight of the resting density φ r when adding to φ (z)

S3 Base CHL profile
Complementarity between observed nitrate and chlorophyll-a profiles is often apparent in the time-series data at all five marine stations in cases of absent subsurface chlorophyll maximum (SCM). This finding is generalized to estimate the CHL concentration of the passively drifting or "immobile" population Chl p constrained by the observed surface chlorophyll concentration Chl 0 . If typical mixing rates by far exceed growth and photo-acclimation rates, both biomass and pigment concentration will be constant throughout the inhabited surface zone.
The mean chemocline slope k N is set to three times the mean light attenuation coefficient (Sec. S1). The CHL penetration depth z N is slightly extended into the chemocline by 10m and in relation to temperature by adding 1.6 m per degree C decrease in temperature. In addition, the observed surface value Chl 0 is reduced by the factor f z in Eq. (4) at shallow chemocline depth, suggesting a contribution of the SCM to surface phytoplankton. However, this modification only applies at in the absence of co-limitation, hence for very small f N in Eq. (4) and within the (summer) low turbulence period (Sec. S5). Under co-limitation (NO 0 3 > 0.3mmol-Nm −3 ) and high surface CHL (>0.3mg-CHLm −3 ), the Chl:C profile of drifters is smoothed by raising the lower threshold c * in Eq. (8) by a factor of two.

S4 Light attenuation
Depth dependent light availability, I(z), is proportional to the surface incident irradiation I 0 , with Chl = Chl p + Chl a (S10) and accounts for attenuation by self-shading of the phytoplankton population determined by the attenuation coefficient k chl and the sum of pigments in passive drifters (Chl p , see Eq. (S9) ) and in active movers (Chl a ).

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Other optically active constituents such as colored dissolved organic matter (CDOM) are here subsumed with coefficient k 0 . The seasonal rise in attenuation 1 is expressed in terms of the time-variable turbulent mixing length δ (Sec. S5): both specific coefficients k chl and k 0 are multiplied with a factor max{1, δ /30m}, thus increase during winterly mixing.
The concentrations of biological constituents such as CDOM vary not only in time but also between sites. We suggest that the concentrations correlate with the long-term average of surface chlorophyll Chl 0 (in units mg-CHLm −3 ), from which we formulate site-specific differences in water attenuation (in units m −1 ): Note that the dependency of k 0 on average CHL levels can be interpreted as an extra contribution of the CHL-specific attenuation coefficient k chl . If k chl is retrieved from correlation analysis, the model value should thus be slightly below the empirical estimate. From the wide scatter in k chl from around 0.01-0.08 mg-CHL −1 m 22 , our setting k chl =0.04 mg-CHL −1 m 2 appears reasonable, especially in light of its seasonal enhancement. Total light attenuation k I near the chemocline can be approximated where the Chl:C ratio is calculated from Eq. (8) inserting q=0.4q max and I=I N =I 0 · exp (−k I z N ).

S5 Seasonality in turbulent mixing
In our study, the turbulent mixing length δ describes the vertical dispersal range during one day. δ is approximated using long-term satellite-derived data for sea surface temperature (SST, or T ) for each station obtained from www.esa-sst-cci.org. The amplitude of seasonal mixing is suggested to reflect the SST seasonality of each site. The seasonal amplitude should be low if the annual standard deviation in SST σ T is below a threshold, and large for high σ T , with a smooth transition. Summer stratification occurs in our simple approximation depending on the current and positive SST deviation from the temporal mean T , thus T =max(T − T , 0), and furthermore on the monthly change in SST, if positive, thus δ T + =max(T (t + 15d) − T (t − 15d, 0). Together, mixing length (m) depends on the SST statistics ( o C) Coefficients were estimated based on physical model results for S1 and HOT 3 . Near tropical sites such as HOT with low seasonality only display a moderate increase in δ during the mixing season, while sites at higher latitudes such as GD and K2 exhibit strong winter mixing as shown in Fig. S1. In case of strong mixing (δ >8m), growth conditions of drifting phytoplankton are adjusted to growth and quota values that are averaged for each layer z from z − δ to z + δ . For example, light I(z) is modified to

S6 Temperature dependencies
Temperature dependency f T of maximum photosynthesis P max (Eq. (1) ) and nutrient uptake rate U (Eq. (2) ) is assumed to follow the Q 10 rule, with Q10 ≈1.
Mortality related processes (see M in Eq. (6) ) are here assumed to predominantly originate from grazing, which typically is more temperature sensitive (higher Q10) (see Tab. S2).

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Figure S1. Annual cycles in mixing length δ reconstructed using SST data for the five time-series stations.    Year, Julian day, and relative RMS deviations of the model are added. The distribution of RMS errors was binned for each station and the number of profiles sampled randomly from each segment was set proportional to the total number in that segment so that the random selection of profiles/dates preserves the error statistics.  To compensate for the reduced growth capability of non-migratory phytoplankton, the specific mortality rate has been lowered by a factor of 5 (thick bluegreen line). The position of the chemocline is indicated with a purple line. Right: Chl:C ratios calculated for different populations, the migratory phytoplankton (green), a non-migratory building the SCM (bluegreen), the entire population (surface immobile + migratory SCM, brown), and for the entire population assuming N-replete cells, thus expressing exclusively photo-acclimation (black).